not included in the model, and it was considered that the cleft free zinc concentration is uniform during the binding process.
2. Methods
2.1 Model equations
The total amount of zinc in the synaptic cleft is, at any one time, partly bound to
a number of sites and partly unbound. We shall refer to the concentration of the
total amount of zinc in the cleft as [Zn] T and to the concentration of the unbound or
free zinc as [Zn]
2+ . These concentrations may increase due to the release of zinc
from the glutamate vesicles when a signal arrives at the presynaptic area and the
vesicles open up and pour their contents into the cleft. They may also decrease due
to the uptake of zinc into the presynaptic region where it will eventually find its
way back into the glutamate vesicles, thus closing the cycle. Therefore, the rate of
change of the total concentration of zinc in the cleft is given by
d Zn
½� T
dt
¼ Rt ðÞ�Ut ðÞ
(1)
where Rt ðÞrepresents the release, i.e., the rate at which zinc is released from the
presynaptic area and enters the cleft, and Ut ðÞrepresents the uptake, i.e., the rate at
which zinc leaves the cleft and is reabsorbed by the presynaptic region.
As zinc and glutamate are assumed to be released simultaneously, it is reasonable
to expect the rates of change of their concentrations to follow the same pattern.
Therefore, it is assumed that the function Rt ðÞwill have a fast rising phase, followed
by a slow decay, as is known to happen for glutamate. In this study, R(t) and U(t)
are described by alpha functions:
Rt ðÞ¼A 1 te
�
t
τ 1
(2)
where A 1 and τ 1 are constant values that define the height and position of the
peak of the release function, and
Ut ðÞ¼A 2 te
�
t
τ 2
(3)
where A 2 and τ 2 are again constant values. We choose τ 2 to be much larger than
τ 1 , as it is well-known that the uptake is usually much slower than the release. These
four constants cannot be all independent from each other, as the total concentration
of zinc must go back to its resting value so that equilibrium is reached again. A 1 , τ 1 ,
and τ 2 are chosen to be the independent parameters and A 2 to depend on them.
Eq. (1) is easily integrated, yielding
Zn
½� T ¼ Zn
½� T r þ A 1 τ 1
τ 1
τ 2
t þ τ 2
ðÞ e
�t=τ 2 � t þ τ 1
ðÞ e
�t=τ 1
(4)
where Zn
½� T r denotes the resting value of the total zinc concentration and
A 2 ¼ A 1
τ 1
τ 2
2
,
(5)
so as to ensure that Zn
½� T will go back to the basal value Zn
½� T r , as t ! ∞.
113
Computer Simulations of Hippocampal Mossy Fiber Cleft Zinc Movements
DOI: http://dx.doi.org/10.5772/intechopen.90094
2. Methods
2.1 Model equations
The total amount of zinc in the synaptic cleft is, at any one time, partly bound to
a number of sites and partly unbound. We shall refer to the concentration of the
total amount of zinc in the cleft as [Zn] T and to the concentration of the unbound or
free zinc as [Zn]
2+ . These concentrations may increase due to the release of zinc
from the glutamate vesicles when a signal arrives at the presynaptic area and the
vesicles open up and pour their contents into the cleft. They may also decrease due
to the uptake of zinc into the presynaptic region where it will eventually find its
way back into the glutamate vesicles, thus closing the cycle. Therefore, the rate of
change of the total concentration of zinc in the cleft is given by
d Zn
½� T
dt
¼ Rt ðÞ�Ut ðÞ
(1)
where Rt ðÞrepresents the release, i.e., the rate at which zinc is released from the
presynaptic area and enters the cleft, and Ut ðÞrepresents the uptake, i.e., the rate at
which zinc leaves the cleft and is reabsorbed by the presynaptic region.
As zinc and glutamate are assumed to be released simultaneously, it is reasonable
to expect the rates of change of their concentrations to follow the same pattern.
Therefore, it is assumed that the function Rt ðÞwill have a fast rising phase, followed
by a slow decay, as is known to happen for glutamate. In this study, R(t) and U(t)
are described by alpha functions:
Rt ðÞ¼A 1 te
�
t
τ 1
(2)
where A 1 and τ 1 are constant values that define the height and position of the
peak of the release function, and
Ut ðÞ¼A 2 te
�
t
τ 2
(3)
where A 2 and τ 2 are again constant values. We choose τ 2 to be much larger than
τ 1 , as it is well-known that the uptake is usually much slower than the release. These
four constants cannot be all independent from each other, as the total concentration
of zinc must go back to its resting value so that equilibrium is reached again. A 1 , τ 1 ,
and τ 2 are chosen to be the independent parameters and A 2 to depend on them.
Eq. (1) is easily integrated, yielding
Zn
½� T ¼ Zn
½� T r þ A 1 τ 1
τ 1
τ 2
t þ τ 2
ðÞ e
�t=τ 2 � t þ τ 1
ðÞ e
�t=τ 1
(4)
where Zn
½� T r denotes the resting value of the total zinc concentration and
A 2 ¼ A 1
τ 1
τ 2
2
,
(5)
so as to ensure that Zn
½� T will go back to the basal value Zn
½� T r , as t ! ∞.
113
Computer Simulations of Hippocampal Mossy Fiber Cleft Zinc Movements
DOI: http://dx.doi.org/10.5772/intechopen.90094