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dynamic model. Reasons for the appearance of noise in neurons are: (1) randomness
in the opening or closing of the ion channels, (2) randomness in the release of
transmitters which in turn leads to random charging and discharging of the neuron's
membrane [
121
,
196
,
197
]. The main effects of the noise in neurons are as follows:
(1) firing of the neurons even under sub-threshold inputs, (2) increase of the pattern
storage capacity.
A basic tool for studying the stochastic dynamics of neurons is a stochastic
differential equation known as Langevin's equation:
dx
D A.x;t/
dt
C B.x;t/
dW
.t/
(7.1)
For the numerical solution of Langevin's equation usually the following solution is
used
x.nC 1/ D x.n/ C hA.x.n/;t
n
/ C B.x.n/;t
n
/
p
h N.0;1/
(7.2)
N.0;1/ is a vector of independent, identically
where h is the discretization step, and
distributed variables.
7.2.2
Wiener Walk and Wiener Process
The Wiener walk and the Wiener process are essential mathematical tools for
describing the stochastic neuron dynamics. Wiener walk will be analyzed and the
Wiener process will be derived as a limit case of the walk [
37
,
90
]. The Wiener walk
describes a simple symmetric random walk. Assume
1
; ;
n
a finite sequence of
independent random variables, each one of which takes the values ˙1 with the same
probability. The random walk is the sequence
s
k
D
1
C
2
CC
k
;0 kn
(7.3)
The n-step Wiener walk is considered in the time interval Œ0;T, where the time step
t is associated with the particle's displacement x
w
.t
k
/ D
1
x CC
k
x
(7.4)
The random function that is described by Eq. (
7.4
) is the Wiener walk [
56
]. A sample
of the Wiener walk in depicted in Fig.
7.1
a.
The Wiener walk is an important topic in the theory of stochastic processes,
is also known as
Brownian motion
and it provides a model for the motion of a
particle under the effect of a potential. The Wiener process is the limit of the
Wiener walk for n!1, and using the central limit theorem (CLT) it can be shown
that the distribution of the Wiener process is Gaussian. Indeed, since the random
variable
w
.t
k
/ of Eq. (
7.4
) is the sum of an infinitely large number of increments
