Biomedical Engineering Reference
In-Depth Information
A
B
C
80
1
1
60
0.75
0.75
40
0.5
0.5
20
0.25
0.25
0
0.32 0.36
0.4
0.44 0.48
0.32 0.36
0.4
0.44 0.48
0
25
50
75
m
C
(nA)
m
C
(nA)
Firing Rate (Hz)
Figure 15.2
Firing rate
(A)
and CV
(B)
of the LIF neuron as a function of the mean current m
C
for
three values of the effective standard deviation in the voltage s
V
1 mV (solid),
1 mV (dashed) and 4 mV (dot-dashed).
(C)
CV as a function of the mean firing
rate when m
C
is varied as in the three curves in
A-B
. The parameters of the cell are
C
m
=
=
0
.
0
.
2nF,
g
L
=
20 nS (t
m
=
10 ms), t
re f
=
2ms,
V
L
=
−
70 mV,
V
th
=
−
50 mV
and
V
r
=
−
60 mV.
/
√
p
x
0
e
u
2
du
.
Once the firing rate is known, the following recursive relationship between the
moments of the first-passage time distribution of the Ornstein-Uhlenbeck process
(see e.g., [117]) can be used to find
where erf
(
x
)=
2
T
fp
>
<
d
2
T
pf
>
dx
2
T
k
fp
>
dx
<
d
<
s
V
2
T
k
−
1
fp
+(
V
ss
−
x
)
=
−
k
<
>,
(15.27)
where
x
=
V
r
. Given that
<
T
fp
>
=
1
/
n in Equation (15.26), the CV of the ISI is
given by [19]
2pn
2
V
th
−
V
ss
dxe
x
2
T
fp
> − <
2
≡
<
T
fp
>
x
s
V
V
r
−
V
ss
s
V
•
dye
y
2
CV
2
=
(
1
+
erf
(
y
))
.
(15.28)
<
T
fp
>
2
−
In
Figures 15.2A-B
we plot the mean firing rate and CV of the LIF neuron as given
by Equations (15.26,15.28), as a function of the mean current m
C
for various values
of the effective standard deviation in the voltage s
V
. The firing rate (Figure 15.2A)
is a monotonic increasing function of the average current. Qualitatively, it starts
to rise when the average current comes within a standard deviation of the current
threshold, defined as
I
th
≡
, and shown as a vertical dashed line in Figure
15.2A. It increases supra-linearly with m
C
for sub-threshold mean currents, and sub-
linearly when the mean current is above threshold, eventually saturating at 1
g
L
(
V
th
−
V
L
)
/
t
re f
.
Therefore, for a wide range of values of s
V
,
m
C
∼
I
th
is close to the point where the
curvature of n
(
m
C
)
changes sign. The deterministic current threshold
I
th
also marks
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