Download Modeling the Gastric Mill Central Pattern Generator of the Lobster

JOURNALOF NEUROPHYSIOLOGY
Vol. 70, No. 3, September 1993. Printed
in U.S.A.
Modeling the Gastric Mill Central Pattern Generator of the Lobster
With a Relaxation-Oscillator
Network
PETER F. ROWAT AND ALLEN I. SELVERSTON
Department of Biology, University of California, San Diego, La Jolla, California
SUMMARY
AND
CONCLUSIONS
1. The gastric mill central pattern generator (CPG) controls the
chewing movements of teeth in the gastric mill of the lobster. This
CPG has been extensively studied, but the precise mechanism
underlying pattern generation is not well understood. The goal of
this research was to develop a simplified model that captures the
principle, biologically significant features of this CPG. We introduce a simplified neuron model that embodies approximations
of
well-known
membrane currents, and is able to reproduce several
global characteristics of gastric mill neurons. A network built with
these neurons, using graded synaptic transmission and having the
synaptic connections of the biological circuit, is sufficient to explain much of the network’s behavior.
2. The cell model is a generalization
and extension of the Van
der Pol relaxation oscillator equations. It is described by two differential equations, one for current conservation and one for slow
current activation. The model has a fast current that may, by
adjusting one parameter, have a region of negative resistance in its
current-voltage (1-v) curve. It also has a slow current with a single
gain parameter that can be regarded as the combination
of slow
inward and outward currents.
3. For suitable values of the fast current parameter and the slow
current parameter, the isolated model neuron exhibits several different behaviors: plateau potentials, postinhibitory
rebound, postburst hyperpolarization,
and endogenous oscillations. When the
slow current is separated into inward and outward fractions with
separately adjustable gain parameters, the model neuron can fire
tonically, be quiescent, or generate spontaneous voltage oscillations with varying amounts of depolarization
or hyperpolarization.
4. The most common form of synaptic interaction in the gastric CPG is reciprocal inhibition.
A pair of identical model cells,
connected with reciprocal inhibition,
oscillates in antiphase if either the isolated cells are endogenous oscillators, or they are quiescent without plateau potentials, or they have plateau potentials
but the synaptic strengths are below a critical level. If the isolated
cells have widely differing frequencies (or would have if the cells
were made to oscillate by adjusting the fast currents), reciprocal
inhibition
entrains the cells to oscillate with the same frequency
but with phases that are advanced or retarded relative to the
phases seen when the cells have the same frequency. The frequency of the entrained pair of cells lies between the frequencies of
the original cells. The relative phases can also be modified by using
very unequal synaptic strengths.
5. A reduced network model was used to study the coordination between the lateral and medial subsets and the effect of deleting a cell from the circuit. The results of killing Int 1 in the model
had effects similar to killing Intl in the biological circuit. This
suggests that Int 1 accomplishes the coordination
of the two subsets by indirectly altering the effective strengths of synapses between them.
6. A network of cells with all the known connections was also
studied. It was found that the network would oscillate and produce
an approximately
biologically correct output pattern over a wide
1030
92093-0322
range of synaptic strengths. This remained true when the individual cells were adjusted to be oscillators or to be quiescent. Random
changes in parameter values of up to 40% had little effect on the
overall pattern. The pattern of phase relationships
remained approximately constant when the model frequency was varied. The
phase lag between the lateral and medial subsets of the gastric
network could be obtained by incorporating
known slow synapses
and by adjusting a slow current parameter. If cells are killed sequentially in the model, the network continues to generate a pattern so long as at least one pair of reciprocal inhibitory cells remains. Changes in the relative phases of slow-wave activity can be
obtained by changing the gains of the slow currents.
7. A cell model that has a fast current with an N-shaped 1-V
curve, and slow inward and outward currents with linear steadystate 1-V curves, captures important
characteristic properties of
gastric neurons, and a network model built by connecting these
cells with instantaneous
graded synaptic transmission
captures
important features of small CPGs. This simple cell model is an
abstraction that delineates a basic mechanism common to all gastric cells and provides a foundation on which to build more comprehensive models of the gastric mill network.
INTRODUCTION
Central pattern generators (CPGs) produce rhythmic
motor outputs that underly repetitive behaviors such as
chewing, swimming, or walking. Many invertebrate CPGs
have very small numbers of neurons and have been intensively studied (Kristan 1980; Selverston and Moulins
1987). The gastric mill CPG controls striated muscles operating three teeth that comprise the gastric mill of the lobster
(Harris-Warrick
et al. 1992). All but two of the 13 neurons
forming the gastric mill CPG lie in the stomatogastric ganglion (STG), a network of -30 neurons located on the dorsal surface of the stomach of the lobster. Most of the nongastric cells in the STG belong to the pyloric CPG. This
controls the pyloric region of the stomach and operates at a
higher frequency.
A great deal is now known about the gastric mill CPG
(Selverston and Moulins 1987). All 13 cells are identifiable.
Many connections are known (Mulloney and Selverston
1974a,b; Selverston and Mulloney 1974) and the results of
many experimental
perturbations,
such as intracellular
current pulse injections and the removal of cells from the
network, have been examined (Selverston et al. 1983). Despite this, the known neuronal interactions are too complex
to easily explain how the overall dynamics of the system
produces the multiphase gastric pattern. To help in understanding the network mechanisms, a rigorous modeling
study was undertaken. The gastric mill network provides an
ideal system with which to attempt a theoretical account of
0022-3077/93 $2.00 Copyright 0 1993 The American Physiological Society
MODEL
OF GASTRIC
MILL
CPG
1031
LPGI
DG
AM
GM
INTI
110mv
5s
Inhibitory
i Functional
Inhibitory
A Excitatory
i Functional
Excitatory
l
-
Elect rotonic
E
J
5mV
1 s.
FIG. 1. A: network diagram for the gastric mill central pattern generator (CPG). Circles represent cells. Overlapping
circles represent multiple cells that are regarded as identical for modeling purposes. All but the 2 E cells are in the stomatogastric ganglion (STG). The key to the types of connections in the network appears below the diagram. A functional inhibitory
or excitatory connection means that not all the tests for monosynapticity, as described in Mulloney (1987), have been carried
out on the connection in question. All cells except Intl and the E cells are motor neurons. B: simultaneous intracellular
membrane-potential recordings of all 7 types of gastric cells in the STG during ongoing activity in an in vitro preparation
(recording courtesy of H.-G. Heinzel). Slow waves are clearly seen. The STG preparation was attached to 3 other ganglia
providing neuromodulatory input necessary for the maintenance of the rhythm. In each cells’ recording, high-frequency
spikes superimposed on the upper parts of the slow waves are action potentials. Signals with frequency between that of the
spikes and the slow waves, particularly prominent in the LPG, DG, AM, and Int 1 traces, are due to inputs from the pyloric
network, another CPG contained in the STG. The small spike amplitude in these recordings is due to the microelectrode
placement being in the soma, at a large electrical distance from the spike initiation zone. C: intracellular recordings from a
GM and an E cell (modified from Selverston et al. 1976). AM, anterior median; DG, dorsal gastric; GM, gastric mill; LG,
lateral gastric; MG, median gastric; LPG, lateral posterior gastric; Int 1, interneuron 1; E, excitatory.
its principles of operation. There have been several previous modeling studies (Friesen and Stent 1978; Selverston
et al. 1976; Thompson 1982; Tsung et al. 1990; Warshaw
and Hartline 1976) of this system. In these studies there
were many free parameters, few of which could be given
unambiguous physiological interpretations, and the models
were not tested with perturbations similar to those possible
experimentally (Selverston et al. 1983). Thus a clear picture
of the basic principles and mechanisms underlying pattern
generation has not emerged.
The cells of the gastric mill network, together with the
currently known interconnections,
are shown in Fig. 1~.
The key to cell name abbreviations is given in the legend for
this figure. There are four identical GM cells, each one connected to the others with electrical junctions, and each playing the same functional role in pattern generation, as far as
is known (Selverston and Mulloney 1974). Because intracellular recordings from individual GM cells are identical,
these cells were modeled as a single unit. The two lateral
posterior gastric (LPG) cells are connected with an electrical junction and were, for the same reasons, modeled as a
single unit. The two E cells are not in the STG. There is one
in each of the bilaterally paired commissural ganglia of the
stomatogastric nervous system (Russell 1976; Selverston et
al. 1976). The E cells are inhibited by interneuron 1 (Int l),
and they excite the GM, lateral gastric (LG), median gastric
(MG), and LPG cells (Fig. IA). The E cells fire in bursts that
are out of phase with Int 1 when the gastric rhythm is active.
They fire tonically if not receiving phasic inhibition from
Intl. Although their role is not clear, the E cells may be
regarded as components of the gastric mill CPG (Russell
1976). The two E cells are functionally identical and were
also modeled as a single unit. We included the E cells in
most of our models of the complete gastric network. Thus
there are seven different types of gastric mill cell within the
stomatogastric ganglion, which together with the E cells
make a total of eight different cell types. Simultaneous intracellular recordings of all seven types in the STG are
shown in Fig. 1B (Heinzel and Selverston 1988). Simultaneous intracellular recordings of a GM and an E cell are
shown in Fig. 1C (Selverston et al. 1976).
The gastric mill CPG generates a pattern only when nonphasic, neuromodulatory,
inputs from other ganglia in the
stomatogastric nervous system are present, or when specific
neuromodulatory
substances have been experimentally applied to the STG. The stomatogastric nerve (Stn) is the sole
input tract from these other ganglia. When these inputs are
removed, by cutting or blocking the Stn, no pattern is generated. We will refer to these different states as cycling and
noncycling states of the gastric mill CPG.
1032
I? F. ROWAT
AND
The gastric mill itself consists of one medial tooth and
two lateral teeth, together with a number of ancillary ossicles and surrounding musculature (see Selverston et al.
1976). Functionally the GM cells cause the medial tooth to
protract in a power stroke, whereas the dorsal gastric (DG)
and anterior median (AM) cells cause it to retract. The LG
and MG cells cause the lateral teeth to come together in a
cut or squeeze movement and are opened by the LPG cells.
The alternate firing of GM with DG and AM, and of LG
and MG with LPG, is seen in the traces of Fig. 1B. Note that
there is a small but distinct phase lag between the outputs of
the cells controlling the lateral teeth (namely LG, MG, and
LPG) and the outputs of cells controlling the medial tooth
(GM, DG, and AM). For example, the bursts of spikes from
the GM cell lag behind the LG and MG bursts, whereas the
DG and AM bursts lag behind the rather poorly defined
LPG bursts. The E cells are synchronous with the GM cells
(Fig. 1C) and thus serve to increase the spiking frequency of
the GM bursts. The E cells therefore modify the force exerted by the medial tooth and provide a means for CNS and
sensory modulation of medial tooth action.
Cell properties
Much is already known about the properties of gastric
mill cells and their synapses (Selverston and Moulins 1987).
All cells, except for the interneurons Int 1 and the E cells, are
motorneurons
having two roles: pattern generation and
muscle activation. The bursts of spikes from each motorneuron activate gastric mill muscles and are not believed to
contribute significantly to the phase-setting mechanisms of
pattern generation. Evidence for this is provided by Elson
and Selverston (1992), who showed that the isolated gastric
mill network, when activated by the presence of pilocarpine, produces slow-wave oscillations that continue with
approximately unchanged phase relationships when spikes
are suppressed by the application of tetrodotoxin (TTX). In
the closely related pyloric CPG, Raper (1979) and Anderson and Barker (198 1) showed that the suppression of
spikes by TTX has little effect on an ongoing pyloric
rhythm. Also in the pyloric CPG, Graubard et al. (1983)
showed that graded and spike-mediated synaptic transmission occur simultaneously
between synaptically coupled
pyloric cells. For the purpose of this work therefore, we
assume that the contribution
of network connectivity to
pattern generation is primarily dependent on graded synaptic transmission and that spike-mediated synaptic transmission makes no significant contribution to the overall pattern-generating process.
The E cells are located in separate ganglia and cannot
communicate
with the gastric mill cells in the STG by
graded synaptic transmission. To simplify the initial modeling studies, it was assumed that the overall contribution to
pattern generation of the excitatory postsynaptic currents
caused by E cell spikes is equivalent to the postsynaptic
effect of excitatory graded transmission from the E cells.
Under normal experimental
conditions the patternforming properties of the gastric mill network appear to be
widely distributed. Several cells exhibit plateau potentials
and postinhibitory rebound. Thus intrinsic cellular proper-
A. I. SELVERSTON
ties appear to play an important role in pattern formation
(Russell and Hartline 1984; Selverston et al. 1976). No single cell has been shown to be a strong endogenous b urster.
In a few preparations, however, the DG cell has been found
to have the capability for endogenous bursting when subjected to injected current or other nonphasic input (Hartline and Russell 1984). This may be a side effect of the
cellular properties of DG that are necessary for the overall
pattern-generating process. When the gastric mill is cycling,
it is possible to remove several cells from the network experimentally, including DG, with little degradation of the remaining rhythm (M. Wadepuhl, unpublished data). Hence
the rhythm is not driven by a single oscillatory cell but must
emerge from the interplay of intrinsic cellular properties
and network connectivity.
Synaptic properties
Neurons linked by reciprocal inhibition have often been
proposed as the basis for the production of alternate bursting in antagonistic pairs of muscles (Brown 19 14; McDougall 1903). A reciprocal inhibitory pair, i.e., two single cells
linked with reciprocal inhibition,
occurs in many invertebrate CPGs (Benjamin 1980; Calabrese et al. 1989; Getting
19 8 1, 19 8 3) and frequently innervates antagonistic muscles
(Robertson and Pearson 1985). Perkel and Mulloney
(1974) using a spike-based model of synaptic interaction,
showed that postinhibitory rebound in each cell of an inhibitory pair was sufficient to cause oscillatory behavior. Satterlie (1985) provided evidence that the swimming pattern in
Clione is based on reciprocal inhibition between two nonoscillatory cells with postinhibitory
rebound. Wang and
Rinzel ( 1992) studied a simplified,
conductance-based
model of reciprocal inhibition. Each cell had a leak current
and a simplified T-type Ca2+ current that activated immediately but inactivated slowly. They found that there were
two distinct modes of operation: “release” of inhibition by
the inhibiting cell or “escape” from inhibition by the inhibited cell. A necessary requirement for oscillation in an inhibitory pair is some time-dependent mechanism to gradually
remove inhibition from one cell so that it can burst. Physiologically, this may take the form of spike adaptation or disinhibition by a hyperpolarization-activated
inward current
in the presynaptic cell (Angstadt and Calabrese 1989).
In the gastric mill CPG it can be seen from the circuit and
traces in Fig. 1 that there are several pairs of neurons, connected by reciprocal inhibitory synapses, that fire approximately out of phase with one another. There are four inhibitory pairs: Int l/LG, Int l/MG, MG/LPG,
and LG/DG. The
LG/MG cells are connected with reciprocal inhibition but
also share an electrical junction and have always been observed to fire approximately
together (Mulloney and Selverston 1974a). Consequently, for the purposes of this
paper, the LG/MG interaction is considered to be dominated by the electrical junction rather than by the inhibitory synaptic connections. The MG/LPG
inhibitory pair
innervates antagonistic muscles that move the lateral teeth.
The cells Intl, MG, and LPG form a three-element chain
with each pair connected by reciprocal inhibition.
Therefore, because MG alternates with Int 1 and with LPG, syn-
MODEL
OF
GASTRIC
chronization of Int 1 and LPG is expected, as can be seen in
Fig. 1B. Thus it is clear that the synaptic arrangement of
pairs of reciprocal inhibitory synapses plays an important
role in the GM CPG.
There are two synapses in the network for which the postsynaptic response is very slow. When Intl is depolarized,
the excitatory synapses from Intl to DG and from Intl to
AM produce a response in the postsynaptic cells that is very
slow and takes on the order of 400 ms to reach its peak
(Figs. 8 and 9 in Selverston and Mulloney 1974; Selverston
et al. 1976, 1983).
Modeling
To investigate the essential mechanisms underlying pattern generation by mathematical modeling, one has to first
address the question of what level of modeling to use.
Ideally the model should be a simplification
of the numerous biophysical events taking place, whereas at the same
time it should be consistent with the experimental data.
One has to find a balance between simplicity, which implies
using the smallest possible number of parameters in a
model, and faithfulness to the experimental data, which
may well require an essentially unlimited number of parameters.
Many sophisticated and complex models of single neurons have been used. Multi-compartmental
models (Edwards and Mulloney 1984; Perkel and Mulloney 1978) and
single compartment models with large numbers of voltageand time-dependent ionic currents (Rose and Hindmarsh
1989a-c; Yamada et al. 1989) have been studied. The
former are ideal for modeling the effect of cell geometry on
network function, whereas the latter are ideal for studying
the detailed interplay of specific types of membrane
currents in endogenous oscillations. We considered both
classes of model to be more complex than needed for models of a small network in which the goal is to elucidate general principles of operation. This is in part because the complexity of a network model constructed with such model
neurons would be too great and in part because membrane
currents in gastric mill cells are not characterized in sufficient detail to permit the construction of cell models that
include accurate representations of ionic currents.
We did not use a cell model based on the Hodgkin-Huxley equations (Hodgkin and Huxley 1952) because these
equations can produce behavior much more complex than
necessary (Carpenter 1979), and because they model action
potentials, which do not play a major role in gastric mill
pattern generation. The basic requirement was for a model
of slow-wave dynamics, not a model for spikes which in the
gastric mill ride on the crests of the slow waves. An even
more crucial reason for not using a conductance-based
model is that there are no data available for membrane
currents in the gastric mill CPG. There are, however, data
available on the membrane currents in cells of the pyloric
CPG (Gola and Selverston 198 1; Golowasch and Marder
1992; Graubard and Hartline 199 1). The Golowasch and
Marder (1992) data were incorporated in a biophysical
model of the LP cell in the pyloric network (Buchholtz et al.
1992). If one hvpothesizes that cells in the same ganglion
MILL
CPG
1033
share similar sets of ionic conductances, then the latter
model may be adaptable for subsequent modeling studies
of the gastric mill.
An important question of principle is whether the gastric
mill CPG is driven by network interactions, by intrinsic
cellular properties, or some combination of the two. Earlier
work shed some light on this matter. Using an abstract formulation, Tsung et al. ( 1990) modeled the gastric mill CPG
with a network of identical neurons. Each neuron was represented as a piece of passive membrane, and the input to a
cell was a weighted sum of the presynaptic potentials. Thus
the pattern-generating
mechanism had, of necessity, to reside in the connection strengths. After parameter adjustment by means of back-propagation (Williams and Zipser
1989), this model captured the approximate phase relations
between the slow membrane-potential
waves seen in Fig.
1B but ceased oscillations when any one cell was removed
from the network. Another problem was that two units connected with reciprocal inhibition would not oscillate. Thus
a pair of reciprocal inhibitory cells, an important constituent of the gastric mill network, could not be modeled. The
failure of this network model tells us that the pattern-generating mechanism cannot reside in the connections alone
but must also depend on intrinsic cellular mechanisms.
Therefore a more sophisticated model was required.
Our goal was to develop a model that was both computationally tractable and physiologically
plausible, in that it
had to be consistent with the most significant features of the
biological data. Specifically, the model had to satisfy the
following requirements.
I) Every model element (every variable and parameter)
had to have a clear and plausible physiological interpretation.
2) It should be easy, by adjusting physiologically meaningful parameters, to cause a model cell to display characteristic properties of gastric cells such as plateau potentials
and postinhibitory
rebound.
3) It should be possible to obtain appropriate behavior
from small networks of model cells such as the reciprocal inhibitory pair and other subnetworks of the gastric
mill CPG.
4) At the network level, the responses of the complete
model to perturbations should correspond to the response
of gastric mill preparations to experimental perturbations.
Other desirable features, not addressed in this paper,
were that the model should provide a useful approximation
for a wide variety of complex biophysical models and that
the model should be mathematically
tractable.
Generalized nonlinear oscillators have been applied to
neurophysiology
by many authors (e.g., Pavlidis and
Pinsker 1977). The application of nonlinear oscillators to
CPGs in particular has been studied by Cohen et al. (1982)
Kopell(1988),
Friesen and Stent (1978), Stein (1977), and
many others. To achieve simplicity, and thus derive general
properties of networks of oscillators, the nonlinear oscillator has often been simplified to a point moving with constant speed on the circumference of a circle. However, the
relation between the circular motion of the point and physiological phenomena such as membrane currents has been
deliberately left undefined. To satisfy 1) above, we considered that another approach was necessary.
1034
P. F. ROWAT
A
AND
C
/ /-
linear
Of = 0.9
FIG. 2. Simulated current-voltage (I-V) curves for the proposed cell
model. A and B: I-Vcurves for the fast current when of = 2.0 (A) and of =
0.9 (B). C steady-state I- Vcurve for the slow current for B, = 3. This curve
can be regarded as the combination of 2 parts: an I- Curve for an inward
current and an I-V curve for an outward current. D: steady-state slow
current I- V curve when the inward and outward parts have different gains
(Oin = 3, lout = 0
The relaxation oscillator has a venerable history. It was
studied in the 19th century by Lord Rayleigh, analyzed further by Van der Pol(l926), used by him and Van der Mark
(1928) to model the heart, and claimed by the physiologist
A. V. Hill (1933) to be the oscillator “with which alone we
are concerned in physiology.” The essential feature of a
relaxation oscillator is that it consists of two opposing forces
acting on different time scales. A generalized CPG model
consisting of a network of relaxation oscillators has been
studied (Bay and Hemami 1987), but its parameters cannot
be interpreted easily in physiological terms. Again, to satisfy 1) above, some other model was needed. We used an
extension of the relaxation oscillator.
METHODS
Cell model
The model for an isolated cell was adapted from the generalization by Lienard (1928) of Van der Pol’s relaxation oscillator
(1926). It is written as two equations
dV
7,-&+F(v)+q=O
(1)
dq
7”dt = -4 + q,(V)
(2)
where F(V) is given by
F(V)
= V - Af tanh [(af/Af) V]
(3)
and qco (V) is given by
q,(V) = %V
An alternative
definition
4*(V)
(4)
is also used
=
I
(TinV
for V-c 0
GoutI/
for V> 0
Here V represents the membrane potential, 7, represents
membrane time constant, F(V) represents the current-voltage
(5)
the
(I-
A. I. SELVERSTON
V) curve of an instantaneous, voltage-dependent
current (Fig. 2, A
and B), and q represents a slow current with time constant 7, and
steady-state I-V curve q, ( V) (Fig. 2, C and 0). F( V), the “fast I- V
curve, ” is N-shaped. Equation 3 allows the degree of N-shape and
the width of the N-shape to be adjusted independently
of one
another by means of the parameters of and Af. of adjusts the degree
of N-shape in Ifast without changing the distance between the
asymptotic values for F(V), V t A, (Fig. 2, A and B), whereas A,
adjusts the width of the N-shape without affecting the degree of
N-shape. Af scales the fast I-V curve without changing its slope at
the origin. The dashed lines in Fig. 2A are the asymptotes for F( V)
when V is large. A, measures one-half the distance between the
asymptotes along the V-axis. 1 - gf is the slope of the 1-V curve at
the origin. When of = 0, F(V) = V so the fast current is purely
ohmic. When 0 < cf 5 1, F(V) has a point of inflexion at V = 0
(Fig. 2B), and when of > 1, F(V) is N-shaped with a region of
negative slope (negative resistance) around the origin. In the latter
case, it crosses the V-axis at three points: on the left of the origin, at
0, and on the right of the origin (Fig. 2A).
qm (V) was usually defined by Eq. 4 (Fig. 2C), so the slow
current could be regarded as the combination
of a slowly activating linear outward current for V > 0 and a slowly activating linear
inward current for V < 0, each with the same gain us. Sometimes
qco (V) was split into inward (V < 0) and outward (V > 0) parts
with separate gains oin and tout (Eq. 5 and Fig. 20). 7, always has a
value smaller than the slow current time constant 7,. Eqs. 1 and 2
become a Lienard equation if the term -q in the second equation
is removed, of > 1, and CT,> cf - 1.
Network connections
The current passing through an electrical junction was modeled
as a fixed conductance times the difference in membrane potentials. The current through the rectifying electrical junction
between GM and LPG is computed as for an electrical junction
when V& > VLpG, otherwise it is zero.
The model of graded chemical synaptic transmission followed
Katz and Miledi ( 1967). Because synaptic transmission time in the
gastric mill CPG is about three orders of magnitude smaller than
the period of the slow waves (Mulloney and Selverston 1974a,b;
Selverston and Mulloney
1974), synaptic delay was ignored. Let
VV be the maximum postsynaptic conductance at a synapse between a presynaptic cell with membrane potential VP,, and a postsynaptic cell with potential Vpost. The proportion
of postsynaptic
channels open at any time due to the binding of transmitter released in graded manner from the presynaptic terminal was assumed to be an instantaneous,
O-l valued, sigmoid function .fof
the presynaptic membrane potential. Hence the postsynaptic conductance is wf( Y,,), and the post-synaptic current S at a synapse
with maximum conductance VV > 0 is given by
s = wf( v,re><vpost
- E,ost)
(6)
where f( V,,) = ( 1 + e-4Vpre)-1 and Epost is the synaptic reversal
potential. The sigmoid function/( Vi,,,) has unit slope at V = 0. We
sometimes refer to the maximum synaptic conductance I/t/as the
synaptic weightor connectionstrength.For an inhibitory synapse
this expression must always be positive (outward current), hence
the value of E’i.,ostmust be below the normal lowest value of the
postsynaptic membrane potential. Similarly, at an excitatory synapse Epost must be above the normal highest value of the postsynaptic membrane potential.
At a slow synapse, the slow response of the postsynaptic membrane potential to a change in the presynaptic potential was assumed to be mediated by a slow process that intervenes between
the binding of transmitter to receptors and the opening of the
MODEL
channel conductances. At a slow synapse the postsynaptic
ductance G is given by
dG
-=
rd dt
and the postsynaptic
-G
OF
GASTRIC
con-
(7)
+ wf( v,,>
s = (-3&OS*
- Epost)
(8)
7d is the time constant for the slow change in postsynaptic conductance in response to a change in the presynaptic membrane potential, V,,,.
Model summary
The model can be summarized in a physiological form as follows. The current equation for a single cell embedded in a network
is obtained from Eq. 1 by adding terms &, lej, and Iinj
dV+r
fast
+I
slow
+
Isyn
+
Iej
=
Iinj
(9)
where V is membrane potential, 7, is membrane time constant,
and 1, is injected current. Ifast = F(V) is the fast current, defined
by Eq. 3. ISlOw
= q is the slow current, which activates by Eq. 2 with
time constant 7,, to the steady-state value qm(V) given by Eq. 4 or
5. Isyn is the sum of all postsynaptic currents, where the current S
from one synapse is given by Eq. 6, or if a slow synapse, by Eqs. 7
and 8. 1ej is the sum of all electrical junction currents where the
current in cell A due to an electrical junction with conductance G
between cells A and B is G( VA - VB).
Simulation
system
An interactive simulation system called “The Preparation”
was
written and used for running all models. The Preparation continually integrates a model and simultaneously
displays model output graphically, while also allowing any parameters of a model to
be changed, without having to stop the model. Integration was
done by the LSODA software package developed by Lawrence
Livermore Laboratories
for integrating large sets of differential
equations with automatic method switching when the system of
equations becomes stiff (Petzold and Hindmarsh
1987).
Phase diagrams
The phase diagram is a mathematical
device used to deduce
qualitative properties of solutions of a system of differential equations without actually solving the system explicitly. The phase
diagram can be used with a system having any number of differential equations but will be described here for a two-equation
system
and applied to the system defined by Eqs. 1 and 2. We will refer to
this system as MC. Figure 3 shows seven phase diagrams. The state
of MC is completely specified by the pair of values (V,q). This is
represented as a point ( V,q) in the phaseplane, a two-dimensional
coordinate system with V on the horizontal
axis and q on the
vertical axis. As the state changes over time, the phasepoint [ V(t),
q(t)] describes a curve in the phase plane. This curve is also referred to as a trajectory or phasepath. An oscillatory solution with
all transients fully decayed will describe a closed curve that does
not intersect itself. This is called a limit cycle. A limit cycle is
attracting if, after a small perturbation that moves the phase point
off the limit cycle, the phase point returns to the limit cycle. Experimentally, only attracting limit cycles are seen. A and F of Fig. 3
have limit cycles. A solution of the system can be viewed as a point
moving on a trajectory in such a manner that the direction and
speed of movement
CPG
at each point of the trajectory
dV
is given by dt’
1035
dq . The V-nukline is the curve on which dtdV = 0, and the q-nulldt
cline is the curve on which dt&I = 0. Every part of Fig. 3 includes
and q-nullclines.
current is then
rm dt
MILL
Nullclines
scribing the movement
V-
are useful because they assist in de-
dV
=0
dt
on the V-nullcline,
any trajectory of MC must cross the V-nullcline vertically. Similarly, any trajectory must cross the q-nullcline
horizontally.
These statements can be checked in every part of Fig.
3 except E. At any point of intersection of the V and q nullclines,
dV dq
-E-E
0, so it is an equilibrium
point (e.p.; see, e.g., Fig. 3E.
dt
dt
the points L, H, and the origin, and D, the starred points and the
origin). The stability of each e.p. can be found by examining the
stability of the linear aproximation
to MC at the e.p. Experimentally, only stable e.p.s are seen. In Fig. 3E, points L and H are
stable e.p.s, whereas the origin is unstable. The origin is an unstable e.p. in A and D-G, but it is stable in B and C (where the
V-nullcline
crosses the V-axis with 0 or negative slope). For the
system MC, the V-nullcline is given by the equation q = -F(V),
and the q-nullcline is given by q = q, (V), or q = a,V. Thus the
V-nullcline is obtained by reflecting the fast I-V curve in the Vaxis, and the q-nullcline is identical with the slow current’s steadystate I-V curve. These two facts make it particularly easy to relate
the I- V curves of our model cell to nullclines in its phase diagram.
Compare Fig. 2, A and C, with the nullclines in Fig. 3A. For instance, injection of a steady depolarizing
current causes the fast
I- V curve to move downward, whereas the V-nullcline moves upward.
When a system is described by two differential equations with
time constants that are significantly different, as in the case here
where 7, < 7,, then the motion of the phase point can be described
approximately
2forz
of the phase point in time. Because -
dV.
as follows. Eq. I for dt is the fast equation
is the slow equation.
and Eq.
Away from the fast, or V-, nullcline,
the changes in the slow variable q are so slow in comparison with
the rate of change of the fast variable V that for all intents and
purposes the phase point is controlled by the fast equation, with
the slow variable essentially constant. Thus the phase point describes a trajectory that is roughly parallel to the fast variable axis,
which means that in our model it is parallel to the V-axis. Once on
or very close to the fast nullcline, however, the movement due to
the fast equation is negligible, because, by definition of the fast
nullcline, the right-hand side of the fast equation is zero or very
close to zero. Instead, the slow equation becomes dominant and
moves the phase point along the fast nullcline. For example, in
Fig. 3, A and F, the roughly horizontal segments of the limit cycle
labeled 1 and 3 are fast movements caused by the fast equation.
The diagonal segments of the limit cycle, labeled 2 and 4, are slow
movements in which the phase point stays close to the fast nullcline while being moved in the vertical (q- or slow) direction by the
slow equation. The fast movements 1 and 3 in the phase diagrams
correspond to the rapid rise and fall in the potential traces in Figs.
3, A and F. The slow movements 2 and 4 in the phase diagram
correspond to the depolarized part and the hyperpolarized
part,
respectively, of the potential traces.
Thus by sketching the nullclines, and taking into account the
presence of widely different time scales in the two equations, one
can obtain a qualitative description of the trajectories of the state
of the model cell MC and hence a description of the form of the
membrane-potential
trace.
When two nullclines are roughly parallel and close together, the
area between the nullclines is referred to as a narrowchannel.If the
P. F. ROWAT
1036
P , w
(7
l
q nullcline
AND A. I. SELVERSTON
Q .
I I
PBH
-
V
FIG. 3. Each box describes a different condition of the model cell. Each box contains a phase portrait, corresponding
traces for membrane potential as a function of time and for applied current if present, and the settings of the parameters of
and us. In the phase portraits, trajectories are shown by consecutive dots drawn at equal-length time intervals. Thus the larger
the dot spacing, the faster the speed of movement of the phase point. The very thick dashed lines are the V-nullclines, the
thinner dashed lines are the q-nullclines. In the B-D and G phase diagrams, a large dot with a surrounding circle marks the
starting point of a trajectory. V, membrane potential; q, slow current. In all boxes A, = 1. Except for G, 7, = 0.1666,~~ = 5. A:
cf = 2, cs = 3. An attracting limit cycle in the phase diagram produces potential traces similar to those often seen in cells LG
and MG in the gastric mill CPG. A, B, and C form a series in which cf is reduced to 0, or equivalently the degree of N-shape
goes to 0, whereas A, D, and E form a series in which cs is reduced to 0. F and G are 2 cases in which a split slow current is
used. B: of = 1. The cell is quiescent but nearly an oscillator, as shown by the slow spiral of the phase point into the stable
equilibrium point (e.p.) at the origin (v, q) = (0,O); the membrane-potential trace shows a damped oscillation. C of = 0. The
model cell is quiescent and exhibits only postinhibitory rebound (PIR) and postburst hyperpolarization (PBH). The steadystate points marked by dot-circles on the q-nullcline arise from the initial hyperpolarizing and depolarizing currents. D: this
case exhibits PIR, PBH, and plateau potentials but does not oscillate. It is intermediate between A and E, with os = 0.5. There
are two stable e.p.s, starred, and an unstable e.p. at the origin. Top pair of traces shows PIR with the cell coming to rest at H,
bottom pair of traces shows PBH with the cell stabilizing at the lower plateau potential L. The last parts of the trajectories,
showing the slow approaches to H and L, have been omitted. E: cell exhibits pure plateau potentials. The slow current q = 0,
so the phase point lies on the V-axis. The phase diagram is effectively 1 dimensional with stable e.p.s at L and H and an
unstable e.p. at the origin. F: a split slow current was used, with the gain of the inward part less than the gain of the outward
part. Thus the traversal of the lefthand downward section of the limit cycle is slower than the traversal of the righthand
upward part. This difference in speeds on different parts of the limit cycle corresponds to the difference in the lengths of the
hyperpolarized and depolarized portions of the potential trace. G: inward slow gain is 0, resulting in a single stable e.p. at L. A
current pulse (arrowhead) moved the phase point from L to a positive potential (dotted circle). The final approach to L is very
slow and stays very close to the fast I/-nullcline; this is not visible in the figure.
MODEL
OF GASTRIC
FIG.
MILL
CPG
1037
5. (continued)
phasepoint enters a narrow channel it is forced to move very
exceeds the threshold value of = 1 + T,/T,. Because T,/T, is
assumed
to be small, of is - 1.0,, the threshold for the slow
slowly becauseit is closeto two lines,with dt = 0 on oneline and
current gain CT~,can be computed from of, T,, and 7,. If the
&I = 0 on the other.
time scales 7,, TV,of the two equations are such that 7, is at
dt
least an order of magnitude smaller than T,, then the wavedV
RESULTS
Our overall strategy was to start by modeling a single cell
and then more complex circuits. We present results for the
single-cell model followed by an inhibitory pair of cells,
then a complete network model, a reduced four-cell network, and finally a complete network model incorporating
slow synapses.
Properties of a single cell
When the fast current 1-V curve F(V) is sufficiently Nshaped (Fig. 24 and the gain (TVof the slow current is above
a certain threshold value 8, > 0 (Fig. 34, the model cell
oscillates (Fig. 34. F(V) is sufficiently N-shaped when cf
F=
l M
,\
z
0
0”
0
10
Slow current gain
20
FIG. 4. Plot of the oscillation frequency of a model cell as a function of
the slow current gain cS.
1038
P. F. ROWAT
AND
form is that of a relaxation oscillator. Often, the intracellular slow waves generated by gastric mill cells LG, DG, or
MG are remarkably similar to relaxation oscillator waveforms, as illustrated by Fig. 5, I and J. If F( V) is deformed
into a non-N-shaped curve by setting cf 5 1 (Fig. 2B), then
the model will not oscillate (Fig. 3B; also Fig. 3C). Thus the
N-shaped feature of F( V) is necessary for the model cell to
oscillate.
The frequency of oscillation of the model cell is primarily
determined by the setting of the gain us of the slow current.
A plot of frequency u against slow current gain as is shown
in Fig. 4. This dia.gram shows how a particular cell model
suddenly changed from the quiescent state to endogenous
oscillations as the parameter gs passed through a threshold
value 8, = 0.49. The amplitude of oscillation of the cell is
primarily determined by the setting of the parameter A, in
Eq. 3, because this determines the width of the N-shape in
the fast current and hence the approximate maximum and
minimum values of the membrane-potential
excursions. If
the cell is oscillatory, changing the value of Af changes the
oscillation amplitude but has no significant effect on the
oscillation frequency. This is because when Af is changed,
the speed of the phase point changes with it, so that although the limit cycle has larger dimensions, the overall
time for one cycle remains approximately the same. Almost
all statements about the qualitative behavior of the cell
model can be verified and made more precise by mathematical analysis of the system MC (Guckenheimer and Holmes
1983).
The steady-state I- Vcurve of the model cell is the sum of
the N-shaped curve F(V) of the fast current and the steadystate 1-V curve of the slow current. The resultant summed
1- Vcurve has a region of negative resistance only when cf >
landl-~f+~,<O,thusonlywhen~,<~f-l.When~,>
cf - 1, the steady-state I-V curve has no region of negative
resistance. If, however, the current is measured in voltage
clamp immediately after the voltage has been stepped from
a holding potential Vh, the instantaneous 1-V curve has a
region of negative resistance. This is because the slow
current has not had time to change significantly, and the
resulting 1-V curve is simply F( V), shifted up or down by
the slow current flowing at the holding potential Vh. Thus
the instantaneous I-V curve will always have a region of
negative resistance when cf > 1. 1-V curves obtained by
measuring current after a time interval sufficiently short
compared with the time constant for the slow current will
also have a region of negative resistance.
The time interval
\
must be less than 7,log, (.,+7-O-,).
If, in the endogenously oscillating model cell, the parameter gf is reduced so that F(V) is no longer sufficiently Nshaped, but (T, remains positive, the model cell does not
oscillate but exhibits postinhibitory
rebound (PIR) and
postburst hyperpolarization
(PBH; Fig. 3, B and C). It does
not have plateau potentials.
If instead, F(V) remains N-shaped but (T, is sufficiently
reduced (Fig. 3, D and E), the model cell does not oscillate
but does exhibit plateau potentials. Provided gs > 0, it will
in addition display PIR and PBH, as shown in Fig. 30. The
top pair of (V, i) traces is a case where a hyperpolarizing
A. I. SELVERSTON
current was stepped to zero; the V trace rebounds too high
then settles down to the high plateau potential H. Similarly
when a depolarizing current was stepped to zero, the Vtrace
rebounded too low then slowly settled up to the low potential L. In Fig. 3E the depolarizing pulse was adjusted to a
critical size barely sufficient to move the phase point past
the unstable e.p. at the origin, similarly for the hyperpolarizing pulse. The existence of the unstable e.p. can be seen as
a brief pause in the rise (fall) of the membrane potential to
H (L)*
Thus according to this model, if an isolated cell oscillates,
then it necessarily also displays plateau potentials and PIR/
PBH. Conversely, if a cell displays plateau potentials, then
it must have a fast current with a region of negative slope,
even if it does not oscillate endogenously. A slow current
may be present but must not be of sufficient magnitude to
cause oscillations. If a cell displays PIR/PBH,
then there
must be slow inward and outward currents present, even if
the fast current does not have an 1- Ycurve with a region of
negative resistance. Thus one would expect that there
would be certain neuromodulatory
conditions such that
both the fast current has a region of negative resistance and
the slow current is of sufficient magnitude to cause endogenous oscillations.
The diagrams of traces from the model cell do not have a
time scale bar, because the time scale is essentially arbitrary.
This is because when the time constants 7, and 7, are both
scaled by a constant factor k, Eqs. 1 and 2 are invariant,
except that time t is also scaled by k. Thus the qualitative
behavior of the model cell does not change when the time
constants are changed, provided their ratio 7,:7, is fixed. As
explained in METHODS,
the difference in time constants, but
not their absolute values, is important to the correct functioning of the model cell. When the ratio 7,:7, becomes 1: 1,
the oscillations cease. However, this value for the 7,:7, ratio
is not physiological. In all model cells considered here it is
1: 10 or greater.
When the slow current is separated into two fractions, a
slow inward and a slow outward current, the parameter us,
which controls the gain of the original slow current, is replaced by two parameters, gin and gout, which control, respectively, the gains of the new slow inward and outward
currents. By adjusting the ratio of gin to gout, the waveform
of the cell can be modified to have larger depolarized or
hyperpolarized parts (Fig. 3E).
In Fig. 5, physiological examples of plateau potentials,
PIR and PBH, and endogenous oscillations, are compared
with model output that demonstrates these properties. The
physiological plateau potential in Fig. 5A is simulated by
the model traces in Fig. 5C. The first two current pulses in
Fig. 5n shows that the pulses must exceed a threshold to
trigger a plateau, and the third pulse shows that a small
hyperpolarizing pulse can terminate a plateau. The model
traces in Fig. 5C simulates this behavior ( 1st pulse and last 2
pulses). Note, however, that the plateau in Fig. 5A continues to rise slowly after initiation, a feature that cannot be
captured by our model. Figure 5B shows a self-terminating
plateau potential, simulated by the model traces in Fig. 5D.
These were obtained by having a split slow current with
zero slow inward gain (gin = 0, coUt > 0), and N-shaped fast
MODEL
A
OF
GASTRIC
v
-2 nA
3 nA
i
A
nn
. .. ... .. .... ..
A
J
/\
ioff
I
CPG
1039
1-V curve (a, > 1). Note the presence of slowly decaying
PBH in B, which is also present in the model potential trace
in D. Figure 5, E and F, shows physiological PIR and PBH.
Figure 5, G and H, shows simulations of these effects, where
the model must have of < 1 (no N-shape) and g’s> 0. Figure
51 shows a physiological trace from gastric cell LG, which is
simulated well by the model trace shown in Fig. 5J. The
latter trace was obtained by having a split slow current with
injected current.
tin
’
gout and a constant depolarizing
These properties of gastric cells can be understood in the
model in terms of movements in the (V, q) phase plane.
Figure 3 illustrates how plateau potentials and PIR are phenomena necessarily associated with an oscillatory cell. For
example, plateau potentials as in Fig. 5C can be interpreted
as movements in the phase diagram of Fig. 3E. There are
two stable e.p.s at L and H and an unstable e.p. at the origin.
If the membrane potential is originally at L and a current
pulse is applied that moves the phase point just to the right
of the origin, then the phase point will continue to move
further right to the high e.p. at H. If the pulse does not move
the phase point as far as the origin then the potential falls
back to L. This occurred for the first two pulses in Fig. 5C.
The third pulse moved the phase point just to the right of
B
2
MILL
I
I
A
q
I
i off
I
FIG. 5. Comparison
of simulated
cell behavior
with physiological
traces from gastric mill cells under various neuromodulatory
conditions.
In the model output traces, Vis membrane
potential,
i is injected current.
Thick dashed lines have been drawn to emphasize
differences
in membrane-potential
values. Thinner vertical bars to right of model traces (C, D,
G, H, and J) are 1 unit in length. A and B show physiological
plateau
potentials;
C and D show model behavior
that simulates A and B. A: gastric
mill cell given small current pulses at the arrowheads.
The 1st pulse failed
to elicit a plateau but the 2nd, slightly larger pulse succeeded (3 nA instead
of 2 nA). The plateau was terminated
by a small hyperpolarizing
pulse.
This was a DG cell, Stn cut, Ba2+ substituted
for Ca2+, 10e7 M tetrodotoxin
(TTX) in bath; 300-ms current pulses at arrowheads.
Calibration:
5 s, 20
mV. Data from Russell and Hartline
(1984, Fig. 1 lA2). B: another gastric
cell, given a single depolarizing
pulse, produces
a plateau potential
that
terminates
endogenously.
Note the small after hyperpolarization.
This was
an MG cell, Stn cut, IO-Hz continuous
Stn stimulation,
-4 nA offset
current.
Calibration:
1 s, 20 mV, 10 nA. Data from Russell and Hartline
(1984, Fig. 8B4). C: model cell was given a succession of depolarizing
or
hyperpolarizing
current pulses. Model parameters
used: slow current gain
of = 1.8. The other parameters
were Af =
O’s = 0 and N-shape parameter
values of these latter parameters
do not
1.o, 7, = 0.2,7, = 4. The particular
affect the production
of plateau potentials.
D: in a series of small current
pulses of increasing amplitude
applied to the cell model, the 3rd elicited a
self-terminating
plateau.
The parameters
used were split slow current,
(T,,~ = 3.0, gin = 0, of = 1.8, 7, = 0.5, r, = 5. E and F: PIR and PBH in
gastric mill cells. The timing dots in E and Fare at 1-s intervals. E: top trace
is the membrane
potential
of an LPG cell when steady hyperpolarizing
current
was suddenly
shut off at the arrowhead.
The trace is not shown
during the hyperpolarizing
current (off scale). Discontinuous
bottom trace
is the current record. Calibration
bar: 10 mV, 10 nA. Data from Selverston
et al. (1976, Fig. 6 1). F: top trace is the membrane
potential of an LPG cell
when steady depolarizing
current was suddenly shut off (arrowhead).
Bottom trace is the current record. Calibration
bar: 10 mV. Data from Selverston et al. (1976, Fig. 59b). G: PIR in the model cell when hyperpolarizing
current was shut off. Top trace is model potential,
bottom trace is model
injected current. H: PBH in the model cell when depolarizing
current was
shut off. The 0 arrow in G and H shows the 0 current level. Critical parameters used for G and H: CT,= 2, cf = 0. I: membrane
potential
trace of an LG
cell in a cycling gastric mill CPG. Calibration:
5 s, 10 mV. Data from
Heinzel
and Selverston
( 1988, Fig. 1OA). J: membrane-potential
trace
from an isolated model cell, with parameters
of = 1.3, oout = 1.2, Gin = 1.O,
7, = 0.06, r, = 4.68, iniected current 0.133.
1040
P. F. ROWAT
AND A. I. SELVERSTON
the unstable e.p. at the origin; consequently, the phase
point continued to move right until it reached H, where its
motion stopped. The fourth pulse, equal and opposite to
the third, moved the phase point back to L. The fifth and
sixth pulses were of exactly the right size to move the phase
point from L to H, and from H to L, without any further
movement. The seventh pulse was a little too large because
it moved the phase point to the right of H; subsequently, it
moved back left to H. Similarly, the eighth pulse was a little
bigger than needed to move the phase point back to L. If the
fast nullcline were moved up slightly by, e.g., injecting inward current, the unstable central e.p. would be closer to
the new low e.p. and further from the new high e.p. In this
case the pulse required to move the potential up from the
low e.p. to the high e.p. would be smaller than the pulse
required to move the membrane potential down from the
high e.p. to the low e.p.
The transition from plateau potentials to oscillations appears in Fig. 3 as the sequence from E to D to A. This
sequence is created by changing the slow current gain us
from 0 to 0.5 to a value, 3, that is greater than the threshold
0,. Geometrically, in Fig. 3, the q-nullcline is being rotated
anticlockwise about the origin. The transition from PIR/
PBH to endogenous oscillations, shown in the Fig. 3 sequence C to B to A, corresponds to the increase of the fast
current parameter cf from zero to a value greater than one.
This transition corresponds to the transition in phase diagrams in Fig. 3, from C to B to A, as cf is increased from 0 to
2. Geometrically, the Knullcline
is being changed from a
linear to an N-shaped form. When the fast and slow nullclines are both linear, as occurs with cf = 0 in Fig. 3C, the
cell exhibits PIR and PBH. Here the trajectory follows a
rapidly decreasing spiral. When cf is increased, the spiral
becomes more open, as in Fig. 3B with gf = 1, and then
becomes a limit cycle as in A.
In Fig. 3F the split slow current has smaller inward gain
gin, and therefore the movement along the fast nullcline at
part 4 of the limit cycle is slower than the movement along
part 2; hence the hyperpolarized portion is longer than the
depolarized portion, as seen in the trace in Fig. 3F. If gin is
considerably less than 8,, for instance gin = 0 in Fig. 3G,
then there is exactly one stable e.p., at L. If a current pulse
moves the phase point to the right of the central unstable
e.p., the phase point follows a trajectory very close to that of
a regular oscillation except that it comes to rest slowly at L:
a self-terminating plateau potential (Fig. 3G). For this case
the ratio of the membrane time constant and slow current
time constant was changed from 1:30 to 1:200. When this
ratio is made very large the rate at which the phase point
moves up or down the fast nullcline is much smaller then
the phase point’s horizontal movements. Consequently, the
downward slope of part 1 of the potential trace in Fig. 3G is
smaller. The trace in Fig. 3G is qualitatively the same as the
trace in Fig. 5D. The slow nullcline need not touch the fast
nullcline at the origin as in this example; any curve that
swings down from upper right and crosses the fast nullcline
at a point L on its left branch will suffice.
Figure 3 outlines some of the different behaviors that can
be obtained from this cell model and how they can be understood by using phase diagrams. Clearly many other be-
haviors can be obtained by using, for example, slow nullclines with extra bends or even dips and bumps that could
be obtained in a cell with enough slow currents.
Reciprocal inhibitory pair
When two identical cells that are endogenous oscillators
are connected with reciprocal inhibition of equal strength,
the two cells oscillate exactly out of phase with each other.
The waveforms of the cells in the reciprocal inhibitory pair
are similar to the waveform of the isolated cells, but with
increased amplitude
due to lower minimum
potential
caused by the inhibition
and higher maximum
potential
due to PIR when the inhibition is suddenly reduced (Fig. 6,
A and B). When the of parameters for both cells are set
below one so that the cells are quiescent in isolation and do
not display plateau potentials but still exhibit PIR/PBH, the
reciprocal inhibitory pair still oscillates in antiphase. If the
of parameters are reduced further, to zero, the inhibitory
pair still oscillates (Fig. 6, C and D), provided the connecting strength IV is large enough. This is in agreement with
the observation by Satterlie ( 1985) that PIR together with
synaptic inhibition is sufficient to cause an inhibitory pair
to oscillate. It also confirms, and extends to graded transmission, the spike-based simulations of Perkel and Mulloney (1974). As the value of cTfis reduced below one, the
synaptic strength Wrequired for the inhibitory pair to oscillate with the same amplitude increases as a linear function
of Us In one model inhibitory pair, the minimum
value of
IV required for oscillations to occur was found to be IV =
0.7 - 0.6a, for 0 < cTf< 1.
Two quiescent model cells, when connected with strong
enough synapses for joint oscillations to occur, may remain
quiescent for a short period before oscillatory behavior begins. In this case the model is sitting at an unstable e.p.
inside a limit cycle, but due to instability, the model will not
remain at this point because the small errors arising in any
numerical integration procedure ensure that the system will
eventually burst into activity.
A partial explanation of the oscillation of the inhibitory
pair, when neither cell is an oscillator, can be given with
reference to its phase diagram as follows. The reciprocal
inhibitory pair whose waveforms are shown in Fig. 6D is
analyzed in Fig. 7. When of = 0 the fast current in each cell
is purely linear as in Fig. 7A. When two cells each having
cf = 0 are connected by reciprocal inhibition, the inhibitory
postsynaptic currents effectively deform the linear fast
currents in each cell into N-shapes, as shown in Fig. 7, B
and C, thus allowing each cell to oscillate as a nearly two-dimensional system with the limit cycle shown in the two-dimensional phase diagram of Fig. 7E. Because the reciprocal
inhibitory pair is described by four differential equations,
its phase diagram is four dimensional. The Vi-nullcline in
Fig. 7E is a projection onto two dimensions of the true
Vi-nullcline in the four-dimensional
phase diagram. The
true Vi-nullcline,
which, from the current conservation
equation is defined by q1 = -0 6) - wf(w 5 - Epos*)9is
a three-dimensional
surface in ( VI, V&ql,q2) space. The reverse N-shape is obtained as the VI- V2 trajectory of Fig. 70
is traversed. In Fig. 700 the
traiectory has been de-
MODEL
OF GASTRIC
MILL
CPG
1041
5050
t
i
I
I
I
I
BB
??i
*L
-1
ViPL
8
4
period
C
1
2
65:35
E
‘I
LL
El1
*J
G
1
-------.
2
-------_
44:56
III
I
i
ii
F
E21
i
H
:
I
i
i
FIG. 6. Potential traces from pairs of reciprocal inhibitory model neurons. A: 2 isolated identical oscillatory cells, by
chance not out-of-phase. of = 1.8, c’s= 3.0,~~ = 0.333, 7, = 5.0. Scale bar extends from + 1 to - 1, centered on 0. B: same cells
as in A, linked by reciprocal inhibition with the synaptic weights set to 0.2. Plot to the right compares the periods and the
voltage ranges of the oscillations in A and B. This has the same scale as the traces, on both axes. C: 2 isolated cells each with
of = 0. No activity is present. The other parameters in each cell were of = 0, (T,= 3.0,~~ = 0.333,7, = 5.0. D: same cells as in
C, linked by reciprocal inhibition. The synaptic connection strengths were 0.8. E: 2 isolated cells each with different periods.
In both, cf = 1.8,~~ = 0.333,~~ = 5.0. In the top cell, (T,= 10.0, whereas the bottom cell had os = 1.5, with ratio of periods ~3.
F: same 2 cells as in E, linked by reciprocal inhibition, with synaptic strengths set to 0.6. The cells have entrained to a
common period between the periods of the original cells and have phases different from the phases of the symmetrical
inhibitory pair. Using the points of maximum potential to define the phases, the cell that was slowest originally has retarded
phase (65% instead of 50%), and the cell that was fastest originally has advanced phase (35% instead of 50%). The plot to the
right compares the periods and voltage excursions of the oscillations before (El and E2) and after (F) introducing the
connections. G: 2 isolated cells that exhibit plateau potentials and PIR/PBH as in Figs. 4E and 5E: of = 3, us = 1.15 in cell 1,
us = 1.11 incell2, 7, = 0.1, 7, = 2.0. I-? same 2 cells as in G, linked by setting the weights to 0.06. The asymmetry in the
relative phase is due to small differences in the slow currents. Dashed lines linking the traces in G and H show that the
opposing plateaus of the cells in the inhibitory pair are approximately the same as when the cells are isolated.
formed by stretching along a diagonal line from lower left to
top right, thus showing the asymmetry between the depolarized and hyperpolarized parts of the waveforms. The reciprocal inhibition moved the V1-V2 trajectory down and left
from the origin (D). In Fig. 7E the dotted trajectory is the
projection of the limit cycle of the reciprocal inhibitory pair
onto the V,-q, plane. Note the similarity between this trajectory and the trajectory of a single endogenous oscillatory
cell in Fig. 3A. The points of steep increase and decrease in
the membrane potential in the traces of Fig. 6D are not
caused by endogenous plateau potentials but by postinhibitory rebound when the inhibition from the other cell is sud-
denly removed. Thus for this parameter setting our model
reciprocal inhibitory pair has behavior similar to the release
mode of operation of the Wang and Rinzel ( 1992) model of
an reciprocal inhibitory pair.
When two nonoscillatory cells that display plateau potentials and PIR/PBH (Fig. 30) are connected with reciprocal inhibition, the inhibitory pair will oscillate, provided the
connecting weights are small enough (Fig. 6, G and k7). The
cells begin to relax to their plateau potentials as in Fig. 30.
Suddenly, the cell at the high potential loses stability due to
the hyperpolarizing
postsynaptic current, and the cells
switch polarities. If the connecting weights are too large, the
1042
P. F. ROWAT
AND A. I. SELVERSTON
fast1
2
I
v2
synl
L
K
c
Vl
fast1
+synl
FqLJ
I
I
/w
ql
nullcline
I
I
91
Vl nullcline
7. Phase diagram for 1 cell of a pair of silent cells linked by reciprocal inhibition. of = 0 in both cells so there is no N-shape in their fast
currents. A: fast current in cell 1, labeled fast 1, as a function of vi. B:
synaptic current syn 1 in cell 1 during 1 oscillation cycle, plotted as a function of vi. The synaptic current as vi increases differs from the synaptic
current as V, decreases because of different values for I$; hence the double
lines shown. C: the total fast current in cell 1 is the sum of the currents in A
and B. D: plot of vi vs. I$ over 1 limit cycle. When vi is low, & is high, so
the synaptic current syn 1 in B is high. DD: data of D have been distorted to
show detail. Twice as many points have been used. (If the points were
plotted on a rubber sheet, and the sheet then uniformly stretched in a
northeast-southwest direction, the resulting data point positions would be
similar to those plotted in DD). E: total fast current in cell 1 is plotted as if it
were the nullcline for I’i . The q,-nullcline and the qi- I$ phaseplot are also
drawn. In each ofA-E the axes meet at (O,O),and the endpoints of the axes
are at + 1 or - 1. In B and C the vertical axis extends from 0 to 1.
FIG.
third, when each cell in isolation is quiescent but exhibits
plateau potentials, the inhibitory connections ensure that
the cells will always be in plateau potentials of opposite
polarities. If the connections are too strong, then one cell
becomes dominant in a high plateau. When the connections are weak enough, the cell in the lower plateau is able
to destabilize the cell in the high plateau, and the polarities
switch.
For an asymmetric reciprocal inhibitory pair, the notation ~1.772 means that the interval between the peak of ceU i
and the following peak of cell 2 is ~1% of the period of the
pair, and the interval between this peak of cell 2 and the
next peak of cell I is ~2% of the period of the pair, where
p2 + pl = 100. This notation has been used in Fig. 6.
When two oscillatory model cells with widely differing
endogenous frequencies are connected with reciprocal inhibition, they immediately
entrain to a common frequency.
The entrained frequency lies between the two endogenous
frequencies, and the entrained cells are no longer exactly
out of phase with each other (Fig. 6, E and F). When compared with the phases of an inhibitory pair with identical
cells, the cell that originally has the slower frequency comes
to a maximum with a slight phase lag, whereas the cell that
originally has the faster frequency comes to a maximum
with a slight phase advance. This effect is more pronounced
when the connections are weak than when they are strong.
By this means it is possible to set phase differences between
cells of up to 15% (Fig. 68). Because the main determinant
of the endogenous frequency of a cell is the gain of the slow
current of the cell, one can state this effect slightly differently: by adjusting the ratios of the gains of the slow
currents in different cells, one can obtain phase differences
between cells of up to 15%.
When the weights of the inhibitory connections are not
identical but are still of the same order of magnitude, there
is no significant difference in phase from the case of identical weights. However, when the weights differ by more than
a factor of 10 (e.g., by a factor of 20), there is a small change
in phase. If cell 2 receives strong inhibition from cell I but
cell 1 receives only weak inhibition from cell 2, then cell 2
will lag behind cell I by >50%, and cell 1 will lag behind cell
2 by 40%.
Complete net work model
cell at the high potential cannot be destabilized, and the
pair does not oscillate. The dashed horizontal lines in Fig. 6
between the traces, Gl and HI, and between the traces G2
and H2, bring out the fact that the plateau potentials in the
reciprocal inhibitory pair, in this case, are approximately
the same as the plateau potentials in the isolated cells.
Thus there are, even for this simple model of an inhibitory pair, three distinct mechanisms of oscillation. In the
first, when both cells are endogenous oscillators, essentially
any nonzero connection strengths, however small, suffice
to set the two cells exactly out of phase with each other. In
the second, when both cells are quiescent in isolation but
show PIR and PBH, the synaptic currents play an essential
role in the oscillation mechanisms, and the connection
strengths are required to be above a minimum value. In the
We constructed a network model of the complete gastric
mill network that included all the connections in Fig. IA. In
view of the large number of independently adjustable parameters, we had assumed that it would be necessary to use
a parameter adjustment algorithm, such as was presented in
Rowat and Selverston (199 l), to find sets of parameter values that would cause the model to generate reasonably
correct output patterns. However, the situation turned out
to be somewhat different.
The network model, with all the cells set as endogenous
oscillators, all the excitatory and inhibitory
synaptic
strengths set to the same small value, and all the electrical
junctions set to a small value, produced approximately
correct output patterns. such as appear in Fig. 8A. Similar
MODEL
OF GASTRIC
I
LG
MG
GM
LPG
DG
AM
lntl
LG
MG
GM
LPG
DG
AM
lntl
C
MG
LPG,
AM
GM,LG
I
A phases
II
DG,
Intl
II
--TI
II
I
B pluses
MG
GM
LG
LPG
DG*
AAi
IC
-I
-i
II
-
lntl
IC
FIG. 8. Pattern generation in the model network, with all connections
present. A: each cell is an endogenous oscillator. B: each cell would be
quiescent if isolated. Dotted lines connect the maximum points in each
trace, to show the phase relationships. In B, the cf value for all the cells has
been reduced so that each cell would be quiescent if isolated. Each cell has
PIR/PBH but does not have plateau potentials. The phase of MG has been
advanced by increasing the gain of the slow current, gs, in MG. This is seen
from the slight rightward tilt in the dotted line connecting the maxima of
MG and LG. All the excitatory weights had the same value, and with 2
exceptions, all the inhibitory weights had the same, slightly larger value.
Exceptions were for the LPG cell: inhibitory weights from DG and AM
were reduced by an order of magnitude. All the electrical junctions had the
same strength. C: comparisons of the phases and amplitudes of the traces
in A and B. Vertical scale is the same in A-C. A horizontal line represents 1
complete cycle. There is a vertical line for each cell, connecting the maximum and minimum values of the cell’s membrane potential, with 0
aligned on the cycle line. Some of the phase lines overlap, e.g., LPG and
AM, so the individual amplitude excursions cannot be seen. IC labels a
dashed line drawn at the relative period of an isolated cell, with symmetrical extent indicating its maximum and minimum.
MILL
CPG
approximately correct patterns were obtained when the parameter values were varied over relatively wide ranges (Fig.
11). By “approximately
correct pattern” we mean that LG
and MG are in phase and are out of phase with Intl and
LPG, AM and DG are in phase together and out of phase
with GM, and that AM and DG are in phase with Intl.
Because there was a large volume of parameter space
containing suitable sets of parameter values, the problem
was transformed from one of using a parameter adjustment
algorithm to find an element in a small volume of parameter space, into a somewhat different problem of understanding the phaselocking properties of networks of relaxation oscillators.
The network model was then altered by making each cell
quiescent, with each cell’s parameter a,just under one (Fig.
3, B and C). The cells did not have plateau potentials but
still had PIR/PBH.
This model also produced approximately correct patterns except for the LPG cell, whose output was greatly reduced in amplitude and sometimes exhibited phase walk-through, i.e., LPG did not have the same
frequency as the other cells of the network, so that the phase
of LPG with respect to the other cells was always changing
(Ermentrout and Rinzel 1984). The phase of LPG always
advanced with respect to the remaining cells of the network,
because the frequency of oscillation of the complete network was slower than the frequency of individual cells (or
potential frequency of an individual cell if it were quiescent
in isolation). This behavior is not entirely unforeseen, because LPG receives antiphase inhibitory input from LG
and MG as well as in-phase inhibitory input from DG and
AM. Because inhibitory input was received at all phases of
the oscillatory cycle, LPG was being subjected to a hyperpolarizing current, whose amplitude
of variation was too
small to produce entrainment. It is known that LPG has
only weak connections back to other cells of the gastric mill
network and has little effect on the overall pattern-generating process. What was needed was synaptic input with large
enough amplitude of variation to entrain LPG. Therefore
we reduced the strength of the connections from DG and
AM to LPG by a factor of 10, so that the LG and MG
connections were the dominant factor in determining the
phase of LPG. With this change, the model produced approximately correct output patterns.
The output of the model in these two cases is shown in
Fig. 8, A and B. In A, each cell is an endogenous oscillator,
whereas in B, each cell would be quiescent in isolation. The
parameter values used in B are given in Table 1. The small
phase advance of the MG signal over LG, corresponding to
the data of Fig. 1B, was set by increasing the gain of the slow
current in the MG model. It can be seen that the output is
approximately correct in both cases except that the characteristic phase lag between the cells controlling the lateral
and medial teeth is not present. Figure 8C shows that the
phases are essentially the same in both cases. The phase
lines extend further below the cycle line than above, showing that the traces are similar to the traces of an isolated cell,
except for being shifted down a small amount by the reciprocal inhibition.
In A, the oscillators were slowed down,
whereas in B, the frequency of the network oscillations is
1044
TABLE
P. F. ROWAT
1.
Parametersfor first network model(Fig. 8)
LG
Network connections
LG
LPG
DG
AM
GM
Intl
E
Cellular parameters
*r?l
9
Af
0s
*S
AND A. I. SELVERSTON
MG
LPG
in 0.06
ej 0.02
in 0.06
ej 0.02
in 0.06
in 0.06
ex 0.05
in 0.06
ej 0.02
in 0.06
DG
AM
in 0.06
in 0.06
GM
Intl
E
in 0.06
ex 0.05
in 0.06
ex 0.05
ej 0.02
in 0.06
in 0.06
ej 0.02
in 0.06
in 0.006
in 0.006
d+ 0.003
ex 0.05
ex 0.05
in 0.06
in 0.06
ej 0.02
in 0.06
in 0.06
ex 0.02
ej 0.02
ex 0.05
d- 0.003
in 0.06
in 0.06
0.066
1.004
1.0
3.0
10.0
0.066
1.004
1.0
5.0
10.0
0.066
0.066
0.066
0.066
0.066
0.066
1.004
1.0
1.004
1.0
1.004
1.0
1.004
1.0
1.004
1.0
1.004
1.0
3.0
10.0
3.0
3.0
3.0
3.0
3.0
10.0
10.0
10.0
10.0
10.0
Synaptic reversal potentials: E,,, = -4.0 (all inhibitory synapses), ,!Zpst= 4.0 (all excitatory synapses). LG, lateral gastric; MG, median gastric; LPG,
lateral posterior gastric; DG, dorsal gastric; AM, anterior median; GM, gastric mill; Int 1, interneuron 1; E, excitatory. Explanation of network connection
entries: “(type) (value)” in column A, row B, means the connection from cell A to cell B is of type (type) with strength (value), where (type) is one of
the following: ex, excitatory; in, inhibitory, ej, electrical junction; d+, source for rectifying gap junction (diode); d-, sink for diode.
faster than the frequency that an isolated cell would have, if
it were made to oscillate by setting of > 1.
Reduced four-cell network model
In the network model just discussed, it became apparent
that the phase changes obtained by adjusting the gains of
the slow currents were insufficient to duplicate the phase
lag observed between the cells that control the lateral teeth
(LG, MG, and LPG) and the cells that control the medial
tooth (DG, AM, and GM; Fig. 84. However, the excitatory
synapses from Intl to DG and to AM produce a very slow
postsynaptic response. Also, it has been observed that one
result of killing cell Int 1 is for the lateral teeth cells and the
medial tooth cells to continue to oscillate but with widely
different frequencies (H.-G. Heinzel, unpublished data; YI=
2). This result is illustrated in Fig. 9, A and B. It appears,
therefore, that Intl serves to coordinate the sets of cells
controlling the lateral and medial teeth.
To determine whether the model was sufficiently adaptable to generate simultaneously the lateral-medial phase lag
and Heinzel’s Intl killing data, with the addition of slow
processes to the Intl to DG and AM synapses, a reduced
four-cell network was used. This consisted of one model cell
to represent the LG/MG pair of cells that normally oscillate
together with no more than - 5% phase difference, one cell
to represent the DG/AM pair of cells that also usually oscillate together with no more than -5% phase difference, one
cell corresponding to Intl, and one cell corresponding to
LPG. The GM cells were ignored because, in general, they
respond passively to inhibition from the DG/AM cells and
have little influence on the remainder of the network (Selverston et al. 1976). The network diagram of Fig. 9 includes
three inhibitory pairs: Intl ++ LG/MG, LG/MG * LPG,
and LG/MG * DG/AM. Because none of the gastric mill
cells, except perhaps DG, are known to be endogenous
oscillators, the cf parameter was set less than one in each of
the cells Int 1, LG/MG, and LPG so that none would oscillate when isolated but would still exhibit PIR/PBH.
The
connection strengths were set strong enough that the inhibitory pairs would oscillate. DG/AM was made to be an endogenous oscillator with frequency three to four times
slower than the frequency of the inhibitory pairs in the network, by setting of larger than one and the gain of the slow
current to be about one-third less than the slow current gain
in the other three cells. The reciprocal inhibition between
DG/AM and LG/MG had to be weak enough that when
Intl was killed, DG/AM would not be entrained 1 to 1 to
the inhibitory pair LG/MG * LPG, but when Intl was
present, DG/AM would be entrained 1 to 1 to the lateral
teeth cells.
The slow synapse was modeled by the use of Eqs. 7 and 8
given above, with the use of a time constant of about onehalf that of the slow currents. The combination of the delayed synapse and the slower endogenous frequency of DG/
AM sufficed to set the phase lag of DG/AM when Intl is
present. An example is shown in Fig. 9C. When Intl was
removed from the network by setting all the pre- and postsynaptic conductances involving Intl to zero, the inhibitory pair LG/MG
++ LPG continued to oscillate, at a
slightly higher frequency than before, whereas the DG/AM
cell now had a frequency three times smaller (Fig. 9D). The
inhibitory pair was phase locked (3:l) to DG/AM by the
weak connections. Note also that in the model DG/AM
trace there is a small secondary peak between the primary
bursts, which may be considered to correspond to the short
secondary bursts in the real AM trace between the long
primary AM bursts. All parameter values used in this reduced model are shown in Table 2. Thus the network
model was capable of generating the lateral-medial
delay
and, simultaneously, providing a mechanism for the coordination of the lateral and medial sets of cells by Int 1.
MODEL
A
LPG
W;
OF
GASTRIC
MILL
CPG
1045
Control
ii
I
10mV
5s
B
0
.
Int 1 killed
LG-
?---
LPG
-
* b
-wwt
e
LG/MG
D
LPG
DG/AM
lntl
2.0
2.0
2*o
I 1
2.0
9. Comparison
of removing
the real Int 1 cell from the physiological
GM circuit, with removing
a model Int 1 cell
from a model network,
using a reduced model gastric network.
A: control traces from the GM. B: GM traces from same
preparation
after cell Int 1 was killed. Traces in A and B are on the same time and voltage scales. Unpublished
data courtesy of
H.-G. Heinzel.
The Intl trace in A, and the MG and GM traces in B, are intracellular
recordings.
All others are nerve
recordings
showing spike trains only. In A, all cells are phase locked. Note the phase lag between Int 1 and DG, AM. In B, cells
LG, LPG, and MG are phase locked at 1 frequency
(O), whereas cells DG, AM, and GM are phase locked at a much slower
frequency
(Cl). Inset: reduced network used; solid circles are inhibitory
synapses, the solid triangle is an excitatory
synapse. C
and D: output from model network.
In C, the phase lag between Int 1 and DG/AM
(vertical dotted lines) was obtained by the
combined
effect of a slow synapse between Intl and DG/AM
and a much reduced slow current gain parameter
in the
DG/AM
cell. In D, Int 1 was “killed”
in the model, and DG/AM
now has a frequency
3 times slower than LG/MG
and LPG.
FIG.
Complete network model incorporating
slow synapses
To obtain a new complete network model that exhibited
the correct lateral-medial
delay, as well as correct phase
relationships among the other cells, we combined the parameter sets from the reduced four-cell model and the previous complete network model as follows. In the new complete network, the Int 1+DG, Int 1-+AM synapses were adjusted to be slow. The parameter set found for the reduced
network of Fig. 9 was extended to the new complete network, as follows: First, the parameter values in the reduced
network were inserted into the complete network. If a connection such as LG/MG to Int 1 in the reduced network had
value X, then the connections LG to Intl and MG to Intl
were both given the value l/2x in the complete network.
Thus the effect of LG and MG on Intl in the complete
network would be approximately the same as the effect of
the single-cell LG/MG on Int 1 in the reduced network, provided LG and MG are approximately synchronous in the
complete network. If a connection such as Int 1 to LG/MG
had value y in the reduced model, then the connections
Intl to LG and Intl to MC were both given the value y in
the complete network. Thus the effect of Int 1 on LG and on
MG would be the same as the effect of Int 1 on the single-cell
LG/MG in the reduced model. Initially, all other connection strengths in the complete network were zero. Then all
these connection strengths were simultaneously slowly increased, in the same proportions as had been established in
1046
TABLE
P. F. ROWAT
2.
Parameters for reduced four-cell network (Fig. 9)
LG/MG
Network
LG/MG
LPG
DG/AM
Intl
Cellular
7rt.l
?f
Af
gs
7s
AND
LPG
DG/AM
in 0.33
in 0.02
Int 1
connections
in 0.66
in 0.02
in 0.2
ex 0.3
rd 0.25
in 0.4
parameters
0.125
1.0
1.0
1.0
1.25
0.125
1.0
1.0
1.0
1.25
0.125
1.4
1.0
0.36
1.25
Synaptic reversal potentials:
Epost = -4.0 (all inhibitory
synapses),
= 4.0 (all excitatory
synapses). rd, time constant for slow excitatory
apse. For other abbreviations
and explanations,
see Table 1.
0.125
1.0
1.0
1.0
1.25
Ei,,,t
syn-
the previous complete network model, to the same order of
magnitude
as the strengths already established in the
current complete network model. The idea was that by this
means the correct overall pattern would still be generated.
When we did this and ran the new complete model, there
was considerable distortion of the waveforms because of the
larger synaptic currents; these in turn were caused by the
larger numbers of synapses per cell in the complete network
than in the reduced network. Therefore we reduced all the
parameter values by an order of magnitude. A further constraint was that the relative connection strengths in the final
set of parameters should be consistent with the relative
functional connection strengths given by Russell ( 1985b).
This has been satisfied. However, we have not yet been able
to do this and, simultaneously, retain the property, exhibited by the reduced circuit, of being able to duplicate Heinzel’s data on removal of Int 1. The output from the resulting
model is shown dotted in Fig. 10, and the parameters used
are given in Table 3.
For comparison we took a set of seven simultaneous intracellular recordings made by H.-G. Heinzel and passed
them through a smoothing filter to remove spikes. In Fig.
A. I. SELVERSTON
10 the smoothed biological traces and the model traces
have been overlaid so that the mutual fit can be seen. The
model traces have been individually vertically scaled so that
each model trace has the same amplitude as its corresponding biological trace. The model LG and DG traces have the
best fit, and the model Intl trace has the worst fit. The
length of the depolarized part of the model MG trace is not
as long as the depolarized part of the biological MG trace.
The interburst potential of the model LPG trace is not sufficiently depolarized and should show a more constant gradient leading to the next burst. The interburst potential of
the model AM trace needs to descend more slowly from the
previous burst. The model GM trace should ascend faster
and descend more slowly. The bursts of the model Intl
trace should be more prolonged and drop more steeply, and
the interburst hyperpolarization
should be shorter and decrease faster when the burst begins.
The need for individual vertical scaling factors (rightmost
scalebars in Fig. 10) in matching the model and biological
traces could have been removed by adjusting the amplitude
of oscillation of the individual cells. This is done by changing the value of the A, parameter for individual cells. The
only significant effect of a change in A, is a change in the
amplitude of the cell’s traces, not a change in its frequency
(or its potential frequency if not oscillatory). The network
effect of changing the A, value in a cell is to change, by Eq.
6, the amplitudes of synaptic currents in that cell and in
cells that are postsynaptic to it. Thus it was expected that
changing the amplitude of oscillation of a single cell would
have little effect on the network phase relationships. This
was confirmed in the model: such adjustments caused insignificant changes in the phase relationships and the slowwave shapes.
The A, parameter is approximately linearly related to the
saturated amplitude of the fast inward current (such as a
persistent Na current, or some combination of several fast
inward currents), which, when summed with a fixed leak
current, gives rise to an overall N-shaped fast 1-V curve. If
we assume a unique solution for the A, factors, their relative
LG
FIG. 10.
Simultaneous
intracellular
recordings
(courtesy
of H.-G.
Heinzel)
were passed through
a
filter to remove spikes. A set of model traces was horizontally
aligned with the biological
traces, then each
trace was individually
vertically
scaled to obtain the
match shown. The 1st set of vertical bars on the right
shows the vertical
scale of the biological
traces,
whereas the 2nd set shows the relative sizes of the
vertical scaling factors used for the model traces.
I
MODEL
TABLE
MG
LPG
in 0.01
ej 0.0066
LPG
DG
in
ej
in
in
AM
ex 0.10
GM
in 0.01
ej 0.0033
in 0.06
Synaptic
and 2.
MILL
CPG
1047
DG
AM
in 0.02
in 0.06
GM
Int 1
E
connections
MG
Intl
E
Cellular
7,
ajAf
OS
7s
GASTRIC
3. Parameters for network model matching Heinzel’s traces (Fig. 10)
LG
Network
LG
OF
0.01
0.0066
0.06
0.02
in 0.06
ex 0.05
in 0.06
ex 0.05
ej 0.0033
in 0.06
in 0.06
in 0.02
in 0.006
in 0.006
ej 0.06
ej 0.0033
d+ 0.003
ej 0.06
in 0.06
in 0.06
ex 0.02
ex 0.10
Td 5.0
ex 0.10
?-d 5.0
in 0.01
in 0.06
ex 0.05
d- 0.003
in 0.06
in 0.06
parameters
0.2
1.004
1.0
3.0
10.0
reversal
potentials:
Epost = -4.0
0.2
1.004
1.0
5.0
10.0
(all inhibitory
0.2
1.004
1.0
3.0
10.0
synapses),
0.2
1.004
1.0
1.0
10.0
Epost = 4.0 (all excitatory
sizes should predict the relative magnitudes of fast inward
currents in different cells. However, the recording site for
the membrane potential of an STG cell is the soma, which
is electrically distant from the slow-wave generation site in
the neurites. Thus the amplitude of membrane-potential
recordings depends on the electrical distance to the neurites
as well as the actual amplitudes. The electrical distance is
not known and probably varies from cell to cell; consequently, the A, parameters cannot be directly related to the
amplitudes of fast currents.
Adjusting the asymptotic values of the fast current (V A,) is not the same as ad hoc scaling of the output after the
model has been run. Scaling A, affects synaptic strengths,
but scaling output does not.
To test the robustness of the model, all of the parameters
except A, had their values randomly changed by as much as
40%. No significant change in the overall output pattern
was found, as shown in Fig. 11. The largest difference in
phase occurred for GM, whose phase relative to MC
changed by 14%. When each individual cell was made to
oscillate with a randomly assigned frequency, and then suddenly the known connections were introduced, as in Fig.
12, the network rapidly entrained to a common period that
was close to the average of the isolated periods. In addition,
the pattern was approximately correct. The period of the
entrained oscillators is approximately
the average of the
periods of the isolated oscillators. The average period of the
isolated cells is 6.19, the period of the entrained network is
6.15, and entrainment of all the cells to a common frequency is complete within about 2 cycles of the new frequency.
When the frequency of all the cells in the network was
uniformly changed by simultaneously decreasing the gain
of the slow current by 50% in all the cells, the phase relationships remained essentially the same, as shown in Fig.
13. The pattern frequency decreased by 67%.
0.2
1.004
1.0
3.0
10.0
synapses).
0.2
1.004
1.0
3.0
10.0
For abbreviations
0.2
1.004
1.0
3.0
10.0
and explanations,
0.2
1.004
1.0
3.0
10.0
see Tables
1
To determine the degree to which the overall computation was distributed, a succession of cells was removed from
the network (Fig. 14). This was accomplished by setting all
of a cell’s pre- and postsynaptic conductances to zero, and
by setting cf = 0. This completely disconnects the cell from
the remainder of the network and changes its phase diagram from one similar to Fig. 3B to one similar to Fig. 3C.
Consequently, after removal, the cell’s membrane potential
smoothly settled to zero with no apparent damped oscillations.
In Fig. 14 neurons were sequentially removed from the
gastric mill network starting with MC (Fig. 14A). MC is
known to play a key role in the formation of the gastric mill
rhythm (Selverston et al. 1983) and is part of the kernel
pattern-generating
circuit under muscarinic modulation
(Elson and Selverston 1992). Although there was no change
in phase relationships, there was an increase in the amplitude of the slow wave for both LG and GM and a decrease
in the amplitude for LPG. The LPG cell makes only weak
connection to the MC cell, and its removal had no effect on
the ongoing rhythm (Fig. 14B). Removal of the AM cell
(Fig. 14C) produced a small decrease in the amplitude of
the GM neurons but had very little effect on the DC cell to
which it is electrically connected. Removal of the GM cell
(Fig. 140), which is connected to the remaining cells only
by an electrical connection, had no noticeable effect. Int 1,
on the other hand, made connections to all three of the
remaining cells, and its removal therefore had significant
effects on the remaining pattern (Fig. 14E). The amplitudes
of LG and DC were reduced considerably almost immediately, and their plateau properties disappeared so that their
waveform was markedly altered. The E cell, whose only
input comes from Int 1, did not stop immediately
but
showed a damped oscillation (Fig. 14, E and F). This sequence of simulated cell kills left only the LG and DC cells,
which are connected to each other by reciprocal inhibitory
1048
P. F. ROWAT
AND
synapses. Removing LG therefore (Fig. 14F) removed the
inhibition to DG, which on its own gradually stopped oscillating and became quiescent. Note that at the right-hand
side of Fig. 14F there are still two cells alive but now quiescent: DG and E. Because there are no connections between
them, they are isolated cells that become quiescent, as in
Fig. 3B. The network continued to oscillate so long as at
least one inhibitory pair remained. In the example shown,
the LG/DG inhibitory pair continued to oscillate after five
other cells had been removed.
The behavior of the subnetwork remaining after a number of cells has been removed does not depend on the order
in which the removals are carried out. Equivalently, every
subnetwork has at most one stable mode of oscillation, in
the parameter ranges used.
A. I. SELVERSTON
A
RANDOM
FREQUENCIES
CONNECTIONS
ON
E
.i
.i
.;
.
.i
.i
LG
MG
GM
LPG
DG
AM
DISCUSSION
We have shown that a network of model neurons, each
with an N-shaped fast current and slow inward and outward currents, together with graded synaptic transmission,
captures several important aspects of pattern generation in
the gastric mill CPG. In particular, the network produced
approximately
correct phase relationships for the in vitro
gastric mill rhythm; it continued to generate an approximately correct pattern even when no constituent cell was an
endogenous oscillator; when cells were serially removed
A
PARAMETER
LG
MG
GM
LPG
DG
AM
lntl
2.0
LG,
E
GM
B
LPG,
AM
I
MG
E
LG
2.0
DG
Phases
before
I
II
I
GM
Inll,
AM.
In!1
Phases
LPG.
PERIODS
B
E
LG
MG
GM
LPGDG
AM
lntl -
I;
I.I.
1.
I,
I
I
1;
f!
2.0
I
Entrained
Ii
1; Average
II
FIG. 12.
Entrainment
of oscillators
with randomized
frequencies
to a
common
rhythm.
All cells were assigned a random frequency
by randomly
setting the parameter
us in the range 1 S-6 for each cell. A: to the left of the
vertical dashed line, the cells are isolated. At the dashed line all the network
connections
were introduced.
B: bar chart of the isolated periods. The
vertical line with long dashes indicates
the entrained
period, the shortdashed line the average period.
RANDOMIZATION
E
MG
lntl
after
DG
FIG. 11.
Randomization
of parameters.
A: at the vertical dashed line,
the connection
and cellular parameters
were given new random
values
within 40% of their original values. In most cases, as illustrated
here, there
is no major change in intercellular
phase relationships.
The period did not
change. Dotted lines join corresponding
maximum
points in each cycle,
thus making clear the phase relationships.
Similar dotted lines are used in
Figs. 12 and 13. All the model cells were endogenous
oscillators. B: comparison of phases before and after randomization,
using same conventions
as
in Fig. 8C.
from the network it continued to oscillate and produced
approximately correct phase relationships in the remaining
cells; and, ignoring spikes, the network produced slow-wave
shapes that corresponded approximately to the shapes of
the slow waves in physiological intracellular recordings of
membrane potential. Small changes in phase relationships
were due to differences in gains of the slow currents,
whereas larger phase differences required other mechanisms such as a synapse with a delayed activation time, as in
the case of the phase lag between the cells controlling the
lateral teeth and the cell controlling the medial tooth.
The most critical point in the development of this model
was the realization that a simple generalization of the wellknown relaxation oscillator of Van der Pol could be given a
physiologically correct interpretation, whereas, at the same
time, several characteristic properties of stomatogastric neurons, such as plateau potentials, PIR, and endogenous oscillations, flowed naturally from this generalization. Additionally, the waveforms produced by this model cell compared
favorably with the typical waveforms of several gastric mill
neurons. Further supporting evidence was the fact that an
N-shaped fast current is necessary for an isolated model cell
to oscillate, consistent with the finding of Wilson and Wachtel(1974) that bursting neurons require a region of negative
resistance in their fast 1-V curve. The model makes
MODEL
50 % DECREASE
IN SLOW
CURRENT
OF GASTRIC
GAIN
LPG
DG
:
:
:
:
:
:
:
:
AM
lntl
2.0
LG,
GM
MG
E
LPG
Pluses
before
AM,
lnl
1
I
B
I
MG
E.
LG.MG
I
2.0
DG
:
:
AM.LPG.lnll.DG
FIG. 13. Constancy of phase relationships when the frequency changes.
A: at the vertical dashed line, the slow current gain oS of each cell was
reduced by 50%, with insignificant change in phase relationships. All the
model cells were endogenous oscillators. B: comparison of phases before
and after the frequency change. Dotted line on the “Phases after” cycle line
is the original period, relative to the period after the frequency change.
precise the statement that if a cell exhibits plateau potentials in one modulatory condition, and PIR and/or PBH in
another condition, then the cell will be an endogenous
burster in a modulatory condition, which allows the expression of plateau potentials and of sufficiently strong PIR.
Physiologically, an N-shaped fast current could arise, for
example, from a combination
of a fixed leak conductance
and a fast, persistent, sodium current. This is the case in
neocortical neurons (Stafstrom et al. 1985). The slow
current can be regarded as the combination of a slow depolarization-activated,
noninactivating,
outward current such
as a calcium-activated potassium current, and a slow hyperpolarization-activated,
non-inactivating,
inward current
similar to the & current in heart cells (DiFrancesco and
Noble 1985) and leech heart interneurons (Angstadt and
Calabrese 1989), or the sag current found in the LP cell in
the pyloric CPG (Golowasch and Marder 1992).
A pair of identical model cells, when connected with reciprocal graded synaptic inhibition,
automatically,
and
very stably, oscillated exactly out of phase with one another. This behavior is seen in reciprocally inhibited pairs
in the gastric mill CPG and other CPGs (Calabrese et al.
1989; Satterlie 1985) in the spike-based modeling study of
Perkel and Mulloney (1974) and in the model of Wang and
Rinzel (1992). Our model reciprocal inhibitory pair can
oscillate in three distinct modes. In the first the connections
merely serve a phase-setting function, because both cells are
endogenous oscillators. In the second the inhibitory connections contribute in an essential way to the oscillations of
MILL
CPG
1049
the system. The cells are not endogenous oscillators and do
not have plateau potentials. In the third the inhibitory connections contribute in an essential way, different from the
preceding, to the oscillations of the system. Both cells have
plateau potentials but do not have large enough slow PIR/
PBH currents for them to be endogenous oscillators. Provided the connections are not too strong, the system continually switches between a high plateau-low plateau configuration and a low plateau-high plateau configuration. If the
connections are too strong, the system locks up and cannot
change configurations.
When a chain of three model cells is constructed with
reciprocal inhibition between the first and second pairs of
cells, the two end cells are in synchrony and out of phase
with the center cell. The three cells Int l-MG-LPG
form a
chain in the gastric mill network (Fig. lA), and, as predicted
by the model, gastric mill recordings (Fig. 1B) show that the
traces of Intl and LPG are synchronous and out of phase
with the trace of MG. By extension of this idea, if six cells
are made into a ring with reciprocal inhibition,
then the
cells numbered 0, 2, and 4 will be in synchrony and out of
phase with the synchronous cells numbered 1, 3, and 5. A
network of this form and with these phase relationships
occurs in the leech heartbeat timing oscillator (Calabrese et
al. 1989).
A pair of model cells with different frequencies (or different potential frequencies if they are not endogenous oscillators) are entrained by reciprocal inhibition to a frequency
lying between the two original frequencies and with small
changes in relative phase from the case of two identical
cells. An example of two cells of widely differing endogenous frequencies entraining to a common frequency occurs
in the isolated gastric mill under the effect of pilocarpine
(Elson and Selverston 1992, Fig(s). 3 and 8). Here the inhibitory pair consists of the cells LG and DG; when DG is
hyperpolarized, LG slows down, but when LG is hyperpolarized, DG speeds up. Thus the reciprocal inhibition between LG and DG serves to entrain two apparently oscillatory cells to a common frequency. Because frequency of a
model cell is controlled primarily by the slow current gain
parameter u,, in a network of cells with many reciprocal
inhibitory connections, the relative phases of different cells
can be adjusted over a small range by adjusting the relative
values of the intracellular g’sparameters. These parameters
correspond to maximal slow current conductances. In our
model the small phase advance of MG over LG was set by
increasing the slow current gain in MG. This is a testable
prediction for the gastric mill network.
Small phase delays can also be set in a model reciprocal
inhibitory pair by using synaptic weights that differ by more
than a factor of 10. Essentially, this mechanism is hypothesized by Williams (1992) to set the intersegmental phase lag
in the spinal cord of the lamprey. This suggests an alternative mechanism, namely pairs of cells linked by reciprocal
inhibitory synapses of very unequal strengths, for establishing phase relations in the gastric mill CPG.
Getting (1989) drew up a list of potential building blocks
of network operation. This network model demonstrates
how, when combined appropriately, the cellular mechanisms of PIR, PBH, and plateau potentials; the synaptic
1050
P. F. ROWAT
a
MG
AND
LPG
KILLED
LG
MG
MG
1
LPG
DG
AM
B
KILLED
LG
LPG
A. I. SELVERSTON
/\
DG
/\
AM
GM
GM
lntl
lntl
E
E
C
AM
//\
D
KILLED
LG
LG
MG
MG
GM
KILLED
LPG
DG
AM
GM
GM
lntl
lntl
E
E
E
Intl
w
w7
F
KILLED
LG KILLED
+
LG
LG
MG
MG
LPG
LPG
DG
DG
AM
AM
GM
GM
lntl
lntl
E
14. Changes in the output
removal of MG. B: removal of LPG.
details.
FIG.
\
\
E
pattern
of the simulated
network
following
sequential
removal
of single neurons.
A:
C: removal of AM. D: removal of GM. E: removal of Int 1. F: removal of LG. See text for
mechanisms of instantaneous and slow graded transmission; and the connectivity mechanisms of reciprocal inhibition and electrical junction all contribute to correct network operation. With the use of a model such as the present
one, the relative importance of different mechanisms to
overall network performance can be computed.
The model presented here can be related to other wellknown simplifying cell models. It is similar to the FitzhughNagumo model (Fitzhugh 196 1; Nagumo et al. 1962) in
that both are defined by two differential equations with
widely differing time constants. However, the form of our
currents is more closely comparable with the physiological
currents believed to underly slow-wave generation in crustacean bursting neurons. It is not surprising that there is a
difference between the forms of the currents, because the
Fitzhugh-Nagumo
equations were intended as a simplification of the Hodgkin-Huxley
equations for action potentials
in the squid axon, not as a model of slow waves in bursting
neurons. It is also comparable with the two-equation Hindmarsh-Rose model (1982), but again the form of our
currents is very different from theirs. They use the narrow
channel property in the phase diagram, in which two nullclines are very close together and thus causes the phase
point to move very slowly in the channel between them, as
the basis for very large and variable interspike intervals.
Kepler et al. ( 1990) and Abbott et al. ( 199 1) used simplified
cell models to study, respectively, control of burst frequency and control of burst duration in the pyloric network
in the STG. Abbott et al. used a two-equation cell model
whose only major difference from ours is the inclusion of
an explicit voltage-dependent
membrane conductance.
Their equations appear more complex, but in fact the dynamics of their model cell is very similar to ours. The cell
model used by Kepler et al. is almost identical to ours; their
N-shaped fast current has an 1-V curve composed of three
linear sections, whereas ours is curvilinear. Wang and Rin-
MODEL
OF
GASTRIC
zel(1992) used a simplified, conductance-based cell model
in their study of the reciprocal inhibitory pair. Their cell
model has a linear leak (fast) current and a slowly inactivating current (“pi? current) that models a T-type calcium
current. Our model has a linear fast current when the parameter of = 0, as in the reciprocal inhibitory pair illustrated in Fig. 6D. However, their pir current cannot be directly compared with our slow current. For instance, its
steady-state 1-v curve is bell-shaped around -8 1 to -65
mV, below the nominal rest potential, whereas our slow
current steady-state 1-V curve is purely linear both above
and below our model’s rest potential. Our model is more
specific than any considered by Kopell and Ermentrout
(1990) in their models of the lamprey spinal cord; their
general theorems on phase locking in chains of coupled
oscillators apply to chains of our model cells.
In summary,
we have constructed
a simplifying,
biologically constrained network model of the gastric mill
CPG, which shows that only a few basic mechanisms are
sufficient to produce the relatively complex patterns characteristic of the cycling gastric mill. Further evidence to
support the validity of this network model is the fact that it
automatically produces approximately
correct gastric mill
output patterns over a wide range of parameter values.
Thus it is a very robust model of the gastric mill mechanism. By this we mean that, because the network model
captures a wide range of characteristic properties of the
gastric mill network, at the cellular, subnetwork, and network levels, and over a broad spectrum of parameter values, then it is probable that any future, more detailed
model of the gastric mill network should be reducible to
this one. Our network model therefore contains some fundamental mechanisms underlying the dynamics of the
gastric mill CPG.
Problem
ofneuromodulation
As we have shown, our model is robust. On the one hand
this is advantageous, in that it produces approximately
correct gastric mill output patterns over a relatively wide
range of parameter values. Thus the network pattern generation mechanism is not sensitive to many parameters and is
not disrupted by small changes in synaptic strengths or in
the intrinsic properties of neurons. On the other hand it
may be disadvantageous because the gastric mill pattern is
highly modulatable (Elson and Selverston 1992; Heinzel
and Selverston 1988; Turrigiano
and Selverston 1989,
1990). To take the simplest example, when the STG is dissected out and disconnected from the other ganglia in the
stomatogastric nervous system, the gastric mill CPG does
not produce a pattern. Some cells fire tonically (Int 1, LPG,
and GM), two are quiescent (LG and MG), and others burst
irregularly (DG and AM) (Selverston and Moulins 1987).
The current model can generate this behavior but requires
drastic changes in the parameter set. It is unlikely that
changes of this sort are an adequate reflection of the neuromodulator-induced
difference between the noncycling state
of the isolated gastric mill CPG and the cycling state that we
have modeled here. This brings into question the usefulness
and validity of our model.
MILL
CPG
1051
The situation can be viewed as follows. There are many
dynamically changing variables in the gastric mill CPG,
many more than are represented in our model or will ever
be represented in any future model. Any model is an approximation to reality. In particular, our model is a gross
approximation
to the true dynamics of the gastric mill, but
it happens to be quite adequate for the neuromodulatory
condition that induces the cycling state we have studied. In
other neuromodulatory
conditions, which induce qualitatively different behaviors, the approximations embedded in
our model break down, and other aspects of the true gastric
mill dynamics must be included to adequately model the
induced behaviors.
To gain a better understanding of the generation of multiple, qualitatively
different behaviors by the single gastric
mill network, a more complex model will ultimately be
required. This model will have many more differential
equations than our model and will have a large parameter
space (e.g., several kinetic parameters for each channel type
in each neuron). This model will have many qualitatively
different behaviors, each associated with a different, welldefined region of its high-dimensional
parameter space.
Each qualitatively different model behavior will correspond
to an experimentally observed gastric mill behavior. In each
region of the high-dimensional
parameter space, a simpler,
low-dimensional
model will be definable that 1) provides
an approximate description of the observed behavior under
the neuromodulatory
conditions defined by this region and
2) whose behavior can be shown to be a reasonable approximation of the behavior of the complex model in this region
of parameter space. A change in neuromodulatory
condition is represented as a movement between regions in the
high-dimensional
parameter space. The motion in parameter space causes a change in qualitative behavior of the
complex model that in turn corresponds to a transition
from one approximating model to another. The mathematical investigation of this topic is the concern of bifurcation
theory.
The model presented in this paper is part of the picture
just sketched, in that it satisfies 1): it approximately
describes the behavior observed in the most well-studied neuromodulatory state, that which is obtained when the STG is
left connected to the other ganglia in the stomatogastric
nervous system. The more complex model for which it is an
approximation
is unknown.
We thank J. E. Lewis and R. L. Calabrese for comments
on the manuscript, I-Teh Hsieh for assistance with the computer
system, and H.-G.
Heinzel for the data used in figures 1,9, and 10. We also thank the referees
for their detailed and perceptive
comments.
This work was supported
by Office of Naval Research Grant NOOO149 l-J-1 720, National
Institutes
of Mental
Health Grant NH-46899,
and
National
Science Foundation
Grant IBN-9 1227 12.
Address for reprint requests: P. F. Rowat, Biology Dept. 0322, University of California
at San Diego, 9500 Gilman
Dr., La Jolla, CA 920930322.
Received
10 December
1992; accepted
in final
form
13 April
1993.
REFERENCES
ABBOTT,
control
L. F., MARDER,
E., AND HOOPER, S. L. Oscillating
networks:
of burst duration
by electrically
coupled neurons. Neural Computation 3: 487-497, 199 1.
ANDERSON,
W. W. AND BARKER, D. L. Synaptic mechanisms
that gener-
1052
P. F. ROWAT
AND
ate network
oscillations
in the absence of discrete postsynaptic
potentials. J. Exp. Zool. 216: 187-191,
1981.
ANGSTADT,
J. D. AND CALABRESE, R. L. A hyperpolarization-activated
inward current in heart interneurons
of the medicinal
leech. J. Neurosci.
9: 2846-2857,
1989.
AYERS, J. L. AND SELVERSTON, A. I. Synaptic perturbation
and entrainment of gastric mill rhythm
of the spiny lobster. J. Neurophysiol 5 1:
113-125,
1984.
BAY, J. S. AND HEMAMI,
H. Modeling
of a neural pattern generator
with
coupled
nonlinear
oscillators.
IEEE Trans. Biomed. Eng. BME-34:
297-306,
1987.
BENJAMIN,
P. R. AND ROSE, R. M. Interneuronal
circuitry
underlying
feeding in gastropod
molluscs.
Trends Neurosci. 3: 272-274, 1980.
BROWN, T. G. On the nature of the fundamental
activity of the nervous
centers together with an analysis of the conditioning
of rhythmic
activity
in progression
and a theory of the evolution
of function
in the nervous
system. J. Physiol. Land. 48: 18-46, 19 14.
BUCHHOLTZ,
F., GOLOWASCH,
J., EPSTEIN, I.R., ANDMARDER,
E.Mathematical model of an identified
stomatogastric
ganglion neuron. J. Neurophysiol. 67: 332-340, 1992.
CALABRESE, R.L., ANGSTADT, J.D., ANDARBAS,
A.A.Aneuraloscillator
based on reciprocal
inhibition.
In: Perspectives in Neural Systems and
Behavior, edited by T. J. Carew and D. Kelly. New York: Liss, 1989, p.
33-50.
CALABRESE, R. L. AND DE SCHUTTER, E. Motor-pattern-generating
networks in invertebrates:
modeling our way toward understanding.
Trends
Neurosci. 15: 439-445,
1992.
CARPENTER, G. A. Bursting phenomena
in excitable membranes.
SIAlM J.
Appl. Math. 36: 334-372, 1979.
COHEN, A. H., HOLMES, P. J., AND RAND, R. H. The nature ofthe coupling
between segmental oscillators
of the lamprey
spinal generator
for locomotion: a mathematical
model. J. Math. Biol. 13: 345-369,
1982.
DIFRANCESCO,
D. AND NOBLE, D. A model of cardiac electrical
activity
incorporating
ionic pumps and concentration
changes. Philos. Trans. R.
Sot. Lond. B Biol. Sci. 307: 353-398, 1985.
EDWARDS, D. H. AND MULLONEY,
B. Compartmental
models of electronic structure
and synaptic integration
in an identified
neurone.
J.
Physiol. Lond. 348: 89-l 13, 1984.
ELSON, R. C. AND SELVERSTON, A. I. Mechanisms
of gastric rhythm generation in the isolated stomatogastric
ganglion of spiny lobsters: bursting
pacemaker
potentials,
synaptic
interactions
and muscarinic
modulation. J. Neurophysiol. 68: 890-907, 1992.
ERMENTROUT,
G. B. AND RINZEL, J. Beyond a pacemaker’s
entrainment
limit: phase walk-through.
Am. J. Physiol. 246 (Regulatory Integrative
Comp. Physiol. 15): R 102-R 106, 1984.
FITZHUGH,
R. Impulses and physiological
states in theoretical
models of
nerve membrane.
Biophys. J. 1: 445-466,
196 1.
FRIESEN, W. 0. AND STENT, G. S. Neural circuits for generating
rhythmic
movements.
Annu. Rev. Biophys. Bioeng. 7: 37-6 1, 1978.
GETTING, P. A. Mechanisms
of pattern generation
underlying
swimming
in Tritonia. I. Network
formed by monosynaptic
connections.
J. Neurophysiol. 46: 65-79, 198 1.
GETTING, P. A. Mechanisms
of pattern generation
underlying
swimming
in Tritonia. II. Network
reconstruction.
J. Neurophysiol. 49: 10 171035, 1983.
GETTING,
P. A. Emerging
principles
governing
the operation
of neural
networks.
Annu. Rev. Neurosci. 12: 185-204, 1989.
GOLA, M. AND SELVERSTON, A. I. Ionic requirements
for bursting activity
in lobster stomatogastric
neurons.
J. Comp. Physiol. 145: 19 l-207,
1981.
GOLOWASCH,
J. AND MARDER, E. Ionic currents of the lateral pyloric neuron of the stomatogastric
ganglion of the crab. J. Neurophysiol. 67: 3 18331, 1992.
GRAUBARD,
K. AND HARTLINE,
D. K. Voltage clamp analysis of intact
stomatogastric
neurons. Brain Res. 557: 24 l-254,
199 1.
GRAUBARD,
K., RAPER, J. A., AND HARTLINE,
D. K. Graded
synaptic
transmission
between identified
spiking neurons. J. Neurophysiol. 50:
508-521,
1983.
GUCKENHEIMER,
J. AND HOLMES, P. Nonlinear Oscillations, Dynamical
Systems, and Bifurcations of Vector Fields. New York: Springer-Verlag,
1983.
HARRIS-WARRICK,
H. W., MARDER,
E., SELVERSTON, A. I., AND MouLINS, M. (Editors)
Dynamic Biological Networks: The Stomatogastric
Nervous System. Cambridge,
MA: MIT Press, 1992.
A. I. SELVERSTON
H ARTLINE, D. K. AND RUSSELL, D. F. Endogenous
burst capability
neuron of the gastric mill pattern generator
of the spiny lobster
in a
Panu-
lirus interruptus. J. Neurobiol. 15: 345-364, 1984.
HEINZEL,
Effects
H. AND SELVERSTON, A. I. Gastric mill activity in the lobster. III.
of proctolin
on the isolated central pattern generator.
J. Neurophysiol. 59: 566-585, 1988.
HILL, A. V. Wave transmission
as the basis of nerve activity.
Cold Spring
Harbor Symp. @ant. Biol. 1: 146- 15 1, 1933.
HINDMARSH,
J. L. AND ROSE, R. M. A model of the nerve impulse using
two first-order
differential
equations.
Nature Land. 296: 162- 164, 1982.
HODGKIN,
A. L. AND HUXLEY,
A. F. A quantitive
description
of membrane current and its application
to conduction
and excitation
in nerve.
J. Physiol. Lond. 117: 500-544,
1952.
KATZ, B. AND MILEDI, R. A study of synaptic transmission
in the absence
of nerve impulses. J. Physiol. Land. 192: 407-436,
1967.
KEPLER, T. B., MARDER,
E., AND ABBOTT, L. F. The effect of electrical
coupling on the frequency
of model neuronal
oscillators.
Science Wash.
DC 248: 83-85,
1990.
KOPELL, N. Toward a theory of modelling
central pattern generators.
In:
Neural Control of Rhythmic Movements in Vertebrates, edited by A. H.
Cohen, S. Rossignol,
and S. Grillner.
New York: Wiley, 1988, p. 369413.
KOPELL, N. AND ERMENTROUT,
G. B. Phase transitions
and other phenomena in chains of coupled oscillators.
SIAM J. Appl. Math. 50: 10 141052, 1990.
KRISTAN, W. B., JR. Generation
of rhythmic
motor patterns. In: Infirmation Processing in the Nervous System, edited by H. M. Pinsker and W.
D. Willis, Jr. New York: Raven, 1980, p. 24 l-26 1.
LI~NARD, A. Etude des oscillations
entretenues.
Revue G&&ale de Z’Electricitd 23, pp. 90 l-9 12, 946-954,
1928.
MAYNARD,
D. M. AND WALTON, K. D. Effects of maintained
depolarization of presynaptic
neurons on inhibitory
transmission
in lobster neuropil. J. Clomp. Physiol. 97: 2 15-243, 1975.
MCDOUGALL,
W. The nature of the inhibitory
processes within the nervous system. Brain 26: 153-19 1, 1903.
MULLONEY,
B. Neural circuits. In: The Crustacean Stomatogastric System, edited by A. I. Selverston and M. Moulins. New York: SpringerVerlag, 1987, p. 57-75.
MULLONEY,
B. AND SELVERSTON, A. I. Organization
of the stomatogastric
ganglion
of the spiny lobster. I. Neurons
driving
the lateral teeth. J.
Comp. Physiol. 9 1: l-32, 1974a.
MULLONEY,
B. AND SELVERSTON, A. I. Organization
of the stomatogastric
ganglion of the spiny lobster. III. Coordination
of the two subsets of the
gastric system. J. Comp. Physiol. 9 1: 53-78,
1974b.
NAGUMO,
J., ARIMOTO,S.,ANDYOSHIZAWA,
S.Anactivepulsetransmission line simulating
nerve axon. Proc. Inst. Radio Eng. 50: 206 l-2070,
1962.
PAVLIDIS, T. AND PINSKER, H. M. Oscillator
theory and neurophysiology.
Federation Proc. 36: 2033-2035,
1977.
PERKEL, D. H. AND MULLONEY,
B. Motor pattern production
in reciprocally inhibitory
neurons
exhibiting
postinhibitory
rebound.
Science
Wash. DC 185: 181-183, 1974.
PERKEL, D. H. AND MULLONEY,
B. Electrotonic
properties
of neurons: the
steady-state
compartmental
model. J. Neurophysiol. 4 1: 62 l-639, 1978.
PETZOLD, L. R. AND HINDMARSH,
A. C. LSODA:
Livermore Solver for
Ordinary D&&erential Equations, with Automatic Method Switching for
Stirand Nonst#Problems. Livermore,
CA 9450: Computing
and Mathematics
Division,
l-3 16, Lawrence
Livermore
National
Laboratory,
1987.
RAPER, J. A. Nonimpulse-mediated
synaptic transmission
during the generation of a cyclic motor program.
Science Wash. DC 205: 304-306,
1979.
ROBERTSON, R. M. AND PEARSON, K. G. Neural circuits in the flight system of the locust. J. Neurophysiol. 53: 110-128,
1985.
ROSE, R. M. AND HINDMARSH,
J. L. The assembly of ionic currents in a
thalamic
neuron. I. The three dimensional
model. Proc. R. Sot. Lond. B
Biol. Sci. 237: 267-288, 1989a.
ROSE, R. M. AND HINDMARSH,
J. L. The assembly of ionic currents in a
thalamic
neuron.
II. The stability
and state diagrams.
Proc. R. Sot.
Lond. B Biol. Sci. 237: 289-3 12, 1989b.
ROSE, R. M. AND HINDMARSH,
J. L. The assembly of ionic currents in a
thalamic neuron. III. The seven-dimensional
model. Proc. R. Sot. Land.
B Biol. Sci. 237: 3 13-334, 1989~.
MODEL
OF GASTRIC
ROWAT, P. F. AND SELVERSTON, A. I. Learning algorithms for oscillatory
networks with gap junctions and membrane currents. Network 2: 17-4 1,
1991.
RUSSELL, D. F. Rhythmic excitatory inputs to the lobster stomatogastric
ganglion. Brain Res. 101: 582-588, 1976.
RUSSELL, D. F. Pattern and reset analysis of the gastric mill rhythm in a
spiny lobster, Panulirus interruptus. J. Exp. Biol. 114: 7 l-98, 1985a.
RUSSELL, D. F. Neural basis of teeth coordination during gastric rhythms
in spiny lobsters, Panulirus interruptus. J. Exp. Biol. 114: 99-l 19,
1985b.
RIJSSELL,D. F. AND HARTLINE, D. K. Synaptic regulation of cellular properties and burst oscillations of neurons in gastric mill system of spiny
lobsters, Panulirus interruptus. J. Neurophysiol. 52: 54-73, 1984.
SATTERLIE, R. A. Reciprocal inhibition and postinhibitory rebound produce reverberation in a locomotor pattern generator. Science Wash. DC
229: 402-404, 1985.
SELVERSTON,A.I., MILLER, J-P., ANDWADEPUHL, M.Cooperativemechanisms for the production of rhythmic movements. In: Neural Origin of
Rhythmic Movements, edited by A. Roberts and B. L. Roberts. Cambridge, UK: Cambridge Univ. Press, 1983, p. 55-87.
SELVERSTON,A. I. AND MOULINS, M. The Crustacean Stomatogastric System. New York: Springer-Verlag, 1987.
SELVERSTON,A. I. AND MULLONEY, B. Organization of the stomatogastric
ganglion of the spiny lobster. II. Neurons driving the medial tooth. J.
Comp. Physiol. 9 1: 33-5 1, 1974.
SELVERSTON,A.I., RUSSELL, D.F., MILLER, J.P., ANDKING, D.G.The
stomatogastric nervous system: structure and function of a small neural
network. Prog. Neurobiol. 7: 2 15-290, 1976.
STAFSTROM, C. E., SCHWINDT, P. C., CHUBB, M. C., AND CRILL, W. E.
Properties of persistent sodium conductance and calcium conductance
of layer V neurons from cat sensorimeotor cortex in vitro. J. Neurophysiol. 53: 153-170, 1985.
STEIN, P. S. G. Application of the mathematics of coupled oscillator systems to the analysis of the neural control of locomotion. Federation
Proc. 36: 2056-2059, 1977.
MILL
CPG
1053
THOMPSON, R. S. A model for basic pattern generating mechanisms in the
lobster stomatogastric ganglion. Biol. Cybern. 43: 7 l-78, 1982.
TSUNG, F.-S., COTTRELL,G.,AND SELVERSTON, A.I. Someexperiments
on learning stable network oscillations. In: Proceedings of the International Joint Conference on Neural Networks. San Diego, CA: IEEE
Neural Networks Council, 1990, p. 169- 174.
TURRIGIANO, G. G. AND SELVERSTON,A. I. Cholecystokinin-like peptide
is a modulator of a crustacean central pattern generator. J. Neurosci. 9:
2486-250 1, 1989.
TURRIGIANO, G. G. AND SELVERSTON, A. I. A cholecystokinin-like hormone activates a feeding-related neural circuit in lobster. Nature Land.
344: 866-868, 1990.
VAN DER POL, B. On “relaxation-oscillations.”
Philos. Mag. 2: 978-992,
1926.
VAN DER POL, B. AND VAN DER MARK, J. The heartbeat considered asa
relaxation oscillator, and an electrical model of the heart. Philos. Mag. 7:
763-775, 1928.
WANG, X.-J. AND RINZEL, J. Alternating and synchronous rhythms in
reciprocally inhibitory model neurons. Neural Computation 4: 84-97,
1992.
WARSHAW, H. S. AND HARTLINE, D. K. Simulation of network activity in
stomatogastric ganglion of the spiny lobster, Panulirus. Brain Res. 110:
259-272, 1976.
WILLIAMS, R. J. AND ZIPSER, D. A learning algorithm for continually
running fully recurrent neural networks. Neural Computation 1: 270280, 1989.
WILLIAMS, T. L. Phase coupling by synaptic spread in chains of coupled
neuronal oscillators. Science Wash. DC 258: 662-665, 1992.
WILSON, W. A. AND WACHTEL, H. Negative resistance characteristic essential for the maintenance of slow oscillations in bursting neurons.
Science Wash. DC 186: 932-934, 1974.
YAMADA, W. M., KOCH, C., AND ADAMS, P. R. Multiple channels and
calcium dynamics. In: Methods in Neuronal Modeling: From Synapses
to Networks, edited by C. Koch and I. Segev. Cambridge, MA: MIT
Press, 1989, p. 97-l 34.