Department of Applied Mathematics, University of Waterloo Waterloo, ON, Canada.
Front Comput Neurosci. 2013 Dec 27;7:184. doi: 10.3389/fncom.2013.00184. eCollection 2013.
We analytically derive mean-field models for all-to-all coupled networks of heterogeneous, adapting, two-dimensional integrate and fire neurons. The class of models we consider includes the Izhikevich, adaptive exponential and quartic integrate and fire models. The heterogeneity in the parameters leads to different moment closure assumptions that can be made in the derivation of the mean-field model from the population density equation for the large network. Three different moment closure assumptions lead to three different mean-field systems. These systems can be used for distinct purposes such as bifurcation analysis of the large networks, prediction of steady state firing rate distributions, parameter estimation for actual neurons and faster exploration of the parameter space. We use the mean-field systems to analyze adaptation induced bursting under realistic sources of heterogeneity in multiple parameters. Our analysis demonstrates that the presence of heterogeneity causes the Hopf bifurcation associated with the emergence of bursting to change from sub-critical to super-critical. This is confirmed with numerical simulations of the full network for biologically reasonable parameter values. This change decreases the plausibility of adaptation being the cause of bursting in hippocampal area CA3, an area with a sizable population of heavily coupled, strongly adapting neurons.
我们分析推导出了全连接网络中异质、自适应、二维积分和点火神经元的平均场模型。我们考虑的模型类包括 Izhikevich、自适应指数和四次积分和点火模型。参数的异质性导致在从大网络的种群密度方程推导出平均场模型时，可以做出不同的矩封闭假设。三个不同的矩封闭假设导致了三个不同的平均场系统。这些系统可用于不同的目的，例如大网络的分岔分析、稳态点火率分布的预测、实际神经元的参数估计以及参数空间的快速探索。我们使用平均场系统来分析在多个参数的实际异质源下引起的爆发式适应。我们的分析表明，异质性的存在导致与爆发式适应相关的 Hopf 分岔从亚临界变为超临界。对于生物合理的参数值，通过对全网络的数值模拟得到了证实。这种变化降低了适应是海马区 CA3 中爆发的原因的可能性，CA3 是一个有大量强耦合、强适应神经元的区域。
