Biomedical Engineering Reference
In-Depth Information
X
and
X
∗
represent inactive and active
Ca
2
+
binding proteins,
V
e
and
V
e
represent inactive and active
vesicles,
T
is a molecule of neurotransmitter, and
n
is the number of neurotransmitters in a vesicle. You
will note that it takes four
Ca
2
+
ions to activate a single binding protein, a number that has been observed
experimentally.There are much more complex models of the
Ca
2
+
modulated release of neurotransmitter
that include ion channels for Calcium influx as well as pumps to return
Ca
2
+
to the synaptic cleft. These
models are often of a similar form to
I
K
and
I
Na
discussed in Ch. 3.
A simpler model can be created by defining an equation for neurotransmitter molecules released
as a function of the pre-synaptic voltage. A typical function is the Bolzmann distribution:
T
max
T(V
pre
=
m
)
(6.1)
e
−
(V
pre
−
V
p
)/k
p
1
+
m
is the maximum concentration that can be released,
V
pre
where
T
max
is the pre synaptic voltage,
V
p
is
m
called the half activation potential, and
k
p
is a slope factor.
While kinematic and function models attempt to capture some aspects of the physiology, two
related and more simple models are sometimes used in computer simulations. First, if an action potential
is detected at the pre-synapse, a short pulse of
T
will be released into the synaptic cleft. Second, the
steps 5-10 of synaptic transmission may be replaced by a simple time delay. In other words, if an action
potential is present at the pre-synapse it will directly cause a depolarization or hyperpolarization, after
some delay, in the post-synapse.
6.3 NEUROTRANSMITTERDIFFUSIONANDCLEARANCE
The diffusion of neurotransmitter across the synaptic cleft is typically not modeled. Instead, it is assumed
that the released
T
molecules raise the concentration in the tiny cleft faster than the reaction time of
the pre or post synapse. Neurotransmitter
clearance
, on the other hand, may be relative slow. Clearance
is accomplished either by reuptake of the molecule by the pre-synapse or by an enzyme that reverts the
neurotransmitter into an inactive state. Both of these mechanisms can be modeled using differential
equations, the most simple of which is a first order decaying exponential.
dT
dt
=−
k
t
.
(6.2)
Typically, the neurotransmitter (either in an active or inactive form) is taken back into the pre-synaptic
axon and transported back to the soma. Here the molecules are reassembled, repackaged into vesicles, and
sent back to the pre-synapse for future release. It is worth noting that if a neuron is undergoing bursting
it is possible for release and uptake of a neurotransmitter to be occurring at the same time.
6.4 THE POST-SYNAPSE
6.4.1 The Post Synaptic Current
The post-synapse translates a concentration of neurotransmitter into a change in the post-synaptic po-
tential (
V
post
m
).The neurotransmitter binds to a post-synaptic docking site and either directly or indirectly
causing ions to rush either into or out of the post-synaptic membrane. The current generated by this ion
flow across the post-synaptic membrane is defined the same way as other membrane currents.
