Biomedical Engineering Reference
In-Depth Information
t
)
≡<
(
t
)
−
exponential two-point correlation function
C
C
(
t
,
I
(
t
)
− <
I
>
)(
I
(
<
I
>
)
>
given by
t
)=(
s
C
/
t
|/
C
C
(
t
,
2t
syn
)
exp
(
−|
t
−
t
syn
)
.
(15.32)
Using once again the diffusion approximation to replace the input to
I
by a Gaus-
sian process with the same mean and correlation function, and defining
(
t
)
d
I
(
t
)=
I
(
t
)
−
m
C
, the Langevin equations of the process now read
t
m
dV
(
t
)
d
I
(
t
)
=
−
(
(
)
−
)+
V
t
V
ss
(15.33)
dt
g
L
d
dt
t
syn
d
I
(
t
)=
−
d
I
(
t
)+
s
C
h
(
t
)
.
(15.34)
(
)
(
)
(
)
Although
V
t
is not Markovian anymore (knowledge of
I
t
, in addition to
V
t
,
(
+
)
(
)
(
)
is needed to determine
V
together
constitute a
bi-variate Markov process [54, 99]. From Equations (15.33, 15.34) one can therefore
derive a Fokker-Planck equation characterizing the evolution in time of the joint
probability of
V
and
I
. However, the presence of temporal correlations in
I
t
dt
probabilistically),
V
t
and
I
t
makes
the calculation of the firing rate much more involved than for the simple Ornstein-
Uhlenbeck case and, indeed, the mean first-passage time can only be obtained in the
case where t
syn
(
t
)
≡
t
syn
/
1
[23, 34, 40, 61, 67]. We present here only the final result: the firing rate is given by
t
m
, using perturbation theory on the parameter
k
t
m
k
as
V
∂n
∂n
∂
V
r
k
2
n
syn
(
k
)=
n
+
∂
V
th
+
+
O
(
)
,
(15.35)
where n is the firing rate of the white noise case, Equation (15.26),
√
2
a
=
−
z
(
1
/
2
)
/
∼
1
.
03
and z is the Riemann zeta function [2]. Note that the firing rate calculated in [23]
does not include the term proportional to ∂n
∂
V
r
, because of the approximation
made in that paper, namely the neuron was assumed to be in the sub-threshold
regime, in which the dependency of the mean firing rate on the reset potential is
very weak.
Another way to write Equation (15.35) is to replace the threshold
V
th
and
V
r
in the
expression for the mean first-passage time obtained for a white noise current (15.26),
by the following effective
k
-dependent expressions
/
V
ef f
th
=
V
th
+
s
V
a
k
(15.36)
V
ef f
r
=
V
r
+
s
V
a
k
.
(15.37)
This first order correction is in good agreement with the results from numerical
simulations for
1 t
m
. To extend the validity of the result to larger values
of t
syn
, a second order correction can be added to the effective threshold, with co-
efficients determined by a fit to numerical simulations with values of t
syn
up to
t
syn
<
0
.
t
m
Search WWH ::
Custom Search