Range, not Independence, Drives Modularity in Biologically Inspired Representations

TL;DR

This paper develops a theory explaining when biologically inspired neural networks form modular representations based on the spread of source data, with implications for neuroscience and machine learning.

Contribution

It introduces necessary and sufficient conditions for modularity in biologically inspired networks, extending beyond independence assumptions to include data spread.

Findings

Sources with sufficiently spread support tend to modularise.

Empirical validation across neural network models supports the theory.

Range independence explains mixing of spatial and reward information in neural data.

Abstract

Why do biological and artificial neurons sometimes modularise, each encoding a single meaningful variable, and sometimes entangle their representation of many variables? In this work, we develop a theory of when biologically inspired networks -- those that are nonnegative and energy efficient -- modularise their representation of source variables (sources). We derive necessary and sufficient conditions on a sample of sources that determine whether the neurons in an optimal biologically-inspired linear autoencoder modularise. Our theory applies to any dataset, extending far beyond the case of statistical independence studied in previous work. Rather we show that sources modularise if their support is ``sufficiently spread''. From this theory, we extract and validate predictions in a variety of empirical studies on how data distribution affects modularisation in nonlinear feedforward and recurrent neural networks trained on supervised and unsupervised tasks. Furthermore, we apply these ideas to neuroscience data, showing that range independence can be used to understand the mixing or modularising of spatial and reward information in entorhinal recordings in seemingly conflicting experiments. Further, we use these results to suggest alternate origins of mixed-selectivity, beyond the predominant theory of flexible nonlinear classification. In sum, our theory prescribes precise conditions on when neural activities modularise, providing tools for inducing and elucidating modular representations in brains and machines.