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Conductance-Based Refractory Density Approach for a Population of Bursting Neurons

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Abstract

The conductance-based refractory density (CBRD) approach is a parsimonious mathematical–computational framework for modelling interacting populations of regular spiking neurons, which, however, has not been yet extended for a population of bursting neurons. The canonical CBRD method allows to describe the firing activity of a statistical ensemble of uncoupled Hodgkin–Huxley-like neurons (differentiated by noise) and has demonstrated its validity against experimental data. The present manuscript generalises the CBRD for a population of bursting neurons; however, in this pilot computational study, we consider the simplest setting in which each individual neuron is governed by a piecewise linear bursting dynamics. The resulting population model makes use of slow–fast analysis, which leads to a novel methodology that combines CBRD with the theory of multiple timescale dynamics. The main prospect is that it opens novel avenues for mathematical explorations, as well as, the derivation of more sophisticated population activity from Hodgkin–Huxley-like bursting neurons, which will allow to capture the activity of synchronised bursting activity in hyper-excitable brain states (e.g. onset of epilepsy).

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Acknowledgements

This research is supported by the Basque Government through the BERC 2018-2021 programme and by the Spanish State Research Agency through BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and through Project RTI2018-093860-B-C21 funded by (AEI/FEDER, UE) and acronym “MathNEURO”. SR would further like to acknowledge the support of Ikerbasque (The Basque Foundation for Science). Moreover, the research of AC was supported by the Russian Science Foundation Grant (Project 16-15-10201). The research of AG was supported by the Spanish Grant MINECO-FEDER-UE MTM-2015-71509-C2-2-R and the Catalan Grant Number 2017SGR1049. Finally, SR, FC and MD would like to acknowledge the support of Inria through the associated team project NeuroTransSF.

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Authors and Affiliations

  1. Ioffe Institute, Politekhnicheskaya str., 26, St. Petersburg, Russia, 194021

    Anton Chizhov

  2. Sechenov Institute of Evolutionary Physiology and Biochemistry of Russian Academy of Sciences, Torez pr., 44, St. Petersburg, Russia, 194223

    Anton Chizhov

  3. BCAM - Basque Center for Applied Mathematics, Bilbao, Bizkaia, Spain

    Anton Chizhov, Fabien Campillo, Mathieu Desroches, Antoni Guillamon & Serafim Rodrigues

  4. Inria Sophia Antipolis Méditerranée, MathNeuro Team, Sophia Antipolis, France

    Fabien Campillo & Mathieu Desroches

  5. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain

    Antoni Guillamon

  6. Ikerbasque, Basque Foundation for Science, Bilbao, Spain

    Serafim Rodrigues

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  1. Anton Chizhov
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Correspondence to Serafim Rodrigues.

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Chizhov, A., Campillo, F., Desroches, M. et al. Conductance-Based Refractory Density Approach for a Population of Bursting Neurons. Bull Math Biol 81, 4124–4143 (2019). https://doi.org/10.1007/s11538-019-00643-8

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