**A Neural Network With Chaotic
Behavior**

*Neurocomputers* (1993)
(in Russian)

Izhikevich E.M. and Malinetskii G.G.

*Department of Computational Mathematics and
Cybernetics,
Moscow State University.
Institute of Applied Mathematics,
Russian Academy of Sciences *

**Abstract. **A chaotic
neural network (NN) is introduced as a model of olfactory system. The network
is an *n*-dimensional mapping that depends upon a parameter. While
the parameter value is high, the NN is equivalent to the classical Hopfield
network. While the value is low, there are many chaotic and regular attractors
in the phase space of the network. Each previously memorized pattern (familiar
odor) has it own attractor.

When there is no input from receptors, the NN is in a chaotic state corresponding to a chaotic attractor having large dimension. During presentation of a previously memorized pattern (inhalation of a familiar odor), the phase space has only one attractor corresponding to the presented pattern. Activity of the NN approaches the attractor after some transient. During presentation of a new pattern (unfamiliar odor), the NN remains in a global chaotic state corresponding to the "I do not know" condition. The more presented pattern resembles a previously memorized pattern, the lower the dimension of the global chaotic attractor.

If a mixture of familiar odors is inhaled, then there is a coexistence of attractors, each for each inhaled odor. The chaotic NN may recognize even a weak odor under the background of a strong one. During recognition of inadequate stimulus there is transition from high-dimensional chaotic attractor corresponding to the dormant state to a lower-dimensional one corresponding to a certain inhaled odor.

This model can illustrate transitions "chaos-order" and "chaos-chaos" seen in the olfactory system of mammals.

*Correspondence to* Eugene
Izhikevich

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