Stable, oscillatory, and chaotic regimes in the dynamics of small neural networks with delay

In this paper we consider simple neural network models consisting of two to three continuous nonlinear neurons, with no intrinsic synaptic plasticity, and with delay in neural signal transmission. We investigate the different dynamic regimes which may exist for these ''minimal'' neural network structures. Examples of stable, oscillatory (periodic or quasi-periodic), and chaotic regimes are reported and analyzed. For chaotic regimes, classical characteristics such as bifurcation diagrams, sensitive dependence on initial conditions, Lyapunov exponents, pseudo phase space attractors, are presented. It is shown that the dynamic regime of a network can be changed through modifications of either internal or external parameters, such as a synaptic weight or an external neuron input. The resulting dynamic regimes offer frameworks to represent various neural functions. For instance, oscillatory regimes provide a mechanism to implement controllable neural oscillators. The sensitive dependence on initial conditions, which is shown to exist even for very small networks, sets a limit to any long term prediction concerning the evolution of the neural system, unless the network adjust its parameters through plasticity in order to avoid chaotic regimes.