##
**Self-excitation of neurons leads to multiperiodicity of discrete-time neural networks with distributed delays.**
*(English)*
Zbl 1227.93074

Summary: We investigate the interesting multiperiodicity of discrete-time neural networks with excitatory self-connections and distributed delays. Due to self-excitation of neurons, we construct \(2^N\) closed regions in state space for \(N\)-dimensional networks and attain the coexistence of \(2^N\) periodic sequence solutions in these closed regions. Meanwhile, we estimate exponential attracting domain for each periodic sequence solution and apply our results to discrete-time analogues of periodic or autonomous neural networks. Under self-excitation of neurons, numerical simulations are performed to illustrate the effectiveness of our results.

### MSC:

93C55 | Discrete-time control/observation systems |

37N35 | Dynamical systems in control |

92B20 | Neural networks for/in biological studies, artificial life and related topics |

PDF
BibTeX
XML
Cite

\textit{Z. Huang} et al., Sci. China, Inf. Sci. 54, No. 2, 305--317 (2011; Zbl 1227.93074)

### References:

[1] | Chua L O, Yang L. Cellular neural networks: theory. IEEE Trans Circ Syst, 1988, 35: 1257–1272 · Zbl 0663.94022 |

[2] | Chua L O, Yang L. Cellular neural networks: Application. IEEE Trans Circ Syst, 1988, 35: 1273–1290 |

[3] | Brucoli M, Carnimeo L, Grassi G. Discrete-time cellular neural networks for associative memories with learning and forgetting capabilities. IEEE Trans Circ Syst I, 1995, 42: 396–399 · Zbl 0846.68085 |

[4] | Wang L, Zou X. Capacity of stable periodic solutions in discrete-time bidirectional associative memory neural networks. IEEE Trans Circ Syst II, 2004, 51: 315–319 |

[5] | Mohamad S, Gopalsamy K. Dynamics of a class of discrete-time neural networks and their continuous-time counterparts. Math Comput Simul, 2000, 53: 1–39 |

[6] | Mohamad S, Gopalsamy K. Exponential stability of continuous-time and discrete-time cellular neural networks with discrete delays. Appl Math Comput, 2003, 135: 17–38 · Zbl 1030.34072 |

[7] | Mohamad S. Exponential stability preservation in discrete-time analogues of artificial neural networks with distributed delays. J Comput Appl Math, 2008, 215: 270–287 · Zbl 1154.39011 |

[8] | Mohamad S, Gopalsamy K. Exponential stability preservation in semi-discretisations of BAM networks with nonlinear impulses. Commun Nonl Sci Num Sim, 2009, 14: 27–50 · Zbl 1221.39007 |

[9] | Huang Z, Wang X, Gao F. The existence and global attractivity of almost periodic sequence solution of discrete-time neural networks. Phys Lett A, 2006, 350: 182–191 · Zbl 1195.34066 |

[10] | Nie X, Cao J. Multistability of competitive neural networks with time-varying and distributed delays. Nonl Anal Real World Appl, 2009, 10: 928–942 · Zbl 1167.34383 |

[11] | Wang L. Stabilizing Hopfield neural networks via inhibitory self-connections. J Math Anal Appl, 2004, 292: 135–147 · Zbl 1062.34087 |

[12] | Cheng C, Lin K, Shih C. Multistability in recurrent neural networks. SIAM J Appl Math, 2006, 66: 1301–1320 · Zbl 1106.34048 |

[13] | Guo S, Huang L, Wang L. Exponential stability of discrete-time Hopfield neural networks. Comput Math Appl, 2004, 47: 1249–1256 · Zbl 1075.34070 |

[14] | Guo S, Huang L. Periodic oscillation for discrete-time Hopfield neural networks. Phys Lett A, 2004, 329: 199–206 · Zbl 1209.82029 |

[15] | Zeng Z, Wang J. Multiperiodicity of discrete-time delayed neural networks evoked by periodic external inputs. IEEE Trans Neural Network, 2006, 17: 1141–1151 |

[16] | Huang Z, Xia Y, Wang X. The existence and exponential attractivity of {\(\kappa\)}-almost periodic sequence solution of discrete time neural networks. Nonl Dynam, 2007, 50: 13–26 · Zbl 1176.92002 |

[17] | Huang Z, Song Q, Feng C. Multistability in networks with self-excitation and high-order synaptic connectivity. IEEE Trans Circ Syst I, 2010, 57: 2144–2155 |

[18] | Huang Z, Wang X, Feng C. Multiperiodicity of periodically oscillated discrete-time neural networks with transient excitatory self-connections and sigmoidal nonlinearities. IEEE Trans Neur Netw, 2010, 21: 1643–1655 |

[19] | Sun C, Feng C. Discrete-time analogues of integrodifferential equations modeling neural networks. Phys Lett A, 2005, 334: 180–191 · Zbl 1123.34339 |

[20] | Sun C, Feng C. Exponential periodicity of continuous-time and discrete-time neural networks with delays. Neur Process Lett, 2004, 19: 131–146 · Zbl 02191601 |

[21] | Liu Y, Wang Z, Serrano A, et al. Discrete-time recurrent neural networks with time-varying delays: Exponential stability analysis. Phys Lett A, 2007, 362: 480–488 |

[22] | Song Q, Wang Z. A delay-dependent LMI approach to dynamics analysis of discrete-time recurrent neural networks with time-varying delays. Phys Lett A, 2007, 368: 134–145 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.