AI 中文总结
该研究分析异构图上反馈伊辛神经网络，发现度异质性通过度矩比调控同步极限环的产生与破坏，揭示了运动学波和相分离态等均匀系统不存在的物理现象，为神经非平衡动力学提供了数学框架。
AI 中文摘要
结构异质性制约着复杂系统的集体动力学，但其非平衡态下的分析可处理性仍十分有限。本研究针对一类由神经元兴奋性与群体放电率间的稳态反馈环驱动至非平衡态的动力学伊辛神经网络展开研究。采用经蒙特卡洛模拟验证的居里-外斯异质平均场近似，我们对异质网络上通过安德罗诺夫-霍普夫分岔产生宏观同步极限环的过程进行了分析表征。我们推导得到了闭式相边界，结果表明振荡的 onset 明确受网络异质性通过度矩比调控。度异质性将单个神经元的放电率 m 与单个突触的放电率 u 解耦，产生了均匀系统中不存在的物理现象，包括：(i) 按度顺序排列的、从网络外围向中枢节点传播的序列激活运动学波；(ii) 经叉形分岔产生的低温相分离态。我们证明，对于高度异质的拓扑结构，该相分离不动点会稳定存在并动态破坏同步极限环。这些结果为理解异质性如何调控神经网络中的宏观振荡与非平衡相变提供了数学框架。
英文摘要
Structural heterogeneity constrains collective dynamics in complex systems. However, its analytical tractability out of equilibrium remains limited. In this work, we study a class of kinetic Ising neural networks driven out of equilibrium by a homeostatic feedback loop between the neuronal excitability and the population firing rate. Using a Curie-Weiss heterogeneous mean-field approximation validated by Monte Carlo simulations, we provide an analytical characterization of how a macroscopic synchronized limit cycle emerges via an Andronov-Hopf bifurcation on heterogeneous networks. We derive closed-form phase boundaries and show that the onset of oscillations is explicitly controlled by network heterogeneity through the degree moment ratio. Degree heterogeneity decouples the spiking rate per neuron m from the spiking rate per synapse u, generating physical phenomena absent in homogeneous systems. These include (i) kinematic waves of sequential, degree-ordered activations propagating from the network periphery to the hubs, and (ii) a low-temperature phase-separated state emerging via a pitchfork bifurcation. We prove that for highly heterogeneous topologies, this phase-separated fixed point stabilizes and dynamically destroys the synchronized limit cycle. These results provide a mathematical framework for understanding how heterogeneity regulates macroscopic oscillations and out-of-equilibrium transitions in neural networks
Comments15 pages, 5 figures
