Advanced Search
Article Contents
Article Contents

Dynamics of large cooperative pulsed-coupled networks

Abstract Related Papers Cited by
  • We study the deterministic dynamics of networks ${\mathcal N}$ composed by $m$ non identical, mutually pulse-coupled cells. We assume weighted, asymmetric and positive (cooperative) interactions among the cells, and arbitrarily large values of $m$. We consider two cases of the network's graph: the complete graph, and the existence of a large core (i.e. a large complete subgraph). First, we prove that the system periodically eventually synchronizes with a natural "spiking period" $p \geq 1$, and that if the cells are mutually structurally identical or similar, then the synchronization is complete ($p= 1$) . Second, we prove that the amount of information $H$ that ${\mathcal N}$ generates or processes, equals $\log p$. Therefore, if ${\mathcal N}$ completely synchronizes, the information is null. Finally, we prove that ${\mathcal N}$ protects the cells from their risk of death.
    Mathematics Subject Classification: Primary: 37NXX, 92B20; Secondary: 34D06, 05C82, 94A17, 92B25.


    \begin{equation} \\ \end{equation}
  • [1]

    E. Accinelli, S. London and E. Sánchez Carrera, A Model of Imitative Behavior in the Population of Firms and Workers, Quaderni del Dipartimento di Economia Politica, 554, University of Siena, Siena, 2009.


    S. Bottani, Synchronization of integrate and fire oscillators with global coupling, Physical Review E, 54 (1996), 2334-2350.doi: 10.1103/PhysRevE.54.2334.


    R. Boulet, B. Jouve, F. Rossi and N. Villa, Batch kernel SOM and related Laplacian methods for social network analysis, Neurocomputing, 71 (2008), 1257-1273.doi: 10.1016/j.neucom.2007.12.026.


    E. Catsigeras and P. Guiraud, Integrate and fire neural networks, piecewise contractive maps and limit cycles, Journ. Math. Biol., 67 (2013), 609-655.doi: 10.1007/s00285-012-0560-7.


    B. Cessac, A discrete time neural network model with spiking neurons. Rigorous results on the spontaneous dynamics, Journ. Math. Biol., 56 (2008), 311-345.doi: 10.1007/s00285-007-0117-3.


    B. Cessac and T. Viéville, On Dynamics of Integrate-and-fire Neural Networks with Conductance Based Synapses, Frontiers In Computational Neuroscience, 2008.doi: 10.3389/neuro.10.002.2008.


    J. R. Chazottes and B. Fernandez (Eds), Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, Lecture Notes in Physics, 671 Springer Berlin, 2005.


    M. Cottrell, M. Olteanu, F. Rossi, J. Rynkiewicz and N. Villa-Vialaneix, Neural networks for complex data, Künstliche Intelligenz, 26 (2012), 373-380.doi: 10.1007/s13218-012-0207-2.


    R. Coutinho, B. Fernandez, R. Lima and A. Meyroneinc, Discrete time piecewise affine models of genetic regulatory networks, Journ. Math. Biol., 52 (2006), 524-570.doi: 10.1007/s00285-005-0359-x.


    AL. Dutot, J. Rynkiewicz, F. Steiner and J. Rude, A 24-h forecast of ozone peaks and exceedance levels using neural classifiers and weather predictions, Environ Model Softw, 22 (2007), 1261-1269.doi: 10.1016/j.envsoft.2006.08.002.


    G. B. Ermentrout and N. Kopell, Oscillator death in systems of coupled neural oscillators, SIAM Journal on Applied Mathematics, 50 (1990), 125-146.doi: 10.1137/0150009.


    G. B. Ermentrout and D. H. Terman, Mathematical Foundations of Neuroscience, Interdisc. Appl. Math., 35, Springer, Dordrecht-Heidelberg-London, 2010.doi: 10.1007/978-0-387-87708-2.


    J. Feng, L. Zhu and H. Wang, Stability of Ecosystem induced by mutual interference between predators, Procedia Environmental Sciences, 2 (2010), 42-48.doi: 10.1016/j.proenv.2010.10.007.


    R. Golamen, Why learning doesn't add up: Equilibrium selection with a composition of learning rules, Int. Jroun. Game Theory, 40 (2011), 719-733.doi: 10.1007/s00182-010-0265-3.


    H. Höglund, Detecting Earnings Management Using Neural Networks, Doctoral Thesis Hanken School of Economics, Economics and Society Series 121, Edita Prima Ltd., Helsinki, 2010. Available from: https://helda.helsinki.fi/handle/10227/742 (Last retrieved 7 Feb. 2013).


    E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, MIT Press, 2007.


    B. Maillet, M. Olteanu and J. Rynkiewicz, Nonlinear analysis of shocks when financial markets are subject to changes in regime, in Proc of XIIth European Symposium on Artificial Neural Networks, (2004), 87-92.


    W. Mass and C. M. Bishop (Eds), Pulsed Neural Networks, MIT Press, Cambridge, 2001.


    I. Milchtaich, Representation of finite games as network of congestion, Int. Journ. Game Theory, 42 (2013), 1085-1096.doi: 10.1007/s00182-012-0363-5.


    R. E. Mirollo and S. H. Strogatz, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math., 50 (1990), 1645-1662.doi: 10.1137/0150098.


    M. A. Jalil and M. Misas, Evaluación de pronósticos de tipo de cambio utilizando redes neuronales y funciones de pérdida asimétricas (Spanish), Revista Colombiana de Estadística, 30 (2007), 143-161.


    M. E. J. Newman, D. J. Watts, and S. H. Strogatz, Random graph models of social networks, Proc. Nal. Acad. Sci. USA, 99 (2002), 2566-2572.doi: 10.1073/pnas.012582999.


    A. Pikovsky and Y. Maistrenko (Editors), Synchronization: Theory and Application, Kluwer Academic Publ, Dordrecht, 2003.doi: 10.1007/978-94-010-0217-2.


    A. Politi and A. Torcini, Stable chaos, in Nonlinear Dynamics and Chaos: Advances and Perspectives, (eds. M. Thiel, J. Kurths, M. C. Romano, G. Károlyi and A. Moura), Understanding Complex Systems, Springer, 2010.doi: 10.1007/978-3-642-04629-2.


    G. M. Ramírez Ávila, J. L. Guisset and J. L. Deneubourg, Synchronization in light-controlled oscillators, Physica D, 182 (2003), 254-273.doi: 10.1016/S0167-2789(03)00135-0.


    V. S. H. Raoa and M. N. Kumarb, Estimation of the parameters of an infectious disease model using neural networks, Nonlinear Analysis: Real World Applications, 11 (2010), 1810-1818.doi: 10.1016/j.nonrwa.2009.04.006.


    N. Rubido, C. Cabeza, S. Kahan, G. M. Ramírez Ávila and A. C. Marti, Synchronization regions of two pulse-coupled electronic piecewise linear oscillators, Europ. Phys. Journ. D, 62 (2011), 51-56.doi: 10.1140/epjd/e2010-00215-4.


    G. T. Stamov and I. Stamova, Almost periodic solutions for impulsive neural networks with delay, Applied Mathematical Modelling, 31 (2007), 1263-1270.doi: 10.1016/j.apm.2006.04.008.


    C. van Vreeswijk, L. F. Abbott and B. Ermentrout, When inhibition not excitation synchronizes neural firing, Journ. Comput. Neuroscience, 1 (1994), 313-321.doi: 10.1007/BF00961879.


    D. A. Vasseur and J. Fox, Phase-locking and environmental fluctuations generate synchrony in a predator-prey community, Nature, 460 (2009), 1007-1010.doi: 10.1038/nature08208.


    D. J. Watts and S. H. Strogatz, Collective Dynamics of Small-World, Nature (London), 393 (1998), 440-440.


    T. Yang and L. O. Chua, Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication, IEEE Trans. Circuits Syst., 44 (1997), 976-988.doi: 10.1109/81.633887.


    Young L.-S, Chaotic phenomena in three setting: Large, noisy and out of equilibrium, Nonlinearity, 21 (2008), T245-T252.doi: 10.1088/0951-7715/21/11/T04.

  • 加载中

Article Metrics

HTML views() PDF downloads(160) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint