Biomedical Engineering Reference
In-Depth Information
[23]. A finite synaptic time constant leads to synaptic filtering of the pre-synaptic
inputs which, in general, leads to a reduction in post-synaptic firing rates [23]. This
effect is more pronounced for sub-threshold mean currents, since in this regime the
neuronal firing results from the fluctuations in the current which can be filtered out
by the synapse. Note that the effect of increasing t
syn
is to spread out in time the
same amount of charge influx into the cell. Since charge is constantly leaking out
of the cell membrane, the longer t
syn
, the lower the overall magnitude of the voltage
fluctuations.
Finally, one can also compute perturbatively the firing rate in the large synaptic
time constant limit [83]. An interpolation between the two limits gives rather accu-
rate results in the whole range of all synaptic time constants. A similar approach has
been used to compute the firing rate of another simple spiking neuron, the quadratic
neuron [22].
15.2.6
Approximate treatment of realistic synaptic dynamics
Real synaptic currents can depart in at least three ways from the currents consid-
ered until now: (i) individual post-synaptic currents can in some circumstances sum
non-linearly, due to receptor saturation; (ii) post-synaptic currents are voltage depen-
dent, because synaptic activation occurs as conductance change rather than current
increase, and because the maximal conductance can itself be voltage-dependent; (iii)
multiple synaptic time scales are present, due to the different kinetics of the AMPA,
GABA
A
, and NMDA receptors. We first describe the standard biophysical model for
describing post-synaptic currents (see also e.g., [33, 120]), and then discuss sepa-
rately how the three issues can be dealt with using approximate treatments.
15.2.6.1
Biophysical models of post-synaptic currents
Synaptic activation opens ion channels in the post-synaptic cell. The amount of
current flowing through these channels is proportional to the product of the number
of open channels times the driving force of the synaptic current:
I
syn
(
t
)=
g
syn
(
V
)
s
(
t
)(
V
(
t
)
−
V
syn
)
,
(15.38)
where
g
syn
(
is a
gating variable measuring the fraction of open channels at the synapse and
V
syn
is the
synaptic reversal potential. The term
V
V
)
is the (possibly voltage-dependent) maximal conductance,
s
(
t
)
V
syn
is the driving force of the synapse, and
it determines its polarity, i.e., whether a synaptic current is depolarizing (
V
−
−
V
syn
<
0) or hyper-polarizing (
V
0). In the presence of a driving force term, all
synaptic inputs are voltage-dependent.
We consider two types of kinetic schemes for the gating variable
s
−
V
syn
>
. If the un-
derlying dynamics of the synaptic channels is fast compared with the typical firing
rates of the spike trains at the synapse, the synapse is usually far from saturation and
a linear kinetic scheme is appropriate. Additionally, in this situation the rise time of
the post-synaptic currents (PSCs) is so fast that it can be considered instantaneous,
(
t
)
Search WWH ::
Custom Search