Download note to users - Bibliothèque et Archives Canada

NOTE TO USERS
The original manuscript received by UMI contains pages with
slanted print. Pages were microfilmed as received.
This reproduction is the best copy available
UMI
Electrical Coupling Mechanisms of Excitable Cells
Edward Joseph Vigrnond
A Thesis submitted in conforrnity with the requirements
for the Degree of Doctor of Philosophy,
Depart ment of Electrical and Computer Engineering and
Tnstitute of Biomedical Engineering, in the
University of Toronto
@Copyright by Edward Joseph Vigmond 1997
1*1
Bibliothèque nationale
National Library
of Canada
du Canada
Acquisitions and
Bibliographie Services
Acquisitions et
services bibliographiques
395 Wellington Street
Ottawa ON K1A O N 4
395, rue Wellington
Ottawa ON KI A O N 4
Canada
Canada
The author has granted a nonexclusive Licence allowing the
National Library of Canada to
reproduce, loan, distribute or sell
copies of this thesis in microform,
paper or electronic formats.
L'auteur a accordé une licence non
exclusive permettant à la
Bibliothèque nationale du Canada de
reproduire, prêter, distribuer ou
vendre des copies de cette thèse sous
la forme de microfiche/fïh, de
reproduction sur papier ou sur format
électronique.
The author retains ownership of the
copyright in this thesis. Neither the
thesis nor substantial extracts f?om it
may be pninted or otherwise
reproduced without the author's
permission.
L'auteur conserve la propriété du
droit d'auteur qui protège cette thèse.
Ni la thèse ni des extraits substantiels
de celle-ci ne doivent être imprimés
ou autrement reproduits sans son
autorisation.
Abstract
ELECTNCAL CO'U'PLING MECHANISMS IN EXCITABLE CELLS
Edward Joseph Vigmond
A thesis submitted for the degree of Doctor of Philosophy,
Graduate Department of Electrical and Computer Engineering and
Institute of Biomedical Engineering, University of Toronto, 1997
Rhythrnic excitable cells can be viewed as noniinear oscillators which are coupled to neighbouring c e h through various mechanisms. Entrainment of oscillators as a result of this coupling is
an important consideration, required for normal functioning of the gut but producing epilepsy in
the brain.
There are two direct electrical coupling mechanisms: gap junctions and electric field coupling.
There are several scenarios where gap junctional coupling is either absent or unable to account for
o b s e ~ e dbehaviour. Electric fields, produced during depolarization, induce voltages in nearby
membranes which cannot be measured direct ly. Modelling must be used to determine if the effects
of these fields are sufficient to bring about entrainment.
The boundary element method (BEM)was used to compute the induced trammembrane
voltage in a celi due to the depolarization of a nearby cell. .4 formulation obtained by differentiating
Green's Theorem to produce a Fredholm equation of the second kind for curent in terms of voltage,
proved to be more computationally efficient and accurate.
Use of BEM to mode1 exterior diffusion was possible for a simple known case, but applicability to morphologically relevant cases was quest ioned due to computationai demands and mapping
of boundary conditions.
Morphological structures between cells, specifically interdigitat ions, limit diffusion and arnpliQ extracellular concentration changes in both amplitude and duration, as well as increase induced
voltage in spherical ce11 models.
The induced voltage produced by a synchronized aggregate of cylindricd cells on a passive
cell was studied. Results found were (1) placing the passive ce11 end-to-end with the aggregate
was most efficacious, (2) decreasing conduction volume decreased volt aga, (3) the dependence of
voltage on source aggregate size, and (4) very low gap junction conductances provide good coupling.
The propagation velocity of depolarization waves is a crucial parameter, yet reinains unknown.
Extending the BEM to use mixed triangular/cylindrical element models allowed modelling
of realistic 3-dimensional neurons with dendritic trees. Biologicdly recorded waveforms called
spikelets, which resemble differentiated action potentials, were reproduced. Based on ampli tude
considerations, we concluded that spikelets are produced by several neurons whose activity is synchronized in part through gap junctions. This supports other biological evidence.
Acknowledgement
First, I would Like to thank Profesor Bardakjian who has guided me through graduate sch001
and shown to me how the desire for learning keeps one young and jubilant. 1 would furthermore
Lke to thank him For his support and confidence in me over the years and to prove to him they were
well founded. There are two things, however, for which 1 would not like to thank him: turning on
the lights and some of his jokes.
1would also like to recognize the collaboration with Dr. Peter Carlen, and the data provided
by Dr. Jose L. Perez Velazquez and Dr. Ta&
Valiante.
1 would like to acknowledge the many people who have helped me in this endeavour. 1 would
like to thank al1 the students in the lab with whom I have commiserated and celebrated over the
years and who have each taught me something, not necessarily academic.
1 must also thank the staff and affiliates at the Engineering Cornputer Facility who have
patiently given me guidance to
Fx my system numerous tirnes.
Lastly, 1would like to thank the person who supports and believes in me and makes al1 of
this possible, the person without whorn this would be an ernpty task: Darlene.
iii
Contents
1 Introduction
1
.........................................
Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CornputerModelhg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Modelling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Motivation
1
1.2
3
1.3
1.4
1.5
1.6
2 Oscillator Theory and Physiology
2.1
Oscillators
2.1.1
2.2
6
.........................
8
9
13
2.2.1
...................................
Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gap Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
Potassium Accumulation .
15
.............................
2.2.4 Nitric Oxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.5 Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Excitable Cells of the Gastrointestinal Tkact . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Electrophysiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rippocampal Neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Physiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Epileptiform Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3
2.4
5
Coupling Mechanisrns
22.2
2.3
4
8
..........................................
Mapped Clock Oscillator Mode1 .
3
14
16
16
16
16
18
24
24
25
2.4.3
2.5
Coupling M d a n i s m s
Summary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
..........................................
3 The Boundary Element Methad
3.1
Mathematical Derivation .
.................................
27
28
29
. . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1.2 Notes on a Galerkin Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1.1
3.2
The Membrane
3.2.2
Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
.................................
Element Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Triangular elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Two Ce11 Problem
Cytindrical Elements
The Electric Field Formulation
4.1
4.2
........................................
Element Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Development
4.4
36
36
38
40
41
42
43
44
44
46
4.2.1
Triangular elernents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.2
Disc Elernents .
4.2.3
Cyündrical Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
...................................
.............................
Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Electric Field Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Uniform Tkansmembrane Voltage . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Point SourceStimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.4 Dynarnic Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.4
4.3
33
Disc Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
................................
3.3.4 Far Field Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Numerical Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Error Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3
3.5
33
3.2.1
3.3.2
3.4
...........................
...................................
Application tu Biological Membranes
3.2.3
3.3
Solution to Poisson's Equat ion
Far Field Approximation
Discussion
..........................................
46
47
47
48
50
51
53
55
5 Ionic Diffusion
59
................................
Finite DiEerence Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
BEMModelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Interpolating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Mathematical Development
60
5.2
60
5.3
5.3.2
Boundary Conditions
5.3.3
Interna1 Poles
5.4
Verification
5.5
Summary
6.2
66
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
....................................
.........................................
69
..........................................
74
6 Spherical Models
6.1
64
70
76
.........................................
Mode1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction
76
77
6.3 Quality of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.3.1
Whole Cell Current
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
.................
6.3.3 Non-Reciprocal Field Effects . . . . . . . . . . . . . . . . . . . . . . . . . . .
Computer Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.1 Effect of Propagation Velocity . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 Validation of Induced Tkansmembrane Voltage
6.4
6.4.2
83
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
..........................................
Effect of Interdigitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Significance of the Induced Tkansmembrane Voltage . . . . . . . . . . . . . .
88
6.5.1
89
7 Cylindrical M o d e l s
7.2
82
Discussion
6.5.2
7.1
82
Induced Trammembrane Voltage . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.4.3 Interdigitation Size
6.5
81
91
93
.....................................
7.1.1 Potent ial Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.2 Extension to N+1 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.3 Electric Field Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1 Effect of Gap Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Two Surface BEM
vi
94
94
95
96
97
97
.............................
99
................................
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
103
7.2.2
7.2.3
7.3
Finite Volume Conductor
Coordinated Sources
8 Hippocampal Neurons
109
112
8.1 Methods
112
8.1.1
...........................................
ModelNeuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
112
8.1.2
Boundary Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.1-3 Electrophysiological recordings . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.2 Results .
8.3
8.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8.2.1 BiologicalRecordings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118
8.2.2 Gap Junctional Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8.2.3 Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
8.2.4 Location of Gap Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
8.2.5 Electrode Placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.2.6 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.2.7 Extracellular Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Suggested Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
9 Conclusions
9.1
Summary
130
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
9.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
A Derivation of Farrnulae for Matrix Entries
A.l niangular Elements
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A2
A.1.2 Double Layer Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A5
Disc Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A6
A.2.1 Single Layer Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A6
A.2.2 Double Layer Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A8
Cyiindrical Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Al1
A.1.1 Single Layer Sources
A.2
A.3
Al
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Al1
4.3.2 Double Layer Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A l 1
A.3.1 Single Layer Sources
..........................
A.3.4 Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Interface of T'riangles and Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.3.3 Cyünder with One Closed End
A.4
A12
A12
A12
B Development of Diffusion Transformation
Bi
B.1 Tkansformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BI
B.2 Matrix Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B2
..................................
B.2.2 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Global Interpolating Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2.1
B.3
Dual Reciprocity
B.4 -4lternative Green's Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B2
B3
B4
B4
B4
List of Figures
2.3
...........................
Fieldcouphg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gap junction coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4
Orientation of srnooth muscle cells
17
2.5
Cell-tece11 contacts
18
2.6
Intrinsic ECA in various regions of the canine stomach
2.1 Schematic of mapped clock oscillator
11
2.2
13
............................
....................................
14
2.8
. . . . . . . . . . . . . . . . . 20
ECA fkom various regions of the cat jejunum . . . . . . . . . . . . . . . . . . . . . . 22
ECA frorn wious positions within colonic canine circula muscle . . . . . . . . . . . 23
2.9
Schematic of hippocampal slice and neuron . . . . . . . . . . . . . . . . . . . . . . . 25
2.7
2.10 Intracellular hippocarnpal recordings . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1
Schernatic of BEM Problern
................................
29
3.2
35
3.5
..................
Two cell computation scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
'Itiangular element definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Solid Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.6
Bifurcation element
.....................................
42
4.1
.................................
Error in computation of electric field of a disk . . . . . . . . . . . . . . . . . . . . . .
Effect of source position on error . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Computation time dependence on number of elements . . . . . . . . . . . . . . . . .
Dynamic cornparison of formulations . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.3
3.4
4.2
4.3
4.4
4.5
Equivalent electrical circuit of a patch of membrane
Boundary Element Models
5.1 Finite difference mode1 of interdigitation
37
39
49
52
54
55
. . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2
Extracellular potassium concentration as a function of time along interdigitation . . 62
5.3
5.8
.......
Effect of separation on peak extracellular concentration and elevation time . . . . . .
Effect of extraceliular potassium concentration on resting voltage . . . . . . . . . . .
Boundary transformation under domain inversion . . . . . . . . . . . . . . . . . . . .
Simplified boundary showing 3 boundary nodes and an interna1 pole. . . . . . . . . .
Analytic solution of time dependent diffusion problem . . . . . . . . . . . . . . . . .
5.9
RMS error for tirne-dependent diffusion problem . . . . . . . . . . . . . . . . . . . . 72
5.4
5.5
5.6
5.7
Effect of interdigitation length on peak concentration and elevation time
5.10 Domain integral over a spherical volume
6.1
6.2
.........................
...................................
Effect of changing 8 on extracellular potential . . . . . . . . . . . . . . . . . . . . .
Spherical BEM models
............
Effect of separation on V&. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Effect of interdigitation length on peak VG and q5e . . . . . . . . . . . . . . . . . . . .
Effect of interdigitation radius on peak VA . and 4e . . . . . . . . . . . . . . . . . . .
Effect of interdigitation length on V& due to fields and [KIe . . . . . . . . . . . . . .
63
63
64
65
69
70
73
79
83
6.3 Effect of propagation velocity profile on extracellular potential
85
6-4
86
6.5
6.6
6.7
81
88
90
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.2 Mode1 ECA waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.3 Gap junctional coupling of SMCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.4 Two surfaces in a bounded volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.5 Effect of finite volume oncylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.6 Bounded current flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.1
Cylindrical BEM model of a SMC
7.7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Effect of orientation on V& . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
EffectofnumberofcellsonV' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708
Effect of number of sources for E coupling . . . . . . . . . . . . . . . . . . . . . . . . 108
7.8
7.9
7.10
Configurations of SMC
8.1 Boundary element mode1 of coupled neurons .
8.2
8.3
. . . . . . . . . . . . . . . . . . . . . . 113
Transitional element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . il5
Patch clamp recordings h m CA1 pyramidal celki . . . . . . . . . . . . . . . . . . . . 119
8.4 Effect of gj on intracellular potential
...........................
8.5 Relationship of action potential to intracellular potential
120
. . . . . . . . . . . . . . . . 121
. . . . . . . . . . . . . . . . . . . . . . . . 121
Effect of separation on qbf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Effect of gap junction position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
E k t of receiving ceU trees on 4:
Effect of conductivity ratio on qbf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Hippocampal coupling schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
8.6 Field induced VA as a function of position
8.7
8.8
8.9
8.10
8.11
A.1 Integration limits for monopole sources on a disk
A.2 Solid angle of disc
. . . . . . . . . . . . . . . . . . . . A7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A10
List of Tables
...
. . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
II Glossary of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
III Super- and Subscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
N Accents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . xv
1
Acronym and Abbrevaitions Used
2.1
Coupling Portalç
................ .. ... ... ....... .......
11
.
45
4.1 Integral forrns of the matrix entries for the potential and electric field formulation
4.2
RMS error in computation of transrnembrane current for point source stimulus . . . 51
4.3
Computational effort required for the potential and electric field formulations.
....
53
5.1 RMS error of local and global approximating functions . . . . . . . . . . . . . . . . . 71
5.2 Intermediate results for time dependent diffusion problem . . . . . . . . . . . . . . . 73
6.1 Relationship between surface voltage gradient and peak extracellular potential
B.1 Global interpolating functions up to order 5
xii
. . . . . . . . . . . . . . .-
. . . 89
. . . . . . . B5
Nomenclature
Matrices are bold uppercase characters and vectors or column matrices are botd lower case
letters. A unboldecl subscript on a matrix or vector denotes a particular elernent. Scalars are
nonbolded. The boundary element notation loosely follows that which is in common usage[l, 21.
Finauy, while electrical engineering denotes current density by the Ietter J, it is common practice
in physiology to denote this quantity by i when referring to membrane current densities.
Table 1: Acronyrns and Abbrevaitions Used
- -
Acronym
BEM
Boundary Element Method
CM
Circula Muscle
ECA
Electrical Control Activity
ERA
Electrical Response Act ivity
EFF
Electric Field Formulation
GI
Gastrointestinal
ICC
Interstitial Cell of Cajal
LM
Longitudinal Muscle
LUD
Lower Upper Decomposition
NO
Nitric Oxide
SMC
Smooth Muscle Ce11
SVD
Single Value Decomposit ion
cPm
cycles per minute
xiii
Table 11: Glossary of Symbols
Symbol
Descript ion
Units
domain of interest
closed surface defining R
normal of T?
conduct ivity
electric potential
point in 3-space
membrane current density
t ransmembrane voltage
Green's function
impressed monopole current source
elemental surface potential vector
impressed electric field
monopole potential matrix
dipole potential matrix
monopole electric field matrix
dipole electric field matrix
membrane capacitance
conductance
surface depolarization wave velocity
ratio of spined to unspined dendritic area
potent ial function
order
Laplacian
inverse Laplacian
xiv
Table III: Super- and Subscripts
Meaning
Superscript
effect on model O
produced by model
applied
boundary
electrode
origin at end
origin at midpoint
receiving mode1
source model
Subscript
Meaning
ext racellular
intracellular
membrane
surface
Table
IV: Accents
Accent
transformed to invertecl domain
unit vector
variable of integration
I
Chapter 1
Introduction
e are electro-diemical machines1, and as such, produce and react to electrical fields.
Measuring these biologically generated fields is inherently difficult, since the systems involved are
t studied in situ. There are too many factors to consider in
very complex, and most c a ~ o be
al1 but the simplest cases. The voltages and currents are minute, on the order of millivolts and
nanoamps respectively. Isolated cells can prove troublesome given that the solitary ce11 may behave
entirely differently than it does in situ. The process of isolation is darnaging due to enzymatic
treatrnent, alteration of environment and mechanical insult. Even if the ce11 is preserved intact,
the input it receives is radically altered, which rnay cause it to display hitherto unseen behaviour.
Blind experimentation at this point will not yield any useful information about functioning of the
tissue unless some theoretical basis is available upon which to interpret results. Modeiling is thus
needed to explain the often seemingly codicting data as well as design experiments in order to
test hypot heses.
1 . Motivation
Modelling biological systems is dficult. We are in effect attempting reverse engineering on Nature,
which has developed extremely complex, nonlinear systerns over billions of years. Nonlinear systems
are troublesome to analyze even when we design them. The basic unit of biological systems is the
' ~ h e o l o ~ i c aarguments
l
notwithstanding
cell, which in itself is a system controlled by hundreds of thousands of genes and yet, we are
attempting to figure out what happens when many cells are assembled together.
It is almost natural that electrical engineers enter the domain of biological modelling given
that excitable cells can be viewed as electricai devices. Excitable cells include striated, smooth
and cardiac muscle, neurons and pancreatic beta cells[3]. Indeed, sorne of the first investigations
into the phenomenon of electricity were directed at living tissue[4]. It was Galvani who looked a t
the voltages produced by fkog muscle. Electrical engineering is also concerned with systems and
their interactions. It has developed many tools to deal with system characterization iike Fourier
analysis[5], communication theory[6] and control theory. TraditionaUy, engineering haç been involved with linear systems given that they are rnuch easier to analyze and design. Unfortunately,
linear systems compose only a minority of what is encountered in the biological realm, and linearization is applicable only within a srnall region about a given point. Nature has deveIoped nonlinear
systems for which no neat general anaiysis is yet available.
Oscillators are an all pervasive phenomenon in our lives. They are occurring within our
bodies on many time scales[7], from microsecond happenings in the cell nucleus, millisecond events
in the brain, one second cycles in the heart, the 15 second breathing cycle, circadian rhythm,
monthly cycles and yearly cycles (such as seasonal affective disorder). Finally, our entire life may
be viewed as but a single period in the life/death cycle.
The relaxation oscillators is a type of nonlinear oscillator which has been used to mode1
biological processes[8]. Relaxation oscillators rely on the periodic storage and discharge of energy
dependent on a nonlinear component. Exarnples of equations describing them are the Van der Pol
oscillator[9], the Fi tzhugh-Nagumo model[lO] and the mapped clock oscillator[l 1]. These oscillators
may display behaviour when coupled together that is entirely dserent frorn any intrinsic behaviour,
and coupling alone is sufficient to bring about this alteration in behaviour. F'requeccy, amplitude
and even waveshape can change drastically for coupled oscillators[l2, 131. Indeed, many diverse
modes are possible making relaxation oscillators an attractive control systern and explaining why
they are so abundant in nature. The amount of coupling - the influence that each oscillator exerts
on a neighbour - required to effect changes is dependent on frequency and resting level differences,
and can be quite small[l4].
Understanding coupling is very important in understanding the etiology of gastrointestinal
motility disorders like tachygastria, ileitis and colitis. Improper contractions will lead to stagnation
2
of luminal contents, result ing in accumulation of waste products and leading to severe abdominal
crarnps. Other complications may arise from the inefficient absorption of nutrients bom food. In
neuronal systems, epileptiform activity is brought about by a change in coupling[l5].
In biological systems, a plethora of coupling modalities is available. These mechanisms
include direct innervation[l6], stretch receptors[l7], changing extracellular ion concentrations[ld
19, 201, exchange of metabolites[21], gap junctions[22], hormones[23], and, finally, it is posited,
electric field coupling[24]-(301. The different types of coupling have different properties. It is the
electrical coupling mechanisms, gap junction and field coupling, that shall be studied herein.
1.2
Scope
This thesis will attempt to determine whether electric field coupling is sufficient for entrainment
of excitable cells. Specifically, electric field coupling of gastrointestinal srnooth muscle cells and
hippocampal neurons will be studied. This type of coupling will be compared to gap junctional
coupling. Induced transmembrane voltages will be viewed in the context of oscillator theory to
determine if they are of consequence. The conditions under which field coupling becomes significant
will be established. The primary method of investigation will be the Boundary EIement Method
to which enhancements will be made.
1.3
Background
Once discovered, gap junctions were assumed to be everywhere; hence, the importance of field
coupling was downplayed[31, 321. Gap junctions are readily identifiable in tissue and have well
known coupling properties[33]. Coupling between cells so connected is easily demonstrable. Field
coupling can be defined as the influence exerted between cells as a result of biologically generated
extracellula. electric fields. This type of coupling camot be rneasureci directly and, therefore, is
difficult to prove, especiaily in situ. Again, the potentialç are very small and may come from any
number of sources. Furthermore, the effect of the fields are not known. They may be insufFicient
to initiate action potentials but strong enough to cause entrainment or alter frequency.
There are two examples in the gut where field coupling has been implicated. The first is
in the proximal intestine where the highest frequency obseweù is higher than the highest intrinsic
fkquency[34]. This is not possible with gap junctions which limit the hequency of coupled cells
3
to the highest intrinsic fiequency. Second, longitudinal muscle has been shown to be well coupled,
yet a convincing demonstration of gap junctions is still forthcoming [32]. Another mechanism must
then be a t work and field coupling is a possible explanation. Metabolic coupüng [21] is another
possibility but that is beyond the scope of this thesis.
Fieid coupling has also been posited as a coupling mechanism in neuronal systems. Measurements of electrical activities of neurons during epileptiform bursts have exhibitecl field coupling[l5].
With single electrode recordings it is not possible to ascertain membrane events fully since there is
a large spatial dependence[35], necessitating modeiiing.
1.4 Computer Modelling
Since the advent of the ever more powerful and p e m i v e computer, m o d e h g has reached new
heights of sophistication. Systerns of increasing complexity and their ensuing computational demands can now be handled in less time than ever before. As a result, rnodelling is used to test and
develop new ideas. Some of the advantages of in silico simulations are:
1. Simultaneous monitoring of every quantity, i.e., an "infinite" number of sensors.
2. Ability to calculate physically unmeasurable quantities.
3. Ease of changing mode1 parameters.
4. Ability to do experimentally difficult procedures.
5. Replacement of costly laboratories wit h less costly computers.
6. Preservation of animal life.
As with al1 things, there is a fly in the ointment. Computer modelling must deal with a
simplified version of the system it is modelling in order to reduce the digital abstraction of reality
to a manageable level. Hence, t here is always some doubt as to whether the assumptions made
are a l i d . One of the fundamental problems of computer modeliing has to do with the validity
and correctness of results. Cornputers have finite accuracy and, thus, even if an algorithm is
correctly implemented, this Iimited accuracy rnay produce erroneous results. This situation is rnost
readily apparent when a denominator is formed from the difference of two very simi1a.r numbers.
An inaccurate qüotient results which will be propagated through the entire solution. It is easy
4
to generate endless data streams whether they are usefui or not. Often, only simple analytical
situations are available with which to compare solutions and extrapolation to more complicated
cases is done on faith. Certain measures may be taken to lend credence to a solution obtained,
such as convergence, as discretization is increased; but the final result can never be proven correct
except in trivial cases. This is not to Say computer modeiiing is not trustworthy. Au contraire, it
is a reminder that results need to be scrutinized lest premature announcements be folly.
1.5
Modelling Approach
The boundary element method
(BEM)is a numerical technique that is used in many fields of engi-
neering to mode1 diverse situations such as fracture medianics in geoloa([36], diffusion and fluid flow
in chemical engineering[37], heat flow in mechanical engineering(381and, finally, electromagnetics in
electrical engineering[39]. Specifically in biomedicd engineering, it is used to compute current flow
in cardiac tissue[28, 401 and the skull[41, 42, 431. BEM is popular since it uses boundary solutions
to determine what is happening over the entire domain of interest, reducing the dimensiondity of
the problem.
The ot her most commoniy used met hod for computing potentids is the finite element met hod
(FEM)[44]. Bot h met hods can solve for arbitrary geometries which allows problems wit h realistic
morphologies to be handled. Compared to the BEM,the advantages of FEM are its ability to deal
with anisotropic media and elemental integrals which are easy to integrate. The BEM may require
the integrat ion of hypersinguiar kernels which can be problematic[45]. TQ its disadvantage, the
FEM requires meshes which are bigger and more finely discretized. The BEM generates smaUer
matrices which are dense. While the
FEM matrices are larger, they are sparse and symmetric, two
qualities which greatly decrease computational costs. The FEM solves for the entire domain a t
once, while the BEM first solves the surface quantities, and then requires additional calculation for
points not on the boundary. Finally, the BEM also has the ability to handle problems in infinite
domains. Thus, each method has its advantages and disadvantages[46].
For this thesis, the BEM was chosen since initially isolated celk in infinite domains are
considered. Another major factor promoting
BEM use
was the ease of generating BEM meshes
without a commercial software package, unlike the FEM,which has complex three dimensional
meshes. No anisotropies were considered nor was the solution required at many domain points.
Membrane quantities were the focus of the study which further emphasized the BEM.
BEM has been used to mode1 cardiac cells where the geometry can be satisfactorily described
by a cylinder[47, 28, 401. In these studies, cylindrical elements have been employed, limiting the
problems to rotational symrnetry. Other attempts at modelling field effects have used lumped
resistors to account for field effects(24, 25, 26, 271.
Modelling of neurons with dendritic trees has examined extraceilular fields but only for
greatly simplified models[48, 49, 501. There is a paucity of modelling studies examining induced
voltages which may be be due to the complexity of the field equations when applied to the neuronal
morphology, or the lad< of recognition of the importance of field effects. There are many fine
campartmental neuronal models which accurately depict the diffusion of chemicak, current flow
and membrane processes which include al1 manner of ionic transport mechanisms[51, 52, 531. What
is lacking in these neuronal models is the accurate depiction of extracellular fields.
1.6 Thesis Organization
The thesis is organized in the following manner:
0
Chapter 2 gives an overview of the physiological systems uncier consideration. Since we are
considering rhythrnic celk, the chapter starts with a description of nonlinear oscillators, the
effect of coupling on their behaviour and coupling mechanisms. The chapter describes the
electrophysiology of the gastrointestinal tract and discusses its behaviour from an oscillator
vantage. Hippocampal neurons are also described.
0
Chapter 3 describes the boundary element method (BEM) and applies it to biological mem-
branes. The boundary element method allows the computation of extracellular fields and
their effects. The different types of elements used in the models are given. The formulae for
the different element types are stated but their development is saved for appendix A. Some
additional aspects of numerical cornputation are elucidated.
0
Chapter 4 describes an enhancement to the BEM which can lead to decreased computation
and greater accuracy. The traditional potential formulation is differentiated to formulate the
problem in terrns of electric field. The new equations are validated and compared to the
potential formulation. Under certain conditions, accuracy is shown to be improved dong
with a reduction in computation costs.
Chapter 5 examines extracellular diffusion of ions. A simple finite dEerence scheme is used
to model diffusion of potassium in an interdigitation and this concentration change is related
to changes in transmembrane voltage. Application of a domain inversion technique to d o w
BEM modelling of exterior diffusion problems is also examined. The technique is verified but
found to be too computationally demanding to be useful.
Chapter 6 is a prelirninary investigation of field coupling between spherical ce&. The effect of
surface depolarization wave velocity profiles is exmined for a nurnber of nonuniform profiles.
The effect of the interdigitation on field coupling is calculated as a function of interdigitation
length and radius. The structure is found to enhance coupling by about 30%.
Chapter 7 explores field coupling between cylindrical cells which are used to model smooth
muscle cells. The e k t s of gap junctions, finite conduction volume, source aggregate and
relative orientation of celk are modelled.
Chapter 8 looks at the phenornenon of spilielets in hippocampal neurons. A novel BEM
method is employed which uses a mixture of triangular and cylindrical elements to model a
realistic 3 dimensional neuron. The roIes of gap junctions and field coupling are elucidated.
The effects of dendritic trees, gap junction location and conductivity are also determined. A
coupling scheme is proposed to account for spikelets.
Chapter 9 gives a summary of the thesis and discusses future work.
Chapter 2
Oscillator Theory and Physiology
It is important that the concept of oscillators be more fully described since the gut and brain
can be considered as interacting populations of oscillators. A theoretical faundation of nonlinear
oscillators in the context of the mapped clock oscillator rnodel[ll] will be given in this chapter.
The eIectrophysiology of the gastrointestinal tract and hippocarnpal neurons will be presented as
well with direct references to oscillator theory.
Oscillations are ubiquitous phenornena of Our existence. Indeed, whether it be the music Ive dance
to, the rotation of celestial bodies or our moods, we are immersed in rhythms. Many of the oscillators can be modelled by relaxation oscillators which rely on nonlinear components to systernatically
release energy after having built up a store, like the tension dispelled by a downpour on a hot day.
Oscillators rnay be broken down, conceptualiy if not physically, into two components: a clock
and a transformer. The clock is the component which does the actual cycling while the transformer
maps the dock state into an observaHe quantity. In this thesis, excitable cells will be considered
as oscillators. The transformer output is the transmembrane voltage of the cell. The rnechanisrn
responsible for the oscillations, the clock, may reside inside the ce11 nucleus, in the ce11 membrane
or outside the cell.
Nonlinear osciilators may interact in various ways depending on the strength of coupling
and the type of coupling[54]. Given two coupled oscillators of different intrinsic frequencies, they
may entrain to kequencies in the range of the two or outside, experience frequency pulling, become
chaotic or cease to oscillate. Thus, while the intrinsic nature of the oscillators is important, the
coupling between oscillators can produce behaviour that is much dserent horn that displayed
intrinsically.
Quasiperiodicity: This state is characterized by changes in fkquency £rom the intrinsic values
where the two oscillators' frequencies are not related. F'requencies wax and wane, not staying
constant. The order of system is the sum of the orders of the oscilators.
Entrainment: Both osciliators have the same period or one's period is a multiple of the other's.
If their periods are equal and there is no phase difference, then it is a special case called
synchrony.
Chaos: Output of the system in this state is seemingly random.
Non-oscillatory: The system cornes to rest at a stable point. This usually occurs when coupling
becornes too strong.
In biological systerns two independent factors contribute to determine which of the coupled
osciliators will lead the other oscillators in phase: frequency and resting level [55]. Higher frequency
oscillators will lead lower hequency oscillators. Since we are dealing wit h biological cells, resting
levek are negative quantities and oscillators with the lowest resting level wiil tend to Iead. These
factors are independent and it is possible to have antagonistic gradients in the same tissue[56].
2.1.1
Mapped Clock Oscillator Mode1
A general model for noniinear oscillaton called the mappeà clock oscillator[ll] has been developed
in o u .lab and used to describe several different osciiiating, biological systems[57, 58, 59, 131. Understanding this model and its decomposition of oscillators provides valuable insight into oscillators
and a way of conceptualizing the systern. There are two components to the model, the clock and
the transformer, as shown in figure 2.1. The clock is a generic set of nonlinear differential equations describing the state space. The state point is then mapped by an appropriate function, the
transformer, to the desired observable output which is the transmembrane voltage. In state space,
there is an unstable attractor at the origin and a stable limit cycle at unit radius and, in polar
coordinates, is described by[ll]:
where
A
a=
radius
(2.4)
4
(2-5)
polar angle
A
r~ = intrinsic
frequency
SC coupling input to portal
(2-6)
(, < E {a,
(2-7)
72. p }
A
y=
transformer output
(2-8)
A
yo =
intrinsic d.c. level of transformer output
f (4) intrinsic a.c. component of transformer output
(2-9)
(2.10)
The different coupling portals represent different directions for furces acting on the state point.
Each portal has a specific effect on the oscillations which are further described in tabIe 2.1 and
graphically displayed in figure 2.1. For a particular portal, the coupling input rnay be further
broken down:
where C is a coupling factor between the input, I , and the oscillator. R is a rekactory function
which reduces the susceptibility of the oscillator to stimuli in a phase dependent manner, and
max f () is the maximum amplitude of the a-c. component of the intrinsic oscillation. The coupling
factor represents the amount of coupting between two oscillators and, in this thesis, is the ratio of
the voltage induced in a cell by an electric field to the voltage in the ce11 producing the field.
Computer simulations by Bardakjian et al.[ll] illustrate the markedly different input levels
required by each portal to produce entrainment to a voltage stimulus. The a portal is incapable
of producing entrainment while the 4 portal entrains only for a small range, above which chaos
is encountered. Stimulating the y portal produces entrainment for input levels very much smaller
than required by the # portal and does not lose entrainment as the input is increased to very large
10
Oscillator :
Figum 2.1: Schematic of mapped clock oscillator. The input (1) is jed throvgh a wupling network
to the input portal3 (shaded) which are then fed to the clock or mapptng unit. The osciilalor output
is fed back through to the wvpling portaks to provide feedback. The influence of the coupling portais
on the state point (dot) is shown in the dock module.
Port al
State Space Force
~ffect
Biological Example
alter amplitude of a.c.
steady current stimulus to
cornponent
SMC[60]
-
-
- - - --
-
-
-
-
application of pentagastrin to
change frequency
SMC[Gl]
change both amplitude transmission
and frequency
s~
through
gap
j unctions [22]
transmission through chemi-
change d.c. level
cal synapses[l6]
Table 2.1: Coupling portais and their effects. Note that there is a 7 portal for each axis.
levels. The type of coupling network determines the portal to which a stirnuius is applied[62]. A
resistive-capacitive network corresponds to qi coupling.
Coupling may be unidirectional, Le., one oscillator inputs to the other with no reciprocal connection, or bidirectional, i.e., the output of each osciUator is fed back into the other. Furthermore,
the coupiing rnay be syrnmetric, i.e. oscillators exert equal influence on each other, or asymmetric.
Tt is important to note that only with symmetnc, bidirectional coupling with resistivecapacitive
coupling is it possible to achieve entrainment of oscillators at a frequency higher than the highest
intrinsic eequency.
lntrinsic properties of the oscillators determine the strength of coupling needed to bring
about entrainment. Both larger intrinsic resting level and kequency differences between cells require
stronger coupiing to bring about entrainment[l2].
As the bidirectional coupling factor is increased bom zero[l2], the oscillators will experience
frequency pulling where frequencies wax and wane as they interact. With larger coupling, the
oscillators will entrain with several difFerent scenarios. There rnay be 1:l coupling where the
fiequency of each is the same but it is also possible to entrain at other ratios whereby one oscillator's
fiequency is a multiple of the other's. Within small regions of the coupling factor it is possible to
produce chaos in the systern. Further increases in coupling factor will bring about the cessation of
oscillation as a stable point is reached. The level of coupling this occurs at is actually quite srnail.
It is important to understand that the oscillators are not driving the oscillations in each
other. They are stightly perturbing the state space trajectories of the oscillators to which they are
coupled. The entrainment rnay take several cycles for the oscillators to reach stable limit cycles.
Use of very small perturbances, on the order of the noise level of the system, to entrain chaotic
systems has been demonstrated[63] as well as the minute changes in coupling factor required to
change the behaviour of coupled neurons fiom that of chaot icity to periodicity [l3]. Ebrt hermorc.
the stabilization of chaotic systems with tiny perturbations has been demonstrated in physical
systems[l4] and in hippocampal slices[64]. In summary, it has been dernonstrated with real systems
that very little coupling is required between oscillators to drastically alter behaviour.
There are many levels at which to view the body and, hence, many levels at which to place
oscillators. The oscillator may be on the scale of the nucleus, the celi, several cells, different ce11
types forrning circuits, tissue or whole body. The mapped clock oscillator mode1 has been applied
to successfully mode1 the colon[57], stomach[58], small intestine[59] and hi ppocampal neurons[l3].
12
(a)Current tiow induced by excited ceII
(b) Ekctrical equivalent
Figure 2.2: Field wupling. (a) A depolarizing ce11 (lzght grey) urill induce c u m n t Jlow in a
volume conductor. A nearby ce11 (dark gray) will ezperàenee a change in membrane voltage due to
the c u m n t pow. (b) Electrically, field wupling can be represented as an resistor-capacttor network.
2.2
Coupling Mechanisms
Coupling in biological systems can be produced by many factors, any subset of which may be
operating concurrently. This section wiil describe some of the mechanisrns.
2.2.1
Electric Fields
Electric field coupling, or ephaptic coupling as it is aiso known, has been proposed as a coupling
its
mechanism in many systems. While it has been demonstrated in neurons in vivo, [65,15,66,67],
role in smooth muscle is still a point of contention. This type of coupling relies on a source ce11 (or
cells) experiencing a surface potential gradient during depolarization which will induce extracellular
current flow and hence, a potential field[29]. This potential field will, in turn, cause a redistribution
of surface charge on nearby cells, depolarizing one portion of the membrane while hyperpolarizing
another. Morphology plays a major role in determining efficacy.
This type of coupling has an extrernely qui& onset and short duration[l5]. A field is developed at the speed at which the transmembrane voltage spreads out over the surface of the
13
cell and lasts the duration of the upstroke. Though brief, it is the lastest acting of the coupling
mechanisrns[i5]. Since fields generated are dependent on the gradient of the source transrnembrane
voltage, which is in turn dependent on the temporal derivative of the voltage and its propagation
velocity over the surface of the ceI1, this type of coupling may be viewed as resistive-capacitive
coupling.
2.2.2
Gap Junctions
Gap junctions are direct electrotonic connections formed by channels between the intracellular regions of adjacent celk (figure 2.3). Each channe1 is formed from two hemichannels, called connexons,
one in each membrane. Each connexon is forrned fkom six protein subunits called connexins (681of
which more than a dozen varieties have been identified. Channels form pores of 1.5 nm of diameter and a conductance ranging fiorn 50-150 pS depending on protein composition. Furthermore,
gap junctions do not appear as isolated channek. Rather, they form a lattice of channels which
resembles a honeycomb when viewed under the microscope. The membranes of the two cells in the
region of the aggregate are separated by only 2-3 nm. In an aggregate, the number of functioning
gap junction channels is less than the number of connexons present in a cell.
(a) Gap Junction
(b) Electrical q u i d e n t
Figure 2.3: Gap junctions: (a) Deptction of gap junction channels in the membrane. Reprinted
/mm [69]. (b) Electtimlly, the gap junction can be represented as a resistor.
Gap junctional conductance is modulated by several agents and responds with a time constant on the order of seconds. A voltage across the channel will close the channel. Since connexons
in each ce11 are oppositely oriented, each connexon wilI see one half of the transjunctional voltage with a polarity opposite to t k t of connexon attached to it. Most gap junctions are formed
from identical connexins so the voltagc+conductance c u v e is symrnetrical about zero dthough it
is possible for each ce11 to express different comexins which will lead to current rectifkation[70].
Intracellular acidification decreases gap j unct ionai conductance while alkalinization increases it .
Alcohols, e.g. octanol, serve to decrease the conductance as weli as neurotransrnitters and second
messengers such as calcium, dopamine, arachidonic acid and phorol esters having the same effect.
Tons and small molecules may pass through the gap junction channels allowing for metabolic
coupling in addition to electrical coupling. Calcium may play an important role as a messenger
since increased calcium produces a cascade of effects inside a cell. Electrical coupling d l be almost
instantaneous by gap junctions[l5]. Hence, gap junctions are well suited for synchronizing activity
of a number of cells. Also, since they are simple resistive pathways, cells so coupled are limi ted
in fi-equency to the fastest intrinsic fiequency of the coupled cells. Higher kequencies are not
possible[62].
This is also a very fast type of coupling, almost instantaneous in onset and lasting as there is
a potential diffaence between the coupled cells. The gap junction channels will eventually close due
to the voltage but this will occur with a tirne constant on the order ofseconds[71] which will be of
consequence over one smooth muscle ce11 osciIlation but not over a neuronal oscillation. This twvpe
of coupling has a slightly longer latency than field coupling since the membrane must be charged
by the current passi ng t hrough the junction while field coupling relies on redist ribut ing the exist ing
membrane charge.
2.2.3
Potassium Accumulation
Due to the restricted diffusional space between smooth muscle cells, especially in regons with intermediate contacts and/or interdigitations, ions rnay accumulate (or be depleted) by an appreciable
level in the narrow intercellular cleh ( s e figure 2.5). Since excitable cells are particularly sensitive
to increases in extracellular potassium concentration which increase excitability, potassium has
been implicated as a coupling mechanism(l8, 19, 721. Computer models of pancreatic beta cells
have also demonstrated this phenomenon of potassium accumulation leading to synchronization[73].
15
Dramatic changes in extracellular ions have aiso b e n noted during epileptiform bursts[20]. This
type of coupling will be a longer lasting modulatory mechanism, not as qui& acting as any of
the other mechanisms mentioned and works synergistically with field coupling since it can increase
membrane susceptibility to electrical stimuli drastically.
2.2.4
Nitric Oxide
Nitric oxide (NO) is a coupling agent in smooth muscle[74, 751. Unlike chernical synapses, there
are no NO receptors on the cell membrane; NO diffuses directly into the intracellular space. It is
difficult to detect directly since its lifetime is very short (2 s) in the extracellular medium. Instead,
the enzyme which produces NO may be detected or blocked t o determine if NO coupling is present.
In smooth muscle, it has been found to be the inhibitory transrnitter in non-adrenergic,
non-cholinergie nerves causing a lessening of muscle tone[74, 751. The effect of NO on colonic
smooth muscle has been to decrease intracellular
ca2+which is a potent second messenger as well
reduction in the contractile response to Ca2+ 1761. The former effect may be c a w d by activation of
uptake mechanisms of the sarcoplasrnic reticulum, activation of K+ channels or a second messenger
cascade. Interstitial Cells of Cajal (ICC7s)increase NO synthesis in response to NO and thus may
serve as NO amplifiers[77].
2.2.5
Other
Srnooth muscle cells contain channels which open under stretch[l7]. If two cells are physically
attached in some way, contraction of one will stretch the surface of the other and activate these
channels, leading to depolarkation.
2.3
2.3.1
Excitable Cells of the Gastrointestinal Tract
Organization
There are three basic layers to the GI tract from outside to inside[78] (see figure 2.4):
Longitudinal muscle layer: Muscle fibres are oriented along the axis of the GI tract with their
contraction produces a local shortening.
Figure 2.4 : Orientation of smooth muscle cells showing the longitudinal (running horizontally)
and circular muscle (shown in cross section) layers. Not observable are mdially oriented cells
running from the myenteric plexus into the circular muscle layer between the Lamellae (outlined).
Reprinted from 1'181.
Circular muscle layer: Celis run perpendicular to the longit udinai layer, circumferentially to the
axis of the tract. Contraction of this layer tends to occlude the GI tract. In humans, circular
muscles are organized into lamellae with narrow, radial spaces in between. In these spaces
are interstitial cells, nerves and smooth muscle ceHs running radidly.
Mucosa: This is the highly vascularized innermost surface of the GI tract through which secretion
of enzymes and absorption of nutrients takes place.
There are two major nerve networks which function to coordinate the operation of the muscle
layers. These two networks comprise the enteric nervous systern with about 108 nerves, 100oth the
number in the brain. The outermost network is the called myenteric plexus and runs between the
two muscle layers while the submucosaI plexus runs between the circular muscle and the mucosa.
The number of celis with which the nerves make contact decreases with distance into the muscle
layer away from the plexus.
The muscle cells of the tract are smooth, i.e. no cross-striations, and are roughly ellipsoidal
with lengths that range fiom 100 to 200 pm and diameter of 5-10 jm. There are several structures (figure 2.5) which appear between smooth muscle celis that may be important for coupiing
reasons[79]. These structures are
(a)Apposition
(b) In termediate
(c) Interdigitation
Figure 2.5: Cell-to-ce11 contacts. The extracellular space is shaded and Iabelled
"E".
Appositions: Two cells have smail processes of limited surface area which come into very close
contact with each other.
Intermediate Contacts: Large portions of neighbouring membranes are pushed close to each
other, separatecl by less than 10 nm. Relatively large surface areas come in very close contact.
Interdigitations: A process of one cell, an extrusion, fits in an indentation of a second cell, an
intrusion, like a finger pushed into dough. The extrusion may be several micrometers long
and undergo a 90 degree bend. The separation between the extrusion and intrusion is again
on the order of a membrane thickness, 10 nm.
2.3.2
Electrophysiology
A major characteristic distinguishing smooth muscle fkom striated muscle is the periodic depolarization smooth muscle undergoes which are myopnic in origin. In contrat, striated muscle must
be stimulated by exogenous means for membrane voltage depolarization and the accompanying
contraction to occur. Smooth muscle cells have a threshold voltage above which mechanical contraction occurs[80]. Generally, this threshold is around the -40 mV but varies among species and
position in the GI tract. Cells distal to the fundus (which is above its excitation point) undergo
periodic depolarization which may or rnay not bring the potential of the ce11 above the contraction
threshold. There is an initial rapid upstroke, possibly followed by a slight repolarization, a plateau
phase lasting several seconds and a repolarization to resting level. The initial depolarization rnay
18
peak several millivolts above the plateau forming an initial spike (see figures 2.6 to 2.8). Contraction will occur if the plateau is above threshold but generally the initial fast spike is of insufficient
duration to initiate contraction. These periodic depolarizations are referred to as slow waves or
electrical control activity (ECA). The frequency of these waves are measured in cycles per minute,
typically about 6 cpm in circular muscle and 20 cpm in longitudinal muscle. The former name arises
to distinguish this activity fiom higher frequency activity while the latter name arises since it is
only during the depolarizations that contractions may occur. Acetylcholine will increase or initiate
contractile activity during the ECA, but not otherwise. Thus, the ECA represents a window of
opportunity for contractions and a means of contmlling their appearance in tirne and space.
Higher fkequency spikes can be observed superimposed on top of the ECA waveform. This
electrical activity is called the electrical response activity (ERA) and represents calcium influx
which leads to depolarization and increased contraction. ERA may be brought about, for exarnple,
by the presence of acetylcholine or oscillator interaction and may be cornposeci of bursts or single
spikes-
Longitudinal Gradients
The kequency of the ECA m i e s along the length of the gut. The fundus of the stomach, the
most proximal region, displays no ECA. Its resting level is above the excitation-contraction level
meaning it maintains a constant contraction, or tone. The rest of the stomach is entrained. ECA
originates in the corpus and traveis distaily to the antrum with an exponentially increasing phase
Iag. In contrast, the resting level decreases exponentially as one proceeds distally. If sectioned, to
aboiish interaction of gastric segments, the intrinsic frequency of the proximal region is found to
be highest (see figure 2.6). The frequency gradient and resting level gradient are thus opposed to
one another, and either end of the stomach is capable of leading the oscillations. Normally, the
contractions start proxirnally to propel food into the duodenum but occasionally, stomach contents
may be required to travel in a retrograde fashion, Le., the emetic response.
After crossing the pylorus, the kequency decreases as one proceeds distally [58]. In the
duodenum of dog and human, there are frequency plateaus. The oscillatory behaviour of the cells
in a plateau region are entrained with an increasing phase lag in the distal direction. Between
plateaus there are very short transition zones where frequencies wax and wane. Transsectioning
the small intestine to determine the intrinsic Çequency profile yields an interesting observation:
19
Fur dur
1-18
Orod Arlrum
u
1
:
:
Orod Tarminol Antrum
Caudad
Trrminal Aatrum
Pyloric
Ring
Figure 2.6: Intrinric ECA in vanous regions of the canine stomach[80].
the proximal intestine is entrained at a kequency above the highest found when transsect ioned [34].
The intrinsic kequency decays exponentiaily in the distal direction.
In the colon, there is a small increasing gradient(801 in the cephalocaudad direction. Tram
sectioning, however, reveals a constant 4 cpm frequency. This is consistent with the colons function
of storage. A gradient is certainly needed to propel the contents of the colon forward but rnost
of the time, this is not required. It is important to keep the contents slowly churning to prevent
solidification. With a smaller gradient, it is easier for coupling to change its behaviour so that
the contents are not always driven forward. ECA records indeed show waves travelling in both
directions[80].
Since sectioning of the gut reveals different frequencies of electrical activity than when intact,
the gut may be considered to consist of a system of coupled nonlinear oscillators[8]. The basic
functiond ascillator, the smallest unit which is capable of displaying rhyt hrnic activity wit hout
input, stiU requires elucidation. It may be a single smooth muscle cell, a group of coupled cells or
a group of cells driven by neuronal input complete with interstitial cells[8].
Transverse Gradients
The circular muscle layers of the antrum of the stomach [8l,821 and small intestine[83, 84, 851
experience similar gradients in resting transmembrane voltage. The outer cells at the myenteric
border exhibit the most depolarized levels, approximately -70 mV while the inner cells at the
submuosal border are hyperpolarized by 20 to 40 mV (see figure 2.7). Also, the more depolarized
the cell, the greater the upstroke of the ECA leading to a plateau gradient that is on the order of
millivolts across the muscle layer, much less than the resting level gradient. Tkansverse sections
are entrained. It is dficult to determine intrinsic frequency gradients in the transverse direction
of small intestine since removal of the nerves kills electrical activity.
The colon displays behaviour opposite to that observed in the stomach and small intestine.
The circular ceils at the submucosal border display the greatest degree of depolarization and greatest
ECA amplitude [85, 86, 87, 88, 891. The dinerence in resting level across the layer is very similar
to that found in the circular layer ( s e figure 2.8).
The muscle layers of the gut have been shown to interact with other. Hara et al [85]demonstrated t hat isolated strips of longitudinal and circular muscle of canine jejunum containing no
ICC's were electrically quiescent though application of acetylcholine produced slow waves. Isolated
21
mitudinal
Circular
outer
Mvenferic
.
Boundary (LW C)
Figum 2.7: ECA frwm
V ~ ~ O regions
U S
middle
of the cal jejunum recorded with cufl electrodes. Reprinted
fmm (841.
inner circular muscle strips still displayed slow waves but t hese may have b e n caused by ICC's
present on the myenteric side of the muscle.
The latter hypothesis is consistent with studies of the canine colon demonstrating the circular
muscle layer is quiescent when detached fiom both ICC layers[89]. The quiescence may be indicative
of a system of oscillators too strongly coupled or a system of labile oscillators without its clock.
However, longitudinal muscle in this preparation exhibited spontaneous slow waves a t a Erequency
of 20 cpm when isolated.
The two muscle layers seem to operate at very different intrinsic Erequencies. When coupled
at the myenteric plexus, the oscillations of each layer seem to propagate into the other layer and
decay with distance[87]. Looking at the circular muscle layer (figure 2.8), spikes at the longitudinal
muscle layer frequency of 20 cpm are superimposecl on the slower 5 cpm plateaus.
R o l e of Interstitial Cells of C a j a l in Smooth Muscle
Nervous input in the gut is restricted to the ICC's and muscle cells a t the submucosal edge of the
circular muscle and the region between the longitudinal and circular muscle. Hence, a very few
number of cells are innervated and some other mechanism must be acting within the muscle layers.
ICC'shave been proposed as being the pacemaker cells of the gut[90,91]. They are small cells
occurring in the myenteric and submucosal plexuses, distinguishable from nerve cells and muscle
Myenteric
100%
-36
Figure 2.8: ECA f r o c various posilions fmm withzn wlonic canine circular muscle. The posiliori
zs defined as a pereentage fmm the submuwsal border. The myentenc border as located at 100%.
Repnnted fmm [87].
cells and share many connections with both including gap junctions. They have been proposed
as s w i n g a pacemaking role in the gut. Evidence for this is that strips of longitudinal and/or
circuiar muscie exhibit slow waves only when ICC's are present in the preparation. Interestingly,
ICC's were once considered electrically quiescent. Their relative scarcity and small size in relation
to the smooth muscle bulk preclude their role in direct electrical driving of the muscle. Instead,
it has been postulated that nitric oxide prcduced by ICC's acts as a coupling agent by lowering
intracellula calciurn[76] in a cyclic manner.
ICC's are considered to act through inhibition of smooth muscle cells. It has been suggested
that one manner in which ICC's achieve this is through nitric oxide (NO). Intracellular calcium
is decreased in smooth muscie cells while it is increased in ICC's in the presence of NO. ICC's
are posited to be NO amplifiers which respond to NO produced by non-adrenergic non-cholinergie
(NANC) nerves[77].
Coupling within the circular muscle is thought to be achieved through gap junctions which
are abundant throughout the tissue and link ICC's to muscle as well as muscle to muscle. However,
in longitudinal muscle gap junctions have not been conclusively demonstrated. Dye coupling has
not been proven nor have gap junctions appeared in keeze fracture although antibody testing has
shown small amount of connexon [32]. Hence, another coupling mechankm is thought to be involved
in longitudinal muscle.
2.4
2.4.1
Hippocampal Neurons
Physiology
The hippocampus is a structure found in the forebrain of mammals. It is implicated in memory
formation as hippocampal trauma interferes with the tramference of short term to long term
memory[lô]. Spatially, the hippocarnpus is weil organized lamina structure with a well defined
circuits (see figure 2.9). The dentate gyrus celk receive input from the perforant pathway. Output
fiom the dentate gyrus layer provide input to the CA3 region of the slice. CA1-CA4 neurons have
roughly pyramidal somata with basal and apical dendritic trees which receive thousands of inputs.
The apical tree is longer than the basilar tree, extending for 200 pm while dendrites run off the
basal side of the soma. The neurons are spatially polarized in the sense that they are parallel to
adjacent cells with the apical trees extending into the center of the hippocampus and the basal trees
24
(b) CA3 Neuron
(a) Hippocampai Slice
Figum 2.9: (a)Schematic of hippocampal sl2ce showing location and orientation of CA 1 and CA3
neumns. D.G.
- dentate granulus, PP - perforant pathway. Reprinted fmm
[92]. (b) Reconstruction
of CA3 neumn. Scale bar represents 50 Pm. Reprànted from [93].
pointing to the edge of the formation. Nerve celis actually reside between the more numerous gliai
cells which have several functions. Glia are well connected by gap junctions and serve regulatory
functions by acting as buffen against changes in the extracellular space[lô].
Pyramidal cells in the CA3 region have a resting level of -70 mV, action potentiais upstrokes
of more than 50 mV, maximum rate of rise of 100 V/s and a total duration of 5 ms (see figure 2.10).
These cells are also capable of spontaneous bursting on the single ce11 level, as opposed to the
pyramidal cells of the CA1 region which only oscillate when part of neuronal circuits.
2.4.2
Epileptiform Activity
Epilepsy is not a single disease but a set of disorders which can be classifieci according to areas of
the brain affected and syrnptoms displayed. The entire brain may be afflicted or only a particular
region. Effects range Born the simple where consciousness is not impaired and the epileptic may
undergo sensory or psychic symptoms to the extreme where consciousness is impaired and muscle
twitches, loss of muscle tone andior automatism are manifest.
Electrophysiologically, epilepsy is marked by the occurrence of regular activity in a normally
25
(a) Action potential
(b) Spikelet
Figure 2.10: Iniracellular hippocarnpal rewrdings
chaotic electroencephalogram
(EEG)[94]. This change is due to regions of the brain entraining to
a single frequency and this kequency imposing itself upon the EEG. DXerent parts of the brain
operate at different hequencies and are usually kept fkom interfering with each other. The brain,
like the gut, may be viewed as a system of coupled nonlinear oscillators. Some isolated neurons
spontaneously spike while others require input to spike. Circuits composed of the latter type
may spontaneously oscillate. In epilepsy, the coupling between neurons somehow changes to bring
about entrainrnent of these oscillators with the resulting impairment of normal functioning. The
hippocampus in an attractive mode1 for epilepsy as epileptiform act ivity can be induced t hrough
various means such as application of penicillin[94] and perfusion with a calcium 6.ee solution[95].
2.4.3
Coupling Mechanisms
Electrical coupiing of. cells has been proposed as a factor in spike entrainrnent of epileptiform
bursts[l5]. This type of coupling works through two distinct mechanisms: gap junctions and
extracellular fields. Glia are known to be extremely weil coupled to other glia by gap junctions
and form many connections with neurons. The dominant connexon is connexin43[22] but between
neurons, gap junctions are much less common and utilize another connexon other than connesin32
or c o ~ e x i n 4 3 .While both field and gap junctional coupling are known to be present in brain[l5].
the relative contribution of each mechanism is not known.
There have been observations of intracellularly recorded small amplitude waveforms w hich
seem to be the product of coupling [96,97,95].These waveforms, referred to as spikelets, are particularly interesting since they appear to be first derivatives of action potentials[95] which is indicative
of field coupling[29]. Furthermore, it was observeci that increased gap junctional conductance lead
to increased spikelet frequency but without a change in amplitude. Similarly, Perez-Velazquez et al
[98] observeci that the synchronization of hippocampal slices in
O-ca2as measured by field poten-
tials was abolished by gap junction blockers. Manipulations that caused intracellula. alkalinization,
which has been shown to increase dye coupling[99], resulted in increased syndironization and, conversely, blocking gap junctional rnechanisrns abolished synchronized epileptiform potentials. These
observations suggest that gap j-mctions play a role in synchronizing action potential firing in the
slice. The relative role of each type of coupling in producing spikelets is unknown at this point.
2.5
Summary
Coupled nonlinear oscillators display behaviour t hat is quite different kom t hat expressed intrinsically. In the extremes, the mode of operation may become chaotic or cease to oscillate. Examination of the underlying electrophysiology of the srnooth muscle cells show a system rife with intrinsic
frequency and resting level gradients that become especially significant when oscillator theory is
brought to bear. Electrical activity of srnooth muscle of the GI tract has been modelled by populations of coupled nonlinear oscillators using the mapped clock oscillator mode1[57, 58, 59, 13, 81.
To achieve entrainment, one oscillator need not drive the other but only perturb its state
space trajectory enough to alter its limit cycle. The extremely small stimuli needed for this has
been demonstrated[l4].
Gap junctions are insufficient to explain coupling in the gut for two reasons: (1) frequencies
above the highest intrinsic hequency are found in human and canine proximal intestine. (2) Failure
to demonstrate gap junctions in longitudinal layer where coupling is present. By elimination, eIectric
field coupling is posited as a strong contender to providing coupling between smooth muscle cells
in the GI tract. Morphology may be favourable to potentiation of any fields produced (which need
not be large) for entrainment.
Furthermore, hippocampal neurons may abo be viewed as an oscillatory system. Intra-
cellular recordings have recorded spikeIets which resemble differentiated action potentials. This
differentiation is indicative of field coupling. The laminar structure of the hippocampus lends itself to field coupling. Elucidation of this phenornenon is required as it rnay help determine the
conditions under which epileptiform activity is initiated or prolonged.
Chapter 3
The Boundary Element Method
The boundary element method is a numerical technique that can be used to solve a differential
equation defined over a region, not necessarily finite, by soiving for values on the boundary of the
region. Specifically, it will be used to solve potential problems formulated using Laplace's and
Poisson's equations.
The chief advantage of the method lies in the ability to completely describe a potential
field over a region by specification of the potential and its normal derivative on its boundary. In
effect, it yields a reduction in dimension. This will Iead to simpler problem formulation compared
to methods such as finite difference and finite elements, which rnust discretize the entire region.
Another consequence of t his method is t hat t here will be fewer variables for which to solve. However,
unlike the other two methods mentioned, the matrices generated are full, not sparse and thus, more
difficult to solve.
The final property that makes BEM attractive is that it rnay be used to solve unbounded
problems. The other two rnethods may be used for the same if a very large region is considered or
through ot her special techniques.
This chapter wili mathematically develop the
BEM, apply it to biological membranes and
outline the solution for a two body problem. Analytic formula were derived for several element
types. The final formulae are stated in this chapter and the full derivations given in Appendix A
.
Figuce 3.1: Schematic of BEM problem showing the region of interest, R, defined by the surface
r, vectors to the surface
(rs), interior (ri) and exterior
(ïe).
The interior and exterior have an
associa~edconductivity denoted by a.
3.1
Let
r
Mat hernatical Derivation
be a boundary surface which divides the domain into two regions, one interior and one
exterior (see figure 3.1). The region of interest shall be denoted 0. All quantities associateci with
the outside be denoted wit h the subscript "en while the subscript "i" shall be reserved for quant hies
holding to the interior of the surface. Let the exterior electric potentid be denoted by 4, (re) which
is a function of position. Laplace's equation may be written:
Define mot her function, g(r, ri), which is a funct ion of two points in space. Using standard vector
identit ies
Subtracting equation (3.2) from (3.3)) we arrive a t the relationship
Integration can now be performed over the region of interest.
which implies
where r, is the field point, ii is the outwardly directly normal of T' and the direction of integration
for the partial derivative. The subscript S denotes a point on î with variables of integration denoted
by a prime (').
The integrals on the right hand side of equation (3.6) may be interpreted as the decompe
sition of sources into two types of surface charge: doubIe layer, i.e. dipole sources, as accounted
for by the left term and single layer, Le. monopole sources, as represented by the right term. The
sources may be real or an quivalent distribution producing the same field as the actual sources.
The choice of the function, g, is still up to our discretion a t this point. The approach taken
is to sirnplifi the integration by choosing g such that it is a Green's function and the foliowing is
sat isfied:
Such a function c m be readily found and for three dimensions is given by
1
d r 7
For any exterior point not on
r t ) = 4+
- 91
r, the potential can then be written as
As the point of observation approaches a smooth portion of the surface, the normal derivative of g
approaches 27r and at the surface it rnay be subtracted from the left hand side to yield
If the surface undergoes a discontinuity, i.e. a corner, the normal derivative of g will approach a
value other than 2 7 ~at that point. Likewise, for the interior, the expression is very similar except the
normal derivative of g undergoes a discontinuity of 47r as the surface is crossed and the expression
becomes
To solve the above equation, ï is discretized into a set of n contiguous elements, each of which
has an associated potential and normal derivative of t he potential. Only zeroth order elements tvhich
have pulse b a i s functions to describe the potential and its derimtive will be considered, i.e., al1
values associated with an element are constant over the element and these values are discontinuous
between elements. As a result of this, the extracellular potential may be written as
where
r k is the portion of surface defined by element k.
Point collocation is the sirnplest method of solution and will be described. A system of
equations may be assembleci by writing equation (3.10)or (3.11) for the potential at the center of
each element. This will yield a system of n equations with Zn variables, n potentials and an q u a 1
nwnber of its normal derivatives. The potentials and their derivatives may be written as colurnn
matrices and the integrals become matrices of order n where the lt%olumn of the k* row describes
the effect at the center of element k of a unit source or source density placed on element k. In
matrix form, equations (3.10) and (3.11) can be written as
where I is the identity matrix, H is the dipole rnatrix and G is the monopole matrix.
There are several types of boundary conditions which may be specified:
Dirichlet The scalar potential on each element is given enabling solution of the normal derivative.
Neumann The normal derivative is given allowing for solution of potentials.
Impedance A relationship between the potential and its derivative is specified.
Different types of boundary conditions may be specified on different elements. Application of the
above boundary conditions reduces the number of unknowns to n, allowing for solution of the
boundary values of the system. Knowing the boundary values, the potential anywhere c m be
calculated from equation (3.9)for exterior points or its interior analog for interior points.
The above equations are Fredholm equations of the second type when solving for potential
given the normal derivative and Fredholm equations of the first kind when solving the normal
31
derivative given the potential. Quations of the second kind are more desirable for numerical
solution since t hey lead to greater diagonal dominance and hence, greater stabili~y[lOO].
3.1.1
Solution to Poisson's Equation
The above formulation assumed no current sources anywhere in the domain. If we remove this
assumption, we get Poisson's equation and write
where ia is the applied current source volume density. In general, this will Iead to a domain
integration which is undesirable. Fortunately, the current sources under considerat ion will eit her be
point sources which are represented by Riemann delta functions thus leading to a trivial integration
or placed on another surface which leads to another surface integral. In the case of point source
stimulation, the integral involving the Laplacian of 4, c m be written as
JJJ
=
g(re, r:)v2+(re)d~
R
Ma
4rue Ire - rai
where a monopole of strength Ma is located at r,. For an intracellular current source, the expression
is the same.
3.1.2
Notes on a Galerkin Formulation
Point coliocation is not the only method by which equation (3.10) or (3.11) may be solved[39]. Let
us introduce a n additional function, w(rl), which is a function of space. Both sides of equation (3.4)
may be multiplied by w and integration performed over S I twice, once for the primed and once for
the double primed coordinates:
The function w may be considered a weighting function which determines how the solution
is applied over the surface. If w is chosen to be the Riemann delta function, the method reduces
to point collocation. The Gaierkin formulation weights the potential over the entire element while
point coliocation only considers the solution at the center of each element. Hence, with the latter
method the potential may vary wildly between solution points. A Galerkin formulation attempts
to minimize this fluctuation between solution points.
32
For a Galerkin formulation, w must be chosen the same as the basis functions which will
lead to elimination of the first order error. However, the cost of computing the necessary matrices
become more expensive as we have to integrate over two areas. In general, Gaussian quadrature
will have to be performed as analytic solutions are not available for these integrations.
An additional advantage of the Galerkin formulation is the creation of symmetric matrices.
This requires approximately one half the nurnber of matrix entries n d be computed and furthermore, rnatrix solution methods rnay be used which take advantage of the symmetry to reduce
computation and storage requirements.
3.2
Application t o Biological Membranes
Unfortunately, surface potentials and their derivatives cannot be measured directly in a biological system. However, examination of the membrane and certain bulk measurements lead to an
appropriate problem formulation.
3.2.1
The Membrane
Biological cells are morphologically bounded by their membrane which is a lipid bilayer with an
approximate thickness of 1 nm[70]. This thickness will be ignored. The membrane divides the
space into intracelluiar and extracelluiar regions. Though each region is electrically neutral, there
are many ions present which make each region conductive.
Due to the insular nature of the protein, a capacitor is formed. Treating the membrane as
a parailel plate capacitor, a capacitance of about 1 p ~ / c m 2is obtained which has been verified
experimentally [l011. While the lipid itself is non-conducting, inserted into t his bilayer are proteins
which allow the transfer of ions from one region into the other. These proteins may be grouped
into three categories:
Channels are composed of protein subunits ringing a central aqueous pore allowing diffusion of
ions through the pore. Modulation of ion movement through the pore can be broken down
into two factors: permeation, the ion flux through the chamel when it is in an open state
and gating, the opening and closing of the channel by conformational changes of the constituent proteins. Permeation rnay be selective to one ion species only and is dependent on
the relative concentration of the ion on either side of the channel and voltage while con33
formational changes can be moddated by transmembrane voltage, pH, ion concentration,
neurotransrnitters, temperature, ATP concentration, mechanical stress and/or other agents.
Pumps move one or more ions across the membrane in a direction counter to the electrcxhemical
gradient thereby requiring the addition of an external energy source, namely adenosine
triphosphate (ATP). Two ionic species may be involved with transport in opposing directions.
Exchangers move one ionic species in one direction while moving another in the opposite direction.
Unlike pumps, there is no net energy required since the movement of one ion against the
gradient is cancelled by another moving with the gradient.
Many modek are available to model the ionic transport mechanisms. The earliest channel model
is the Hodgkin-Huxley model[lOl]. Later descriptions include the Goldman-Hodgkin-Katz current
equation, Eyring Rate Theory[70] and Statistical Rate Theory[l02].
3.2.2
Boundary Conditions
Of particular interest for voltage gated channels is the transmembrane voltage. This is the voltage
which exists between a point on the inside of the surface and the point on the extracellular surface
directly across from it. While this c a ~ obe
t measured, impaiement of a cell by an electrode aIlows
recording of the intracellular voltage which is a weighted average of the transmembrane voltage
over the whole cell. The transmembrane voltage is defined as
The other quantity of interest is the current density of the transmembrane current and is
denoted by i,.
This quantity is continuous across the membrane being the sum of the ionic and
capacitive current and can be expressed in terrns of the normal derivative of the potential:
where a is the conductivity, ç,, is the membrane capacitance per unit area, and iimic is the current
density due to ionic transport mechanisms. Note this latter quantity as previously stated can be a
Figure 3.2: Equiualent electn'cal circuit of membrane showing polarity convei.;.tions. V, is the
tmnsmembrane voltage, ,i
is the transmembrune currenl, c, is the
membrane cupacitance and
iionicis the current due to the ton tmnsport mechanisms.
function of the transmernbrane voltage as well as time. A complete description of al1 voltages and
currents is given in figure 3.2.
Often, electrodes are introduced intracellularly to inject currents or extracellularly to induce
electric fields. Their presence can be modelled by point sources.
Substituting the definition of ,i
and V, into equations (3.13) and (3.14) and assurning an
extracellular current source the system c m be expressed as
where
is a vector describing the potential produced on each element by any extracellular current
sources whet her they be caused by ekctrodes or nearby celk. Wit h simple algebraic manipulation
of the above equations along with (3.18),a relationship between v, and ,i
can be found:
Hence, when solving equation (3.231, the solution is valid for al1 space, both intracellular and extracellular. The conditions at the membrane link the two domains. Further algebraic manipulation
of equations (3.13) and (3.14) allow one to derive the values of qbi and &.,
Computation of the time course of the trammembrane voltage is possible if the current-
voltage relationship is known. A system of first order differential equations can be constructed:
h m
dt n =,2 - lionic
Ç
3.2.3
(3.24)
Two Ceii Problem
The object of these simulations is to compute the coupling between two celk. One cell is assumed
to be active and is denoted the source ce11 (see figure 3.3). A transmembrane voltage waveform is
assigned to the source cell which describes the transmembrane voltage at any point on the surface
of the ce11 as a function of time. An ongin of electrical activity is selected on the surface of the
cell horn which the transmembrane voltage waveform propagates as a surface depolarization wave
over the cell. The transmembrane voltage waveform is chosen in accordance with biological data.
Aence, the transmembrane voltage, over the source cell, v k , is known for ail time.
A second cell is assumed passive and is referred to as the receiving cell. An initial resting
transmembrane voltage is assigned to the receiving cell. The course of the transmembrane voltage
in this ce11 will be determined from field calculations.
Calculations begin with the source ceU. The transmembrane current density in the source
celi,
ik,can be calculated from (3.23)
by setting
c#ia
to zero. Knowledge of the current density
allows one to compute @: hom (3.22).
The transrnembrane current and extracellular surface
potential can then be used to compute
on the surface of the receiving cell. R o m
+a
and v k ,
the transmembrane current in the receiving cell, ik,can be calculated. In addition, knowledge of
u& allows computation of the ionic current density, iionjc.Finally, Erom (3.24), the ionic current
densities can be subtracted from the transmembrane current density to arrive at the rate of change
of the transmembrane voltage in the receiving cell, dvmldt. This derivative is then fed to a
differential equation solver. In summary, the known transmembrane voltage in the source ce11 allows
the computation of the derivative of the transmembrane voltage in the receiving cell. Knowiedge
of this derivative dong with the initial transrnembrane voltage in the receiving ceU allows one to
compute the evolution of the voltage.
3.3
Element Types
When discret izing surfaces, t here are several considerat ions:
Source
Receiving
Figure 3.3: Two ce12 computation scheme. The source ce11 (dark gray) has an origin of elect~cal
activàty (0)from whtch a surfice depolarization wave spfeads ouer the surface. The receiving ceIl
(light gray) is passive. The flow chart below the cells shows the computation steps for calculating
the rate of change of the receiving transmembmne voltage. h'nown values are shaded gray and the
destred quantity ut each tirne step is s h o w with a thick outline.
Elernent type: Many dEerent shapes of elements are possible with which to discretize the surface.
These include triangles, squares, curvilinear squares and cylinders. Tt is also possible to mix
types provided that the nodes which define the elements are properly connected.
Element order: By increasing the order of the element, the values associated with an element
vary over the element, i.e., with a second order element, a quadratic function approximates
the potentid over the surface. The higher the order, the more nodes that are required and
the higher the computation cost.
Element size: Finer discretization is better able to deal with regions of high spatial gradient but
a t a cost of storage and computation. Generally, the storage increases as 0(n2)and solution
time increases as 0(n3).
Ultimately, a tradeoff must be reached between the accuracy of solution and computational power at
hand. Techniques such as increasing discretization in regions of interest and using larger curvilinear
elements as opposed to more Bat elements over non-planar regions help reduce computation while
maintainhg accuracy. Generally, using fewer higher order elements is advantageous over more
lower order elements. A notable exception is the better performance of zeroth order over first order
elements.
While using analytic functions to compute matrix entries is desirable, they are not always
available. The following sections will describe the various elements used in this thesis and state the
associated formulae. Detailed derivations and explanation of all terms are available in Appendix A.
Al1 field points are assumed to be located at the origin and the element rotated so its normal or
axis in the case of cylindrical elements, only has a component in the z direction.
3.3.1
Xkiangular elements
The simplest three dimensional surface element is the planar triangle. We will consider the triangle
to be lying in an xy plane at a height z . Referring to figure 3.4 for labelling convention, and noting
that
< is a modulo 3 quantity, the sides of the triangle may be written in polar coordinates and are
defined by the start and stop angles, rpçTi, the minimum distance to the origin, Pt,and the angle
at which the minimum occurs,
Consequently,
Figure 3.4: îEongular element definitions. Side 3 is defined by vertices Ri and
R2 which are
located at (pi, <pi,z ) und ('fi,e,
z ) in cyiindrical cwrdinutes respectively. The closest the side
oppmaches the origin is
and occurs at angle @a. The stde is defined by p ( 9 ) = f i / COS(^ - 6 ) -
Similar quantitaes exbt for the other sides.
where
f!(a,p)
= min-'
=
l
z sin CY
dG2-
if the z-axis intercepts the element
27r
p{+2
O
- V(+I
if the z axis intercepts vertex 4
(3.23)
otherwise
A simple expression for the double layer potential produced by a triangular element was developed[l03]
and is given by
where ri, r2 and r3 are the vertices of the triangle.
39
Disc Elements
3.3.2
An analytic expression was not available for the potential produced by a disk. Given a disk of
radius r and centered a distance of po from the z-ais, it was possible to d u c e the dimension o l
the integral by one and then use Gaussian quadrature. For very close elements, 10 points proved
satisfactory. The exact form used to compute the rnatrix entries depend on the relationship between
r and po:
B = 2po cos < p \ l t 2
-
sin2p
(3.33)
The potential produced by a double layer on a disc is equimlent to the solid angle of an right
elliptical cone whose intersection with the z plane is a circle. Considering the elliptical cross
section of the cone, let the angle between the central axis and the vertices by denoted by a and
the angle between the central axis and the points on the rninor axis be denoted by 0. The rnatrix
entries can t hen be expressed analytically:
H l =
where
r
~
-21 tan -a2 sin p
and
1 rM2 are vectors Fom the field point to the vertices and r,l
and r,z
are vectors
to the points on the minor axis.
3.3.3
Cylindrical Elements
For a cylindrical element with two open ends, the monopole entry is done in a straightforward
marner. The openings are located at
away Eiom the z-axis.
zl
and
Q
with radii of r and the axis if the cylinder is located
DifTerentiating the above to anive at the dipole expression for Hkl is messy. By noting that
the solid angle of a closed surface is zero, it can be reasoned that the solid angle of a cylinder is
the negative of the sum of the solid angles of the two ends for which we can use equation (3.34).
See figure 3.5.
Figure 3.5: The solid angle of a surface is the sum of the solid angles of the portions which
comprise tt. The solzd angle o j a closed object (on the 1eJt) is identically zem.
Closed End
If one end of the cylinder is closed, it may be treated for monopolar calculations as a cylinder and
a disk. Dipolar calculations are simpler as oniy the disk defining the open end need be considered.
Bifurcation Elements
When modelling a tree structure, e.g. dendrites, it is important to be able to mode1 the bifurcation
it undergoes. To this end a Y-shaped element as seen in figure 3.6 can be used a t the intersection
of three cylindrical elements. For the single layer calculations it may be treated as three cylinders
and equation (3.37) and (4.10) used for each. There will be overlap of the three cylinders but this
is minimal if the cylinders are much longer than the overlapping region. By again noting that the
solid angle of an unclosed surface is the negative of the solid angle of the open ends, the solid angle
of such an oddly shaped structure simplifies greatly. The double layer matrix entry is simply the
negative of the surn of the solid angles of the three discs where the cylindrical elements attach.
3.3.4
Far Field Approximation
As the distance between the field point and the source element increases, the element increasingly
resembles a point. For monopole calculations, the error present in such a n approximation does not
lead to disastrous results and in general, the matrix entries can be approximated as
Figum 3.6: Bifurcation element composed of three cylinders with M e r i n g radii and whose ends
face a cammon point denoted by dot.
where At is the area of
rl.
For entries involving double layer sources, far field approximations were not made. This was
due to a property of the matrices that need be presemed: the sum of any row be zero. It was
possible to get cancellation of in at l e s t 8 and up to 14 decimal points by not using the far field
approximation. When using a far field approximation, it was not possible to get cancellation in as
many decimai places.
3.4
Numerical Computation
Given a system of equations to soive
There are several approaches to solve for x given b. The traditional one is Gauss-Jordan elimination
which proves to be the least efficient method when the same system is solved many times[t00].
Better methods involve d.xomposing the A matrix into several matrices with specific properties
that facilitate solution. Decomposition techniques also have the advantage that if one needs to
multiply by the original matrix after its decomposition, it can be done using the decomposed form
reducing storage requirernents.
A11 matrices generated will be full. For nonsingular, non-symmetric matrices, Lower Upper
decornposition
(LUD)ïs the method of choice while for nonsingular symmetric matrices Cholesky
decornposition works well. Fiaally, for singular or non-square matrices, Single Value Decomposition
(SVD) can be used[100]. As more limitations are removed bom the rnatrix to be decomposed.
more memory and computation are needed. Thus, SVD requires the most memory and the most
computation time. The other two decomposed matrices can be put inside their original matrix but
SVD requires a vector and matrix in addition to the original matrix space. Only square, nonsingular
matrices possess a true inverse. SVD generalizes the concept of inverses to include what are caUed
pseudo inverses. If A is of order M by
N,where M # hl, we would like to to find A-' su& that
where the order of the identity matrix in (3.41) is M while in (3.42) is Pd. There are MN entries in
A-' to solve M? + F"P entries in the identity matrices. In one of the preceding equations, depending
on whether M > N, there will be less entries to fit than equations describing them and only an
approximate identity rnatrix can be solved. SVD wilI optimize to find a pseudo inverse which will
give a least squares minimal error to the problem[l04]. If M = N,(3.41) will be unsolvable but a
least squares error approximation to it can still be made. Numerically, SVD is very important.
For solving differential equations, the LSODE[105] differential equation solver was used.
This solver is a variable step predictor corrector method with the advantage of being able to solve
stiff problems, i.e., ones in which components of the solution vector change at vastly different
rates. Many solvers are unable to handle stiff problems efficiently. Furt hermore, time step size
is automatically handIed by the routine ensuring that the largest possible steps are taken while
maintaining accuracy.
3.5
Error Measurement
Where possible, the answer produced by the BEM will be compared with an analytic solution. One
way of measuring the performance of the BEM is with a root mean square (RMS) error taken over
the boundary which alilows modefs of different discretizations to be validated and compared. If
there are W elements, with analytic solution q5(rS),and a BEM solution vector
is computed as
+, the RMS error
Chapter 4
The Electric Field Forrnulat ion
Previous integral formulation studies [47, 281 have applied Green's t heorem which was formulated
in terrns of potential, not electric field, and a Fredholm equation of the first kind was obtained.
Unfortunately, this class of equation does not possess as strong a diagonal dominance as Fredholm
equations of the second kind[100]. To achieve better numerical results, the latter type of equation
is sought and is the focus of this chapter.
A new formulation for electric field computation is presented which took the gradient of
Green's Theorern and used it as a b a i s to formulate the problem directly in terms of the electric field. In this manner, a FYedholm equation of the second kind was produced. Formulae for
computing the field produced by several element types were developed for both single and double
layer sources. This technique can be applied to determine the tirne course of the transrnernbrane
voltage for a ce11 as well as computing the extracellular electric field at any point. In addition. the
equations can be easily extended to include any number of celis, being Limited only by cornputer
speed and storage. Exarnples are given to illustrate this tedinique and to compare the electric field
formulation (EFF) with the potential formulation. The new method is shown to perform better.
In addition, formulae had to be derived to compute the necessary matrices. The formulae
are stated in this chapter with the full derivations given in Appendix A.
4.1
Development
As previously stated, equations (3.11) and (3.10) are Fredholm equations of the second kind for the
solution of the potential. It may be desirable to solve for the normal derivative, which is a scaled
44
version of the transmembrane current, and in such an instance, an equation of the second kind would
dso be desirable. This can be accomplished by taking the normal derivative of equations (3.10)
and (3.11)[106]:
These equations can be written in a matrix form when
r
is discretized into elements indexed by
the subscripts of the matrix entries:
with
corresponds to the monopoles and A corresponds to the dipoles. The matrix entries of the
EFF are contrasted with those of the potential formulation in the following table.
Potent ial
Source Type
Monopole
Gki =
/pl
Elect ric Field
g(.r, rst)fll
Dipole
Table 4.1: lntegml forms of the mat*
entries for the potential and electn'c field formulation.
Note that for the diagonal entries, Hkk= *kk.
Applying the definitions of transmembrane voltage (equation 3.18) and current (equation 3.19)
a simple expression is obtained:
where ea is the colurnn matrix of the normd component of the electric field due to any applied
current sources. The time course of the voltage may be computed through use of equation (3.24).
There is one less matrix multiplication on each side. The dxerence in performance wiil be investigated in section 4.3.
4.2
Element Types
Formulae for the sarne element types as given in 93.3 are presented here. Any variables used will
ais0 have been previously defined in that section. Equations in this section have been derived by
differentiating those of 53.3 and only the final results stated. Detailed derivations are available in
Appendix A.
4.2.1
'Itiangular elements
Referring to figure 3.4 for labeiling convention, and noting that E is a rnodulo 3 quantity,
and
1 V V N - NVV
Akl = 27r
v2 +Ar2
4-2.2
'
fik
Disc Elements
Calculation of the monopole matrix entry for disk eiements was performed by numerical integration.
It was found that ten point Gaussian quadrature is sufficient for the monopole computation.
4.2.3
Cylindrical Elements
Besides cylinders, cylindrical elements can be used to build branching tree-like structures such as
dendrites.
Open Ends
For a cylindrical element with two open ends, the monopole entry is done in a straightforward
manner. The openings are located at zl and 22 wit h radii of r and the axis if the cylinder is located
Given that the sum of the solid angles of the surfaces defining a closed surface is zero, the
surn of the deritvatives of the solid angles must also be zero. Hence, we may use the negative of
sum of equation (4.9) as applied to the open ends for calculation of Aki over the cylinder.
Closed End
If one end of the cylinder is closed, it may be treated for monopolar calculations as a cylinder and
a disk. Dipolar calculations are simpler as only the disk defining the open end need be considered.
Bifurcation Elements
Bifurcation elements are treated the same way as by the potential formulation, three cylinders for
monopole operations and three disks for dipole operations.
4.2.4
Far Field Approximation
For monopole calculations, the matrix entries can be approximated as
No far field approximations were used for the double layer sources.
4.3
Performance
Four computer simulations were run to demonstrate the accuracy of the formulae stated in the
previous section as well as the accuracy of the EFF. Known analytic solutions will be used to
validate the stated formulae. Firstly, the electric field of a disk, seen in Figure 4.la, was calculated.
Secondly, the trammembrane current was computed for a uniform double layer source distribution
47
and thirdly, for a spherical cell, seen in Figure 4.lb, the transmernbrane current density induced
by a point source stimulus was computed. The surface of the spherical ce11 was discretized into
704 triangular elements. The interior of the sphere had a conductivity of
medium in which it was imrnersed had a conductivity of a..
gi
while the infinite
Fourthly, a dynarnic simulation of
a passive membrane was performed for a biologically scaled sphere. The details of the potential
formulation, against which the proposed electric field formulation is compared, are given in the
appendix. Computation times given are for unoptimized C code compiled and run on a Silicon
Graphics 4D/20workstation running a t 12 MHz with 24 MB memory ( r a t 4 13 MIPs).
Figure 4.1: Boundary element models. A: Dzsk composed o j 50 elements. B: Sphere wmposed of
704 elernents.
4.3.1
Electric Field Calculation
A simpIe example with a closed form solution was used to test the eIectric field calculations. Three
methods were compared: the analytic expression for 9 (equation 4.6), 7 point two-dimensional
Gauss quadrature[l07] and a numerical expression for Q:
where 6 represents an incremental distance and was chosen to be 1/100 of the magnitude of the
distance vector. The last method is included to demonstrate the accuracy of equation (3.25). A
disk of radius one lying in the xy plane and centered at the origin was discretized into 50 triangular
48
elernents with a uniform monopolar density of 1 ~ / r n The
~ . electric field along the z a i s is given
b~
The errors in the three coordinate directions are given in figure 4.2 as the field point approaches
the disk along the z mis.
r
............
-
--'.
-- '--
7PtGQr
Numerical r
\
,
\
Figut-e 4.2: Error in computation of electric field of a disk lying in the x y plane centered at origin.
The field for each axial direction was computed using 7 point Gaussion quadrature (7 Pt GQJ, a
numerical diflerence of the potential (Numerical) or the analytic elemental expression (Analytical).
In the x and y directions, there should be no field. At far distances, al1 three methods
produced similar results. Moving doser to the disk, the difference rnethod started to break down.
The error grew as the disk was approached. At the closest distance used, the error was about 5
orders of magnitude worse than the analytic expression. The quadrature method error was only an
49
order of magnitude larger than the analytic method. It should be noted that the closest approach
given (10-~ m) is much closer than what would normally be required. Such a distance represents
less than 1/30 000 the length of an eiement edge.
In the z direction, the electric field calculated by all three methods was in good agreement
with theory until the 0.1 meter mark. It was a t this distance that the quadrature method started
to diverge and started producing results that were much too small. At distances less than 0.01 m,
the error was ciose to 100%. The other two methods continued to provide excellent answers, al1
the way down to the minimum distance but again, compared to the analytic rnethod, the difference
met hod was worse.
The analytic expression, as expected, gave the best overall performance. It gave a smaller
error in the x and y directions than the quadrature method, as well as a much better
;component.
Seven point Gaussian quadrature is quicker and gives answers comparable to the analytic expression
for points that are no closer than three time the Iength of an element edge. Closer than this distance
will result in greatly erroneous computations. Obviously, many difFerent choices for the number of
quadrature points exist. At farther distances fewer points are needed, and one may be sufficient
[log]. In these cases, the advantage of quadrature over the analytic expression is a savings in
computation. The simple clifference performed surprisingly well and is testimony to the accuracy
of equation (3.25) which is used in the potential formulation.
4.3.2
Uniform Sransmembrane Voltage
A sphere, as shown in figure 4.1B,with a uniform transmembrane voltage of one volt was used as
the first test to compare the field and potential formulations. The electric field at the surface of
the celi should be zero for this configuration. I, was set to zero in this case, so only the dipole
calculation was diecked. The EFF gave a result of 1.76 x 10-15 V/m in 192 s while the potential
formulation gave a field of 1.54 x 10-~V/m in 2649 S. It is evident that the EFF gives a better
estimate, producing an electric field that was eight orders of magnitude srnaller. The electric field
calculated by the potential formulation rnay seem at first glance a sfi-ciently good estimate. If
the geometry of the problem is scaled down, however, the ermr will be scaled that much larger.
Instead of a field of tens of nanovolts per meter, it may become millivolts per meter since biological
cell dimensions are on the order of micrometers.
Using the
EFF,the time to compute the electric field normal to each surface was almost
50
14
times quicker than that for the potential formulation. The potential formulation required computation of two matrices, squaring of one of them and then performing decomposition while the EFF
only had to compute one matrix.
4.3.3
Point Source Stimulation
A point current source of 1 A was placed at the center of the sphere. 1, was radially symmetric
and equal to 1/47r A/m2, independent of the conductivities. The transmembrane current was
calculated for the sphere and then integrated to determine the total current passing through the
sphere. Idealiy, this should be 1 A. Results are given in table 4.2. The EFF provided estimates
that were consistently better by a t least an order of magnitude.
Table 4.2: Relative ewor (96;1 in wrnputation of total tmnsmembmne current for point source
stimulus wtng the EFF and potential Jmnulatzon
(a).
A source of 1A was placed at the center of
the spherical ce21 and 1, was àntegrated ouer the surface for vanous conductivity ratios.
EFF
0.162
0.132
0.099
0.066
0.000
0.066
0.098
0.131
0.160
Source Position
The effect of moving the point source was considered. For a conductivity ratio of 0.1, the current
stimulus was moved dong the line with direction vector (1,1,1). In Figure 4.3, the total transmembrane current was computed for various points along the Line. Over the central region, the
EFF was much better. However, as the source approached the ce11 surface, the error in the electric
field formulation began to approach that of the potential formulation. Placing the stimulus close
51
-0.9
-0.6
-0.3
0.0
0.3
0.6
0.9
Source position ( m )
Figum 4.3: Effect of source position on error. A monopole was moued along a diagonal and the
total transmembrane current was calculated for the EFF und potential forrnulation
(a).
to the membrane means that a larger gradient must be devebped on the membrane. The spatial
discretization limits how well the puise basis functions can approximate the continuous gradient.
Because the electric field produced by the stimulus is more sensitive to distance than the potential,
the EFF will be more affectecl as the source approaches the surface.
Equal versus Unequd Conductivities
The computation time for the cases of equal and unequal conductivities were exptored. When the
conductivity ratio is unity, the sets of equations for both the electric field and potential formulations
reduce. The EFF reduces to
while the potential formulation expression
The operations affected are given in table 4.3.
A series of runs were performed for spherical cells of unit radius having a monopole source
placed at their centers. The nurnber of elements for each celi was different. For each cell, the
52
Table 4.3: Computational eflort requized for the potential
(a} and electric field formulations. The
upper portion of the table gives the eflort to genemte and decompose the required matrices (N=704)
whzch need only be done once. The time to perjonn al1 these colculations is the time to perjonn
a static solution and is rejemd to as the setup time. The lower portion details the wmputation
which mvst be perfonned ut mch time step of a dynamic simulation.
times to solve the transmembrane current by each method was calculated for conductivity ratios
of 1 and 2. Assuming that the computation time for each method could be described by a third
order polynomial since matrix operations are of order N* and IV3, four spheres were used and a
polynomial determined. Two additional spheres were then used to test the predictive value of the
equation. The actual computation times dXered from the predicted times by less than two percent.
The relative computation time (of the potential to the field formulation) is depicted in figure 4.4
for different nurnbers of elements. The times for the particular case of 704 elements is given in
table 4.3.
The EFF is slower at unequal conductivities for l e s than 132 elements. Above this point the
EFF becomes faster and reaches a maximum of 4.95 times faster than the potential formulation.
Again, equal conductivities are a special case and there is a separate curve in figure 4.4 for this
case. Here we note that the EFF is always faster and the ratio increases linearly.
4.3.4
Dynamic Simulation
An analytical expression for the transmembrane voltage induced in a spherical ce11 by a point
source stimulus has been derived[l09]. A ce11 of 10 pm radius with a membrane thickness of 10
nm, membrane conductivity of 3 x IO-* S/m, membrane capacitance of l C i ~ / c r nand
2 intracellular
53
Number of elements
Figure
4.4: Computation time dependence on number of elements for EFF and potential formu-
lation. The cases of equal and unequal conductivity ratio are shown. The vertical lzne indicated Ihe
number of elements for which computation times are equal for unequal conductivities.
conductivity 0.7 S/m was discretized into 704 elements (figure 4.1B). The extracellular conductivity
was chosen to be the same as the intracellular conductivity. A curent source of 1 nA placed at the
center of the sphere was abruptly turned on at time O. The transmembrane voltage time course was
computed for the first 0.02 s (about five membrane time constants) which represents sufficient time
for the membrane voltage to reach steady state. A variable step size predictor-corrector differential
equation solver was used to solve the system of first order differential equations [105].
The results are shown in figure 4.5. For bath methods, the RMS percentage difference
between the analytic solution and the numerical rnethod remained nearly constant over the entire
interval, changing only in the third decimat place and was about one percent. The EFF appears to
give a slightly larger error than the potential formulation; however, this is not the case. The solution
was compared to that obtained Çom a perfect sphere. Even with 704 elements, there will be a slight
discrepancy with the perfect sphere solution. This alone does not justify saying that the EFF really
gave a better solution. An even more important indicator of quality of solution is the nurnber of
function calk which calculate the vector of time derivatives given the transmembrane voltages and
currents. The EFF required 3553 function calls while the potential formulation required 5668 calls.
The srnoother (and more predictable) the function is, the larger the time steps the differential
54
O
5
10
15
20
Time ( ms )
Figure 4.5: Dynamic wmparison of formulations. The EFF and potential formulation
(a)
both
computed the time course of the tmnsrnembrane voltage of a ce11 undergoing a constant current
injection for which the analytic solution is avaiLable.
equation solver can take. The potential formulation required 60% more function calls signifying
that the solver detected more erratic behaviou in the solution and had to take smaller time steps.
Also, the potential formulation required computation in double precision. In single precision (4
bytes per number), the solver had crippling convergence problerns. For the EFF, double precision
produced better results more quickly but was not necessary.
The EFF required 6780 seconds of computation time while the potential formulation required
58380 seconds. Thus, the time to compute the transmembrane voltage was reduced by almost a
factor of 9 using the EFF. This is a very significant saving in time. Not only are the individual
function calls quicker in the case of the EFF,but fewer are required. Hence, an equally if not more
accurate solution was produced in much less time with the EFF which required half the storage.
4.4
Discussion
The new formulation[lOô] in terms of electric field results in a fiedholrn equation of the second
kind. This is desirable since the system to solve has strong diagonal dominance, which results in
better numerical stability and, consequently, better estimates, An indirect approach implies that
an extra operation rnust be performed to arrive at the desired quantity. With each additional
matrix multiplication, errors introduced into the rnatrix entries accumulate and grow. Thus, it
is not surprising that the results are worse for the potential formulation as it must perform a
differentiation operation of sorts on any potential through multiplication by the (112 - R) matrix.
If the intracellular and extracellular conduct ivity are equal, t hen equations (4.5) and (3.Z3)
greatly simplify to equations (4.14) and (4.15). The terms involving the single layer sources drop
out and it is not necessary to solve a set of equations to obtain the electric field or temporal
derivative of voltage. Simple multiplication of a rnatrix by a vector is al1 that is required. This is
clearly an advantage over the potential formulation which must solve a set of equations, regardless
of the conductivity values.
A cornparison of the cornputation requirements is given in table 3. The time to perform
a full stat ic solution which includes matrix generation, decomposition and multiplications, if any,
for the potential formulation is roughly 3600 seconds while for the electric field it is about 1600
seconds. This solution time for a static solution also represents the setup time for a dynamic
solution. The matrices need be assembled and manipulated but only once for a particular geometry.
Thus, if considering dynamic problems, it takes only 45% of the time to set up the problem.
F'urthermore, the EFF need only perform one matrix-vector muItiplication at each time step, not
two. If the conductivities are equai, the EFF saves even more computation as a back-substitution
is unnecessary.
The ratio of setup times for the two formulations is given in figure 4.4. For unequal conductivities, the potential formulation is quicker for systems with fewer than 132 elements. The
time to do the matrix multiplication is offset by the lower matrix generation tirne, resulting in an
overall quicker computation. With more elements, however, the EFF is quicker. As the number of
elements increases, the ratio of setup times converges to 4.95. Hence, this is the best possible reduction in set up time, notwit hstanding computer paging penalties due to increased memory access
of the potential formulation in cases where physical memory is exceeded. For equal conductivities,
the ratio of computation is unbounded, indicating the EFF problem grows at a slower rate. For the
EFF with equal conductivities, the problem is of order lV2, since only mztrix generation is required.
With the potential formulation, a linear set of equations must be solved which is an operation of
order N ~ Thus,
.
a s the number of elements increases, the ratio of setup times increases with an
order N.
56
Setup time can be further reduced by using a hybrid scheme to compute rnatrix entries.
h t e a d of only using the analytic solution, Gaussian quadrature could b e used for points which
are more than three times the element length away fkom the field point. Seven point quadrature
requires only 38% of the time to compute the
I' matrix compared to the analytic expression.
13
point quadrature requires 64% of the time. Depending on the distance between the field point and
the source element, differing degrees of quadrature, fkom one and up, allow for a trade-off between
cornputation time and accuracy. At larger distances, fewer points need b e used. Most elements are
separated by more than t h e e elernents, meaning solution time can be reduced further from that
given.
The cost of matrix operations in a function call at a particular time can also be analyzed. A
back substitution, required to solve a decomposed system of equations, requires approximately N~
multiplications. This means for unequal conductivities, the
EFF perforrns only 67% of the rnatriv
operations performed by the potential formulation and for equal conduct ivi ties, only 33%. Thus,
each time step is quicker using the EFF.
With a time-dependent problem, calculating the computation time ratio of the two methods
as above is not possible for several reasons. The above calculations include the costs of matrix
generation and decomposition which become less of a factor as the time of simulation is increased
and the cost of function calls dominate. Secondly, the cost of computing the transmembrane current
may be high and may be the rnost expensive component of function calls. Most importantly, there is
a dependence on the d8erentiaI equation solver. Variable step size solvers take the largest possible
time step wit hin an error tolerance. How the solver determines this is unique to the solver method.
The more a solut ion changes, the smaller the time step that can be taken. Computation errors add
"noise" to the system and result in a less stable solution. This error then reduces the possible step
size since it is adding an inconsistency to the solution. Hence, the number of function calls is a
more appropriate measure of the efficiency of computation. The dynamic simulation showed that
the EFF required only about 60% of the function calls of the potential formulation. Also, each
function call was much simpler, leading to an even greater savings in computation.
Using a potential formulation to arrive at a relationship between transmembrane curent and
voltage, one must either multiply fundamental rnatrices[47] or solve a system that is twice as large
since the solution vector is in terms of the surface potentials on the intracellular and extracellular
sides [28] (see appendix). Formulating the problem in terms of eiectric field is ûee fiom both of
57
these shortcomings. Considering a system of order N, matrix multiplication is of order N3 which
can get very costly since there are typicaiiy hundreds or thousands of elements in a model. In
fact, a n N 3 multiplication takes 4 times longer than a matrix decomposition (2055s vs. 519s).
Integral methods, which are a subset of the method of moments [110], tend to be limited by the
size of the matrices they generate. By using a system of twice the order to eliminate the matrix
multiplication, storage costs grow as the square of the order. Thus, four times the matrix entries
must be calculateci and four times the arnount of storage must be aliocated for the matrices. The
cost of solving the system is also of order N~ so it will take 8 times longer to determine the electric
field or do matrix decomposition. The new formulation is superior in these respects since it either
results in srnaller matrices or does not require square matrix multiplication.
To spare the need to compute the matrices for each simulation, matrices may be stored after
an initial computation. The potential formulation always requires two matrices to be stored (R and
G). For multiple ce11 problems, G-' is needed as well. With the EFF, two matrices need be stored
(A and
r) for unequal conductivities and multiple celis, but only one (A) for equal conductivities.
These matrices may easily be on the order of several megabytes and if several geometries are considered, storage becomes an issue. Intermediate calculations may require temporary matrices placing
further demands on rnemory. The matrix entries of the new method are slightly computationally
more expensive than the potential formulation but by no more than a factor of 1.5 for dipole sources
and 1.3 for monopole sources. These additional costs are still much less than the penalties paid for
matrix multiplication or doubling the system order.
The EFF oniy offers advantages over the potential formulation when only the transrnembrane voltage is desired and when the electric field is a quantity of interest. If surface potentials
are computed as well, the amount of computation required by each method is comparabIe. The
disadvantage of the EFF is only being able to calculate surface potentials within a constant. Looking at equations (4.3) and (4.4), kncwledge of ,i
is singular. SVD can be used to determine
requires inversion of A to determine 4i,, but A
within a constant which is suficient to compute
values at domain points since this dictates multiplication by another A matrix. This multiplication will ignore the effect of any bias in the potential vector. In conclusion, for best accuracy, the
EFF should be used to compute transmembrane voltages and the potential formulation to compute
surface potentials. If memory is a n issue, one method must be chosen based on the particular
quantities to be determined.
58
Chapter 5
Ionic Diffusion
Many morphological structures, e.g. interdigitat ions and intermediate contacts[79], bring regions
of the membrane within very close proxirnity to each other. If these structures do not prove
to be electrically significant then another pathway for coupling may be operating. It has been
suggested[l9, 72, 73) that potassium accumulation may play a significant role in coupling of cells.
The transmembrane voltage of a ceIl is a function of the ratios of the intra- to extracellular
ion concentrations[70]. Since the membrane is most permeable t o potassium at rest, the ratio
of the intra- to extracellular concentration is large (>IO), and the extracellular concentration of
potassium is low (3-10 mM), small changes in extracellular potassium can have significant effects
on the transmembrane voltage[70]. These concentrat ion changes can be direct ly mapped to voltage
changes[lll]and, in effect, are a type of electrical coupling.
Extracellular potassium increases lead to depolarization and increased excitability of the ce11
membrane. When depolarized, cells will be even more susceptible to small voltages induced by field
effects. Such structures as discussed, would limit the diffusional space and thus, prolong the effects
of any substance introduced into the extracellular space. The observation that smooth muscle cells
are capable of synthesizing nitric oxide means NO may be an important myotransmitter. It has
been demonstrated[76] that intracellular concentrations of calcium are reduced in smooth muscle
cefls when NO is introduced. NO has a very short lifetime which lirnits its diffusion distance as
well as confines its path for effective coupling. One structure that is of pai-ticular interest is the
interdigitation. Examples can be found where the intruding process makes a 90 degree bend and
continues to travel for several more micrometers [112].
This chapter examines the effect of interdigitations on the extracellular diffusion of ions. An
59
initial finite difference appraach is used with results showing that the rise in extracellular potassium
is significant. By extending a domain inversion technique to three dimensions,
BEM is explored as
a method to model diffusion as well. The results indicate it is possible but probkmatic.
5.1
Mathematical Development
For a potential field, whether electrical, diemical or thermal, the continuity equation is given by[113]
where J is the flux of the potential u. If ions are considered and electrical fields are present as weIl
as concentration gradients, the flux can be writ ten[lOl]
zFu
J = -D(VU + -V+)
RI'
(5.2)
where u is the concentration of the ionic species, D is the diffusion coefficient and 4 is the scalar
electric potential. It follows
where the dot product term is e q u i d e n t to a convection term. The last term on the right hand
side is zero since Laplace's equation is observed ele~trically~
To examine diffusion, a prelirninary model using a finite difference scheme was investigated.
A simple model of an interdigitation was used and preliminary calculations made. The BEhd was
then evaluated to determine its applicability to solving time-dependent diffusion equations.
5.2
Finite Difference Method
A preliminary investigation of the effects of the interdigitation on diffusion of potassium was undertaken using a simple one dimensional finite difference scheme. A very small gap was assumed
to exist between the intrusion and extrusion. Fbtational symmetry was assumed with a nominal
interdigitation length of 2.1 pm and extrusion radius of 0.3 p m with a gap of 10 nm between the two
ce11 membranes. The concentration of potassium at the open end of the structure (see figure 5.1)
was held constant at 10 rnM. The transmembrane flux of the ion was assumed to be triangular with
n ~
1 ms and declining to zero a t 2 rns and constant over the
time, peaking with a value of 9 ~ / r at
60
separation
End
I
1 End
Fàguve 5.1: Finite difference mode1 of interdigitation. a: The interdigitatfon fmnz the side. b:
End view showing radius of extrusion held constant ut 0.3 pm while separation distance was changed.
over the entire interdigitation. This waveform is an approximation of that produced by computer
simulations which took into account the sodium-potassium pump. This method is in agreement
with the method used for electrical field computations where the transmembrane voltage is assigned
on the source ce11 and takes into account al1 the unspecified ionic transport mechanisms. The interdigitation was divided into 40 extracellular compartments with the change of concentration in
the jth compartment given by:
where V is the volume of the compartment, N, is the number of particles introduced through the
membrane of the compartment, D is the diffusion coefficient, and A is the cross-sectional area of
the compartment. Finer discretization did not alter the resuIts indicating convergence had been
reached.
The results at three dXerent points for the nominally sized interdigitation are shown in
figure 5.2. The concentration at the closed end rises the most while that a t the open end hardly
changes. The point halfway along the interdigitation experiences an appreciable increase in concentration as well. As the interdigitation length is increased (see figure 5.3), the extracellular
concentration rises as well as lasts longer. The duration of the extracellular accumulation was measured by measuring how long the extracellular potassium stayed above 10.5 mM and it is denùted
61
Time (ms )
Figure 5.2: ExtmcelLular potassium concentration as a junction of time al three points dong the
interdigitation.
by td,,. As the length increased, the peak potassium concentration seemed to saturate. The time
of this elevated concentration, however, increased wit h length.
The two membranes of the cells were assumed to be a fixed distance apart over the interdigitation. The effect of changing this distance was next studied over the range 3-20 nm. The lower
lirnit is as close as two cells approach each other. This separation distance has a large effect on the
extracellular potassium (see figure 5.4). Both the peak concentration and
tdeV are
inversely pro-
portional to the separation. Unlike the length effect, the values did not saturate as they increased.
Also, 29 n m was the maximum separation that produced a 0.5 mM increase in the concentration.
Cornputer simulations of smooth muscle cells[lll] have shown the resting transmembrane
voltage to be sensitive to the extracellular potassium concentration (see figure 5.5). The change in
concentration can accordingly be converted into a change in transmembrane voltage. A change of Z
rnM will result in a membrane voltage change of more than one millivolt. Hence, if the concentration
changes by 4 mM as shown with decreasing membrane separation, the voltage change could be 6
mV.
This preliminary study indicates that the interdigitation can significantly affect diffusion of
1
O
2
Length ( prn )
3
4
Figure 5.3: Eflèct of interdigitation lengt A on peak eztracellular concentration and elevation time
for a membrane sepamtion of 10 nm.
10
'
'
4
P
,
I
8
12
16
20 '
Separation ( nm )
Figure 5.4: Eflect of membrane separntion on peak extracellular concentration and elevation time
for an interdigitation of length 2.1 Pm.
Figure 5.5: E f f e t of eztmcellular potassium concentration o n resting voltage. Adapled /rom [ I I 11.
substances introduced therein. However, the finite ?ifFerence method does not allow for the type
of modelling we wish to perform. For this, we must again consider BEM.
5.3
BEM Modelling
Starting with equation (5.3) and multiplying by a fundamental solution and integrating over al1
space as before, the following relat ionship is derived
where the normal derivative of the ionic concentration may be reiated to the transmembrane current
due to all ionic mechanisms
(5.6)
where F is Faraday's constant, ru is the valence of the ion and Ju is the current density. The caveat
associated with equation (5.3) is not this term however. It is the domain integral that must be
performed on the right hand side. Since we are dealing with an exterior problem, the integration
takes place over infinity. This is problematic since many BEM techniques like the Dual Reciprocity
Method (DRM) rely on approximat ing functions which only work in finite volumes[ll4].
An approach to handle this integral was put forward by Zhu and Zhang [115] based on
domain inversion. For two dimensions, the technique was shown to work. They proposed that in
64
Figure 5.6: ?kaansformation of original boundary (solid line) to new boundary (dashed line) under
domain inversion (or vice versa).
spherical coordinates, the radial coordinate r be mapped to its inverse:
The result of this mapping is that infinity is mapped to the origin and, hence, an exterior problem
becomes an interior one. An example of such a transformation is given in figure 5.6. Quantities
which have been transformed are denoted by a tilde accent
(3. In
the new coordinate system the
original problem is transformed into
After application of Green's Theorem, it is found
The development is given in greater detail appendix B. Dual Reciprocity, which is a reapplicatiori
of Green's Second t heorem, was applied to the terrn on the left. The effect of this on the temporal
derivative is as follows:
where V2represents the inverse of the Laplacian operator. The domain integral is now reduced
to a trivial domain integral and two surface integrals of unknown functions. At this point, DRM
65
dictates a known and nicely behaving function be fitted to approximate the function
and then
analytic operations can be performed on the approxirnating function. The potential function is
sirnilarly approximated. The choice of interpolation function is discussed in the next section. It is
essentid that a set of interpolating functions be used for whom an inverse Laplacian can be found.
5.3.1
Interpolating Functions
The choice of interpolating function will have an impact on the solution obtained. The function
may eit her be global, i-e. depending on globd coordinates only, or local, i.e. dependent on distances
between points. Examples of each of these types will be considered.
Local
Ernpirically it has been found[ll4] that interpolating functions of the forrn f =
x,dk work well
where d is the distance from a discretization point to a soIution point (see figure 5.7). After BEM
discretization, there will exist n points at which a solution is desired. Each solution point will have
an associated interpolation function and the solution at one of these points, j, is approximated by
where
Un
are coefficients to determined and dj,, is the distance from point j to point n. With n
equations and n unknowns, matrix notation becomes easier
A class of interpolation functions has also been found which provide good solutions[ll4] and
have the form
where w, the surnmation index, refers to an element in any subset of whole numbers. Furthermore,
the choice fn(dj,n) = 1 + djVnis the simplet while ensuring completeness of expansion. Little is
gained from adding more terms[ll4]. The resulting inverse Laplacian and its normal derivat ive in
3 dimensions are
where di,, is the displacement vector fkom point j to point n.
Global Interpolation Funct ions
Cheng et a1[116] r a i d the possibility of using an approximation based on a set of global interpoMing functions. In two dimensions it was shown to work better than local functions of the type
already discussed. In general, the functions are part of the series 1,x,y,z,x2,y2,~2,xy,xz,Yz,x3~y3,z3,1
x2y,x2z,y2x,y2z,z2xlz2Y,xYz,.
.. . The required number of functions are simply taken from the s e
ries, starting at 1. By finding the inverse Laplacian of the first terms in the series it was found by
inspection that for a function of the form f (1, ml n) = x' ymrnwith 1, rn and n integers, the inverse
Laplacian is given by
where terms not satis&ing the following are discarded
Integer division is performed for the summation limits, i.e. remainders are ignored. It has been
assumed 1
> m 2 n in the above. If this is not true, the coordinates may be permuted to order the
powers appropriately. Results were later confirmed with Mathematica[l04] to verify the validity of
equation (5.16). The inverse Laplacian is not unique for a given function. Due to differentiation
of the Laplacian, constants and terms which only have a first order dependence on a coordinate,
e.g. 3x, wilI disappear and thus, there are an infinite number of inverse Laplacians. Exarnples of
functions and the inverse Laplacians as computed by equation (5.16) are given in appendix (B).
Also, the assumption regarding the ordering of ml n and 1 is not necessary but leads to the inverse
Laplacian with the srnallest number of terrns. If you follow the pattern of the inverse Laplacian
presented, you will find that the series increases the x exponent while decreasing the exponents of
y and z, and when these latter exponents reach zero, the series stops. Ergo, decreasing the smallest
exponents will lead to the smallest series.
I t is also required that the normal derivative of the inverse Laplacian be computed. Direct
different iation of equat ion (5.16) yields:
Operations on Interpolation Functions
Two matrices Ü and
O are definecl by
It is now possible to approximate any necessary gradients as
VU = VFa
In matrix notation, the system to solve can be written (see appendix B)
with
Note that the equations from 54.2 can be used to calculate the gradients of H and G . Tlie
expressions are not dotted with a normal as before, but with the gradient of the concentration.
5.3.2
Boundary Conditions
Inverting the domain requires transforming the boundary conditions as well. The potential is
undianged by the transformation but the normal gradient is changed. The full derivation is in
68
appendix B with the final result
where the approximation of equation (5.22) has been used to determine the gradient of u. The
value of the potential at the origin is known since we know the value at i d n i t y of the unknown
function which yields another interpolation point. This datum may be used to help produce a
better approximation.
The approximation scheme as presented only makes use of boundary values. This may lead to
inaccurate results since the interior potential field is ignored. If based only on the boundary, the
approximating function rnay vary wildly in the interior with a large error yet match the boundary
points well. To overcome this inadequacy, solutions at points other than the boundary can be
used ( s e figure 5.7). These points are termed internal poles and lead to greater accuracy of the
approximating function at the cornputational cost of tracking the solution at the poles. There will
be an additionai interpolating function, f,, introduced at each pole which will provide a srnoother
overall approximating function since in general, more interpolating functions will provide a better
approximation. The initial conditions must be known at the poies as well. If there are n boundary
points and 1 poles, the resulting system will be O(n + 1 ) .
BN
Figure 5.7: Simplifieà boundary ( ï j showing 3 boundary nodes (BN) and internal pole (IP).
Distance between the points are used to calculate local interpolatang functions and approximate the
potential function at a point (open dot).
O
1
2
3
4
Time ( s )
Figure 5.8: Analytic solution of potential as a function of time for a sphere with mdzative bounday
condition and initial condation of zero.
5.4
Verification
To verify the efficacy and efficiency of the preceding development, a sarnple problem was attempted
for which the analytic solution was available. Given a sphere of unit radius, which is an eigensurface
under dornain inversion, with the boundary and initial conditions
the potential is given by
The solution is plotted in figure 5.8. Under the domain inversion, the boundary condit ion is simply
(5.30)
=u-1
Integration in time was carried out by a linear time interpolation scheme 121:
(T
- &S)
u(t
+ A t ) = GR"
ân
"'
(
T i - -5
where T is defined in table 5.2 and At is the time increment.
70
.'t)
u(t)
The number of elements was varied as weIl as the number of internal poles. Internal poles
were not added one at a time but in shells with at least 8 poles per shell. A sheU was defined by its
radius with points evenly spaced its surface. Generally, more shells were placed near the boundary
since the solution had the greatest gradient there.
Before a dynamic situation was at tempted, a test was made to deterrnine which interpolat ing
functions produced the best results. Given u and u, the normal derivative was computed. Rom
table 5.1, it is obvious that the local interpolating functions produce much better results. The
global functions proved unusable since the error level was much too large even at 704 elements'. To
achieve an approximation comparable to the local functions, over 1000 elements would be required.
Table 5.1:
RMS error
O/
local and global approximating functions as a function of discretiration.
1
n
1 Local
~iobal
Results of the dynamic simulation are given in figure 5.9. The RMS error is displayed as
a function of time far the minimum nurnber of po1es required to achieve a solution. As expected,
the initial error was large and then decreased as time progressed, converging to a final value.
The solution converged quicker as the number of model was discretized finer and reached a Iower
RMS error. If an insuficient nurnber of interna1 poles were used, the solution failed to converge,
regardless of the discretization of the surface. This minimum number of poles for a solution was
approximately the same as the number of elements in the model. Increasing the number of poles
did not irnprove the solution.
Domain inversion with dual reciprocity is also cornputationally very complex. Examination
of equation (5.23) shows that 6 matrix multiplications and one matrix inversion need to be performed. These are the costliest operations since they are all order O ( ( n+ z ) ~ ) The
.
addition of
the internal poles therefore increases the set up time of the problem by a factor of 8 and increases
memory consumption by a factor of 4.
'Using gIobai interpolating functions, it was not possibie to solve the dynamic systern.
71
Figum 5.9: RMS error for the-dependent diflwion problem. The number of elements used to
discretire the sphere and the minimum number of interna1 poles to achieve a solution for a given
surface discretization is indicated in the legend.
The intermediate matrix results were also analyzed at one point in time to determine how
well the approximating functions held. Using the analytic values of the potential function and its
derivatives, the various matrix products were computed. Due to the symmetry of the problem,
the domain integrals (5.10) could a t least be reduced to one dimension. By considering a sphere
centered at the origin (see figure 5-10), the domain integral for field points on the boundary could
be computed for functions with only radial dependencies, 3,(r), as foilows:
r23, (r) sin 0
dû dr
r q RR'
- 2Rr cos 8
Depending on F,(r), equation (5.34) was either computed analyticaily or numerically using the
Mathematica software package [104].The errors for various vectors are given in table 5.2.
As
expected, increasing the number of elements decreased the error in most of the vectors. For the
latter two columns, however, the error decreased and then increased again.
Figum 5.10: Domain integrul over a spherical volume of mdius R. Given the observation point
('lled dot), the distance ( d ) as a function of r and polar angle (8) only need be wnsidered over al1
domain points (open dot) due to symrnetry.
Table 5.2: Intemediate results ut time 0.1 for ttme dependent diflusion problem with nurnber
of interna1 poles egual to number of discretzzation points (n). D = IO(R~@'-'
T =H R ~~
G R- SD.
~ EThe numbers shown are the average o j the vector.
n
Du
SDu
(HR4 -SD)u
Tu
Tu-Gq
SY
68
9.2
-0.89
0.89
0.43
-0.22
-0.23
+ 2 ~ * and
)
5.5
Summary
While domain inversion 2nd dual reciprocity may have been sufficient to solve the example problem,
its applicability to other problems is severely questioned. This is for several reasons:
1. The domain d i s c r e t i d was the simplest one possible in three dimensions: the sphere. The
biological features of note represent local regions of interest and much more complicated morphology. If the technique required such fine discretization for a radially symmetric problem,
how will it fair for a highly irregular one?
2. The computational requirement of the method is large. Only a single celi was considered and
cornputations had to be performed in double precision. There are many matrices which must
be computed which rnay result in a trading off between memov and cornputation time to
conform to available computing power.
3. The use of interna1 poles will make an already large problem even larger.
4. The unit sphere is an eigensurface with the sirnplest transformation of boundary conditions
possible, a change of sign. Using more compiicated surfaces will necessitate more elaborate
mappings of boundaq conditions which, unlike the sample problem, rely on an approximation
of the radial component of t h e potential which will add more error to the solution.
5. If electrical and concentration fields are to be solved simultaneously, it is desirable to only
compute one boundary mesh. Domain inversion creates a second boundary on which the
electric potential problem may be solved but at reduced accuracy. If the both the interior
and exterior domains are to be solved simultaneously, there is no way to avoid solving two
meshes .
One way of perhaps improving the performance of this rnethod would be to use a more
appropriate Green's function to elirninate the troublesome domain terrns and reduce reliance on the
duai reciprocity approximation. Other kerneis have been derived for a number of other systems[l14,
117, 371. For example, it would be desirable to solve
10 ag(r, r')
~ ~ ~
r') +
( r ,
r
dr
+ F30g ( r ,
ri) = 6(r - r')
since this would leave only the domain integral associated with the time derivative.
74
In conclusion, it is possible to use domain inversion with the dual reciprocity boundary
element method to mode1 the convection-diffusion equation. Aowever, cornputation and storage
costs are large and mapping of boundary conditions may be problematic. It was this limit of
computational power that deterred further investigation of applying this technique to more reaiistic
probiems. More powerfui cornputers may be able to overcome this problem but the mapping of
boundary conditions remains theoreticaily more problematic.
Chapter 6
Spherical Models
6.1
Introduction
In this chapter, the extracellular potential due to a depolarization wave travelling on the surface
of a spherical, isolated, excitable ceU were investigated. A sphere was initially chosen since only its
radius was needed to describe its shape, and analytic solutions were available for certain situations
which allowed comparison with numericd results to gauge the accuracy of solution before more
complex geometries were rnodelled. More of an emphasis was placed on quantitative, rather than
qualitative results although dimensions chosen were within the biological ce11 realm.
The transmembrane voltage waveform was chosen in accordance with reported biological
data[ll8] of the ECA in the canine antrum. The depolarization of the surface of the ce11 was
assumed to start a t a single site and then spread at a velocity which was not necessarily unilornl.
Such depolarization was associated with intracellular and extracellular potentids. Thus, the tinw
course of the transmembrane depolarizat ion implied a time course for the extracellular potent ial
measured at a point just outside the ce11 where an adjacent ce11 might be located[29].
As mentioned previously, several structures are found between smooth muscle cells whicli
may play a mle in coupling[79]. One of these structures, the interdigitation, is a small rod-likc
appendage of one cell that intrudes into a pouch in a neighbouring cell and forms a region with a
very narrow cleft between the two cells. Interdigitations have d s o been observed in neuromuscular
junctions [119]. The effects of this structure upon coupling was examined by computing extracellular
fields and transmembrane voltages induced in neighbouring cells. The resul ts were compared to
simulations of ce11 pairs wit hout an interdigitation[30].
76
Model Description
6.2
The transmembrane voltage waveform is not assumed to spread a t a constant rate across the cell's
surface. Factors such as curvature[l20] and radius[lll] lead to nonuniform propagation velocit ies
and several propagation velocity profiles were considered. The velocities at three sites, the origin
(O), the point furthest fiom the origin, the terminus (T), and the point midway between the two,
the rnidpoint(M), can be used to fit a quadratic function to the velocity profile (see figure 6.1).
Thus, the propagation velocity, 8,can be expressed as a function of distance from the origin, x ,
such that
where a, b and c are the parameters of the velocity profile. Knowing the distance from the origin,
the time delay,
7,at
that site can be deterrnined as follows:
This allows the transmembrane voltage to be written as
Rearranging equat ions (3.21) and (3.22) produce an expression for the extracellular potent ial:
where
kr
is the ratio of conductivities,
Equation (6.5) enables computation of the exterior
surface pot ential given the transmembrane voltage, independent of the t ransmembrane current .
A quasi-static solution was obtained for the system. The system was solved at each instant in
time by using the current values of +e and v,on
the right hand side of equation (6.5) to determine
the new value of 9, on the left hand side and the time is incremented to compute the t ime course.
77
A time increment of 0.5
mç
gave good results as solutions were the same for successiveIy srnalier
tirne increments and ringing of the system about the final value was not observable on the plots,
unlike larger increments.
Knowing 9,,its derivative may be computed by rearranging equation (3.21). LUD was then
used to solve for the exterior surface potential derivative in the rearranged expression. Knowing
the values of 4, and its derivative on ï, the extracellular potential &(r) may be computed using
equation (3.9) for any site in the extracellular space,
The potential above also becomes the coupiing potential for an adjacent ce11
where 4:' is the vector of potentials produced at the center of elements of ce11 1 by ce11 O using
expression (6.6) and the superscript "r" denotes the receiving cell, i.e. the passive one.
Also, the propagation velocity of the intrusion could be set independently fiom the quadratic
fit to gauge the sensitivity of extracellular potentials to the changes of velocity in the intrusion. A
change in ionic channel density or radius of the intrusion may contribute to a change in propagation
velocity.
A sphere of 10 p m radius was used as the basic ce11 shape for all modeis in the cornputer
simulations. The sphere, intrusion and extrusion are shown in figure 6.1.
The intrusion was
produced by removing a small cylindrical volume from the cell. A standard size was used for most
simulations uniess otherwise noted with the length chosen a s
& the radius of the cell, l p m , and
the radius chosen to be 0.2pm with an area of 1.382 ,um2 . The extrusion was created by adding a
small cylindrical volume to the cell that was designed to fit inside the intrusion with a gap in the
order of a membrane thickness, 10 nm. The surfaces of the models were meshed into triangular
elements of zeroth order. For the source ceU models, the sphere model consisted of 704 elements,
the intrusion model 1112 elements and the extrusion model was composed of 1316 elements.
The interdigitation models contained so many elements because of the tight fit required
between models. The extrusion fit inside the intrusion with a smail gap. The circular opening was
approxirnated with straight line segments which connected nodes on the desired opening radius.
Chords connecting the nodes cut across the inside of the circle and reduced its effective radius.
78
Figum 6.1: Sphericcll
BEM models. a: Eztrusion. b: Intrusion. c: Sphere. d : Detail of
digitatzon structums. The origzn (O), midpoint (M) and teminus
propagation are indicated.
inîer-
(T)of transmembmne voltage
The relative amount that the effective radius is decreased is given by 1 - cos(?r/n) where n is the
nurnber of discretization points. Thus to decrease this error to l a s than IO%, 7 points needed
to be used to discretize the circular opening and this lead to a large nurnber of elements in the
interdigitation region. If there are not enough nodes used, the minimum intrusion radius will be
less than the maximum extrusion radius meaning the intracellular domains intersect, a nonsensical
situation. The extrusion need not be discretized as finely as the intrusion since if its effective radius
is decreased, it will still fit inside the intrusion but increasing the discretization produced a more
uniform gap.
The origin of electrical activity was chosen to maximize any field effects. Previous studies
have shown that for a sphere, points directly above the terminus and origin experience the greatest
fields[29] Thus, the origin of electrical activity for the intrusion and extrusion were the points on
the far side of the model, exactly opposite the structure of note. For the sphere, the extracellular
potential was computed on the far side of the ceU, away from the origin.
Source and receiving cells were positioned so that the closest parts of the ce11 were 10 nm
from each other. In the case of the interdigitation structure, this meant piacing the extrusion
inside the intrusion with a gap of 10 nrn on all sides. Receiving cell models did not contain as
many elements a s the source ce11 modeis, but the eIements were meshed much finer in the regions
near the source cell, sometimes containing as many as a quarter of the model elements in the
intrusion or extrusion structures. Thus, accuracy is maintaineci while reducing computation time.
The minimum number of elements used per receiving ce11 model was 370 in the case of a sphere.
Results frorn models with the same shape but different numbers of elements were used to check for
convergence. Cornputer simu1ation.s were performed on a Silicon Graphics IRIS workstation, with
run times which ranged from minutes for simple extracellular field computations to several hours
for determining induced voltages.
The extracelluiar potentials were computed at a distance of 10 n m bom the cell surface
because the closest distance between two cells was assumed to be on the order of a membrane
t hickness. Intracellular and extracellular conduct ivities of 0.75 S/m were used t hroughout the simulations. Only the upstroke phase of the transmembrane voltage was considered as it corresponded
to the production of the only significant extracellular potential over the time course of the depolarization waveform[29]. The upstroke was approximated by a linear rise of 1.4 V/s lasting 33 ms in
accordance wit h reported biological data [118]. Peak potent ials given refer to the spatietemporal
80
peak amplitude experienced a t a given point over the course of the upstroke. If no position for the
field point k expiicitly specified, then it is the terminus.
6.3
Quality of Solution
Gauging the accuracy of solution and verifying the correctness of solution is of paramount importance to the modeller. Several efforts were made in an attempt towards this end.
A measure of the quality of the solution is the instantanmus whole ceIl current. Since we are
solving Laplace's equation, the following must be obeyed since the intra- and extracellular media
are source free:
Examination of several runs showed that whole ce11 currents peaked in the picoarnp range. The
current, which should have been zero, showed ringing at the beginning and end of the voltage rise,
indicating that the method of solution was responsible for the transient error. The steady state
error was much lower than the peak, again in the picoamp range. Distributed over the whole cell,
the voltage produced by the current would be close to 10 nV. Hence, the error current was very
low and did not contribute a significant potential.
6.3.2
Validation of Induced Tkanemembrane Voltage
Analytical solutions to the problem of a dielectric spherical sheil placed in a uniform electric field
have been found based on the solution to Laplace's equation and have been experimentally veri-
fied for lipid bilayers[lP]. Since the extracellular medium conductivity is much greater than the
membrane conduct ivity, the analyt ical solut ion simplifies to:
VA = l.SRE(1 - e - 5 ) cos 0
w here
The electric field strength is given by E, R is the radius of the sphere and 8 is the polar angle with
respect to the electric field orientation. Assuming the electric field is orient4 in the z direction,
this situation can be duplicated by making 4, proportional to the z ordinate of the center of the
element. The results can be seen to be in excellent agreement wit h the analytical solution, being
The
. simulation ran with
within 1.2%. The time constant of the system is calculated to be 0 . 2 ~ ~
a time constant of 0.23~swhich agreed with the calculated time constant of the system.
6.3.3
Non-Reciprocal Field Effects
Equation (6.7) is one sided in the respect t hat the field produced by the receiving ce11 does not affect
that produced by the source cell, ignoring reciprocity. The complete solut ion (equations 16-20) was
compared to the non-reciprocal solution for the cases of adjacent spheres and the interdigitation
structure with the intrusion as the source and the extrusion as the source. Peak induced transmembrane voltage errors were 4.18%, 5.60 % and 4.00% respectively. The error is thus reasonably smail
and the use of the approximate solution can be justified when speed and simplicity are required.
Savings in computation time were very great as the complete solution required 2 CPU hours on the
faster computer and twice as much (at least 10 megabytes more for a reasonable model) runtime
memory which may incur a large page swapping penalty if mernory is exceeded.
6.4
Cornputer Simulations
R o m previous studies[29], it is known that the extracellular potentials are linearly dependent on
the surface voltage gradient which can be expressed as
The results in this section will vary linearly with the rate of rise of the waveform describing the
transmembrane voltage and inversely with propagation velocity. The peak extracellular potential
allows indirect cornparison of the induced transmembrane voltages to be expected since larger
potentials will produce larger gradients. As larger gradients produce larger extracellular potentials,
so larger applied extracellular potentials produce larger voltage gradients in the receiving cell.
F i g u e 6.2: Eflect of changing 8 on &racellular potential.
6.4.1
Effect of Propagation Velocity
T h e Sphere Mode1
A non-uniform velocity profile was considered. Three velocities specifjr the propagation velocity
profile of the sphere: the velocity at the origin, midway point and terminus. The effect of each
parameter was investigated while the other two parameters were kept constant at 1 cm/s. Changing
the origin propagation velocity did not have a very large effect on the extracellular potential.
Decreasing the velocity, increased the potential but not significantly. Varying the rnidpoint velocity
(figure6.2) had the largest effect on the cuve fitting equation and, consequently, the entire velocity
profile. Thus, it is to be expected that it would have the greatest effect on the extracellular
potential. Doubüng the midpoint velocity caused more than a 30% reduction in potential while
halving the velocity resulted in a 60% increase. The effect of the terminus propagation velocity
on the extracellular potential (figure 6.2) was such that doubling the terminus velocity resulted
in a 20% decrease in potential while halving it yielded a 44% increase in the peak extracellular
potential.
The Intrusion Mode1
Five dserent propagation velocity profiles were examined for the intrusion mode1 (Figure 6.3a).
Determinhg the exact profile is difficult and requires full knowledge of the geornetrical and mem-
brane properties. Traveling waves in excitable media tend to decrease speed in regions of negative
cunature and increase speed in regions of positive curvature[l20] which would result in a mon*
tonically increasing velocity profile when applied to a sphere (profiles 4 and 5 in figure 6.3a).
However, if this were the only factor, nerve impulses would decrease in velocity afier traversing
regions of negative curvature at the transition between srnalier and larger radius axonal segments.
ModeUing studies[l21] have shown that there is an initial slowdown followed by abrupt increase
brought about by the change in radius. Downplaying curvature effects, radius considerations would
dictate a slow-fast-slow profile (profile 1). Another study modelling short fibres[l23] suggested
that a more realistic profile may be an initial high veIocity, foliowed by a slow down and increase
near the end (profile 3). FinaUy, constant membrane properties have been assumed with all the
modelling studies. With nonuniform charnel distributions, a new factor is entered as membrane
properties become spatiaily dependent and maybe radically so[124]. With difFering cunature and
membrane propert ies, the velocity wili not be constant. nonet heless, a constant velocity profile
is also presented for cornparison purposes (profile 2). Al1 profiles shared an average velocity of 1
crn/s.
These profiles pertained to the spherical part of the mode1 and not the intrusion structure.
Along the intrusion, the propagation velocity was constant and could be chosen independently
A reference potential was defined to be the peak extracelrequiring a fourth parameter, einbUsim.
lular potential produced by a velocity profile when the intrusion velocity was equal to the velocity
at the sphere-cylinder interface, 8intmfam.
The reference potentials are shown in figure 6.3b for
the five velocity profiles. Figures 6.3 indicate that a decrease in the velocity near the intrusion
(field point), causes an increase in peak extracellular potential. A11 potentials were within 9% of
the constant velocity profile, suggesting that the total propagation time is much more important
than the exact profile.
The sensitivity of the potential to the intrusion velocity was also investigated (figure 6.3b).
Lowering the intrusion velocity (figure 6.3b) increased the peak extracellular potential with a 60%
reduction in velocity causing a maximal 30% increase in the extracellular potential. The profiles
which produced the greatest reference potentials were also the most sensitive to changes in the
intrusion propagation velocity.
(a) Propagation velocity profiles of spherical region of int.rcision morl~l
(b) V,uiat.iori in peak
O,
w i ~ hC3
Figure 6.3: Eflect oj propagation i~elocityprofile on extracellzrlar poteritial. ( a ) The propagation
rrelocity as a /unclion of position d o n g Ihe surface. ( h ) The peak ext~mcellzrlaryolential nolnznlited
iuith respecl to reference polentials (indicalecl hy arrows on the righl m i s ) ,versus the propugalion
uelocit?y in the interà2gitation normalited with respect to the uelocitg irnmediatelq adjacent the inI m ~ i o n(the rightmost unlzres o j the curves in ( a ) ) .
85
Figure 6.4: E B c t of separation on VA. Sepamlion is measumd as the closest distance between
points on each model.
Induced Tkammembrane Voltage
6.4.2
The effect of distance between cells is seen for the case of interdigitation with the extrusion as the
source in figure 6.4. For small distances, the induced voltage is much greater for the interdigitation
than the spheres. As the two cells are pulled apart, the induced voltage rapidly drops to that of
the sphere-sphere case. It is only at the very small separat ion distances t hat the large voltages are
induced.
The configuration which produced the largest induced voltage was the interdigitation with
the extrusion as the source . Compared to the sphere coupling case, a t a length of 1 pm, the voltage
was increased by over 30% to 1.32 mV and compared to the interdigitation with the intrusion as
a source, there was approximately a 10% irnprovement over the induced voltage of 1.2 mV. This
may seem surprising as the extrusion produced extracellular potentials that were far smaller than
the intrusion and not significantly different from the sphere. Clearly, the electric field and receiving
ceil geometry are very important, not just the magnitude of the potential sensed a t the membrane.
6.4.3
Interdigitation Size
Both the radius and the length of the interdigitation structure were varied with both source configurations. As the length was increased from O to 2 pm, with the radius constant at 0.2 p m
(figure 6.5), the peak extracellular potential increased &om 0.7 mV to 1.33 mV for the intrusion
86
,
O
'
100
~ntnrsionsource 1
I
Figure 6.5: Effect of interdigitation length on peak VA and 4,. Both models were considered as
the source.
source. The induced voltage increased also, albeit at a slower rate than the potential. The induced
voltage increased hom 1 mV to 1.4 rnV for the intrusion source, a 40% increase, while the potential
increased by almost 90%. With no interdigitation, the induced voltage is 50% higher than the
extracellular potential. This increase, however, decreases wit h increasing intrusion lengt h unt il at
2 Pm,the increase is only 5%. The induced voltage is therefore, not simply dependent on the peak
extracellular potent ial.
With the extrusion as the source, the peak extracellular potential did not increase with
increasing length. The field of the extrusion is the essentially the sarne as the sphere for al1 points
not contained in the extrusion. A simulation was run with the sphere as the source and the intrusion
as the receiving ce11 to verify this. This result differed by 6% from the extrusion/intrusion case.
The induced voltage in the intrusion is, however, greater than in the extrusion, As the length
of interdigitation is increased, this difference grows until at 2 Pm, there is a 15% increase. It is
interesting to note that the induced voltage in the intrusion mode1 follows the same shape as the
peak extracellular potential generated by the intrusion.
Changing the radius size at a length of 1 pm had a small effect on the induced voltage with
the smaller radii inducing siightly larger voltages than the larger radii (Figure 6.6). The increase
was not significant, however
and may be attributable to discretization error.
0.6
0.10
L
I
I
0.15
0.20
0.25
0.6
Radius ( pm )
Figum 6.6: Eflect of tnterdigitation radius on peak VA and 4, for an tnterdigitation length of
1 Pm.
6.5
Discussion
To achieve greater field COupling between cells, the ext racellular pot ent ial generated by cells sho uld
be rnaximized. Cells utilizing this form of coupling would be expected to have developed certain
characteristics that increase the voltage gradient along the surface of the cell. These characteristics
include a) slow propagation velocities especially near sites of coupling b) fast rates of depolarization
and c) low extracellular conductivities as decreasing the extracellular conductivity increases the
extraceliular potential because the exterior surface potential depends the ratio of the intracellular
to extraceliular conductivities. As the external conductivity approaches zero, the external current
must sornehow limit itself or the extracellular potential will approach infinity. This in turn limi ts the
intracellular current and the surface potentials. The gradient effect is verified in table 6.1. Identical
gradients give near identical extracellular potentials while a halving of the gradient produced a
having of the potential.
Two regions of the ce11 seem the most suitable sites at which field coupling is feasible: the
origin and terminus. It is here t hat the extracellular potential is proportional to the first derivative
of V, and the potential is at a maximum both in duration and magnitude. Compared with the
end points, the midpoint pulse lasts only 1/10 as long and is 1/5 of the magnitude. In addition,
Table 6.1: Relationshzp between surface voltage gradient and peak eztmcellular potential for a
sphere (rc = 1).
certain geornetrical features like an intrusion and close proximity might also be indicative of field
coupling.
Nonuniform propagation velocity profiles produced po tentials that were greater or less t han
those produced by a uniform velocity q u a 1 to the average velocity. The region near the field point,
in this case above the intrusion and near the terminus, was the most important. Profiles with lower
velocit ies in this region produced greater potent ials. The largest reference potential, produced by
profile 1, was 0.95 rnV while the smallest reference potential, produced by profile 5, was 0.844 mV.
The profiles producing the largest reference potentials were also the most sensitive t O changes
in the intrusion propagation velocity. These profiles produced the largest increases in extracellular
potential when the intrusion velocity was decreased. This is to be expected as there is an inverse
dependence on the propagation velocity. Dserentiat ion of this term to determine sensit ivity yields
a derivative proportional to the inverse squared which wiIl increase as the propagation velocity
decreases. The profiles producing the largest potential have the lowest propagation velocities and,
hence, the highest sensitivity to changes in the propagation velocity. By lowering the intrusion
velocity and adjusting the propagation velocity profile while keeping the average velocity constant,
the extracellular potential can be significantly increased.
6.5.1
Effect of Interdigitations
The extracellular potential produced by the intrusion mode1 was much greater t han t hat prod uced
by the sphere and extrusion models. Even though the intrusion represented a distance of
$J of
the diameter of the sphere, it produced an extracellular potential that was close to double that of
the sphere. Noting that away h m the structures, the intrusion and extrusion, the extracellular
89
F i g u e 6.7: Estimate oj e&t
o j interdigitation length on peak induced voltage considering both
electric field wupltng and potassium accumulation. The contribution of each mechanzsm is shown.
potentials were identical to those produced by the sphere, it seerns that what is important is the
proximity to the center of the sphere. Any equivalent source would be placed in the center of the
sphere and the intrusion structure is the closest to the source while the extrusion is the farthest
away. This point is further illustrated in figure 6.6 where the peak potential is plotted as a function
of intrusion radius. The potential rernains fairly constant as the radius is varied at a constant
length. It should still be noted that the small radii produce slightly higher potentials. Taken to
the limit, as the radius is made large, eventually the edge of the sphere is reached, at which point
the sphere becomes truncated, and a srnailer field is obviously produced.
The effect of potassium accumulation as discussed in $5.2 shouId also be incorporated into
the efféct of the interdigitation. Using graph 5.5 to translate the concentration into voltage. an
estimate of the induced voltage can be produced which includes both electric field and potassiuiii
effects (figure 6.7). The induced voltage produced by an interdigitation is drastically greater tlian
that produced by a sphere when the two effects are summed.
Besides the interdigitation, there are several other structures in smooth muscle cells whose
chief characteristic is a very narrow cleft between two muscle cells. These structures include intermediate contacts and appositions. If field effects are to be important, they will only be so at very
close distances.
90
6.5.2
Significance of the Induced Tkansmembrane Voltage
The extracellular potentials produced transmembrane voltages of larger magnitude. These peak
voltages were in a region very close to the source cells. The receiving ce11 must set up a reaction field,
by redistribution of surface charge, to counteract the current which is flowing into the relatively
non-conductive membrane. The reaction field must divert most current away from the ce11 which
requires a reaction field directly opposite to the source field and stronger.
Starting with equation (6.7) and assurning the time constant is fast enough to be ignored,
allowing aV&/ût to be set to 0, and assurning the term involving conductivity is negligible (a
reasonable approximation), a simple expression results:
In the case of no electric fieid, signified by al1 & entries being the same, the resulting VA wiU
simply be the negative of
.:9
For VA to be rnaximized, the change in
must be maximized.
The surface voltage gradient of the source cell will produce an electric field which will, in turn,
produce a surface voltage gradient on the receiving cell. The transmembrane voltage at the end
of the interdigitation structure will be the Line integral of the surface voltage gradient. Thus, the
source transmembrane voltage will be differentiated, attenuated, and t hen reintegrated to obtain
the peak induced voltage.
An aspect of reciprocity is illustrated in figure 6.5 where the intrusion model both produces
the greatest extracellular potential and experiences the greatest transmembrane voltages. While
the extrusion produces neither a greater extracellular potential, nor a greater electric field as the
interdigitation is elongated, the intrusion converts the greater length of electric field sensed into
a larger transmembrane voltage. As the length of the interdigitation is increased, the electric
field at bottom of the intrusion becomes weaker explaining why the derivative with respect to
length of the induced voltage becomes smaller: the incremental increase in length produces a
region of increasingly smaller electric field strength which will add less to the total voltage than
the preceding section. The extrusion model, conversely, experiences an increased eIectric field as
the interdigitation length is increased. Countering this increased field is a reduced sensit ivity to
the field at the tip of the extrusion, where the field is most intense, as the extrusion is lengthened
resulting in the extrusion model not producing as great a voltage when subjected to an electric
field as the intrusion. As it generates a smaller extracellular potential, so it experiences a smaller
91
induced voltage.
The coupling is not symmetric at the intrusion/extrusion structure. The extrusion will have
a greater effect on the intrusion than vice versa. The effect of non-symmetric coupling has not been
studied in detail. As well as a magnitude difference, there may be a polarity difference depending
on the origin of the source transmembrane voltage wave.
The longer the interdigitation structure, the greater the induced voltage will be. The sensitivity of the peak induced transmembrane voltage to the radius is small, but, if peak voltage is
the measure chosen, smaller radii are preferable. Looking at just the peak value does not take into
account the notion that a certain area of membrane must be stirnulated for an effect to manifest.
Therefore, while smaller radii produce larger voltages, their region of affect is that much smaller
and rnay be insufficient to elicit a change in neighbouring oscillators.
Cornputer simulations of gastric ECA oscillators 1121 have shown that smali coupling strengths
are needed to entrain two gastric oscillators. Coupling strength is defined as the ratio of the induced transmembrane voltage to the transmit ted voltage upstroke. Rom the simulations of gastric
oscillators[l2], the peak received voltage needed for entrainment was in the range of 0.2-1.1 mV,
representing coupling factors in the range of 0.01-0.03, a relatively small quantity dependent upon
the intrinsic period difference between the oscillators. For these small period differences, required
coupling factors are also smail. Considering the number of cells which comprise the frequency
gradient of the stomach, one would expect adjacent cells to have similar frequencies. With received
voltages of up to 1.4 mV the entire regime of nonlinear system behaviour (entrainment, periodicity,
quasiperiodicity and chaos) would be available.
There is sorne difficulty, however, in comparing the induced voltages to those required for
entraining the gastric oscillator models. The induced voltage is significant in only a very smali
region. Only if this region were specialized in some manner could it affect the whole cell. Also, the
propagation velocity is unknown and could be greater than that used.
Chapter 7
C ylindrical Models
The previous chapter dealt with spherical cells which are poor approximations to smooth muscle
ce&. Cylinders are much closer to the actual ellipsoidal shapes of the cells and are used in this
chapter.
The BEM equations for both the potential and electric field formulations are given for two
cells coupled by both gap junctions and electric fields. In addition, the potential formulation
equations are extended to a n arbitrary nurnber of cells. Extension of the EFF can be made by
inspection. These equations are wed to determine the effect of gap junctions on smooth muscle
coupling.
Next, the hitherto unexamined assumption of an infinite volume conductor is tested. The
effect on field coupling of placing two cells in a bounded volume is examined initially for sphercs
and then cylinders with similar results obtained for the differing geornetries.
Lastly, a synchronized group of source cells will be considered as opposed to only one as
has previously been done. This also raises the issue of the relative orientation between the source
and receiving cells that was not possible using spherical models. With the differing orientatioiis
of the circular and longitudinal muscle found in w'vo, many coupling configurations are operating.
Four different coupling configurations are identified and investigated. The relative efficacy of thc
configurations is determined as weli as the effect of the number of source cells on induced voltage.
7.1 Two Surface BEM
7.1.1
Potential Formulation
Given two surfaces, labeiled O and 1, and using the potential formulation, the following set of
equations which relate transrnembrane potent ial and current may be written out:
The superscript 01 refers to the effect of ce11 1 on ce11 O and vice versa.
t$y is a term representing
the effect of gap junctions on ce11 O. It can be viewed as an applied current source placed on the
membrane directly. If element rn of ce11 O is connected to element n of ce11 1 through a gap junction
of conductance g j , the effect seen at element 1 of ce11 O is
while the effect on element k of cell 1 is
where A is the elemental area.
Solving the above system to isolate the potential in terms of the transmembrane voltage, it
is found that
This equation takes into account bot h field and gap junctional coupling and represents a Fredholm
equation of the second kind for finding the intracellular surface potential kom the transmembrane
voltage. If the number of matrices stored and computed is to be minimized, a Fredholm equation
of the first kind must be solved to determine the transmembrane current density:
Note that diagonal elements from the matrix multiplying v, in (7.7) can be used when determining
the current density in (7.8). If memory requirements are not a scarce resource, the EFF rnay be
used to obtain a Fredholm equation of the second kind for :,i
The derivative of the transmembrane voltage for each ce11 is found in the regular way.
For the element which contains the gap junction, iianic is equal to the current through the gap
junction plus the m e n t through any non gap junction charnels.
Given a system with & elements in model O and W1 elements in model 1, the system required
to solve 4i will be O( (No+ NI)?) in size and computation time at each time step. Finding ,i
simpler as the problem may now be solved on a cell by ce11 basis with O(@
+
is
) requirements in
totaI, less than solving for c $ ~ .
7.1.2
Extension to N+l Surfaces
Though not undertaken, it is possible to couple more than two cells together with this scheme.
By writing down equations similar to equations (7.1) to (7.4) for each cell, equation (7.7) is easily
extended. For N+1 cells, the system to solve becornes
7.1.3
Electric Field Formulation
By writing the EFF analogues to equations (7.1) to (7.4), the EFF can also be used to solve the
problem wit h the final result
There is one complication in this formulation which is Iacking in the potential formulation. The
A matrix is singular and, hence, standard inversion cannot be used. Instead, SVD must be used
to find the pseudo inverse of A. This situation arises since the A matrices essentially difFerentiate
the surface potential and remove any d.c. bias from the potential vector in the process. Thus,
the inverses of these matrices are singular since the original value can only be determined within a
constant. This is not a problem, however, as the pseudo inverse is always premultiplied by another
A matrix which eliminates any constants. The cost of finding a pseudo inverse is several times more
computationally expensive as finding the inverse of a non-singular matrix since SVD is much more
demanding than LUD or Cholesky decornposition for symmetric matrices[lOO]. As the potential
formulation was extended to more than two surfaces, so can the EFF.
96
7.2 Results
The basic mode1 used for SMCs in this chapter is the cylinder with a radius that is one twentieth
of the length (figure Tl).
Figure 7 . 1 : Cylindrical BEM mode1 of a SMC composed of 132 elements and shown at a slzghtly
oblique angle to display the discretitation ut the end (on lefi).
The EFF, potential and mixed formulation were used to solve the problems. There was
no significant difference in solution for the three methods. For the gap junction results, the EFF
was compared with the potential formulation to solve for
4iand
using the EFF to solve for .,i
The EFF required 16022 function evaluations to solve the system while the potent ial formulation
required 18966, 12.5% more than the EFF. There was not a very large difference in perfarmance.
7.2.1
Effect of Gap Junctions
Since gap junctions are present at the borders of circuIar muscle, the effect of gap junctions were
considered. Gap junctions have a closing time constant on the order of seconds when a voltage is
applied across the gap[71]. Since the depolarization plateau of the ECA lasts seconds, the closing
of the gap must be taken into account. The effects of gap junctions on 200 prn long cylindrical
ceus with radii of 10 pm placed end to end were modelled (see figure 7.3). The transmembrane
vdtage (figure 7.2) had the following characteristics which fall within the typical range of srnooth
muscle parameters: voltage rise, 1V/s; plateau amplitude, 30 mV; plateau duration, 6s; rate of €au,
0.25 V/s. The receiving ce11 was passive with a conductance of 0.025 s/rn2[125] and a membrane
capacitance of 0.01 F/rn2.
As the gap junction conductance is increased, the closing of the channel impacts less upon
the received voltage. For a very large conductance (100 nS), it had very little effect. The field
efTect can be seen as a negative going spike preceding the upstroke. Essentially, the position of
the channel does not matter nor do the relative position and orientation of the cells affects gap
junction induced voltages, unlike field coupling. The induced voltage is also relatively constant
-70
'
2
O
1
6
4
Time ( s )
Figure 7.2: Mode1 ECA w a v e f o n used to describe transrnembrane voltage in source cells for al1
simulatOons in thzs chapter.
O
2
4
6
8
Time ( s )
Figum 7.3: Gap junctional coupling of SMCs for various conductance values. The tmnsrnembrane
voltage of the recezving ce11 due to both fields and a gap junction with a closing tirne wnstant of 2
seconds i s s h o w .
over the surface of the cell, unlike field coupling.
7.2.2
Finite Volume Conductor
Up to this point, al1 cens modelled have been immersed in an infinite medium. The body, on the
other hand, is quite weil bounded in space, with an abrupt discontinuity in conductivity and, hence,
the approximation made up to this point may not be valid. The body can be said to fit inside a
sphere of 1 m radius [4]. The d e c t of a finite volume on the induced voltage was determined by
adding an enclosing boundary surface. The superscript "b" denotes quantities associated with this
surface. Referring to figure 7.4, we see space is divided into four regions:
1. The infinite exterior domain,
O,, for which we require no solution.
2. The domain inside the ceii O,
no,bounded by r0 with conductivity 0;.
3. The domain inside the ce11 1, R1, bounded by î1with conductivity
4. The bounded domain enclosing the cells,
nb,with conductivity
0
i
a
.
.
This has always been
placed at oo until now.
rb,the validity of the infinite approach can now be gauged.
no current Rows across rb,Green's theorern for the bounding surface can be
Taking into account
Assuming
written as
and this can be incorporated into equation (7.7)
i,
S M must be used to solve the system since the matrix H ~ +which comprises the lower
right portion of the left hand side matrix, is singular since the results are not unique. This arises
Figum 7.4: Two surfaces (ro and
rl) in a volume bounded by rb,two intracellular domains, Ro
and Ri, one extrucellular domain, Rb (shaded), and one unbounded domain, 0,. The wnductivity
of the intracellular m e d b fs cq while that of the eztracellular space is a,. Surface normals are
indzcated (ri).
since interior current flow is dependent on potential differences existing between portions af the
interior surface. If the surface was at a constant potential, no currerit would flow. Hence, an idinite
number of potential distributions will produce any given current flow, only differing by a constant.
In this case, only one singular value will be set to zero.
Two cylinders were placed end to end and enclosed by a box (see figure 7.5). The origin of
depolarization on the source ce11 was the center of the end cap farthest away from the receiving
cell. The depolarization wave travelled at 1 cm/s, producing a 100 V/m transmembrane voltage
gradient along the surface. The effect of the box on the induced voltage is seen in figure 7.5. The
induced voltage, remained constant for small gaps then it increased with increasing gap length
in a logarithmic fashion. For small gaps, the voltage remained constant because of the corner
regions of the box. Even though the distance between the center of the sides of the box and the
cylinder decreased, the corner regions remained relatively unaffected and current flowed t hrough
these regions. As the gap becarne bigger, the corner areas becarne insignificantl
'~imulationswere performed with spheres with a bounding surface which did not have any residual corners like
the box.
The voltage continued to decrease as the gap was decreased.
(a) Induced voltage
(b) Bounding box
Figure 7.5: (a)Effect of finite volume on cylinders enclosed in a bos. The induced trunsmem-
bmne voltage is shown as a function of the mtnimum gap between the cylinders and the enclosing
mctangular box. (b) The cells and bounding box.
These results can be explained by an equivalent circuit ( s e figure 7.6) : The ind uced transmembrane voltage of the receiving ce11 is proportional to the voltage gradient, modelled by RR,
caused by the current flowing around it. This pathway, &, is in parallel with the return path
directly to the source cell, R,. As the gap is decreased, R, grows and Iess current flows through
RR resulting in a decreased voltage. Hence, a smaller transmembrane voltage is induced with a
decreasing gap.
The above represents the worst case scenario for volume conduction. T h e boundary, un-
like the one here, is not a perfect insulator and impedance boundary conditions will allow more
current to flow. &hermore,
the potential boundary conditions may be different than the ones
here. If electrical activity is propagating through a tissue, one region of the tissue will be at a
slightly different potential. This will promote current flow and allow more current to flow dong
the membrane of the receiving cell, increasing the voltage gradient. Nonetheless, the finite volume
will reduce the induced voltage but it may be somewhat offset by potential shifts and impedance
boundary conditions.
- a + - - - - - - - - - -
I
Cell
I
I
Ce11
I
(a) Current flow (dashed Iine) in bounded volume
Source
(b) Equivalent circuit
Figum 7.6: Bounded current Born for cylindrical cells. (a) Path of bounded current jlow.
(b)
Equivalent circuit of the current Jow. Current from the source cell is distributed between the shortest
return path to the source cell,
Rs, and the path
around the receiving cell.
Rc
is the resistance of
the intefcellular coupling pathway whàle resistor RR detemines the induced tnznsmembrane voltage
in the receiving cell.
7.2.3
Coordinated Sources
If multiple sources undergo synchronous behaviour, their fields will combine to enhance their effects. The organization of cells at the myenteric and submucosal borders make such an assumption
plausible. Nerve ceils make contact with several ICC's and SMCs while ICC's are coupled by gap
junctions to other SMCs. Gap junctions act rather quickly and seem suited to synchronize activity amongst populations of ceIIs. Electrical activity starting at these nerve boundaries (at the
myenteric border in the stomach and small intestine and at the submucosal border in the colon)
could be initiated simultaneously across a large area, creating a planar wavefiont which wouId
spread transversely t hrough the tissue. Assuming t his plane of activity, several consequences make
extending the formulation to more than one source ceil simple. The transmembrane voltage is
assumeci identical over al1 N source cells as well as the transmembrane current. Wherrnore, if
the effect of the source cells on each other are ignored, which is reasonable since voltage waveforms
were recorded fkom tissue preparations, the coupling potential on the receiving cell, ce11 1, can be
determined:
Now, al1 the matrices (Haq and Gaq) in equation (7.16) need only be computed once and then
summed into two final matrices, HO'and GO'.
Storage requirements are no larger t han considering
two celk and while the initial setup must now calculate 2(N - 1) more matrices, the computation
at each time step is not any more demanding.
The effect of the source cells upon thernselves was also approximated by computing the
effect on the ce11 a t the center of the aggregate and assuming al1 other cells behaved as the center
one. Furthermore, an additional simplification was made: The potential produced by the source
monopoles were ignored on the center source cell. Examination of other trials indicated that for
cylindrical cells, the monopolar potential was at least 50 times weaker than the dipolar potential.
The e f k t on ce11 O fkom the other source ceüs was incorporated into equation (7.4):
The source cells were packed into a reguiar formation to form a source aggregate which
is seen in figure 7.7. Sources cens in the same row were separated on centre by a distance
JZd.
Adjacent rows were placed in parallel and offset Çom other such that cells in adjacent rows are
separated by d. Four basic arrangements of source aggregate to receiving cell are possible and occur
in the GI tract (figure 7.7).
T - tee. The axes of the cells are perpendicular with the receiving ce11 exposed to the end of the
source aggïegate. This occurs where the radial muscle cells run between the lamellae and
contact the circular muscle.
P - perpendicular. This configuration describes the orientation a t the myenteric border where
longitudinal and circulas muscle meet. The cells are beside each other with orthogonal axes.
S
- side-teside.
The receiving ceil is parailel and beside the source cells. This occurs when ECA
propagates perpendicularly to the orientation of the muscle cells, i-e. circumferentially in
longitudinal muscle, distally in circular muscle and radially in both layers.
E - end-to-end. The axis of the receiving ceIl is parallel to the axis of the source cells and is below
the source cells. This orientation is found in the muscle layers where the ECA propagates in
the direction of the muscle cells.
The source cells were oriented in the z direction. Activity in all xy planes was synchronous with
the origin of electricd activity at the ends of the ceils or at the midpoint. If activity originatd
at the midpoint, it spread to both ends. Orientations with activity originating in the middle are
designated with the superscript m, while the superscript e denoted activity originating at the end.
The source aggregate was cornposed of cells 600 pm long and 20 pm diameter. This was an
attempt to simulate three layers of cells. The activity was assumed to propagate from one ce11 to
the next without pause. The number of source mode1 cells in the aggregate was w i e d from 1 to 86,
Figure 7.7: Configumtions of SMC. The various coupling configurations for source (unshaded) and
receiving (shaded) cells: E , end-to-end; S , side-by-side;
P,perpendicular; 7, tees. The spacàng
of
cells in the source aggregate is shown on the right where d is the minimum center-to-center distance
of two cetls. Dots indicate the top of the receiving c d . The dashed line on the left indicates the
middle level fmrn which the surface depalarization wave may originate i n the source aggregate.
meaning that fkom 3 to 258 biologieal cells were considered. The activity for E and 7 was always
started at the end. Results for the induced voltages with respect to time are given in figure 7.10.
The induced transmembrane voltages could either be s d e d versions of the first derivative or second
O
20
40
60
O
80
20
40
60
80
100
Time ( ms )
Time ( ms )
Figure 7.8: Efect of source aggregate-receitn'ng ce12 orientation on induced tmnsmembrane volt-
age. The voltage ut the two ends of the receiving ce11 and along the midpoznt are giuen for a source
aggregate of 86 cells. See jigure 7.7 for definitions of orientation and asson'ated tops. The superscript refers to whether the voltage depolarizatzon wave was initiated at the en& (e) or midpoints
(m)
of the cylinders comprising the source aggregate.
derivative. The h t derivative effect resembled a shark's fin in the figures. The upstroke is iinear
and the true derivative is a step. It takes time for the wave to propagate over the surface of this
cell and this results in the smoothing of the step. The second derivative effect is seen as two bumps
106
whidi correspond to two smeared delta functions. In some instances, e.g. S,
cornponents of both
derivatives may be present in the sarne waveform.
The flow of curent through the ce11 can ako be deduced from the differences in c w e s .
Current entering the ceil produces a negative voltage while current leaving the cell has a depolarizing
influence. The voltage is also proportional to the c w e n t density. Exarnining the records for the
various configurations, the current flow c m be deduced. For E , the current clearly leaves the ce11
at the top and flows out in a distributed fashion dong the rest of the cell. With
Y coupling, the
current enters in kom the ends and leaves out from the middle.
The depolarization wave takes 30 ms to travel down one half of the source cell. Since the
upstroke lasts 30 ms, the activity wiil last for 60 ms if activity originates in the middle of the
aggregate and 90 rns if it originates fkom the end.
The effect of the nwnber of source cells on induced voltage is depicted in figure 7.9 on a
log-linear scale for each configuration. As previously noted, E coupling produces the greatest effect
for al1 numbers of cells while the second strongest was the Sm configuration. With the other configurations, the dependence on the number of source cells was clearly different for each configuration.
The 7configuration peaked at 41 ceus, decreasing slightly after t hat. The P configurations levelled
off aRer 40 cells, unlike the side-to-side and end configurations which continued to increase.
The difference in coupling when only one cell is considered is drastic. The E configuration
was more than a millivolt greater than the rest, many times greater than the next largest one. As
the number of source celis was increased, the E configuration was still on the order of a millivolt
S, and P configurations
greater; it did not preserve the proportionality found with one cell. The 7,
only seem to produce significant voltages if the solwce aggregate contains many cells.
Since & coupling produced the greatest voltages, it was studied in more detail. The source
cells were reduced in length fkom the previous simulations to t hat of a single cell. The size of t hc
cells as well as the number of layers of source cells were varied (figure 7.10). As the nurnber of cells
increased, so did the induced voltage. The increase however was less with each source ce11 added.
The curve was very linear when plotted on a log-linear scale. Two layers offered no irnprovement
in induced potential over one layer. This indicates the field produced by a ce11 decays within a
distance of one ce11 along the axial direction and accounts for the delay in the potential seen in
figure 7.8.
1
10
100
Number of source cells
Figure 7.9: Effect of number of cells on V& for various configurations. The source cells were 600
pm long to simulate three layers of SMCs.
Number of celldiayer
Figure 7.10: E&ct of number o j source cells on induced voltage as a function of size for the &
configuration. A standard mode1 of length 1 m and radius 0.5 m was scaled by the number indzcated
on the graph. Note that for a scaling by 5 x 10-~, o d y one layer wos modelled.
7.3 Discussion
The assumption that all source celis in the aggregate act the same as the center one, was tested
by pefiorming several runs with the assurnption that the source cells do not affect each other.
The true case would lie somewhere between these two extremes. The resuits obtained with the
assumption that the cells do not affect each other only resulted in a slight ( ~ 5 % )increase in the
induced voltage. Hence, the original assumpt ion is reasonable.
The induced transmembrane voltage depends upon which of the various smooth muscle ce11
configurations are being
CO nsidered.
The end-teend coupling experiences the greatest induced
voltages, even at the single source ce11 level. This type of orientation seems most likely for eiectric
field coupling. This suggests that, it is coupling of the longitudinal muscle of the proximal intestine
which produces the high observai frequencies. This is expected since circular muscles are connected
by gap junctions and thus cannot produce the high o b s e ~ e dfrequencies. By elinrination, the
longitudinal muscle must be responsible.
The other configurations are capable of producing significant voltages, but only if the fields
are produced by cells whose electrical activity is organized both spatially and temporally. The
side-to-side configuration produced the largest potentials out of the remaining configurations. The
parallel configuration produced low voltages but field coupling rnay not be needed in the region
where this occurs. At the LM-CM border, t here are ICC's and nerve cells which synapse on both
LM and CM. Thus, interlayer coupling may occur through these intermediary nonmuscle celis.
Conversely, the cells in this region might be the most highly coordinated since they are receiving
the direct nervous and ICC input and thus, might form the largest source aggregates.
The spread of the activity through the tissue was also important for certain configurations,
especially S and P. Activity rnay originate in the middle of the aggregate a nerve or ICC synapses
with the muscles a t that point. The field produced by the wave travelling along the cylinder can
be envisioned as that produced by a dipole oriented in the direction of the axis. A dipole field
will change polarity as the observation point changes kom being closer to the positive pole to be
closer to the negative pole. Thus, the field produced by one part of the cell will cancel the field
produced by another part of the ce11 if the equivalent dipole of each section is oriented in the same
direction and one is between them. When the activity starts in the middle and propagates to each
end, the dipoles are oriented in opposite directions and the fields add. This occurs in the S" and
Pm configurations.
The tortuosity will tend to increase the field &ects as the effective conductivity of the
extracellular volume decreaseç. As ions travel away fkom the cell, they will be squeezed into a
srnaller conducting space, increasing the current density which leads to a greater potential drop in
the direction of current flow. If there is a boundary preventing ion flow, the current will tend to
flow near the source ce11 and not over the surface of the receiving cell. This boundary effect is only
important for very small gaps.
Interdigitations do not seem to be very important eiectric field coupling structures in the
context of cylindrical cells. An interdigitation serves to enhance coupling between two cells. For
al1 configurations except El the voltages induced by one source cell was very small. Enhancing
this by a factor of 10% as in the sphere interdigitation case, would still produce a very smali
voltage. The & configuration would benefit from such a structure but su& structures have not
been observed in locations congruent with this coupling configuration. Many cells are needed
for non-E constellations, defeating the purpose of having an interdigitation for coupling means.
Perhaps the interdigitation is associated with other types of coupling, as suggested by diffusion
considerations, or is a mechanical fixation point between cells which are contracting.
The rate of propagation of slow waves through canine circular muscle has been measured in
stomach[82, 1261 and colon[127]. It was found that propagation through circular muscle parallel to
the axis occurred in the range of 2-7 cm/s while in perpendicular directions it was on the order
of 1 cm/s. Assurning that the cells are approximately cylinders of length 150 p m and diameter 10
Pm,that would give rise to an intercellular delays of about 4 rns in the axial direction and about 1
rns in the other directions on average. This delay could be less than the tirne it takes for the ECA
to propagate over the surface of the ce11 if 8 is less than the spread of activity in the tissue. It is
also possible that aggregates depolarize nearly synchronously and there are longer delays between
aggregates. The tissue propagation velocity cannot be used to accurately determine the cellular
propagation velocity, although it can give a rough estimate of the surface depolarization velocity,
especially if the cells are well connected by gap junctions and form a syncytium.
The quantitative aspect of this work relies on the transmembrane voltage gradient which is
unknown up to this point. This gradient is dependent on the the rate of rise of voltage and the
velocity a t which this activity spreads acras the surface. The first quantity, the temporal derivative
can be obtained kom patch clamp recordings and results are available from several studies[l28,
110
81, 85, 87, 891. The propagation velocity is not known. Even modelling of this is troublesome.
Unlike nerve cells and pancreatic beta cells which are membrane oscillators as evidenced by their
prepotential leading to action potentials, SMCs do not seem to be membrane oscillators for several
reasons:
1. They do not exhibit prepotentials before firing. There is no slow buildup of voltage until a
threshold is reached. The membrane depolarizes rather abruptly from a flat resting level.
2. Application of current pulses do not produce slow waves but can affect the kequency and
amplitude of slow waves
3. EIimination of ICC's kili ECA
Biological estimation of the propagation velocity seerns the best approach at this point in time. Volt-
age sensitive dyes are available which have been used to study single ceUs of cardiac myocytes[l29]
and neurons[l30]. Perhaps, t hese dyes could be appüed to smooth muscle preparat ions.
Chapter 8
Hippocampal Neurons
In this chapter, field coupling of hippocampai neurons is examined for its role in the appearance of
epileptiforrn bursts. Computer simulations were performed to mode1 electrical interaction between
two neurons. The boundary element method was used with a novel implementation employing a
mixture of triangular somatic elemerits and cylindricai dendritic elements to couple 3-dimensional
models of neurons. Assuming an active source c d , the effeets of gap junctions and extracellular
fields on transmembrane voltage and intracellular potential in a neighbouring passive ce11 were
computed. These results were compareci with biological data and a generation scheme for spikelets
was hypot hesized [35].Biological experirnents to veri fy the simulation are presented at the end.
8.1
Methods
To mode1 a neuron by the BEM, an accurate t h e dimensional representation of the ceil including
its dendritic tree is needed. Using data £kom CA3 neurons[93], such a mode1 was constructed
(Figure 8.1). The soma was represented as a cylinder which slightly tapered near the ends, with
a central radius 6 pm and length of 79 Pm. A total ceil length of 400 Pm was used. Membrane
. somatic conductance was 0.4
capacitance was set to 1 I r ~ / c m 2The
s/m2 while the dendrites were
assumed to have a 0.17 S/m2 conductance.
The entire dendritic tree was not replicated but rather, several branches whose physical
length and diarneter were consistent with the biological data as well as occupying sirnilar volumes.
Both basai and apical branches were considered. A random process was used to generate bifurcation
112
Source Ce11
Receiving Ce11
F'igum 8.1: Boundary element model of wupled neurons. The membrane conductance of the
receiving model is indicated. The lightening bolt represents field coupling whtch ucts in pamllel with
gap junctzonal wuplzng which is represented by the conductance,
on the left near the base of the basal tree.
gj.
The action wauefonn is shown
points which had a spatial probability distribution simiIar to that found biologically. To imitate
the d e c t of an entire tree, the effective dendritic capacitance and conductance were appropriately
scaled as is done to incorporate the increased surface area due to dendritic spines[l31]. If F is the
ratio of the spined branch area to the unspined branch area and NB is the density adjustment, the
ratio of actual branches to model branches, both the conductance and capacitance are multiplied
by F x NB. A value of 2 was used for F while NB was set to 3. After adjustment, the ce11 surface
area was 74 900 cm2 with 69 dendritic tips. The electrotonic length[51] of the farthest tip of the
apical branch was 0.516 and this nurnber was cornputeci by
electrotonic length = max
aii paths
lk
& path
/z
pkai
A
where k is the cylindrical element index, g, is the dendritic conductance, Ir. is the length of the
element and
pk
is its radius.
The surface of the cetl was discretized into a set of contiguous elements. The somatic
surface was discretized into a set of 96 triangles. Such a strategy applied to the dendrites wouId
have lead to an unwieldy number of elements. Thus, to reduce the number of elernents in the
trees, cylindrical elements were employed. Cylindrical elements have been previously employed to
rnodel cardiac cells[28, 401 but a radially symmetric field has been wurned. Here, it is assumed
that the diameters of the dendrites are small enough such that potential variation frorn one side
to the other may be neglected. Since the soma has a much larger diameter, substantial gradients
in potential may exist circumferentially precluding the use of cylindrical elements. Three types of
cylindrical elements were used to build the dendritic trees: open ended cylinders, cylinders with
one closed end for terminal segments and branch points comprised of three open ended cylinders
in a Y-configuration (see figure 3.6). A total of 454 elements were used to model the dendrites.
When triangles and cylinders are used in the sarne boundary element model, there will be
an interfacial cylinder which wili join the two element types. Thus, there must be an opening in
the triangularly teçsellated section which approxirnates a circie. Regardless of how many triangles
are used, the opening will never by truly round and flux will escape through the gap between
the cyünder and straight line segments defining the opening leading to numerical disaster. For
monopolar calculations, this effect is insignificant and the element may be treated as a cylinder
but not for the dipolar calculations. In the latter case, the interfacial cylindrical element must be
treated specially. The end away fiom the triangular elements may be treated normally but the
114
Figure 8.2: rrcinsitional element. The Ieft end attaches to the soma with the opening trilated as a
set of tfiangular elements.
cylinder end at the junction of the two types rnust be treated as a set of triangular elements, not
a disc. This will remedy the geometrical mismatch between the triangles and cyiinder by closing
any gaps at the interface.
To determine the dendritic segmentd diameters, the terminal segment of each branch was
set a constant and the 3/2 power appiied to each bifurcation point to determine the diameter of
each parent. Knowing the diameters of the two daughters of a bifurcation, the diameter of the
parent was determined and this procedure was applied recursively until the root of the tree was
reached. AU segments were t hen scaled by the same arnount to merge the root segment with the
opening in the soma at the point of attachment. The terminal segments were then verified to be
wit hin biological limits.
Q
8.1.2
Boundary Element Method
Two mode1 neurons were placed next to each other in an infinite medium. One was considered active
and called the source ce11 while the second was passive and called the receiving cell. Quantities
associated with the former are denoted by the superscript "s" while those associated with the latter
are denoted by the superscript 9".The effect of the receiving ce11 on the source was ignored as
it has been shown to be small[30]. The source cell is assigned a transmembrane voltage waveforrn
which is a function of time taken from bioIogica1 data and smoothed with a 5-point moving average
window (see figure 8.5). A surface depolarization wave was assumed to start a t a specific spot on
the membrane and travel with a constant propagation velocity (0.1 m/s) over the surface of the
soma, into the dendrites with an amplitude reduced by 30%[132].
A Galerkin formulation was used since solution at only one point of the cylindrical elements
provided inaccurate results, especially for nearby elements as the potential produced by a single
element can vary considerably over dimensions similar to itself. Analytic expressions for the cornputation of potential for triangular elements have been derived[l06] and the expressions for cylindrical
elements are derived in appendix A. To generate G and H, these expressions were integrated over
the field point element by three point Gaussian quadrature for triangular elements [107] and 12
point quadrature for cylinders. Computations involving cylindrical elements were compared against
those involving trianguiar elements for the simple case of a 10 prn radius sphere with 200 p m long
collapsed dendrite. The results of the two simulations gave identical results indicating the correctness of the cylindrical element computation. The triangular element computation has already been
verified. When modelling more complex trees, the G and H matrices were checked. The system
was solved and the elements of x examined. If any elements were negative then the matrix was
rejected. H was verified by summing the entries of the rows. If the sum of a row was greater than
1 0 - ~ ,the matrix was rejected. If the matrix was rejected, additional quadrature points were tried
and if these matrices failed the test, a new tree was generated.
The receiving cell was assumed to be passive and its membrane was modelled as an RC
network with the ionic current modelled as a simple resistive current. Having knowledge of ik,the
rate of change of
vk
can be computed from
where Cm is the diagonal matrix of elemental membrane capacitances ( ~ / r n and
~) ,
g
diagonal matrix of elemental membrane conductances
(s/n2).
Again,
is the
a first order differential
equation must be solved.
The potential recorded fiom an intracellular electrode, @f, can be computed once
,i
+i
and
are known by application of Green's theorem[29]. The field point is taken to be the center of
the ceil and it follows:
8.1.3
Electrophysiological recordings
Electrophysiological recordings were made from rat hippocampal slices which exhibit epileptiform
bursting under zero extracellular Ca++ conditions. Besides producing bursting, the la& of calcium
also inhibits any synaptic activity since Ca++ uptake is required at synaptic terminais to allow for
exocytosis of neurotransmitter fiiled vesicles. The recordings were made by Jose L. Perez Velaquez
at the Toronto Hospital-Western DivisionL
Wistar rats (20-30 days old) were anaesthetised with halothane (Fluothane, Ayerst Laboratories, Montreal) and decapitated. 'Ttansverse brain slices (400 PM) were obtained using a Vibratome (Series 1000, Technical Products International) and maintained in artificial cerebrospinal
fluid (ACSF) which contained in mM: NaCl. 125; KCI, 5 ; NaH2P04, 1.25; MgS04, 2; CaC12, 2;
NaHC03, 25; glucose, 10, pH 7.4 when aerated with 95% 02/5 % CO2. Osmolarity was 300 f
5 mOsm. Calcium tiee ACSF was the sarne but without added CaC12, and with ImM EGTA4to
reduce contaminating calcium. The internal solution in the recording e k t r o d e contained in mM:
potassium gluconate, 150; HEPES, 10; Mg-ATP,2; KCI, 5; pH 7.2 adjusted wi th KOH, osmolarity
265 =t 5 mOsm.
For recordings, slices were transferred to a superfusion chamber maintained at 35 C (Medical Systems Corp., Mode1 PDMI- 2). Neuronal recordings were performed using the whole-ce11
configuration of the patch-clamp technique (Harnill et al., 1981). Patch pipettes were pulled fiom
borosilicate capillary tubing (World precision Instruments, New Haven). Electrodes had tip resistances ranging fkom 4 to 6 MR when filled with internal solution. Signals were filtered at IkHz,
digitized at 88 kHz, and stored on video tape using a digi ta1 data recorder ML-10 (Instrutech Corp.,
NY) for later playbadc and analysis.
For spikelet analysis, 60-120 seconds of neuronal activity were digitized at 10 kHz using
Fetchex (Axon Instruments). Spikelets were detected and analyzed using PCLAMP6 software
(Axon Instruments). Cells were hyperpolarized by 10 mV during spikelet recording to prevent
spontaneous action potentials from occurring in the recording cell.
' ~ h work
e
was done in lab space provided by Dr. Peter L. Carlen, Director, Playfair Neuroscience Unit, Toronto
Hospital-Western Division.
Results
8.2
Biological Recordings
8.2.1
Perfusion of hippocampal neurons pyramidal neurons in &Ca++ ACSF causes cells to spontaneously
fie. Waveforms recorded (figure 8.3) by patch electrodes can be divided into three categories:
Action Potendials
: These may either appear singly or in bursts of 3 to
5. The amplitude of
the spike is about 40 mV. When occurring in bursts, the bursts appear on top of a slight
depolarization plateau with each successive burst smaller t han the preceding one. The spikes
are monophasic and 1 s t about 4 rns.
Spikelets : Small amplitude short duration potentials, or spikelets, are bipbasic with a duration
similar to that of the action potentials. They also appear singly or in bursts of 3 to 5 . They
bear a strong resemblance to differentiated action potentials and
vaïy
in amplitude from 0.5
to 5 mV.
Small amplitude long duration potentials
:
These waveforrns are monophasic, have a n initial
peak and a decay much longer than an action potential duration. These are typically in the
5 mV amplitude range.
8.2.2
Gap Junctional Conductance
The simulated intraceiiular potential is given in figure 8.4. In the absence of gap jünctions, the
received potential is a biphasic waveform, lasting as long as the action potential with a peak
amplitude of 80 p V . As the gap junction conductance is increased, the received potential increases
in amplitude and seems to be composed of two components: a monophasic potential which rises
for the duration of the source action potential and then decreases with the field induced potential
superimposed. As the gap junction conductance is increased, the monophasic potential readies a
higher value but also decays slightly more rapidly. This is due to the increased shunting of currcnt
through the gap junction in addition to the membrane flow. The fieid effects are always present,
superimposed on the gap junction effects. As the gap junction conductance increases, the field
effects becomes insignificant .
Integrating the field induced potential with respect to time does indeed produce a waveform
which is very similar to the source action potential (figure 8.5). In contrast, integrating the gap
118
Figure 8.3: Patch clamp recordangs /mm CA1 pyramidal cells. A: Action potentiak; occur in
bursts or singly. B: Spikelets with inset showing ezparrded tirne scale. C : Long duration potentials
with inset showing ezpanded time scale.
Figure 8.4: Effect of gj on zntracelluiar potential. The anset shows a detail of the initial response.
junction induced potentials does not produce a waveform which resembles the source action potential. Conversely, differentiating the source action potential produces a waveform which resembles
the intracellular potential.
The transmembrane voltage is the quantity of interest which will affect voltage-dependent
membrane processes such as ionic channels. The transmembrane voltage may be relatively uniform
or differ drastically over the surface of the ce11 depending on whether the coupling is prirnarily gap
junction or field induced.
The effects of field coupling on transmembrane voltage are shown in figure 8.6. The transmembrane voltage is a highly spatially-dependent quantity. Depending on which point of the
membrane is being observed, the voltage will either be depolarizing or hyperpolarizing and resemble either first or second derivatives. Also, the sirnulated transmembrane voltage is an order
of magnitude higher in amplitude than potentials measured at the electrode. The peak transmembrane voltage was 400 pV while the intracellular potential peak was only 40 pV. Hence, the
intracellular or whole ce11 patch recording electrode c a n o t accurately m e s u r e field coupling nor
does it represent what is happening across most of the neuronal membrane.
With gap junctional coupling, the junctional curent distributes itself evenly over the receiving cell and the transmembrane voltage is uniform across the ceU (figure 8.8b). Furtherrnore,
the transmembrane voltage and the intracellular potent ial are identical and the recording electrode
accurately depicts the situation across the ent ire membrane.
Differentiated Source
Action Potential
Source Action Potential
fi:./
\
'----
lntegrated Gap Junction Effect
_ _ - - - - --
_ c - - -
/
-
-
lntegrated Field Effect
/
- - - - - -\
--_____--
#
Figure 8.5: Relationship of action potential to intracellular potentials produced by gap junctionr
and field egBcts. A: Intracellular potentials are compared to a diflerentioted action potential.
B:
Intmcellular potentaal. are integrated and compared to an action polential. A 1 n S gap junction
was used for these simulations.
\
Central
Basal
Figuw 8.6: Field induced VA as a function of position. AI1 points are on the soma located ut the
buse of the apo'cal tree (Apical), base of the basal tree (Basal) or at the midpoint (Central).
Separation ( Fm )
Figure 8.7: Elgect of separation on
#f. Inset shows recordings with respect to tzme for two
distances.
8.2.3
Separation
The effect of distance on the received potential is seen in figure 8.7. The transmembrane voltage induced in a ce11 decreases as the celb are separated with the voltage decreased to one half maximum
at 27 Pm. Regression fitting to an exponential function yielded a decay rate for the intracellular
potential that was approximately the inverse of the distance, r (actually .-'-O3).
T h e transmem-
brane voltage decayed a t a similar rate since it is linearly related to the electrode potential (or
intracellular). It should also be noted that the peaks of the received potentials al1 occurred at the
same tirne irrespective of distance.
8.2.4
Location of Gap Junction
Dendro-dendritic gap junctions have been found and may be more numerou, than soma-somatic
gap junctions[l5]. Apical and basal dendro-dendritic connections were bot h investigated. Apical
junctions at two electrotonic lengths from the soma, 0.25 and 0.432, were simulated as well as a
basal junction located 0.13 from the soma. A conductance of 1 nS was used in each case. The
results are plotted in figure 8.8. Moving the gap junction farther away fiom the soma reduces the
electrode potential as well as smoothes out the waveform and introduces a noticeable time delay.
The effects on uC, are the same as those for #f
122
. In figure 8.8b,a major difference between
(a) 4f due to gap junctions located at various denciritic positions
basal
Soma Position
(b)
at three different somatic locations due to gap junctions at different
locations. inset details the boxed region.
Figure 8.8: E m t of gap junction position. A 1 nS gap junction was positaoned at three diflerent
places in the dendrites.
gap junctional and field coupling can be seen. Initially, field effects dominate and different points on
the surface of the cell experience different transmembrane voltages. As the field effect dies off and
the gap junction efféct takes effect, the ce11 becomes more isopotential as the gap junction current
distributes itself rat her uniformly around the cell. Initially, the potentials fkom each site depend
on the recording position on the soma but merge into one curve as tirne progresses depending on
the location of the gap junction.
8.2.5
Electrode Placement
The electrode has been assumed to be at the center of the cell. Generally this will not be the case.
For patch electrodes, the electrode is usually somewhere on the soma or large proximal dendrite.
The effect of different receiving measuring sites within the soma was considered. The intracellular
potential was sampled at various points dong the axis of the soma. Waveforrns from the various
positions were indist inguishable (data not shown). Since the intracellular conductivity is large, the
potential will not va.ry greatly within the ce11 and exact electrode placement seems to be irrelevant.
Hence, the electrode potential was assumed to be the intracellular potential.
8.2.6
Trees
The effect of the dendritic trees on the receiving potential was determined in the absence of gap
junctions (figure 8.9). The source ce11 remained unchanged with both apical and basal trees while
the receiving ceil had no trees, only one tree or two trees. Depending on the tree structure, a
very different potential is observed. With the apical tree only, the receiving potential is reversed
in polarity kom the control case of two trees. The potential occurs slightly sooner and is larger in
magnitude as well. M'ith only the basal tree, the received potential is triphasic. The initial portion
is hyperpolarizing, followed by a larger depolarizing portion and finaiiy a smaller and longer Lasting
hyperpolarization. In an approximate way, it resembles the negative derivative of the two tree case,
making it the negative second derivative of the source action potential. Finally, with no trees, i.e.,
a slightly t a p e r 4 cylinder, the electrode potential again resembles a second derivative although
reversed in polarity from the basai tree case and slightly increased in amplitude. Thus, depending
on tree structure, the electrode will measure positive or negative f i s t or second derivatives of the
source action potential.
Basal
.,
,\
Figure 8.9: E ' t of receivzng ce11 trees on c#f.
The source cell was lefl intact uthale the receivzng
cell had no trees (none), only the apical (Apicul), only the basal (Basal) o r both (Botla) trees.
8.2.7
Extraceliuiar Conductivity
Decreasing the conductivi ty ratio ( u i / o e )decreases the electrode potential and transmembrane
voltage (figure 8.10). The effect is to simply scale the waveforrns. Assessing the effective conductivity of the extracellular medium is difficult since although the specific conductivity of the
extracellular medium is known, the effect of the restricted extracelhdar space must be considered.
Dense packing of neurons and glia such as occws in DG and pyramidal layers of the hippocampus
lead to a reduction of the effective conductivity as the tortuosity is increased[20].
8.3 Discussion
Electrotonic gap junctional coupling is very different Som field coupling. Field coupling is faster
acting, shorter in duration, depolarizes regions of the membrane while simultaneously hyperpolarizing other regions and produces waveforms resembling first and second derivatives of the source
action potential. Fbrthermore, intracellular recordings represent a bulk recording which is very
different from the membrane events. ?I.ansmembrane events are an order of magnitude larger
than what is measured by the recording electrode. For events mediated through the gap junction
conductance, the electrode conveys an accurate representation of transmembrane events since the
125
Figure 8.10: E&t
of conductiuaty mtio on #f.
current injected through the gap spreads itself out rather uniformly through the membrane. Field
effects do not inject current into the receiving cell, but rather cause a redistribution of charge dong
the intracellular and extracellular surfaces of the membrane. This accounts for the very qui& time
constant of the electrical activity caused by this type of coupling as charge does not have to flow
through the membrane which has a time constant on the order of tens of miliiseconds.
Gap junctional coupling lasts longer than the duration of the action potential for several
reasons. Current will continue to flow through the gap junct ion as long as the intracellular potential
of the source ce11 is higher than in the receiving cell. This condition is met during alrnost al1 of the
action potential since the depolarization in the receiving cell is only a couple of millivolts. While
gap junctions are modulated by voltage, the opening and closing time constants are on the order
of seconds(331, a time scale two orders of magnitude larger than the one with which we are dealing.
Hence, the static resistor is an acceptable mode1 when only one action potential is being considered.
The membrane voltage of receiving ce11 will decay passively with a time constant on the order of
fifteen milliseconds, again longer than an action potential. Hence, as spikelets are observed to only
last as long as the action potential, it seems unlikely they are mediated electrotonically through
gap junctions whose effects last longer than an action potential.
Another possible source of differentiation of transrnembrane voltage is the capacitance of the
gap junctional aggregate. Assuming there is no conductance between membranes in the region of
126
the aggregate, a capacitor is formed with a unit area capacitance half that of only one membrane.
Using the recorded action potential to determine the derivative and taking a n area of 0.28
for
the aggregate[66], the current would be on the order of 0.2 PA. This is one five hundredth of the
current flowing through a I nS conductance. Note that this is an overestirnation since the capacitor
is leaky and current will be shunted into the extracellular medium. Hence, this current is too small
to cause spikelets.
While both capacitive coupling as described above and field coupling produce difierentiated
versions of the source waveform, they rnay be fundamentally distinguished in that the former is the
result of a temporal derivative while the latter is dependent on a spatial derivative. There may be
a linkage of the two when propagation of the waveform over the surface is considered to be uniform
but, it is possible to get one without the other. If the entire membrane were to simultaneously
depolarize, there would be no extracellular field produced but capacitive coupling would be possible.
Likewise, a transmembrane voltage gradient that did not change with time would produce a static
electric field but no induce no current flow through a capacitor to which the cell was connected.
Furthermore, a capacitive current wodd share the properties of the resistive curent. The entire
receiving ce11 membrane would tend to be depolarized spontaneously and the effective time constant
would be that of the membrane, not that of the medium.
The intracellular potential calculated in the receiving cell is consistent with spikelets in
both shape and duration when the gap junction is ignored- The simulated amplitude is much
smaller than that measured, however. This suggests spikelets are the product of several celis firing
synchronously if electric field effects are to be responsible for spikelet generation. Based on Our
results, the number of cells in such a cluster would range from 5 to 40 to account for the values
of potential measured. This consistent with reports of 0.4 mV population spikes being produced
by groups of 5-7 synchronously firing ceils [133, 1341 as verified by quanta1 analysis of population
spike magnitudes and dye coupling.
The potential produced by a single ce11 is srnail, below the noise level of the recording set up.
Individual events will go unnoticed. To reach a measurable level there must be a synchrony of several
cells. Based on the decay with distance, synchronously firing neurons three soma widths away would
still have a significant effect compared to adjacent neurons. Gap junctions seem ideally suited to
mediating this synchrony as they provide a pathway for almost instantaneous coupling. This
is consistent with the observations of pharmacological manipulations which altered gap junction
127
Figum 8.11: Proposed coupizng scheme for pyramidal cells of the hippocampus. Conductances
represent gap junctions while the lightening bolt represents field wupling.
conductance[98]. Increasing the conductance through intraceliular alkalinization would lead to a
greater probability of action potential propagation across the gap and hence, greater synchrony.
Note, since spikelet potentials are biphasic, a small time delay between neighbouring firing neurons
can lead to one cell producing a depolarizing pulse while the other produces a hyperpolarizing pulse
and the total effect is diminished. Conversely, intracellular acidification or application of octanol
will lead to a loss of synchrony through a decrease gap junction conductance [98, 951.
The pyramidal ce11 layers display a highly laminar structure. The neurons are organized in
a strip with the somata in a thin layer and trees a l oriented in the same direction. This is ideal for
field coupling as e l ~ t n fields
c
produced by the neurons will be oriented in the same direction and
the resultant field a simple superposition. A firing cell may be considered a dipole source oriented
in the direction of action potential propagation over the ce11 surface. If the cells were randomly
placed, the spatial dependence of the dipole fields would lead t o cancellation and synchrony would
be meaningless. The laminar structure of the CA3 region leads to greater field effects.
Results of this simulation suggest a role for both types of electrical coupling in the h i p
pocampus (figure 8.11). Individual neurons induce intracellular potentials in neighbouring cells
through extracellular fields which are too small to measure. There exist aggregates of neurons
whose electrical activity is synchronized by gap junctions. Such synchronized behaviour produces
fields which are the superposition of the fields produced by the individual neurons and result in
measurable intracellular potentials in nearby cells. These field induced potentials may cause cells
near threshold to fire and bring about entrainment, exacerbating or inducing epileptiform activity
in the slice.
8.4
Suggested Experiments
Results of this simulation extend beyond what has been rneasured in vivo and fiom them, one
should be able to draw conclusions and design experiments to test the conclusions. Whether field
coupling is truly the phenomenon being observed can be clarified by the following experiments:
1. Increasing extracellular conductivity by altering extracellular ionic concentration should increase spikelet amplitude.
2. Increasing tortuosity by causing ce11 sweliing should increase spikelet . amplitude
3. Double patch clamp experirnents where a neuron is stimulated and activity in a neighbouring
ce11 is recorded. Acidification of the receiving ce11 will close gap junctions, if any, that exist
between celis and allow for measurement of field effect. Ensembling may be necessary to
detect small voitage fluctuations
4. Spikelets recorded after removal of basal and/or apical dendritic trees by attaching patch
electrode and pulling away should correspond to (figure 8.9).
This suite of experirnents should determine if spikelets are indeed the product of field coupling.
Chapter 9
Conclusions
9.1 Summary
Two biological systerns have been anaiyzed, smooth muscle cells of the gut and pyramidal cells
of the hippocampus, using the boundary element method. The BEM was extended to include a
formulation based on electric field directly. This approach can lead to reduced computation and
increased accuracy. Analytic forrnulae were developed for zeroth order triangular and cylindrical
elements. A method to model branching structures was also presented. It employed models comprised of both cylindrical and triangular elements with special bifurcation and interface elements.
An analytic formula were developed for the solid angle of an arbitrarily oriented disc to help with
double layer cornputations. Finally, the applicability of the BEM to exterior convection-diffusion
problems was studied. Using domain inversion, the BEM approach was found t o be computationally expensive, relying on too many approximat.ions and having problematic boundary conditions.
Further analytic work, particularly a new Green's function to eliminate domain integrais, may yield
a more workable system.
For field coupling in general, the two most significant factors which determines the extracellular field produced are the number of cells which synchronously fire and the surface voltage
gradient which is a function of the rate of rise of the transmembrane voltage and the propagation
velocity. I t is experimentally possible to get an estimate of the number of ceiis coupled by gap
junctions by dye injection. The rate of rise can be directly measured making it the most reliable
parameter. Estimating the propagation velocity is a much more difficult task. The velocity relies on ce11 geometry as well as nonlinear membrane properties which have a nonuniforrn spatial
130
distribution.
Comparing the two biological systems, neurons seem capable of producing larger fields than
smooth muscle cells due to a rate of rise that is about 100 times faster than that of SMC's, and
dendrites with diameters 10 times smaller than SMC which produce slower propagation velocities.
There is also direct evidence for field coupling in neurons (the "spikelets"), unlike SMC's. Gap
junctions between smooth muscle ceils will produce greater effects as the depolarkation duration is
a thousand times longer than that of neurons, enswing entrainment even for few low conductances.
Gap junctional coupling by itself is insuficient to explain electrical activity in the gut. If
field coupling is to be significant in gastrointestinal srnooth muscle and bring about entrainment,
certain condit ions must apply:
1. Small intrinsic fkequency differences be tween adjacent cells.
2. Slow surface propagation velocity.
3. Highly concentrated channel density in high field regions.
4. Highly coordinated activity in the source region.
The first condition would seem to be met. Gradients in electrical parameters of smooth muscle
vary smoothly over several rnillirneters in the transverse direction and over a greater distance in
the axial direct ion. The voltages induced under the simuIated conditions assumed a relatively slow
propagation velocity which may be off by an order of magnitude. This can have a great impact on
results as the fields are inversely proportional this quantity. Nonetheless, even srnall perturbations
can bnng about entrainment of nonlinear oscillators so their effect cannot be precluded at this
point.
Since this study suggests that field coupling in both neurons and smooth muscle cells is only
significant if there is a source aggregate, morphologie structures between cells will not contribute
significantly to the induced transrnembrane voltage. An interdigitat ion was found to increase the
voltage by 30%, but this is if a several hundred per cent increase over a single cell's effect is
required. There stili exists the poçsibility that nonelectrical mechanisms may be acting in these
morphological structures. Potassium was shown to remain elevated in an interdigitation and stretch
activated charnels may be present as well.
Finally, even if rnasked by other coupling mechanisms, field coupling may still be present,
acting before other rnedianisms and being rnasked aftenvord. The difficulty in dismissing field
coupling lies in the small amounts of coupling needed for entrainment. For neurons, field coupling
seems to be of sufficient magnitude, several millivolts, to affect the exact timing of action potentials.
Whether this coupling alone can bring about the entrainment of the whole hippocampus, experie n c d during epileptiform bursts, is still unknown. In smooth muscle ceils, with large aggregates
and a sufficiently slow propagation velocity, field coupling rnay contribute to the entrainment of
the electrical activity of the functioning gut.
9.2
Future Work
There still remains much work to be done on the modelling of the effect of electric fields. The
membranes of the cells were considered passive. The extension to active membranes is an obvious
step. For neurons, there exist many membrane models from which to choose[51]. For smooth
muscle cells, membrane models may be insufficient to fully describe behaviour, requiring a clocking
unit which represents some intracellular or extracellular process.
In summary, the foremost tasks suggested by this work are
Make the membranes of both neurons and smooth muscle ceils excitable
Mode1 extracelluiar ion concentrations
0
Biologically measure the surface propagation velocity with voltage sensitive dyes
0
Investigation of non-electrical coupling
Find better Green's function for domain inversion
Bibliography
[l] C. A. Brebbia, Boundary Element Methods in Heat Ransfer. Boston: Computational Me-
chanics Publications, 1992.
[2] P. W. Partridge and C. 4. Brebbia, "The dual reciprocity boundary element method for the
diffusion convection equation," in Computational Methodr in Wuter &sources (T. E. Russel,
ed.), New York: Elsevier, 1992.
[3] J. D. Bronzino, ed., The B i o m d c a l Engineering Handbook. Boca Raton: CRC Press, 1993.
[4] J. Malmivuo and R. Plonsey, Bioelectmmagneth. New York: Oxford University Press Inc.,
1995.
[5] A. Papoulis, The Fourier Integral and Its Applimtions. New York: McGraw-Hill, 1962.
[6] S. Haykin, Communication Systems. New York: Wiley, 2 ed., 1983.
[7] A. T . Winhee, The Geometry of Bzological Time. Biomathernatics 8 , New York: Springer-
Verlag, 1980.
[8] E. E. Daniel, B. L. Bardakjian, J. D. Huizinga, and N. E. Diamant, "Relaxation oscillator
and core conductor models are needed for understanding of GI electrical activities," Am. J.
Physiol., vol. 266, pp. G33%349, 1994.
[9] B. Van Der Pol, "Biological rhythms considered as relaxation oscillators," Acta Med. Scand.
Suppl., vol. 108, pp. 76-88, 1940.
[IO]
R. FitzHugh, uMathematical models of excitation and propagation in nerve," in Biological
Engineering
(H.P.Schwan, ed.), ch. 1, pp. 1-85, New York: McGraw-Hill, 1969.
[Il] B. L. Bardakjian and N. E. Diamant, "A mapped ciock oscillator mode1 for transmembrane
electrical rhythmic activity in excitable cells," J. Theor. Biol., vol. 166, pp. 225-35, 1994.
[12] P. Hanson, B. L. Bardakjian, and N. E. Diamant, 'Chaos in coupled nonlinear gastric oscillators," Math. Comput. Model., vol. 14, no. 586-91, 1990.
[13] R Aschenbrenner-Schiebe, "Chaosand entrainment in hippocampal neurons," Master1s thesis, University of Toronto, 1995.
[14] T. Shinbrot, C. Grebogi, E. Ott, and J. A. Yorke, "Wsing small perturbations to control
chaos,* Nature, vol. 363, pp. 411-417, 1993.
[15] F. E. Dudek, R. W. Snow, and C. P. Taylor, "Role of electrical interactions in synchmnization
of epileptiform bursts," Aduances in Neumlugy, vol. 44, pp. 5 9 3 4 17,1986.
[16] E. R. Kandel, J. H. Schwartz, and T. M. Jessell, eds., EssentiaLs of Neuml Science and
Behaviour. Toronto: Prentice Hall Canada, Inc., 19%.
[17] M. T. Kirber, J. V. Walsh, Jr., and J. J. Singer, "Stretch-activated ion channels in smooth
muscle: A mechanism for initiation of stretch-induced contraction," Pflugers Arch., vol. 412,
pp. 339-345, 1988.
[18] Y. Yarom and M. E. Spira, &Extracellularpotassium ions mediate specific neuronal interaction," Nature, vol. 216, pp. 80-82, 1982.
[19] M. E. Spira, D. Zeldes, and E. Krasner, "Specific and efficient synaptic interactions rnediated
by potassium ions," Israel J. Med. Sci., vol. 23, 1987.
[20] R. Pumain and U. Heinemann, 'Changes in extracellular ions associated with epilepti-
form discharges," in Genemllted Epilepsy: Neurobiologicul Appmaches (M. Avoli, P. Gloor,
G. Kostopoulos, and R. Naquet, eds.), ch. 6, p. 481, Boston: Birkhauser Inc., 1990.
[21]J. D. Huizinga, L. W. Liu, M. G. Ble~erhassett,L. Thuneberg, and A. MoUernan, Yntercellular communication in smooth muscle," Eqerientia, vol. 48, no. 10, pp. 932-941, 1992.
[22] R. Dermietzel and D. Spray, "Gap junctions in the brain: Where, what type, how many and
why?," Zhnd Neurosci., vol. 16, no. 5, pp. 186-192, 1993.
[23] S. 1. Fox, Human Physiology. Dubuque, Iowa: Wm. C. Brown, 1987.
[24] D. B. Heppner and R. Plonsey, "Simulation of electrical interaction of cardiac ceUsln Biophys.
J., vol. 10, pp. 1057-1075, 1970.
[25] J. E. Mann, N. Sperelakis, and J. A. Ruffner, "Alteration in sodium channel gate kinetics
of the Hodgkin-Huxley equations applied to an electric field model for interaction between
excitable cells," IEEE %ns. Biomed. Eng., vol. 28, pp. 655461, 1981.
[26] N. Sperelakis, R. A. Marschall, and J. E. Mann, "Propagation down a chain of excitable
cells by electric field interactions in the junctional clefts: Effect of variation in extracellular
resistances, including a "sucrose gapn simulation," IEEE 2Yan.s Biomed Eng., vol. 30, pp. 658664, 1983.
[27] J. H. Sperelakis, "Elettrical field model: An alternative mechanism for cell-tecell propagation
in cardiac muscle and smooth muscle," J. Gastmintestinal Motility, vol. 3, pp. 63-84, 1991.
[28] H. Hogues, L. J. Leon, and F. A. Roberge, "A mode1 study of electric field interactions
between cardiac myocytes," IEEE ïYcans. Biomed. Eng., vol. 39, no. 12, pp. 1232-1243, 1992.
[29] B. L. Bardakjian and E. J. Vigmond, &Effectsof a surface depolarization wave on the extracellular potential of an excitable cell," IEEE h n s . Biomed. Eng., vol. 41, no. 5, pp. 432-439,
1994.
[30] E. J. Vigmond and B. L. Bardakjian, "The eifect of morphological interdigitation on field
coupling between srnooth muscle cells," IEEE Dans. Biomed. Eng., vol. 42, no. 2, pp. 162-
171, 1995.
[31] E. E. Daniel, D. G. Duchon, R. E. Garfield, M. Nichols, S. K. Malhotra, and M. Oki, -1s
the nexus necessary for cell-tecell coupling of smooth muscle," Membrane Biol., vol. 28.
pp. 207-239, 1976.
[32] R.
E. Garfield and G. Thilander, "Are gap junctions necessary for cell-to-cell coupling of
smooth muscle?: An update," Can. J. Physiol. Pharmawl., vol. 70, pp. 481-490, 1992.
[33] A. P. Moreno, M. B. Rook, G. 1. Fishman, and D. C. Spray, "Gap junction charnels: Distinct
voltage-sensitive and -insensitive states," Btophys. J., vol. 67, pp. 113-119, July 1994.
135
[MlS. K. Sarna, E. E. Daniel, and Y. K. Kingma, "Simulation of slow wave activity of smail
intestine," Am. J. Physiol., vol. 221, pp. 16ô-75, 1971.
(351 E. J. Vigmond, J. L. Perez Velazquez, T. A. Valiante, B.
L. Bardakjian, and P. L. Carlen,
"Mechanisms of electrical coupling between pyramidal cells." Submitted to J. Neurophysiol.,
October 1996.
[36] S. Crouch and A. M. Starfield, Boundary Element Methodc in Solid Mechanics. Cambridge,
MA: Unwin Hyman Inc., 1990.
1371 H. Power and P. W. Partridge, Time-dependent stokes flow for interior and exterior domains
via dual reciprocity boundary element method," in Boundary Elements XV ( C . A. Brebbia
and J. J. Rencis, eds.), pp. 331-342, New York: EIsevier Science PubIishers, 1993.
[38] C. A. Brebbia, Boundary Eiement Techniques: Theory and Applications in Engineering. New
York: Springer-Verlag, 1984.
[39] C. W. Steele, Numen'ml Computation of Electric and Magnetic Fields. New York: Van
Nostrand Reinhold Company, 1987.
[40] L. J. Leon and F. A. Roberge, "A model study of extracellular stimulation of cardiac cells,"
IEEE Rans. Biomed. Eng., vol. 40, no. 12, pp. 1307-1319, 1994.
(41)
H. A. Schlitt, L. Heller, R. Aaron, E. B a t , and D. M. Ranken, 'Evaluation of boundary
element methods for the EEG forward problem: Effect of linear interpolation," IEEE %ns.
Biomed. Eng., vol. 42, no. 1, pp. 52-58, 1995.
[42] B. N. Cuffin, 'A method for localizing EEG sources in realistic head models," IEEE Thlis.
Biomed. Eng., vol. 42, no. 1, pp. 68-71, 1995.
[43] D. J. Fletcher, A. Amir, D. L. Jewett, and G. Fein, "Improved method for computation of
potentials in a realistic head shaped model," IEEE IFans. Biomed. h g . , vol. 42, no. 11,
pp. 1094-1104, 1995.
[44] G . Beer, Introduction to Finite and Boundary Element Methods for Engineers. New York:
Wiley, 1992.
[45] D. Rosen and D. E. Cormack, "The continuation approach: A general Earnework for the
analysis of singular and near-singular integrds," SIAM J. Appl. Math., vol. 55, no. 3, pp. 723762, 1995.
[46] A. Pullan, 'A high-order coupled finite element/boundary element torso model," IEEE ?tans.
Biomed. Eng., vol. 43, no. 3, pp. 292-298, 1996.
[47] L. J. Leon and F. A. Roberge, "A new cable model based on Green's theorem," Ann. Biomed.
Eng., vol. 18, pp. 1-17> 1990.
[48]W. RaU, "Electrophysiology of a dendritic neuron model," Bzophys. J., vol. 2, no. 2, part 2,
pp. 145-167, 1962.
[49] W. Rall, "Distinguishing theoretical synaptic potentials computed for different soma-dendritic
distributions of synaptic input," J. Neurophystol., vol. 30, pp. 1138-1168, 1967.
(501 M. Klee and W. Rall, "Computed potentials of cortically arranged populations of neurons,"
J. Neurophysiology, vol. 40, no. 3, pp. 6474566, 1977.
[51] J. M. Bower and D. Beeman, The Book of Genesis. New York: Springer-Verlag, 1994.
[52]
R D. Traub, R. K. S. Wong, R. Miles, and H. Michelson, "A model of a CA3 hippocarnpal
pyramidal neuron incorporating voltage clamp data on intrinsic conductances," J. Neuro-
physiol., vol. 66, no. 2, pp. 635450, 1991.
[53] R. D. Traub, J. G. Jefferys, R. Miles, M. A. Whittington, and K. Toth, "A branching dendritic
model of a rodent CA3 pyramidal neurone," J. Physiol., vol. 481, no. Pt 1, pp. 79-85, 1994.
[54] P. Kaiser, "Nonlinear oscillations (limit cycles) in physical and biological systems," in Non-
linear Electrornagnetics (P. E. L. Uslenghi, ed.), pp. 343-389, New York: Academic Press
Inc., 1980.
[55] B. L. Bardakjian and S. K. Sarna, "Mathematical investigation of populations of coupled
synthesized relaxation oscillators representing biological rhyt hrns ," IEEE 7kans. Biomed.
Eng., vol. 28, no. 1, pp. 10-15, 1981.
[56] B. L. Bardakjian, T. Y. El-Sharkawy, and
N. E. Diamant, Ynteraction of coupled nonlinear
oscillators having difTerent intrinsic resting levels," J. Theor. Biol., vol. 106, pp. 9-23, 1984.
[57] B. L. Bardakjian and S. K. Sama, "A computer mode1 of human colonic electrical activity,"
IEEE Tkans. Biomed. Eng., vol. 27, no. 4, pp. 193-202, 1980.
[58] B. L. Bardakjian, uComputer models of gastrointestinal rnyoelectric activity," Automedica,
vol. 7, pp. 261-276, 1987.
[59] K. Ahrned, "A computer mode1 of a neural pattern generator for the canine s m d l intestine,"
Master's thesis, University of Toronto, 1992.
[60] T. Tomita, uElectrical activity (spikes and slow waves) in gastrointestinal smooth muscles,"
in Smooth Muscles, A n Assessrnent of Current Knowledge (E. Rulbring, A. F. Brading, A. W.
Jones, and T. Tornita, eds.), ch. 6, London: Edward Arnold, 1981.
[61] J. H. Szurszewski, "Mechanism of action o l pentagastrin and acetylcholine on the longitudinal
muscle of canine antrurn," J. Physiol., vol. 252, pp. 335-361, 1975.
[62] B. L. Bardakjian, "Models of interaction between nonhear oscillators," in Modellnig and
Contml in Biomediml Systems (C. Cobelli and L. Mariani, eds.), pp. 251-255, Oxford: Pergamon Press Inc., 1989.
[63] Y. Lai and C. Grebogi, "Synchronization of chaotic trajectories using control," Physicol Rev.
E,vol. 47, no. 4, pp. 2357-2360, 1993.
[64] S.
J. Schiff, K. Jerger, D. H. Duong, T. Chang, M. L. Spano, and W. L. Ditto, UControlling
chms in the brain," Nature, vol. 370, pp. 615-620, 1994.
[65] R. Turner, T. L. Richardson, and J. J. Miller, "Ephaptic interactions contribute to paired
pulse faciütation and £requency potentiation of hippocampal field potentials," Ezperimental
Bruin mearch, vol. 54, pp. 567-570, 1984.
[66] F. E. Dudek, EL D. Andrew, B. A. MacVicar, R. W. Snow, and C. P. Taylor, "Recent
evidence for and possible significance of electrotonic synapses in the rnarnmalian brain," in
Basic Mechanisms o j Neuronal Hyperezcitabtltty, ch. 2, pp. 31-73, New York: Alan R. Liss,
Inc., 1983.
[67] T.L. Richardson, R W. Turner, and J. J. Miller, "Extracellular fields induce transmembrane potentids and synchronization of hippocampal neuronal activity," Bmin Res., vol. 294,
pp. 255-262, 1984.
[68] A.-Z. Wang, J. Li, L. F. Lemanski, and R
D. Veenstra, "Gating of mammalian cardiac gap
junction channels by transjunctional voltage," Biophys. J., vol. 63, pp. 139-151, July 1992.
[69] L. Makowski, D. L. D. Casper, W. C. Philiips, and D. A. Goodenough, "Gap junction structure II. Analysis of X-ray difiaction data," J. Cell Biol., vol. 74, pp. 629-645, 1977.
1701
B. Hille, Ionic Channels of Excitable Membranes. Sunderland, MA: Sinauer Associates Inc.,
second ed., 1992.
[71]A. P. Moreno, J. G. Laing, E. C. Beyer, and D. C. Spray, uProperties of gap junction channels
formed of connexon 45 endogenously expressed in human hepatoma SKHepl cells," Am. J.
Physiol., vol. 268, pp. C356-C365, 1995.
[72]
N. Sperelakis, B. LoBracco, Jr., J. E. Mann, and R. Marschall, "Potassium accumulation
in intercellular junctions combined with electric field interactions for propagation in cardiac
muscle," Innou. Tech. Biol. Meth., vol. 6, pp. 25-43, 1985.
[73] C. L. Stokes and J. L. Rinzel, ''Diffusion of extracellular K+ can synchronize bursting oscillations in a mode1 islet of Langerhans," Biophys. J., vol. 65, pp. 597-607, 1993.
[74] L. Bartho and A. Lefebvre, R., "Nitric oxide-mediated contraction in enteric smooth muscle,"
A x h Int Phannacodyn Ther., vol. 329, no. 1, pp. 53-66, 1995.
[75] A. Postorino,
R. Serio, and F. Mute, "Nitric oxide is involved in non-adrenergic, non-
cholinergie inhibitory neurotransmission in rat duodenum," J Auton Pharmacul., vol. 15,
no. 2, pp. 65-71, 1995.
[76] C. W. R. Shuttleworth and K. M. Sanders, uInvolvement of nitric oxide in neuromuscular
transmission in canine proximal colon," froc. Soc. Ezp. Bio. Med., pp. 16-23, 1996.
(771 N. G.Publicover, E.M. Hammond, and K. M. Sanders, "Amplification of nitric oxide signaling
by interstitial cells isolated from canine colon," Pmc. Natl. Aaid. Sci. USA, vol. 90, pp. 2087-
2091, 1993.
139
[78] G. Gabeiia, "Structure of muscles and nerves in the gastrointestinal tract," in Physiology of
the Gastmintestinal Z h c t (L. R. Johnson, ed.), ch. 11, pp. 335-381, New York: Raven Press,
1987.
[79]
R. M. Henderson, 'Cell-to-ce11 contacts," Methods in Phannacology, vol. 3, pp. 47-77? 1975.
[80] J. H. Szurszewski, "Electrical basis for gastrointestinal motility," in Physiology of the Gastrointestinal h c t (L. R. Johnson, ed.), ch. 12, pp. 383-422, New York: Raven Press, 1987.
[al] A. J. Bauer, J. B. R e d , and K. M. Sanders, "Slow wave heterogeneity within the circular
muscle layer of the canine gastric antrum," J. Physiol, vol. 366, pp. 221-232, 1985.
[82] A.
J. Bauer, N. G.Publicover, and K. M. Sanders, "Origin and spread of slow waves in canine
antral circular muscle," Am. J. Physiol., vol. 249, pp. G800-43806, 1983.
1831 M. Kobayoshi, T. Nagal, and C. L. Prosser, uElectrical interaction between muscle layers of
cat intestine," Am. J. Physiol., vol. 211, no. 6,pp. 1281-1291, 1966.
[84] N. Suzuki, C. L. Prosser, and V. Dahms, "Boundary cells between longitudinal and circular
layers: Essential for electrical slow waves in cat intestine," Am. J. Physiol., vol. 250, pp. G287G294, 1986.
[85] Y. Hara, M. Kubota, and J. H. Szurszewski, uElectrophysiologyof smooth muscle of the smali
intestine of some marnmals," J. Physiol., vol. 372, pp. 501-520, 1986.
[86]T.K. Smith, J. B. Reed, and K. M. Sanders, "Origin and propagation of electrical slow waves
in circular muscle of canine proximal colon," Am. J. Physiol., vol. 252, pp. C2154224, 1987.
(871 T. K. Smith, J. B. R d , and K. M. Sanders, "Interaction of two electrical pacemakers in
muscdaris of canine proximal colon," Am. J. Physiol., vol. 252, pp. C29û-C299, 1987.
[88] T. K. Smith, J. B. Reed, and K. M. Sanders, "Effects of membrane potential on electrical
slow waves of canine proximal colon," Am. J. Physiol., no. 255, pp. C828-(2834, 1988.
[89] L. W. C. Liu and J. D. Huizinga, "Electrical coupling of circular muscle t o longitudinal muscle
and interstitial cells of Cajal in canine colon," J. Physiol., vol. 470, pp. 445-461, 1993.
[go]
L. Thuneberg, uInterstitial cells of Cajal: Intestinal pacemaker cells?," Adv. A n d . Embryol.
Cell. Biol., vol. 71, pp. 1-130, 1982.
140
[91] L. Thuneberg, "Interstitial cells of Cajal," in Handbook of Physiology. The Gastmintestinal
System (J. D. Wood, ed.) , pp. 349-386, Bethesda, MD: American Physiological Society, 1989.
[92]
T.W. Berger, G. B. Robinson, R. L. Bort, and R. J. Sclabassi, "Nonlinear systerns analysis of
the functional properties of the hippocampal formation," in Advanced Methods of Physiologi d Syslem Modeling (V. 2. Marmarelis, ed.), pp. 73-103, University of Southern California,
Lm Angeles: Biomedical Simulations Resource, 1987.
[93] G. Major, A. U. Larkman, P. Jones, B. Sakmann, and J. J. B. Jack, UDetailed passive
cable properties of whole-ce11 recorded CA3 pyramidal neurons in rat hippocampal slices," J.
Neuroscience, vol. iS, no. 8, pp. 46134638, 1994.
[94] H.
H. Jasper, "Historical introduction," in Generalized Epzlepsy: Neurobiological Appmaches
(M. A. et al., ed.), ch. 1, Boston: Birkhauser, 1990.
[95] T. A. Valiante, J. L. Perez Velazquez, S. S. Jahromi, and P. L. Carlen, "Coupling potentiais
in CA1 neurons during calcium-free-induced field burst activity," J. Neuroscience, vol. 15,
no. 10, pp. 6946-6956, 1995.
[96] B. A. MacVicar and F. E. Diidek, "Electrotonic coupling between pyramidal celis: A direct
demonstration in rat hippocampal dices," Science, vol. 213, pp. 782-785, 1981.
[97] C. P. Taylor and F. E. Dudek, "A physiological test for electrotonic coupling between CA1
pyramidal ceils in rat hippocampal slices," Brain Res., vol. 235, pp. 351-357, 1982.
[98] J. L. Perez-Velaqua, T. A. Valiante, and P. L. Carlen, "Modulation of gap junctional rnechanisms during calcium-free induced field burst activity: A possible role for electrotonic coupiing
in epileptogenesis," J. Neuroscience, vol. 14, no. 7, pp. 4308-17, 1994.
[99] J. Church and K. Baimbridge, "Exposure to high-ph medium increases the incidence and ex-
tent of dye coupling between rat hippocampal CA1 pyramidal neurons in vitro," J. Neurosci.,
vol. 3, pp. 773-782, 1991.
[100] W. A. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numericul Recipes in C:
The A r t of Scientific Computing. New York: Cambridge University Press, second ed., 1994.
[101] R. C.Plonsey and
R. C. Barr, Bioelectricity: A Quantitative Appmach. New York: Plenum
Press, 1988.
[IO21 F. K.Skinner, C. A. Ward, and B. L. Bardakjian, "Permeation in ionic channels: A statistical
rate theory approach," Biophys. J., vol. 65, 1993.
11031
A. van Oosterom and J. Strackee, 'The solid angle of a plane triangle," IEEE rrcins. Biomed.
Eng., vol. 30, pp. 125-6, 1983.
[IO41 S. Wolfram, Mathematim: A System for Doing Mathematics by Cornputer. Redwood City:
Addison-Wesley, 1991.
[105] K. Radhakrishnan and A. C. Hindmarsh, "Description and use of LSODE, the Livermore
solver for ordinary differential equations," tech. rep., NASA Reference Publication 1327,
1993.
[lûû] E. J. Vigmond and B. L. Bardakjian, "Efficient and accurate computation of the electric
fields of excitable cells," Annals Biomed. Eng., vol. 24, no. 1, pp. 168-179, 1996.
11071 G. R. Cowper. "Gaussian quadrature formulas for triangles," Numerical Methods in Eng.,
vol. 7, no. 3, pp. 405-408, 1973.
[108] C.H. Chan and R A. Kipp, YAnirnproved irnplementation of triangular basis functions for
the analysis of microstrip interconnects," J. Electm. Waues, vol. 4 1 , pp. 781-797, 1994.
[log] A. Peskoff and R. S. Eisenberg, 'Potential in a spherical ce11 using matched asymptotic
expressions," J. Math. Biol., vol. 2 , pp. 277-300, 1975.
[Il01 R. F. Harrington, Field Computation by Methwl Moments. Malabar, FL: R. E. Krieger
Publishing Co., 1983.
[ I l l ] F. K.Skinner, Ionic ltnnsport Mechanisms in a Mode1 of Gastric Srnooth Muscle. PhD thesis,
University of Toronto, 1992.
[Il21 L. Thuneberg. Personal communication, 1994.
[Il31 W. H.Hayt, Jr., Engineering Electmmagnetics. New York: McGraw-Hill Inc., fifth ed., 1989.
[Il41
P. W. Partridge, C. A. Brebbia, and L. C. Wrobel, The Dual Recipmcity Boundary Element
Method. New York: Elsevier Science Publishing Company Inc., 1992.
[115] S. Zhu and Y. Zhang, "Solving problem in infinite domains with dual reciprocity boundary
element method," Eng. Anal. with Boundary Elements, vol. 12, pp. 241-250, 1993.
[116] A. H.-D. Cheng, S. Grilli, and 0. Lafe, "Dual reciprocity boundary element based on complete set global shape functions," in Boundary Elements XV, vol. 1, pp. 342-357, New York:
Elsevier Science Publishers, 1993.
[117] 2. H. Qiu, L. C. Wrobel, and H. Power, "An evahation of boundary element schemes for
c~nvection-diKusionproblems," in Boundary Elements X V ( C . A. Brebbia and J. J. Rencis,
eds.), vol. 1, pp. 26-37, New York: Elsevier Science Publishers, 1993.
[118] T. Y. El-Sharkaway, "Electrical activities of the muscle layers of the canine colon," J. Physiol.,
pp. 67-83, 1983.
[Il91 G. Pilar, J. nittle, and K. Vaca, "Functional maturation of motor nerve terminals in the
avian iris: Ultrastructures, transmitters, metabolism and synaptic reliability," J. Physiol.,
vol. 321, pp. 175-93, 1981.
[120] J. J. Tyson and J. P. Keener, "Singular perturbation theory of traveling waves in excitable
media: A review," Physica Dl vol. 32, pp. 327-61, 1988.
[121] R. W. Joyner, M. Westerfield, and N. Stockbridge, "A numerical method to mode1 excitable
ceils," Bfophys. J., vol. 22, pp. 155-70, 1978.
[122] 2. Lojewska, D. L. Farkas, B. Ehrenberg, and L. Loew, "Analysis of the effect of medium
and membrane conductance on the amplitude and kinetics of membrane potentials induced
by externally applied electric fields," Biophys. J., vol. 56, pp. 121-28, 1989.
[123] N. A. Dirnitrova and G. V. Dimitrov, " E k t of electrical parameters on the development
and propagation of action potentials in short excitable fibres," Electmencephalogmphy and
Clinical Neumph ysiology, vol. 70, pp. 453-9, 1988.
[124] 0. V. Brazil and M. D. Fontana, "Toxins as tools in the study of sodium channel distribution
in the muscle fibre membrane," Tozicon, vol. 31, pp. 1085-1098, 1993.
143
[125] T. Tomita, "Electrical properties of mammalian smooth muscle," in Smooth Muscle
(E. Buhlbring, ed.), ch. 7, pp. 197-243, London: E.Arnold, 1970.
[126] N. G. Publicover and K. M. Sanders, "Myogenic regdation of propagation in gastric smooth
muscle," Am. J. Physiol., vol. 248, pp. G5124520, 1985.
(1271 K. M. Sanders, R Stevens, E. Burke, and S. W. Ward, "Slow waves actively propagate at
submucosal surface of circuIar layer in canine colon," Am. J. Physiol., vol. 259, no. 2 P t 1,
pp. G25SG263, 1990.
[128] T. Y. El-Sharkawy, K. G. Morgan, and J. H. Szurszewski, "Intracellular electrical activity of
canine and human gastric smooth muscle," J. Physiol., pp. 291-307, 1978.
[129] H. Windischand, H. Ahammer, P. Schaffer, W. Muller, and D. Platzer, "Optical multisite
monitoring of ceil excitation phenornena in isolated cardiomyocytes.," Pfigers Arch., vol. 430,
no. 4, pp. 508-518, 1995.
[130] P. Gogan, 1. Schmiedel-Jakob, Y. Chitti, and S. Tyc-Dumont, "Fluorescence imaging of local
membrane electric fields during the excitation of single neurons in culturelq Biophys J.,vol. 69,
no. 2, pp. 299-310, 1995.
[131] M. Rapp, 1. Segev, and Y. Yarom, "Physiology, morphology and detailed passive structures
of guinea-pig cerebellar Purkinje cells," J. Physiol., vol. 474, pp. 101-118, 1993.
[132] G. J . Stuart and B. Sakrnann, "Active propagation of somatic action potentials into neocortical pyramidal cell dendrites," Nature, vol. 367, pp. 69-72, 1994.
[133] P. Anderson, "Unit analysis of hippocampal population spikes," Ezp. Bmin Res., vol. 13,
pp. 208-221, 1971.
[134] 0. Herrera, J. M. Solis, R. M. del Rio, and J. Lerma, "Characteristics of CA1 activation t hrough the hippocarnpal trisynaptic pathway in the unanaest hetized rat," Brain Res.,
vol. 413, pp. 7586, 1987.
[135] E. J. Vigmond, uElectrical field coupüng of smooth muscle cells in the ptrointestinal tract,"
Master's thesis, University of Toronto, Toronto, 1991.
Appendix A
Derivat ion of Formulae for Matrix
Ent ries
The formulae developed are applicable to zeroth order boundary elernents, meaning that any quantities associated with an element are constant over that element. A matrix entry in column 1 of
row k describes the effect of the lth element on the kth element. There are two types of sources
to consider: monopoles and dipoles. With the potential formulation, the potentials produced by
these sources must be computed while, the electric fields must be computed for the electric field
formulation.
The double layer potential calculation for zeroth order elements is equivalent to the solid
angle cdculation. This can be considered a measure of an object's "f'ootprintn. It has the following
propert ies:
1. For a closed surface, the solid angle is zero for points outside the surface.
2. For a closed surface, the solid angle is 47r for points inside the surface.
3. For a closed surface, the solid angle is 2n for points on continuous portions of the surface.
4. The solid angle of a non-closed surface may be determined by the contour of the opening.
Thus, as long as the opening is not disturbed, the suxface may be distorted in any way without
afkting the solid angle.
The electric field matrices are related to the potential formulation matrices by the relation-
A.1
Triangular Elements
A.1.1
Single Layer Sources
We will consider the triangle to be lying in an q plane at a height z. Referring to figure 3.4
for labelling convention, and noting that { is a modulo 3 quantity, the sides of the triangle may
be written in polar coordinats and are defined by the start and stop angles, OcTl, the minimum
distance to the origin, FE,and the angle a t which the minimum occurs, & (see figure fg:trimonodefs-
An analytic expression for the monopolar potential has been developed [135]:
w here
if the z-axis intercepts the element
2?r
q + 2
- e+l if the r axis intercepts vertex <
ot herwise
O
ALI other quantities are defined in figure 3.4. Function f2 is written in a form that takes
advantage of the IEEE recommended
C language function l o g l p o which computes log(1 + x )
accurately for srnail x. The surnmation index,
E, should be taken modulo 3 , and refers t o the side
of the triangle over which integration is being perfomed. Taking the gradient of the potential is
possible, albeit somewhat messy.
w here
vf3(wl-pc,P)
=
v f l ( ( ~-p<ip)
~ 1 1+ p E v f 2 ( ~ < & l
-F,f,p)
(A.10)
The gradient of @ is zero since it is a constant of integration. Now, pc is the angle at which a
radial line intercepts edge ( of the element at a right angle. As can be seen in figure 3.4, this
angle is dependent soleIy on the dope of the side, not its absolute position. Thus, the derivative
under a simple translation in the x or y direction is zero. For differentiation of the angle
ipc,
we
start off with its definition in Cartesian coordinates and convert to polar coordinates, initially only
considering the x direction for clxi ty:
where we apply the basic definition of the radial ordinate
and i t follows:
The last gradient required is that of &, the minimum distance of edge
Erom the origin occurring
at angle
vt. To dxerentiate, we utilize equations defining the edge:
Substituting equation A S 4 into equation A.11, it is found that
Note in the evaluation of equation A.21 there is no dependence on the edge parameters,
p and p.
Thus, performing the integration over the sides depends only on the vertex d u e s and will result
in cancellation when the contour of integration is closed. Hence, the contribution of Vf3 can be
ignored.
For calculation of the y component of the gradient, only V q is needed. Substitution of y
for x in the numerator of equation A.16 results in
The derivative in the z direction is the solid angle divided by 47r since the normal direction
is the direction of dzerentiation and we can use equation A.25 which is developed in the next
section. Equation 8.9 can also be used by considering the gradient in the z direction. The angular
and radial quantities do not change with
2.
The necessary derivatives are computed from
It is quicker to use the solid angle formula than the preceding since fewer trigonometric function
calls are required for the solid angle.
The three directional derivatives now form a complete bais set and we need only project
the gradient onto the desired normal to get the necessary matrix entry. For the case where the
field point is at the center of the source element, only the solid angle need be considered. The solid
angle evaluates to 27r but this is removed from the rnatrix to be subtracted from the left hand side
accounting for the one half factor in the limit as the point of observation approaches the surface.
Finally, in surnrnary (and utiiizing equation A.25),
A.1.2
Double Layer Sources
As mentioned previously, with zeroth order elernents, the potential produced by dipole sources is
equivalent to the solid angle of the element. An efficient analytic solution for the solid angle of a
triangular plane has been derived[l03]:
where rc = (x - xc,y - yc,z
- q),(t = 1,2 or 3), is the vector from the field point,
(2,y,r),
on
element k, to a vertex of the source elernent, 1. Denoting the numerator of the arctan argument as
N and the denominator as V ,the EFF matrix is obtained as follows:
First, the numerator will be considered.
Taking the gradient and noting the subscript t cycles modulo 3 results in:
To differentiate the denorninator, V , it is helpful to note that
r€
Vq = -
lrcl
and
V (q,r h ) = q, + rc2.
After application of the chain rule on the denorninator, the derivative unfolds:
Thus, ail the quantities necessary for equation A.26 are determined and the gradient is projected
on to the required normal. The entire computation for one matrix entry requires 62 additions,
69 multiplications and 3 square roots. This is just less than twice the solid angle calculation and
requires no arctangent computation.
A.2
Disc Elements
A.2.1
Single Layer Sources
Assurning the field point is at the origin and a disc of radius r is lying with its normal in the r
direction and its center at (po,O, z). In cylindrical coordinates, the integral can be expressed:
Therefore
Figum A . 1: Integration Zirnits for monopole sources on a disk. Two cases to constder for a disk
with radius r centered at po ( solid lines) and ph (dashed lines). For ph
[O, 2n). For po
> r , the limits
< r , the intenial for
p' is
of integrution for 9 are [- arcsin r/po, arcsin r/*].
Referring to figure A . l , we see the limits on p and cp depend whether the z-axis intercepts the dix.
Thus, we have
Unfortunately, due to the complicated nature of the ü m i t s expressed in equation A.37, further
analytic integration is not possible and Gaussian quadrature is required.
where
For the electric field, we must
&O
consider the derivative of the limits:
but
Hence,
wit h
A.2.2
Double Layer Sources
Consider a disc of radius p which lies in the t-plane and is centered on the origin. To determine
Hkl
, it is important to recall that the solid angle is invariant under any distortions of the surface
as long as the contour defining the surface is unchangeci. The simplest case to consider is when the
field point lies on the z-ais. To sirnplih the computation, the surface to attach to the contour is
the portion of a sphere centered on the field point which will then align the surface normal with
the distance vector. The equation becornes trivial:
= 27r(1
- cos O )
Thus, the integral is determined by the angular spread of the cone in the polar direction, 8. If a
right elliptical cone1 is considered, it may be transformed to a right cylindrical cone by a simple
scaling of an axis. Thus, given such a cone which has a value of k as the ratio of its minor to major
axes, the previous formula may simply be scaled by k. Furthermore, the ratio k rnay be expressed
in terms of the angular spread of the cone in the major and minor axes directions. At a given
distance, z, dong the central a i s , let the angular spread in the major direction be
ct
while the
spread in the minor direction is ,O. Hence, the length of the major axis will be z s i n a and that of
the minor axis will b e z sin ,û and k is the ratio of the two, sin a/sin 4.
The problem of finding the potential due to double layer sources on a disc at any point (not
only at those on the z-mis) can now be solved by restating the problem in a difFerent way: What
is the solid angle of the right ellipticol cone having its origin ut the field point and intersecting the
z = O plane with the circular contour defied by
F? The answer is refreshingly simple.
Since the field is rotationauy symmetric, the field point may be considered to Lie on the
origin with the disk lying in a Z plane centered at (x, O, 2) (see figure A.2). Utilizing the minimum
and maximum x and y points on the disc, the vectors from the field point to the points on opposite
ends of the major axis are
r~1=(x+p,O,z)
and
and to points on opposite ends of the minor axis are
rrnl
= (x,p, 4
Hkl
and
=
1 - cosck
sin 0
2 sin ct
which can be written in a slightly sirnpler form:
cos2a =
rM2
and
cos 2P = rml " rm2
l2
km1
l f ~ l l l ~ 2 1
+
' A right ellipticai cone is a volume deûned by the equation ( ~ / a ) (y/b)2
~
to the centrai axis, the cross section is elliptical.
A9
< (kz)'.
When cut perpendicularly
Note that no explicit trigonometric function calls need be evaluated since half angle formulae ailow
one to mite
sinx =
4'-2''
and
COS 5
=
J1+2"
(b) XY plane view
(a) Oblique view
Figure A.2: Solid angle of disk of mda'us p as descràbed by an ellipsoidal cone whose azis is
indicated by the dashed line. The major and minor axk of the ellipse are indicated on the disc in
figure b.
Computation of the electric field is straight forward. The gradient of equation A S 2 is taken
and dotted with the appropriate normal.
1
9 Q
hki= - (sec- - sin
4
w here
2
CY
va:+ 2 tan cos fl VP) . fik
2
and
A.3
Cylindrical Elements
A.3.1
Single Layer Sources
For a cylindrical element with two open ends, the monopole entry is done in a straightfonvard
manner. The openings are located at rl and r:! with radii of r and the axis if the cylinder is located
PO away hom the z-axis. For a point at the origin, the distance vector to a point on the cylinder
is
(m + r cos q ,r sin cp, 2 ) . Therefore,
The gradient is straightforward:
Double Layer Sources
The solid angle of a cylinder is equal to the negative of the surn of the solid angle of the open ends.
This must hold since the solid angle of a closed cylkder is zero. Ergo, the solid angle of the tube
is cancelled by the solid angle of the ends. Hence, equation A.52 is sirnply applied to each end for
computation of
Hkl.
If the sclid angle of a closed object is zero, the gradient of the same must be also zero, and by
an argument sirnilar to the one just given, Akl is equal to the negative of the sum of equation A.55
applied to each end of the cyIinder.
Al1
A.3.3
CyLinder with One Closed End
For monopole calculations, the element may be treated as a disc and a n open cylinder. For dipole
calculations, it is simply the negative of the result for the disc defining the open end.
A.3.4
Branches
Branches are simply treated as three separate cylinders which share a common end point for
monopolar calculations. As long as the radii of the cylinders is small compared to the length of
the cylinders, the region of overlap will be small and will introduce little error into the monopole
calculations. Hence, the region of overlap can be ignored. For dipole calculations, we need only
consider the three open ends away fiom the comrnon point.
A.4
Interface of Triangles and Cylinders
When triangles and cylinders are used in the same boundary element model, there will be an
interface cylinder which will join the two element types. Thus, there must be an opening in the
triangular section which approximates a circle. Regardless of how many triangles are used, the
opening will never by truly round and flux will escape through the gap between the cylinder and
straight line segments defining the opening. For monopolar calculations, this effect is insignificant
but not for the dipolar calculations. In this case, the interfacial cylindrical element must be treated
in especial. The end away from the triangular elements rnay be treated normally but the end at
the junction of the two types must be treated as a set of triangular elernents, not a disc. This will
remedy the geometrical mismatch between the triangles and cylinder.
Appendix B
Development of Diffusion
Transformation
B.1 Transformation
The object is to solve Poisson's equation for a time and space dependent source term:
The domain inversion transformation maps the radial component of the original coordinate system,
r to its inverse, F,where a tilde accent (') signifies a transforrned quantity. The method presented
by Zhu[ll5] will be expandecl to three dimensions. We start with the transformation:
r - r/r
7
(B-2)
The other coordinates, $ and O , are invariant under this mapping. For the ensuing development, the
derivative of the transforrned coordinate with respect to the untransformed coordinate is required
The original Laplacian is now converted to the inverted domain coordinates.
Now, the Laplacian of the original potential function muItiplied by a coordinate dependent function
is taken
-0
= SOT-u+ IO?-
au
+
&-
P a%
d2u
P d s i d aau
P
+
-s +Ti??
sinede
sin 0 d&
Once a solution for the above is obtained, the desired potential may be extracted from it.
B.2
Matrix Notation
The transformed integral equation, obtained after multiplying by a Green's function and applying
Green's f i s t identity, is given by
The product rule is applied to the normal derivative of P u :
f?
TT
B.2.1
u ~ P - ~ ( ~ T s u - + P * ) ~ </ /~/ T( 6' += 1 0 i ~ ~ + 2 0 i 2 ~ ) ~ d ~ '
-r
an
Dual Reciprocity
Green's fust identity can be stated as
Now, the functions are chosen such that
an
n
ai=
(~3.13)
where v is an arbitrary function. The domain integral which we wish to e d u a t e c m be isolated
on the left hand side and expressed as
The arbitrary function is now approximated by the interpolation functions:
Thus, the effect of the dornain integral on the boundary can be expressed in matrix notation as
below:
B.2.2
Approximations
The interpolating functions can also be used to approximate the radial derivative of u given u:
Dual reciprocity is now applied to all terms on the right hand side of equation B.13 and the
the entire equation put into matrix form.
The terms can now be gmuped for simplification.
B.2.3
Boundary Conditions
The boundary conditions must be mapped frorn the original domain to the inverted one. The
gradient in the unrnapped domain is expressed in terrns of the gradient in the inverted domain:
We really need to know
We again apply the interpolating function approximation
B.3
Global Interpolating Function
Global interpolation functions of the form x'ymzn were considered to approximate a known func-
tion. This is an extension to three dimensions of a set of two dimensional functions proposed by
Cheng[llG]. If the order of the interpolating function is given by IV, the surn of the exponents,
the nurnber of function in a given order is given by p / 2
+ 3N/2 + 1. The table below (table B.l)
lists ail the global interpolating functions up to order 5 and their inverse Laplacian. The unlisteci
functions are obtained by permuting z,y and z.
B .4
Alternative Green's Functions
The Green's function used until now has been the solution to Laplace's equation. It may be possible
to achieve better results by using more complicated functions to eliminate dornian integral terrns.
Starting with the differential form of equation B.11 and ignoring b, it follows:
Table B. 1: Global interpolating functions up to order 5
No. Permutations
Now, we wish to eliminate the partial derivative of u with respect to r
This can now be substituted back into equation B.30
The surface integral on the right can be readily cornputed and the domain integral wili be trivial
if a g(r, r') can be found which satisfies
where the positional dependencies have been included.
IMAGE EVALUATlON
TEST TARGET (QA-3)
-
APPLIED
IMAGE.lnc
= 1653 East Main Street
-.
Rochester. NY 14609 USA
-----
Phone: 71W482-0300
Fax: 716/200-5989
O 1993. Applied Image. Inc..Ail Righls R e s e ~ e d