Precise Dynamics of Spiking Neural Networks -- Unstable Attractors, Synchronization and Natural Computation
Dr. Marc Timme
Max Planck Institute for Dynamics and Selforganization, Goettingen, Germany


Precisely coordinated spatio-temporal spiking dynamics have been observed experimentally in different neuronal systems and are discussed to be an essential part of computation in the brain. Their dynamical origin, however, remains unknown. Here we study the dynamics of neural network models that reveal basic mechanisms underlying the neurons' precise temporal coordination. Special emphasis is given on delayed interactions, complicated network connectivity and computation. First, we present unstable attractors: invariant periodic orbits with a positive measure basin that are locally unstable. These occur in networks of neural oscillators due to delayed interactions. They are enclosed by basins of attraction of other attractors but are remote from their own basin volume such that arbitrarily small noise leads to a switching among attractors. These switching phenomena precisely coordinate spike timing. They may be useful for artificial neural network computation and operate in biological neural networks, such as the olfactory system, as well. Second, we elaborate on the exact dynamics of neural networks exhibiting a complicated topology. In such networks, an irregular, balanced state coexists with a synchronous state of regular activity. Using a random matrix approach, introduced by Wigner in the 1950s to characterize energy spectra of atomic nuclei, we predict the speed of synchronization in such networks in dependence of neuron and network properties. We find that the speed of synchronization is limited by the network connectivity and remains finite, even if the coupling strengths between neurons becomes infinite.

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