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Wilson–Cowan Equations for Neocortical Dynamics

To analyze such effects we need to look more closely at the attractor dynamics of Eq. (7). There are two cases to consider. In case 1, the attractor is either an asymptotically stable node or focus, or else a limit cycle. In case 2, the attractor is only marginally stable. In nonlinear dynamics this is a bifurcation point, e.g. a Bogdanov–Takens point, or a saddle node or Andronov–Hopf point. In statistical mechanics this is the critical point of a phase transition. 5.1 The System-Size Expansion The system-size expansion was introduced by van Kampen [36] to analyze the effects of intrinsic fluctuations in case 1. The intuition behind this approach comes from the idea that if neurons are independently activated, then the total activity in a excitatory neural network in such a…