Chaotic phenomenon in both an integrate-and-fire circuit with periodic pulse-train input and a discrete neural network model

We first introduce the working principle of a pacemaker neuron type integrate-and-fire circuit having two states with a periodic pulse-train input, which is described by a standard impulsive differential equation and whose dynamics are always applied to simulate the evolution of the pulse-coupled neural networks. Furthermore, by applying the Marotto theorem, we theoretically prove that the circuit does not exhibit chaotic dynamics with some little period of the pulse-train input but it appears chaotic phenomenon with the increase of the period. The numerical simulations and corresponding calculation, as illustrative examples, reinforce our theoretical proof and theory. Also, we investigate the complex structure in a discrete neural network model, and we also theoretically prove that it exhibits chaotic phenomenon in the sense of Marotto