Period-adding bifurcation of neural firings induced by inhibitory autapses with time-delay

Neural firing rhythm plays an important role in achieving the function of a nervous system. Neurons with autapse, which starts and ends in the same cell, are widespread in the nervous system. Previous results of both experimental and theoretical studies have shown that autaptic connection plays a role in influencing dynamics of neural firing patterns and has a significant physiological function. In the present study, the dynamics of a neuronal model, i.e., Rulkov model with inhibitory autapse and time delay, is investigated, and compared with the dynamics of neurons without autapse. The bifurcations with respect to time-delay and the coupling strength are extensively studied, and the time series of membrane potentials is also calculated to confirm the bifurcation analysis. It can be found that with the increase of time-delay and/or the coupling strength, the period-adding bifurcation of neural firing patterns can be induced in the Rulkov neuron model. With the increase of the period number of the firing rhythm, the average firing frequency increases. When time-delay and/or coupling strength are/is greater than their/its corresponding certain thresholds/threshold, the average firing frequency is higher than that of the neuron without autapse. Furthermore, new bursting patterns, which appear at suitable time delays and coupling strengths, can be well interpreted with the dynamic responses of an isolated single neuron to a negative square current whose action time, duration, and strength are similar to those of the inhibitory coupling current modulated by the coupling strength and time delay. The bursts of neurons with autapse show the same pattern as the square negative current-induced burst of the isolated single neuron when the time delay corresponds to the phase. The bifurcation structure of the neural firing rhythm of the neuron without autapse can be obtained with the fast-slow dissection method. The dynamic responses of the isolated bursting neuron to the negative square current are acquired by using the fast-slow variable dissection method, which can help to recognize the new rhythms induced by the external negative pulse current applied at different phases. The new rhythm patterns are consistent with those lying in the period-adding bifurcations. The results not only reveal that the inhibitory autapse can induce typical nonlinear phenomena such as the period-adding bifurcations, but also provide the new phenomenon that the inhibitory autapse can enhance the firing frequency, which is different from previous viewpoint that inhibitory effect often reduces the firing frequency. These findings further enrich the understanding of the nonlinear phenomena induced by inhibitory autapse.