The ability of the brain to retain, work with, and recall information over time is believed to be underwritten by circuit processes that extend the short time-scales on which its units operate, and codes that mitigate the effects of noise and maximize capacity.

I will discuss how mechanistic but simplified models are helping to unravel the architectures underlying such circuits. As a particular example, I will focus on how the cortical grid cell response may emerge from neural interactions. A tight loop between theory, experiment, and quantitative data analysis, together with the fortuitous properties of the grid cell system, have allowed us to reach a level of understanding about mechanism which rivals that for orientation tuning in V1, one of the best-studied cortical systems in neuroscience.

I will next discuss how different neural codes can dramatically influence the capacity and fidelity of neural representations. I will describe a qualitatively new class of population codes that may enable N neurons to robustly represent an analog variable with a dynamic range that grows exponentially with N, in contrast to previously characterized neural codes, whose dynamic range grows only linearly with N. I will conclude by describing a recent theoretical construction for Hopfield networks, using expander graphs, that exhibits exponentially many, robust stable states, again in contrast to previous constructions that achieved only sub-exponential performance.