Cortical neural circuits display highly irregular spiking in individual neurons but variably sized collective firing, oscillations and critical avalanches at the population level, all of which have functional importance for information processing. Theoretically, the balance of excitation and inhibition inputs is thought to account for spiking irregularity and critical avalanches may originate from an underlying phase transition. However, the theoretical reconciliation of these multilevel dynamic aspects in neural circuits remains an open question. Herein, we study excitation-inhibition (E-I) balanced neuronal network with biologically realistic synaptic kinetics. It can maintain irregular spiking dynamics with different levels of synchrony and critical avalanches emerge near the synchronous transition point. We propose a novel semi-analytical mean-field theory to derive the field equations governing the network macroscopic dynamics. It reveals that the E-I balanced state of the network manifesting irregular individual spiking is characterized by a macroscopic stable state, which can be either a fixed point or a periodic motion and the transition is predicted by a Hopf bifurcation in the macroscopic field. Furthermore, by analyzing public data, we find the coexistence of irregular spiking and critical avalanches in the spontaneous spiking activities of mouse cortical slice in vitro, indicating the universality of the observed phenomena. Our theory unveils the mechanism that permits complex neural activities in different spatiotemporal scales to coexist and elucidates a possible origin of the criticality of neural systems. It also provides a novel tool for analyzing the macroscopic dynamics of E-I balanced networks and its relationship to the microscopic counterparts, which can be useful for large-scale modeling and computation of cortical dynamics.

中文翻译：

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平均场中的 Hopf 分岔解释了兴奋抑制平衡神经元网络中的临界雪崩：多尺度变异性的机制


皮层神经回路在单个神经元中表现出高度不规则的尖峰，但在群体水平上表现出不同大小的集体放电、振荡和临界雪崩，所有这些对于信息处理都具有重要的功能。理论上，激发和抑制输入的平衡被认为是尖峰不规则性的原因，而临界雪崩可能源于潜在的相变。然而，神经回路中这些多级动态方面的理论协调仍然是一个悬而未决的问题。在此，我们研究具有生物学真实突触动力学的兴奋抑制（EI）平衡神经元网络。它可以保持不同程度的同步的不规则尖峰动态，并且在同步转变点附近出现临界雪崩。我们提出了一种新颖的半解析平均场理论来推导控制网络宏观动力学的场方程。它揭示了网络的EI平衡状态表现出不规则的个体尖峰，其特征是宏观稳定状态，可以是固定点，也可以是周期性运动，并且通过宏观场中的Hopf分岔来预测过渡。此外，通过分析公开数据，我们发现在体外小鼠皮质切片的自发尖峰活动中，不规则尖峰和临界雪崩并存，表明所观察到的现象的普遍性。我们的理论揭示了允许不同时空尺度的复杂神经活动共存的机制，并阐明了神经系统关键性的可能起源。 它还提供了一种新的工具来分析 EI 平衡网络的宏观动力学及其与微观对应网络的关系，这对于皮质动力学的大规模建模和计算非常有用。