Mechanism for neuronal spike generation by small and large ion channel clusters
Shangyou Zeng and Peter Jung
Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA
(Received 20 November 2003; published 7 July 2004)
Neuronal action potentials are generated by clusters of ion channels between the Hillock and the first
segment. If the clusters comprise a large number of sodium and potassium channels, action potentials are
generated if the membrane potential exceeds a threshold of about -55 mV. Such behavior is well described
by an excitable model such as, for example, the Hodgkin-Huxley equations. In this paper we show through
stochastic modeling that if the size of the generating ion channel cluster is small, action potentials are gener-
ated regardless of whether the membrane potential is below or above the excitation threshold. Action potential
generation is then determined by single-channel kinetics. We further show that this switch in generation
mechanism manifests itself in peculiar statistical properties of the generated spike trains at small cluster sizes.
DOI: 10.1103/PhysRevE.70.011903 PACS number(s): 87.16.Uv, 87.15.Ya
I. INTRODUCTION
Conductance-based models for the transmembrane volt-
age of neurons—pioneered in the seminal paper by Hodgkin
and Huxley [1]—are the cornerstone of modern computa-
tional neuroscience. The essential idea is that the conduc-
tance of the membrane is determined by the conductance of
the potassium and sodium systems which in turn is deter-
mined by the membrane potential. The nonlinear dependence
of the sodium and potassium conductance on the membrane
potentials generates action potentials that travel down the
axon to contact other neurons. The conductance of sodium
and potassium through the membrane is facilitated by spe-
cific ion channels that individually switch stochastically be-
tween the open and the closed state as demonstrated by Ne-
her and Sakman [2]. Experiments show that individual ion
channels open and close randomly with membrane-voltage
dependent opening and closing rates [2,3]. The deterministic
Hodgkin-Huxley equations [1] describe the dynamics of the
membrane potential if the number of ion channels is very
large, i.e., when conductance fluctuations are negligible. If
the action potentials are generated by a cluster of sodium and
potassium channels that comprises few channels only, sto-
chastic effects become important, giving rise to spontaneous
spiking [4,5]. In such situations, stochastic Hodgkin-Huxley
equations have to be employed to describe the transmem-
brane potential [6–11]. When the ion channel number is
large, the stochastic Hodgkin-Huxley equations will ap-
proach the conventional Hodgkin-Huxley equations [10,11].
The effects of channel noise (as a function of the size of the
ion channel cluster) have been studied recently in the context
of the coherence of the generated neuronal spike train
[12,13]. Besides channel noise, other sources of noise are
important. Synaptic noise is generated by stochastic effects
in the transport of neurotransmitter through the synaptic cleft
as well as by the relative small number of postsynaptic re-
ceptors. Furthermore, a neuron is often contacted by a large
number of other neurons whose signals can act like a noise
source [14]. Other sources of noise are ligand-gated ion
channels [15]. In this paper we report on the differences of
the mechanism of action potential generation by small and
large ion channel clusters and how these differences are ex-
pressed in the statistical properties of the neuronal spike
train. We further explore the role of synaptic noise on the
generation of action potentials by a small and large clusters
of ion channels in the neuronal membrane. Since synaptic
noise is extrinsic to the ion channel processes that generate
the action potentials, it appears as noise term in the equation
for the membrane voltage. Intrinsic channel noise appears in
the equations for the gating variables [10,11]. In Sec. II, we
describe the stochastic Hodgkin-Huxley model in the pres-
ence of channel noise and synaptic noise. In Sec. III, four
algorithms are described that are commonly used to evaluate
the stochastic Hodgkin-Huxley model, the straightforward
simulation of each individual gate of each channel, a Markov
process mimicking transitions between occupation-number
states of the ion channel cluster, the Gillespie method, and a
Langevin approach. In Sec. IV we describe results for the
spiking rates, variability of the spiking and temporal coher-
ence of the generated spike trains. In Sec. V our results are
summarized.
II. MODEL
We adopt the classic model for the ion channels intro-
duced by Hodgkin and Huxley that models the potassium
channel by four identical gates that stochastically switch be-
tween an open state and a closed state. The open probabili-
ties p
n
for the four gates n =1,2,3,4 are described by the
rate equations
p ˙
n
t =-
K
v +
K
v p
n
t +
K
v , 1
where
K
v and
K
v are the membrane-voltage v depen-
dent opening and closing rates
K
v =
0.0110 - v
exp10 - v/10 -1
,
K
v = 0.125 exp -
v
80
.
2
The transmembrane voltage v is measured here and in all
equations below in millivolt with respect to the physiologic
cellular resting potential of -65 mV. The potassium chan-
nel is open only when all four gates are open, i.e., with
probability p
1
p
2
p
3
p
4
.
PHYSICAL REVIEW E 70, 011903 (2004)
1539-3755/2004/70(1)/011903(8)/$22.50 ©2004 The American Physical Society 70 011903-1