Origin of finite pulse trains: Homoclinic snaking in excitable media

Abstract

Many physical, chemical, and biological systems exhibit traveling waves as a result of either an oscillatory instability or excitability. In the latter case a large multiplicity of stable spatially localized wavetrains consisting of different numbers of traveling pulses may be present. 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 customized 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.

Catalytic Membrane Reactor Model as a Laboratory for Pattern Emergence in Reaction‐diffusion‐advection Media

“…Similarly to RD systems [63,67,72], and as also indicated by direct numerical integration, the sub-critical regime Da < Da − W also supports a multiplicity of secondary solutions with λ = λ ± W . While we portray only two of such secondary branches [Fig.…”

“…Similarly to RD systems,,, and as also indicated by direct numerical integration, the sub‐critical regime also supports a multiplicity of secondary solutions with . While we portray only two of such secondary branches [Figure (a)], there are infinitely many solutions, each of which corresponds to a different period.…”

“…Spatial dynamics is a powerful methodology that allows exploiting the tools developed for ordinary differential equations (ODE) for analysis of nonuniform states in the spatially extended contexts to reveal the coexisting solutions in the parameter space of the problem; the stability properties of these solutions are obtained in the next stage by solving a temporal eigenvalue problem. This theoretical approach was shown to be useful for uncovering a number of complex nonlinear mechanisms, such as pattern formation in the presence of homoclinic snaking, both in dissipative [62][63][64][65][66][67] and variational [68,69] model equations.…”

Why a Large-Scale Mode Can Be Essential for Understanding Intracellular Actin Waves

“…On the other hand, bifurcation analysis can also be a tedious task as there may be many local and global bifurcations that coexist in a given parameter range (as an example we refer the reader to a systematic extension of excitable media by Champneys et al [ 143 ]). Nevertheless, utilizing recent advances in nonlinear perturbations [ 83 , 144 ] and numerical path continuation methods [ 145 , 146 , 147 , 148 , 149 ] it might be possible to navigate between coexisting bifurcations and a multiplicity of emerging stable and unstable solutions [ 144 , 150 ]. Next, we turn to conservation and ask whether it may prescribe a fundamental and robust qualitative change, as compared to typical local RD modeling in the absence of conserved quantities.…”

Origin of Jumping Oscillons in an Excitable Reaction-Diffusion System

“…The system is a three-variable FHN system with one activator and two inhibitors acting on distinct timescales, and has broad applicability in studies of dissipative solitons in excitable RD systems in both one (1D) and two (2D) spatial dimensions (pulse interactions in 1D [33][34][35][36][37] and time-dependent spots in 2D [31,[38][39][40]). The closely related models studied in [41][42][43][44][45][46] exhibit similarly rich dynamics.…”