Probing the structure–function relationship with neural networks constructed by solving a system of linear equations

Neural network models are an invaluable tool to understand brain function, since they allow to connect the cellular and circuit levels with behaviour. Neural networks usually comprise a huge number of parameters, which must be chosen carefully such that networks reproduce anatomical, behavioural and neurophysiological data. These parameters are usually fitted with off-the-shelf optimization algorithms that iteratively change network parameters and simulate the network to evaluate the changes and improve fitting. Here we propose to invert the fitting process by proceeding from the network dynamics towards network parameters. Firing state transitions are chosen according to the transition graph followed by an agent when solving a given behavioural task. Then, a system of linear equations is constructed from the network firing states and membrane potentials, in such a way that system consistency in guarantee. This allows to uncouple the activity features of the model, like its neurons firing rate and correlation, from the connectivity features and from the task-solving algorithm implemented by the network, allowing to fit these three levels separately. We employed the method to probe the structure-function relationship in a stimuli sequence memory task, finding solution networks where commonly employed optimization algorithms failed. The constructed networks showed reciprocity and correlated firing patterns that recapitulated experimental observations. We argue that the proposed method is a complementary and needed alternative to the way neural networks are constructed to model brain function.

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