Journal of Statistical Physics

, Volume 151, Issue 5, pp 896–921 | Cite as

Infinite Systems of Interacting Chains with Memory of Variable Length—A Stochastic Model for Biological Neural Nets

  • A. Galves
  • E. LöcherbachEmail author


We consider a new class of non Markovian processes with a countable number of interacting components. At each time unit, each component can take two values, indicating if it has a spike or not at this precise moment. The system evolves as follows. For each component, the probability of having a spike at the next time unit depends on the entire time evolution of the system after the last spike time of the component. This class of systems extends in a non trivial way both the interacting particle systems, which are Markovian (Spitzer in Adv. Math. 5:246–290, 1970) and the stochastic chains with memory of variable length which have finite state space (Rissanen in IEEE Trans. Inf. Theory 29(5):656–664, 1983). These features make it suitable to describe the time evolution of biological neural systems. We construct a stationary version of the process by using a probabilistic tool which is a Kalikow-type decomposition either in random environment or in space-time. This construction implies uniqueness of the stationary process. Finally we consider the case where the interactions between components are given by a critical directed Erdös-Rényi-type random graph with a large but finite number of components. In this framework we obtain an explicit upper-bound for the correlation between successive inter-spike intervals which is compatible with previous empirical findings.


Biological neural nets Interacting particle systems Chains of infinite memory Chains of variable length memory Hawkes process Kalikow-decomposition 



We thank D. Brillinger, B. Cessac, S. Ditlevsen, M. Jara, M. Kelbert, Y. Kohayakawa, C. Landim, R.I. Oliveira, S. Ribeiro, L. Triolo, C. Vargas and N. Vasconcelos for many discussions on Hawkes processes, random graphs and neural nets at the beginning of this project.

This work is part of USP project “Mathematics, computation, language and the brain”, FAPESP project “NeuroMat” (grant 2011/51350-6), USP/COFECUB project “Stochastic systems with interactions of variable range” and CNPq project “Stochastic modeling of the brain activity” (grant 480108/2012-9). A.G. is partially supported by a CNPq fellowship (grant 309501/2011-3), A.G. and E.L. have been partially supported by the MathAmSud project “Stochastic structures of large interacting systems” (grant 009/10). E.L. thanks Numec, USP, for hospitality and support.


  1. 1.
    Berbee, H., Chains with infinite connections: uniqueness and Markov representation. Probab. Theory Relat. Fields 76, 243–253 (1987) MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Bollobás, B.: Random Graphs, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 73. Cambridge University Press, Cambridge (2001) zbMATHCrossRefGoogle Scholar
  3. 3.
    Brémaud, P., Massoulié, L.: Stability of nonlinear Hawkes processes. Ann. Probab. 24(3), 1563–1588 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bressaud, X., Fernández, R., Galves, A.: Decay of correlations for non-Hölderian dynamics. A coupling approach. Elect. J. Prob. 4, 1–19 (1999) Google Scholar
  5. 5.
    Brillinger, D.: Maximum likelihood analysis of spike trains of interacting nerve cells. Biol. Cybern. 59(3), 189–200 (1988) zbMATHCrossRefGoogle Scholar
  6. 6.
    Cessac, B.: A discrete time neural network model with spiking neurons: II: Dynamics with noise. J. Math. Biol. 62, 863–900 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Comets, F., Fernández, R., Ferrari, P.A.: Processes with long memory: regenerative construction and perfect simulation. Ann. Appl. Probab. 12(3), 921–943 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Doeblin, W., Fortet, R.: Sur les chaînes à liaisons complétes. Bull. Soc. Math. Fr. 65, 132–148 (1937) MathSciNetGoogle Scholar
  9. 9.
    Fernández, R., Maillard, G.: Chains with complete connections and one-dimensional Gibbs measures. Electron. J. Probab. 9(6), 145–176 (2004) MathSciNetGoogle Scholar
  10. 10.
    Fernández, R., Ferrari, P.A., Galves, A.: Coupling, renewal and perfect simulation of chains of infinite order. Notes for a minicourse given at the Vth Brazilian School of Probability. Manuscript (2001).
  11. 11.
    Fernández, R., Ferrari, P.A., Garcia, N.L.: Loss network representation of Peierls contours. Ann. Probab. 29(2), 902–937 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Fernández, R., Ferrari, P.A., Garcia, N.L.: Perfect simulation for interacting point processes, loss networks and Ising models. Stoch. Process. Appl. 102(1), 63–88 (2002) zbMATHCrossRefGoogle Scholar
  13. 13.
    Ferrari, P.A., Maass, A., Martínez S., Ney, P.: Cesàro mean distribution of group automata starting from measures with summable decay. Ergod. Theory Dyn. Syst. 20(6), 1657–1670 (2000) zbMATHCrossRefGoogle Scholar
  14. 14.
    Gallo, S.: Chains with unbounded variable length memory: perfect simulation and visible regeneration scheme. Adv. Appl. Probab. 43, 735–759 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Galves, A., Garcia, N., Löcherbach, E., Orlandi, E.: Kalikow-type decomposition for multicolor infinite range particle systems (2013) Google Scholar
  16. 16.
    Gerstner, W., Kistler, W.M.: Spiking Neuron Models. Single Neurons, Populations, Plasticity. Cambridge University Press, Cambridge (2002) zbMATHCrossRefGoogle Scholar
  17. 17.
    Goldberg, J.M., Adrian, H.O., Smith, F.D.: Response of neurons of the superior olivary complex of the cat to acoustic stimuli of long duration. J. Neurophysiol. 27, 706–749 (1964) Google Scholar
  18. 18.
    Harris, T.E.: On chains of infinite order. Pac. J. Math. 5, 707–724 (1955) zbMATHCrossRefGoogle Scholar
  19. 19.
    Hawkes, A.G.: Point spectra of some mutually exciting point processes. J. R. Stat. Soc. B 33, 438–443 (1971) MathSciNetADSzbMATHGoogle Scholar
  20. 20.
    Johansson, A., Öberg, A.: Square summability of variations of g-functions and uniqueness of g-measures. Math. Res. Lett. 10(5–6), 587–601 (2003) MathSciNetzbMATHGoogle Scholar
  21. 21.
    Krumin, M., Reutsky, I., Shoham, S.: Correlation-based analysis and generation of multiple spike trains using Hawkes models with an exogenous input. Front Comput Neurosci. 4, 147 (2010). doi: 10.3389/fncom.2010.00147 CrossRefGoogle Scholar
  22. 22.
    Møller, J., Rasmussen, J.G.: Perfect simulation of Hawkes processes. Adv. Appl. Probab. 37(3), 629–646 (2005) CrossRefGoogle Scholar
  23. 23.
    Nawrot, M.P., Boucsein, C., Rodriguez-Molina, V., Aertsen, A., Grün, S., Rotter, S.: Serial interval statistics of spontaneous activity in cortical neurons in vivo and in vitro. Neurocomputing 70, 1717–1722 (2007) CrossRefGoogle Scholar
  24. 24.
    Rissanen, J.: A universal data compression system. IEEE Trans. Inf. Theory 29(5), 656–664 (1983) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Spitzer, F.: Interaction of Markov processes. Adv. Math. 5(2), 246–290 (1970) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.AGM, CNRS-UMR 8088Université de Cergy-PontoiseCergy-PontoiseFrance
  2. 2.Instituto de Matemática e EstatisticaUniversidade de Sao PauloSao PauloBrazil

Personalised recommendations