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In-Depth Information
@
@V
.V
R
;
R
/
@
@
.V
R
;
R
/
@g
!
M D
(2.48)
@V
.V
R
;
R
/
@
@
.V
R
;
R
/
which results into the matrix
@I
ion
@V
.V
R
;
R
/=C
M
g
k
.V
R
E
k
/=C
M
0
1
=
.V
R
/
J D
(2.49)
=
.V
R
/
The eigenvalues of matrix M define stability round fixed points (stable or
unstable fixed point). To this end, one has to find the roots of the associated
characteristic polynomial that is given by det.I J/ D 0 where I is the identity
matrix.
2.4
Phase Diagrams and Equilibria of Neuronal Models
2.4.1
Phase Diagrams for Linear Dynamical Systems
The following autonomous linear system is considered
x D Ax
(2.50)
The eigenvalues of matrix A define the system dynamics. Some terminology
associated with fixed points is as follows.
A fixed point for the system of Eq. (
2.50
) is called hyperbolic if none of the
eigenvalues of matrix A has zero real part. A hyperbolic fixed point is called a
saddle if some of the eigenvalues of matrix A have real parts greater than zero and
the rest of the eigenvalues have real parts less than zero. If all of the eigenvalues
have negative real parts, then the hyperbolic fixed point is called a stable node or
sink. If all of the eigenvalues have positive real parts, then the hyperbolic fixed point
is called an unstable node or source. If the eigenvalues are purely imaginary, then
one has an elliptic fixed point which is said to be a center.
Case 1
: Both eigenvalues of matrix A are real and unequal, that is
1
¤
1
¤0.
For
1
<0and
2
<0the phase diagram for
z
1
and
z
2
is shown in Fig.
2.7
:
In case that
2
is smaller than
1
,theterme
2
t
decays faster than e
1
t
.
For
1
>0>
2
the phase diagram of Fig.
2.8
is obtained:
In the latter case there are stable trajectories along eigenvector
v
1
and unstable
trajectories along eigenvector
v
2
of matrix A. The stability point .0;0/ is said to be
a saddle point.
When
1
>
2
>0then one has the phase diagrams of Fig.
2.9
:
