American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles
  • DCDS Home
  • This Issue
  • Next Article
    Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations
December  2010, 28(4): 1555-1588. doi: 10.3934/dcds.2010.28.1555

Stable and unstable periodic orbits in complex networks of spiking neurons with delays

Raoul-Martin Memmesheimer 1, and  Marc Timme 2,

1. 

Center for Brain Science, Faculty of Arts and Sciences, Harvard University, Cambridge, MA, United States
2. 

Network Dynamics Group, Max Planck Institute for Dynamics & Self-Organization (MPIDS), Göttingen, Bernstein Center for Computational Neuroscience (BCCN) Göttingen, Department of Physics, Georg August University, Göttingen, Germany

Received  October 2009 Revised  February 2010 Published  June 2010

Is a periodic orbit underlying a periodic pattern of spikes in a heterogeneous neural network stable or unstable? We analytically assess this question in neural networks with delayed interactions by explicitly studying the microscopic time evolution of perturbations. We show that in purely inhibitorily coupled networks of neurons with normal dissipation (concave rise function), such as common leaky integrate-and-fire neurons, all orbits underlying non-degenerate periodic spike patterns are stable. In purely inhibitorily coupled networks with strongly connected topology and normal dissipation (strictly concave rise function), they are even asymptotically stable. In contrast, for the same type of individual neurons, all orbits underlying such patterns are unstable if the coupling is excitatory. For networks of neurons with anomalous dissipation ((strictly) convex rise function), the reverse statements hold. For the stable dynamics, we give an analytical lower bound on the local size of the basin of attraction. Numerical simulations of networks with different integrate-and-fire type neurons illustrate our results.
Keywords: synchronization, Spiking neural network, hybrid dynamical system, stability, periodic orbit, local cortical circuits, attractor neural networks..
Mathematics Subject Classification: 92B20, 94C99, 34B37, 34C2.
Citation: Raoul-Martin Memmesheimer, Marc Timme. Stable and unstable periodic orbits in complex networks of spiking neurons with delays. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1555-1588. doi: 10.3934/dcds.2010.28.1555
[1]

Hiroaki Uchida, Yuya Oishi, Toshimichi Saito. A simple digital spiking neural network: Synchronization and spike-train approximation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020374
[2]

Jianfeng Feng, Mariya Shcherbina, Brunello Tirozzi. Stability of the dynamics of an asymmetric neural network. Communications on Pure & Applied Analysis, 2009, 8 (2) : 655-671. doi: 10.3934/cpaa.2009.8.655
[3]

Karim El Laithy, Martin Bogdan. Synaptic energy drives the information processing mechanisms in spiking neural networks. Mathematical Biosciences & Engineering, 2014, 11 (2) : 233-256. doi: 10.3934/mbe.2014.11.233
[4]

Pierre Guiraud, Etienne Tanré. Stability of synchronization under stochastic perturbations in leaky integrate and fire neural networks of finite size. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5183-5201. doi: 10.3934/dcdsb.2019056
[5]

Ying Sue Huang, Chai Wah Wu. Stability of cellular neural network with small delays. Conference Publications, 2005, 2005 (Special) : 420-426. doi: 10.3934/proc.2005.2005.420
[6]

Ndolane Sene. Fractional input stability and its application to neural network. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 853-865. doi: 10.3934/dcdss.2020049
[7]

Tingting Su, Xinsong Yang. Finite-time synchronization of competitive neural networks with mixed delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3655-3667. doi: 10.3934/dcdsb.2016115
[8]

Benedetta Lisena. Average criteria for periodic neural networks with delay. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 761-773. doi: 10.3934/dcdsb.2014.19.761
[9]

Ricai Luo, Honglei Xu, Wu-Sheng Wang, Jie Sun, Wei Xu. A weak condition for global stability of delayed neural networks. Journal of Industrial & Management Optimization, 2016, 12 (2) : 505-514. doi: 10.3934/jimo.2016.12.505
[10]

Leong-Kwan Li, Sally Shao, K. F. Cedric Yiu. Nonlinear dynamical system modeling via recurrent neural networks and a weighted state space search algorithm. Journal of Industrial & Management Optimization, 2011, 7 (2) : 385-400. doi: 10.3934/jimo.2011.7.385
[11]

Zhuwei Qin, Fuxun Yu, Chenchen Liu, Xiang Chen. How convolutional neural networks see the world --- A survey of convolutional neural network visualization methods. Mathematical Foundations of Computing, 2018, 1 (2) : 149-180. doi: 10.3934/mfc.2018008
[12]

K. L. Mak, J. G. Peng, Z. B. Xu, K. F. C. Yiu. A novel neural network for associative memory via dynamical systems. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 573-590. doi: 10.3934/dcdsb.2006.6.573
[13]

Ying Sue Huang. Resynchronization of delayed neural networks. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 397-401. doi: 10.3934/dcds.2001.7.397
[14]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367
[15]

Yong Zhao, Qishao Lu. Periodic oscillations in a class of fuzzy neural networks under impulsive control. Conference Publications, 2011, 2011 (Special) : 1457-1466. doi: 10.3934/proc.2011.2011.1457
[16]

Sanjay K. Mazumdar, Cheng-Chew Lim. A neural network based anti-skid brake system. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 321-338. doi: 10.3934/dcds.1999.5.321
[17]

Sylvia Novo, Rafael Obaya, Ana M. Sanz. Exponential stability in non-autonomous delayed equations with applications to neural networks. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 517-536. doi: 10.3934/dcds.2007.18.517
[18]

Ivanka Stamova, Gani Stamov. On the stability of sets for reaction–diffusion Cohen–Grossberg delayed neural networks. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020370
[19]

Ruoxia Li, Huaiqin Wu, Xiaowei Zhang, Rong Yao. Adaptive projective synchronization of memristive neural networks with time-varying delays and stochastic perturbation. Mathematical Control & Related Fields, 2015, 5 (4) : 827-844. doi: 10.3934/mcrf.2015.5.827
[20]

Jianping Zhou, Yamin Liu, Ju H. Park, Qingkai Kong, Zhen Wang. Fault-tolerant anti-synchronization control for chaotic switched neural networks with time delay and reaction diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020357

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences