# A Markovian event-based framework for stochastic spiking neural networks

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## Abstract

In spiking neural networks, the information is conveyed by the spike times, that depend on the intrinsic dynamics of each neuron, the input they receive and on the connections between neurons. In this article we study the Markovian nature of the sequence of spike times in stochastic neural networks, and in particular the ability to deduce from a spike train the next spike time, and therefore produce a description of the network activity only based on the spike times regardless of the membrane potential process. To study this question in a rigorous manner, we introduce and study an event-based description of networks of noisy integrate-and-fire neurons, i.e. that is based on the computation of the spike times. We show that the firing times of the neurons in the networks constitute a Markov chain, whose transition probability is related to the probability distribution of the interspike interval of the neurons in the network. In the cases where the Markovian model can be developed, the transition probability is explicitly derived in such classical cases of neural networks as the linear integrate-and-fire neuron models with excitatory and inhibitory interactions, for different types of synapses, possibly featuring noisy synaptic integration, transmission delays and absolute and relative refractory period. This covers most of the cases that have been investigated in the event-based description of spiking deterministic neural networks.

## Keywords

Stochastic network Linear integrate-and-fire neurons Event-based model Event-based simulation## Notes

### Acknowledgements

The authors warmly acknowledge Romain Brette for very insightful discussions on the concepts, Philippe Robert for interesting discussions and for reading suggestions, Olivier Rochel for his introduction to MVA Spike and for sharing his code, and Renaud Keriven and Alexandre Chariot for developing a GPU simulation code (not presented here). This work was partially supported by the ERC advanced grant NerVi number 227747.

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