Stochastic Models of Biological Neuron Dynamics - Advanced Models of Neural Networks - page 154

Assume that .x;t/ is the probability density function for the stochastic variable x.
It is noted that

EfU.x;t/gD R U.x;t/.x;t/ dx

(7.20)

Using Taylor series expansion one gets

dt R h.x/.x;t/ dx D R Œh 0 .x/f.x;t/ C h 00 g 2 .x;t/=2.x;t/ dx

d

(7.21)

which is also written as

dt R h.x/.x;t/ dx D R Œ.f.x;t/.x;t// 0 C .g 2 .x;t/=2.x;t// 00 h.x/ dx

d

(7.22)

and since the relation described in Eq. ( 7.22 ) should hold for all h.x/ one gets

@
@t D

@ 1

2

@

@x Œf.x;t/.x;t/ C @x Œg 2 .x;t/.x;t/

(7.23)

This is Fokker-Planck equation.

7.3.2 First Passage Time

This is a parameter which permits to compute the firing rate of the neurons. In such
a case, the aim is to compute the statistical distribution that is followed by the time
instant T in which stochastic variable x exits the interval Œa;b (Fig. 7.2 ).
The probability for x to belong in the interval Œa;b at time instant t is denoted
as G.x;t/. Thus if t is the time instant at which x exits the interval Œa;b, then
G.x;t/ D prob.T t/.
In the case of neurons, the first passage time corresponds to the time instant where
the membrane's potential exceeds a threshold and a spike is generated. It holds that
G.x;t/ satisfies Fokker-Planck equation, that is

2
2

@ 2 G.x;t/
@x 2

@G.x;t/
@t

D f.x;t/ @G.x;t/
@x

(7.24)

C

Regarding boundary conditions it holds that G.x;0/ D 1 if ax<b.

7.3.3 Meaning of the First Passage Time

In biological neuron models, such as the Morris-Lecar model, a spike is generated
if variable w .t/ exceeds a threshold w . One can write the complete Fokker-Planck
equations for a wide domain, compute the outgoing flux (rate in time that the voltage
exceeded certain boundaries) and thus compute the firing rate.