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Title: Dynamical stability and fast algorithms of neuronal networks
周栋焯(Douglas Zhou)教授
Address: Institute of Natural Sciences & School of Mathematical Sciences, Shanghai Jiao Tong University

It has been shown that a single integrate-and-fire (I&F) neuron under a general time-dependent stimulus cannot possess chaotic dynamics despite the firing-reset discontinuity. However, whether the dynamics of I&F neuronal networks can be chaotic was an open question. Through correct renormalization and augmented dynamics, we extend the classical Lyapunov exponents (LEs) theory, which is established for smooth dynamical systems, to the I&F like network dynamics and provide a stable and accurate numerical algorithm to compute the LEs of these non-smooth dynamical systems. Inspired by the low computational cost of I&F models, we further present an efficient library-based numerical method for simulating the Hodgkin–Huxley (HH) neuronal networks. Numerical simulations show that our library-based HH model can well capture the dynamical regimes, which are characterized by LEs, of the original HH model. In addition, our model can break the numerical stability requirement of Runge-Kutta methods for the original HH model, thus leading to much higher computational efficiency.