Analytical dynamics of nerve impulse oscillation

A space clamped neural axon along which flows a constant current exhibits current dependent oscillations of the action potential. With increasing current a characteristic Hopf bifurcation scenario can be observed: approximately harmonic small amplitude oscillations arise and bifurcate subsequently into relaxation oscillations of relatively large amplitude. Amplitude and period of the oscillations depend on critical relationships between the magnitude of current and the control parameters of the system. These behaviors are encapsulated by the FitzHugh-Nagumo equations, which were developed as a reduction and simplification of the Hodgkin-Huxley equations. It is shown that the FitzHugh-Nagumo equations can be reduced in close approximation to a van der Pol-like oscillator externally driven by the current that functions as an external constant force. Generalizing previous studies on the van der Pol system a weighted step model for neuron response is developed. This model allows for analytic solutions which predict, in terms of the FitzHugh-Nagumo equation parameters, the critical values of the current at which the bifurcations occur as well as the periods of the oscillations as a function of current input. The analytic simplicity of the solutions suggests an applicability of the approach to network dynamics of coupled neurons.