Dynamical behaviors for discontinuous and delayed neural networks in the framework of Filippov differential inclusions.

This paper is concerned with the periodic dynamics of a class of delayed neural networks with discontinuous neural activation functions. Under the Filippov framework, the cone expansion and compression fixed point theorems of set-valued maps are successfully employed to derive the existence of the ω-periodic positive solution. However, before the discussion of the periodicity, there still remains a fundamental issue about viability to be solved due to the presence of general mixed time-delays involving both time-varying delays and distributed delays. This difficulty can be overcome by a transformation and the continuation theorem. Then, for the discontinuous and delayed neural network system with time-periodic coefficients, the uniqueness and global exponential stability of the periodic state solution are proved by using non-smooth analysis theory with generalized Lyapunov approach. Furthermore, the global convergence in measure of the periodic output is also investigated. The obtained results are a very good extension and improvement of previous works on discontinuous dynamical neuron systems with a broad range of neuron activations dropping the assumption of boundedness or monotonicity. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.