Bastien Lelan

Humans get faster and more accurate with experience, a fact that casts the brain as a statistical learner. This principle underpins normative theories framing perception as inference. However, turning this idea into testable predictions remains challenging without additional constraints. As a descriptive counterpart, the neural-manifold literature characterizes population activity as evolving on a smooth, low-dimensional, and geometrically constrained manifold. My PhD project combines statistical learning and manifold geometry to derive a normative, yet testable theory of cognition. I introduce the hypothesis that cognition—from perception to decision-making—follows a path of least cognitive effort: the `shortest path" (a geodesic) through a mental space continuously shaped by our experiences. To make this mental space measurable, I propose a novel framework that uses Riemannian geometry to model this mental landscape. The framework provides a "local ruler" that dynamically recalibrates with learning: in familiar conceptual regions, mental distances" are compressed, allowing for faster processing, while in unfamiliar regions they are expanded. Using state-of-the-art generative models—especially Energy-Based Models—we learn a familiarity field that estimates statistical exposure. This familiarity field directly sets the local ruler, which scales distances inversely with exposure. Our geodesics law of cognition yields testable predictions: at the behavioral level, geodesic length tracks reaction time (RT) and geodesic endpoints predict response accuracy & variability; at the neural level, geodesics map onto population dynamics. Early in the thesis, I will test these predictions at the behavioral level. In mental-rotation tasks, I will estimate how statistical exposure reshapes reaction-time profiles as participants rotate a query shape to match a target. Later, I will analyze neural recordings, including monkey electrophysiology, to benchmark our experience-shaped manifold against traditional neural manifold approaches. We expect our model to better capture space warping when statistical exposure is causally manipulated. By offering a unifying geometry for perception, learning, and cognition, my project will not only advance our fundamental understanding of the brain but also introduce new tools to probe the mind and a path toward more robust, human-aligned AI.