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

Dynamics of large cooperative pulsed-coupled networks

Eleonora Catsigeras 1,

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.
Keywords: synchronization, impulsive ODE, Pulse-coupled networks, information., cooperative evolutive game.
Mathematics Subject Classification: Primary: 37NXX, 92B20; Secondary: 34D06, 05C82, 94A17, 92B2.
Citation: Eleonora Catsigeras. Dynamics of large cooperative pulsed-coupled networks. Journal of Dynamics and 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, 554, University of Siena, Siena, 2009.

[2]

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

[3]

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

[4]

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.

[5]

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.

[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.

[7]

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.

[8]

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.

[9]

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

[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), 1261-1269. doi: 10.1016/j.envsoft.2006.08.002.

[11]

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.

[12]

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.

[13]

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.

[14]

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.

[15]

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).

[16]

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

[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), 87-92.

[18]

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

[19]

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.

[20]

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.

[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), 143-161.

[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), 2566-2572. doi: 10.1073/pnas.012582999.

[23]

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

[24]

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.

[25]

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.

[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), 1810-1818. doi: 10.1016/j.nonrwa.2009.04.006.

[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), 51-56. doi: 10.1140/epjd/e2010-00215-4.

[28]

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.

[29]

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.

[30]

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.

[31]

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

[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), 976-988. doi: 10.1109/81.633887.

[33]

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.

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, 554, University of Siena, Siena, 2009.

[2]

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

[3]

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

[4]

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.

[5]

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.

[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.

[7]

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.

[8]

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.

[9]

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

[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), 1261-1269. doi: 10.1016/j.envsoft.2006.08.002.

[11]

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.

[12]

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.

[13]

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.

[14]

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.

[15]

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).

[16]

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

[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), 87-92.

[18]

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

[19]

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.

[20]

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.

[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), 143-161.

[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), 2566-2572. doi: 10.1073/pnas.012582999.

[23]

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

[24]

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.

[25]

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.

[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), 1810-1818. doi: 10.1016/j.nonrwa.2009.04.006.

[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), 51-56. doi: 10.1140/epjd/e2010-00215-4.

[28]

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.

[29]

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.

[30]

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.

[31]

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

[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), 976-988. doi: 10.1109/81.633887.

[33]

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.

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