[论文解读] On the dynamics of mean-field equations for stochastic neural fields with delays

Jonathan Touboul|arXiv (Cornell University)|Aug 11, 2011

Neural dynamics and brain function参考文献 37被引用 4

一句话总结

本文研究了在发放率神经元中具有延迟的随机神经场方程的动力学，表明由于解的高斯性质，复杂的平均场方程恰好简化为前两阶矩的延迟积分微分方程。其主要贡献在于非线性地整合了内在噪声，揭示了噪声幅值如何驱动级联霍普夫分岔，并诱导出波和团块分裂等复杂时空模式。

ABSTRACT

The cortex is composed of large-scale cell assemblies sharing the same individual properties and receiving the same input, in charge of certain functions, and subject to noise. Such assemblies are characterized by specific space locations and space-dependent delayed interactions. The mean-field equations for such systems were rigorously derived in a recent paper for general models, under mild assumptions on the network, using probabilistic methods. We summarize and investigate general implications of this result. We then address the dynamics of these stochastic neural field equations in the case of firing-rate neurons. This is a unique case where the very complex stochastic mean-field equations exactly reduce to a set of delayed differential or integro-differential equations on the two first moments of the solutions, this reduction being possible due to the Gaussian nature of the solutions. The obtained equations differ from more customary approaches in that it incorporates intrinsic noise levels nonlinearly and make explicit the interaction between the mean activity and its correlations. We analyze the dynamics of these equations, with a particular focus on the influence of noise levels on shaping the collective response of neural assemblies and brain states. Cascades of Hopf bifurcations are observed as a function of noise amplitude, for noise levels small enough, and delays, in a finite-population system. The presence of spatially homogeneous solutions in law is discussed in different non-delayed neural fields and an instability, as noise amplitude is varied, of the homogeneous state, is found. In these regimes, very complex irregular and structured spatio-temporal patterns of activity are exhibited including in particular wave or bump splitting.

研究动机与目标

• 理解受噪声和空间延迟相互作用影响的大规模神经集合的集体动力学。
• 研究内在噪声水平如何在延迟平均场模型中非线性地塑造神经种群的响应。
• 分析同质活动状态的稳定性以及在不同噪声和延迟参数下复杂时空模式的出现。
• 建立随机平均场方程向发放率神经元矩方程的严格约化。

提出的方法

• 该研究使用概率方法，在较弱的网络假设下严格推导具有延迟的随机神经场的平均场方程。
• 重点研究发放率神经元，其解的高斯性质使得完整随机方程能精确约化为前两阶矩的延迟积分微分方程系统。
• 利用分歧理论分析这些矩方程的动力学，特别关注噪声幅值和延迟作为函数的霍普夫分歧。
• 分析包括在分布意义下的空间同质解的稳定性，以及波和团块分裂等结构化、不规则模式的出现。
• 应用数值与解析技术，探讨噪声对集体脑状态和模式形成的影响。

实验结果

研究问题

• RQ1 内在噪声如何非线性地影响延迟随机神经场中神经集合的集体动力学？
• RQ2 延迟和噪声幅值在有限种群神经系统中如何诱导霍普夫分歧？
• RQ3 在何种条件下，分布意义下的同质态会变得不稳定，从而导致模式形成？
• RQ4 噪声与延迟如何共同塑造波或团块分裂等复杂时空模式的出现？

主要发现

• 随着噪声幅值增加，观察到霍普夫分歧的级联现象，对于足够小的噪声水平，表明集体振荡活动的出现。
• 当噪声幅值变化时，分布意义下的同质态变得不稳定，导致复杂、不规则且结构化的时空模式出现。
• 波和团块分裂现象是系统中噪声诱导不稳定性所直接导致的结果。
• 由于解的高斯性质，完整随机平均场方程对发放率神经元的约化是精确的。
• 所推导的方程明确捕捉了平均活动与其相关性的非线性相互作用，从而与标准方法区分开来。