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Center for Nonlinear Studies |
The reliability and predictability of neuronal network dynamics is a central question in neuroscience. We present a numerical analysis of the dynamics of all-to-all pulsed-coupled Hodgkin-Huxley (HH) neuronal networks. Since this is a non-smooth dynamical system, we propose a pseudo-Lyapunov exponent (PLE) that captures the long-time predictability of HH neuronal networks. The PLE can capture very well the dynamical regimes of the network. Furthermore, we present an efficient multiscale library-based numerical method for simulating HH neuronal networks. Our pre-computed high resolution data library can allow us to avoid resolving the spikes in detail and to use large numerical time steps for evolving the HH neuron equations. By using the library-based method, we can evolve the HH networks using time steps one order of magnitude larger than the typical time steps used for resolving the trajectories without the library, while achieving comparable resolution in statistical quantifications of the network activity. Moreover, our large time steps using the library method can overcome the stability requirement of standard ODE methods for the original dynamics. In the time permits, we will present a multiscale method for epitaxial growth, where we couple kinetic Monte-Carlo (KMC) simulations on the microscale with the island dynamics model based on the level set method and a diffusion equation on the macroscale. Numerical results for the step edge evolutions show substantially improved efficiency over pure KMC simulations.
Host: Shengtai Li, T-5