Exact linear theory of perturbation response in a space- and feature-dependent cortical circuit model

What are the principles that govern the responses of cortical networks to their inputs and the emergence of these responses from recurrent connectivity? Recent experiments have probed these questions by measuring cortical responses to two-photon optogenetic perturbations of single cells in the mouse primary visual cortex. A robust theoretical framework is needed to determine the implications of these responses for cortical recurrence. Here we propose a novel analytical approach: a formulation of the dependence of cell-type-specific connectivity on spatial distance that yields an exact solution for the linear perturbation response of a model with multiple cell types and space- and feature-dependent connectivity. Importantly and unlike previous approaches, the solution is valid in regimes of strong as well as weak intra-cortical coupling. Analysis reveals the structure of connectivity implied by various features of single-cell perturbation responses, such as the surprisingly narrow spatial radius of nearby excitation beyond which inhibition dominates, the number of transitions between mean excitation and inhibition thereafter, and the dependence of these responses on feature preferences. Comparison of these results to existing optogenetic perturbation data yields constraints on cell-type-specific connection strengths and their tuning dependence. Finally, we provide experimental predictions regarding the response of inhibitory neurons to single-cell perturbations and the modulation of perturbation response by neuronal gain; the latter can explain observed differences in the feature-tuning of perturbation responses in the presence vs. absence of visual stimuli.