Synchronization control for reaction-diffusion FitzHugh-Nagumo systems with spatial sampled-data

Abstract To gain a better understanding the synchronization mechanism of networked neurons, this paper studies the synchronization control for a class of reaction–diffusion FitzHugh–Nagumo systems, associated with a digraph containing at least one directed spanning tree. A novel control method adopting spatial sampling strategies is proposed, in which the control inputs are constructed directly on the spatial means of system state variables on sampling subsets. After discussing the existence and uniqueness of classical solutions, we show analytically that the synchronization of the controlled systems is equivalent to that of the corresponding FitzHugh–Nagumo systems under the corresponding control inputs. Based on the study on general algebraic connectivity, a sufficient condition for the system synchronization is given, together with case simulations to illustrate the effectiveness and potential of the new control method.

[1]  Quanxin Zhu,et al.  Synchronization of reaction–diffusion neural networks with time-varying delays via stochastic sampled-data controller , 2014, Nonlinear Dynamics.

[2]  Kexue Zhang,et al.  Pinning Impulsive Synchronization of Reaction–Diffusion Neural Networks With Time-Varying Delays , 2017, IEEE Transactions on Neural Networks and Learning Systems.

[3]  Qunjiao Zhang,et al.  Robust synchronization of FitzHugh–Nagumo network with parameter disturbances by sliding mode control , 2014 .

[4]  Keum Shik Hong,et al.  Synchronization of Coupled Different Chaotic FitzHugh-Nagumo Neurons with Unknown Parameters under Communication-Direction-Dependent Coupling , 2014, Comput. Math. Methods Medicine.

[5]  Jeff Morgan.  Global existence for semilinear parabolic systems , 1989 .

[6]  Shuichi Jimbo,et al.  Spectral comparison and gradient-like property in the FitzHugh?Nagumo type equations , 2015 .

[7]  Guanghui Wen,et al.  Consensus in multi‐agent systems with communication constraints , 2012 .

[8]  W. Strauss,et al.  Partial Differential Equations: An Introduction , 1992 .

[9]  Tingwen Huang,et al.  Passivity-based synchronization of a class of complex dynamical networks with time-varying delay , 2015, Autom..

[10]  Qing-Long Han,et al.  A distributed event-triggered transmission strategy for sampled-data consensus of multi-agent systems , 2014, Autom..

[11]  M. A. Aziz-Alaoui,et al.  Synchronization and control of a network of coupled Reaction-Diffusion systems of generalized FitzHugh-Nagumo type , 2013 .

[12]  H. Weinberger,et al.  An optimal Poincaré inequality for convex domains , 1960 .

[13]  Wenwu Yu,et al.  Second-Order Consensus for Multiagent Systems With Directed Topologies and Nonlinear Dynamics , 2010, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[14]  Lihua Xie,et al.  Sampled-data based average consensus with logarithmic quantizers , 2012, Proceedings of the 31st Chinese Control Conference.

[15]  Feng Zhilin,et al.  STATE FEEDBACK CONTROL FOR UNCERTAIN NETWORKED CONTROL SYSTEMS WITH DISTURBANCES , 2017 .

[16]  Wenwu Yu,et al.  Consensus in Directed Networks of Agents With Nonlinear Dynamics , 2011, IEEE Transactions on Automatic Control.

[17]  Wei Xing Zheng,et al.  Impulsive Synchronization of Reaction–Diffusion Neural Networks With Mixed Delays and Its Application to Image Encryption , 2016, IEEE Transactions on Neural Networks and Learning Systems.

[18]  S. Yoshizawa,et al.  An Active Pulse Transmission Line Simulating Nerve Axon , 1962, Proceedings of the IRE.

[19]  Richard A. Brualdi,et al.  Combinatorial matrix theory , 1991 .

[20]  Tingwen Huang,et al.  Passivity and Synchronization of Linearly Coupled Reaction-Diffusion Neural Networks With Adaptive Coupling , 2015, IEEE Transactions on Cybernetics.

[21]  M. A. Aziz-Alaoui,et al.  Synchronization and control of coupled reaction-diffusion systems of the FitzHugh-Nagumo type , 2012, Comput. Math. Appl..

[22]  Romain Brette,et al.  Computing with Neural Synchrony , 2012, PLoS Comput. Biol..

[23]  Zhidong Teng,et al.  Exponential synchronization for reaction-diffusion networks with mixed delays in terms of p-norm via intermittent driving , 2012, Neural Networks.

[24]  Lei Guo,et al.  Novel Adaptive Strategies for Synchronization of Linearly Coupled Neural Networks With Reaction-Diffusion Terms , 2014, IEEE Transactions on Neural Networks and Learning Systems.

[25]  Henry C. Tuckwell,et al.  Analytical and Simulation Results for Stochastic Fitzhugh-Nagumo Neurons and Neural Networks , 1998, Journal of Computational Neuroscience.

[26]  Bao-quan Ai,et al.  Interaction of multiarmed spirals in bistable media. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  Hongye Su,et al.  Distributed consensus of multi-agent systems with nonlinear dynamics via adaptive intermittent control , 2015, J. Frankl. Inst..

[28]  Huai-Ning Wu,et al.  Pinning Control Strategies for Synchronization of Linearly Coupled Neural Networks With Reaction–Diffusion Terms , 2016, IEEE Transactions on Neural Networks and Learning Systems.

[29]  D. Hoff,et al.  LARGE TIME BEHAVIOR OF SOLUTIONS OF SYSTEMS OF NONLINEAR REACTION-DIFFUSION EQUATIONS* , 1978 .

[30]  Peng Shi,et al.  Sampled-Data Synchronization of Complex Networks With Partial Couplings and T–S Fuzzy Nodes , 2018, IEEE Transactions on Fuzzy Systems.

[31]  Henk Nijmeijer,et al.  Network synchronization using invariant-manifold-based diffusive dynamic couplings with time-delay , 2015, Autom..