The traditional Wilson-Cowan model of excitatory and inhibitory meanfield interactions in neuronal populations considers a weak Gamma distribution of time delays when processing inputs, and is obtained via a time-coarse graining technique that averages the population response. Previous analyses of the stability of the Wilson-Cowan model focused on more simplified cases, where the delays were either not present, constant or were of a specific type. Since these simplifications may significantly alter the behavior of the model, we focus on understanding the behavior of the system before time-course graining, and for a wider range of delay distributions.

For these generalized delay equations, we perform stability and bifurcation analyses with respect to parameters that capture both the coupling profile, and the time delay. The investigation is done through the examination of the system’s associated characteristic equation. Under mild assumptions, we give complete mathematical proofs of our theoretical results, for the model with general delay distributions and prove the transversality condition for the possible Hopf bifurcations, in a generalized context. The stability region in this parameter space is described theoretically for several types of delay kernels, and numerical simulations are presented to substantiate the theoretical results.

We found that the stability regions and bifurcations differ significantly between different types of delay distributions: weak Gamma distributions promote stable firing rates, while strong Gamma distributions are associated with regular oscillations, and Dirac distributions appear to facilitate more complex aperiodic patterns. This supports the unexplored possibility of different delay distributions being used as the substrate for different functional behaviors, and emphasizes the importance of a careful choice of the delay kernel in the mathematical model.

We illustrate these theoretical principles in an application to a basal ganglia circuit, in which $\beta$-band oscillations have been associated with Parkinson’s Disease.

神经元群体中兴奋性和抑制性平均场相互作用的传统 Wilson-Cowan 模型在处理输入时考虑了时间延迟的弱 Gamma 分布，并且是通过平均群体响应的时间粗粒度技术获得的。之前对 Wilson-Cowan 模型稳定性的分析侧重于更简化的情况，其中延迟要么不存在，要么是恒定的，要么是特定类型的。由于这些简化可能会显着改变模型的行为，因此我们专注于在时程粒度化之前了解系统的行为，以及更广泛的延迟分布。

对于这些广义延迟方程，我们针对捕获耦合分布和时间延迟的参数执行稳定性和分岔分析。调查是通过检查系统的相关特征方程来完成的。在温和的假设下，我们对具有一般延迟分布的模型的理论结果给出了完整的数学证明，并在广义上下文中证明了可能的 Hopf 分岔的横向条件。该参数空间中的稳定区域针对几种类型的延迟内核进行了理论上的描述，并提供了数值模拟来证实理论结果。

我们发现不同类型的延迟分布之间的稳定性区域和分岔有显着差异：弱 Gamma 分布促进稳定的发射率，而强 Gamma 分布与规则振荡相关，而 Dirac 分布似乎促进更复杂的非周期性模式。这支持了将不同延迟分布用作不同功能行为的基础的未探索可能性，并强调了在数学模型中谨慎选择延迟内核的重要性。

我们在基底神经节回路的应用中说明了这些理论原理，其中 $\beta$带振荡与帕金森氏病有关。