This article is part of the Physical Review Research collection titled Physics of Neuroscience.

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 nonzero 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.

• Received 15 December 2021
• Accepted 25 May 2022

DOI:https://doi.org/10.1103/PhysRevResearch.4.023255

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

This article appears in the following collection:

Physics of Neuroscience

Physics of Neuroscience has emerged as an area of research at the edge of a breadth of multiple disciplines, ranging from statistical physics and computer science all the way to biomedical research. As a venue committed to multi- and interdisciplinary work with a connection to Physics, Physical Review Research is delighted to start a Collection, organized by guest editors David Schwab and Carina Curto, to host the outstanding research that is being produced in this field and to invite this burgeoning community to partner with us in our mission to disseminate novel and significant content to the public.