Mean-field theory links the physiological properties of individual neurons to the emergent dynamics of neural population activity. These models provide an essential tool for studying brain function at different scales; however, for their application to neural populations on large scale, they need to account for differences between distinct neuron types. The Izhikevich single neuron model can account for a broad range of different neuron types and spiking patterns, thus rendering it an optimal candidate for a mean-field theoretic treatment of brain dynamics in heterogeneous networks. Here, we derive the mean-field equations for networks of all-to-all coupled Izhikevich neurons with heterogeneous spiking thresholds. Using methods from bifurcation theory, we examine the conditions under which the mean-field theory accurately predicts the dynamics of the Izhikevich neuron network. To this end, we focus on three important features of the Izhikevich model that are subject here to simplifying assumptions: (i) spike-frequency adaptation, (ii) the spike reset conditions, and (iii) the distribution of single-cell spike thresholds across neurons. Our results indicate that, while the mean-field model is not an exact model of the Izhikevich network dynamics, it faithfully captures its different dynamic regimes and phase transitions. We thus present a mean-field model that can represent different neuron types and spiking dynamics. The model is comprised of biophysical state variables and parameters, incorporates realistic spike resetting conditions, and accounts for heterogeneity in neural spiking thresholds. These features allow for a broad applicability of the model as well as for a direct comparison to experimental data.

中文翻译：

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具有异构尖峰阈值的神经网络的宏观动力学

平均场理论将单个神经元的生理特性与神经群体活动的涌现动态联系起来。这些模型为研究不同尺度的大脑功能提供了重要工具；然而，为了将它们大规模应用于神经群体，它们需要考虑不同神经元类型之间的差异。Izhikevich 单神经元模型可以解释广泛的不同神经元类型和尖峰模式，从而使其成为异构网络中大脑动力学平均场理论处理的最佳候选者。在这里，我们推导出具有异质尖峰阈值的全对全耦合 Izhikevich 神经元网络的平均场方程。使用分岔理论的方法，我们检查了平均场理论准确预测 Izhikevich 神经元网络动力学的条件。为此，我们专注于 Izhikevich 模型的三个重要特征，这些特征在这里受到简化假设的影响：（i）尖峰频率适应，（ii）尖峰重置条件，以及（iii）单细胞尖峰阈值的分布跨神经元。我们的结果表明，虽然平均场模型不是 Izhikevich 网络动力学的精确模型，但它忠实地捕捉了其不同的动态状态和相变。因此，我们提出了一个平均场模型，可以代表不同的神经元类型和尖峰动力学。该模型由生物物理状态变量和参数组成，结合了现实的尖峰重置条件，并解释了神经尖峰阈值的异质性。