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Neuronal activity correlations are key to understanding how populations of neurons collectively encode information. While two-photon calcium imaging has created a unique opportunity to record the activity of large populations of neurons, existing methods for inferring correlations from these data face several challenges. First, the observations of spiking activity produced by two-photon imaging are temporally blurred and noisy. Secondly, even if the spiking data were perfectly recovered via deconvolution, inferring network-level features from binary spiking data is a challenging task due to the non-linear relation of neuronal spiking to endogenous and exogenous inputs. In this work, we propose a methodology to explicitly model and directly estimate signal and noise correlations from two-photon fluorescence observations, without requiring intermediate spike deconvolution. We provide theoretical guarantees on the performance of the proposed estimator and demonstrate its utility through applications to simulated and experimentally recorded data from the mouse auditory cortex.

The field of neuroscience has striven for more than a century to understand how the brain functionally coordinates billions of neurons to perform its many tasks. Recent advancements in neural data acquisition techniques such as multi-electrode arrays, two-photon calcium imaging, and high-speed light-sheet microscopy have significantly contributed to this endeavor's progression by facilitating concurrent observation of spiking activity in large neuronal populations. However, existing methods for network-level inference from these data have several shortcomings: including undermining the non-linear dynamics, ignoring non-stationary brain activity, and causing error propagation by performing inference in a multi-stage fashion. The goal of this dissertation is to close this gap by developing models and methods to directly infer the dynamic spectral and temporal network organizations in the brain, from these ensemble neural data. In the first part of this dissertation, we introduce Bayesian methods to infer dynamic frequency-domain network organizations in neuronal ensembles from spiking observations, by integrating techniques such as point process modeling, state-space estimation, and multitaper spectral estimation. Firstly, we introduce a semi-stationary multitaper multivariate spectral analysis method tailored for neuronal spiking data and establish theoretical bounds on its performance. Building upon this estimator, we then introduce a framework to derive spectrotemporal Granger causal interactions in a population of neurons from spiking data. We demonstrate the validity of these methods through simulations, and applications on real data recorded from cortical neurons of rats during sleep, and human subjects undergoing anesthesia. Finally, we extend these methods to develop a precise frequency-domain inference method to characterize human heart rate variability from electrocardiogram data. The second part introduces a methodology to directly estimate signal and noise correlation networks from two-photon calcium imaging observations. We explicitly model the observation noise, temporal blurring of spiking activities, and other underlying non-linearities in a Bayesian framework, and derive an efficient variational inference method. We demonstrate the validity of the resulting estimators through theoretical analysis and extensive simulations, all of which establish significant gains over existing methods. Applications of our method on real data recorded from the mouse primary auditory cortex reveal novel and distinct spatial patterns in the correlation networks. Finally, we use our methods to investigate how the correlation networks in the auditory cortex change under different stimulus conditions, and during perceptual learning. In the third part, we investigate the respiratory network and the swimming-respiration coordination in larval zebrafish by applying several spectro-temporal analysis techniques, on whole-brain light-sheet microscopy imaging data. Firstly, using multitaper spectrotemporal analysis techniques, we categorize brain regions that are synchronized with the respiratory rhythm based on their distinct phases. Then, we demonstrate that zebrafish swimming is phase-locked to breathing. Next, through the analysis of neural activity and behavior under optogenetic stimulations and two-photon ablations, we identify the brain regions that are key for this swimming-respiration coordination. Finally, using the Izhikevich model for spiking neurons, we develop and simulate a circuit model that replicates this swimming-respiration coupling phenomenon, providing new insights into the possible underlying neural circuitry.

Extracting the spectral representations of the neural processes that underlie spiking activity is key to understanding how the brain rhythms mediate cognitive functions. While spectral estimation of continuous time-series is well studied, inferring the spectral representation of latent non-stationary processes based on spiking observations is a challenging problem. In this paper, we address this issue by developing a multitaper spectral estimation methodology that can be directly applied to multivariate spiking observations in order to extract the evolutionary spectral density of the latent non-stationary processes that drive spiking activity, based on point process theory. We establish theoretical bounds on the bias-variance trade-off of the proposed estimator. Finally, we compare the performance of our proposed technique with existing methods using simulation studies and application to real data, which reveal significant gains in terms of the bias-variance trade-off.

Neurons exhibit substantial trial-to-trial variability in response to repeated stimuli, posing a major challenge for understanding the information content of neural spike trains. In visual cortex, responses show greater-than-Poisson variability, whose origins and structure remain unclear. To address this puzzle, we introduce a continuous, doubly stochastic model of spike train variability that partitions neural responses into a smooth stimulus-driven component and a time-varying stochastic gain process. We applied this model to spike trains from four visual areas (LGN, V1, V2, and MT) and found that the gain process is well described by an exponentiated power law, with increasing amplitude and slower decay at higher levels of the visual hierarchy. The model also provides analytical expressions for the Fano factor of binned spike counts as a function of timescale, linking observed variability to underlying modulatory dynamics. Together, these results establish a principled framework for characterizing neural variability across cortical processing stages.

Abstract Neuronal activity correlations are key to understanding how populations of neurons collectively encode information. While two-photon calcium imaging has created a unique opportunity to record the activity of large populations of neurons, existing methods for inferring correlations from these data face several challenges. First, the observations of spiking activity produced by two-photon imaging are temporally blurred and noisy. Secondly, even if the spiking data were perfectly recovered via deconvolution, inferring network-level features from binary spiking data is a challenging task due to the non-linear relation of neuronal spiking to endogenous and exogenous inputs. In this work, we propose a methodology to explicitly model and directly estimate signal and noise correlations from two-photon fluorescence observations, without requiring intermediate spike deconvolution. We provide theoretical guarantees on the performance of the proposed estimator and demonstrate its utility through applications to simulated and experimentally recorded data from the mouse auditory cortex.