Polynomial approximation of derivatives by the constrained mock-Chebyshev least squares operator

The constrained mock-Chebyshev least squares operator is a linear approximation operator based on an equispaced grid of points. Like other polynomial or rational approximation methods, it was recently introduced in order to defeat the Runge phenomenon that occurs when using polynomial interpolation on large sets of equally spaced points. The idea is to improve the mock-Chebyshev subset interpolation, where the considered function $f$ is interpolated only on a proper subset of the uniform grid, formed by nodes that mimic the behavior of Chebyshev--Lobatto nodes. In the mock-Chebyshev subset interpolation all remaining nodes are discarded, while in the constrained mock-Chebyshev least squares interpolation they are used in a simultaneous regression, with the aim to further improving the accuracy of the approximation provided by the mock-Chebyshev subset interpolation. The goal of this paper is two-fold. We discuss some theoretical aspects of the constrained mock-Chebyshev least squares operator and present new results. In particular, we introduce explicit representations of the error and its derivatives. Moreover, for a sufficiently smooth function $f$ in $[-1,1]$, we present a method for approximating the successive derivatives of $f$ at a point $x\in [-1,1]$, based on the constrained mock-Chebyshev least squares operator and provide estimates for these approximations. Numerical tests demonstrate the effectiveness of the proposed method.Comment: 17 pages, 23 figure

LFP Analysis of Brain Injured Anesthetized Animals Undergoing Closed-Loop Intracortical Stimulation.

Activity dependent stimulation (ADS) is a closed loop stimulation technique whose neurophysiological effects have not been deeply investigated. Here we explored how Local field Potentials (LFP) are impacted by a focal ischemic lesion and, subsequently, by ADS treatment. Intracortical microelectrode arrays were implanted in the rostral forelimb area (RFA) and in the primary somatosensory area (S1) of anaesthetized rats. An ischemic injury was induced in the caudal forelimb area through microinjections of Endothelin-1. The lesion induced an acute depressive trend in LFP power in RFA (evaluated in 6 bands of interest: Delta (1-4Hz), Theta (4-8Hz), Alpha (8-11Hz), Beta (11-30Hz), LowGamma (30-55Hz) and HighGamma (55-80)) followed by a noticeable significant rebound in both areas. Applying ADS induced an overall decrease of power. The lesion impacted the connectivity in a frequency specific manner, resulting in widespread increase in connectivity in Delta both between and within areas. Two hours after the lesion, without stimulation, correlated activity between areas increased in Beta and Gamma. After stimulation, inter-area connectivity increased in Delta, Theta and Alpha, while considerably dropping within RFA in highGamma. By computing phase-amplitude coupling, we found that the lesion produced an incremental increase in the coupling between (Theta) Alpha phase and (lowGamma) highGamma amplitude within RFA, while S1 had a more generalized increase. Likewise, coupling between Theta phase and lowGamma/highGamma amplitudes increased between areas after lesion. ADS induced a similar increase, but greater in magnitude both within and between RFA and S1. These results have important implications on the emerging field of closed-loop adaptive stimulation promoting ADS as an innovative tool for the treatment of neurological disorders