Advances in neural recording technologies allow for the exciting opportunity to study large scale neural dynamics in the living brain. The focus of this thesis is to infer connectivity in the brain network from modern high-dimensional spike train data. These data, which represent the firing times of individual neurons, provide the gold standard for measuring localised neural activity. Statistically, neural spike trains can be modelled in continuous time, using the raw spikes, i.e., the exact time of neural firing, or in discrete time, when the spikes are transformed into binary data. The contributions of this thesis are threefold, each addressing a distinct challenge in neural spike train analysis. In the first instance, we focus on advancing techniques in spectral analysis, to infer the dependence structure of a multivariate and high-dimensional point process. We propose novel methodology that combines a Whittle pseudo-likelihood with ridge or Lasso style penalties, improving the efficiency of spectral estimation in the point process framework. We establish both asymptotic and large sample properties for our proposed estimators, and assess their performance on synthetic data simulated from a multivariate Hawkes process. Finally, we apply our methodology to neural spike train data, demonstrating its ability to infer connectivity in the brain network. As a second focus of this thesis, we propose a novel method to identify the effective (directional) connectivity of a population of neurons, under a binary time series representation of the neural data. In this framework, we directly account for non-stationary firing rates and propose an inference procedure to quantify the uncertainty associated with our estimated networks of neural interactions. We empirically validate the performance of our model and the inferential procedure, illustrating its ability to detect both excitatory and inhibitory neural interactions. Lastly, we extend our discussion to the multi-subject setting, exploring the variability of functional and effective connectivity estimates across experimental subjects. We also make comparisons between our proposed methods and existing approaches for neuronal network estimation, in both the single and multi-subject setting. Ultimately, this thesis contributes to the field of high-dimensional statistics, for the analysis of neural spiking data. In a world where neural recording technologies continue to advance, it is imperative that the statistics literature evolves in parallel, to enable the best possible insights from these complex datasets.
Carla Pinkney (Thu,) studied this question.