Multistability of switched neural networks with sigmoidal activation functions under state-dependent switching

This paper presents theoretical results on the multistability of switched neural networks with commonly used sigmoidal activation functions under state-dependent switching. The multistability analysis with such an activation function is difficult because state–space partition is not as straightforward as that with piecewise-linear activations. Sufficient conditions are derived for ascertaining the existence and stability of multiple equilibria. It is shown that the number of stable equilibria of an n-neuron switched neural networks is up to 3ⁿ under given conditions. In contrast to existing multistability results with piecewise-linear activation functions, the results herein are also applicable to the equilibria at switching points. Four examples are discussed to substantiate the theoretical results.