A Three-Scale Model of Spatio-Temporal Bursting

Abstract

Abstract. We study spatio-temporal bursting in a three-scale reaction diffusion equation organized by the winged cusp singularity. For large time-scale separation the model exhibits traveling bursts, whereas for large space-scale separation the model exhibits standing bursts. Both behaviors exhibit a common singular skeleton, whose geometry is fully determined by persistent bifurcation diagrams of the winged cusp. The modulation of spatio-temporal bursting in such a model naturally translates into paths in the universal unfolding of the winged-cusp.Key words. spatio-temporal bursting, reaction-diffusion, singularity theory, geometric singular perturbation theory AMS subject classifications. 37N25,37G99,34E13,35K57,35B361. Introduction. In the recent paper [12], we proposed that temporal bursting can be conveniently studied in a three-time-scale model with a single scalar winged-cusp nonlinearity, a basic organizing center in the language of singularity theory. Because of its particular structure, the model is readily interpreted as a three-scale generalization of the celebrated FitzHugh-Nagumo model, a two-time-scale model similarly organized by a single scalar hysteresis nonlinearity.With enough time-scale separation, the (temporal) patterns exhibited by the trajectories of those models are robust and modulable in the parameter space, because they are in oneto-one correspondence with the persistent bifurcation diagrams of the universal unfolding of the hysteresis and winged-cusp, respectively.We showed that this feature is of prime interest in neurophysiology, for instance, because it provides insight in the physiological parameters that can regulate the continuous transition between distinct temporal patterns as observed in experimental neuronal recordings.The present paper pursues this investigation by adding diffusion in the three-scale temporal model and exploring the resulting three-scale spatio-temporal patterns: traveling bursts and standing bursts.The close relationship of the model with its two-dimensional counterpart is of great help in the analysis because we extensively rely on the existing theory for two-scale models with a hysteresis nonlinearity. We study the existence of traveling bursts and standing bursts in our three-scale model by mimicking the analysis of traveling pulses and standing pulses in earlier two-scale models. This geometric analysis exploits the singular limit to construct the skeleton

Feedback design of spatially-distributed filters with tunable resolution

“…The black trace is the filter response a(x) scaled by s −5/2 . Despite perturbations becoming evident at small scales, the scaled filter response robustly converge to the signal's second derivative as predicted by wavelet-zoom theory (15).…”

“…Biological neural dynamics are intrinsically nonlinear, both in time and space. The role of nonlinearities is to transform linear instability into multi-scale excitability and pattern formation [15], [16]. These all-or-none responses robustify information processing by providing neuronal communication with a mixed analog/digital nature [17], [18].…”