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Bioelectric modelling – Subthreshold, 2008
Johannes J. Struijk
Subthreshold Phenomena
Subthreshold phenomena may be interesting in themselves, but the active behavior of a nerve or a
muscle fiber is, seen from a functional point of view, the most important part of the physiological
description of nerve and muscle. However, in the context of bioelectricity, and up to about 80% of
the threshold, the neural membrane can be very adequately described as a passive RC-network. This
implies that, even when studying electrical stimulation, where the goal is to activate, in particular,
neurons, study of the passive behavior can give a very good insight in the behavior of the neurons
and the effect of several stimulus parameters. Roughly speaking, the passive behavior of the cell
explains 80% of the phenomena during electrical activation.
1. I-t curve (based on passive model of membrane patch)
The simplest model of a cell is a spherical passive membrane. If a stimulation electrode, carrying a
current, Is, with duration T, would be placed in the center of such a cell, together with a reference
electrode far away from the cell, then the membrane voltage due to the current would be spherically
symmetrical. Therefore, this whole cell membrane can be represented by a lumped RmCm network
(figure 2).
Is
Cm
Rm
T
t
Figure 2 left: schematic view of a stimulation electrode in the center of a spherical cell;
the arrows indicate the current flow through the cell membrane;
right: RC-network as an electric circuit model of the cell
The total current, Is, through the membrane is then divided into a capacitive (displacement) current
and an ohmic (ionic) current:
(1.)
I s  I c  I i  Cm
dVm Vm

dt
Rm
The membrane potential Vm=Vm(t) can then be solved for the duration of the pulse (0  t  T) as:
(2.)


Vm (t )  I s Rm 1  e t /  m ,
0tT
where m is the membrane time constant: m =RmCm. This function is monotonically increasing with
t. Thus, for a pulse with length T, the maximum voltage will be reached at the end of the pulse, t=T.
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Bioelectric modelling – Subthreshold, 2008
Johannes J. Struijk
If we assume that the threshold for excitation is simply a constant voltage Vth, then the lowest
current needed to reach Vth will be obtained when the duration of the current pulse is infinitely long.
The minimum current to reach threshold with the infinitely long pulse is called the rheobase, Irh.
(3.)



lim Vm (t )  Vth  lim I rh Rm 1  e t /  m  I rh Rm
t 
t 
which gives
(4.)
Irh=Vth/Rm
Then, for pulse durations T< the threshold Vth will be reached at the end of the pulse (t=T) when
the stimulus current is:
(5.)
I s, th 
I rh
1  e T /  m
The graph of the threshold current Is,th as a function of the pulse duration, T, is called the strengthduration curve, or in short, I-t curve.
threshold current
5
4
3
Rheobase
2
Chronaxy
1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
pulse duration
Figure 3 Strength duration curve for the spherical cell of figure 2.
Rheobase is defined as the minimum threshold for infinite pulse duration;
chronaxy is the minimum pulse duration needed to excite the cell with a current
that is twice the rheobase.
A useful parameter is the chronaxy of the cell. Chronaxy, Tchr, is defined as the minimum pulse
duration needed to reach threshold if the current is twice the rheobase: Is,th = 2Irh. From (5.) it
follows directly that
(6.)
Tchr = m ln2  0.69m
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Bioelectric modelling – Subthreshold, 2008
Johannes J. Struijk
It is important to note that for a given membrane thickness and membrane material, Rm depends
only on the size of the spherical cell: Rm= c/r2, where c is a constant and r is the radius of the cell.
According to (4.) the rheobase will then increase with the square of the radius of the cell.
The chronaxy, however, depends on both Rm and Cm. And because Cm=k.r2, with k a constant, we
see that the membrane time constant, m, is independent of the size of the cell. With (6.) this means
that the chronaxy is independent of the size of the cell. Chronaxy can therefore, be used to
characterize the membrane independent of the size of the cell. We will see later that this is true only
for a spherical cell, but not for the general case of stimulaton of nervous tissue (even though
generally it is true that chronaxy is much less dependent on parameters such as cell size and
electrode-tissue distance than rheobase.
The derivation above was made with an electrode in the center of the cell. Because the intracellular
fluid has a much higher conductivity than the cell membrane, it is almost irrelevant what the exact
location of the electrode is. Even if the electrode is not an electrode but a synaptic transmission,
which can be modeled as a small current injected through the membrane into the cell, the
considerations as given above still hold: larger cells have a lower input impedance, which means
that the cell needs higher currents (i.e., more synaptic inputs) than smaller cells to be excited.
Note that a model is just a model. In reality larger cells tend to have higher membrane time
constants and smaller cells have lower values for the membrane time constant, which indicates that
the constants c and k are not really constant or that the non-spherical shape of the cell plays a role
as well.
For practical stimulation purposes, not only current is of interest, but also the charge injection is
important. The threshold charge, associated with the threshold current Is,th is given by
(7.)
Qth = T . Is,th
From (5.) and (7.) it follows that
(8.)
Qth (T ) 
T  I rh
1  e T /  m
This function has a minimum for T=0: Qth(T=0)=Irh/m which means that short pulses give the best
conditions in terms of injected charge (relatively low charge for short pulses with high currents as
compared with the injected charge for longer pulses with lower currents).
The spherical cell as described above serves as a paradigm for other situations. Especially, the
nomenclature rheobase, chronaxy, and I-t-curve are derived from it.
2. Passive axon model (unmyelinated axon)
Where the resistance and capacitance of the membrane of a perfectly spherical passive cell in the
case of current injection in the center of the cell can be lumped into a single RC network, current
injection in an axon demands a more elaborate description of the membrane.
A very popular, highly stylized model of an axon is the description of the membrane as a cable
network. Let us first consider the unmyelinated axon. The axon membrane is considered to be a
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Bioelectric modelling – Subthreshold, 2008
Johannes J. Struijk
perfect, long cylinder and the electric current and membrane potential are assumed to be perfectly
cylindrically symmetrical, as for example, in the case of a point current injection at the axis of the
cylinder (figure) or a long wire electrode at the axis of the axon.
Figure 4
Current injection in a cylindrical cell. The line thickness of the
current flow schematically indicates the current density.
In this case the membrane can be collapsed into a one-dimensional cable structure consisting of
resistances and capacitances: a resistance times unit length, rm (m), and a capacitance per unit
length, cm (F/m). The core of the cylinder (the intracellular space) is modeled as a resistive onedimensional medium with a resistance per unit length (ri, in /m) as is the extracellular space (re, in
/m).
ie(x)
e(x)
ie(x+dx)
redx
Vm
+
cmdx
rm
dx
extracellular space
membrane
im(x)
i(x)
ii(x)
ridx
ii(x+dx)
intracellular space
x
Figure 5
RC-cable model of the cylindrical cell
The differential equation describing the membrane potential in this cable model is obtained as
follows.
(9.)
(10.)
(11.)
(12.)
Vm ( x, t )  i ( x, t )  e ( x, t )
i
 ri ii
x
 e
 re ie
x
i
i
V
V
im   i  e  m  cm m
x x rm
t
Differentiating (9.) twice with respect to x, and using (10.) and (11.) and subsequently (12.) yields
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Bioelectric modelling – Subthreshold, 2008
Johannes J. Struijk
(13.)
 2Vm
x 2

2
x
  e  
2 i
r r
V

reie  ri ii   e i Vm  (re  ri )cm m
x
rm
t
or
(14.)
 2
 2Vm
x 2
 m
Vm
 Vm  0
t
rm
is the square of the so-called length constant and  m  rm cm is the time
re  ri
constant of the membrane. The interpretation of the length constant can be highlighted by
considering a steady state situation, in which the membrane potential does not change as a function
of time. In that case the second term in (14.) vanishes, leaving a simple second order ordinary
differential equation. With boundary condition Vm(0)=V0 and a vanishing potential at infinity, the
solution then is
where 2 
(15.)
Vm ( x)  V0  e
 x /
In other words: the length constant is the distance from the site x=0 of a (steady state) disturbance in
the membrane potential to the position where the influence of the disturbance is reduced to e-1 =0.37
(x=).
For an interpretation of the time constant a similar exercise can be made. Assume the membrane
potential is independent of x. Then (14.) reduces to
(16.)
m
Vm
 Vm  0
t
which is of the same form as (1.). Without sources, but with the initial condition Vm(0)=V0, we
have
(17.)
Vm (t )  V0 e t /  m
The time constant thus gives an indication of how quickly the membrane potential changes after a
disturbance.
These two cases, the membrane potential either being independent of time or independent of the
spatial coordinate, have relatively simple solutions. If we do not assume these simplifications, then
the solution of (14.) becomes quite complicated indeed. Suppose we have a sudden intracellular
current injection in x=0 at t=0 with a current I0: Is(x,t)=I0(x,t). In this case the solution of (14.) can
be written as
(18.)

 
Vm, I  ( x, t )  rm I 0
exp   m
2
s
 t
4

m / t
2 
 t 2
 x  


      2   
 m 

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Bioelectric modelling – Subthreshold, 2008
Johannes J. Struijk
which shows a rather complicated interaction between t and x.
Because (18.) is the impulse response of the fiber, the membrane potential for an infinitely long
fiber, due to a current injection with a different current waveform, Is=Is(x,t), in time, t, and a
distribution in space, x, can be obtained by a convolution of Vm as given in (18.) with the current
Is(x,t) with respect to both x and t:

(19.)
Vm ( x, t ) 

  I s (, )Vm, I s  ( x  , t  ) dd
     0
A detailed analysis of (18.) or even more complex situations (19.) is beyond the scope of this text.
The time constant, m= rmcm, is independent of axon diameter since rm is reciprocal with diameter
and cm is proportional with diameter. The length constant, 2=rm/(ri+re), depends on axon diameter
because ri is reciprocal with diameter squared, whereas re is negligible under normal circumstances,
where the extracellular space is much larger than the intracellular space. therefore 2 is linear with
fiber diameter: larger fibers have greater length constants. In other words a change in membrane
potential in a thick fiber is spread over a longer distance than in a thin fiber. This is the key to
understanding why thick fibers unmyelinated fibers have a higher conduction velocity than thin
unmyelinated fibers.
Typical length constants are in the order of 0.1 – 1 mm. Typical time constants are in the order of 1
ms.
3. Myelinated axon
The simplest model for the myelinated axon is to consider each node of Ranvier as a discrete RCnetwork, the axonal cylinder between adjacent nodes as a single resistor, and the extracellular space
between adjacent nodes as a single resistor as well. Labelling the membrane potentials for node of
Ranvier number n as Vm,n(t), the membrane potential can be written as:
(20.)


 2 Vm, n 1  2Vm, n  Vm, n 1   m
Vm, n
t
 Vm, n  0
similar to (14.), but with a second order difference term with respect to the discrete space
coordinate n, instead of the second order derivative with respect to the space coordinate x.
m=RmCm, and 2=Rm/(Ri+Re), where Rm () is the nodal membrane resistance (inverse proportional
with the axon diameter), Cm (F) is the nodal membrane capacitance (proportional with axon
diameter, Ri is the intraaxonal resistance between adjacent nodes (inverse proportional with axon
diameter), and Re, the extracellular resistance is negligible because it represents a much larger space
than Ri. The proportionalities mentioned are valid only under the assumptions that the length of the
node of Ranvier (1-2 m) is independent of fiber diameter, and that the distance between adjacent
nodes of Ranvier (internodal distance) is proportional to fiber diameter (proportionality constant
approximately 100). In that case it turns out that both m and  are independent of fiber diameter,
which implies that (20.), and thus the membrane potential, is independent of fiber diameter.
However, experiments show that the conduction velocity is close to linear with fiber diameter and
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Bioelectric modelling – Subthreshold, 2008
Johannes J. Struijk
that excitation thresholds for electrical stimulation also have a strong dependence on fiber diameter.
The key is that the distance between nodes n and n+1 is proportional to fiber diameter, which
explains the linearity of conduction velocity with fiber diameter.
4. Extracellular stimulation
Within the framework of neuroprostheses we are particularly interested in the situation where the
stimulating electrodes are not inside the cell, but at some distance outside the cell, and in the case of
nerve stimulation we are often interested in axons rather than cell bodies. If the field is created by
external electrodes at some distance from the cell, it is the larger cell, or the larger axon that has the
lower threshold, opposite to the situation of intracellular current injection.
Spherical cells
Consider an extremely simplified spherical cell, with a membrane having a very high resistivity and
a plasma having a very high conductivity. If such a cell is placed in a flat electric field E, for
example, because of a stimulation electrode at some distance from the cell, then the cell membrane
will become depolarized on one side of the cell and hyperpolarized at the other side, with
membrane potential:
3
(21.) Vm   E r cos 
2
where r is the cell’s radius and  is the angle with the field axis as shown in figure 6.
r
Figure 6


E
Left: Spherical cell with the definition of the angle  relative to the direction
of the electric field E. Right: current flow around a spherical cell
with very high membrane resistivity.
(from K.S. Cole, “Membranes Ions and Impulses”, 1968.)
The mathematics used to derive (21.) involves the Laplace equation in polar coordinates with
suitable boundary conditions (   ()  0 ) and is beyond the scope of this text. The importance
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Bioelectric modelling – Subthreshold, 2008
Johannes J. Struijk
of (21.) is that it shows that the membrane potential is linear with the size of the cell. Or,
equivalently, that the stimulus threshold is inversely proportional with the cell radius, which is the
opposite of the result that we obtained for intracellular stimulation of a spherical cell. In summary:
Intracellular stimulation  stimulus threshold proportional to the square of the cell radius
Extracellular stimulation  stimulus threshold inversely proportional to cell radius
Nerve fibers
For cylindrical cells, and for a homogeneous field perpendicular to the fiber, the same approach as
for the spherical cell could be used, now based on Laplace’s equation in cylindrical coordinates,
resulting in
(22.)
Vm  2E r cos 
However, for a cylindrical cell the field along the cylinder can usually not be considered to be
homogeneous, but a clear nonhomogeneous profile along the fiber will usually exist, together with a
longitudinal component of the field. It turns out that the longitudinal component becomes the
dominating factor in virtually all cases of interest, making (22.) useless. A different approach has
thus to be used.
Moreover, the models for the unmyelinated and myelinated nerve fibers as presented above, are
difficult to use for the case of electrical stimulation with an electrode at some distance from the
fiber.
Ve,n
Rm
-
Cm
Vm,n
+
Im,n
Vi,n
Figure 7
extracellular space
Ri
membrane
intracellular space
McNeal’s cable model of a myelinated fiber for extracellular stimulation.
For this situation a modification of the models was made by McNeal (1976), who used the
extracellular potential field Ve, due to electrical stimulation, as the driving source for the membrane
potential. Thus the external potential field at the nodes of Ranvier serve as ideal voltage sources in
the cable model.
For the passive model the governing equation for the membrane potential can be derived as follows.
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Bioelectric modelling – Subthreshold, 2008
Johannes J. Struijk
(23.)
Vm, n  Vi, n  Ve, n ,
where Ve,n is the known, impressed, extracellular potenital at node n.
(24.)
I m, n 
(25.)
I m, n 
Vi, n 1  Vi, n
Ri

Vi, n  Vi, n 1
Ri

1
Vi, n 1  2Vi, n  Vi, n 1 
Ri
dVm, n
1
Vm , n  C m
Rm
dt
Combining (24.) and (25.) and subsequently substituting (23.) finally gives:
(26.)


 2 Vm, n 1  2Vm, n  Vm, n 1   m
Vm, n
t

 Vm, n  2 Ve, n 1  2Ve, n  Ve, n 1

Note that the left hand side of (26.) is identical to (20.), whereas the right hand side now is the
second order difference of the extracellular potentials at the nodes of Ranvier.
A similar derivation for the extracellular stimulation of unmyelinated fibers yields:
(27.)
 2
 2Vm
x 2
 m
Vm
 2Ve
 Vm  2
t
x 2
which is similar to (14.) but with the right hand side being the second derivative of the extracellular
potential along the fiber.
The right hand side in (27.) was termed “activating function” by Rattay (1986), and this activating
function is very useful to see what initially happens to the membrane potential when a stimulus
pulse is applied. Suppose that a fiber is stimulated with a rectangular stimulus pulse, and that
initially the membrane potential Vm(0)=0. Then Ve will be constant for the duration of the stimulus
pulse, and zero before and after the pulse, and so will
m
 2Ve
x 2
. Then (27.) reduces to
Vm
 2Ve
 2
with a solution
t t  0
x 2
(28.)
2  2Ve
Vm (t ) 
 t , for small values of t (relative to m).
 m x 2
Thus, a negative activating function will decrease the membrane potential (hyperpolarize the
membrane) and a positive activating function will increase the membrane potential (depolarize the
membrane). In other words: if activation of the fiber occurs, it will be there where the activating
function is positive.
For a long fiber in a large homogeneous medium and a monopolar point electrode in x=0 at some
distance, h, from the fiber, the extracellular potential at the fiber is given by
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Bioelectric modelling – Subthreshold, 2008
Johannes J. Struijk
(29.)
Ve ( x, t ) 
I s (t )
4 x 2  h 2
The second order derivative with respect to x then is
(30.)
 2Ve ( x, t )
x 2
I (t ) 2 x 2  h 2
 s 
5/ 2
4
x2  h2


Figure 8 shows the extracellular potential profile along the fiber and the activating function
(normalized to their peak values) for a monopolar cathode at 1 mm from the fiber.
We see that the activating function is positive near the cathode (x=0) where the membrane
depolarizes, corresponding to an outward current from the fiber towards the negative electrode. The
relatively limited region of outward current is flanked by regions of more diffuse inward current,
which is shown in the activating function as sidelobes with sidelobe amplitudes that are
approximately 20% of the main lobe amplitude.
For an anodal electrode the signs of Ve and d2Ve/dx2 would be reversed, meaning that close to the
electrode there would be a region of hyperpolarization and somewhat away from the electrode there
would be regions of depolarization with a much lower amplitude. This is confirmed by
experimental data that show that direct anodal stimulation requires much higher currents than
cathodal stimulaton of nerves.
Ve
0
-0.2
-0.4
-0.6
-0.8
-1
-5
-4
-3
-2
-1
0
1
2
3
4
5
x (mm)
2
d Ve
dx 2
1
0.5
0
-0.5
-5
-4
-3
-2
-1
0
1
2
3
4
+
-
+
-
+
-
5
x (mm)
Figure 8 Left upper: extracellular potential along the nerve fiber;
left lower: 2nd order derivative of the extracellular potential;
right: schematic drawing of the current flow towards the negative electrode (cathode):
the high density current leaves the fiber close to the cathode,
strongly depolarizing the membrane, whereas the current is more diffuse
where it enters the fiber.
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Bioelectric modelling – Subthreshold, 2008
Johannes J. Struijk
For longer pulses the transmembrane potential becomes a smeared version of the activating
function, but even in the steady state case (infinitely long pulse) the activating function is a useful
indicator for the transmembrane potential.
5. Rheobase and Chronaxy for the case of external stimulation
Where for the round cell chronaxy was found to be related to the membrane time constant in a very
straightforward way (6.), for the axon the matter is much more complicated. One would not like to
do an analysis similar to the one made for the spherical cell, on basis of, for example, (18.) and
(19.). For extracellular stimulation of the stylized axon as described above, chronaxy turns out to
depend on, for example, electrode-fiber distance, in such a way that a point source close to the fiber
gives the lowest value for the chronaxy, which monotonically increases by up to a factor two for
increasing electrode-fiber distance.
Of course, rheobase values are dependent on the size of the target cell, electrode-cell distance,
electrode configuration, surrounding tissue, cell orientation, and the value of the rheobase can vary
over several decades of magnitude for different cases. Chronaxy is much less variable, and, even
though chronaxy is not completely independent of the stimulation conditions, it makes sense to give
chronaxy values to classify various tissues.
Tissue
Skeletal muscle
Cardiac muscle
Smooth muscle
Myelinated nerve fiber
Chronaxy (ms)
0.1 - 1
1 - 3
100
0.1 - 0.3
Note the importance for the choice of pulse width: it doesn’t make sense to try to stimulate smooth
muscle with 100 s pulses, whereas for myelinated nerve fibers this would be a perfectly sensible
thing to do.
Some typical value ranges for
membrane capacitance: 0.05 – 0.2 F/m2
membrane resistivity: 0.1 – 1 m2 (values for nerve are in the lower range, for muscle in the
higher values)
intracellular resistivity: 0.5 - 2 m (values for nerve are in the lower range, for muscle in the
higher values)
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