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 .
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 , 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 rectified-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 .
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the solution space of
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We then use this analytical characterization to rigorously analyze
A generalization accounting for noise further reveals that
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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
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Numerical simulations of feedforward and recurrent networks verify our analytical results.
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. It thus provides novel opportunities for discerning what model features are required to accurately relate neural network structure and function