In this study, we integrated neural encoding and decoding into a unified framework for spatial information processing in the brain. Specifically, the neural representations of self-location in the hippocampus (HPC) and entorhinal cortex (EC) play crucial roles in spatial navigation. Intriguingly, these neural representations in these neighboring brain areas show stark differences. Whereas the place cells in the HPC fire as a unimodal function of spatial location, the grid cells in the EC show periodic tuning curves with different periods for different subpopulations (called modules). By combining an encoding model for this modular neural representation and a realistic decoding model based on belief propagation, we investigated the manner in which self-location is encoded by neurons in the EC and then decoded by downstream neurons in the HPC. Through the results of numerical simulations, we first show the positive synergy effects of the modular structure in the EC. The modular structure introduces more coupling between heterogeneous modules with different periodicities, which provides increased error-correcting capabilities. This is also demonstrated through a comparison of the beliefs produced for decoding two- and four-module codes. Whereas the former resulted in a complete decoding failure, the latter correctly recovered the self-location even from the same inputs. Further analysis of belief propagation during decoding revealed complex dynamics in information updates due to interactions among multiple modules having diverse scales. Therefore, the proposed unified framework allows one to investigate the overall flow of spatial information, closing the loop of encoding and decoding self-location in the brain.

Neural systems are inherently noisy. One well-studied example of a noise reduction mechanism in the brain is the population code, where representing a variable with multiple neurons allows the encoded variable to be recovered with fewer errors. Studies have assumed ideal observer models for decoding population codes, and the manner in which information in the neural population can be retrieved remains elusive. This letter addresses a mechanism by which realistic neural circuits can recover encoded variables. Specifically, the decoding problem of recovering a spatial location from populations of grid cells is studied using belief propagation. We extend the belief propagation decoding algorithm in two aspects. First, beliefs are approximated rather than being calculated exactly. Second, decoding noises are introduced into the decoding circuits. Numerical simulations demonstrate that beliefs can be effectively approximated by combining polynomial nonlinearities with divisive normalization. This approximate belief propagation algorithm is tolerant to decoding noises. Thus, this letter presents a realistic model for decoding neural population codes and investigates fault-tolerant information retrieval mechanisms in the brain.

Graphical models and related algorithmic tools such as belief propagation have proven to be useful tools in (approximately) solving combinatorial optimization problems across many application domains. A particularly combinatorially challenging problem is that of determining solutions to a set of simultaneous congruences. Specifically, a continuous source is encoded into multiple residues with respect to distinct moduli, and the goal is to recover the source efficiently from noisy measurements of these residues. This problem is of interest in multiple disciplines, including neural codes, decentralized compression in sensor networks, and distributed consensus in information and social networks. This letter reformulates the recovery problem as an optimization over binary latent variables. Then we present a belief propagation algorithm, a layered variant of affinity propagation, to solve the problem. The underlying encoding structure of multiple congruences naturally results in a layered graphical model for the problem, over which the algorithms are deployed, resulting in a layered affinity propagation (LAP) solution. First, the convergence of LAP to an approximation of the maximum likelihood (ML) estimate is shown. Second, numerical simulations show that LAP converges within a few iterations and that the mean square error of LAP approaches that of the ML estimation at high signal-to-noise ratios.