In this work, we establish the existence of traveling fronts in a fractional-order formulation of the Amari neural field model. Fractional-order models act as a memory index of the underlying dynamical system. Therefore, in a fractional-order neural field model, we potentially incorporate the effect of neuronal collective memory. Considering Caputo’s fractional derivative framework and a fractional-order of $0\le \alpha \le 1$, we establish explicit front solutions that allow us to analyze frontspeed and frontshape features directly. Furthermore, considering an exponential synaptic connectivity kernel, we find a bifurcation on the effect of fractional-order on front features. In particular, we find the existence of a critical synaptic threshold, $k_{\star }$, that qualitatively modifies the effect of fractional order on frontspeed. Below this critical threshold, fractional-order increases frontspeed whereas, above this threshold, fractional-order decreases frontspeed. In particular, less fractional-order implies a more substantial impact on frontspeed (either by increasing or decreasing frontspeed). Also, we find that lower fractional orders imply, in general, a slower power-law tendency towards the excited state. Therefore, our results establish the presence of different dynamics in the propagation of spatio-temporal patterns on neural fields due to the incorporation of a fractional-order framework and a potential memory index.

在这项工作中，我们在 Amari 神经场模型的分数阶公式中建立了行进前沿的存在。分数阶模型充当底层动力系统的记忆索引。因此，在分数阶神经场模型中，我们可能会结合神经元集体记忆的影响。考虑 Caputo 的分数阶导数框架和分数阶$0\le \alpha \le 1$，我们建立了明确的前沿解决方案，使我们能够直接分析前沿速度和前沿形状特征。此外，考虑到指数突触连接核，我们发现分数阶对前端特征的影响存在分歧。特别是，我们发现存在一个关键的突触阈值，${ķ}_{\star }$，这定性地修改了分数阶对前速的影响。低于这个临界阈值，分数阶会增加前速，而高于这个阈值，分数阶会降低前速。特别是，较少的分数阶意味着对前端速度的更大影响（通过增加或降低前端速度）。此外，我们发现较低的分数阶通常意味着激发态的幂律趋势较慢。因此，由于分数阶框架和潜在记忆指数的结合，我们的结果确定了在神经场上的时空模式传播中存在不同的动态。