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Neuronal Synchronisation in High-Dimensions

Understanding the complex dynamics of the human brain is a challenging task that has captured the interest of many scientists. The study of brain activity requires a multidisciplinary approach, involving researchers in both neuroscience and statistics. An active area of research concerns the characterisation of dependence between neurons as evidenced via their firing patterns and rates. Traditionally, the time-varying activity of neurons has been measured at an aggregate scale via methods such as functional magnetic resonance imaging (fMRI) and electroencephalography (EEG). However, due to recent technological advances, electrical activity can now be measured at an individual neuron level, for example via electrodes implanted directly into the brain or via calcium fluorescence imaging methods. These direct measurements provide the gold standard for quantifying localised activity.

After adequate pre-processing of the signals, measurements known as spike-trains can be obtained, which represent when a given neuron is firing. Statistically, these can be thought of as observations from a marked multivariate point process. While existing statistical methodology can capture the behaviour of a handful of neurons, the development of new technologies has enabled the recording of activity from hundreds of neurons. Therefore, there is both an opportunity, and a need, to develop new statistical methods capable of handling this increase in dimensionality.

In this project, we are primarily interested in characterising dependencies between neuronal point processes. To do so, fundamental techniques based on spectral analysis are explored, with a particular focus on obtaining smoothed spectral estimates. We focus our study in the spectral domain because it has several advantages over the time domain. Firstly, subtle structural differences can be detected with frequency domain estimators that are difficult to observe in the time domain estimators. Secondly, the spectral domain provides a rich environment for characterising the second order structure of point processes. Specifically, reasonably accurate confidence intervals may be placed on estimates of the second order properties, thus enabling the significance of features to be assessed. Finally, a measurement known as coherence provides a normalised measure of correlations between processes, unlike time-domain cross correlations that cannot be easily normalised. Despite this, the spectral analysis of point processes has received limited attention in the statistics literature in recent years. However, it is hoped that by exploring fundamental spectral techniques we might be able to extract some meaningful dependencies in the signalling dynamics of neurons, and consequently provide an enhanced understanding of connectivity in the brain network.