MODEL NEURONS: FROM HODGKIN-HUXLEY TO HOPFIELD
L.E Abbott" and Thomas B. Kepler**
Physics Department', Biology Department** and
Center for Complex Systems
Brandeis University Waltham, MA 02254
Model neural networks are built of model neurons. While real biological neurons exhibit
extremely complex and rich behavior, neuronal dynamics must be considerably simplified to
make networks analytically and computafionally tractable. The most complete and realistic
descriptions of neuronal behavior are based on analytic fits of detailed voltage clamp measurements
of the voltage and time dependence of cell membrane currents. Many different
membrane currents may be involved and the resulting models can be extremely complex.
The simplest model of this type, and the one which will be considered here, is the classic
Hodgkin-Huxley model [1] of the squid giant axon. Even in this case the relative complexity
of the model makes it rather intimidating to imagine studying large networks built from
Hodgkin-Huxley cells. On the other extreme is the binary, on/off ('McCulloch-Pitts, Little or
Hopfield) model cell used in most studies of attractor neural networks [2]. The simplicity
of the binary neuronal model makes extremely detailed studies of network behavior possible
but leaves much of the biological complexity behind. Clearly a first priority in improving
the biological accuracy of network modeling is to consider networks built from more realistic
model neurons.
Neuronal models falling between the extremes represented by the Hodgkin-Huxley and
binary, Hopfield-like models do of course exist. Most prominent are the integrate and fire
models [3] and the FitzHugh-Nagumo model [4]. These were constructed phenomenologically
to match the basic behavior displayed by the Hodgkin-Huxley model and to a lesser degree by
real neurons. The absence of a more logical derivation of these models makes it difficult to
see their exact relationship to more complete descriptions and to understand the nature of the
approximations which have tacitly been used. Here we will present a step by step reduction
of the classic Hodgldn-Huxley model based on work done in collaboration with Eve Marder.
The advantage of this derivation is that it is mathematically explicit and that it provides a
variety of descriptions of increasing simplicity as more and more simplifying approximations
are made.
The starting point of the derivation is the Hodgkin-Huxley model. Through a reduction
procedure the Hodgkin-Huxley equations are approximated by a system of two first-order
differential equations. This results in a model similar in spirit to that of FitzHugh and
Nagumo, but considerably different from it in detail. Our derivation provides missing links
between the Hodgkin-Huxley description and more phenomenological models while at the
same time producing a simplified model which is more accurate than the FitzHugh-Nagumo
model. A reduction similar in some ways to ours has been given previously [5] but our
methods are both more accurate and more general.
Once the two-dimensional model has been derived there are two possible approaches for
further reduction and simplification. One of these leads to either a linear or nonlinear integrate
and fire model while the other gives rise to a binary model with a time-dependent threshold
[6,7]. Through our reduction procedure all of the parameters arising in these simplified
models will be directly obtained from parameters of the original Hodgkin-Huxley description
and all steps and approximations in the derivation will be explicit. In particular, we will
arrive at a binary model of the general Hopfield type except that it includes hysteresis and
a time-dependent threshold factor with dynamics completely determined by the underlying
Hodgkin-Huxley description.
This work was supported by Department of Energy Contract DE-AC0276-ER03230, NIH
Training Grant NIH-T32-HS0792 and NIMH grant NIMH-MH-46742.
THE HODGKIN-HUXLEY MODEL
The basic equation governing the dynamics of any neuron with spatially constant membrane
potential V is given by conservation of electric charge,
C ~v = -F + I (2.1)
dt
where C is the cell capacitance, F the membrane current and I the sum of external and
synaptic currents entering the cell, each per unit of cell membrane area. If the potential is not
spatially constant an additional Laplacian term enters this equation making the analysis much
more complicated. We will restrict our attention to the spatially constant or space clamped
case although our results can be applied to the more general situation as well. The membrane
capacitance C is typically 1 #Farad per square centimeter. In the Hodgkin-Huxley model
the membrane current F arises primarily from the conduction of sodium and potassium ions
through voltage dependent channels in the membrane. In addition, other ionic currents are
described by an Ohmic leakage contribution. F is a function of V and of three time- and
voltage-dependent conductance variables m, h and n,
F(V, rn, h, n) = gL(V -- VL) + gKn4(V -- IRA') + gg:hma(V - VNa) (2.2)
where gL -- 0.3 mmho/cm2, gg = 36 mmho/cm~, gNa = 120 mmho/cm2, VL = -54.402 mvolt,
VI~-= -77 mvolt and VNa = 50 mvolt.
The conductance variables m, h and n are both voltage and time dependent and by
definition take values between zero and one. They approach asymptotic values ~(V), h(V)
and ~(V) with time constants rm(V), rh(V) and r,~(V) respectively,
r V dm dh r V dn = n(V) - n. (2.3)m( )W = - m -- h ,,( )-j-;
The six functions rm(V), vh(V), r,~(Y), ~(V), h(V) and ~(V) are given by fairly complicated
formulas which are the result of detailed fits to experimental data. For all three
variables,
1 (r~, h, ~) = a(m,h,.) (2.4)
r(~,h,.) = a(m,h,~) + ~(~,h,.) a(~,h,~) + ~(.~,h,.)
where
.I(V + 40)
c~.~ = c~h = .07 exp[-.OS(V + 65)1 (2.5)
1 - exp[-.l(V + 40)]
.01(v + 55) ~= = 4exp[-.055o(v + 85)]
*° = 1 - exp[-.l(v + 55)]
1
~h = 1 + exp[-.l(v + 35)] ~= = .125 exp[-.0195(v + 65)].
Here and in the following we suppress the units in formulas with the understanding that all
potentials are in mvolts, all times in msec and all currents in #amps per cm2. Although the
Hodgkin-Huxley equations are quite complicated they mix in a rather minimal way. The cell
potential V is affected by all three of the other dynamic variables ra, h and n. However, m,
h and n do not directly couple to each other, they only interact through V. This property
will allow us to approximate the dynamics through the introduction of an auxiliary potential
variable.
The functions ~, h, ~ and all three r's are plotted in Figure 1. Note that the time constant
1.0
0.8
0.6
0.4
0.2
0.£
-100
: •
,~ •
: 7
•." i"
./ .J
-50
°.,SIm I
msec
V (mvolt)
1°t ,.
'I ..."-.-,=
-100 -50 0
i i t,.,
0 50 V (mvolt)
50
Figure 1
which governs the behavior of m is much smaller than those for h and n. Furthermore the
time constants for h and n are roughly the same over most of the voltage range. These
properties will be important for our reduction of the model.
The result of integrating the Hodgkin-Huxley equations are as follows. For I = 0, V
remains at a resting potential of -65 mvolt. A positive current I of sufficient strength and
duration will induce a sudden depolarization of the cell to about 50 mvolts followed rapidly
by a hyperpolarization and then a slower recovery back to the resting level. This is of course
an action potential. The time course of a typical action potential is shown in Figure 2. Note
that the rise of the action potential is extremely rapid and that its fall has two parts, an initial
downward ramp followed by a more rapid drop to a hyperpolarized potential. There is then a
relatively long recovery back to the resting potential. The sudden rise of the action potential
is caused by a rapid increase of the variable m which turns on a positive inward sodium
current. The actionpotential is terminated as the variables h and n adjust more slowly to the
change in membrane potential. The variable h decreases shutting off the sodium current which
drove the upward swing of the action potential. At the same time an increase in n initiates a
positive outward potassium current which hyperpolarizes the cell. The final recovery involves
the readjustment of h and n back to their resting values.
V (mvolt)
50J
-50
.100 . . . . . . .
-6 -4 -2 0 2 4 6 8
t (msec)
Figure 2
Three essential features are exhibited by the Hodgkin-Huxley model and these ideally
should be present in any serious model of neuronal dynamics. They are 1) the action potential
itself, 2) the refractory period after an action potential during which the slow recovery of the
potassium and sodium conductance has a drastic impact on electrical properties and 3) the
ability of the model cell to capacitively integrate incoming current pulses. These last two
features are essential if a model cell is to react appropriately to synaptic inputs. The reduced
two-dimensional model we discuss retains all three properties while the integrate and fire or
binary models involve either some or complete loss of accuracy in describing property 2) or
3) respectively.
REDUCTION OF THE HOGDKIN-HUXLEY MODEL
The Hodgkin-Huxley model involves four time-dependent dynamical variables V, m,
h and n. The four-dimensional nature of the phase space makes it difficult to visualize
and intuitively understand the workings of this model. There is a tremendous conceptual
advantage in reducing the model to two dimensions so that the phase space can be depicted
in a straightforward manner. We therefore propose a procedure for reducing the number of
dynamical variables from four to two. We do this in two steps. First, as discussed above the
time scale associated with changes in m, r,~, is much smaller than those associated with h
and n. Thus, m will reach its asymptotic value ~(V) much more rapidly than other changes
in the model. If we are willing to give up some of the accuracy of the model over very
short time scales we can replace m by its asymptotic value ~(V) and ignore the differential
equation for m entirely writing
m ~ ~(V) (3.1)
and
F(V,m, h,n) " .F. (3.2)
The last equivalence signifies that when the symbol F is used below it stands for the function
F(V,m(V), h,n). The instantaneous approximation for m reduces the number of dynamic
variables from four to three.
It would simplify things considerably if we could also replace h and n by their asymptotic
values. However, if we did this we would destroy the ability of the model to generate action
potentials since h and n would terminate the action potential as quickly as rn could initiate
it. Instead because of their longer time constants, these variables should lag behind, reaching
their asymptotic values more slowly. This effect can be simulated by introducing an auxiliary
voltage variable U and replacing h and n by their asymptotic values not at the potential V
but rather at U. The dynamics of the U variable will cause it to lag behind V but to approach
it asymptotically. Thus we write,
h ~ h(U) n ~ ~(U) (3.3)
so that
F(V,m, h,n) .~ F(V,m(V),'h(U),-~(U)) - f(V, U). (3.4)
Again the last equivalence indicates that f refers to the function F(v,-~(V),-h(U),~(U)).
Because both h and ~ are monotonic functions the replacement of either one of the variables
h or n by its asymptotic value at U would correspond to a simple change of variables. Indeed
our reduction procedure is exact in this case. However, because we are replacing both h and
n by a single variable U the reduction is approximate. In order to minimize the impact
of this approximation we must choose the variable U carefully. Clearly we want the time
dependence of U in f to mimic the time-dependence induced into F in the full model by the
changing values of h and n. Thus, we equate time derivatives of F at constant V in the full
and reduced models,
OFdh(Y) Ordn(V) _ (Of d'h(U) Of d'ff(U)~ dV (3.5)
Oh dt + ~n -~ \O-h dU +-~n -~ ] "~"
We now use the original formulas for dh/dt and dn/dt incorporating the approximations
h ~ h(U) and n ~ g(U) so that
dh dn .~ "~(V ) - -~(U). (3.6)Rv) - Ru) n(v)7;
Using these result we can solve the equal time-derivative condition for dU/dt in terms of V
and U. This gives us a reduced two-dimensional version of the Hodgkin-Huxley model,
C dV
d---t= -f(Y, U) + I (3.7)
and
where
with
dU
dt = g(< U) (3.S)
A
g(V,U) = ~ (3.9)
(3.10)
10
and
Of dh(~) af d~(~r) (3.11)
13=0-~ d---U--+OndU
where OF/Oh and OF/On are to be evaluated at h = h(U) and n = ~(U). Note that when
U = V, g(V, U) = 0 so U will approach V asymptotically if it can.
Simulations have shown that this reduced model is a good approximation of the full
model. The easiest way to envision the dynamics given by Eqs. (3.7)-(3.11) is to plot the
curves (isoclines) dV/dt = -f + I = 0 and dU/dt = g = 0 in the U - V plane and to
indicate the flows of U and V. This is done for I = 0 on the left side of Figure 3 below.
The straight line corresponds to U = V which makes dU/dt = 0. Along the curved line
dV/dt = 0. The point V = -65 and U = -65 where these two curves intersect is the resting
equilibrium point. The arrows indicate the flows toward these two curves. The right side of
Figure 3 shows dV/dt = 0 isoclines for various positive values of the current I. The lowest
curve corresponds to I = 0 and the left-hand portion of the curve rises as I is increased.
-25
U -50
-75
-100
I
T,,¢....-
,,q[-50
0 50
V
-25
U -50
-75
-I00 -50 0 50
V
Figure 3
The model we have derived is, at least in spirit, similar to the FitzHugh-Nagumo model
[4] which also involves two first-order differential equations. However, it has some essential
and important differences. In the FitzHugh-Nagumo model g(V,U) is taken to be linear in both
U and V while f(V, U) is cubic in V and linear in U. The linear form for g is not such a bad
approximation but the form assumed by FitzHugh and Nagumo for f considerably distorts
the dynamics. In particular, as shown on the right side of Figure 3 the low voltage section of
the dV/dt = 0 isocline moves up with positive current I. However, the high voltage portion
of this curve is almost completely insensitive to external current. In the FitzHugh-Nagumo
model the entire dV/dt = 0 isocline moves up by the same amount as a function of I and this
causes some fairly severe misrepresentations of cell behavior. For example, the amplitude of
the action potentials in the full Hodgkin-Huxley model and in our reduced model decrease
-25
U .5o,
when the cell is pushed to a high firing rate by large injected current. In the FitzHugh-Nagumo
model the amplitude is fairly independent of firing frequency. Furthermore no direct relation
between parameters of the FitzHugh-Nagumo model and the underlying Hodgkin-Huxley
model exists.
To indicate the behavior of the two-dimensional model we first consider the response of
the system to constant positive current. Figure 4 shows the phase plane for zero current
(lower curve) and for a positive current large enough to cause repetitive firing (I=10). The
circular point on the lower curve is the resting point of the cell with zero current. This stable
equilibrium lies at the point where the two isoclines shown intersect. When positive current
is applied the low-voltage portion of the dV/dt = 0 isocline moves up until the intersection
of the two isoclines falls well within the portion of the dV/dt = 0 isocline with positive
slope. This point is unstable so the system goes into a limit cycle as shown by the arrows in
Figure 4. This produces a train of action potentials.
-75
-100
. . • , - • , - . ,
-50 0 50
11
V
Figure 4
Another way to exhibit the dynamics of the reduced model is to subject a model cell
which is initially at the resting potential V = -65 and U = -65 to (square) current pulses
of various amplitudes and durations. Figures 5 and 6 show responses to such pulses. Along
with the dV/dt = 0 and dU/dt = 0 isoclines, the phase-space trajectory is shown with data
points marked at equal time intervals and arrows indicating the sense of the motion. The
initial line of data points in all the figures shown indicates the behavior of V and U during
the application of the current pulse while the curved trajectories give the subsequent response.
The left side of Figure 5 shows the effect of a small pulse of current. This causes an upward
shift in V that is insufficient to move V out of the region where dV/dt < 0 (compare with
Figure 3). As a result V moves back down to its resting value and no action potential is
fired. The loop in the return trajectory indicates that the U variable has also responded to the
current pulse although this response is less dramatic than that for V. The right side of Figure
5 shows the effects of a stronger current pulse which is nevertheless still too weak to produce
an action potential. Here, the dynamics of the U variable plays an essential role in preventing
the firing of an action potential. Note that the current pulse is sufficiently strong to move V
12
into the region to the right of the dV/dt = 0 isocline where dV/dt > 0. However, no action
potential results because the increase in the U variable causes the trajectory to curve back
into the region where dV/dt < O.
U
-64.0
-64.5
-65.0~
-65.5~
-67 -65
/
-62
/
U -64
-6~ /
-63 -6t -~8 -h -60
V V
Figure 5
Figure 6 shows the effects of a current pulse sufficiently strong to produce an action
potential. Here, V is pushed far enough into the unstable dV/dt > 0 region so that it climbs
all the way up to the high-voltage portion of the dV/dt = 0 isocline. The initial trajectory is
shown at left and the complete loop from the resting potential, through the action potential
and back to the resting potential again is shown at right. It is interesting to note that the
phase plane trajectory stays quite close to the dV/dt = 0 isocline except initially near the
resting potential.
-62
: °
U .6~
'-./)
-66
-70 -60 -50
V
-25
U -50
-75
-I00
. . . . . ° .
-50 0 50
V
Figure 6
13
FURTHER REDUCTION OF THE MODEL
We will now show how the two-dimensional model can be further reduced to yield some
even simpler models. Clearly with two differential equations in the model an obvious simplification
is to ignore either one or the other. The differential equation for V reflects the
capacitive properties of the cell, as the presence of the capacitance C would suggest. The
differentialequation for U on the other hand reproduces the time dependence of the membrane
conductance. The capacitive behavior reflected in the V equation is responsible for the integrative
behavior of the cell while the time-dependent conductances given by the U equation
produce the refractory period. Thus, to proceed we must be willing to give up an accurate
description of one or the other of these two essential features. As we will see, integrate and
fire models ignore the U equation and must incorporate refractoriness in an ad hoc and crude
manner. Binary models on the other hand ignore the V equation and thus do not simulate the
integrative and time-delay behavior which capacitance provides. Nevertheless, each of these
models rather accurately reproduces the remaining dynamics and thus provides an extremely
useful if partial description of neuronal behavior.
A standard procedure for reducing a set~of two nonlinear equations like (3.7)-(3.11) (if
such a reduction is in fact possible) is to treat one of the variables as instantaneous. For
example if dV/dt is much greater than dU/dtthe V variable will rapidly move to a value
which makes dV/dt = 0 and the longer time scale dynamics of the system will be governed
by the constraint dV/dt = -f + I = 0 and the slower behavior given by the differential
equation for U. Indeed this is the case through most of the dynamic range of V and U. This
is evident in Figure 6 where we noted that the phase-space trajectory stays quite close the
dV/dt = 0 isocline throughout the action potential. In fact throughout most of the phase
plane If[ >> tgt, so that V is constrained to stay close to the dV/d~= 0 isocline most of the
time. However, to accurately predict the firing pattern of a cell we are most interested in what
happens near the resting potential and firing threshold and here the situation is more complex.
Figure 7 shows -f(V, -65) and g(V, -65) in the relevant region of V. In this region f and
revolt
msec
-75 -70 -65 -60
¥
Figure 7
14
g are of comparable magnitude. Below the resting potential the magnitude of f tends to be
bigger. Thus in this region V can be considered instantaneous and the dynamics is determined
by the isocline constraint -f + I = 0 and the equation for dU/dt. However for potentials
slightly above the resting potential f is actually smaller the g. This region is essential for
triggering an action potential and we see that ignoring the capacitive effects incorporated in
the equation for dV/dt here would be a mistake. A reduction of the two differential equations
involves a trade-off. If the region between the resting potential and the firing threshold is of
most interest then the V dynamics dominates. This is the approach taken by integrate and
fire models as we discuss next. The approximation dV/dt = -f + I = 0 which removes
V as an independent dynamical variable leaving only the differential equation for U is valid
throughout most of the range of V except in the threshold region. This approximation gives
rise to the binary models.
INTEGRATE AND FIRE MODELS
Integrate and fire models can approximately duplicate the capacitive behavior of a cell, but
they treat the refractory properties very roughly at best. To derive an integrate and fire model
from the reduced model of the last section we ignore the dynamics of the U variable. This
is essentially a large C approximation of Eqs. (3.7)-(3.11). Since the time-dependence of the
membrane conductance is not reproduced adequately the model cannot itself produce action
potentials. Rather it is assumed that when the membrane potential exceeds some threshold
value the result is the firing of an action potential. Likewise, since the U dynamics is ignored
no refractory period arises in the model but this is sometimes included crudely by specifying
that the V dynamics freezes for a certain time period following an action potential.
The elimination of the dynamics of the U variable can be done in two different ways.
First we can set U = V permanently making dU/dt = 0. The remaining dynamics is then
just CdV
dt = -f(V, V) + I. (5.1)
As can be seen from the left side of Figure 8 f(V, V) is roughly linear and this further
approximation is often made. The result is the linear integrate and fire model. To write it we
-f(V,V) 0
-5 . . . . . •
-70 -65 -60
V
2.0"
1.0"
-f(V,-65)
0.0"
-1.0
-70 -65 -60
V
Figure 8
15
re-express V as
so that
v = , - 65 (5.2)
cdv v
dt = --R + Z (5.3)
where by fitting the curve f(V, V) we find R = 0.8 kOhm-cm2.
The approximation U = V is actually a pretty terrible one to make. As can be seen in
the phase-space curves for the two-dimensional model, U is never very close to V except at
the resting value of -65. A more sensible approximation is to set U = -65 and leave it there.
Then, V is governed by the equation
ceVdt = -f(V, -65) + I. (5.4)
The curve f(V, -65) is given in the right side of Figure 8. This curve is quite far from linear,
in particular it passes through zero twice. The higher value of V for which f = 0 defines
the threshold potential since for all higher potentials up to the peak of the action potential
dV/dt > 0. Fitting this curve in the relevant region gives a nonlinear integrate and fire model
again written in terms of v,
C dv = -.250v + .083v2 + .008v3 + I (5.5)
dt
with Vthrcshold = 2.5. Note that the parameters for both these integrate and fire models are
given directly by the underlying Hodgkin-Huxley model through the reduction procedure and
simple curve fitting.
BINARY MODELS
As discussed above the membrane potential V tends to stay quite close to the dV/dt = 0
isocline most of the time throughout most of the phase space. Thus, another approach to
further reduction is to assume that the neuron has small enough capacitance C so that the
differential equation for V can be replaced with the constraint equation
f(V, U) = I. (6.1)
In other words, we assume that the capacitance is small enough so that changes of V caused
by changes in the current I and in U can be approximated as instantaneous. We can then
solve the constraint equation for V as a function of U. The resulting values of V versus U
are plotted at the left in Figure 9. Note that V is a multi-valued function of U. To correctly
define the inverse we can introduce a binary variable S which keeps track of which branch
of the curve is being used. In Figure 9 an upper region given by S = +1 and a lower
region given by S = -1 are indicated. For this figure I = 0. The correspondence between
this S variable and the binary variable in Hopfield-type models should be obvious. ,.q = +1
when the membrane potential takes high values characteristic of an action potential while
S = -1 when the cell is not firing and near its resting potential. The region between those
indicated by S = +1 and S = -1 does not need to be considered because V will never
enter this region. As the arrows indicate V will jump instantaneously between the S = 4-1
16
regions. This is an instantaneous approximation to the beginning and end (specifically the
second rapidly dropping part of the end) of an action potential and is a result of our ignoring
the cell capacitance. Since these processes are extremely rapid this is actually quite a good
approximation.
V
50
0
-50
---+-
S---I
-z0c 4
-75 -50 -25
dU
dt
40
30
20
10"
0"
-10
-70 -60 -50 -40
U U
Figure 9
From the curve shown on the left side of Figure 9, V can be eliminated in favor of S and
U. Then dU/dt depends on U and S rather than on U and V. This is shown as the hatched
line in the right-hand panel of Figure 9. Also shown by solid lines in this figure are linear fits
to dU/dt in the regions S = +1 and S = -1 which are used to further simplify the model.
As can be seen these linear fits are quite good. A non-zero current can be included by noting
that for S = +1 current has no appreciable impact while for S = -1 it effectively shifts the
U variable (see the right side of Figure 3). In particular we find that to a good approximation
dU
d't" = -.3(U + 65 - .61) (6.2)
for S = -1 and
dU
- 1.3U- 5O (6.3)
dt
for S = +1. The worst part of this approximation is the fact that the differential equation for
U is linear in I when in fact the response becomes nonlinear if the current is too large.
The final binary model involves two equations, one to maintain the value of the S variable
and the other a differential equation for U obtained by combining the two equations for dU/dt
given above. We will keep track of S by demanding that it flip from -1 to +1 when U goes
through the value -66 from above and from +1 to -1 when U passes through -43 from below.
To simplify the final formulas we make the change of variables
U = 11.5u - 65 i = .03• (6.4)
Then the binary model is given by
S = sign[S + .9 - u + (I - S)i] (6.5)
17
with
du
-- (.8 + .5S)u+ 1.5(1 + S) + .3(1 - S)i (6.6)
dt
The equation (6.5) is similar to that of a binary, Hopfield model with three important
extra properties. 1) The presence of the S variable in the sign function provides hysteresis.
2) The u variable acts as a time-dependent threshold determined by the differential equation
(6.6). 3) The current i only couples when S = -1. The addition of these three features lifts
the binary model to a level where it is accurately reflecting the underlying dynamics of the
full Hodgkin-Huxley model except of course that capacitive effects are not included. The
additional complications introduced to achieve this are really quite minimal. In particular
the differential equation for u is linear so it can be solved easily and analytic results can be
obtained. A model similar to (6.5) and (6.6) has been studied extensively [7]. It is possible to
derive analytic expressions for firing rates, phase response curves, firing delays, phase locking
regions and many other responses of the cell to constant and time varying external currents.
A discrete time version of the model can be generated by turning Eq. (6.5) into a dynamic
map and by integrating Eq. (6.6) over one time step At,
S(t + At) = sign[S(t) + .9 - u(t) + (1 - S(t))i(t)] (6.7)
with
where
u(t + At) = ~ + (~(t) - ~) exp[-(.8 + .5s(t))zxt] (6.8)
= 1.5(1 + S(t)) + .3(1 - S(t))i(t)
.8 + .5S(t) (6.9)
Systems of binary model cells can be constructed by using the standard expression
1~ Jj(Sj + 1)i=~__ (6.10)
for the synaptic current where the sum is over other cells coupled to the neuron being studied
and the Jj are synaptic weights. This has also been done in a related model [7].
CONCLUSIONS
Following a well-defined reduction procedure we have constructed three types of reduced
models. The two-dimensional model incorporates both the effects of cell capacitance and of
time-dependent membrane conductances and so is the most accurate of the three. Integrate
and fire models can be derived only at the expense of weakening the treatment of timedependent
conductances and most importantly refractoriness. Nevertheless these provide quite
a good description near the resting and threshold potentials. By ignoring cell capacitance we
obtained an interesting binary model which incorporates quite accurately aspects of the timedependent
conductances through hysteresis and through a time-dependent threshold factor.
An addition difference between this derived binary model and those more commonly in use
is that synaptic current only couples to the model cell when it is in the non-firing S = -1
state. In all three cases, parameters of the reduced models were obtained directly from the
underlying full Hodgkin-Huxley model. An advantage of our reduction approach is that it
18
allows us to include the effects of other membrane currents besides those included in the
original Hodgkin-Huxely description. For example we can determine how additional currents
affect the time-dependent threshold of the binary model. This is presently being done.
REFERENCES
1. Hodgkin, A.L. and Huxley, A.E (1952) J. Physiol. (London) 117. 500.
2. McCulloch, W.S. and Pitts, W. (1943) Bull. Math. Biophys. 5, 115; Little, W.A. (1975)
Math Biosci. 19, 101; Hopfield, J.J. (1982) Proc. Natl. Acad. USA 79, 2554.
3. For a review see: Tuckwell (1988) H. Introduction to Theoretical Neurobiology (Cambridge
University Press, Cambridge).
4. FitzHugh, R. (1961) Biophys. J. 1, 445; Nagumo, J.S., Arimoto, S. and Yoshizawa, S.
(1962) Proc. IRE 50, 2061.
5. Krinskii, V.I. and Kokoz, Y.M. (1973) Biofizika 18, 533 and 937.
6. Horn, D. and Usher, M. (1989) Phys. Rev. A40, 1036.
7. Abbott, L.E (1990) Y. Phys. A (in press); Proc. Natl. Acad. Sci USA (submitted).