Neurons at rest can exhibit diverse firing activities patterns in response to various external deterministic and random stimuli, especially additional currents. In this paper, neuronal firing patterns from bursting to spiking, induced by additional direct and stochastic currents, are explored in rest states corresponding to two values of the parameter VK in the Chay neuron system. Three cases are considered by numerical simulation and fast/slow dynamic analysis, in which only the direct current or the stochastic current exists, or the direct and stochastic currents coexist. Meanwhile, several important bursting patterns in neuronal experiments, such as the period-1 ＂circle/homoclinic＂ bursting and the integer multiple ＂fold/homoclinic＂ bursting with onc spike per burst, as well as the transition from integer multiple bursting to period-1 ＂circle/homoclinic＂ bursting and that from stochastic ＂Hopf/homoclinic＂ bursting to ＂Hopf/homoclinic＂ bursting, are investigated in detail.
Based on actual neuronal firing activities, bursting in the Chay neuronal model is considered, in which VK, reversal potentials for K+, VC, reversal potentials for Ca2+, time kinetic constant λn and an additional depolarized current I are considered as dynamical parameters. According to the number of the Hopf bifurcation points on the upper branch of the bifurcation curve of fast subsystem, which is associated with the stable limit cycle corresponding to spiking states, different types of bursting and their respective dynamical behavior are surveyed by means of fast-slow dynamical bifurcation analysis.
