Cross-frequency coupling (CFC) has been proposed to coordinate neural dynamics across spatial and temporal scales. Despite its potential relevance for understanding healthy and pathological brain function, the standard CFC analysis and physiological interpretation come with fundamental problems. For example, apparent CFC can appear because of spectral correlations due to common non-stationarities that may arise in the total absence of interactions between neural frequency components. To provide a road map towards an improved mechanistic understanding of CFC, we organize the available and potential novel statistical/modeling approaches according to their biophysical interpretability. While we do not provide solutions for all the problems described, we provide a list of practical recommendations to avoid common errors and to enhance the interpretability of CFC analysis.

Cross-frequency coupling (CFC) has been proposed to coordinate neural dynamics across spatial and temporal scales. Despite its potential relevance for understanding healthy and pathological brain function, the standard CFC analysis and physiological interpretation come with fundamental problems. For example, apparent CFC can appear because of spectral correlations due to common non-stationarities that may arise in the total absence of interactions between neural frequency components. To provide a road map towards an improved mechanistic understanding of CFC, we organize the available and potential novel statistical/modeling approaches according to their biophysical interpretability. While we do not provide solutions for all the problems described, we provide a list of practical recommendations to avoid common errors and to enhance the interpretability of CFC analysis.. CC-BY-NC-ND 4.0 International license It is made available under a (which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.The copyright holder for this preprint . http://dx.doi.org/10.1101/005926 doi: bioRxiv preprint first posted online Jun. 4, 2014; 2 HighlightsFundamental caveats and confounds in the methodology of assessing CFC are discussed.Significant CFC can be observed without any underlying physiological coupling.Non-stationarity of a time-series leads to spectral correlations interpreted as CFC.We offer practical recommendations, which can relieve some of the current confounds.Further theoretical and experimental work is needed to ground the CFC analysis.. CC-BY-NC-ND 4.0 International license It is made available under a (which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.The copyright holder for this preprint . http://dx.doi.org/10.1101/005926 doi: bioRxiv preprint first posted online Jun. 4, 2014; 3 Cross-frequency coupling: How much is that in real money?One of the central questions in neuroscience is how neural activity is coordinated across different spatial and temporal scales. An elegant solution to this problem could be that the activity of local neural populations is modulated according to the global neuronal dynamics. As larger populations oscillate and synchronize at lower frequencies and smaller ensembles are active at higher frequencies [1], cross-frequency coupling would facilitate flexible coordination of neural activity simultaneously in time and space. In line with this proposal, many studies have reported such crossfrequency relationships [2][3][4]. Especially phase-amplitude CFC, where the phase of the low frequency component modulates the amplitude of the high frequency activity, has been claimed to play important functional roles in neural information processing and cognition, e.g. in learning and memory [4][5][6][7][8] Furthermore, changes in CFC patterns have been linked to certain neurological and mental disorders such as Parkinson's disease [9][10][11], schizo...

We study certain classes of local sets of the two-dimensional Gaussian free field (GFF) in a simply-connected domain, and their relation to the conformal loop ensemble CLE4 and its variants. More specifically, we consider bounded-type thin local sets (BTLS), where thin means that the local set is small in size, and bounded-type means that the harmonic function describing the mean value of the field away from the local set is bounded by some deterministic constant. We show that a local set is a BTLS if and only if it is contained in some nested version of the CLE4 carpet, and prove that all BTLS are necessarily connected to the boundary of the domain. We also construct all possible BTLS for which the corresponding harmonic function takes only two prescribed values and show that all these sets (and this includes the case of CLE4) are in fact measurable functions of the GFF.

We introduce the first passage set (FPS) of constant level −a of the two-dimensional continuum Gaussian free field (GFF) on finitely connected domains. Informally, it is the set of points in the domain that can be connected to the boundary by a path on which the GFF does not go below −a. It is, thus, the two-dimensional analogue of the first hitting time of −a by a one-dimensional Brownian motion. We provide an axiomatic characterization of the FPS, a continuum construction using level lines, and study its properties: it is a fractal set of zero Lebesgue measure and Minkowski dimension 2 that is coupled with the GFF Φ as a local set A so that Φ + a restricted to A is a positive measure. One of the highlights of this paper is identifying this measure as a Minkowski content measure in the non-integer gauge r → | log(r)| 1/2 r 2 , by using Gaussian multiplicative chaos theory.2010 Mathematics Subject Classification. 60G15; 60G60; 60J65; 60J67; 81T40.