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

被引:70
|
作者
Galves, A. [1 ]
Loecherbach, E. [2 ]
机构
[1] Univ Sao Paulo, Inst Matemat & Estat, BR-05315970 Sao Paulo, Brazil
[2] Univ Cergy Pontoise, CNRS UMR 8088, AGM, F-95000 Cergy Pontoise, France
基金
巴西圣保罗研究基金会;
关键词
Biological neural nets; Interacting particle systems; Chains of infinite memory; Chains of variable length memory; Hawkes process; Kalikow-decomposition; PERFECT SIMULATION; HAWKES PROCESSES; POINT-PROCESSES; SPIKE TRAINS; REPRESENTATION; CONNECTIONS; UNIQUENESS; NEURONS;
D O I
10.1007/s10955-013-0733-9
中图分类号
O4 [物理学];
学科分类号
0702 ;
摘要
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 Erdos-R,nyi-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.
引用
收藏
页码:896 / 921
页数:26
相关论文
共 5 条