14 April 2015 – Arik Yochelis (BGU)

Title:  Origin of finite pulse trains ​in ​reaction-diffusion PDEs: Homoclinic snaking in excitable media

Abstract:  Spatially localized and stationary states have recently brought to the spotlight of mathematical analysis of nonlinear PDEs via analysis of models exhibiting free energy and/or conservation. However, many chemical and biological systems exhibit rather localized traveling pulses, such as action potentials in axons and cardiac muscles. We identify and describe a qualitative novel property of excitable media that enables us to generate a sequence of traveling pulses of any desired length, using a one-time initial stimulus. The existence of these states is related here to the presence of homoclinic snaking in the vicinity of a subcritical, finite wavenumber Hopf bifurcation. The pulses are organized in a slanted snaking structure resulting from the presence of a heteroclinic cycle between small and large amplitude traveling waves. Connections of this type require a multivalued dispersion relation. This dispersion relation is computed numerically and used to interpret the profile of the pulse group. The different spatially localized pulse trains can be accessed by appropriately customised initial stimuli thereby blurring the traditional distinction between oscillatory and excitable systems. The results reveal a new class of phenomena relevant to spatiotemporal dynamics of excitable media, particularly in chemical and biological systems with multiple activators and inhibitors.