Pattern formation in the Fitzhugh–Nagumo neuron with diffusion relaxation

We examine the spatiotemporal dynamics of the Fitzhugh-nagumo neuron taking into account the effects of relaxation induced by finite speeds of propagation. Stability analysis indicates the presence of Hopf bifurcations induced by relaxation as well as Pitchfork bifurcations due to by diffusion, and independent of the relaxation time. Analysis of the dispersion relation of the oscillatory waves demonstrates that the system, unlike the classical models, allows for finite speeds of propagation for non-negligible values of the relaxation time. Using the center manifold theorem, we reduce the system to its normal form representation both in the strong and weakly coupled limits. From the restricted dynamics, the direction of the Hopf bifurcation is computed, and the collective dynamics inferred. Numerical simulations of the nonlinear wave states of the system show that the uniform oscillatory state is stable against long wave perturbations, indicating full synchronization. The current model might be suitable to describe the dynamics of intracortical neurons, where lack of myelination leads to lower propagation velocities and ultimately larger delays.