類神經網路的數學模型與分析

We plan to introduce feedforward networks, recurrent networks, as well as Hopfield neural network, Cohen-Grossberg network, and cellular neural network. These networks are discrete-time or continuous time. One of the focusses is to discuss multistability of recurrent neural networks (RNN). Multistability indicates coexistence of stable multiple equilibria of the system. The applications of multistability include memory storage, image denoising, feature extraction, and character recognition. For example, in designing an RNN-based associative memory, the templates of many patterns to be stored are encoded as the stable equilibria of the RNN. We will also introduce transiently chaotic neural network which takes parameter values from the chaotic regime. Such networks can be used in the annealing process in solving combinatorial optimization problems. In addition to these networks, we also plan to introduce neuroscience and neuronal models, including Integrate-and-Fire models and the Hudgkin-Huxley model. Mathematical theories involved in these discussions include Fixed-Point Theorem (Mathematical Analysis), Transition Matrix (Linear Algebra), LaSalle's Invariance Principle (ODE), Monotone Dynamics Theory (ODE), Marotto Theorem (Chaos in Dynamical Systems)

Differential Equations

將使用E3教學平台:補充說明、補充教材、討論與回應將置於E3

作業 (數學、MATLAB數值計算) 50 % 報告: 期中報告 (20%)、期末報告 (30%)

Reference Books: Theoretical Neuroscience, Computational and Mathematical Modeling of Neural Systems, by P. Dayan and L. F. Abbott, MIT Press Neuronal Dynamics, From Single Neurons to Networks and Models of Cognition, by W. Gerstner, W. Kistler, R. Naud, L. Paninski, Cambridge Mathematical Foundations of Neuroscience, by G. B. Ermentrout, D. H. Terman, Springer Introduction to the Theory of Neural Computation, Volume I, by A. Hertz, A. Krogh, R. Palmer Reference Papers: Liu, Wang, Zeng, An overview of the stability Analysis of recurrent neural networks with multiple equilibria, IEEE Transaction on neural networks and learning systems, 2023 C.-Y. Cheng, K.-H. Lin, C.-W. Shih, J.-P. Tseng, Multistability for delayed neural networks via sequential contracting, IEEE Transactions on Neural Networks and Learning Systems, 2015. C.-W. Shih, J. P. Tseng, Convergent dynamics for multistable delayed neural networks, Nonlinearity, 2008 C.-Y. Cheng, K.-H. Lin, C.-W. Shih, Multistability and convergence in delayed neural networks, Physica D, 2007 C.-Y. Cheng, K.-H. Lin, C.-W. Shih, Multistability in recurrent neural networks, SIAM J. Applied Math., 2006 S.-S. Chen , C.-W. Shih, Transversal homoclinic orbits in a transiently chaotic neural network, Chaos, 2002