Abstract
Some recent papers relate the criticality of complex systems to their maximal capacity of information processing. In the present paper, we consider high dimensional point processes, known as age-dependent Hawkes processes, which have been used to model spiking neural networks. Using mean-field approximation, the response of the network to a stimulus is computed and we provide a notion of stimulus sensitivity. It appears that the maximal sensitivity is achieved in the sub-critical regime, yet almost critical for a range of biologically relevant parameters.
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Notes
In fact, exponential convergence of the solution of the PDE system (1) to the unique solution of its stationary version (2) can be proved in the weak (\(\alpha <\alpha _{\mathrm{weak}}\)) and the strong (\(\alpha >\alpha _{\mathrm{strong}}\)) connectivity regime [20]. However, the result are not quantitative: \(\alpha _{\mathrm{weak}}\) and \(\alpha _{\mathrm{strong}}\) are unknown. We do not know how they compare to \(\alpha _{c}\).
It is dimensionless since \(\mu \) is a frequency and \(\delta \) a duration.
This can be proved thanks to the cluster representation of linear Hawkes processes [7] and similar results available on Galton–Watson trees.
For all \(t\ge 0, u(t,\cdot )\) remains a probability density and so it is for \(u_{\infty }\).
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Acknowledgements
The author would like to thank Eva Löcherbach for inspiring discussions on this topic. This research was supported by the project Labex MME-DII (ANR11-LBX-0023-01) and mainly conducted during the stay of the author at Université de Cergy-Pontoise.
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Chevallier, J. Stimulus Sensitivity of a Spiking Neural Network Model. J Stat Phys 170, 800–808 (2018). https://doi.org/10.1007/s10955-017-1948-y
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DOI: https://doi.org/10.1007/s10955-017-1948-y