Weak resonant Double Hopf bifurcations in an Inertial Four-Neuron Model with Time Delay

In this paper, a four-neuron delayed bidirectional associative memory (BAM) model with inertia is considered. Weak resonant double Hopf bifurcations are completely analyzed in the parameter space of the coupling weight and the coupling delay by the perturbation-incremental scheme (PIS). Numerical simulations are given for justifying the theoretical results. To the best of our knowledge, the paper is the first one to introduce inertia to a four-neuron delayed system and clarify the relationship between system parameters and dynamical behaviors.

[1]  Zheru Chi,et al.  Improvement of Image Classification Using Wavelet Coefficients with Structured-Based Neural Network , 2008, Int. J. Neural Syst..

[2]  Hojjat Adeli,et al.  A neural dynamics model for structural optimization—Theory , 1995 .

[3]  Hojjat Adeli,et al.  A new supervised learning algorithm for multiple spiking neural networks with application in epilepsy and seizure detection , 2009, Neural Networks.

[4]  Changyin Sun,et al.  Lmi-Based asymptotic Stability Analysis of Neural Networks with Time-Varying Delays , 2008, Int. J. Neural Syst..

[5]  Hojjat Adeli,et al.  Dynamic fuzzy wavelet neuroemulator for non‐linear control of irregular building structures , 2008 .

[6]  Kwok-wai Chung,et al.  An Efficient Method for Studying Weak Resonant Double Hopf Bifurcation in Nonlinear Systems with Delayed Feedbacks , 2008 .

[7]  F. Dodge,et al.  Subthreshold Behavior and Phenomenological Impedance of the Squid Giant Axon , 1970, The Journal of general physiology.

[8]  Adnan Khashman,et al.  Document Image binarisation Using a Supervised Neural Network , 2008, Int. J. Neural Syst..

[9]  Xuyang Lou,et al.  Robust Exponential Stability of Markovian Jumping Neural Networks with Time-Varying Delay , 2008, Int. J. Neural Syst..

[10]  Diego Andina,et al.  Artificial Metaplasticity Neural Network Applied to Credit Scoring , 2011, Int. J. Neural Syst..

[11]  R. Westervelt,et al.  Dynamics of simple electronic neural networks , 1987 .

[12]  Kazuyuki Aihara,et al.  Chaotic simulated annealing by a neural network model with transient chaos , 1995, Neural Networks.

[13]  Hojjat Adeli,et al.  A probabilistic neural network for earthquake magnitude prediction , 2009, Neural Networks.

[14]  X. Liao,et al.  Stability of bifurcating periodic solutions for a single delayed inertial neuron model under periodic excitation , 2009 .

[15]  Teresa Bernarda Ludermir,et al.  Hybrid neural systems for pattern recognition in artificial noses , 2005, Int. J. Neural Syst..

[16]  Uwe Reuter,et al.  Artificial Neural Networks for Forecasting of Fuzzy Time Series , 2010, Comput. Aided Civ. Infrastructure Eng..

[17]  Jian Xu,et al.  Bursting Near Bautin bifurcation in a Neural Network with Delay Coupling , 2009, Int. J. Neural Syst..

[18]  Hojjat Adeli,et al.  TOWARD INTELLIGENT VARIABLE MESSAGE SIGNS IN FREEWAY WORK ZONES: NEURAL NETWORK MODEL , 2004 .

[19]  Christof Koch,et al.  Cable theory in neurons with active, linearized membranes , 2004, Biological Cybernetics.

[20]  Adnan Khashman.  Blood Cell Identification Using a Simple Neural Network , 2008, Int. J. Neural Syst..

[21]  Hojjat Adeli,et al.  Distributed neural dynamics algorithms for optimization of large steel structures , 1997 .

[22]  Shangjiang Guo,et al.  Two-parameter bifurcations in a network of two neurons with multiple delays , 2008 .

[23]  Nicolas Le Bihan,et al.  Polarized Signal Classification by Complex and quaternionic Multi-Layer Perceptrons , 2008, Int. J. Neural Syst..

[24]  ASHIF PANAKKAT,et al.  Neural Network Models for Earthquake Magnitude Prediction Using Multiple seismicity Indicators , 2007, Int. J. Neural Syst..

[25]  K. Y. Zhu,et al.  Biomechanical Stability Analysis of the lambda-Model Controlling One Joint , 2007, Int. J. Neural Syst..

[26]  Manolis A. Christodoulou,et al.  Collision Avoidance in Commercial Aircraft Free Flight via Neural Networks and Non-Linear Programming , 2008, Int. J. Neural Syst..

[27]  Xuyang Lou,et al.  New Criteria on Global Exponential Stability of BAM Neural Networks with Distributed Delays and Reaction-Diffusion Terms , 2007, Int. J. Neural Syst..

[28]  Sue Ann Campbell,et al.  Stability, Bifurcation, and Multistability in a System of Two Coupled Neurons with Multiple Time Delays , 2000, SIAM J. Appl. Math..

[29]  Adnan Khashman,et al.  A Neural Network Model for Credit Risk Evaluation , 2009, Int. J. Neural Syst..

[30]  Lei Wang,et al.  Hopf bifurcation and stability analysis on discrete-time Hopfield neural network with delay , 2008 .

[31]  Shengyuan Xu,et al.  Delay-Dependent Robust Exponential Stability for Uncertain Recurrent Neural Networks with Time-Varying Delays , 2007, Int. J. Neural Syst..

[32]  J. Xu,et al.  The nonresonant double Hopf bifurcation in delayed neural network , 2008, Int. J. Comput. Math..

[33]  Kreangkri Ratchagit,et al.  Asymptotic Stability of Delay-Difference System of Hopfield Neural Networks via Matrix Inequalities and Application , 2007, Int. J. Neural Syst..

[34]  R. Westervelt,et al.  Stability of analog neural networks with delay. , 1989, Physical review. A, General physics.

[35]  Dora E. Angelaki,et al.  Models of membrane resonance in pigeon semicircular canal type II hair cells , 1991, Biological Cybernetics.

[36]  Jinde Cao,et al.  Delay-dependent multistability in recurrent neural networks , 2010, Neural Networks.

[37]  Hojjat Adeli,et al.  Optimization of space structures by neural dynamics , 1995, Neural Networks.

[38]  Wei Wu,et al.  LP Approximation Capabilities of Sum-of-Product and Sigma-pi-Sigma Neural Networks , 2007, Int. J. Neural Syst..

[39]  Jinde Cao,et al.  Stability Analysis of impulsive Cohen-Grossberg Neural Network with Unbounded Discrete Time-Varying Delays , 2007, Int. J. Neural Syst..

[40]  D. Attwell,et al.  Models for electrical tuning in hair cells , 1985, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[41]  Tipu Z. Aziz,et al.  Prediction of Parkinson's Disease tremor Onset Using a Radial Basis Function Neural Network Based on Particle Swarm Optimization , 2010, Int. J. Neural Syst..

[42]  G. Samaey,et al.  DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations , 2001 .

[43]  H. Adeli,et al.  A spatio-temporal wavelet-chaos methodology for EEG-based diagnosis of Alzheimer's disease , 2008, Neuroscience Letters.

[44]  Hojjat Adeli,et al.  Dynamic Wavelet Neural Network for Nonlinear Identification of Highrise Buildings , 2005 .

[45]  Diek W. Wheeler,et al.  Stability and chaos in an inertial two-neuron system , 1997 .

[46]  Hojjat Adeli,et al.  Cost Optimization of Prestressed Concrete Bridges , 2005 .

[47]  J. Hale.  Theory of Functional Differential Equations , 1977 .

[48]  H. Adeli,et al.  Dynamic Fuzzy Wavelet Neural Network Model for Structural System Identification , 2006 .

[49]  J J Hopfield,et al.  Neural networks and physical systems with emergent collective computational abilities. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

[50]  X. Liao,et al.  Hopf bifurcation and chaos in a single inertial neuron model with time delay , 2004, nlin/0411027.

[51]  Jinde Cao,et al.  Stability and Hopf Bifurcation in a Simplified BAM Neural Network With Two Time Delays , 2007, IEEE Transactions on Neural Networks.

[52]  Dimitris C. Theodoridis,et al.  Indirect Adaptive Control of Unknown Multi Variable nonlinear Systems with Parametric and Dynamic Uncertainties Using a New Neuro-Fuzzy System Description , 2010, Int. J. Neural Syst..

[53]  Francisco Javier Díaz Pernas,et al.  Wavelet‐Based Denoising for Traffic Volume Time Series Forecasting with Self‐Organizing Neural Networks , 2010, Comput. Aided Civ. Infrastructure Eng..

[54]  JUN CHEN,et al.  New Criteria of Almost Periodic Solution for BAM Neural Networks with Delays and impulsive Effects , 2007, Int. J. Neural Syst..

[55]  Hojjat Adeli,et al.  Dynamic Wavelet Neural Network Model for Traffic Flow Forecasting , 2005 .

[56]  Hojjat Adeli,et al.  Spiking Neural Networks , 2009, Int. J. Neural Syst..

[57]  Hojjat Adeli,et al.  Recurrent Neural Network for Approximate Earthquake Time and Location Prediction Using Multiple Seismicity Indicators , 2009, Comput. Aided Civ. Infrastructure Eng..

[58]  Haiyan Hu,et al.  Hopf bifurcation analysis of a four-neuron network with multiple time delays , 2009 .

[59]  Mohammed Al-Shalalfa,et al.  Adaptive Machine Learning Technique for Periodicity Detection in Biological Sequences , 2009, Int. J. Neural Syst..

[60]  Jinn-Wen Wu,et al.  Stability of Neural Field , 2007, Int. J. Neural Syst..

[61]  Hojjat Adeli,et al.  Optimum cost design of reinforced concrete slabs using neural dynamics model , 2005, Eng. Appl. Artif. Intell..

[62]  Jinde Cao,et al.  Stability and Hopf bifurcation analysis on a four-neuron BAM neural network with time delays , 2006 .

[63]  Hojjat Adeli,et al.  Neuro‐genetic algorithm for non‐linear active control of structures , 2008 .

[64]  David H Timm,et al.  AXLE LOAD SPECTRA CHARACTERIZATION BY MIXED DISTRIBUTION MODELING , 2005 .

[65]  Junjie Wei,et al.  Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays , 2005 .

[66]  Sharene D. Bungay,et al.  Patterns of oscillation in a Ring of Identical Cells with Delayed Coupling , 2007, Int. J. Bifurc. Chaos.

[67]  Amir F. Atiya,et al.  How delays affect neural dynamics and learning , 1994, IEEE Trans. Neural Networks.

[68]  X. Liao,et al.  Dynamics of an inertial two-neuron system with time delay , 2009 .