# Activity propagation with Hebbian learning

## TL;DR

This paper explores how Hebbian learning influences infection and activity spreading models, revealing complex behaviors like phase transitions, immune regions, and Griffiths effects driven by local reinforcement mechanisms.

## Contribution

It introduces a novel extension of the contact process incorporating Hebbian learning, demonstrating diverse emergent phenomena and global effects from local incentives.

## Key findings

- Negative reinforcement promotes activity spreading and immune regions.
- Positive reinforcement can cause Griffiths effects and non-universal scaling.
- Emergent behaviors include phase transitions and complex activity patterns.

## Abstract

We investigate the impact of Hebbian learning on the contact process, a paradigmatic model for infection spreading, which has been also proposed as a simple model to capture the dynamics of inter-regional brain activity propagation as well as population spreading. Each of these contexts calls for an extension of the contact process with local learning. We introduce Hebbian learning as a positive or negative reinforcement of the activation rate between a pair of sites after each successful activation event. Learning can happen either in both directions motivated by social distancing (mutual learning model), or in only one of the directions motivated by brain and population dynamics (source or target learning models). Hebbian learning leads to a rich class of emergent behavior, where local incentives can lead to the opposite global effects. In general, positive reinforcement (increasing activation rates) leads to a loss of the active phase, while negative reinforcement (reducing activation rates) can turn the inactive phase into a globally active phase. In two dimensions and above, the effect of negative reinforcement is twofold: it promotes the spreading of activity, but at the same time gives rise to the appearance of effectively immune regions, entailing the emergence of two distinct critical points. Positive reinforcement can lead to Griffiths effects with non-universal power-law scaling, through the formation of random loops of activity, a manifestation of the ``ant mill" phenomenon.