Optimal Decisions: From Neural Spikes, through Stochastic Differential Equations, to Behavior*This paper is an expanded version of a plenary lecture delivered at the International Symposium on Nonlinear Theory and its Applications (NOLTA2004), Fukuoka, Japan, Nov. 29--Dec. 3, 2004.

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 neurons in oculo-motor areas 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. We review 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. We then briefly describe a biophysically-based model of a pool of neurons in locus coeruleus, a brainstem nucleus implicated in widespread norepinephrine release. This neurotransmitter can effect transient gain changes in cortical circuits of the type that the abstract drift-diffusion analysis requires. We also describe how optimal gain schedules can be computed in the presence of time-varying noisy signals. We argue that a rational account of how neural spikes give rise to simple behaviors is beginning to emerge.