Relaxation, closing probabilities and transition from oscillatory to chaotic attractors in asymmetric neural networks

Attractors in asymmetric neural networks with deterministic parallel dynamics present a 'chaotic' regime at symmetry eta less than or equal to 0.5 where the average length of the cycles increases exponentially with system size, and an oscillatory regime at high symmetry. where the average length of the cycles is 2. We show, both with analytic arguments and numerically, that there is a sharp transition, at a critical symmetry eta(c) = 0.33, between a phase where the typical cycles have length 2 and basins of attraction of vanishing weight and a phase where the typical cycles are exponentially long with system size, and the weights of their attraction basins are distributed as in a random map with reversal symmetry. The timescale after which cycles are reached grows exponentially with system size N, and the exponent vanishes in the symmetric limit, where T proportional to N(2/3). The transition can be related to the dynamics of the infinite system (where cycles are never reached), using the closing probabilities as a tool. We also study the relaxation of the function E(t) = -1/N Sigma(l)\h(i)(t)\, where h(i) is the local field experienced by the neuron i. In the symmetric system, it plays the role of a Ljapunov function which drives the system towards its minima through steepest descent. This interpretation survives, even if only on the average, also for small asymmetry. This acts like an effective temperature: the larger the asymmetry, the faster the relaxation, and the higher the asymptotic value reached. E reaches very deep minima in the fixed points of the dynamics, which are reached with vanishing probability, and attains a larger value on the typical attractors, which are cycles of length 2.