Wave propagation phenomena in nonlinear hierarchical neural networks with predictive coding feedback dynamics

TL;DR

This paper develops a mathematical framework to analyze wave propagation in nonlinear hierarchical neural networks with predictive coding feedback, identifying conditions for propagation, failure, and long-term behavior under various inputs.

Contribution

It introduces a systematic approach to determine propagation conditions in hierarchical predictive coding neural models, linking theoretical analysis with neural perception phenomena.

Findings

Conditions for upward, downward, or failed propagation identified.

Threshold behavior for external input amplitude demonstrated.

Parameter regions linked to potential perceptual dysfunctions.

Abstract

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.