Geometric framework to predict structure from function in neural networks

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.

arXiv comment46 pages, 12 figures, final version, corrected typos, minor clarifications, new Discussion material

References in corpus (9)

• Empirical Evaluation of Rectified Activations in Convolutional Network
• Weak pairwise correlations imply strongly correlated network states in a neural population
• The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks
• Structural Properties of the Caenorhabditis elegans Neuronal Network
• Brain-wide 3D imaging of neuronal activity in Caenorhabditis elegans with sculpted light
• Parameter Space Compression Underlies Emergent Theories and Predictive Models
• Deconstructing Lottery Tickets: Zeros, Signs, and the Supermask
• Statistical mechanics of complex neural systems and high dimensional data
• A Connectome Based Hexagonal Lattice Convolutional Network Model of the Drosophila Visual System

Cited by in corpus (2)

• Core motifs predict dynamic attractors in combinatorial threshold-linear networks
• Linear Classification of Neural Manifolds with Correlated Variability