Biomedical Engineering Reference
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network model of Integrate-and-Fire (IF) type called “generalized Integrate-and-
Fire” (gIF) and introduced by Rudolph and Destexhe [
56
]. This section summarizes
the paper [
7
].
8.4.4.1
The gIF Model
We consider the evolution of a set of
N
neurons. Here, neurons are considered as
“points” instead of spatially extended and structured objects. As a consequence,
we define, for each neuron
k
,avariable
V
k
(
t
) called the “membrane
potential of neuron
k
at time
t
” without specification of which part of a real neuron
(axon, soma, dendritic spine, ...) it corresponds to.
Fix a firing threshold
θ
. The sub-threshold dynamics of the model is:
∈{
1
...N
}
C
k
dV
k
dt
+
g
k
(
t, ω
)
V
k
=
i
k
(
t, ω
)
.
(8.34)
C
k
is the membrane capacity.
g
k
(
t, ω
) is the integrated conductance at time
t
given
the past spike activity encoded in the raster
ω
. It is defined in the following way.
Denote
α
kj
(
t
τ
) the synaptic response of neuron
k
, at time
t
, to a pre-synaptic
spike from neuron
j
that aroused at time
τ
. Classical examples of synaptic responses
are
α
kj
(
t
)=
e
−
−
t
τ
kj
H
(
t
) or
α
kj
(
t
)=
e
−
t
t
τ
kj
τ
kj
H
(
t
),where
H
the Heaviside
function (that mimics causality) and
τ
kj
is the characteristic decay times of the
synaptic response. In gIF model the conductance
g
k
(
t, ω
) integrates the synaptic
responses of neuron
k
to spikes arising in the past. Call
t
(
r
j
(
ω
) the
r
-th spike-time
emitted by neuron
j
in the raster
ω
, namely
ω
j
(
n
)=1if and only if
n
=
t
(
r
)
(
ω
)
j
for some
r
.Then
N
g
k
(
t, ω
)=
g
L,k
+
G
kj
α
kj
(
t, ω
)
,
(8.35)
j
=1
sums up the spike responses
of post-synaptic neuron
k
to spikes emitted by the pre-synaptic neuron
j
at times
t
(
r
j
(
ω
)
<t
.
g
L,k
is the leak conductance of neuron
k
.
Returning to Eq. (
8.34
)theterm
i
k
(
t, ω
) is a current given by
i
k
(
t, ω
)=
g
L,k
E
L
+
j
=1
W
kj
α
kj
(
t, ω
)+
i
(
ext
)
(
ω
)
<t
α
kj
t
where
α
kj
(
t, ω
)=
t
(
r
)
j
t
(
r
)
j
−
(
ω
)
(
t
)+
σ
B
dB
k
dt
,where
E
L
is the Nernst
leak potential,
W
kj
is the synaptic weight from
j
to
k
,
i
(
ext
k
(
t
) the (time-dependent)
external stimulus received by neuron
k
,and
dB
k
is a Brownian noise whose variance
is controlled by
σ
B
.
As in all IF models, when
V
k
reaches the firing threshold
θ
, it is reset. Here,
however, it is not instantaneously reset to a constant value, but to a Gaussian random
variable with mean zero and variance
σ
R
, after a time delay
δ>
0 including the
depolarization-repolarization and refractory period. We call
τ
k
(
t, ω
) the
last time
before
t
where neuron
k
has been reset
.
k
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