Geometric framework to predict structure from function in neural networks

TL;DR

This paper introduces a geometric framework to determine the synaptic connectivity patterns necessary for specific steady-state responses in neural networks, linking structure to function analytically and accounting for noise effects.

Contribution

The authors develop a novel geometric approach to identify required synaptic connections for network responses, providing analytical solutions and topological insights.

Findings

Solution space of connectivity matrices can be analytically characterized.

Solution space topology changes with increasing allowed error.

Certainty conditions for non-zero synapses are derived.

Abstract

Neural computation in biological and artificial networks relies on the nonlinear summation of many inputs. The structural connectivity matrix of synaptic weights between neurons is a critical determinant of overall network function, but quantitative links between neural network structure and function are complex and subtle. For example, many networks can give rise to similar functional responses, and the same network can function differently depending on context. Whether certain patterns of synaptic connectivity are required to generate specific network-level computations is largely unknown. Here we introduce a geometric framework for identifying synaptic connections required by steady-state responses in recurrent networks of threshold-linear neurons. Assuming that the number of specified response patterns does not exceed the number of input synapses, we analytically calculate the solution space of all feedforward and recurrent connectivity matrices that can generate the specified responses from the network inputs. A generalization accounting for noise further reveals that the solution space geometry can undergo topological transitions as the allowed error increases, which could provide insight into both neuroscience and machine learning. We ultimately use this geometric characterization to derive certainty conditions guaranteeing a non-zero synapse between neurons. Our theoretical framework could thus be applied to neural activity data to make rigorous anatomical predictions that follow generally from the model architecture.