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Abstract
Neural activity in the brain is stochastic in nature, with variable responses across trials under the same experimental conditions. It is also high-dimensional, with thousands of neurons in a network working in concert to fulfill a computational role. This dissertation presents three models that each employ different mathematical techniques to produce theoretical insights on neural variability without reducing the dimensionality of the system as is classically done in mean-field theories.
In the first project, we prove in a recurrent circuit model that the more heterogeneous the firing rates of neurons in a population, the lower the effective dimension of their trial-to-trial covariability. This was achieved by using operator-valued free probability theory to analyze the interaction between external inputs and recurrent dynamics.
The second project addresses a long-standing question in neuroscience about how the neural code for a sensory, motor, or cognitive variable should be organized to optimize its discriminability. We analytically minimize the average binary classification error of a circular variable in the function space of all population tuning curves for various noise models by solving a nonlocal variational problem. We obtained the solution by viewing the space of neural response distributions as a Riemannian manifold in the sense of information geometry and utilizing a result from knot energy theory. The first two projects both make novel predictions that are verified in experimental data.
The third project also concerns discriminability, but addresses the classification of discrete classes instead of continuous variables. Neural manifold capacity is a recently introduced measure of representational geometry that quantifies how many objects a large population can represent while ensuring the feasibility of linear classifications with high probability. We present novel derivations of three versions of the manifold capacity formula based on integral geometry, which introduce a new mathematical perspective to the problem compared to the original derivation based on statistical mechanics techniques.