Wave Propagation Phenomena in Nonlinear Hierarchical Neural Networks with Predictive Coding Feedback Dynamics

We propose a mathematical framework to systematically explore the propagation properties of a class of continuous in time nonlinear neural network models comprising a hierarchy of processing areas, mutually connected according to the principles of predictive coding. We precisely determine the conditions under which upward propagation, downward propagation or even propagation failure can occur in both bi-infinite and semi-infinite idealizations of the model. We also study the long-time behavior of the system when either a fixed external input is constantly presented at the first layer of the network or when this external input consists in the presentation of constant input with large amplitude for a fixed time window followed by a reset to a down state of the network for all later times. In both cases, we numerically demonstrate the existence of threshold behavior for the amplitude of the external input characterizing whether or not a full propagation within the network can occur. Our theoretical results are consistent with predictive coding theories and allow us to identify regions of parameters that could be associated with dysfunctional perceptions.