Self-synchronization of networks with a strong kernel of integrate and fire excitatory neurons

This a reduced preliminary version of the paper. It only includes the abstract,introduction, figure, conclusion and references. The complete version is published inOpen Access mode, by World Science and Engineering Academy and Society in thepeer reviewed journal:WSEASTransactions on Mathematics (ISSN 1109-2769),Volume 12, Issue 7 , year 2013, pages 786–797.URL address of the journal:http://wseas.org/cms.action?id=6946URL address of the final version of this paper (free access):http://www.wseas.org/multimedia/journals/mathematics/2013/145706-180.pdfAbstractWe study the global dynamics of networks of pulsed-coupled neuronsthat are modeled as integrate and fire oscillators. We focus on excita-tory networks with a strong kernel. We prove the synchronization of thewhole network from any initial state, and find a bound from above ofthe transients until the full synchronization is achieved. The methodol-ogy of research is by exact mathematical definitions and statements andby deductive proofs, from standard arguments of the mathematical theoryof abstract dynamical systems. We include examples of applications todiverse fields, and also a brief review of other mathematical methods ofresearch on general networks of dynamically interacting units.Key–Words: Pulse-coupled networks, synchronization, neuralnetworks,integrate and fire oscillators.

[1]  C. Volos,et al.  Synchronization Phenomena in Coupled Nonlinear Systems Applied in Economic Cycles , 2012 .

[2]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[3]  Thomas P. Witelski,et al.  On Spiking Models for Synaptic Activity and Impulsive Differential Equations , 2008, SIAM Rev..

[4]  J. Fox,et al.  Phase-locking and environmental fluctuations generate synchrony in a predator–prey community , 2009, Nature.

[5]  Russell Golman.  Why learning doesn’t add up: equilibrium selection with a composition of learning rules , 2011, Int. J. Game Theory.

[6]  Zhang Lingling,et al.  The solutions of initial value problems for nonlinear fourth-order impulsive integro-differential equations in banach spaces , 2012 .

[7]  Eugene M. Izhikevich,et al.  Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting , 2006 .

[8]  L. Gardini,et al.  The dynamics of the NAIRU model with two switching regimes , 2010 .

[9]  Jean-Louis Deneubourg,et al.  Synchronization in light-controlled oscillators , 2003 .

[10]  Evaluation of the Lyapunov exponent for stochastic dynamical systems with event synchronization , 2012 .

[11]  Lai-Sang Young,et al.  Shear-induced chaos , 2007, 0705.3294.

[12]  P. Subashini,et al.  Optimization of image processing techniques using neural networks: a review , 2011 .

[13]  An-Pin Chen,et al.  Applying self-organizing mapping neural network for discovery market behavior of equity fund , 2010 .

[14]  Christos Volos,et al.  Inverse lag synchronization in mutually coupled nonlinear circuits , 2010, ICC 2010.

[15]  Jürgen Kurths,et al.  Nonlinear dynamics and chaos : advances and perspectives , 2010 .

[16]  B. Cessac.  A discrete time neural network model with spiking neurons , 2007, Journal of mathematical biology.

[17]  A. Khashman,et al.  Image compression using neural networks and haar wavelet , 2008 .

[18]  Henrik Höglund,et al.  Detecting earnings management with neural networks , 2012, Expert Syst. Appl..

[20]  S. Strogatz,et al.  Synchronization of pulse-coupled biological oscillators , 1990 .

[21]  N. Rubido,et al.  Synchronization regions of two pulse-coupled electronic piecewise linear oscillators , 2011 .

[22]  Ammar Belatreche,et al.  Challenges for large-scale implementations of spiking neural networks on FPGAs , 2007, Neurocomputing.

[23]  L. Young.  Chaotic phenomena in three settings: large, noisy and out of equilibrium , 2008 .

[24]  Jun-Guo Lu,et al.  Global asymptotical synchronization of chaotic neural networks by output feedback impulsive control: An LMI approach , 2009 .

[25]  Global solutions for second order impulsive integro-differential equations in Banach spaces , 2012 .

[26]  Bottani.  Synchronization of integrate and fire oscillators with global coupling. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[27]  Manuel J. López,et al.  Computer simulation and method for heart rhythm control based on ECG signal reference tracking , 2010 .

[28]  Xiaodi Li,et al.  The LMI method for stationary oscillation of interval neural networks with three neuron activations under impulsive effects , 2013 .

[29]  Louis M. Pecora,et al.  Synchronization of oscillators in complex networks , 2008 .

[30]  Jonathan L. F. King.  Entropy in Ergodic Theory , 2009, Encyclopedia of Complexity and Systems Science.

[31]  Fabrice Rossi,et al.  Batch kernel SOM and related Laplacian methods for social network analysis , 2008, Neurocomputing.

[32]  Ivanka M. Stamova,et al.  Almost periodic solutions for impulsive neural networks with delay , 2007 .

[33]  Aminah Robinson Fayek,et al.  Adaptive Learning of Contractor Default Prediction Model for Surety Bonding , 2013 .

[34]  W. Hongwu.  Chaos Control and Synchronization of a Fractional-order Autonomous System , 2012 .

[35]  Eugene M. Izhikevich,et al.  Weakly pulse-coupled oscillators, FM interactions, synchronization, and oscillatory associative memory , 1999, IEEE Trans. Neural Networks.

[36]  Bursting Oscillations of Neurons and Synchronization , 2012 .

[37]  B Mashayekhi,et al.  DETECTING EARNINGS MANAGEMENT USING NEURAL NETWORKS , 2012 .

[38]  A Model of Imitative Behavior in the Population of Firms and Workers , 2009 .

[39]  Dynamics of stage-structured population models with harvesting pulses , 2012 .

[40]  Marie Cottrell,et al.  Neural Networks for Complex Data , 2012, KI - Künstliche Intelligenz.

[41]  Igal Milchtaich,et al.  Representation of finite games as network congestion games , 2011, International Conference on NETwork Games, Control and Optimization (NetGCooP 2011).

[42]  A. Fukasawa,et al.  Activity of a Neuron and Formulation of a Neural Group for Synchronized Neural Systems , 2012 .

[43]  J. C. Quadrado,et al.  Wind speed prediction using artificial neural networks , 2005 .

[44]  Stephen Coombes,et al.  Limitations of perturbative techniques in the analysis of rhythms and oscillations , 2011, Journal of mathematical biology.

[45]  Nicolas Rubido,et al.  Experimental results on synchronization times and stable states in locally coupled light-controlled oscillators , 2009, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[46]  Junhai Ma,et al.  Research on delayed complexity based on nonlinear price game of insurance market , 2011 .

[47]  Georgi S. Medvedev,et al.  Synchronization of coupled stochastic limit cycle oscillators , 2009, 0911.5520.

[48]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[49]  Hongli Wang,et al.  Stability of Ecosystem Induced by Mutual Interference between Predators , 2010 .

[50]  Quanxin Zhu,et al.  Synchronization of switched neural networks with mixed delays via impulsive control , 2011 .