Global asymptotic stability of neural networks with discontinuous activations

Without assuming the boundedness and monotonicity of the neuron activations, we discuss the dynamics of a class of neural networks with discontinuous activation functions. The Leray–Schauder theorem of set-valued maps is successfully employed to derive the existence of an equilibrium point. A Lyapunov-like approach is applied to differential equations with discontinuous right-hand sides modeling the neural network dynamics, which yields conditions for global convergence or convergence in finite time. The obtained results extend previous works on global stability of neural networks with continuous neuron activations or discontinuous neuron activations.