Transition from regular to complex behaviour in a discrete deterministic asymmetric neural network model

TL;DR

This paper investigates how a discrete asymmetric neural network transitions from ordered to complex transient behavior as a parameter varies, revealing similarities to chaos despite the system's finite state space.

Contribution

It identifies a transition point where transient dynamics shift from short to long, complex behavior, linking it to changes in transient length scaling and entropy measures.

Findings

Transition point $k_c$ marks change from power-law to exponential transient length scaling.

Complex transient behavior exhibits decay of correlations and positive entropy.

Transition resembles intermittent chaos with scaling laws and entropy fluctuations.

Abstract

We study the long time behaviour of the transient before the collapse on the periodic attractors of a discrete deterministic asymmetric neural networks model. The system has a finite number of possible states so it is not possible to use the term chaos in the usual sense of sensitive dependence on the initial condition. Nevertheless, at varying the asymmetry parameter, $k$, one observes a transition from ordered motion (i.e. short transients and short periods on the attractors) to a ``complex'' temporal behaviour. This transition takes place for the same value $k_{\rm c}$ at which one has a change for the mean transient length from a power law in the size of the system ($N$) to an exponential law in $N$. The ``complex'' behaviour during the transient shows strong analogies with the chaotic behaviour: decay of temporal correlations, positive Shannon entropy, non-constant Renyi entropies of different orders. Moreover the transition is very similar to that one for the intermittent transition in chaotic systems: scaling law for the Shannon entropy and strong fluctuations of the ``effective Shannon entropy'' along the transient, for $k > k_{\rm c}$.