Traveling wave solutions of diffusive Hindmarsh–Rose-type equations with recurrent neural feedback

From the perspective of bifurcation theory, this study investigates the existence of traveling wave solutions for diffusive Hindmarsh–Rose-type (dHR-type) equations with recurrent neural feedback (RNF). The applied model comprises two additional terms: 1) a diffusion term for the conduction process of action potentials and 2) a delay term. The delay term is introduced because if a neuron excites a second neuron, the second neuron, in turn, excites or inhibits the first neuron. To probe the existence of traveling wave solutions, this study applies center manifold reduction and a normal form method, and the results demonstrate the existence of a heteroclinic orbit of a three-dimensional vector for dHR-type equations with RNF near a fold–Hopf bifurcation. Finally, numerical simulations are presented.