Ioffe Institute, Politekhnicheskaya str., 26, St. Petersburg, Russia, 194021.
Sechenov Institute of Evolutionary Physiology and Biochemistry of Russian Academy of Sciences, Torez pr., 44, St. Petersburg, Russia, 194223.
Bull Math Biol. 2019 Oct;81(10):4124-4143. doi: 10.1007/s11538-019-00643-8. Epub 2019 Jul 16.
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).
基于电导的不应期密度（CBRD）方法是一种用于模拟规则放电神经元相互作用群体的简约数学计算框架，但尚未扩展到突发神经元群体。经典的 CBRD 方法允许描述由噪声区分的不耦合 Hodgkin-Huxley 样神经元（uncoupled Hodgkin-Huxley-like neurons）统计集合的放电活动，并已证明其与实验数据的有效性。本文将 CBRD 方法推广到突发神经元群体；然而，在这项初步的计算研究中，我们考虑了最简单的情况，其中每个神经元都由分段线性突发动力学控制。所得到的群体模型利用快慢分析，这导致了一种新的方法，该方法将 CBRD 与多时间尺度动力学理论相结合。主要的前景是，它为数学探索开辟了新的途径，以及从 Hodgkin-Huxley 样突发神经元推导出更复杂的群体活动，这将允许捕获超兴奋脑状态（如癫痫发作的开始）中的同步突发活动。
