April  2014, 1(2): 255-281. doi: 10.3934/jdg.2014.1.255

Dynamics of large cooperative pulsed-coupled networks

1. 

Instituto de Matemática y Estadística Rafael Laguardia, Universidad de la República, Av. Herrera y Reissig 565, C.P.11300, Montevideo

Received  February 2013 Revised  December 2013 Published  March 2014

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.
Citation: Eleonora Catsigeras. Dynamics of large cooperative pulsed-coupled networks. Journal of Dynamics & Games, 2014, 1 (2) : 255-281. doi: 10.3934/jdg.2014.1.255
References:
[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, (2009).   Google Scholar

[2]

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

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

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

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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.  Google Scholar

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

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

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

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G. B. Ermentrout and D. H. Terman, Mathematical Foundations of Neuroscience,, Interdisc. Appl. Math., (2010).  doi: 10.1007/978-0-387-87708-2.  Google Scholar

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

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

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H. Höglund, Detecting Earnings Management Using Neural Networks,, Doctoral Thesis Hanken School of Economics, (2010).   Google Scholar

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

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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.   Google Scholar

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

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

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

[21]

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.   Google Scholar

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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.  doi: 10.1073/pnas.012582999.  Google Scholar

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

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A. Politi and A. Torcini, Stable chaos,, in Nonlinear Dynamics and Chaos: Advances and Perspectives, (2010).  doi: 10.1007/978-3-642-04629-2.  Google Scholar

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

[26]

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.  doi: 10.1016/j.nonrwa.2009.04.006.  Google Scholar

[27]

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.  doi: 10.1140/epjd/e2010-00215-4.  Google Scholar

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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.  doi: 10.1109/81.633887.  Google Scholar

[33]

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

show all references

References:
[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, (2009).   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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.  doi: 10.1016/j.envsoft.2006.08.002.  Google Scholar

[11]

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

[12]

G. B. Ermentrout and D. H. Terman, Mathematical Foundations of Neuroscience,, Interdisc. Appl. Math., (2010).  doi: 10.1007/978-0-387-87708-2.  Google Scholar

[13]

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

[14]

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

[15]

H. Höglund, Detecting Earnings Management Using Neural Networks,, Doctoral Thesis Hanken School of Economics, (2010).   Google Scholar

[16]

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

[17]

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.   Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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.   Google Scholar

[22]

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.  doi: 10.1073/pnas.012582999.  Google Scholar

[23]

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

[24]

A. Politi and A. Torcini, Stable chaos,, in Nonlinear Dynamics and Chaos: Advances and Perspectives, (2010).  doi: 10.1007/978-3-642-04629-2.  Google Scholar

[25]

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

[26]

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.  doi: 10.1016/j.nonrwa.2009.04.006.  Google Scholar

[27]

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.  doi: 10.1140/epjd/e2010-00215-4.  Google Scholar

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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.  doi: 10.1109/81.633887.  Google Scholar

[33]

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

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