Chaotic dynamics of an integrate-and-fire circuit with periodic pulse-train input

In this brief, we introduce the working principle of a pacemaker neuron type integrate-and-fire circuit having two states with a periodic pulse-train input, first proposed by Nakano et al. (1999). The dynamics of this circuit can be described by a standard impulsive differential equation and applied to simulate the evolution of the pulse-coupled neural networks. By applying the Marotto theorem (1978), we theoretically prove that the circuit becomes chaotic as the parameters enter some regions. We find that the circuit does not exhibit chaotic dynamics with a small period of the pulse-train input but that a chaotic phenomenon appears with increase of the period. The relations between this circuit and that of symbolic dynamics are further investigated. Numerical simulations and corresponding calculation, as illustrative examples, reinforce our theoretical proof and theory.