Inhibition and modulation of rhythmic neuronal spiking by noise
Henry C. Tuckwell and Jürgen Jost
Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany
Boris S. Gutkin
Group for Neural Theory, Départment des Etudes Cognitives, Ecole Normale Supérieure, 5 rue d’Ulm, 75005 Paris, France
Received 24 December 2008; revised manuscript received 11 June 2009; published 18 September 2009
We investigated the effects of noise on periodic firing in the Hodgkin-Huxley nonlinear system. With mean
input current as a bifurcation parameter, a bifurcation to repetitive spiking occurs at a critical value
c
6.44. The firing behavior was studied as a function of the mean and variance of the input current, firstly with
initial resting conditions. Noise of a small amplitude can turn off the spiking for values of close to
c
, and
the number of spikes undergoes a minimum as a function of the noise level. The robustness of these phenom-
ena was confirmed by simulations with random initial conditions and with random time of commencement of
the noise. Furthermore, their generality was indicated by their occurrence when additive noise was replaced by
conductance-based noise. For long periods of observation, many frequent transitions may occur from spiking
to nonspiking activity when the noise is sufficiently strong. Explanations of the above phenomena are sought
in terms of the underlying bifurcation structure and the probabilities that noise shifts the process from the basin
of attraction of a stable limit cycle to that of a stable rest state. The waiting times for such transitions depend
strongly on the values of and and on the forms of the basins of attraction. The observed effects of noise
are expected to occur in diverse fields in systems with the same underlying dynamical structure.
DOI: 10.1103/PhysRevE.80.031907 PACS numbers: 87.19.lc, 05.40.a
I. INTRODUCTION
Models for nerve cell activity often take the form of a
nonlinear system of differential equations 1. The effects of
noise on such dynamical systems have been investigated
with many models and preparations 2–11. In the majority
of cases, the effects have been facilitatory; that is, neurons
tend to fire more rapidly when their input processes have a
stochastic component 12–14, even if the latter has zero
mean 15. In some cases, there is a maximal response at a
particular noise level—a phenomenon called stochastic reso-
nance, which arises in several biological and other applica-
tions 16–22. This may occur even when the input signal is
nonperiodic 23 or without external forcing 24. Of interest
also is the phenomenon of coherence resonance 25,26,
which has been studied for noise-induced firing in a
Hodgkin-Huxley model with Ornstein-Uhlenbeck synaptic
input 28.
In recent articles 27,29,30, we have reported and ana-
lyzed a case where noise, instead of having a facilitatory
effect, could inhibit the spiking activity of coupled pairs of
type 1 31,32 model neurons, typified by quadratic integrate
and fire or theta-neuron model cells 1,32. Here, we report
on the occurrence of the inhibitory effects of noise on spik-
ing activity in a single type 2 Hodgkin-Huxley HH model
neuron. This model, which consists of a system of four ordi-
nary or partial differential equations, is basic in neurophys-
ics as it was the first to provide a theoretical framework for
action potentials or spikes. Thenceforth, it has often been
employed to ascertain the effects of noise on spiking activity
5,13,26,28,41.
The inputs we consider are both of the additive current
type, with fixed and random initial conditions, and the con-
ductance type. When the HH neuron is driven by mean cur-
rents close to the critical value for the onset of repetitive
firing, a small amount of noise can dramatically reduce the
firing activity. This phenomenon has indeed been hinted at
experimentally 33. In addition, we have found that there is
a minimum in the firing rate at a particular noise amplitude.
Additionally, we briefly consider long-term periods of noise,
which, particularly when the noise amplitude is large, may
lead to rapid intermittent switching from spiking to nonspik-
ing states.
II. HH NEURONS WITH ADDITIVE NOISE
For a single space-clamped HH-model neuron 34 with
additive or “current” noise we have for the depolarization
Vt at time t
dV =
1
C
+ g ¯
K
n
4
V
K
- V + g ¯
Na
m
3
hV
Na
- V
+ g
L
V
L
- Vdt + dW , 1
and for the dimensionless auxiliary variables
dn =
n
1- n -
n
ndt 2
dm =
m
1- m -
m
mdt 3
dh =
h
1- h -
h
hdt , 4
where C is the membrane capacitance per unit area, , which
may depend on t, is the mean input current density, g ¯
K
, g ¯
Na
,
and g
L
are the maximal constant potassium, sodium, and
leak conductances per unit area with corresponding equilib-
rium potentials V
K
, V
Na
, and V
L
, respectively. The noise en-
ters as the derivative of a standard Wiener process W and has
amplitude . The auxiliary variables are nt, the potassium
PHYSICAL REVIEW E 80, 031907 2009
1539-3755/2009/803/0319078 ©2009 The American Physical Society 031907-1