Markov chains, neural responses, and optimal temporal computations

Recent experimental data showing that neurons and neural networks can be multi-state systems has raised two important questions: (1) what theoretical frameworks should be employed to model how neurons respond to their inputs in the multi-state setting; and (2) what kinds of computations can be performed by neural systems with multiple states? Addressing the former, this thesis proposes a new multi-state neural response model which is based on the standard hidden Markov model framework, but is expanded to allow both the neural spiking behavior and the hidden state dynamics to be driven by some time-varying external stimulus. The model is then used to show that neural network states may be defined by state-specific dynamics. This result suggests that multiple states may be much more common than previously believed since earlier techniques are unable to recover these dynamically defined states from experimental data. To address the second question, a specific computation for which multi-state systems are well suited is analyzed: namely, the estimation of an interval of time. It is shown analytically that there is a unique system architecture which maximally reduces the error in the time estimate, and is shown numerically that this architecture seems to be robust to the kinds of real-world constraints that physical systems might be expected to encounter. Thus, it is concluded that any physical multi-state system, for which temporal reliability is paramount, would be expected to conform to this optimal architecture.