Efficient Solutions to High-Dimensional and Nonlinear Neural Inverse Problems

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Development of various data acquisition techniques has enabled researchers to study the brain as a complex system and gain insight into the high-level functions performed by different regions of the brain. These data are typically high-dimensional as they pertain to hundreds of sensors and span hours of recording. In many experiments involving sensory or cognitive tasks, the underlying cortical activity admits sparse and structured representations in the temporal, spatial, or spectral domains, or combinations thereof. However, current neural data analysis approaches do not take account of sparsity in order to harness the high-dimensionality. Also, many existing approaches suffer from high bias due to the heavy usage of linear models and estimation techniques, given that cortical activity is known to exhibit various degrees of non-linearity. Finally, the majority of current methods in computational neuroscience are tailored for static estimation in batch-mode and offline settings, and with the advancement of brain-computer interface technologies, these methods need to be extended to capture neural dynamics in a real-time fashion. The objective of this dissertation is to devise novel algorithms for real-time estimation settings and to incorporate the sparsity and non-linear properties of brain activity for providing efficient solutions to neural inverse problems involving high-dimensional data. Along the same line, our goal is to provide efficient representations of these high-dimensional data that are easy to interpret and assess statistically.

First, we consider the problem of spectral estimation from binary neuronal spiking data. Due to the non-linearities involved in spiking dynamics, classical spectral representation methods fail to capture the spectral properties of these data. To address this challenge, we integrate point process theory, sparse estimation, and non-linear signal processing methods to propose a spectral representation modeling and estimation framework for spiking data. Our model takes into account the sparse spectral structure of spiking data, which is crucial in the analysis of electrophysiology data in conditions such as sleep and anesthesia. We validate the performance of our spectral estimation framework using simulated spiking data as well as multi-unit spike recordings from human subjects under general anesthesia.

Next, we tackle the problem of real-time auditory attention decoding from electroencephalography (EEG) or magnetoencephalography (MEG) data in a competing-speaker environment. Most existing algorithms for this purpose operate offline and require access to multiple trials for a reliable performance; hence, they are not suitable for real-time applications. To address these shortcomings, we integrate techniques from state-space modeling, Bayesian filtering, and sparse estimation to propose a real-time algorithm for attention decoding that provides robust, statistically interpretable, and dynamic measures of the attentional state of the listener. We validate the performance of our proposed algorithm using simulated and experimentally-recorded M/EEG data. Our analysis reveals that our algorithms perform comparable to the state-of-the-art offline attention decoding techniques, while providing significant computational savings.

Finally, we study the problem of dynamic estimation of Temporal Response Functions (TRFs) for analyzing neural response to auditory stimuli. A TRF can be viewed as the impulse response of the brain in a linear stimulus-response model. Over the past few years, TRF analysis has provided researchers with great insight into auditory processing, specially under competing speaker environments. However, most existing results correspond to static TRF estimates and do not examine TRF dynamics, especially in multi-speaker environments with attentional modulation. Using state-space models, we provide a framework for a robust and comprehensive dynamic analysis of TRFs using single trial data. TRF components at specific lags may exhibit peaks which arise, persist, and disappear over time according to the attentional state of the listener. To account for this specific behavior in our model, we consider a state-space model with a Gaussian mixture process noise, and devise an algorithm to efficiently estimate the process noise parameters from the recorded M/EEG data. Application to simulated and recorded MEG data shows that the {proposed state-space modeling and inference framework can reliably capture the dynamic changes in the TRF, which can in turn improve our access to the attentional state in competing-speaker environments.