Bifurcation and Stability Analysis of Bistable Neuromodules

TL;DR

This paper analyzes the stability and bifurcations of simple neuromodules, revealing complex dynamics like chaos and hysteresis, with bifurcation diagrams generated via feedback mechanisms, applicable to neural networks and brain development.

Contribution

First-time plotting of bifurcation diagrams using feedback, highlighting the importance of simultaneous stability and bifurcation analysis for understanding neuromodule dynamics.

Findings

Bifurcation diagrams reveal hysteresis, chaos, and quasiperiodic behavior.

System stability depends on synaptic weights, biases, and transfer function gradients.

Dynamics are history-dependent, influencing neural network behavior.

Abstract

This paper presents a stability analysis of simple neuromodules displaying fold bifurcations (leading to hysteresis), flip bifurcations (period doubling and undoubling to and from chaos) and Neimark-Sacker bifurcations (quasiperiodic and periodic bifurcations). For the first time, bifurcation diagrams are plotted using a feedback mechanism. It is shown that the stability curves and bifurcation diagrams must be dealt with simultaneously in order to fully understand the dynamics of the systems involved. Synaptic weights, biases and gradients of transfer functions are varied and the system is shown to be history dependent. The work can be applied to artificial neural networks and developing brains and gives a very important generalization of previous work in this field.