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Fig. 7.2
First passage time for a wiener process
For example, in case of a 2D neuron model one has
dV
D f.V;
w
/
dt
C
dw
dw
D g.V;
w
/
dt
(7.25)
and the associated Fokker-Planck equation is written as
2
2
@
2
P
@P
(7.26)
@t
D
@V
Œf.V;
w
/P
V
Œg.V;
w
/P
w
Assume that the stationary solution of the Fokker-Planck equation is computed,
denoted as P.V;
w
/. This variable defines the probability distribution for the values
of V and
w
. Then one has firing of a spike if
w
exceeds threshold
w
. The firing rate
is computed as
F D
R
V
2
(7.27)
V
1
J
w
.V;
w
/
dV
where J
w
.V;
w
/ D g.V;
w
/P.V;
w
/.
7.4
Stochastic Modelling of Ion Channels
By introducing stochastic dynamics to the neurons' model (i.e, to the Morris-Lecar
model), one obtains equations of the Fokker-Planck or Langevin type. Assume that
the number of ion channels in the membrane is N, ˛ is the transition rate from open
