Bursting and spiking due to additional direct and stochastic currents in neuron models

Neurons at rest can exhibit diverse firing activitiespatterns in response to various external deterministic and randomstimuli, especially additional currents. In this paper, neuronalfiring patterns from bursting to spiking, induced by additionaldirect and stochastic currents, are explored in rest statescorresponding to two values of the parameter $V_{rm K}$ in the Chayneuron system. Three cases are considered by numerical simulationand fast/slow dynamic analysis, in which only the direct currentor the stochastic current exists, or the direct andstochastic currents coexist. Meanwhile, several important burstingpatterns in neuronal experiments, such as the period-1``circle/homoclinic" bursting and the integer multiple``fold/homoclinic" bursting with one spike per burst, as well asthe transition from integer multiple bursting to period-1``circle/homoclinic" bursting and that from stochastic``Hopf/homoclinic" bursting to ``Hopf/homoclinic" bursting, areinvestigated in detail.