Biomedical Engineering Reference
In-Depth Information
The Pinsky-Rinzel model is defined by
⎧
⎨
C
m
dV
s
(
t
)
/
dt
=
−
I
Leak
(
V
s
)
−
I
Na
(
V
s
,
h
)
−
I
K
−
DR
(
V
s
,
n
)
g
c
V
d
(
t
)
−
V
s
(
t
)
+
p
C
m
dV
d
(
t
)
/
dt
=
−
I
Leak
(
V
d
)
−
I
Ca
(
V
d
,
s
)
−
I
K
−
AHP
(
V
d
,
q
)
(1.4)
⎩
I
g
c
V
s
(
t
)
−
V
d
(
t
)
−
I
K
−
C
(
V
d
,
Ca
,
c
)+
p
+
1
−
1
−
p
]
=
−
[
Ca
0
.
002
I
Ca
−
0
.
0125
[
Ca
]
All parameters and other equations of ionic channels can be found in [56],
]
is the calcium concentration (see
Chapter 3
for a detailed account of modelling
calcium activity).
[
Ca
1.2.3
Phase model
It is relatively easy to characterize deterministic neuronal models, in comparison
with models with stochastic behaviour which are described in the next section. One
of the popular ways to carry out such an analysis is by means of the so-called
phase
model
[15, 42, 69].
Models
When the spiking output of a cell is approximately periodic, the under-
lying dynamics may be described by a single variable known as the phase, usually
denoted by q
(
)
∈
[
,
]
.Asq changes from 0 to 2p, the neuronal oscillator pro-
gresses from rest to depolarization to spike generation to repolarization and around
again over the course of one period.
Within the phase description framework, sometimes we are able to calculate and
understand how the detailed description of the synaptic interactions among neurons
can effect their spiking timing and, thus, lead to the formation of spatially and tem-
porally patterned electrical output. More specifically,
t
0
2p
Â
j
=
1
d
q
i
dt
=
w
i
+
G
ij
(
q
j
−
q
i
)
,
i
=
1
, ··· ,
N
.
(1.5)
where q
i
is the phase of the
ith
neuron, w
i
is the initial phase and G is the interaction
between neurons. Even though the reduction to a phase model represents a great
simplification, these equations are still too difficult to analyze mathematically, since
the interaction functions could have arbitrarily many Fourier harmonics, which is
always the case in an actual situation, and the connection topology is unspecified
and largely unknown in biology.
The
Kuramoto model
corresponds to the simplest possible case of equally weighted,
all-to-all, purely sinusoidal coupling
K
N
sin
G
ij
(
q
j
−
q
i
)=
(
q
j
−
q
i
)
.
(1.6)
Many theoretical results are known for the Kuramoto model, as briefly described
below.
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