The firing dynamics of excitable systems are critical to understand organized responses in cortical networks. In this paper, we examine a fractional-order Wilson–Cowan (W–C) neural network model composed of excitatory and inhibitory neuron populations, utilizing Caputo’s fractional-order derivative formalism to explore the influence of fractional-order dynamics on firing behavior. The significance of extending to the fractional-order domain lies in the model’s theoretical framework, which inherently retains memory and hereditary characteristics. We investigate memory-dependent response functions and average neuronal characteristics, enabling us to formulate a fractional-order model that incorporates past dynamics into the neuronal populations’ features. This generalized model is capable of producing alternations between spiking and bursting phenomena, including mixed-mode oscillations (MMOs). Using stability and bifurcation analyses, we delineate the parameter space within which variations in firing patterns emerge. One notable impact of the model’s memory property is the potential stabilization of neuronal activity. We demonstrate that our numerical findings are in alignment with the analytical predictions and that the memory trace depends on the fractional-order dynamics.

中文翻译：

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分数阶 Wilson-Cowan 神经网络系统中的涌现动力学

可兴奋系统的放电动力学对于理解皮质网络中的有组织反应至关重要。在本文中，我们研究了由兴奋性和抑制性神经元群体组成的分数阶威尔逊-考恩（W-C）神经网络模型，利用卡普托的分数阶导数形式主义来探索分数阶动力学对放电行为的影响。扩展到分数阶域的意义在于模型的理论框架，它本质上保留了记忆和遗传特征。我们研究记忆依赖性反应函数和平均神经元特征，使我们能够制定一个分数阶模型，将过去的动态纳入神经元群体的特征中。这种广义模型能够产生尖峰和爆发现象之间的交替，包括混合模式振荡 (MMO)。使用稳定性和分岔分析，我们描绘了发射模式变化出现的参数空间。该模型的记忆特性的一个显着影响是神经元活动的潜在稳定性。我们证明我们的数值结果与分析预测一致，并且记忆轨迹取决于分数阶动力学。