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process, i.e. once membrane potential V exceeds a threshold value V
T
,aspikeis
assumed and the membrane potential is reset to V
R
where V
R
V
L
V
T
.Aneven
more elaborated integrate-and-fire neuron model has been proposed, which takes
the form
V D
1
C
Œg
L
.V
L
V/C s.t/ C ˛.V
R
V/ˇ
(7.37)
The spiking period T varies according to
T D 1 ˛
TH
.V/; ˇ D exp
2
2
T
2
(
1VV
T
0V <V
T
(7.38)
H.V/ D
The integration of Eq. (
7.36
) provides also the spiking period of the neuron
R
V
T
V
R
D
R
dT
C
g
L
.V
L
V/Cs.t/
dV
D T
(7.39)
When s.t/ is deterministic then the neuron spikes with a specific period. On the
other hand, when the input s.t/ of the neuron is a stochastic variable then the
neuron spikes with a random period. In stochastic neurons there are stochastic flows
of charges and stochastic currents s.t/ which cause spiking at a random rate. The
models of Eqs. (
7.35
) and (
7.37
) can be also extended to include various types of
ion-channel dynamics, and to model accordingly spike-rate adaptation and synaptic
transmission [
75
].
7.5.2
Stochastic Integrate-and-Fire Neuron Model
and the Fokker-Planck Equation
The dynamics of the previously described integrate-and-fire stochastic neuron can
be associated with Fokker-Planck equation [
75
]. Now, input s.t/ to the neuron's
model is a random variable (e.g., there is randomness in the flow of ions and
randomness in the associated currents).
As a consequence, the spiking period in the stochastic neuron cannot be
analytically defined [see Eq. (
7.36
)], but is sampled from a probability distribution
[
88
]. Using the stochastic input s.t/ in the equations of neuron dynamics, i.e.
Eqs.(
7.35
) and (
7.37
), the spatio-temporal variations of the neuron's voltage will no
longer be associated with deterministic conditions, but will be given by a probability
density function .x;t/.In[
75
] it has been proposed to associate ions density
function .x;t/ with a Fokker-Planck equation (or an advection-diffusion equation)
of the form
