The geometry of neural representation: From neural tuning to manifold geometry

Neural activity is inherently high-dimensional, yet it often self-organizes into low-dimensional neural manifolds that capture relational structures among visual stimuli and support perception and cognition. Although this framework provides a quantitative basis for object representation, the mechanisms by which neuronal tuning generates such manifolds and determines their impact on behavioral performance remain poorly understood. Here, we propose a unified analytical framework that integrates theoretical modeling, multi-objective optimization, and large-scale simulations to elucidate object representation under the neural manifold theory. By systematically manipulating neuronal tuning parameters, this framework reveals how neuronal tuning gives rise to neural manifolds and how these manifolds, in turn, shape the information distribution and behavioral performance of object representations, thereby explaining key neural functions such as continuous-variable encoding, categorical discrimination, and invariant representation. Building on this framework, we show that under structural constraints neuronal tuning patterns naturally emerge as optimal solutions of multi-objective optimization, balancing representational precision, metabolic cost, and informational redundancy. Extending these principles to artificial systems, we further apply geometric constraints to deep neural networks and find that they spontaneously develop biologically plausible tuning, which markedly improves both the precision and the fidelity of continuous representations. Together, these findings establish a geometry-based analytical framework that elucidates how neural systems achieve efficient and adaptive representations under competing functional demands.