Information Technology Reference
In-Depth Information
2
2
@
2
@x
2
@
@t
D
@
u
.x/
@x
C
(7.40)
where
u
.x/ is an external potential function (drift function) and
2
is a diffusion
constant. This is in accordance with the Fokker-Planck equation and can be used to
describe the stochastic neuron dynamics [
149
,
156
].
7.5.3
Rate of Firing for Neural Models
Most of the integrate-and-fire neural models activate a reset condition when the
voltage of the membrane exceeds a specific threshold denoted as V
spike
. Since
voltage is reset when the threshold value is reached, from that point on the
probability to detect this voltage level at the neuron's membrane becomes zero.
What can be detected instead is the generated current.
A generic scalar model of the voltage of the neuron's membrane is used first
(7.41)
dV
D f.V;t/
dt
C
dw
.t/
When the voltage V.t/reaches the threshold value V
spike
then it is reset to the initial
value V
reset
.Iff.V;t/D˛V C I, then one has the leaky integrate-and-fire model
of the neuron. If f.V;t/ D ˛V
2
C I, then one has the quadratic integrate-and-fire
neuron's model.
As explained in the previous subsections, to compute the mean firing rate one
should compute the solution of the Fokker-Planck equation in steady state, that is
@
2
G.x;t/
@x
2
2
2
@G.x;t/
@t
D f.x;t/
@G.x;t/
@x
(7.42)
C
7.6
Stochasticity in Neural Dynamics and Relation
to Quantum Mechanics
7.6.1
Basics of Quantum Mechanics
In quantum mechanics the state of an isolated quantum system Q is represented by
a vector j .t/ > in a Hilbert space. This vector satisfies Schrödinger's diffusion
equation [
34
,
133
]
d
dt
j .t/ >D H .t/
i„
(7.43)
