Effects of high-frequency cyclic input and noise on interspike intervals in the coupled Bonhoeffer-van der Pol (BVP) model are studied with computer simulation. When two BVP elements are weakly coupled and cyclic input or noise is added to the first element, the interspike intervals of the second element decrease non-monotonically as the amplitude of the input increases. Further, complicated bifurcations in the interspike intervals are caused by cyclic input in the coupled BVP model in the oscillating state. Effects of the coupling on small rotations due to noise and the interruption of oscillations due to cyclic input, which occur when the equilibrium point is close to the critical point, are also studied. The non-monotonic changes and bifurcations in the interspike intervals are attributed to the phase locking of the coupled elements.

We propose a minimal model of neuronal burst-firing that can be considered as a modification and extention of the Bonhoeffer-van der Pol (BVP) model. By using linear stability analysis we show that one of the equilibrium points of the fast subsystem is a saddle point which divides the phase plane into two regions. In one region all phase trajectories approach a limit cycle and in the other they approach a stable equilibrium point. The slow subsystem describes a slowly varying inward current. Various types of bursting phenomena are presented by using bifurcation analysis. The simplicity of the model and the variety of firing modes are the biggest advantages of our model with obvious applications in understanding underlying mechanisms of generation of neuronal firings and modeling oscillatory neural networks.

Responses in the Nagumo neural circuit to pulse-train stimulation are studied using the time sequence, phase diagram, Poincare section, return map, firing rate, Lyapunov number and bifurcation diagram. For the mono-stable neuron with an equilibrium point deeper than the maximal point of a tunnel diode curve, main responses are periodic or all-or-none and chaotic responses are rarely observed. For the neuron with an equilibrium point located near the maximal point, the response to one input pulse oscillates after the undershoot and responses to pulse-trains make complex bifurcation structure in the threshold diagram. The ranges of periodic responses are stratified in the diagram. There exist broad regions of chaotic responses and chaos is not a special response of the Nagumo circuit, but it often comes out. The results are different from those obtained from Hodgkin-Huxley equations and the BVP model.