Multi-mode attractors and spatio-temporal canards

Abstract In this article, we report the numerical discovery of multi-mode attractors for reaction–diffusion systems in which the kinetics feature slow/fast dynamics. Multi-mode attractors (MMAs) are a class of attractors in which different regions of the spatial domain exhibit different modes of (temporal) oscillation. These modes include spiking modes, bursting modes of many different types with s small-amplitude oscillations at the end of each burst event, as well as alternating modes in which various sequences of spiking and bursting are exhibited in alternation. We present the numerical discovery of MMAs in the context of a spatially-extended pituitary cell model with diffusive coupling and a spatially inhomogeneous applied current. We demonstrate that the MMAs are robust, occurring on large open parameter sets and for a variety of biophysically-relevant spatially-inhomogeneous currents, including Gaussian and mollified step profiles. Also, we provide evidence that the MMAs exhibit new types of maximal spatio-temporal canards. These lie in the transition intervals between adjacent regions in which the MMA exhibits distinct modes of oscillation, and they are necessary for the smooth and gradual transitions between bursting and spiking, as well as between bursting modes with different numbers of small oscillations. Furthermore, we study how the structures of the MMAs change as the amplitude of the diffusivity decreases and the PDE model limits on a family of uncoupled ODEs, one for each point in the domain. Also, we show that the MMAs, which are spatially non-uniform, can coexist in the reaction–diffusion system with other types of attractors which are spatially-uniform. Finally, we report that the MMAs discovered here are also present in numerical simulations of other reaction–diffusion systems, especially those that arise in neural and cardiac models.