, 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.