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Bump attractors and waves in networks of leaky integrate-and-fire neurons

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ADIW02 - Mathematical and Computational Modelling of Anti-Diffusive Phenomena

Coauthors: Daniele Avitabile (VU Amsterdam), Joshua L. Davis (DSTL) Bump attractors are wandering localised patterns observed in in vivo experiments of spatially-extended neurobiological networks. They are important for the brain’s navigational system and speci c memory tasks. A bump attractor is characterised by a core in which neurons  re frequently, while those away from the core do not  re. These structures have been found in simulations of spiking neural networks, but we do not yet have a mathematical understanding of their existence, because a rigorous analysis of the nonsmooth networks that support them is challenging. We uncover a relationship between bump attractors and travelling waves in a classical network of excitable, leaky integrate-and- re neurons. This relationship bears strong similarities to the one between complex spatiotemporal patterns and waves at the onset of pipe turbulence. Waves in the spiking network are determined by a  ring set, that is, the collection of times at which neurons reach a threshold and  re as the wave propagates. We de ne and study analytical properties of the voltage mapping, an operator transforming a solution’s ring set into its spatiotemporal pro le. This operator allows us to construct localised travelling waves with an arbitrary number of spikes at the core, and to study their linear stability. A homogeneous \laminar” state exists in the network, and it is linearly stable for all values of the principal control parameter. Su ciently wide disturbances to the homogeneous state elicit the bump attractor. We show that one can construct waves with a seemingly arbitrary number of spikes at the core; the higher the number of spikes, the slower the wave, and the more its pro le resembles a stationary bump. As in the  uid-dynamical analogy, such waves coexist with the homogeneous state, are unstable, and the solution branches to which they belong are disconnected from the laminar state; we provide evidence that the dynamics of the bump attractor displays echoes of the unstable waves, which form its building blocks.

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