Delay-induced spiking dynamics in integrate-and-fire neurons

Experiments showed that a neuron can fire when its membrane po tential (an intrinsic quality related to its membrane electrical charge) reaches a specific threshold. On theoretical studies, there are two crucial issues in ex- ploring cortical neuronal dynamics: (i) what model describes spiking dynamics of each neuron, and (ii) how the neurons are connected [E. M. Izhikevich, IEEE Trans. Neural Networks, 15 (2004)]. To study the first issue, we propose the time delay effect on the well-known integrate-and-fire (IF) model which is clas- sically introduced to study the spiking behaviors in neural systems by using the spike-and-reset procedure. Under the consideration of delayed adaptation on the membrane potential, the parameter range for the IF model with spiking dynamics becomes wider due to undergoing subcritical Hopf bifurcation and the existence of an unstable orbit. To study the second issue, we consider the system with two coupled identical IF units where time delay takes place in the coupling structure. We also demonstrate spiking behaviors in the coupled system when the delay time is large enough, and it contributes an original viewpoint of the connection between neurons. In contrast with the emergence of delay-induced spiking in a single-neuron system, a coupled two-neuron sys- tem involve both emergence and death of spiking according to different values of delay times. We also discuss the ranges of different parameters in which it allows occurrence of spiking behaviors.