Oscillatory dynamics for a coupled membrane-bulk diffusion model with fitzhugh{nagumo membrane kinetics

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 di- mension 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 ux 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 oscil- lations 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, Ana- lyzing, 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 os- cillatory 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.