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To understand single neuron computation, it is necessary to know how specific physiological parameters affect neural spiking patterns that emerge in response to specific stimuli. Here we present a computational pipeline combining biophysical and statistical models that provides a link between variation in functional ion channel expression and changes in single neuron stimulus encoding. More specifically, we create a mapping from biophysical model parameters to stimulus encoding statistical model parameters. Biophysical models provide mechanistic insight, whereas statistical models can identify associations between spiking patterns and the stimuli they encode. We used public biophysical models of two morphologically and functionally distinct projection neuron cell types: mitral cells (MCs) of the main olfactory bulb, and layer V cortical pyramidal cells (PCs). We first simulated sequences of action potentials according to certain stimuli while scaling individual ion channel conductances. We then fitted point process generalized linear models (PP-GLMs), and we constructed a mapping between the parameters in the two types of models. This framework lets us detect effects on stimulus encoding of changing an ion channel conductance. The computational pipeline combines models across scales and can be applied as a screen of channels, in any cell type of interest, to identify ways that channel properties influence single neuron computation.

The human visual cortex is organized in a hierarchical manner. Although previous evidence supporting this hypothesis has been accumulated, specific details regarding the spatiotemporal information flow remain open. Here we present detailed spatiotemporal correlation profiles of neural activity with low-level and high-level features derived from an eight-layer neural network pretrained for object recognition. These correlation profiles indicate an early-to-late shift from low-level features to high-level features and from low-level regions to higher-level regions along the visual hierarchy, consistent with feedforward information flow. Additionally, we computed three sets of features from the low- and high-level features provided by the neural network: object-category-relevant low-level features (the common components between low-level and high-level features), low-level features roughly orthogonal to high-level features (the residual Layer 1 features), and unique high-level features that were roughly orthogonal to low-level features (the residual Layer 7 features). Contrasting the correlation effects of the common components and the residual Layer 1 features, we observed that the early visual cortex (EVC) exhibited a similar amount of correlation with the two feature sets early in time, but in a later time window, the EVC exhibited a higher and longer correlation effect with the common components (i.e., the low-level object-category-relevant features) than with the low-level residual features-an effect unlikely to arise from purely feedforward information flow. Overall, our results indicate that non-feedforward processes, for example, top-down influences from mental representations of categories, may facilitate differentiation between these two types of low-level features within the EVC.

Because local field potentials (LFPs) arise from multiple sources in different spatial locations, they do not easily reveal coordinated activity across neural populations on a trial-to-trial basis. As we show here, however, once disparate source signals are decoupled, their trial-to-trial fluctuations become more accessible, and cross-population correlations become more apparent. To decouple sources we introduce a general framework for estimation of current source densities (CSDs). In this framework, the set of LFPs result from noise being added to the transform of the CSD by a biophysical forward model, while the CSD is considered to be the sum of a zero-mean, stationary, spatiotemporal Gaussian process, having fast and slow components, and a mean function, which is the sum of multiple time-varying functions distributed across space, each varying across trials. We derived biophysical forward models relevant to the data we analyzed. In simulation studies this approach improved identification of source signals compared to existing CSD estimation methods. Using data recorded from primate auditory cortex, we analyzed trial-to-trial fluctuations in both steady-state and task-evoked signals. We found cortical layer-specific phase coupling between two probes and showed that the same analysis applied directly to LFPs did not recover these patterns. We also found task-evoked CSDs to be correlated across probes, at specific cortical depths. Using data from Neuropixels probes in mouse visual areas, we again found evidence for depth-specific phase coupling of primary visual cortex and lateromedial area based on the CSDs., Author summary To better understand information processing in the brain, it is important to identify situations in which neural activity is coordinated across populations of neurons, including those in distinct layers of the cortex. Bulk population activity results in voltage changes across extracellular electrodes, but in raw form such voltage recordings can be hard to analyze and interpret. In this paper, we develop a novel framework for locating the sources of currents that produce the measured voltages, and decomposing those currents into interpretable components. We use this flexible framework to develop statistical methods of analysis, and we show that our methodology can be more effective than existing techniques for current source localization. We apply the method to extracellular recordings in two contexts: primate auditory cortex in response to tone stimuli, and mouse visual cortex in response to visual stimuli. In both cases, we get results that are useful for understanding cross-population activity while being difficult or impossible to obtain using the raw voltage signals. We thereby demonstrate the broad utility of our approach for identifying coordinated neural activity based on extracellular voltage recordings.

The 2019 Society for Neuroscience Professional Development Workshop on Teaching reviewed current tools, approaches, and examples for teaching computation in neuroscience. Robert Kass described the statistical foundations that students need to properly analyze data. Pascal Wallisch compared MATLAB and Python as programming languages for teaching students. Adrienne Fairhall discussed computational methods, training opportunities, and curricular considerations. Walt Babiec provided a view from the trenches on practical aspects of teaching computational neuroscience. Mathew Abrams concluded the session with an overview of resources for teaching and learning computational modeling in neuroscience.

The problem of identifying functional connectivity from multiple time series data recorded in each of two or more brain areas arises in many neuroscientific investigations. For a single stationary time series in each of two brain areas statistical tools such as cross-correlation and Granger causality may be applied. On the other hand, to examine multivariate interactions at a single time point, canonical correlation, which finds the linear combinations of signals that maximize the correlation, may be used. We report here a new method that produces interpretations much like these standard techniques and, in addition, 1) extends the idea of canonical correlation to 3-way arrays (with dimensionality number of signals by number of time points by number of trials), 2) allows for nonstationarity, 3) also allows for nonlinearity, 4) scales well as the number of signals increases, and 5) captures predictive relationships, as is done with Granger causality. We demonstrate the effectiveness of the method through simulation studies and illustrate by analyzing local field potentials recorded from a behaving primate. NEW & NOTEWORTHY Multiple signals recorded from each of multiple brain regions may contain information about cross-region interactions. This article provides a method for visualizing the complicated interdependencies contained in these signals and assessing them statistically. The method combines signals optimally but allows the resulting measure of dependence to change, both within and between regions, as the responses evolve dynamically across time. We demonstrate the effectiveness of the method through numerical simulations and by uncovering a novel connectivity pattern between hippocampus and prefrontal cortex during a declarative memory task.

Point process regression models, based on generalized linear model (GLM) technology, have been widely used for spike train analysis, but a recent paper by Gerhard et al. described a kind of instability, in which fitted models can generate simulated spike trains with explosive firing rates. We analyze the problem by extending the methods of Gerhard et al. First, we improve their instability diagnostic and extend it to a wider class of models. Next, we point out some common situations in which instability can be traced to model lack of fit. Finally, we investigate distinctions between models that use a single filter to represent the effects of all spikes prior to any particular time t, as in a 2008 paper by Pillow et al., and those that allow different filters for each spike prior to time t, as in a 2001 paper by Kass and Ventura. We re-analyze the data sets used by Gerhard et al., introduce an additional data set that exhibits bursting, and use a well-known model described by Izhikevich to simulate spike trains from various ground truth scenarios. We conclude that models with multiple filters tend to avoid instability, but there are unlikely to be universal rules. Instead, care in data fitting is required and models need to be assessed for each unique set of data.

Developing models of the dynamic and complex patterns of information processing that take place during behavior is a major thrust of systems neuroscience. An underlying assumption of many models is that the same set of rules applies across different conditions. This has been the case for directional tuning during volitional movement; a single cosine function has been remarkably robust for describing the encoding of movement direction in different types of neurons, in many locations of the nervous system, and even across species. However, detailed examination of the tuning time course in motor cortex suggests that direction coding may be labile. Here, we show that there are discrete time epochs within single reaches, between which individual neurons change their tuning. Our findings suggest that motor cortical activity patterns may reflect consistent changes in the state of the control system during center-out reaching. These transitions are likely linked to different behavioral components, suggesting that the task defines changes in the operational structure of the control system.

Many functional neuroimaging-based studies involve repetitions of a task that may require several phases, or states, of mental activity. An appealing idea is to use relevant brain regions to identify the states. We developed a novel change-point methodology that adapts to the repeated trial structure of such experiments by assuming the number of states stays fixed across similar trials while allowing the timing of change-points to change across trials. Model fitting is based on reversible-jump MCMC. Simulation studies verified its ability to identify change-points successfully. We applied this technique to data collected via functional magnetic resonance imaging (fMRI) while each of 20 subjects solved unfamiliar arithmetic problems. Our methodology supplies both a summary of state dimensionality and uncertainty assessments about number of states and the timing of state transitions. Copyright © 2016 John Wiley & Sons, Ltd.

Populations of cortical neurons exhibit shared fluctuations in spiking activity over time. When measured for a pair of neurons over multiple repetitions of an identical stimulus, this phenomenon emerges as correlated trial-to-trial response variability via spike count correlation (SCC). However, spike counts can be viewed as noisy versions of firing rates, which can vary from trial to trial. From this perspective, the SCC for a pair of neurons becomes a noisy version of the corresponding firing rate correlation (FRC). Furthermore, the magnitude of the SCC is generally smaller than that of the FRC and is likely to be less sensitive to experimental manipulation. We provide statistical methods for disambiguating time-averaged drive from within-trial noise, thereby separating FRC from SCC. We study these methods to document their reliability, and we apply them to neurons recorded in vivo from area V4 in an alert animal. We show how the various effects we describe are reflected in the data: within-trial effects are largely negligible, while attenuation due to trial-to-trial variation dominates and frequently produces comparisons in SCC that, because of noise, do not accurately reflect those based on the underlying FRC.

The human visual cortex is organized in a hierarchical manner. Although a significant body of evidence has been accumulated in support of this hypothesis, specific details regarding the spatial and temporal information flow remain open. Here we present detailed spatio-temporal correlation profiles of neural activity with low-level and high-level features derived from a “deep” (8-layer) neural network pre-trained for object recognition. These correlation profiles indicate an early-to-late shift from low-level features to high-level features and from low-level regions to higher-level regions along the visual hierarchy, consistent with feedforward information flow. To refine our understanding of information flow, we computed three sets of features from the low-and high-level features provided by the neural network: object-category-relevant low-level features (the common components between low-level and high-level features), low-level features roughly orthogonal to high-level features (the residual Layer 1 features), and unique high-level features that were roughly orthogonal to low-level features (the residual Layer 7 features). Contrasting the correlation effects of the common components and the residual Layer 1 features, we observed that the early visual cortex exhibits a similar amount of correlation with the two feature sets early in time (60 to 120 ms), but in a later time window, the early visual cortex exhibits a higher and longer correlation effect with the common components/low-level task-relevant features as compared to the low-level residual features—an effect unlikely to arise from purely feedforward information flow. Overall, our results indicate that non-feedforward processes, for example, top-down influences from mental representations of categories, may facilitate differentiation between these two types of low-level features within the early visual cortex.

The problem of multiple testing arises in many contexts, including testing for pairwise interaction among a large number of neurons. Recently a method was developed to control false positives when covariate information, such as distances between pairs of neurons, is available. This method, however, relies on computationally-intensive Markov Chain Monte Carlo (MCMC). Here we develop an alternative, based on Sequential Monte Carlo, which only requires one pass of the data. This scheme considers data items sequentially, with relevant probabilities being updated at each step. Simulation experiments demonstrate that the proposed algorithm delivers results as accurately as the previous MCMC method. We illustrate the method by using it to analyze neural recordings from extrastriate cortex in a macaque monkey.

A major challenge in contemporary neuroscience is to analyze data from large numbers of neurons recorded simultaneously across many experimental replications (trials), where the data are counts of neural firing events, and one of the basic problems is to characterize the dependence structure among such multivariate counts. Methods of estimating high-dimensional covariation based on $\ell_{1}$-regularization are most appropriate when there are a small number of relatively large partial correlations, but in neural data there are often large numbers of relatively small partial correlations. Furthermore, the variation across trials is often confounded by Poisson-like variation within trials. To overcome these problems we introduce a comprehensive methodology that imbeds a Gaussian graphical model into a hierarchical structure: the counts are assumed Poisson, conditionally on latent variables that follow a Gaussian graphical model, and the graphical model parameters, in turn, are assumed to depend on physiologically-motivated covariates, which can greatly improve correct detection of interactions (nonzero partial correlations). We develop a Bayesian approach to fitting this covariate-adjusted generalized graphical model and we demonstrate its success in simulation studies. We then apply it to data from an experiment on visual attention, where we assess functional interactions between neurons recorded from two brain areas.

In vivo calcium imaging through microendoscopic lenses enables imaging of previously inaccessible neuronal populations deep within the brains of freely moving animals. However, it is computationally challenging to extract single-neuronal activity from microendoscopic data, because of the very large background fluctuations and high spatial overlaps intrinsic to this recording modality. Here, we describe a new constrained matrix factorization approach to accurately separate the background and then demix and denoise the neuronal signals of interest. We compared the proposed method against previous independent components analysis and constrained nonnegative matrix factorization approaches. On both simulated and experimental data recorded from mice, our method substantially improved the quality of extracted cellular signals and detected more well-isolated neural signals, especially in noisy data regimes. These advances can in turn significantly enhance the statistical power of downstream analyses, and ultimately improve scientific conclusions derived from microendoscopic data.

Pairs of active neurons frequently fire action potentials or “spikes” nearly synchronously (i.e., within 5 ms of each other). This spike synchrony may occur by chance, based solely on the neurons’ fluctuating firing patterns, or it may occur too frequently to be explicable by chance alone. When spike synchrony above chances levels is present, it may subserve computation for a specific cognitive process, or it could be an irrelevant byproduct of such computation. Either way, spike synchrony is a feature of neural data that should be explained. A point process regression framework has been developed previously for this purpose, using generalized linear models (GLMs). In this framework, the observed number of synchronous spikes is compared to the number predicted by chance under varying assumptions about the factors that affect each of the individual neuron’s firing-rate functions. An important possible source of spike synchrony is network-wide oscillations, which may provide an essential mechanism of network information flow. To establish the statistical link between spike synchrony and network-wide oscillations, we have integrated oscillatory field potentials into our point process regression framework. We first extended a previously-published model of spike-field association and showed that we could recover phase relationships between oscillatory field potentials and firing rates. We then used this new framework to demonstrate the statistical relationship between oscillatory field potentials and spike synchrony in: 1) simulated neurons, 2) in vitro recordings of hippocampal CA1 pyramidal cells, and 3) in vivo recordings of neocortical V4 neurons. Our results provide a rigorous method for establishing a statistical link between network oscillations and neural synchrony., Author Summary Spike synchrony, which is widely reported in neural systems, may contribute to information transmission within and across brain regions. Critical to this theory is the potential link between oscillatory activity and synchronous spiking. In this article we provide a method for establishing the statistical association of spike synchrony with an oscillatory local field potential. We demonstrate the value of this technique by numerical simulation together with application to both in vitro and in vivo neural recordings.

Mathematical and statistical models have played important roles in neuroscience, especially by describing the electrical activity of neurons recorded individually, or collectively across large networks. As the field moves forward rapidly, new challenges are emerging. For maximal effectiveness, those working to advance computational neuroscience will need to appreciate and exploit the complementary strengths of mechanistic theory and the statistical paradigm., National Science Foundation (U.S.) (Grant DMS-1127914), National Institute of Mental Health (U.S.) (Grant MH064537)

Multiple and unpredictable numbers of actions are often required to achieve a goal. In order to organize behavior and allocate effort so that optimal behavioral policies can be selected, it is necessary to continually monitor ongoing actions. Real-time processing of information related to actions and outcomes is typically assigned to the prefrontal cortex and basal ganglia, but also depends on midbrain regions, especially the ventral tegmental area (VTA). We were interested in how individual VTA neurons, as well as networks within the VTA, encode salient events when an unpredictable number of serial actions are required to obtain a reward. We recorded from ensembles of putative dopamine and non-dopamine neurons in the VTA as animals performed multiple cued trials in a recording session where, in each trial, serial actions were randomly rewarded. While averaging population activity did not reveal a response pattern, we observed that different neurons selectively tuned to low, medium, or high numbered actions in a trial. This preferential tuning of putative dopamine and non-dopamine VTA neurons to different subsets of actions in a trial allowed information about binned action number to be decoded from the ensemble activity. At the network level, tuning curve similarity was positively associated with action-evoked noise correlations, suggesting that action number selectivity reflects functional connectivity within these networks. Analysis of phasic responses to cue and reward revealed that the requirement to execute multiple and uncertain numbers of actions weakens both cue-evoked responses and cue-reward response correlation. The functional connectivity and ensemble coding scheme that we observe here may allow VTA neurons to cooperatively provide a real-time account of ongoing behavior. These computations may be critical to cognitive and motivational functions that have long been associated with VTA dopamine neurons.

It is now common to record dozens to hundreds or more neurons simultaneously, and to ask how the network activity changes across experimental conditions. A natural framework for addressing questions of functional connectivity is to apply Gaussian graphical modeling to neural data, where each edge in the graph corresponds to a non-zero partial correlation between neurons. Because the number of possible edges is large, one strategy for estimating the graph has been to apply methods that aim to identify large sparse effects using an [Formula: see text] penalty. However, the partial correlations found in neural spike count data are neither large nor sparse, so techniques that perform well in sparse settings will typically perform poorly in the context of neural spike count data. Fortunately, the correlated firing for any pair of cortical neurons depends strongly on both their distance apart and the features for which they are tuned. We introduce a method that takes advantage of these known, strong effects by allowing the penalty to depend on them: thus, for example, the connection between pairs of neurons that are close together will be penalized less than pairs that are far apart. We show through simulations that this physiologically-motivated procedure performs substantially better than off-the-shelf generic tools, and we illustrate by applying the methodology to populations of neurons recorded with multielectrode arrays implanted in macaque visual cortex areas V1 and V4.

We propose a Multivariate Gaussian Process Factor Model to estimate low dimensional spatio-temporal patterns of finger motion in repeated reach-to-grasp movements. Our model decomposes and reduces the dimensionality of variation of the multivariate functional data. We first account for time variability through multivariate functional registration, then decompose finger motion into a term that is shared among replications and a term that encodes the variation per replication. We discuss variants of our model, estimation algorithms, and we evaluate its performance in simulations and in data collected from a non-human primate executing a reach-to-grasp task. We show that by taking advantage of the repeated trial structure of the experiments, our model yields an intuitive way to interpret the time and replication variation in our kinematic dataset.

Much attention has been paid to the question of how Bayesian integration of information could be implemented by a simple neural mechanism. We show that population vectors based on point-process inputs combine evidence in a form that closely resembles Bayesian inference, with each input spike carrying information about the tuning of the input neuron. We also show that population vectors can combine information relatively accurately in the presence of noisy synaptic encoding of tuning curves.

Magnetoencephalography (MEG) has a high temporal resolution well-suited for studying perceptual learning. However, to identify where learning happens in the brain, one needs to ap- ply source localization techniques to project MEG sensor data into brain space. Previous source localization methods, such as the short-time Fourier transform (STFT) method by Gramfort et al.([Gramfort et al., 2013]) produced intriguing results, but they were not designed to incor- porate trial-by-trial learning effects. Here we modify the approach in [Gramfort et al., 2013] to produce an STFT-based source localization method (STFT-R) that includes an additional regression of the STFT components on covariates such as the behavioral learning curve. We also exploit a hierarchical L 21 penalty to induce structured sparsity of STFT components and to emphasize signals from regions of interest (ROIs) that are selected according to prior knowl- edge. In reconstructing the ROI source signals from simulated data, STFT-R achieved smaller errors than a two-step method using the popular minimum-norm estimate (MNE), and in a real-world human learning experiment, STFT-R yielded more interpretable results about what time-frequency components of the ROI signals were correlated with learning., Comment: Author manuscript accepted to 4th NIPS Workshop on Machine Learning and Interpretation in Neuroimaging (2014), (in press on Lecture Notes in Computer Science, by Springer)

Several authors have previously discussed the use of log-linear models, often called maximum entropy models, for analyzing spike train data to detect synchrony. The usual log-linear modeling techniques, however, do not allow time-varying firing rates that typically appear in stimulus-driven (or action-driven) neurons, nor do they incorporate non-Poisson history effects or covariate effects. We generalize the usual approach, combining point-process regression models of individual neuron activity with log-linear models of multiway synchronous interaction. The methods are illustrated with results found in spike trains recorded simultaneously from primary visual cortex. We then assess the amount of data needed to reliably detect multiway spiking.

Identifying brain regions with high differential response under multiple experimental conditions is a fundamental goal of functional imaging. In many studies, regions of interest (ROIs) are not determined a priori but are instead discovered from the data, a process that requires care because of the great potential for false discovery. An additional challenge is that magnetoencephalography/electroencephalography sensor signals are very noisy, and brain source images are usually produced by averaging sensor signals across trials. As a consequence, for a given subject, there is only one source data vector for each condition, making it impossible to apply testing methods such as analysis of variance. We solve these problems in several steps. (1) To obtain within-condition uncertainty, we apply the bootstrap across trials, producing many bootstrap source images. To discover ‘hot spots’ in space and time that could become ROIs, (2) we find source locations where likelihood ratio statistics take unusually large values. We are not interested in isolated brain locations where a test statistic might happen to be large. Instead, (3) we apply a clustering algorithm to identify sources that are contiguous in space and time where the test statistic takes an ‘excursion’ above some threshold. Having identified possible spatiotemporal ROIs, (4) we evaluate global statistical significance of ROIs by using a permutation test. After these steps, we check performance via simulation, and then illustrate their application in a magnetoencephalography study of four-direction center-out wrist movement, showing that this approach identifies statistically significant spatiotemporal ROIs in the motor and visual cortices of individual subjects.

We provide a short overview of importance sampling—a popular sampling tool used for Monte Carlo computing. We discuss its mathematical foundation and properties that determine its accuracy in Monte Carlo approximations. We review the fundamental developments in designing efficient importance sampling (IS) for practical use. This includes parametric approximation with optimization-based adaptation, sequential sampling with dynamic adaptation through resampling and population-based approaches that make use of Markov chain sampling. Copyright © 2009 John Wiley & Sons, Inc. For further resources related to this article, please visit the WIREs website.

Neuroprosthetic devices such as a computer cursor can be controlled by the activity of cortical neurons when an appropriate algorithm is used to decode motor intention. Algorithms which have been proposed for this purpose range from the simple population vector algorithm (PVA) and optimal linear estimator (OLE) to various versions of Bayesian decoders. Although Bayesian decoders typically provide the most accurate off-line reconstructions, it is not known which model assumptions in these algorithms are critical for improving decoding performance. Furthermore, it is not necessarily true that improvements (or deficits) in off-line reconstruction will translate into improvements (or deficits) in on-line control, as the subject might compensate for the specifics of the decoder in use at the time. Here we show that by comparing the performance of nine decoders, assumptions about uniformly distributed preferred directions and the way the cursor trajectories are smoothed have the most impact on decoder performance in off-line reconstruction, while assumptions about tuning curve linearity and spike count variance play relatively minor roles. In on-line control, subjects compensate for directional biases caused by non-uniformly distributed preferred directions, leaving cursor smoothing differences as the largest single algorithmic difference driving decoder performance.

Neurons in vitro and in vivo have epochs of bursting or "up state" activity during which firing rates are dramatically elevated. Various methods of detecting bursts in extracellular spike trains have appeared in the literature, the most widely used apparently being Poisson Surprise (PS). A natural description of the phenomenon assumes (1) there are two hidden states, which we label "burst" and "non-burst," (2) the neuron evolves stochastically, switching at random between these two states, and (3) within each state the spike train follows a time-homogeneous point process. If in (2) the transitions from non-burst to burst and burst to non-burst states are memoryless, this becomes a hidden Markov model (HMM). For HMMs, the state transitions follow exponential distributions, and are highly irregular. Because observed bursting may in some cases be fairly regular--exhibiting inter-burst intervals with small variation--we relaxed this assumption. When more general probability distributions are used to describe the state transitions the two-state point process model becomes a hidden semi-Markov model (HSMM). We developed an efficient Bayesian computational scheme to fit HSMMs to spike train data. Numerical simulations indicate the method can perform well, sometimes yielding very different results than those based on PS.

Perception, memory, learning, and decision making are processes carried out in the brain. The performance of such intelligent tasks is made possible by the communication of neurons through sequences of voltage pulses called spike trains. It is of great interest to have methods of extracting information from spike trains in order to learn about their relationship to behavior. In this article, we review a Bayesian approach to this problem based on state-space representations of point processes. We discuss some of the theory and we describe the way these methods are used in decoding motor cortical activity, in which the hand motion is reconstructed from neural spike trains.

When correlation is measured in the presence of noise, its value is decreased. In single-neuron recording experiments, for example, the correlation of selectivity indices in a pair of tasks may be assessed across neurons, but, because the number of trials is limited, the measured index values for each neuron will be noisy. This attenuates the correlation. A correction for such attenuation was proposed by Spearman more than 100 yr ago, and more recent work has shown how confidence intervals may be constructed to supplement the correction. In this paper, we propose an alternative Bayesian correction. A simulation study shows that this approach can be far superior to Spearman's, both in accuracy of the correction and in coverage of the resulting confidence intervals. We demonstrate the usefulness of this technology by applying it to a set of data obtained from the frontal cortex of a macaque monkey while performing serial order and variable reward saccade tasks. There the correction results in a substantial increase in the correlation across neurons in the two tasks.

Information estimates such as the direct method of Strong, Koberle, de Ruyter van Steveninck, and Bialek (1998) sidestep the difficult problem of estimating the joint distribution of response and stimulus by instead estimating the difference between the marginal and conditional entropies of the response. While this is an effective estimation strategy, it tempts the practitioner to ignore the role of the stimulus and the meaning of mutual information. We show here that as the number of trials increases indefinitely, the direct (or plug-in) estimate of marginal entropy converges (with probability 1) to the entropy of the time-averaged conditional distribution of the response, and the direct estimate of the conditional entropy converges to the time-averaged entropy of the conditional distribution of the response. Under joint stationarity and ergodicity of the response and stimulus, the difference of these quantities converges to the mutual information. When the stimulus is deterministic or nonstationary the direct estimate of information no longer estimates mutual information, which is no longer meaningful, but it remains a measure of variability of the response distribution across time.

Efforts to study the neural correlates of learning are hampered by the size of the network in which learning occurs. To understand the importance of learning-related changes in a network of neurons, it is necessary to understand how the network acts as a whole to generate behavior. Here we introduce a paradigm in which the output of a cortical network can be perturbed directly and the neural basis of the compensatory changes studied in detail. Using a brain-computer interface, dozens of simultaneously recorded neurons in the motor cortex of awake, behaving monkeys are used to control the movement of a cursor in a three-dimensional virtual-reality environment. This device creates a precise, well-defined mapping between the firing of the recorded neurons and an expressed behavior (cursor movement). In a series of experiments, we force the animal to relearn the association between neural firing and cursor movement in a subset of neurons and assess how the network changes to compensate. We find that changes in neural activity reflect not only an alteration of behavioral strategy but also the relative contributions of individual neurons to the population error signal.

Over the past few decades, developments in technology have significantly improved the ability to measure activity in the brain. This has spurred a great deal of research into brain function and its relation to external stimuli, and has important implications in medicine and other fields. As a result of improved understanding of brain function, it is now possible to build devices that provide direct interfaces between the brain and the external world. We describe some of the current understanding of function of the motor cortex region. We then discuss a typical likelihood-based state-space model and filtering based approach to address the problems associated with building a motor cortical-controlled cursor or robotic prosthetic device. As a variation on previous work using this approach, we introduce the idea of using Markov chain Monte Carlo methods for parameter estimation in this context. By doing this instead of performing maximum likelihood estimation, it is possible to expand the range of possible models that can be explored, at a cost in terms of computational load. We demonstrate results obtained applying this methodology to experimental data gathered from a monkey.

We consider the problem of comparing noisy functions, here trial-averaged neuronal firing-rate curves, across multiple experimental conditions. Of interest are comparisons both within neurons and also among populations of individually recorded neurons. We propose likelihood ratio tests to perform comparisons either pointwise or globally over the entire experimental time. A simulation study of power demonstrates the strength of these tests even for moderate sample sizes. We implement these tests on a group of 233 neurons recorded from primate frontal oculomotor cortex, first, to screen for condition-related differential activity and, second, to search for neurons displaying interesting time-locked features that vary with condition.

Oscillations are observed at various frequency bands in continuous-valued neural recordings like the electroencephalogram (EEG) and local field potential (LFP) in bulk brain matter, and analysis of spike-field coherence reveals that spiking of single neurons often occurs at certain phases of the global oscillation. Oscillatory modulation has been examined in relation to continuous-valued oscillatory signals, and independently from the spike train alone, but behavior or stimulus triggered firing-rate modulation, spiking sparseness, presence of slow modulation not locked to stimuli and irregular oscillations with large variability in oscillatory periods, present challenges to searching for temporal structures present in the spike train. In order to study oscillatory modulation in real data collected under a variety of experimental conditions, we describe a flexible point-process framework we call the Latent Oscillatory Spike Train (LOST) model to decompose the instantaneous firing rate in biologically and behaviorally relevant factors: spiking refractoriness, event-locked firing rate non-stationarity, and trial-to-trial variability accounted for by baseline offset and a stochastic oscillatory modulation. We also extend the LOST model to accommodate changes in the modulatory structure over the duration of the experiment, and thereby discover trial-to-trial variability in the spike-field coherence of a rat primary motor cortical neuron to the LFP theta rhythm. Because LOST incorporates a latent stochastic auto-regressive term, LOST is able to detect oscillations when the firing rate is low, the modulation is weak, and when the modulating oscillation has a broad spectral peak., Author summary Oscillatory modulation of neural activity in the brain is widely observed under conditions associated with a variety of cognitive tasks and mental states. Within individual neurons, oscillations may be uncovered in the moment-to-moment variation in neural firing rate. This, however, is often challenging because many factors may affect fluctuations in neural firing rate and, in addition, neurons fire irregular sets of action potentials, or spike trains, due to an unknown combination of meaningful signals and extraneous noise. We have devised a statistical Latent Oscillatory Spike Train (LOST) model with accompanying model-fitting technology, that is able to detect subtle oscillations in spike trains by taking into account both spiking noise and temporal variation in the oscillation itself. The method couples two techniques developed for other purposes in the literature on Bayesian analysis. Using data simulated from theoretical neurons and real data recorded from cortical motor neurons, we demonstrate the method’s ability to track changes in the modulatory structure of the oscillation across experimental trials.

We propose a Multivariate Gaussian Process Factor Model to estimate low dimensional spatio-temporal patterns of finger motion in repeated reach-to-grasp movements. Our model decomposes and reduces the dimensionality of variation of the multivariate functional data. We first account for time variability through multivariate functional registration, then decompose finger motion into a term that is shared among replications and a term that encodes the variation per replication. We discuss variants of our model, estimation algorithms, and we evaluate its performance in simulations and in data collected from a non-human primate executing a reach-to-grasp task. We show that by taking advantage of the repeated trial structure of the experiments, our model yields an intuitive way to interpret the time and replication variation in our kinematic dataset.

It has been observed that spike count correlation between two simultaneously recorded neurons often increases with the length of time interval examined. Under simple assumptions that are roughly consistent with much experimental data, we show that this phenomenon may be explained as being due to excess trial-to-trial variation. The resulting formula for the correlation is able to predict the observed correlation of two neurons recorded from primary visual cortex as a function of interval length.

The joint peristimulus time histogram (JPSTH) and cross-correlogram provide a visual representation of correlated activity for a pair of neurons, and the way this activity may increase or decrease over time. In a companion paper we showed how a Bootstrap evaluation of the peaks in the smoothed diagonals of the JPSTH may be used to establish the likely validity of apparent time-varying correlation. As noted in earlier studies by Brody and Ben-Shaul et al., trial-to-trial variation can confound correlation and synchrony effects. In this paper we elaborate on that observation, and present a method of estimating the time-dependent trial-to-trial variation in spike trains that may exceed the natural variation displayed by Poisson and non-Poisson point processes. The statistical problem is somewhat subtle because relatively few spikes per trial are available for estimating a firing-rate function that fluctuates over time. The method developed here decomposes the spike-train variability into a stimulus-related component and a trial-specific component, allowing many degrees of freedom to characterize the former while assuming a small number suffices to characterize the latter. The Bootstrap significance test of the companion paper is then modified to accommodate these general excitability effects. This methodology allows an investigator to assess whether excitability effects are constant or time-varying, and whether they are shared by two neurons. In data from two V1 neurons we find that highly statistically significant evidence of dependency disappears after adjustment for time-varying trial-to-trial variation.

SUMMARY In many applications of functional data analysis, summarising functional variation based on fits, without taking account of the estimation process, runs the risk of attributing the estimation variation to the functional variation, thereby overstating the latter. For example, the first eigenvalue of a sample covariance matrix computed from estimated functions may be biased upwards. We display a set of estimated neuronal Poisson-process intensity functions where this bias is substantial, and we discuss two methods for account ing for estimation variation. One method uses a random-coefficient model, which requires all functions to be fitted with the same basis functions. An alternative method removes the same-basis restriction by means of a hierarchical Gaussian process model. In a small simulation study the hierarchical Gaussian process model outperformed the random coefficient model and greatly reduced the bias in the estimated first eigenvalue that would result from ignoring estimation variability. For the neuronal data the hierarchical Gaussian process estimate of the first eigenvalue was much smaller than the naive estimate that ignored variability due to function estimation. The neuronal setting also illustrates the benefit of incorporating alignment parameters into the hierarchical scheme.

SUMMARY A fundamental methodology in neurophysiology involves recording the electrical signals associated with individual neurons within brains of awake behaving animals. Traditional statistical analyses have relied mainly on mean firing rates over some epoch (often several hundred milliseconds) that are compared across experimental conditions by analysis of variance. Often, however, the time course of the neuronal firing patterns is of interest, and a more refined procedure can produce substantial additional information. In this paper we compare neuronal firing in the supplementary eye field of a macaque monkey across two experimental conditions. We take the electrical discharges, or ‘spikes’, to be arrivals in a inhomogeneous Poisson process and then model the firing intensity function using both a simple parametric form and more flexible splines. Our main interest is in making inferences about certain characteristics of the intensity, including the timing of the maximal firing rate. We examine data from 84 neurons individually and also combine results into a hierarchical model. We use Bayesian estimation methods and frequentist significance tests based on a nonparametric bootstrap procedure. We are thereby able to conclude that a substantial fraction of the neurons exhibit important temporal differences in firing intensity across the two conditions, and we quantify the effect across the population of neurons.

Humans are experts at face individuation. Although previous work has identified a network of face-sensitive regions and some of the temporal signatures of face processing, as yet, we do not have a clear understanding of how such face-sensitive regions support learning at different time points. To study the joint spatio-temporal neural basis of face learning, we trained subjects to categorize two groups of novel faces and recorded their neural responses using magnetoencephalography (MEG) throughout learning. A regression analysis of neural responses in face-sensitive regions against behavioral learning curves revealed significant correlations with learning in the majority of the face-sensitive regions in the face network, mostly between 150–250 ms, but also after 300 ms. However, the effect was smaller in nonventral regions (within the superior temporal areas and prefrontal cortex) than that in the ventral regions (within the inferior occipital gyri (IOG), midfusiform gyri (mFUS) and anterior temporal lobes). A multivariate discriminant analysis also revealed that IOG and mFUS, which showed strong correlation effects with learning, exhibited significant discriminability between the two face categories at different time points both between 150–250 ms and after 300 ms. In contrast, the nonventral face-sensitive regions, where correlation effects with learning were smaller, did exhibit some significant discriminability, but mainly after 300 ms. In sum, our findings indicate that early and recurring temporal components arising from ventral face-sensitive regions are critically involved in learning new faces.

We address the problem of segmenting a multi-dimensional time series into stationary blocks by improving AutoSLEX [1], which has been successfully used for this purpose. AutoSLEX finds the best basis in a library of smoothed localized exponentials (SLEX) basis functions that are orthogonal and localized in both time and frequency. We introduce DynamicSLEX, a variant of AutoSLEX that relaxes the dyadic intervals constraint of AutoSLEX, allowing for more flexible segmentation while maintaining tractability. Then, we introduce RandSLEX, which uses random projections to scale-up SLEX-based segmentation to high dimensional inputs and to establish a notion of strength of splitting points in the segmentation. We demonstrate the utility of the proposed improvements on synthetic and real data.

The notion ofhypothesis is fundamental to science. Typically it refers to an idea that might plausibly be true, and that is to be examined or “tested” with some experimental data. Sometimes, the expectation is that the data will conform to the hypothesis. In other situations, the hypothesis is introduced with the goal of refuting it.

In Chapter 1 we said that a measurement is determined in part by a “signal” of interest, and in part by unknown factors we may call “noise.” Statistical models introduce probability distributions to describe the variation due to noise, and thereby achieve quantitative expressions of knowledge about the signal—a process we will describe more fully in Chapters 7 and 10.

In Section 7.2.1 we showed how the method of moments may be used to estimate the parameters of a \(Gamma(\alpha ,\beta )\) distribution, and we immediately stated that the method of maximum likelihood provides a better solution. How do we know this? In general, how should alternative methods of estimation be compared? In this chapter we lay out a series of principles that serve as guides to practice.

Many experiments examine the effects of multiple experimental conditions. When each measured response from a subject is a single-number, the data are usually analyzed with analysis of variance (ANOVA) .

At the beginning of this book we said that we wanted to lay out the key features of what we called, “the statistical paradigm,” which consists of broadly applicable concepts that guide reasoning from data in diverse contexts. One of its foundations is the idea that data may be used to express knowledge and uncertainty about unknown values of model parameters, especially through confidence intervals.