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In-Depth Information
2.3
Systems Theory and Neurodynamics
Basic features that are important in the study of neurodynamics are (1) equilibria
(fixed points), (2) limit cycles, (3) phase diagrams, (4) periodic orbits, and (5)
bifurcations of fixed points [
92
,
209
]. The definition of these features will be given
through examples in the case neuron models.
2.3.1
The Phase Diagram
One can consider the Morris-Lecar model with the two state variables, V and .
The dynamics of the Morris-Lecar model can be written as
dV
dt
D f.V;t/
(2.33)
d
dt
D g.V;/
The phase diagram consists of the points on the trajectories of the solution of the
associated differential equation, i.e. .V.t
k
/;.t
k
//.
At a fixed point or equilibrium it holds f.V
R
;
R
/ D 0 and g.V
R
;
R
/ D 0.
The closed trajectories are associated with periodic solutions. If there are closed
trajectories, then 9T>0such that .V.t
k
/;.t
k
// D .V.t
k
C T/;.t
k
C T//.
Another useful parameter is the nullclines. The V -nullcline is characterized by
the relation V D f.V;/ D 0.The-nullcline is characterized by the relation
P D g.V;/ D 0. The fixed points (or equilibria) are found on the intersection of
nullclines.
2.3.2
Stability Analysis of Nonlinear Systems
2.3.2.1
Local Linearization
A manner to examine stability in nonlinear dynamical systems is to perform local
linearization around an equilibrium. Assume the nonlinear system x D f.x;
u
/ and
f.x
0
;
u
/ D 0 that is x
0
is the local equilibrium. The linearization of f.x;
u
/ with
respect to
u
round x
0
(Taylor series expansion) gives the equivalent description
x D
Ax
C
Bu
(2.34)
