Abstract
For large fully connected neuron networks, we study the dynamics of homogenous assemblies of interacting neurons described by time elapsed models. Under general assumptions on the firing rate which include the ones made in previous works (Pakdaman et al. in Nonlinearity 23(1):55–75, 2010; SIAM J Appl Math 73(3):1260–1279, 2013, Mischler and Weng in Acta Appl Math, 2015), we establish accurate estimate on the long time behavior of the solutions in the weak and the strong connectivity regime both in the case with and without delay. Our results improve (Pakdaman et al. 2010, 2013) where a less accurate estimate was established and Mischler and Weng (2015) where only smooth firing rates were considered. Our approach combines several arguments introduced in the above previous works as well as a slightly refined version of the Weyl’s and spectral mapping theorems presented in Voigt (Monatsh Math 90(2):153–161, 1980) and Mischler and Scher (Ann Inst H Poincaré Anal Non Linéaire 33(3):849–898, 2016).
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Acknowledgements
The research leading to this paper was (partially) funded by the French “ANR blanche” project Kibord: ANR-13-BS01-0004. C. Q. acknowledges the funding of CONICYT Postdoctorado 2017 No. 3170435 as well as the CEREMADE at University Paris-Dauphine for the invitation in February 2018. The authors also acknowledge the anonymous referees for the conscientious reading and the many suggestions they formulate for improving the present work.
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Mischler, S., Quiñinao, C. & Weng, Q. Weak and Strong Connectivity Regimes for a General Time Elapsed Neuron Network Model. J Stat Phys 173, 77–98 (2018). https://doi.org/10.1007/s10955-018-2122-x
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DOI: https://doi.org/10.1007/s10955-018-2122-x