A Differential Quadrature proposed here can be used to solve boundary-value and initial-value differential equations with a linear or nonlinear nature. Unlike the classic Differential Quadrature Method (DQM), the newly proposed Differential Quadrature chooses the function values and some derivatives wherever necessary as independent variables. Therefore, the d-type grid arrangement used in the classic DQM is exempt while applying the boundary conditions exactly. Most importantly, the explicit weighting coef®cients can be obtained using the proposed procedures. The present method is used to solve two types of differential equations which are the singlespan Bernoulli±Euler beam's buckling equation and the one-degree-of-freedom solid dynamic equation. Excellent results were obtained.

SUMMARYThe generalized di erential quadrature rule (GDQR) proposed here is aimed at solving high-order di erential equations. The improved approach is completely exempted from the use of the existing -point technique by applying multiple conditions in a rigorous manner. The GDQR is used here to static and dynamic analyses of Bernoulli-Euler beams and classical rectangular plates. Numerical error analysis caused by the method itself is carried out in the beam analysis. Independent variables for the plate are ÿrst deÿned. The explicit weighting coe cients are derived for a fourth-order di erential equation with two conditions at two di erent points. It is quite evident that the GDQR expressions and weighting coe cients for two-dimensional problems are not a direct application of those for one-dimensional problems. The GDQR are implemented through a number of examples. Good results are obtained in this work.

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Previous simulation works concerned with the mechanism of non-invasive neuromodulation has isolated many of the factors that can influence stimulation potency, but an inclusive account of the interplay between these factors on realistic neurons is still lacking. To give a comprehensive investigation on the stimulation-evoked neuronal activation, we developed a simulation scheme which incorporates highly detailed physiological and morphological properties of pyramidal cells. The model was implemented on a multitude of neurons; their thresholds and corresponding activation points with respect to various field directions and pulse waveforms were recorded. The results showed that the simulated thresholds had a minor anisotropy and reached minimum when the field direction was parallel to the dendritic-somatic axis; the layer 5 pyramidal cells always had lower thresholds but substantial variances were also observed within layers; reducing pulse length could magnify the threshold values as well as the variance; tortuosity and arborization of axonal segments could obstruct action potential initiation. The dependence of the initiation sites on both the orientation and the duration of the stimulus implies that the cellular excitability might represent the result of the competition between various firing-capable axonal components, each with a unique susceptibility determined by the local geometry. Moreover, the measurements obtained in simulation intimately resemble recordings in physiological and clinical studies, which seems to suggest that, with minimum simplification of the neuron model, the cable theory-based simulation approach can have sufficient verisimilitude to give quantitatively accurate evaluation of cell activities in response to the externally applied field.