Information Technology Reference
In-Depth Information
Chapter 2
Systems Theory for the Analysis of Biological
Neuron Dynamics
Abstract
The chapter analyzes the basics of systems theory which can be used in
the modelling of biological neurons dynamics. To understand the oscillatory behav-
ior of biological neurons benchmark examples of oscillators are given. Moreover,
using as an example, the model of biological neurons the following properties are
analyzed: phase diagram, isoclines, attractors, local stability, bifurcations of fixed
points, and chaotic dynamics.
2.1
Characteristics of the Dynamics of Nonlinear Systems
Main features characterizing the stability of nonlinear dynamical systems are
defined as follows [
92
,
209
]:
1.
Finite escape time
: It is the finite time within which the state-vector of the
nonlinear system converges to infinity.
2.
Multiple isolated equilibria
: A linear system can have only one equilibrium
to which converges the state vector of the system in steady-state. A nonlinear
system can have more than one isolated equilibria (fixed points). Depending on
the initial state of the system, in steady-state the state vector of the system can
converge to one of these equilibria.
3.
Limit cycles
: For a linear system to exhibit oscillations it must have eigenvalues
on the imaginary axis. The amplitude of the oscillations depends on initial condi-
tions. In nonlinear systems one may have oscillations of constant amplitude and
frequency, which do not depend on initial conditions. This type of oscillations is
known as
limit cycles
.
4.
Sub-harmonic, harmonic, and almost periodic oscillations
: A stable linear sys-
tem under periodic input produces an output of the same frequency. A nonlinear
system, under periodic excitation can generate oscillations with frequencies
which are several times smaller (subharmonic) or multiples of the frequency
of the input (harmonic). It may also generate almost periodic oscillations with
