The Wilson-Cowan neural field equations describe the dynamical behavior of a 1-D continuum of excitatory and inhibitory cortical neural aggregates, using a pair of coupled integro-differential equations. Here we use bifurcation theory and small-noise linear stochastics to study the range of a phase transitions-sudden qualitative changes in the state of a dynamical system emerging from a bifurcation-accessible to the Wilson-Cowan network. Specifically, we examine saddle-node, Hopf, Turing, and Turing-Hopf instabilities. We introduce stochasticity by adding small-amplitude spatio-temporal white noise, and analyze the resulting subthreshold fluctuations using an Ornstein-Uhlenbeck linearization. This analysis predicts divergent changes in correlation and spectral characteristics of neural activity during close approach to bifurcation from below. We validate these theoretical predictions using numerical simulations. The results demonstrate the role of noise in the emergence of critically slowed precursors in both space and time, and suggest that these early-warning signals are a universal feature of a neural system close to bifurcation. In particular, these precursor signals are likely to have neurobiological significance as early warnings of impending state change in the cortex. We support this claim with an analysis of the in vitro local field potentials recorded from slices of mouse-brain tissue. We show that in the period leading up to emergence of spontaneous seizure-like events, the mouse field potentials show a characteristic spectral focusing toward lower frequencies concomitant with a growth in fluctuation variance, consistent with critical slowing near a bifurcation point. This observation of biological criticality has clear implications regarding the feasibility of seizure prediction.

中文翻译：

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随机的Wilson-cowan模型中由噪声引起的状态转换前兆。

威尔逊-科万（Wilson-Cowan）神经场方程使用一对耦合的积分微分方程描述了一维连续的兴奋性和抑制性皮质神经聚集体的动力学行为。在这里，我们使用分叉理论和小噪声线性随机性研究了从分叉可访问到Wilson-Cowan网络的动力系统的状态突然发生质变的相变范围。具体来说，我们研究了鞍结，Hopf，Turing和Turing-Hopf不稳定性。我们通过添加小幅度时空白噪声来引入随机性，并使用Ornstein-Uhlenbeck线性化分析由此产生的亚阈值波动。该分析预测了从下面接近分叉过程中神经活动的相关性和频谱特征的变化。我们使用数值模拟验证了这些理论预测。结果表明，噪声在空间和时间上都显着减慢了前体的出现，并表明这些预警信号是接近分叉的神经系统的普遍特征。特别是，这些前体信号可能具有神经生物学意义，作为即将发生的皮质状态改变的预警。我们通过对小鼠脑组织切片中记录的体外局部场电位进行分析来支持这一主张。我们表明，在导致自发性癫痫样事件出现的时期，小鼠场电位显示出一个特征频谱，该频谱集中于较低的频率，并伴随着波动方差的增长，这与分叉点附近的临界减慢相一致。