Solvability and spectral properties of integral equations on the real line: I. Weighted spaces of continuous functions

We consider in this paper the solvability of linear integral equations on the real line, in operator form (λ−K)φ=ψ, where and K is an integral operator. We impose conditions on the kernel, k, of K which ensure that K is bounded as an operator on . Let Xa denote the weighted space as |s|→∞}. Our first result is that if, additionally, |k(s,t)|⩽κ(s−t), with and κ(s)=O(|s|−b) as |s|→∞, for some b>1, then the spectrum of K is the same on Xa as on X, for 01. As an example where kernels of this latter form occur we discuss a boundary integral equation formulation of an impedance boundary value problem for the Helmholtz equation in a half-plane.

On the stability and convergence of the finite section method for integral equation formulations of rough surface scattering

We consider the Dirichlet and Robin boundary value problems for the Helmholtz equation in a non-locally perturbed half-plane, modelling time harmonic acoustic scattering of an incident field by, respectively, sound-soft and impedance infinite rough surfaces.Recently proposed novel boundary integral equation formulations of these problems are discussed. It is usual in practical computations to truncate the infinite rough surface, solving a boundary integral equation on a finite section of the boundary, of length 2A, say. In the case of surfaces of small amplitude and slope we prove the stability and convergence as A→∞ of this approximation procedure. For surfaces of arbitrarily large amplitude and/or surface slope we prove stability and convergence of a modified finite section procedure in which the truncated boundary is ‘flattened’ in finite neighbourhoods of its two endpoints. Copyright © 2001 John Wiley & Sons, Ltd.

Padé approximants for the acoustical properties of rigid frame porous media with pore size distributions

Expressions for the viscosity correction function, and hence bulk complex impedance, density, compressibility, and propagation constant, are obtained for a rigid frame porous medium whose pores are prismatic with fixed cross-sectional shape, but of variable pore size distribution. The lowand high-frequency behavior of the viscosity correction function is derived for the particular case of a log-normal pore size distribution, in terms of coefficients which can, in general, be computed numerically, and are given here explicitly for the particular cases of pores of equilateral triangular, circular, and slitlike cross-section. Simple approximate formulae, based on two-point Pade´ approximants for the viscosity correction function are obtained, which avoid a requirement for numerical integration or evaluation of special functions, and their accuracy is illustrated and investigated for the three pore shapes already mentioned

Scattering by rough surfaces: the Dirichlet problem for the Helmholtz equation in a non-locally perturbed half-plane

We consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane, this problem arising in electromagnetic scattering by one-dimensional rough, perfectly conducting surfaces. We propose a new boundary integral equation formulation for this problem, utilizing the Green's function for an impedance half-plane in place of the standard fundamental solution. We show, at least for surfaces not differing too much from the flat boundary, that the integral equation is uniquely solvable in the space of bounded and continuous functions, and hence that, for a variety of incident fields including an incident plane wave, the boundary value problem for the scattered field has a unique solution satisfying the limiting absorption principle. Finally, a result of continuous dependence of the solution on the boundary shape is obtained.

Padé approximants for the acoustical characteristics of rigid frame porous media

In this paper it is shown that a number of theoretical models of the acoustical properties of rigid frame porous media, especially those involving ratios of Bessel functions of complex argument, can be accurately approximated and greatly simplified by the use of Padé approximation techniques. In the case of the model of Attenborough [J. Acoust. Soc. Am. 81, 93–102 (1987)] rational approximations are produced for the characteristic impedance, propagation constant, dynamic compressibility, and dynamic density, as a function of frequency and the material parameters. The model proposed by Stinson and Champoux

Concurrent recordings of bladder afferents from multiple nerves using a microfabricated PDMS microchannel electrode array

In this paper we present a compliant neural interface designed to record bladder afferent activity. We developed the implant's microfabrication process using multiple layers of silicone rubber and thin metal so that a gold microelectrode array is embedded within four parallel polydimethylsiloxane (PDMS) microchannels (5 mm long, 100 μm wide, 100 μm deep). Electrode impedance at 1 kHz was optimized using a reactive ion etching (RIE) step, which increased the porosity of the electrode surface. The electrodes did not deteriorate after a 3 month immersion in phosphate buffered saline (PBS) at 37 °C. Due to the unique microscopic topography of the metal film on PDMS, the electrodes are extremely compliant and can withstand handling during implantation (twisting and bending) without electrical failure. The device was transplanted acutely to anaesthetized rats, and strands of the dorsal branch of roots L6 and S1 were surgically teased and inserted in three microchannels under saline immersion to allow for simultaneous in vivo recordings in an acute setting. We utilized a tripole electrode configuration to maintain background noise low and improve the signal to noise ratio. The device could distinguish two types of afferent nerve activity related to increasing bladder filling and contraction. To our knowledge, this is the first report of multichannel recordings of bladder afferent activity.

High sensitivity recording of afferent nerve activity using ultra-compliant microchannel electrodes: an acutein vivovalidation

Neuroprostheses interfaced with transected peripheral nerves are technological routes to control robotic limbs as well as convey sensory feedback to patients suffering from traumatic neural injuries or degenerative diseases. To maximize the wealth of data obtained in recordings, interfacing devices are required to have intrafascicular resolution and provide high signal-to-noise ratio (SNR) recordings. In this paper, we focus on a possible building block of a three-dimensional regenerative implant: a polydimethylsiloxane (PDMS) microchannel electrode capable of highly sensitive recordings in vivo. The PDMS 'micro-cuff' consists of a 3.5 mm long (100 µm × 70 µm cross section) microfluidic channel equipped with five evaporated Ti/Au/Ti electrodes of sub-100 nm thickness. Individual electrodes have average impedance of 640 ± 30 kΩ with a phase angle of −58 ± 1 degrees at 1 kHz and survive demanding mechanical handling such as twisting and bending. In proof-of-principle acute implantation experiments in rats, surgically teased afferent nerve strands from the L5 dorsal root were threaded through the microchannel. Tactile stimulation of the skin was reliably monitored with the three inner electrodes in the device, simultaneously recording signal amplitudes of up to 50 µV under saline immersion. The overall SNR was approximately 4. A small but consistent time lag between the signals arriving at the three electrodes was observed and yields a fibre conduction velocity of 30 m s−1. The fidelity of the recordings was verified by placing the same nerve strand in oil and recording activity with hook electrodes. Our results show that PDMS microchannel electrodes open a promising technological path to 3D regenerative interfaces.

Evaluation of an elastomer based gold microelectrode array for neural recording applications

We are reporting on the fabrication and electrical characterization of a novel elastomer based micro-cuff neural interface. Electrodes are gold (Au) tracks of sub-100nm thickness and are thermally evaporated on a 0.5 mm thick polydimethylsiloxane (PDMS) substrate. We investigate how electrode area and immersion in phosphate buffered saline (PBS) at 37°C influence electrode impedance. A microfluidic channel is bonded to the electrode array to form the cuff. In an acute, in-vivo, proof-of-principle recording, the device is capable of detecting light stroking and pinch of a hind leg of an anaesthetized rat.

Compliant Multi-electrode Micro-channel Array for Recording Afferent Nerve Activity

We have fabricated a compliant neural interface to record afferent nerve activity. Stretchable gold electrodes were evaporated on a polydimethylsiloxane (PDMS) substrate and were encapsulated using photo-patternable PDMS. The built-in microstructure of the gold film on PDMS allows the electrodes to twist and flex repeatedly, without loss of electrical conductivity. PDMS microchannels (5mm long, 100μm wide, 100μm deep) were then plasma bonded irreversibly on top of the electrode array to define five parallel-conduit implants. The soft gold microelectrodes have a low impedance of ~200kΩ at the 1kHz frequency range. Teased nerves from the L6 dorsal root of an anaesthetized Sprague Dawley rat were threaded through the microchannels. Acute tripolar recordings of cutaneous activity are demonstrated, from multiple nerve rootlets simultaneously. Confinement of the axons within narrow microchannels allows for reliable recordings of low amplitude afferents. This electrode technology promises exciting applications in neuroprosthetic devices including bladder fullness monitors and peripheral nervous system implants.

Is the Helmholtz equation really sign-indefinite?

The usual variational (or weak) formulations of the Helmholtz equation are sign-indefinite in the sense that the bilinear forms cannot be bounded below by a positive multiple of the appropriate norm squared. This is often for a good reason, since in bounded domains under certain boundary conditions the solution of the Helmholtz equation is not unique at wavenumbers that correspond to eigenvalues of the Laplacian, and thus the variational problem cannot be sign-definite. However, even in cases where the solution is unique for all wavenumbers, the standard variational formulations of the Helmholtz equation are still indefinite when the wavenumber is large. This indefiniteness has implications for both the analysis and the practical implementation of finite element methods. In this paper we introduce new sign-definite (also called coercive or elliptic) formulations of the Helmholtz equation posed in either the interior of a star-shaped domain with impedance boundary conditions, or the exterior of a star-shaped domain with Dirichlet boundary conditions. Like the standard variational formulations, these new formulations arise just by multiplying the Helmholtz equation by particular test functions and integrating by parts.

Magnetic resonant modes in a wireless-powered circuit

A pair of circular loops of the same radius set at a distance form a closed induction circuit. Coupled to a capacitor, these loops form a resonant circuit wherein energy is transported from one to the other. When a current is introduced into one of the loops, it is received by its companion. This is due to the propagation of magnetic waves through the medium, in this instance, atmospheric air of a characteristic impedance.

Fractional order and non-linear system identification algorithms for biomedical applications

We discuss the modelling of dielectric responses of amorphous biological samples. Such samples are commonly encountered in impedance spectroscopy studies as well as in UV, IR, optical and THz transient spectroscopy experiments and in pump-probe studies. In many occasions, the samples may display quenched absorption bands. A systems identification framework may be developed to provide parsimonious representations of such responses. To achieve this, it is appropriate to augment the standard models found in the identification literature to incorporate fractional order dynamics. Extensions of models using the forward shift operator, state space models as well as their non-linear Hammerstein-Wiener counterpart models are highlighted. We also discuss the need to extend the theory of electromagnetically excited networks which can account for fractional order behaviour in the non-linear regime by incorporating nonlinear elements to account for the observed non-linearities. The proposed approach leads to the development of a range of new chemometrics tools for biomedical data analysis and classification.

Pharmacological actions of nobiletin in the modulation of platelet function

Background and Purpose The discovery that flavonoids are capable of inhibiting platelet function has led to their investigation as potential antithrombotic agents. However, despite the range of studies on the antiplatelet properties of flavonoids, little is known about the mechanisms by which flavonoids inhibit platelet function. In this study, we aimed to explore the pharmacological effects of a polymethoxy flavonoid, nobiletin in the modulation of platelet function. Experimental Approach The ability of nobiletin to modulate platelet function was explored by using a range of in vitro and in vivo experimental approaches. Aggregation, dense granule secretion and spreading assays were performed using washed platelets. The fibrinogen binding, α-granule secretion and calcium mobilisation assays were performed using platelet-rich plasma and whole blood was used in impedance aggregometry and thrombus formation experiments. The effect of nobiletin in vivo was assessed by measuring tail bleeding time using C57BL/6 mice. Key Results Nobiletin was shown to supress a range of well-established activatory mechanisms, including platelet aggregation, granule secretion, integrin modulation, calcium mobilisation and thrombus formation. Nobiletin was shown to extend bleeding time in mice and reduce the phosphorylation of Akt and PLCγ2 within the collagen receptor (GPVI) - stimulated pathway, in addition to increasing the levels of cGMP and phosphorylation of VASP, a protein whose activity is associated with inhibitory cyclic nucleotide signalling. Conclusions and Implications This study provides insight into the underlying molecular mechanisms through which nobiletin modulates haemostasis and thrombus formation. Therefore nobiletin may represent a potential antithrombotic agent of dietary origins.

Root elongation rate is correlated with the length of the bare root apex of maize and lupin roots despite contrasting responses of root growth to compact and dry soils

Background We investigated interacting effects of matric potential and soil strength on root elongation of maize and lupin, and relations between root elongation rates and the length of bare (hairless) root apex. Methods Root elongation rates and the length of bare root apexwere determined formaize and lupin seedlings in sandy loam soil of various matric potentials (−0.01 to −1.6 MPa) and bulk densities (0.9 to 1.5 Mg m−3). Results Root elongation rates slowed with both decreasing matric potential and increasing penetrometer resistance. Root elongation of maize slowed to 10 % of the unimpeded rate when penetrometer resistance increased to 2 MPa, whereas lupin elongated at about 40 % of the unimpeded rate. Maize root elongation rate was more sensitive to changes in matric potential in loosely packed soil (penetrometer resistances <1 MPa) than lupin. Despite these differing responses, root elongation rate of both species was linearly correlated with length of the bare root apex (r2 0.69 to 0.97). Conclusion Maize root elongation was more sensitive to changes in matric potential and mechanical impedance than lupin. Robust linear relationships between elongation rate and length of bare apex suggest good potential for estimating root elongation rates for excavated roots.

A role for adhesion and degranulation-promoting adapter protein in collagen-induced platelet activation mediated via integrin α(2) β(1)

Platelet aggregation and phosphorylation of phospholipase Cγ2 induced by collagen were attenuated in ADAP(-/-) platelets. However, aggregation and signaling induced by collagen-related peptide (CRP), a GPVI-selective agonist, were largely unaffected. Platelet adhesion to CRP was also unaffected by ADAP deficiency. Adhesion to the α(2) β(1) -selective ligand GFOGER and to a peptide (III-04), which supports adhesion that is dependent on both GPVI and α(2) β(1), was reduced in ADAP(-/-) platelets. An impedance-based label-free detection technique, which measures adhesion and spreading of platelets, indicated that, in the absence of ADAP, spreading on GFOGER was also reduced. This was confirmed with non-fluorescent differential-interference contrast microscopy, which revealed reduced filpodia formation in ADAP(-/-) platelets adherent to GFOGER. This indicates that ADAP plays a role in mediating platelet activation via the collagen-binding integrin α(2) β(1). In addition, we found that ADAP(-/-) mice, which are mildly thrombocytopenic, have enlarged spleens as compared with wild-type animals. This may reflect increased removal of platelets from the circulation.