On the connection weight space structure of a two-neuron discrete neural network: Bifurcations of the fixed point at the origin

Abstract

The dynamics of a two neuron artificial recurrent neural network depends on six parameters: four synaptic weights and two external inputs. There are complicated relationships between these parameters and the behavior of the system. The full parameter space has not been studied yet and has been limited to detect behaviors for specific configurations, i.e., when some parameters are pre-set, either for two or three neurons. In this study we analyze the nature of the fixed point at the origin in a two-neuron discrete recurrent neural network by plotting the bifurcation manifolds in the full weights space which is a 4-dimensinal one that gives a clear view of what the dynamics of the system can be around this fixed point, which is a very influent point in the global dynamics of the system. The possible bifurcations at the origin are Saddle, Period Doubling and Neimark-Sacker. We found, among other results, that the Neimark- Sacker bifurcation is only possible when the synaptic connections between neurons are one excitatory and one inhibitory.