In 1972-1973 Wilson and Cowan introduced a mathematical model of the population dynamics of synaptically coupled excitatory and inhibitory neurons in the neocortex. The model dealt only with the mean numbers of activated and quiescent excitatory and inhibitory neurons, and said nothing about fluctuations and correlations of such activity. However, in 1997 Ohira and Cowan, and then in 2007-2009 Buice and Cowan introduced Markov models of such activity that included fluctuation and correlation effects. Here we show how both models can be used to provide a quantitative account of the population dynamics of neocortical activity.We first describe how the Markov models account for many recent measurements of the resting or spontaneous activity of the neocortex. In particular we show that the power spectrum of large-scale neocortical activity has a Brownian motion baseline, and that the statistical structure of the random bursts of spiking activity found near the resting state indicates that such a state can be represented as a percolation process on a random graph, called directed percolation.Other data indicate that resting cortex exhibits pair correlations between neighboring populations of cells, the amplitudes of which decay slowly with distance, whereas stimulated cortex exhibits pair correlations which decay rapidly with distance. Here we show how the Markov model can account for the behavior of the pair correlations.Finally we show how the 1972-1973 Wilson-Cowan equations can account for recent data which indicates that there are at least two distinct modes of cortical responses to stimuli. In mode 1 a low intensity stimulus triggers a wave that propagates at a velocity of about 0.3 m/s, with an amplitude that decays exponentially. In mode 2 a high intensity stimulus triggers a larger response that remains local and does not propagate to neighboring regions.

中文翻译：

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新皮质动力学的威尔逊-考恩方程。

1972-1973 年，Wilson 和 Cowan 引入了新皮质中突触耦合的兴奋性和抑制性神经元群体动态的数学模型。该模型仅处理激活和静止的兴奋性和抑制性神经元的平均数量，而没有提及此类活动的波动和相关性。然而，Ohira 和 Cowan 在 1997 年以及随后在 2007-2009 年 Buice 和 Cowan 引入了此类活动的马尔可夫模型，其中包括波动和相关效应。在这里，我们展示了如何使用这两种模型来定量描述新皮质活动的群体动态。我们首先描述马尔可夫模型如何解释新皮质静息或自发活动的许多最新测量结果。特别是，我们表明，大规模新皮质活动的功率谱具有布朗运动基线，并且在静息状态附近发现的尖峰活动随机爆发的统计结构表明，这种状态可以表示为渗透过程其他数据表明，静息皮层表现出相邻细胞群之间的成对相关性，其幅度随着距离而缓慢衰减，而受刺激的皮层表现出成对相关性，随着距离而快速衰减。在这里，我们展示了马尔可夫模型如何解释配对相关性的行为。最后，我们展示了 1972-1973 年威尔逊-考恩方程如何解释最近的数据，这些数据表明皮层对刺激的反应至少有两种不同的模式。在模式 1 中，低强度刺激会触发以约 0.3 m/s 的速度传播的波，其幅度呈指数衰减。在模式 2 中，高强度刺激会触发更大的反应，该反应仍然是局部的，不会传播到邻近区域。