Self-organization processes in chaotic neural networks
under external periodic force
Vladimir E. Bondarenko
The Institute of Biochemical Physics,
Russian Academy of Sciences,
Moscow, Russia
The particular attention to the research of chaos in human brain and artificial neural networks is explained by the hope that the understanding of human memory and consciousness will be found just in this direction.
For recent years have been shown that human and animal EEG are the
"deterministic chaotic processes" which are produced by the strongly
non-equilibrium low-dimensional systems. The number of degrees of
freedom in different functional states of the brain is changed from
in awake state to 4 during sleep or to 2 in epileptic seizure
state. The largest Lyapunov exponent
value of EEG time series are in the range from 0.028 
to
2.9 , as were obtained in different works.
In this paper the processes of self-organization in asymmetric analog neural network model with time delay and under external sinusoidal force are examined. The changes in quantitative characteristics of neural network outputs such as the correlation dimension, largest Lyapunov exponent, Shannon entropy and normalized Shannon entropy are studied as functions of amplitude and frequency of the external force at one-neuron and all-neurons action.
The asymmetric analog neural network model with time delay under the
study is described by the set of ordinary differential equations.
It is obtained from the solution of the differential equations
that the neural network model produces signal which have no
significant differences from the human 
-rhythm.
The calculation of the correlation dimension 
shows that its value
can be the same for all neurons simultaneously or can be varied from one
neuron to another.
To compare chaotic solutions of the differential equations with the human
EEG we make the time normalization so that the main frequency of numerical
solution to be equal to the frequency of human -rhythm. In this
case the largest Lyapunov exponents obtained from the numerical solution
achieve the values 
s 
which are equal to experimental
values (up to 2.9 s ). The calculated correlation dimension
also are in the range of the experimental values.
Thus, the results of this work have demonstrated controlling degree of chaos by the external sinusoidal force in the neural networks with relatively high correlation dimensions comparable to that for human EEG. The self-organization processes can occur in such neural systems when both the amplitude and the frequency of the external force are varied. Decreasing of the correlation dimension values and changing of the sign and value of the largest Lyapunov exponent are obtained in this case. Irregular variations of the Shannon entropy are observed at different neuron outputs.
The neural network under external force can be considered as a qualitative model of the infra-frequencies action on the brain and epilepsy.