Neural Dynamics as the Composition of Quantized Units

Abstract

Deep learning is commonly interpreted at two levels: the macroscopic, through aggregate trends in loss summarized by scaling laws, and the microscopic, through neurons, features, and circuits. A central challenge is understanding how these levels connect, so that we can explain how elementary computations compose and collectively shape macroscopic behavior. To this end, we study an intermediate abstraction in which training is described as the ordered acquisition of quanta: reusable computations acquired suddenly and binary-activated across examples to reduce loss. By approximating population-gradient updates, we derive quanta's acquisition dynamics. This yields an acquisition priority governed by demand, how frequently a computation is required across examples, and conditional complexity, how difficult that computation is to acquire given those already available. In a Boolean compositional task, we derive predictions for acquisition order and show how staggered discrete acquisitions can produce smooth aggregate loss and, under certain geometries of quanta composition, give rise to scaling laws. We then train a Transformer to map numerals to English number names and recover candidate quanta from its checkpoint trajectory. From these units, we construct a model that preserves much of the Transformer's behavior while exposing interpretable latent computations and acquisition dynamics consistent with the theory. Separately, the quanta structure can serve as training targets to improve transformer generalization. Together, these results suggest the quanta abstraction can provide useful computational atoms for studying a variety of macroscopic phenomena.