The seminar meets regularly on wednesday at 4pm in Eckhart 202. We also have special seminars during other days.
In the light of pathological diseases, such as Alzheimer’s, more and more significance has been attributed to cellular calcium signaling in neurons in the last years. Cellular calcium is regulated by a complex system of channels which allow the exchange of calcium between extracellular space and the cytosol and the exchange between cytosol and endoplasmic reticulum (ER) (Berridge, 2002) (or other organelles such as mitochondria). Further sources of intracellular calcium are active synapses which regulate NMDA-gated calcium levels and produce enzymatically regulated inositol-3-phosphate (IP3). This molecule interacts with the calcium relevant IP3-receptor embedded in the ER-membrane (Foskett et al. 2007). Pathologies like Alzheimer’s can disrupt this delicate system at multiple levels. We are currently developing a cellular calcium model including the realistic three-dimensional morphology of neurons (using automatic reconstruction methods (Broser et al., 2004, Queisser et al., 2008) and their calcium-relevant components, such as calcium-regulating channels on the plasma membrane, calcium-exchange mechanisms on the endoplasmic membrane, the biophysics of cytosolic and endoplasmic calcium, as well as the cell nucleus. The biophysical model is based on continuum mechanics derived systems of partial differential equations which are solved on the simulation platform uG (Bastian & Wittum, 1994). This work is based on previous calcium studies within neuron nuclei, (Wittmann et al., 2010). This cellular model is intended to describe the nuclear response to a wide range of cellular configurations. In particular one is able to extract differences in nuclear responses between healthy and pathological neurons.
Many phenomena in biology involve both reactions and chemotaxis. These processes can clearly influence each other, and chemotaxis can play an important role in sustaining and speeding up the reaction. However, to the best of our knowledge, the question of reaction enhancement by chemotaxis has not yet received extensive treatment either analytically or numerically. We consider a model with a single density function involving diffusion, advection, chemotaxis, and absorbing reaction. The model is motivated, in particular, by studies of coral broadcast spawning, where experimental observations of the efficiency of fertilization rates significantly exceed the data obtained from numerical models that do not take chemotaxis (attraction of sperm gametes by a chemical secreted by egg gametes) into account. We prove that in the framework of our model, chemotaxis plays a crucial role. There is a rigid limit to how much the fertilization efficiency can be enhanced if there is no chemotaxis but only advection and diffusion. On the other hand, when chemotaxis is present, the fertilization rate can be arbitrarily close to being complete provided that the chemotactic attraction is sufficiently strong. Moreover, an interesting feature of the estimates on fertilization rate and timescales in the chemotactic case is that they do not depend on the amplitude of the reaction term.
We study the incompressible Navier--Stokes equations with potential body forces on the three-dimensional torus. We show that the normalization introduced in the paper [Ann. Inst. H. Poincare Anal. Non Lineaire, 4(1):1--47, 1987] produces a Poincare-Dulac normal form which is obtained by an explicit change of variable. This change is the formal power series expansion of the inverse of the normalization map. Each homogeneous term of a finite degree in the series is proved to be well-defined in appropriate Sobolev spaces and is estimated recursively by using a family of homogeneous gauges which is suitable for estimating homogeneous polynomials in infinite dimensional spaces. This is joint work with Ciprian Foias and Jean-Claude Saut.
We study numerically the regularity problem of the $3D$ Navier-Stokes equations by estimating some integrals that appear in the Caffarelli-Kohn-Nirenberg theory. In particular, the theory states that the Hausdorff dimension of possible singular sets of velocity does not exceed $1$ in $(3+1)$ dimensional space-time. The key quantity involved is written (in standard notations) $\delta(r)=1/(\nu r) \int_Q |\nabla u|^2 dx dt,$ a non-dimensional local space-time integral of squared velocity gradient over a parabolic cylinder Q. We quantify how regular the Navier-Stokes flows can be by determining the scaling behaviour of $\delta(r)$ in direct numerical simulations of the Navier- Stokes equations. In moderately high Reynolds number turbulent flows (around $Re=100$) which have a short Kolmogorov inertial subrange, we have found that $\delta(r) \propto r^4$ for small $r$. This suggests that the simulated flow is far from a singularity. The same exponent is obtained when $\delta(r)$ is estimated by the Burgers vortex model, an exact steady solution of the Navier-Stokes equations. (This is a joint work with Mark Dowker.)
We discuss the large deviations principle and the problem of designing asymptotically optimal importance sampling schemes for stochastic differential equations with small noise and fast oscillating coefficients. There are three possible regimes depending on how fast the intensity of the noise goes to zero relative to the homogenization parameter. We use weak convergence methods which provide us with convenient representations for the action functional for all three regimes, and then we use these representations to study their similarities and differences. Furthermore, we derive a control that nearly achieves the large deviations lower bound at the prelimit level. This control is useful for designing efficient importance sampling schemes. Standard Monte Carlo methods perform poorly in these kind of problems in the small noise limit, and apart from the smallness of the noise, an additional reason for this is the presence of the fast oscillating coefficients. These results have applications in chemical physics and biology. Examples will be provided. Joint work with P. Dupuis and H. Wang.
We will consider three one dimensional models which arise in the context of fluid mechanics. These models are nonlocal and nonlinear partial differential equations. They are motivated by the Surface Quasi-Geostrophic equation and by the vortex patch problem. We will be interested in the formation of singularities in finite time.
Despite much recent (and not so recent) attention to solutions of integro-differential equations of elliptic type, it is surprising that a fundamental result such as a comparison theorem which can deal with only measure theoretic norms of the right hand side of the equation ($L^n$ and $L^\infty$) has gone unexplored. For the case of second order equations this result is known as the Aleksandrov-Bakelman-Pucci estimate (and dates back to circa 1960s), which says that for supersolutions of uniformly elliptic equation $Lu=f$, the supremum of u is controlled by the $L^n$ norm of $f$ ($n$ being the underlying dimension of the domain). We discuss extensions of this estimate to fully nonlinear integro-differential equations and present a recent result in this direction. (Joint with Nestor Guillen, available at arXiv:1101.0279v3 [math.AP] )
I present work modeling the motion of slender bodies in fluid flow. The flow is slow and on a small length scale so that some or all of the inertial forces may be neglected. Slender bodies are characterized by a large length-to-diameter ratio and are modeled in my work by a distribution of regularized fundamental solutions along the body centerline. A small parameter governing the width of the regularization can be related to the body diameter. I apply this method to biological examples including arrays of oscillating cricket sensory hairs. I characterize the amount of fluid-mediated coupling that occurs between the slender bodies in the array and comment on the implications for biological function.
