Hawkes processes with variable length memory and an infinite number of\n components

2014/10/20 by Pierre Hodara, Eva Löcherbach, Hodara, Pierre +1 · 2 citations
Mathematics · Biochemistry, Genetics and Molecular Biology · #Point processes and geometric inequalities #Diffusion and Search Dynamics

paper · pdf · doi:10.48550/arxiv.1410.5235

Abstract

In this paper, we build a model for biological neural nets where the activity\nof the network is described by Hawkes processes having a variable length\nmemory. The particularity of this paper is to deal with an infinite number of\ncomponents. We propose a graphical construction of the process and we build, by\nmeans of a perfect simulation algorithm, a stationary version of the process.\nTo carry out this algorithm, we make use of a Kalikow-type decomposition\ntechnique.\n Two models are described in this paper. In the first model, we associate to\neach edge of the interaction graph a saturation threshold that controls the\ninfluence of a neuron on another. In the second model, we impose a structure on\nthe interaction graph leading to a cascade of spike trains. Such structures,\nwhere neurons are divided into layers can be found in retina.\n

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• Efficient Simulation of Hawkes Processes using their Affine Volterra Structure
• Hawkes Processes with Variable Length Memory: Existence, Inference and Application to Neuronal Activity

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