Department of Applied Mathematics and Centre for Theoretical Neuroscience, University of Waterloo, Waterloo, N2L 3G1, ON, Canada.
J Comput Neurosci. 2022 Nov;50(4):445-469. doi: 10.1007/s10827-022-00825-9. Epub 2022 Jul 14.
Networks of spiking neurons with adaption have been shown to be able to reproduce a wide range of neural activities, including the emergent population bursting and spike synchrony that underpin brain disorders and normal function. Exact mean-field models derived from spiking neural networks are extremely valuable, as such models can be used to determine how individual neurons and the network they reside within interact to produce macroscopic network behaviours. In the paper, we derive and analyze a set of exact mean-field equations for the neural network with spike frequency adaptation. Specifically, our model is a network of Izhikevich neurons, where each neuron is modeled by a two dimensional system consisting of a quadratic integrate and fire equation plus an equation which implements spike frequency adaptation. Previous work deriving a mean-field model for this type of network, relied on the assumption of sufficiently slow dynamics of the adaptation variable. However, this approximation did not succeed in establishing an exact correspondence between the macroscopic description and the realistic neural network, especially when the adaptation time constant was not large. The challenge lies in how to achieve a closed set of mean-field equations with the inclusion of the mean-field dynamics of the adaptation variable. We address this problem by using a Lorentzian ansatz combined with the moment closure approach to arrive at a mean-field system in the thermodynamic limit. The resulting macroscopic description is capable of qualitatively and quantitatively describing the collective dynamics of the neural network, including transition between states where the individual neurons exhibit asynchronous tonic firing and synchronous bursting. We extend the approach to a network of two populations of neurons and discuss the accuracy and efficacy of our mean-field approximations by examining all assumptions that are imposed during the derivation. Numerical bifurcation analysis of our mean-field models reveals bifurcations not previously observed in the models, including a novel mechanism for emergence of bursting in the network. We anticipate our results will provide a tractable and reliable tool to investigate the underlying mechanism of brain function and dysfunction from the perspective of computational neuroscience.
具有适应能力的尖峰神经元网络已被证明能够再现广泛的神经活动，包括突发种群爆发和尖峰同步，这是大脑疾病和正常功能的基础。从尖峰神经网络中得出的精确平均场模型非常有价值，因为这些模型可用于确定单个神经元及其所在网络如何相互作用以产生宏观网络行为。在本文中，我们为具有尖峰频率适应的神经网络推导出并分析了一组精确的平均场方程。具体来说，我们的模型是一个伊泽基维奇神经元网络，其中每个神经元由一个二维系统建模，该系统由一个二次积分和点火方程和一个实现尖峰频率适应的方程组成。以前为这种类型的网络推导出平均场模型的工作依赖于适应变量的足够慢动力学的假设。然而，这种逼近在建立宏观描述与现实神经网络之间的精确对应关系方面并没有成功，特别是当适应时间常数不大时。挑战在于如何在包含适应变量的平均场动力学的情况下获得一组封闭的平均场方程。我们通过使用洛伦兹ansatz 并结合矩封闭方法来解决这个问题，从而在热力学极限下得到一个平均场系统。得到的宏观描述能够定性和定量地描述神经网络的集体动力学，包括个体神经元表现出异步紧张发射和同步爆发的状态之间的转变。我们将该方法扩展到两个神经元种群的网络，并通过检查推导过程中施加的所有假设，来讨论我们的平均场近似的准确性和有效性。我们的平均场模型的数值分岔分析揭示了以前在模型中未观察到的分岔，包括网络中突发出现的新机制。我们预计我们的结果将为从计算神经科学的角度研究大脑功能和功能障碍的潜在机制提供一个易于处理和可靠的工具。
