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

Journal of Statistical Physics

Volumn 151, Issue 5, 2013, Pages 896-921

  Galves, A ^b   Locherbach E ^a  

^a UNIVERSITÉ DE CERGY PONTOISE   (France)

^b UNIVERSITY OF SÃO PAULO   (Brazil)

Author keywords

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

Indexed keywords

EID: 84877608573     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-013-0733-9     Document Type: Article

References (25)

1. Berbee, H., Chains with infinite connections: uniqueness and Markov representation. Probab. Theory Relat. Fields 76, 243-253 (1987).
2. Bollobás, B.: Random Graphs, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 73. Cambridge University Press, Cambridge (2001).
3. Brémaud, P., Massoulié, L.: Stability of nonlinear Hawkes processes. Ann. Probab. 24(3), 1563-1588 (1996).
4. Bressaud, X., Fernández, R., Galves, A.: Decay of correlations for non-Hölderian dynamics. A coupling approach. Elect. J. Prob. 4, 1-19 (1999).
5. Brillinger, D.: Maximum likelihood analysis of spike trains of interacting nerve cells. Biol. Cybern. 59(3), 189-200 (1988).
6. Cessac, B.: A discrete time neural network model with spiking neurons: II: Dynamics with noise. J. Math. Biol. 62, 863-900 (2011).
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).
8. Doeblin, W., Fortet, R.: Sur les chaînes à liaisons complétes. Bull. Soc. Math. Fr. 65, 132-148 (1937).
9. Fernández, R., Maillard, G.: Chains with complete connections and one-dimensional Gibbs measures. Electron. J. Probab. 9(6), 145-176 (2004).
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). http://www. univ-rouen. fr/LMRS/Persopage/Fernandez/resucoup. html.
11. Fernández, R., Ferrari, P. A., Garcia, N. L.: Loss network representation of Peierls contours. Ann. Probab. 29(2), 902-937 (2001).
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).
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).
14. Gallo, S.: Chains with unbounded variable length memory: perfect simulation and visible regeneration scheme. Adv. Appl. Probab. 43, 735-759 (2011).
15. Galves, A., Garcia, N., Löcherbach, E., Orlandi, E.: Kalikow-type decomposition for multicolor infinite range particle systems (2013).
16. Gerstner, W., Kistler, W. M.: Spiking Neuron Models. Single Neurons, Populations, Plasticity. Cambridge University Press, Cambridge (2002).
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).
18. Harris, T. E.: On chains of infinite order. Pac. J. Math. 5, 707-724 (1955).
19. Hawkes, A. G.: Point spectra of some mutually exciting point processes. J. R. Stat. Soc. B 33, 438-443 (1971).
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).
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.
22. Møller, J., Rasmussen, J. G.: Perfect simulation of Hawkes processes. Adv. Appl. Probab. 37(3), 629-646 (2005).
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).
24. Rissanen, J.: A universal data compression system. IEEE Trans. Inf. Theory 29(5), 656-664 (1983).
25. Spitzer, F.: Interaction of Markov processes. Adv. Math. 5(2), 246-290 (1970).