Integral inequalities for algebraic and trigonometric polynomials

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Progression of auditory discrimination based on neural decoding predicts awakening from coma.

Auditory evoked potentials are informative of intact cortical functions of comatose patients. The integrity of auditory functions evaluated using mismatch negativity paradigms has been associated with their chances of survival. However, because auditory discrimination is assessed at various delays after coma onset, it is still unclear whether this impairment depends on the time of the recording. We hypothesized that impairment in auditory discrimination capabilities is indicative of coma progression, rather than of the comatose state itself and that rudimentary auditory discrimination remains intact during acute stages of coma. We studied 30 post-anoxic comatose patients resuscitated from cardiac arrest and five healthy, age-matched controls. Using a mismatch negativity paradigm, we performed two electroencephalography recordings with a standard 19-channel clinical montage: the first within 24 h after coma onset and under mild therapeutic hypothermia, and the second after 1 day and under normothermic conditions. We analysed electroencephalography responses based on a multivariate decoding algorithm that automatically quantifies neural discrimination at the single patient level. Results showed high average decoding accuracy in discriminating sounds both for control subjects and comatose patients. Importantly, accurate decoding was largely independent of patients' chance of survival. However, the progression of auditory discrimination between the first and second recordings was informative of a patient's chance of survival. A deterioration of auditory discrimination was observed in all non-survivors (equivalent to 100% positive predictive value for survivors). We show, for the first time, evidence of intact auditory processing even in comatose patients who do not survive and that progression of sound discrimination over time is informative of a patient's chance of survival. Tracking auditory discrimination in comatose patients could provide new insight to the chance of awakening in a quantitative and automatic fashion during early stages of coma.

Decoding decisions in healthy subjects and auditory discrimination in comatose patients through single-trial topographic EEG analyses

Neuroimaging studies analyzing neurophysiological signals are typically based on comparing averages of peri-stimulus epochs across experimental conditions. This approach can however be problematic in the case of high-level cognitive tasks, where response variability across trials is expected to be high and in cases where subjects cannot be considered part of a group. The main goal of this thesis has been to address this issue by developing a novel approach for analyzing electroencephalography (EEG) responses at the single-trial level. This approach takes advantage of the spatial distribution of the electric field on the scalp (topography) and exploits repetitions across trials for quantifying the degree of discrimination between experimental conditions through a classification scheme. In the first part of this thesis, I developed and validated this new method (Tzovara et al., 2012a,b). Its general applicability was demonstrated with three separate datasets, two in the visual modality and one in the auditory. This development allowed then to target two new lines of research, one in basic and one in clinical neuroscience, which represent the second and third part of this thesis respectively. For the second part of this thesis (Tzovara et al., 2012c), I employed the developed method for assessing the timing of exploratory decision-making. Using single-trial topographic EEG activity during presentation of a choice's payoff, I could predict the subjects' subsequent decisions. This prediction was due to a topographic difference which appeared on average at ~516ms after the presentation of payoff and was subject-specific. These results exploit for the first time the temporal correlates of individual subjects' decisions and additionally show that the underlying neural generators start differentiating their responses already ~880ms before the button press. Finally, in the third part of this project, I focused on a clinical study with the goal of assessing the degree of intact neural functions in comatose patients. Auditory EEG responses were assessed through a classical mismatch negativity paradigm, during the very early phase of coma, which is currently under-investigated. By taking advantage of the decoding method developed in the first part of the thesis, I could quantify the degree of auditory discrimination at the single patient level (Tzovara et al., in press). Our results showed for the first time that even patients who do not survive the coma can discriminate sounds at the neural level, during the first hours after coma onset. Importantly, an improvement in auditory discrimination during the first 48hours of coma was predictive of awakening and survival, with 100% positive predictive value. - L'analyse des signaux électrophysiologiques en neuroimagerie se base typiquement sur la comparaison des réponses neurophysiologiques à différentes conditions expérimentales qui sont moyennées après plusieurs répétitions d'une tâche. Pourtant, cette approche peut être problématique dans le cas des fonctions cognitives de haut niveau, où la variabilité des réponses entre les essais peut être très élevéeou dans le cas où des sujets individuels ne peuvent pas être considérés comme partie d'un groupe. Le but principal de cette thèse est d'investiguer cette problématique en développant une nouvelle approche pour l'analyse des réponses d'électroencephalographie (EEG) au niveau de chaque essai. Cette approche se base sur la modélisation de la distribution du champ électrique sur le crâne (topographie) et profite des répétitions parmi les essais afin de quantifier, à l'aide d'un schéma de classification, le degré de discrimination entre des conditions expérimentales. Dans la première partie de cette thèse, j'ai développé et validé cette nouvelle méthode (Tzovara et al., 2012a,b). Son applicabilité générale a été démontrée avec trois ensembles de données, deux dans le domaine visuel et un dans l'auditif. Ce développement a permis de cibler deux nouvelles lignes de recherche, la première dans le domaine des neurosciences cognitives et l'autre dans le domaine des neurosciences cliniques, représentant respectivement la deuxième et troisième partie de ce projet. En particulier, pour la partie cognitive, j'ai appliqué cette méthode pour évaluer l'information temporelle de la prise des décisions (Tzovara et al., 2012c). En se basant sur l'activité topographique de l'EEG au niveau de chaque essai pendant la présentation de la récompense liée à un choix, on a pu prédire les décisions suivantes des sujets (en termes d'exploration/exploitation). Cette prédiction s'appuie sur une différence topographique qui apparaît en moyenne ~516ms après la présentation de la récompense. Ces résultats exploitent pour la première fois, les corrélés temporels des décisions au niveau de chaque sujet séparément et montrent que les générateurs neuronaux de ces décisions commencent à différentier leurs réponses déjà depuis ~880ms avant que les sujets appuient sur le bouton. Finalement, pour la dernière partie de ce projet, je me suis focalisée sur une étude Clinique afin d'évaluer le degré des fonctions neuronales intactes chez les patients comateux. Des réponses EEG auditives ont été examinées avec un paradigme classique de mismatch negativity, pendant la phase précoce du coma qui est actuellement sous-investiguée. En utilisant la méthode de décodage développée dans la première partie de la thèse, j'ai pu quantifier le degré de discrimination auditive au niveau de chaque patient (Tzovara et al., in press). Nos résultats montrent pour la première fois que même des patients comateux qui ne vont pas survivre peuvent discriminer des sons au niveau neuronal, lors de la phase aigue du coma. De plus, une amélioration dans la discrimination auditive pendant les premières 48heures du coma a été observée seulement chez des patients qui se sont réveillés par la suite (100% de valeur prédictive pour un réveil).

Decoding sequence learning from single-trial intracranial EEG in humans.

We propose and validate a multivariate classification algorithm for characterizing changes in human intracranial electroencephalographic data (iEEG) after learning motor sequences. The algorithm is based on a Hidden Markov Model (HMM) that captures spatio-temporal properties of the iEEG at the level of single trials. Continuous intracranial iEEG was acquired during two sessions (one before and one after a night of sleep) in two patients with depth electrodes implanted in several brain areas. They performed a visuomotor sequence (serial reaction time task, SRTT) using the fingers of their non-dominant hand. Our results show that the decoding algorithm correctly classified single iEEG trials from the trained sequence as belonging to either the initial training phase (day 1, before sleep) or a later consolidated phase (day 2, after sleep), whereas it failed to do so for trials belonging to a control condition (pseudo-random sequence). Accurate single-trial classification was achieved by taking advantage of the distributed pattern of neural activity. However, across all the contacts the hippocampus contributed most significantly to the classification accuracy for both patients, and one fronto-striatal contact for one patient. Together, these human intracranial findings demonstrate that a multivariate decoding approach can detect learning-related changes at the level of single-trial iEEG. Because it allows an unbiased identification of brain sites contributing to a behavioral effect (or experimental condition) at the level of single subject, this approach could be usefully applied to assess the neural correlates of other complex cognitive functions in patients implanted with multiple electrodes.

The algebraic equality of two asymptotic tests for the hypothesis that a normal distribution has a specified correlation matrix

It is proved the algebraic equality between Jennrich's (1970) asymptotic$X^2$ test for equality of correlation matrices, and a Wald test statisticderived from Neudecker and Wesselman's (1990) expression of theasymptoticvariance matrix of the sample correlation matrix.

Decoding stimulus-related information from single-trial EEG responses based on voltage topographies

Neuroimaging studies typically compare experimental conditions using average brain responses, thereby overlooking the stimulus-related information conveyed by distributed spatio-temporal patterns of single-trial responses. Here, we take advantage of this rich information at a single-trial level to decode stimulus-related signals in two event-related potential (ERP) studies. Our method models the statistical distribution of the voltage topographies with a Gaussian Mixture Model (GMM), which reduces the dataset to a number of representative voltage topographies. The degree of presence of these topographies across trials at specific latencies is then used to classify experimental conditions. We tested the algorithm using a cross-validation procedure in two independent EEG datasets. In the first ERP study, we classified left- versus right-hemifield checkerboard stimuli for upper and lower visual hemifields. In a second ERP study, when functional differences cannot be assumed, we classified initial versus repeated presentations of visual objects. With minimal a priori information, the GMM model provides neurophysiologically interpretable features - vis à vis voltage topographies - as well as dynamic information about brain function. This method can in principle be applied to any ERP dataset testing the functional relevance of specific time periods for stimulus processing, the predictability of subject's behavior and cognitive states, and the discrimination between healthy and clinical populations.

The Brauer group and the second Abel-Jacobi map for 0-cycles on algebraic varieties

This paper focuses on the connection between the Brauer group and the 0-cycles of an algebraic variety. We give an alternative construction of the second l-adic Abel-Jacobi map for such cycles, linked to the algebraic geometry of Severi-Brauer varieties on X. This allows us then to relate this Abel-Jacobi map to the standard pairing between 0-cycles and Brauer groups (see [M], [L]), completing results from [M] in this direction. Second, for surfaces, it allows us to present this map according to the more geometrical approach devised by M. Green in the framework of (arithmetic) mixed Hodge structures (see [G]). Needless to say, this paper owes much to the work of U. Jannsen and, especially, to his recently published older letter [J4] to B. Gross.

A note on quadrics through an algebraic curve

In this note we describe the intersection of all quadric hypersur- faces containing a given linearly normal smooth projective curve of genus n and degree 2n + 1

Algebraic decay of velocity fluctuations in a confined fluid

Computer simulations of a colloidal particle suspended in a fluid confined by rigid walls show that, at long times, the velocity correlation function decays with a negative algebraic tail. The exponent depends on the confining geometry, rather than the spatial dimensionality. We can account for the tail by using a simple mode-coupling theory which exploits the fact that the sound wave generated by a moving particle becomes diffusive.

Periodic orbits of integrable birational maps on the plane: blending dynamics and algebraic geometry, the Lyness' case

Contingut del Pòster presentat al congrés New Trends in Dynamical Systems

Algebraic decay of velocity fluctuations near a wall

Computer simulations of the dynamics of a colloidal particle suspended in a fluid confined by an interface show that the asymptotic decay of the velocity correlation functions is algebraic. The exponents of the long-time tails depend on the direction of motion of the particle relative to the surface, as well as on the specific nature of the boundary conditions. In particular, we find that for the angular velocity correlation function, the decay in the presence of a slip surface is faster than the one corresponding to a stick one. An intuitive picture is introduced to explain the various long-time tails, and the simulations are compared with theoretical expressions where available.

A game theoretical approach to the algebraic counterpart of the Wagner hierarchy

La hiérarchie de Wagner constitue à ce jour la plus fine classification des langages ω-réguliers. Par ailleurs, l'approche algébrique de la théorie de langages formels montre que ces ensembles ω-réguliers correspondent précisément aux langages reconnaissables par des ω-semigroupes finis pointés. Ce travail s'inscrit dans ce contexte en fournissant une description complète de la contrepartie algébrique de la hiérarchie de Wagner, et ce par le biais de la théorie descriptive des jeux de Wadge. Plus précisément, nous montrons d'abord que le degré de Wagner d'un langage ω-régulier est effectivement un invariant syntaxique. Nous définissons ensuite une relation de réduction entre ω-semigroupes pointés par le biais d'un jeu infini de type Wadge. La collection de ces structures algébriques ordonnée par cette relation apparaît alors comme étant isomorphe à la hiérarchie de Wagner, soit un quasi bon ordre décidable de largeur 2 et de hauteur ω. Nous exposons par la suite une procédure de décidabilité de cette hiérarchie algébrique : on décrit une représentation graphique des ω-semigroupes finis pointés, puis un algorithme sur ces structures graphiques qui calcule le degré de Wagner de n'importe quel élément. Ainsi le degré de Wagner de tout langage ω-régulier peut être calculé de manière effective directement sur son image syntaxique. Nous montrons ensuite comment construire directement et inductivement une structure de n''importe quel degré. Nous terminons par une description détaillée des invariants algébriques qui caractérisent tous les degrés de cette hiérarchie. Abstract The Wagner hierarchy is known so far to be the most refined topological classification of ω-rational languages. Also, the algebraic study of formal languages shows that these ω-rational sets correspond precisely to the languages recognizable by finite pointed ω-semigroups. Within this framework, we provide a construction of the algebraic counterpart of the Wagner hierarchy. We adopt a hierarchical game approach, by translating the Wadge theory from the ω-rational language to the ω-semigroup context. More precisely, we first show that the Wagner degree is indeed a syntactic invariant. We then define a reduction relation on finite pointed ω-semigroups by means of a Wadge-like infinite two-player game. The collection of these algebraic structures ordered by this reduction is then proven to be isomorphic to the Wagner hierarchy, namely a well-founded and decidable partial ordering of width 2 and height $\omega^\omega$. We also describe a decidability procedure of this hierarchy: we introduce a graph representation of finite pointed ω-semigroups allowing to compute their precise Wagner degrees. The Wagner degree of every ω-rational language can therefore be computed directly on its syntactic image. We then show how to build a finite pointed ω-semigroup of any given Wagner degree. We finally describe the algebraic invariants characterizing every Wagner degree of this hierarchy.

A mathematical study of fuzzy logic; an algebraic approach

Fuzzy set theory and Fuzzy logic is studied from a mathematical point of view. The main goal is to investigatecommon mathematical structures in various fuzzy logical inference systems and to establish a general mathematical basis for fuzzy logic when considered as multi-valued logic. The study is composed of six distinct publications. The first paper deals with Mattila'sLPC+Ch Calculus. THis fuzzy inference system is an attempt to introduce linguistic objects to mathematical logic without defining these objects mathematically.LPC+Ch Calculus is analyzed from algebraic point of view and it is demonstratedthat suitable factorization of the set of well formed formulae (in fact, Lindenbaum algebra) leads to a structure called ET-algebra and introduced in the beginning of the paper. On its basis, all the theorems presented by Mattila and many others can be proved in a simple way which is demonstrated in the Lemmas 1 and 2and Propositions 1-3. The conclusion critically discusses some other issues of LPC+Ch Calculus, specially that no formal semantics for it is given.In the second paper the characterization of solvability of the relational equation RoX=T, where R, X, T are fuzzy relations, X the unknown one, and o the minimum-induced composition by Sanchez, is extended to compositions induced by more general products in the general value lattice. Moreover, the procedure also applies to systemsof equations. In the third publication common features in various fuzzy logicalsystems are investigated. It turns out that adjoint couples and residuated lattices are very often present, though not always explicitly expressed. Some minor new results are also proved.The fourth study concerns Novak's paper, in which Novak introduced first-order fuzzy logic and proved, among other things, the semantico-syntactical completeness of this logic. He also demonstrated that the algebra of his logic is a generalized residuated lattice. In proving that the examination of Novak's logic can be reduced to the examination of locally finite MV-algebras.In the fifth paper a multi-valued sentential logic with values of truth in an injective MV-algebra is introduced and the axiomatizability of this logic is proved. The paper developes some ideas of Goguen and generalizes the results of Pavelka on the unit interval. Our proof for the completeness is purely algebraic. A corollary of the Completeness Theorem is that fuzzy logic on the unit interval is semantically complete if, and only if the algebra of the valuesof truth is a complete MV-algebra. The Compactness Theorem holds in our well-defined fuzzy sentential logic, while the Deduction Theorem and the Finiteness Theorem do not. Because of its generality and good-behaviour, MV-valued logic can be regarded as a mathematical basis of fuzzy reasoning. The last paper is a continuation of the fifth study. The semantics and syntax of fuzzy predicate logic with values of truth in ana injective MV-algerba are introduced, and a list of universally valid sentences is established. The system is proved to be semanticallycomplete. This proof is based on an idea utilizing some elementary properties of injective MV-algebras and MV-homomorphisms, and is purely algebraic.

Decoding auditory EEG responses in healthy and clinical populations: A comparative study.

BACKGROUND: Analyses of brain responses to external stimuli are typically based on the means computed across conditions. However in many cognitive and clinical applications, taking into account their variability across trials has turned out to be statistically more sensitive than comparing their means. NEW METHOD: In this study we present a novel implementation of a single-trial topographic analysis (STTA) for discriminating auditory evoked potentials at predefined time-windows. This analysis has been previously introduced for extracting spatio-temporal features at the level of the whole neural response. Adapting the STTA on specific time windows is an essential step for comparing its performance to other time-window based algorithms. RESULTS: We analyzed responses to standard vs. deviant sounds and showed that the new implementation of the STTA gives above-chance decoding results in all subjects (in comparison to 7 out of 11 with the original method). In comatose patients, the improvement of the decoding performance was even more pronounced than in healthy controls and doubled the number of significant results. COMPARISON WITH EXISTING METHOD(S): We compared the results obtained with the new STTA to those based on a logistic regression in healthy controls and patients. We showed that the first of these two comparisons provided a better performance of the logistic regression; however only the new STTA provided significant results in comatose patients at group level. CONCLUSIONS: Our results provide quantitative evidence that a systematic investigation of the accuracy of established methods in normal and clinical population is an essential step for optimizing decoding performance.

Quantifying the time for accurate EEG decoding of single value-based decisions.

BACKGROUND: Recent neuroimaging studies suggest that value-based decision-making may rely on mechanisms of evidence accumulation. However no studies have explicitly investigated the time when single decisions are taken based on such an accumulation process. NEW METHOD: Here, we outline a novel electroencephalography (EEG) decoding technique which is based on accumulating the probability of appearance of prototypical voltage topographies and can be used for predicting subjects' decisions. We use this approach for studying the time-course of single decisions, during a task where subjects were asked to compare reward vs. loss points for accepting or rejecting offers. RESULTS: We show that based on this new method, we can accurately decode decisions for the majority of the subjects. The typical time-period for accurate decoding was modulated by task difficulty on a trial-by-trial basis. Typical latencies of when decisions are made were detected at ∼500ms for 'easy' vs. ∼700ms for 'hard' decisions, well before subjects' response (∼340ms). Importantly, this decision time correlated with the drift rates of a diffusion model, evaluated independently at the behavioral level. COMPARISON WITH EXISTING METHOD(S): We compare the performance of our algorithm with logistic regression and support vector machine and show that we obtain significant results for a higher number of subjects than with these two approaches. We also carry out analyses at the average event-related potential level, for comparison with previous studies on decision-making. CONCLUSIONS: We present a novel approach for studying the timing of value-based decision-making, by accumulating patterns of topographic EEG activity at single-trial level.