Abstract
The main objective of this article is to study the dynamic transitions of the FitzHugh-Nagumo equations on a finite domain with the Neumann boundary conditions and with uniformly injected current. We show that when certain parameter conditions are satisfied, the system undergoes a continuous dynamic transition to a limit cycle. A mixed type transition is also found when other conditions are imposed on the parameters. The main method used here is Ma & Wang's dynamic transition theory, which can be used generally on different set-ups for the FitzHugh-Nagumo equations.
Highlights
Propagated signaling in axons is an important phenomenon in neural science, various models have been put forward to explain this behavior, the Hodgkin-Huxley (HH) equations [8] is a satisfactory and experimentally supported model
The complex transition shows that when γ is fixed, whenever is small enough, the system will undergo a continuous type dynamic transition from the basic state to a spatio-temporal oscillating pattern (Hopf bifurcation). This corresponds to the behavior of repetitive firing of neurons when there is a large enough persistent stimulating current
Though in experiment this repetitive firing is damped after several spikes [5], it was thought this damping was due to poor condition of the axon
Summary
Propagated signaling in axons is an important phenomenon in neural science, various models have been put forward to explain this behavior, the Hodgkin-Huxley (HH) equations [8] is a satisfactory and experimentally supported model. This corresponds to the case when current I is injected uniformly into an axon with length L, both ends of which are sealed (no current going through) Equations of this form allow for simpler mathematical manipulations, and we will consider this setup in current paper. This represents a more realistic case, exciting current I is injected at one end of an axon, while the other end being sealed (forming synapses).
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