Sensory perception (e.g. vision) relies on a hierarchy of cortical areas, in which neural activity propagates in both directions, to convey information not only about sensory inputs but also about cognitive states, expectations and predictions. At the macroscopic scale, neurophysiological experiments have described the corresponding neural signals as both forward and backward-travelling waves, sometimes with characteristic oscillatory signatures. It remains unclear, however, how such activity patterns relate to specific functional properties of the perceptual apparatus. Here, we present a mathematical framework, inspired by neural network models of predictive coding, to systematically investigate neural dynamics in a hierarchical perceptual system. We show that stability of the system can be systematically derived from the values of hyper-parameters controlling the different signals (related to bottom-up inputs, top-down prediction and error correction). Similarly, it is possible to determine in which direction, and at what speed neural activity propagates in the system. Different neural assemblies (reflecting distinct eigenvectors of the connectivity matrices) can simultaneously and independently display different properties in terms of stability, propagation speed or direction. We also derive continuous-limit versions of the system, both in time and in neural space. Finally, we analyze the possible influence of transmission delays between layers, and reveal the emergence of oscillations at biologically plausible frequencies.

翻译：感知（例如视觉）依赖于皮层区域的分层结构，其中神经活动沿双向传播，不仅传递关于感觉输入的信息，还传递关于认知状态、期望和预测的信息。在宏观尺度上，神经生理学实验已将相应的神经信号描述为前向和后向传播波，有时伴有特征性振荡标志。然而，这些活动模式如何与感知装置的具体功能特性相关联仍不清楚。在此，我们提出一个受预测编码神经网络模型启发的数学框架，系统研究分层感知系统中的神经动力学。我们展示系统的稳定性可由控制不同信号（与自下而上输入、自上而下预测及误差修正相关）的超参数值系统推导得出。类似地，可以确定神经活动在系统中的传播方向及速度。不同的神经集群（反映连接矩阵的不同特征向量）可同时且独立地在稳定性、传播速度或方向方面展现不同特性。我们还推导出系统在时间和神经空间中的连续极限版本。最后，我们分析层间传输延迟的可能影响，并揭示在生物可解释频率下振荡的出现。