Oscillatory Dynamics for a Coupled Membrane-Bulk Diffusion Model with Fitzhugh--Nagumo Membrane Kinetics

Source abstract

Oscillatory dynamics associated with the coupled membrane-bulk PDE-ODE model of Gomez-Marin, Garcia-Ojalvon, and Sancho [Phys. Rev. Lett., 98 (2007), 168303] in one spatial dimension is analyzed using a combination of asymptotic analysis, linear stability theory, and numerical bifurcation software. The mathematical model consists of two dynamically active membranes with Fitzhugh--Nagumo kinetics, separated spatially by a distance $L$, that are coupled together through a diffusion field that occupies the bulk region $0<x<L$. The flux of the diffusion field on the membranes at $x=0$ and $x=L$ provides feedback to the local dynamics on the membranes. In the absence of membrane-bulk coupling the membrane kinetics have a stable fixed point. The effect of bulk diffusion is to trigger either synchronous and asynchronous oscillations in the two membranes. In the singular limit of slow-fast membrane dynamics, and with only one diffusing species in the bulk, phase diagrams in parameter space showing where either synchronous or asynchronous oscillations occur, together with the corresponding Hopf frequencies at onset, are provided analytically. When the membrane kinetics is not of slow-fast type, a numerical study of the stability problem together with the numerical bifurcation software XPPAUT [G. B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, SIAM, Philadelphia, 2002] is used to to construct global bifurcation diagrams of steady-states and the bifurcating periodic solution branches for the case of either one or two diffusing species in the bulk. Overall, our results show the existence of a wide parameter range where stable synchronous oscillatory dynamics in the two membranes can occur. Predictions from the analytical and bifurcation theory are confirmed with full numerical simulations of the PDE-ODE system.