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Barbara Zubik-Kowal, "Brain dynamics systems and their numerical solutions."" Short abstract: In this talk we investigate thalamo-cortical systems, which describe brain dynamics. The model is formulated as a system of nonlinear Volterra integro-differential equations. A new algorithm is proposed for numerical solutions of the systems. The idea of the new algorithm is based on the properties of the kernels that are applied to the systems. A moderate value t_0 > 0 is determined according to these properties and the interval [0,T] of the time integration of the systems is divided into two parts, [0,t_0] and [t_0,T]. Arbitrarily large values of T are investigated with T significantly larger than t_0. Classical methods are applied to solve the systems in the first short subinterval [0,t_0]. Then, the new algorithm is applied on the remaining arbitrarily long subinterval [t_0,T]. The new algorithm brings three important advantages: (1) the thalamo-cortical systems can be efficiently integrated in a parallel computing environment, (2) the parallel algorithm converges for arbitrarily long time intervals, and (3) its convergence is fast. We introduce new theorems, which prove its rapid convergence on arbitrarily long time intervals. The theoretical results are conrmed by numerical experiments. |