A matrix recurrence for systems of Clifford algebra-valued orthogonal polynomials

Recently, the authors developed a matrix approach to multivariate polynomial sequences by using methods of Hypercomplex Function Theory ("Matrix representations of a basic polynomial sequence in arbitrary dimension". Comput. Methods Funct. Theory, 12 (2012), no. 2, 371-391). This paper deals with an extension of that approach to a recurrence relation for the construction of a complete system of orthogonal Clifford-algebra valued polynomials of arbitrary degree. At the same time the matrix approach sheds new light on results about systems of Clifford algebra-valued orthogonal polynomials obtained by Guerlebeck, Bock, Lavicka, Delanghe et al. during the last five years. In fact, it allows to prove directly some intrinsic properties of the building blocks essential in the construction process, but not studied so far.Fundação para a Ciência e a Tecnologia (FCT

Three-term recurrence relations for systems of Clifford algebra-valued orthogonal polynomials

Recently, systems of Clifford algebra-valued orthogonal polynomials have been studied from different points of view. We prove in this paper that for their building blocks there exist some three-term recurrence relations, similar to that for orthogonal polynomials of one real variable. As a surprising byproduct of own interest we found out that the whole construction process of Clifford algebra-valued orthogonal polynomials via Gelfand-Tsetlin basis or otherwise relies only on one and the same basic Appell sequence of polynomials.This work was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications of the University of Aveiro, the CMAT - Research Centre of Mathematics of the University of Minho and the FCT - Portuguese Foundation for Science and Technology (“Fundação para a Ciˆencia e a Tecnologia”), within projects PEst-OE/MAT/UI4106/2014 and PEst-OE/MAT/UI0013/2014.info:eu-repo/semantics/publishedVersio

Harmonic analysis and hypercomplex function theory in co-dimension one

Fundamentals of a function theory in co-dimension one for Clifford algebra valued functions over ℝn+1 are considered. Special attention is given to their origins in analytic properties of holomorphic functions of one and, by some duality reasons, also of several complex variables. Due to algebraic peculiarities caused by non-commutativity of the Clifford product, generalized holomorphic functions are characterized by two different but equivalent properties: on one side by local derivability (existence of a well defined derivative related to co-dimension one) and on the other side by differentiability (existence of a local approximation by linear mappings related to dimension one). As important applications, sequences of harmonic Appell polynomials are considered whose definition and explicit analytic representations rely essentially on both dual approaches.The work of the first, second and fourth authors was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e Tecnologia”), within project PEst-OE/MAT/UI4106/2013. The work of the second author was supported by Portuguese funds through the CMAT - Centre of Mathematics and FCT within the Project UID/MAT/00013/2013

Matrix representations of a special polynomial sequence in arbitrary dimension

This paper provides an insight into different structures of a special polynomial sequence of binomial type in higher dimensions with values in a Clifford algebra. The elements of the special polynomial sequence are homogeneous hypercomplex differentiable (monogenic) functions of different degrees and their matrix representation allows to prove their recursive construction in analogy to the complex power functions. This property can somehow be considered as a compensation for the loss of multiplicativity caused by the non-commutativity of the underlying algebra.Fundação para a Ciência e a Tecnologia (FCT

Hypercomplex polynomials, vietoris’ rational numbers and a related integer numbers sequence

This paper aims to give new insights into homogeneous hypercomplex Appell polynomials through the study of some interesting arithmetical properties of their coefficients. Here Appell polynomials are introduced as constituting a hypercomplex generalized geometric series whose fundamental role sometimes seems to have been neglected. Surprisingly, in the simplest non-commutative case their rational coefficient sequence reduces to a coefficient sequence S used in a celebrated theorem on positive trigonometric sums by Vietoris (Sitzungsber Österr Akad Wiss 167:125–135, 1958). For S a generating function is obtained which allows to derive an interesting relation to a result deduced by Askey and Steinig (Trans AMS 187(1):295–307, 1974) about some trigonometric series. The further study of S is concerned with a sequence of integers leading to its irreducible representation and its relation to central binomial coefficients.The work of the first and third authors was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e Tecnologia”), within project PEstOE/MAT/UI4106/2013. The work of the second author was supported by Portuguese funds through the CMAT - Centre of Mathematics and FCT within the Project UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

Carbamazepine reduces memory induced activation of mesial temporal lobe structures: a pharmacological fMRI-study

BACKGROUND AND PURPOSE: It is not known whether carbamazepine (CBZ; a drug widely used in neurology and psychiatry) influences the blood oxygenation level dependent (BOLD) contrast changes induced by neuronal activation and measured by functional MRI (fMRI). We aimed to investigate the influence of CBZ on memory induced activation of the mesial temporal lobes in patients with symptomatic temporal lobe epilepsy (TLE). MATERIAL AND METHODS: Twenty-one individual patients with refractory symptomatic TLE with different CBZ serum levels and 20 healthy controls were studied using BOLD fMRI. Mesial temporal lobe (MTL) activation was induced by a task that is based on the retrieval of individually familiar visuo-spatial knowledge. The extent of significant MTL fMRI activation was measured and correlated with the CBZ serum level. RESULTS: In TLE patients, the extent of significant fMRI activation over both MTL was negatively correlated to the CBZ serum level (Spearman r = -0.654, P < 0.001). Activation over the supposedly normal MTL, i.e. contralateral to the seizure onset of TLE patients, was smaller than the averaged MTL activation in healthy controls (P < 0.005). Age, duration of epilepsy, side of seizure onset, and intelligence were not correlated to the extent of the significant BOLD-response over both MTL in patients with TLE. CONCLUSIONS: In TLE patients, carbamazepine reduces the fMRI-detectable changes within the mesial temporal lobes as induced by effortful memory retrieval. FMRI appears to be suitable to study the effects of chronic drug treatment in patients with epilepsy

Symmetric Sensorimotor Somatotopy

BACKGROUND: Functional imaging has recently been used to investigate detailed somatosensory organization in human cortex. Such studies frequently assume that human cortical areas are only identifiable insofar as they resemble those measured invasively in monkeys. This is true despite the electrophysiological basis of the latter recordings, which are typically extracellular recordings of action potentials from a restricted sample of cells. METHODOLOGY/PRINCIPAL FINDINGS: Using high-resolution functional magnetic resonance imaging in human subjects, we found a widely distributed cortical response in both primary somatosensory and motor cortex upon pneumatic stimulation of the hairless surface of the thumb, index and ring fingers. Though not organized in a discrete somatotopic fashion, the population activity in response to thumb and index finger stimulation indicated a disproportionate response to fingertip stimulation, and one that was modulated by stimulation direction. Furthermore, the activation was structured with a line of symmetry through the central sulcus reflecting inputs both to primary somatosensory cortex and, precentrally, to primary motor cortex. CONCLUSIONS/SIGNIFICANCE: In considering functional activation that is not somatotopically or anatomically restricted as in monkey electrophysiology studies, our methodology reveals finger-related activation that is not organized in a simple somatotopic manner but is nevertheless as structured as it is widespread. Our findings suggest a striking functional mirroring in cortical areas conventionally ascribed either an input or an output somatotopic function

Phosphodiesterase Inhibition Increases CREB Phosphorylation and Restores Orientation Selectivity in a Model of Fetal Alcohol Spectrum Disorders

Background: Fetal alcohol spectrum disorders (FASD) are the leading cause of mental retardation in the western world and children with FASD present altered somatosensory, auditory and visual processing. There is growing evidence that some of these sensory processing problems may be related to altered cortical maps caused by impaired developmental neuronal plasticity. Methodology/Principal Findings: Here we show that the primary visual cortex of ferrets exposed to alcohol during the third trimester equivalent of human gestation have decreased CREB phosphorylation and poor orientation selectivity revealed by western blotting, optical imaging of intrinsic signals and single-unit extracellular recording techniques. Treating animals several days after the period of alcohol exposure with a phosphodiesterase type 1 inhibitor (Vinpocetine) increased CREB phosphorylation and restored orientation selectivity columns and neuronal orientation tuning. Conclusions/Significance: These findings suggest that CREB function is important for the maturation of orientation selectivity and that plasticity enhancement by vinpocetine may play a role in the treatment of sensory problems in FASD

A Functional Architecture of Optic Flow in the Inferior Parietal Lobule of the Behaving Monkey

The representation of navigational optic flow across the inferior parietal lobule was assessed using optical imaging of intrinsic signals in behaving monkeys. The exposed cortex, corresponding to the dorsal-most portion of areas 7a and dorsal prelunate (DP), was imaged in two hemispheres of two rhesus monkeys. The monkeys actively attended to changes in motion stimuli while fixating. Radial expansion and contraction, and rotation clockwise and counter-clockwise optic flow stimuli were presented concentric to the fixation point at two angles of gaze to assess the interrelationship between the eye position and optic flow signal. The cortical response depended upon the type of flow and was modulated by eye position. The optic flow selectivity was embedded in a patchy architecture within the gain field architecture. All four optic flow stimuli tested were represented in areas 7a and DP. The location of the patches varied across days. However the spatial periodicity of the patches remained constant across days at ∼950 and 1100 µm for the two animals examined. These optical recordings agree with previous electrophysiological studies of area 7a, and provide new evidence for flow selectivity in DP and a fine scale description of its cortical topography. That the functional architectures for optic flow can change over time was unexpected. These and earlier results also from inferior parietal lobule support the inclusion of both static and dynamic functional architectures that define association cortical areas and ultimately support complex cognitive function

Integrating Functional and Diffusion Magnetic Resonance Imaging for Analysis of Structure-Function Relationship in the Human Language Network

The capabilities of magnetic resonance imaging (MRI) to measure structural and functional connectivity in the human brain have motivated growing interest in characterizing the relationship between these measures in the distributed neural networks of the brain. In this study, we attempted an integration of structural and functional analyses of the human language circuits, including Wernicke's (WA), Broca's (BA) and supplementary motor area (SMA), using a combination of blood oxygen level dependent (BOLD) and diffusion tensor MRI.Functional connectivity was measured by low frequency inter-regional correlations of BOLD MRI signals acquired in a resting steady-state, and structural connectivity was measured by using adaptive fiber tracking with diffusion tensor MRI data. The results showed that different language pathways exhibited different structural and functional connectivity, indicating varying levels of inter-dependence in processing across regions. Along the path between BA and SMA, the fibers tracked generally formed a single bundle and the mean radius of the bundle was positively correlated with functional connectivity. However, fractional anisotropy was found not to be correlated with functional connectivity along paths connecting either BA and SMA or BA and WA. for use in diagnosing and determining disease progression and recovery