Coding theory was born in 1948 with the work of Claude Shannon, who proved that for every information rate $R$ up to channel capacity, there exists a code of rate $R$ that guarantees a vanishing probability of decoding error. Shannon, however, did not tell us how to find such codes nor how to decode them. It was recognized early on that codes with good Hamming distance can correct many errors, while codes endowed with algebraic structure admit efficient algebraic decoding algorithms. This has led to over 50 years of research in algebraic and combinatorial coding theory. We will survey several key problems and new results in this area. In particular, we'll elaborate upon a new asymptotic improvement of the Gilbert-Varshamov bound and upon recent methods for decoding Reed-Solomon codes using bivariate polynomial interpolation.

About 10 years ago, the field of coding theory was transformed by the discovery of codes defined on certain graphs, with no algebraic structure, that perform extremely close to the Shannon capacity under probabilistic message-passing decoding. We will briefly review this exciting development, and point out the challenges that lie ahead in the area of "probabilistic" coding theory.

There is increasing evidence from in vivo recordings in monkeys trained to respond to stimuli by making left- or rightward eye movements, that firing rates in certain groups of `visual' neurons mimic drift-diffusion processes, rising to a (fixed) threshold prior to movement initiation. This supplements earlier observations of psychologists, that human reaction time and error rate data can be fitted by random walk and diffusion models, and has renewed interest in optimal decision-making ideas from information theory and statistical decision theory as a clue to neural mechanisms.

I will review some results from decision theory and stochastic ordinary differential equations, and show how they may be extended and applied to derive explicit parameter dependencies in optimal performance that may be tested on human and animal subjects. I will then describe a biophysically-based model of a pool of neurons in a brainstem organ - locus coeruleus - that is implicated in widespread norepinephrine release. This neurotransmitter can effect transient gain and response threshold changes in cortical circuits of the type that the abstract drift-diffusion analysis requires. I will argue that, in spite of many gaps and leaps of faith, a rational account of how neural spikes give rise to simple behaviors is beginning to emerge.

This work is in collaboration with Eric Brown, Rafal Bogacz, Jeff Moehlis and Jonathan Cohen (Princeton University), and Ed Clayton, Janusz Rajkowski and Gary Aston-Jones (University of Pennsylvania). It is supported by the National Institutes of Mental Health.