Communicated
by
John Rinzel
Reduction
of
the Hodgkin-Huxley Equations to
a
Single-Variable Threshold Model
Werner
.M.
Kistler
Wulfram Gerstner
J.
Leo van Hemmen
lnstitutfur Theoretische Physik, Physik-Department der TUMiinchen,
0-85747
Garch-
ing bei Miinchen, Germany
It is generally believed that a neuron
is
a threshold element that fires
when some variable
u
reaches a threshold. Here we pursue the question
of whether this picture can be justified and study the four-dimensional
neuron model of Hodgkin and Huxley as a concrete example. The model
is approximated by a response kernel expansion in terms of a single vari-
able, the membrane voltage. The first-order term is linear in the input and
its kernel has the typical form of an elementary postsynaptic potential.
Higher-order kernels take care of nonlinear interactions between input
spikes. In contrast to the standard Volterra expansion, the kernels depend
on the firing time of the most recent output spike. In particular, a zero-
order kernel that describes the shape of the spike and the typical after-
potential is included. Our model neuron fires if the membrane voltage,
given by the truncated response kernel expansion, crosses a threshold.
The threshold model is tested on a spike train generated by the Hodgkin-
Huxley model with a stochastic input current. We find that the threshold
model predicts
90
percent of the spikes correctly. Our results show that, to
good approximation, the description of a neuron as a threshold element
can indeed be justified.
1
Introduction
Neuronal activity
is
the result of a highly nonlinear dynamic process that
was first described mathematically by Hodgkin and Huxley (1952), who
proposed a set of four coupled differential equations. The mathematical
analysis of these and other coupled nonlinear equations is known to be a
hard problem, and an intuitive understanding of the dynamics is difficult to
obtain. Hence, a simplified description of neuronal activity is highly desir-
able and has been attempted repeatedly, for example, by FitzHugh (1961);
Rinzel(l985); Abbott and Kepler (1990); Av-Ron, Pamas, and Segel(l991);
Kepler, Abbott, and Marder (1992); and Joeken and Schwegler (1995), to
mention only a few.
An
example of a simple model of neuronal spike dy-
namics is the integrate-and-fire neuron, which dates back to Lapicque (1907)
Neural
Computation
9,1015-1045
(1997)
@
1997 Massachusetts Institute
of
Technology
1016
Werner M. Kistler, Wulfram Gerstner, and
J.
Leo van Hemmen
and has become quite popular in neural networks modeling
(see,
e.g., Usher,
Schuster, and Niebur, 1993; Abbott
&
van Vreeswijk, 1993; Tsodyks, Mitkov,
&
Sompolinsky, 1993; Hopfield
&
Herz, 1995).
In this article we aim at a generalized, and realistic, version of the inte-
grate-and-fire model. In order to reduce the four nonlinear equations
of
Hodgkin and Huxley to a simplified model with a single, scalar variable
u,
we use the spike response method, which has been introduced previously
(Gerstner, 1991,1995; Gerstner &van Hemmen, 1992).
A
spike is triggered at
a time
tf
if the membrane potential
u
approaches a threshold
0
from below.
For
f
>
tf
and
tf
being the most recent firing time, the membrane voltage
u
in the reduced description is given by an expansion of the form,
0
The following spike is generated if
u(t)
=
0
and
$u(t)
>
0,
which defines
the next firing time
tf,
and
so
on. The kernels
q
and
E
have a direct neuronal
interpretation. The first term
v(t
-
tf)
in the right-hand side
of
equation
1.1
describes the form
of
a spike and the typical afterpotential following it. The
action potential is triggered at
t
=
tf,
it reaches its maximum after a time
6,
and it is followed by a period of hyperpolarization that extends over
approximately 15 ms (see Figure la). From a different point of view, we
can think of
v(t
-
tf)
as the neuron’s response to the threshold crossing at
tf.
Similarly, the kernel
6
describes the response of the membrane potential
to an input current
1.
More precisely,
t;”(cm;s)
as a function of
s
is the
linear response to a small current pulse at time
s
=
0
given that the neuron
did not fire in the past. In biological terms,
E(’)
describes the form of the
postsynaptic potential evoked by an input spike
(see
Figure lb). Since the
responsiveness of the membrane to input pulses is reduced during or shortly
after an action potential, the form of the postsynaptic potential depends also
on the time
f
-
tf
that has passed since the last output spike. Higher-order
terms in equation
1.1
describe the nonlinear interaction between several
input pulses.
We concentrate here on the four-dimensional set of equations proposed
by Hodgkin and Huxley to describe the spike activity in the giant axon of the
squid (Hodgkin
&
Huxley, 1952). We consider it as a well-studied standard
model of spike activity, even though it does not provide a correct description
of spiking in cortical neurons of mammals. The methods introduced below
are, however, more general and can also be applied to more detailed neuron
models, which often involve tens or hundreds of variables (Yamada, Koch,
&
Adams, 1989; Wilson, Bhalla, Uhley,
&
Bower, 1989; Traub, Wong, Miles,
&
Michelson, 1991; and Ekeberg et al., 1991).
Reduction
of
the Hodgkin-Huxley Equations
1
0
5
10
15
20
1017
1
0.8
-
0.6
0.4
-
0.2
0
-0.2
u
.-
+
?I
w
3
-3
w
0
5
10
15
20
tlms
(b)
Figure
1:
Kernels corresponding to the Hodgkin-Huxley equations and ob-
tained by the methods described in this article. (a) The kernel
ql
describes the
typical shape of a spike and the afterpotential. The spike has been triggered at
time
t
=
0.
(b)
The first-order kernel
ti’’
describes the form of the postsynaptic
potential. The solid line in
(b)
shows the postsynaptic potential evoked by a
short pulse of unit strength that has arrived at time
t
=
0
while
no
spikes had
been generated in the past.
If
there was an output spike
At
=
6.5
ms (dotted line)
or
At
=
10.5
ms (dashed line) before the arrival
of
the input pulse, the response
is significantly reduced. The solid line corresponds to
At
=
co.
1018
Werner
M.
Kistler, Wulfram Gerstner, and
J.
Leo van Hemmen
2
Theoretical Framework
2.1
Standard Volterra Expansion.
In order to approximate the mem-
brane voltage
u
of the Hodgkin-Huxley equations, we start with a Volterra
expansion (Volterra,
1959),
~(t)
=
ds
cA1)(S)Z(f
-
S)
i
0
eA2'(s,
s')Z(t
-
s)Z(t
-
s')
+
.
.
.
(2.1)
00
The first-order term
cA')(s)
describes the membrane's linear response to
an external input current
Z(f),
the second-order term
cA2)(s,
s')
takes into
account nonlinear interactions between inputs at two different times, and
the following terms take care of higher-order nonlinearities. We would like
to stress that equation
2.2
is an ansatz whose usefulness we still have to
verify. It is by no means evident that the series converges reasonably quickly.
2.2
Expansion During and After Spikes.
When dealing with series ex-
pansions, we have to examine their convergence.
As
we will
see,
the expan-
sion
2.1
converges as long
as
the input is
so
weak that no spikes are gen-
erated.
If
a spike is triggered at time
ff
by some strong input, the neuronal
dynamics jumps onto a trajectory in phase space that is always (nearly) the
same-the action potential. During the spike, the evolution is determined
by nonlinear intrinsic processes and is hardly influenced by external input.
We therefore replace equation
2.1
by
x
+-
ds ds'
ci2'(t
-
tf;
s,
s')Z(t
-
s)Z(t
-
s')
+.
.
.
(2.2)
2!
'J
s
00
Here
ql(t
-
tf)
describes the evolution of the membrane potential during
the spike and the relaxation process thereafter. In biological terms,
ql
(t
-
tf)
gives the form of
a
spike and the afterpotential.
As
in equation
2.1,
the
term
6;')
describes the membrane's linear response to an input
Z(t).
Since
inputs are much less effective during and shortly after a spike, the term
6;')
depends on the time
t
-
tf
that has passed since the spike was triggered
Reduction of the Hodgkin-Huxley Equations 1019
at
tf.
The
E'S
are called response kernels. The formal arguments justifying
equation 2.2 are discussed in the appendix.
The lower index of
ci')
indicates that we take into account the effects
of the most recent spike only. We can systematically increase the accuracy
of our description by including more and more spikes that have occurred
in the recent past. Taking into account the most recent
p
spikes, we would
have:
u(t)
=
qp(t
-
4.
.
.
.
,
f
-
$)
Tx)
I
]....,
t-$;s)l(t-s)
0
I
+-
ds ds'
cj2)(f
-
{,
.
.
.
,
k
-
fp;
s,
s')l(f
-
s)l(t
-
s')
2!
'J J
00
+...
(2.3)
In general,
qp(al,
. . .
,
ap)
is a complicated function of the
p
arguments.
Since we expect that adaptation is the dominant effect, we make the ansatz
U
(2.4)
k=l
Here
q1
(a)
describes a single spike and its afterpotential.
A
linear summa-
tion of the afterpotentials of several preceding spikes is often sufficient to
describe the well-known adaptation effects of neuronal activity (Gerstner
&
van Hemmen, 1992; Ekeberg et al., 1991; Kernel1
&
Sjoholm,
1973).
We will show that for the Hodgkin-Huxley model, excellent results are
achieved with equation 2.2, that is, with
p
=
1.
In other words, only the most
recent output spike affects the dynamics. This is not too surprising since the
Hodgkin-Huxley equations show no pronounced adaptation effects.
2.3
Relation to Integrate-and-Fire Models.
In passing we note that the
integrate-and-fire model is a special case of equation 2.3 with the response
kernels (Gerstner, 1995)
where
uo
is the voltage to which the system
is
reset after a spike,
T
is
the mem-
brane time constant, and
o(.)
is the Heaviside step function with
@(s)
=
1
for
s
1
0
and
0
otherwise. Since the standard integrate-and-fire model is lin-
ear except at firing, the determination of the kernels
is
simple. In particular,
all terms beyond
6:')
vanish.