A geometric framework to predict structure from function in neural networks

Neural computation in biological and artificial networks relies on nonlinear synaptic integration. The structural connectivity matrix of synaptic weights between neurons is a critical determinant of overall network function. However, 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 rectified-linear neurons. Assuming that the number of specified response patterns does not exceed the number of input synapses, we analytically calculate all feedforward and recurrent connectivity matrices that can generate the specified responses from the network inputs. We then use this analytical characterization to rigorously analyze the solution space geometry and derive certainty conditions guaranteeing a non-zero synapse between neurons. Numerical simulations of feedforward and recurrent networks verify our analytical results. Our theoretical framework could be applied to neural activity data to make anatomical predictions that follow generally from the model architecture. It thus provides novel opportunities for discerning what model features are required to accurately relate neural network structure and function.