Biomedical Engineering Reference
In-Depth Information
In
in vivo
experiments, it was estimated that, even when neurons fire at low rates
(a few hertz),
g
ef f
L
is at least 3-5 times larger than
g
L
[32], therefore t
ef f
is 3
−
5
m
10 ms, then t
ef f
shorter than t
m
. For example, if t
m
=
3 ms. When neurons
fire at higher rates (leading to larger synaptic conductances), the value of
g
ef f
L
2
−
m
would
be significantly larger and t
ef
m
would be even smaller.
The steady-state voltage
V
ss
now becomes
V
ef f
E
C
E
g
ef f
V
ss
=[
g
L
V
L
+(
C
E
g
AMPA
s
AMPA
)
V
E
+(
NMDA
s
NMDA
)
g
ef f
L
+(
C
I
g
GABA
s
GABA
)
V
I
]
/
.
(15.61)
Note that the steady state potential
V
ss
is bounded between the highest and the low-
est reversal potentials of the four currents to the neuron. In particular, it can never
become lower than
V
I
. Thus, no matter how strong inhibition is, in this model the
average membrane potential will fluctuate around a value not lower than the reversal
potential of the inhibitory synaptic current, e.g., at approximately
70 mV.
Since Equations (15.57) and (15.58) can be mapped identically to Equations (15.33)
and (15.34), one can now use equation (15.26) to compute the firing rate of a neuron,
−
⎡
⎤
−
1
V
ef f
th
t
ef
m
√
p
−
V
ss
s
ef f
V
ef f
r
e
x
2
⎣
⎦
=
t
re f
+
(
+
(
))
.
n
post
1
erf
x
(15.62)
−
V
ss
s
ef f
where t
ef f
th
and
V
ef
r
are given
by Equations (15.36-15.37). Note that now, the average voltage in the steady state
V
plays a role in determining the firing rate, through both
V
ss
and s
ef f
.Since
V
is
related linearly to the firing rate (Equation (15.52)), the firing rate is not an explicit
function of the synaptic input. Even if the inputs are entirely external (feedforward),
and all the synaptic conductances are fixed,
V
still depends on the post-synaptic firing
rate n itself. Therefore, n must be determined self-consistently.
Equation (15.62) constitutes a non-linear input-output relationship between the
firing rate of our post-synaptic neuron and the average firing rates n
E
and n
I
of the
pre-synaptic excitatory and inhibitory neural populations. This input-output func-
tion is conceptually equivalent to the simple threshold-linear or sigmoid input-output
functions routinely used in firing-rate models. What we have gained from all these
efforts is a firing-rate model that captures many of the underlying biophysics of the
real spiking neurons. This makes it possible to quantitatively compare the derived
firing-rate model with detailed numerical simulations of the irregularly firing spiking
neurons, an important step to relate the theory with neurophysiological data.
and
V
ss
are given by Equations (15.59-15.61);
V
ef f
m
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