Systems Theory for the Analysis of Biological Neuron Dynamics - Advanced Models of Neural Networks - page 46

2.5 Bifurcations in Neuronal Dynamics

2.5.1 Bifurcations of Fixed Points of Biological Neuron

Models

As the parameters of the nonlinear model of the neuron are changed, the stability
of the equilibrium point can also change and also the number of equilibria may
vary. Values of these parameters at which the locus of the equilibrium (as a function
of the parameter) changes and different branches appear are known as critical or
bifurcation values. The phenomenon of bifurcation has to do with quantitative
changes of parameters leading to qualitative changes of the system's properties
[ 36 , 54 , 76 , 97 , 113 ].
Another issue in bifurcation analysis has to do with the study of the segments
of the bifurcation branches in which the fixed points are no longer stable but either
become unstable or are associated with limit cycles. The latter case is called Hopf
bifurcation and the system's Jacobian matrix has a pair of complex imaginary
eigenvalues.

2.5.2 Saddle-Node Bifurcations of Fixed Points

in a One-Dimensional System

The considered dynamical system is given by x D x 2 . The fixed point s o f the
system result from the co ndition x D 0 which for >0gives x D˙ p .T he
first fixed point x D p is a stable one whereas the second fixed point x D p
is an unstable one. The phase diagram of the system is given in Fig. 2.17 . Since
there is one stable and one unstable fixed point the associated bifurcation (locus of
the fixed points in the phase plane) will be a saddle-node one.
The bifurcations diagram is given next. The diagram shows how the fixed points
of the dynamical system vary with respect to the values of parameter . In the above
case it represents a parabola in the x plane as shown in Fig. 2.18 .
For >0the dynamical system has two fixed points located at ˙ p .The
one fixed point is stable and is associated with the upper branch of the parabola.
The other fixed point is unstable and is associated with the lower branch of the
parabola. The value D 0 is considered to be a bifurcation value and the point
.x;/ D .0;0/ is a bifurcation point. This particular type of bifurcation where one
branch is associated with fixed points and the other branch is not associated with
any fixed points is known as saddle-node bifurcation.