Biomedical Engineering Reference
In-Depth Information
g
syn
O(t)
V
post
E
syn
I
syn
(t)
=
−
(6.3)
m
where
V
pos
m
is the post-synaptic membrane potential,
g
syn
is the maximum conductance,
O
is the prob-
ability of an open channel, and
E
syn
is the reversal potential.
6.4.2 Excitatory, Inhibitory and Silent Synapses
Given the formulation of Eq. (6.3),
E
syn
will determine whether the synapse will be
excitatory
,
inhibitory
,
or
silent
. To classify
E
syn
, we will assume that the post synaptic potential is at rest (
V
post
V
rest
m
=
). If
m
E
syn
>V
post
m
,
I
syn
will be negative. Remember that a negative membrane current will depolarize the
membrane so
I
syn
will
excite
the cell membrane. The resulting depolarization is called a often called an
Excitatory Post Synaptic Potential
(EPSP). Likewise, if
E
syn
<V
post
m
,
I
syn
will be positive, hyperpolarize
V
pos
m
and
inhibit
the cell membrane. The hyperpolarization of the post-synapse is called an
Inhibitory
Post Synaptic Potential
(IPSP). If
E
syn
=
V
rest
m
the synapse is said to be
silent
.The role of the silent synapse
will be explored further in Sec. 6.4.5.
6.4.3 Neurotransmitter Gating
The gating mechanism for a synaptic current,
O
in Eq. (6.3), may be described in the same way as the
gating of any other ionic channel (see Ch. 3). The exception is that instead of being dependent upon the
membrane voltage, synaptic gating is typically dependent upon the concentration of neurotransmitter in
the synaptic cleft. There are many possible formulations for
O
so we will cover only the most significant.
A generic neurotransmitter gated reaction is
C
+
nT
→
αβO
governed by a differential equation
dO
dt
=
α
[
T
]
(
1
−
O)
−
βO
(6.4)
where
is the concentration of neurotransmitter present. Similar to other gating mechanisms we could
rewrite Eqs. (6.4)
[
T
]
dO
dt
=
O
∞
−
O
τ
O
(6.5)
αT
max
αT
max
+
O
∞
=
(6.6)
β
1
αT
max
+
τ
O
=
.
(6.7)
β
More simply,
O(t)
could be described by an analytic function, for example a bi-exponential
O(t
−
τ
s
)
=
K
e
−
(t
−
t
s
)/τ
2
−
e
−
(t
−
t
s
)/τ
1
(6.8)