A central goal in neuroscience is to understand how orchestrated computations in the brain arise from the properties of single neurons and networks of such neurons. Answering this question requires theoretical advances that shine a light on the ‘black box’ of representations in neural circuits. In this talk, we will demonstrate theoretical approaches that help describe how cognitive task implementations emerge from the structure in neural populations and from biologically plausible neural networks. We will introduce a new statistical mechanical theory that connects geometric structures that arise from neural responses (i.e., neural manifolds) to the neural representation’s efficiency in implementing a task. In particular, this theory describes how many neural manifolds can be represented (or ‘packed’) in the neural activity space while they can be linearly decoded by a downstream readout neuron. The intuition from this theory is remarkably simple: like a sphere packing problem in physical space, we can encode many “neural manifolds” into the neural activity space if these manifolds are small and low-dimensional, and vice versa. ext, we will describe how such an approach can, in fact, open the ‘black box’ of distributed neuronal circuits in a range of settings, such as experimental neural datasets and artificial neural networks. In particular, our method overcomes the limitations of traditional dimensionality reduction techniques, as it operates directly on the high-dimensional representations rather than relying on low-dimensionality assumptions for visualization. Furthermore, this method allows for simultaneous multi-level analysis, by measuring geometric properties in neural population data and estimating the amount of task information embedded in the same population. This geometric approach is general and can be used across different brain areas and task modalities, as demonstrated in our works and others. Finally, we will discuss our recent efforts to fully extend this multi-level description of neural populations by (1) investigating how single neuron properties shape the representation geometry in early sensory areas and by (2) understanding how task-implementing neural manifolds emerge in downstream areas in biological and artificial networks. By extending our mathematical toolkit for analyzing representations underlying complex neuronal networks, we hope to contribute to the long-term challenge of understanding the neuronal basis of tasks and behaviors.
