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If the currents I
L
;I
K
;I
Na
which are applied to the membrane are lower than a
specific threshold, then the variation of the membrane's potential returns fast to
the equilibrium. If the currents I
L
;I
K
;I
Na
which are applied to the membrane are
greater than a specific threshold, then the variation of the membrane's potential
exhibits oscillations.
1.5
The FitzHugh-Nagumo Model of Neurons
The FitzHugh-Nagumo model is a two-dimensional simplified form of the
Hodgkin-Huxley model and in several cases represents efficiently the membrane's
voltage dynamics in neurons [
16
,
55
]. The model captures all the qualitative
properties of the Hodgkin-Huxley dynamics, in the sense that it results in the same
bifurcation diagram (loci of fixed points with respect to model's parameters). In
Eq. (
1.54
) variable V stands for the membrane's voltage, while variable
w
is known
as recovery variable and is associated with the ionic currents of the membrane.
Finally, variable I is an external input current.
The FitzHugh-Nagumo model comprises two differential equations
dV
dt
D V.V a/.1 V/
w
C I
d
w
dt
(1.54)
D .V
w
/
where typically 0<a<1, >0, and 0. The FitzHugh-Nagumo model is
equivalent to Van Der Pol oscillator, the latter being described by
C
dV
dt
DF.V/ J
(1.55)
L
dJ
dt
D V
(1.56)
As shown in Fig.
1.9
the model of the FitzHugh-Nagumo neuron, for specific
values of its parameters or suitable feedback control input I may exhibit sustained
oscillations in its state variables and in such a case limit cycles will also appear in
the associated phase diagram. The detailed stability analysis of models of biological
neurons will be given in Chap.
3
.
1.6
The Morris-Lecar Model of Neurons
The dynamics of a neuron (cell's membrane and the associated potential) can
be also studied using not cable's equation for the Hodgkin-Huxley model but a
simpler model that is called Morris-Lecar model [
16
,
112
]. The Morris-Lecar model
