Rational Function Interpolation of Electromagnetic Transfer Functions of High-Speed Interconnect Systems from Discrete Time-Domain and Frequency-Domain Data

We present new methodologies to generate rational function approximations of broadband electromagnetic responses of linear and passive networks of high-speed interconnects, and to construct SPICE-compatible, equivalent circuit representations of the generated rational functions. These new methodologies are driven by the desire to improve the computational efficiency of the rational function fitting process, and to ensure enhanced accuracy of the generated rational function interpolation and its equivalent circuit representation. Toward this goal, we propose two new methodologies for rational function approximation of high-speed interconnect network responses. The first one relies on the use of both time-domain and frequency-domain data, obtained either through measurement or numerical simulation, to generate a rational function representation that extrapolates the input, early-time transient response data to late-time response while at the same time providing a means to both interpolate and extrapolate the used frequency-domain data. The aforementioned hybrid methodology can be considered as a generalization of the frequency-domain rational function fitting utilizing frequency-domain response data only, and the time-domain rational function fitting utilizing transient response data only. In this context, a guideline is proposed for estimating the order of the rational function approximation from transient data. The availability of such an estimate expedites the time-domain rational function fitting process. The second approach relies on the extraction of the delay associated with causal electromagnetic responses of interconnect systems to provide for a more stable rational function process utilizing a lower-order rational function interpolation. A distinctive feature of the proposed methodology is its utilization of scattering parameters. For both methodologies, the approach of fitting the electromagnetic network matrix one element at a time is applied. It is shown that, with regard to the computational cost of the rational function fitting process, such an element-by-element rational function fitting is more advantageous than full matrix fitting for systems with a large number of ports. Despite the disadvantage that different sets of poles are used in the rational function of different elements in the network matrix, such an approach provides for improved accuracy in the fitting of network matrices of systems characterized by both strongly coupled and weakly coupled ports. Finally, in order to provide a means for enforcing passivity in the adopted element-by-element rational function fitting approach, the methodology for passivity enforcement via quadratic programming is modified appropriately for this purpose and demonstrated in the context of element-by-element rational function fitting of the admittance matrix of an electromagnetic multiport.

Molecular mechanisms involved in cell response to mechanical forces

New devices were designed to generate a localized mechanical vibration of flexible gels where human umbilical vein endothelial cells (HUVECs) were cultured. The stimulation setups were able to apply relatively large strains (30%~50%) at high temporal frequencies (140~207 Hz) in a localized subcellular region. One of the advantages of this technique was to be less invasive to the innate cellular functions because there was no direct contact between the stimulating probe and the cell body. A mechanical vibration induced by the device in the substrate gel where cells were seeded could mainly cause global calcium responses of the cells. This global response was initiated by the influx of calcium across the stretch-activated channels in the plasma membrane. The subsequent production of inositol triphosphate (IP3) via phospholipase C (PLC) activation triggered the calcium release from the endoplasmic reticulum (ER) to cause a global intracellular calcium fluctuation over the whole cell body. This global calcium response was also shown to depend on actomyosin contractility and F-actin integrity, probably controlling the membrane stretch-activated channels. The localized nature of the stimulation is one of the most important features of these new designs as it allowed the observation of the calcium signaling propagation by ER calcium release. The next step was to focus on the calcium influx, more specifically the TRPM7 channels. As TRPM7 expression may modulate cell adhesion, an adhesion assay was developed and tested on HUVECs seeded on gel substrates with different treatments: normal treatment on gels showed highest attachment rate, followed by the partially treated gels (only 5% of usual fibronectin amount) and untreated gels, with the lowest attachment rate. The trend of the attachment rates correlated to the magnitude of the calcium signaling observed after mechanical stimulation. TRPM7 expression inhibition by siRNA caused an increased attachment rate when compared to both control and non-targeting siRNA-treated cells, but resulted in an actual weaker response in terms of calcium signaling. It suggests that TRPM7 channels are indeed important for the calcium signaling in response to mechanical stimulation. A complementary study was also conducted consisting in the mechanical stimulation of a dissected Drosophila embryo. Although ionomycin treatment showed calcium influx in the tissue, the mechanical stimulation delivered as a vertical vibration did not elicited calcium signaling in response. One possible reason is the dissection procedure causing desensitization of the tissue due to the scrapings and manipulations to open the embryo.