, we misunderstood the method described in Barnett and Seth (25) to compute the conditional Granger causality. We realize now that the spectral factorization method they describe can be used to obtain the conditional Granger causality with a single model fit, which would avoid the computational problems associated with separate model fits. We apologize for this error." Granger causality methods were developed to analyze the flow of information between time series. These methods have become more widely applied in neuroscience. Frequency-domain causality measures, such as those of Geweke, as well as multivariate methods, have particular appeal in neuroscience due to the prevalence of oscillatory phenomena and highly multivariate experimental recordings. Despite its widespread application in many fields, there are ongoing concerns regarding the applicability of Granger causality methods in neuroscience. When are these methods appropriate? How reliably do they recover the system structure underlying the observed data? What do frequency-domain causality measures tell us about the functional properties of oscillatory neural systems? In this paper, we analyze fundamental properties of Granger-Geweke (GG) causality, both computational and conceptual. Specifically, we show that (i) GG causality estimates can be either severely biased or of high variance, both leading to spurious results; (ii) even if estimated correctly, GG causality estimates alone are not interpretable without examining the component behaviors of the system model; and (iii) GG causality ignores critical components of a system's dynamics. Based on this analysis, we find that the notion of causality quantified is incompatible with the objectives of many neuroscience investigations, leading to highly counterintuitive and potentially misleading results. Through the analysis of these problems, we provide important conceptual clarification of GG causality, with implications for other related causality approaches and for the role of causality analyses in neuroscience as a whole.Granger causality | time series analysis | neural oscillations | connectivity | system identification G ranger causality is a statistical tool developed to analyze the flow of information between time series. Neuroscientists have applied Granger causality methods to diverse sources of data, including electroencephalography (EEG), magnetoencephalography (MEG), functional magnetic resonance imaging (fMRI), and local field potentials (LFP). These studies have investigated functional neural systems at scales of organization from the cellular level (1-3) to whole-brain network activity (4), under a range of conditions, including sensory stimuli (5-7), varying levels of consciousness (8-10), cognitive tasks (11), and pathological states (12, 13). In such analyses, the time series data are interpreted to reflect neural activity from a particular source, and Granger causality is used to characterize the directionality, directness, and dynamics of influence between sources.Oscillations are a ...

Electrodermal activity (EDA) is a measure of physical arousal, which is frequently measured during psychophysical tasks relevant for anxiety disorders. Recently, specific protocols and procedures have been devised in order to examine the neural mechanisms of fear conditioning and extinction. EDA reflects important responses associated with stimuli specifically administrated during these procedures. Although several previous studies have demonstrated the reproducibility of measures estimated from EDA, a mathematical framework associated with the stimulus-response experiments in question and, at the same time, including the underlying emotional state of the subject during fear conditioning and/or extinction experiments is not well studied. We here propose an ordinary differential equation model based on sudomotor nerve activity, and estimate the fear eliciting stimulus using a compressed sensing algorithm. Our results show that we are able to recover the underlying stimulus (visual cue or mild electrical shock). Moreover, relating the time-delay in the estimated stimulation to the visual cue during extinction period shows that fear level decreases as visual cues are presented without shock, suggesting that this feature might be used to estimate the fear state. These findings indicate that a mathematical model based on electrodermal responses might be critical in defining a low-dimensional representation of essential cognitive features in order to describe dynamic behavioral states.

Transient oscillatory events in the sleep electroencephalogram represent short-term coordinated network activity. Of particular importance, sleep spindles are transient oscillatory events associated with memory consolidation, which are altered in aging and in several psychiatric and neurodegenerative disorders. Spindle identification, however, currently contains implicit assumptions derived from what waveforms were historically easiest to discern by eye, and has recently been shown to select only a high-amplitude subset of transient events. Moreover, spindle activity is typically averaged across a sleep stage, collapsing continuous dynamics into discrete states. What information can be gained by expanding our view of transient oscillatory events and their dynamics? In this paper, we develop a novel approach to electroencephalographic phenotyping, characterizing a generalized class of transient time-frequency events across a wide frequency range using continuous dynamics. We demonstrate that the complex temporal evolution of transient events during sleep is highly stereotyped when viewed as a function of slow oscillation power (an objective, continuous metric of depth-of-sleep) and phase (a correlate of cortical up/down states). This two-fold power-phase representation has large intersubject variability—even within healthy controls—yet strong night-to-night stability for individuals, suggesting a robust basis for phenotyping. As a clinical application, we then analyze patients with schizophrenia, confirming established spindle (12–15 Hz) deficits as well as identifying novel differences in transient non-rapid eye movement events in low-alpha (7–10 Hz) and theta (4–6 Hz) ranges. Overall, these results offer an expanded view of transient activity, describing a broad class of events with properties varying continuously across spatial, temporal, and phase-coupling dimensions.

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Detection of spectral peaks and estimation of their properties, including frequency and amplitude, are fundamental to many applications of signal processing. Electroencephalography (EEG) of sleep, in particular, displays characteristic oscillations that change continuously throughout the night. Capturing these dynamics is essential to understanding the sleep process and characterizing the heterogeneity observed across individuals. Most sleep EEG analyses rely on either time-averaged spectra or bandpassed amplitude/power. Unfortunately, these approaches obscure the time-variability of peak properties, require specification of a priori criteria, and cannot distinguish power from nearby oscillations. More sophisticated approaches, using various spectral models, have been proposed to better estimate oscillatory properties, but these too have limitations. We present an improved approach to spectrogram decomposition, tracking time-varying parameterized peak functions and dynamically estimating their parameters using a modified form of the iterated extended Kalman filter (IEKF) that incorporates discrete On/Off-switching of peak combinations and a sampling step to draw the initial reference trajectory. We evaluate this approach on two types of simulated examples—one nearly within the model class and one outside. We find excellent performance, in terms of spectral fits and accuracy of estimated states, for both simulation types. We then apply the approach to real EEG data of sleep onset, obtaining quality spectral estimates with estimated peak combinations closely matching the expert-scored sleep stages. This approach offers not only the ability to estimate time-varying parameters of spectral peaks but, moving forward, the potential to estimate the governing dynamics and analyze their variability across nights, subjects, and clinical groups.

In this paper we present a strategy for reducing the number of false-positives in computer-aided mass detection. Our approach is to only mark "consensus" detections from among the suspicious sites identified by different "stage-1" detection algorithms. By "stage-1" we mean that each of the Computer-aided Detection (CADe) algorithms is designed to operate with high sensitivity, allowing for a large number of false positives. In this study, two mass detection methods were used: (1) Heath and Bowyer's algorithm based on the average fraction under the minimum filter (AFUM) and (2) a low-threshold bi-lateral subtraction algorithm. The two methods were applied separately to a set of images from the Digital Database for Screening Mammography (DDSM) to obtain paired sets of mass candidates. The consensus mass candidates for each image were identified by a logical "and" operation of the two CADe algorithms so as to eliminate regions of suspicion that were not independently identified by both techniques. It was shown that by combining the evidence from the AFUM filter method with that obtained from bi-lateral subtraction, the same sensitivity could be reached with fewer false-positives per image relative to using the AFUM filter alone.

We would like to thank the authors Faes et al. [1], as well as Barnett et al. [2], for their thoughtful commentary on our paper [3]. The main points of our work were to 1) characterize statistical properties of the traditional computation of Granger-Geweke (GG) causality, and 2) to analyze how the dynamics of the system are represented in the GG-causality measure. Barnett et al. [2] and Faes et al.[1] both point out that the issues with bias and variance in the conditional GG-causality can be addressed using a state space approach and a single model fit. This is clearly the case, and is demonstrated in the simulation studies in Faes et al. [1]. We regret that we were unaware of the earlier papers on GG-causality using state space models by Barnett and Seth [4] and Solo [5]. The original submission of our paper was in October of 2014, and a considerable time elapsed before our re-submitted manuscript was ultimately accepted, and our coverage of the literature during that span of time was not up to date. We also described the state space solution to these problems in Dr. Stokes' Ph.D. thesis [6] in January 2015, but felt it was important to first characterize and describe the problem, before laying out a solution to that problem. Along those lines, we believe our paper makes an important contribution by illustrating the form of the bias and variance, particularly in frequency domain, when separate model fits are employed. As Faes et al. [1] suggest, and as we have noted in other discussions surrounding our paper, many analysts continue to use separate model fits while performing GG-causality analysis, likely unaware of the problems with this approach. Again, these computational issues can be avoided by using a single model fit, preferably using the state space approach [4, 5, 1, 6]. 1Barnett et al.[2] emphasize that Granger causality reflects a "directed information flow." But how does one meaningfully interpret that information flow? Our analysis suggests that the receiver independence property is highly problematic for neuroscience studies, where the objective is typically to identify and/or characterize the mechanism of some observed effect. As we have shown, the dynamics of the effect nodes are absent in GG-causality. Oscillations play an important role in systems neuroscience, and interpretation of causality measures appears particularly problematic in systems with strong frequency-dependent structure. Studies of oscillatory phenomena are invariably geared towards understanding the factors that contribute to oscillations observed at specific frequencies. Ignoring these observed dynamics is simply not compatible with the goal of understanding them.As Barnett et al.[2] state, one can make a distinction between physiological or "physical causal mechanisms" and "directed information flow." However, we perceive that in practice, the need to interpret and ascribe meaning to data analyses would tend to lead investigators to interpret "directed information flow" in mechanistic terms. So the notions of "information...

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