Title: Methods for studying the neural code in high dimensions
Description: Over the last two decades technological developments in multi-electrode arrays and fluorescence microscopy have made it possible to simultaneously record from hundreds to thousands of neurons. Developing methods for analyzing these data in order to learn how networks of neurons respond to external stimuli and process information is an outstanding challenge for neuroscience. In this dissertation, I address the challenge of developing and testing models that are both flexible and computationally tractable when used with high dimensional data. In chapter 2 I will discuss an approximation to the generalized linear model (GLM) log-likelihood that I developed in collaboration with my thesis advisor. This approximation is designed to ease the computational burden of evaluating GLMs. I will show that our method reduces the computational cost of evaluating the GLM log-likelihood by a factor proportional to the number of parameters in the model times the number of observations. Therefore it is most beneficial in typical neuroscience applications where the number of parameters is large.

I then detail a variety of applications where our method can be of use, including Maximum Likelihood estimation of GLM parameters, marginal likelihood calculations for model selection and Markov chain Monte Carlo methods for sampling from posterior parameter distributions. I go on to show that our model does not necessarily sacrifice accuracy for speed. Using both analytic calculations and multi-unit, primate retinal responses, I show that parameter estimates and predictions using our model can have the same accuracy as that of generalized linear models. In chapter 3 I study the neural decoding problem of predicting stimuli from neuronal responses. The focus is on reconstructing zebra finch song spectrograms, which are high-dimensional, by combining the spike trains of zebra finch auditory midbrain neurons with information about the correlations present in all zebra finch song. I use a GLM to model neuronal responses and a series of prior distributions, each carrying different amounts of statistical information about zebra finch song.

For song reconstruction I make use of recent connections made between the applied mathematics literature on solving linear systems of equations involving matrices with special structure and neural decoding. This allowed me to calculate \textit{maximum a posteriori} (MAP) estimates of song spectrograms in a time that only grows linearly, and is therefore quite tractable, with the number of time-bins in the song spectrogram. This speed was beneficial for answering questions which required the reconstruction of a variety of song spectrograms each corresponding to different priors made on the distribution of zebra finch song. My collaborators and I found that spike trains from a population of MLd neurons combined with an uncorrelated Gaussian prior can estimate the amplitude envelope of song spectrograms. The same set of responses can be combined with Gaussian priors that have correlations matched to those found across multiple zebra finch songs to yield song spectrograms similar to those presented to the animal.

The fidelity of spectrogram reconstructions from MLd responses relies more heavily on prior knowledge of spectral correlations than temporal correlations. However the best reconstructions combine MLd responses with both spectral and temporal correlations.