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

In this paper we consider simple neural network models consisting oftwo to three continuons nonlinear neurons, with no intrinsic synaptic plasticity, and with delay in neural signal transmission. We investigate thé différent dynamic régimes which may exist for thèse "minimal" neural network structures. Examples of stable, oscillatory (periodic or quasi-periodic), and chaotic régimes are reported and analyzed. For chaotic régimes, classical characteristics such as bifurcation diagrams, sensitive dependence on initial conditions, Lyapunov exponents, pseudo phase space attractors, are presented. Il is shown thaï thé dynamic régime of a network can be changea through modifications ofeither internai or external parameters, such as a synaptic weight or an external neuron input. The resulting dynamic régimes offer frameworks to represent varions neural functions. For instance, oscillatory régimes provide a mechanism to implement controllable neural oscillators. The sensitive dependence on initial conditions, which is shown to exist evenfor very small networks, sets a limit to an y long term prédiction concerning thé évolution ofthe neural System, unless thé network adjttst ils parameters through plasticity in order to avoid chaotic régimes. Keywords—Neural network, Dynamics, Stability, Unstability, Oscillator, Quasi-periodicity. Chaos, Delay.