Systems Theory for the Analysis of Biological Neuron Dynamics - Advanced Models of Neural Networks - page 51

2.6 Chaos in Neurons

2.6.1 Chaotic Dynamics in Neurons

Nonlinear dynamical systems, such as neurons, can exhibit a phenomenon called
chaos, by which it is meant that the system's output becomes extremely sensitive
to initial conditions. The essential feature of chaos is the unpredictability of the
system's output. Even if the model of the chaotic system is known, the system's
response in the long run cannot be predicted. Chaos is distinguished from random
motion. In the latter case, the system's model or input contain uncertainty and as a
result, the time variation of the output cannot be predicted exactly (only statistical
measures can be computed). In chaotic systems, the involved dynamical model
is deterministic and there is little uncertainty about the system's model, input,
and initial conditions. In such systems, by slightly varying initial conditions or
parameters values, a completely different phase diagram can be produced [ 214 , 216 ].
Some known chaotic dynamical systems are the Van der Pol oscillator which
as explained in Sect. 1.5 . is equivalent to the model of the FitzHugh-Nagumo
neuron [ 207 , 224 ]. Other chaotic models are the Duffing oscillator which is
considered to be equivalent to variants of the FitzHugh-Nagumo neuron and which
has been proposed to model the dynamics of neuronal groups recorded by the
Electroengephalogram (EEG).

2.6.2 Chaotic Dynamics in Associative Memories

The variations of the parameters of associative memories (elements of the weight
matrix) within specific ranges can result in the appearance of chaotic attractors or in
the appearance of limitcycles. The change in the branches of the locus where these
attractors reside, which is due to changes in the values of the dynamic model's
parameters, is also noted as bifurcation. An example about chaotic dynamics
emerging in Hopfield associated memories is shown in Fig. 2.23 .

2.7 Conclusions

In this chapter, the main elements of systems theory are overviewed, thus providing
the basis for modelling of biological neurons dynamics. To explain oscillatory phe-
nomena and consequently the behavior of biological neurons benchmark examples