Numerical computation of an Evans function for travelling waves

We demonstrate a geometrically inspired technique for computing Evans functions for the linearised operators about travelling waves. Using the examples of the F-KPP equation and a Keller-Segel model of bacterial chemotaxis, we produce an Evans function which is computable through several orders of magnitude in the spectral parameter and show how such a function can naturally be extended into the continuous spectrum. In both examples, we use this function to numerically verify the absence of eigenvalues in a large region of the right half of the spectral plane. We also include a new proof of spectral stability in the appropriate weighted space of travelling waves of speed $c \geq 2 \sqrt{\delta}$ in the F-KPP equation.Comment: 37 pages, 11 figure

On the stability of isothermal shocks in black hole accretion disks

Most black holes possess accretion disks. Models of such disks inform observations and constrain the properties of the black holes and their surrounding medium. Here, we study isothermal shocks in a thin black hole accretion flow. Modelling infinitesimal molecular viscosity allows the use of multiple-scales matched asymptotic methods. We thus derive the first explicit calculations of isothermal shock stability. We find that the inner shock is always unstable, and the outer shock is always stable. The growth/decay rates of perturbations depend only on an effective potential and the incoming--outgoing flow difference at the shock location. We give a prescription of accretion regimes in terms of angular momentum and black hole radius. Accounting for angular momentum dissipation implies unstable outer shocks in much of parameter space, even for realistic viscous Reynolds numbers of the order $\approx 10^{20}$.Comment: 26 page

The dynamics underlying pseudo-plateau bursting in a pituitary cell model.

This is the final version of the article. Available from BioMed Central via the DOI in this record.Pituitary cells of the anterior pituitary gland secrete hormones in response to patterns of electrical activity. Several types of pituitary cells produce short bursts of electrical activity which are more effective than single spikes in evoking hormone release. These bursts, called pseudo-plateau bursts, are unlike bursts studied mathematically in neurons (plateau bursting) and the standard fast-slow analysis used for plateau bursting is of limited use. Using an alternative fast-slow analysis, with one fast and two slow variables, we show that pseudo-plateau bursting is a canard-induced mixed mode oscillation. Using this technique, it is possible to determine the region of parameter space where bursting occurs as well as salient properties of the burst such as the number of spikes in the burst. The information gained from this one-fast/two-slow decomposition complements the information obtained from a two-fast/one-slow decomposition.This work was supported by NSF grant DMS 0917664 to RB and NIH grant DK 043200 to RB and JT

Canards existence in the Hindmarsh-Rose model

In two previous papers we have proposed a new method for proving the existence of "canard solutions" on one hand for three and four-dimensional singularly perturbed systems with only one fast variable and, on the other hand for four-dimensional singularly perturbed systems with two fast variables [J.M. Ginoux and J. Llibre, Qual. Theory Dyn. Syst. 15 (2016) 381-431; J.M. Ginoux and J. Llibre, Qual. Theory Dyn. Syst. 15 (2015) 342010]. The aim of this work is to extend this method which improves the classical ones used till now to the case of three-dimensional singularly perturbed systems with two fast variables. This method enables to state a unique generic condition for the existence of "canard solutions" for such three-dimensional singularly perturbed systems which is based on the stability of folded singularities (pseudo singular points in this case) of the normalized slow dynamics deduced from a well-known property of linear algebra. Applications of this method to a famous neuronal bursting model enables to show the existence of "canard solutions" in the Hindmarsh-Rose model

Canards existence in the Hindmarsh-Rose model

In two previous papers, we have proposed a new method for proving the existence of "canard solutions" on one hand for three- and four-dimensional singularly perturbed systems with only one fast variable and, on the other hand, for four-dimensional singularly perturbed systems with two fast variables; see [4, 5]. The aim of this work is to extend this method, which improves the classical ones used till now to the case of three-dimensional singularly perturbed systems with two fast variables. This method enables to state a unique generic condition for the existence of "canard solutions" for such three-dimensional singularly perturbed systems which is based on the stability of folded singularities (pseudo singular points in this case) of the normalized slow dynamics deduced from a well-known property of linear algebra. Applications of this method to a famous neuronal bursting model enables to show the existence of "canard solutions" in the Hindmarsh-Rose model

Peripheral infusion of rat bone marrow derived endothelial progenitor cells leads to homing in acute lung injury

<p>Abstract</p> <p>Background</p> <p>Bone marrow-derived progenitors for both epithelial and endothelial cells have been observed in the lung. Besides mature endothelial cells (EC) that compose the adult vasculature, endothelial progenitor cells (EPC) are supposed to be released from the bone marrow into the peripheral blood after stimulation by distinct inflammatory injuries. Homing of <it>ex vivo </it>generated bone marrow-derived EPC into the injured lung has not been investigated so far. We therefore tested the hypothesis whether homing of EPC in damaged lung tissue occurs after intravenous administration.</p> <p>Methods</p> <p>Ex vivo generated, characterized and cultivated rat bone marrow-derived EPC were investigated for proliferation and vasculogenic properties in vitro. EPC were tested for their homing in a left-sided rat lung transplant model mimicking a severe acute lung injury. EPC were transplanted into the host animal by peripheral administration into the femoral vein (10⁶cells). Rats were sacrificed 1, 4 or 9 days after lung transplantation and homing of EPC was evaluated by fluorescence microscopy. EPC were tested further for their involvement in vasculogenesis processes occurring in subcutaneously applied Matrigel in transplanted animals.</p> <p>Results</p> <p>We demonstrate the integration of intravenously injected EPC into the tissue of the transplanted left lung suffering from acute lung injury. EPC were localized in vessel walls as well as in destructed lung tissue. Virtually no cells were found in the right lung or in other organs. However, few EPC were found in subcutaneous Matrigel in transplanted rats.</p> <p>Conclusion</p> <p>Transplanted EPC may play an important role in reestablishing the endothelial integrity in vessels after severe injury or at inflamatory sites and might further contribute to vascular repair or wound healing processes in severely damaged tissue. Therapeutic applications of EPC transplantation may ensue.</p

Differential Mammary Gland Development in FVB and C57Bl/6 Mice: Implications for Breast Cancer Research

A growing body of research suggests a linkage between pubertal mammary gland development and environmental factors such as diet as modifiers of long term breast cancer risk. Much of this research is dependent upon mouse models, which may vary between studies. However, effects may be strain dependent and further modified by diet, which has not been previously examined. Therefore, the objective of the present study was to determine whether mammary gland development differs between FVB and C57Bl/6 strains on diets containing either n-6 or n-3 polyunsaturated fats. Developmental measures related to onset of puberty and mammary gland development differed between strains. Mice fed the n-3 polyunsaturated fatty acids (PUFA) diet were shown to have lower numbers of terminal end buds, a marker of mammary gland development. This study helps to further clarify differences in development and dietary response between FVB and C57Bl/6 mice in order to more appropriately relate mammary gland research to human populations