Optimizing population codes for distributional representation

Animals learn to reason and act successfully based on internal estimates of the state of their environment and their relationship to it. Such perceptual estimation is complicated by noise and ambiguity in sensory signals, but many experiments demonstrate behaviour (in humans and other animals) that accounts near optimally for the resulting uncertainty. Theories of how distributional uncertainty may be represented in neuronal activity have largely focused on competing views of the parametric or stochastic form of the belief representation, and less on how individual codes might be adapted to the nature and shape of uncertainty that an animal encounters. Here, we present a framework within which the optimal encoding functions associated with a distributed distributional code (DDC) can be derived, based on the family of distributions experienced. In particular, we consider encoding functions that constrain as tightly as possible a set of encoded belief distribution functions that are sparse within a fixed basis. This hypothesis leads to a simple optimisation principle for neural codes of uncertainty, leading to an efficient representation of distributional beliefs. The principle determines the shapes of the encoding functions, and provided a normative account for a range of tuning curve shapes. In the context of hippocampal place representations, the framework predicts the adaptive distribution of place fields in different environments, and particularly the accumulation of place fields around frequently occupied locations. Finally, the optimal code alters with statistics and experienced sequence of beliefs, offering a potential account for dynamical changes in tuning curves associated with representational drift.