Local Analysis of Inverse Problems: H\"{o}lder Stability and Iterative Reconstruction

We consider a class of inverse problems defined by a nonlinear map from parameter or model functions to the data. We assume that solutions exist. The space of model functions is a Banach space which is smooth and uniformly convex; however, the data space can be an arbitrary Banach space. We study sequences of parameter functions generated by a nonlinear Landweber iteration and conditions under which these strongly converge, locally, to the solutions within an appropriate distance. We express the conditions for convergence in terms of H\"{o}lder stability of the inverse maps, which ties naturally to the analysis of inverse problems

Theory of Spike Spiral Waves in a Reaction-Diffusion System

We discovered a new type of spiral wave solutions in reaction-diffusion systems --- spike spiral wave, which significantly differs from spiral waves observed in FitzHugh-Nagumo-type models. We present an asymptotic theory of these waves in Gray-Scott model. We derive the kinematic relations describing the shape of this spiral and find the dependence of its main parameters on the control parameters. The theory does not rely on the specific features of Gray-Scott model and thus is expected to be applicable to a broad range of reaction-diffusion systems.Comment: 4 pages (REVTeX), 2 figures (postscript), submitted to Phys. Rev. Let

Difference factorizations and monotonicity in inverse medium scattering for contrasts with fixed sign on the boundary

We generalize the factorization method for inverse medium scattering using a particular factorization of the difference of two far field operators. While the factorization method has been used so far mainly to identify the shape of a scatterer's support, we show that factorizations based on Dirichlet-to-Neumann operators can be used to compute bounds for numerical values of the medium on the boundary of its support. To this end, we generalize ideas from inside-outside duality to obtain a monotonicity principle that allows for alternative uniqueness proofs for particular inverse scattering problems (e.g., when obstacles are present inside the medium). This monotonicity principle indeed is our most important technical tool: It further directly shows that the boundary values of the medium's contrast function are uniquely determined by the corresponding far field operator. Our particular factorization of far field operators additionally implies that the factorization method rigorously characterizes the support of an inhomogeneous medium if the contrast function takes merely positive or negative values on the boundary of its support independently of the contrast's values inside its support. Finally, the monotonicity principle yields a simple algorithm to compute upper and lower bounds for these boundary values, assuming the support of the contrast is known. Numerical experiments show feasibility of a resulting numerical algorithm

Ancient Chinese medicine and mechanistic evidence of acupuncture physiology

Acupuncture has been widely used in China for three millennia as an art of healing. Yet, its physiology is not yet understood. The current interest in acupuncture started in 1971. Soon afterward, extensive research led to the concept of neural signaling with possible involvement of opioid peptides, glutamate, adenosine and identifying responsive parts in the central nervous system. In the last decade scientists began investigating the subject with anatomical and molecular imaging. It was found that mechanical movements of the needle, ignored in the past, appear to be central to the method and intracellular calcium ions may play a pivotal role. In this review, we trace the technique of clinical treatment from the first written record about 2,200 years ago to the modern time. The ancient texts have been used to introduce the concepts of yin, yang, qi, de qi, and meridians, the traditional foundation of acupuncture. We explore the sequence of the physiological process, from the turning of the needle, the mechanical wave activation of calcium ion channel to beta-endorphin secretion. By using modern terminology to re-interpret the ancient texts, we have found that the 2nd century b.c. physiologists were meticulous investigators and their explanation fits well with the mechanistic model derived from magnetic resonance imaging (MRI) and confocal microscopy. In conclusion, the ancient model appears to have withstood the test of time surprisingly well confirming the popular axiom that the old wine is better than the new

Does the Potential for Chaos Constrain the Embryonic Cell-Cycle Oscillator?

Although many of the core components of the embryonic cell-cycle network have been elucidated, the question of how embryos achieve robust, synchronous cellular divisions post-fertilization remains unexplored. What are the different schemes that could be implemented by the embryo to achieve synchronization? By extending a cell-cycle model previously developed for embryos of the frog Xenopus laevis to include the spatial dimensions of the embryo, we establish a novel role for the rapid, fertilization-initiated calcium wave that triggers cell-cycle oscillations. Specifically, in our simulations a fast calcium wave results in synchronized cell cycles, while a slow wave results in full-blown spatio-temporal chaos. We show that such chaos would ultimately lead to an unpredictable patchwork of cell divisions across the embryo. Given this potential for chaos, our results indicate a novel design principle whereby the fast calcium-wave trigger following embryo fertilization synchronizes cell divisions

Diffusive coupling can discriminate between similar reaction mechanisms in an allosteric enzyme system

<p>Abstract</p> <p>Background</p> <p>A central question for the understanding of biological reaction networks is how a particular dynamic behavior, such as bistability or oscillations, is realized at the molecular level. So far this question has been mainly addressed in well-mixed reaction systems which are conveniently described by ordinary differential equations. However, much less is known about how molecular details of a reaction mechanism can affect the dynamics in diffusively coupled systems because the resulting partial differential equations are much more difficult to analyze.</p> <p>Results</p> <p>Motivated by recent experiments we compare two closely related mechanisms for the product activation of allosteric enzymes with respect to their ability to induce different types of reaction-diffusion waves and stationary Turing patterns. The analysis is facilitated by mapping each model to an associated complex Ginzburg-Landau equation. We show that a sequential activation mechanism, as implemented in the model of Monod, Wyman and Changeux (MWC), can generate inward rotating spiral waves which were recently observed as glycolytic activity waves in yeast extracts. In contrast, in the limiting case of a simple Hill activation, the formation of inward propagating waves is suppressed by a Turing instability. The occurrence of this unusual wave dynamics is not related to the magnitude of the enzyme cooperativity (as it is true for the occurrence of oscillations), but to the sensitivity with respect to changes of the activator concentration. Also, the MWC mechanism generates wave patterns that are more stable against long wave length perturbations.</p> <p>Conclusions</p> <p>This analysis demonstrates that amplitude equations, which describe the spatio-temporal dynamics near an instability, represent a valuable tool to investigate the molecular effects of reaction mechanisms on pattern formation in spatially extended systems. Using this approach we have shown that the occurrence of inward rotating spiral waves in glycolysis can be explained in terms of an MWC, but not with a Hill mechanism for the activation of the allosteric enzyme phosphofructokinase. Our results also highlight the importance of enzyme oligomerization for a possible experimental generation of Turing patterns in biological systems.</p

A Compact Multiphoton 3D Imaging System for Recording Fast Neuronal Activity

We constructed a simple and compact imaging system designed specifically for the recording of fast neuronal activity in a 3D volume. The system uses an Yb:KYW femtosecond laser we designed for use with acousto-optic deflection. An integrated two-axis acousto-optic deflector, driven by digitally synthesized signals, can target locations in three dimensions. Data acquisition and the control of scanning are performed by a LeCroy digital oscilloscope. The total cost of construction was one order of magnitude lower than that of a typical Ti:sapphire system. The entire imaging apparatus, including the laser, fits comfortably onto a small rig for electrophysiology. Despite the low cost and simplicity, the convergence of several new technologies allowed us to achieve the following capabilities: i) full-frame acquisition at video rates suitable for patch clamping; ii) random access in under ten microseconds with dwelling ability in the nominal focal plane; iii) three-dimensional random access with the ability to perform fast volume sweeps at kilohertz rates; and iv) fluorescence lifetime imaging. We demonstrate the ability to record action potentials with high temporal resolution using intracellularly loaded potentiometric dye di-2-ANEPEQ. Our design proffers easy integration with electrophysiology and promises a more widespread adoption of functional two-photon imaging as a tool for the study of neuronal activity. The software and firmware we developed is available for download at http://neurospy.org/ under an open source license

Calcineurin Interacts with PERK and Dephosphorylates Calnexin to Relieve ER Stress in Mammals and Frogs

Background: The accumulation of misfolded proteins within the endoplasmic reticulum (ER) triggers a cellular process known as the Unfolded Protein Response (UPR). One of the earliest responses is the attenuation of protein translation. Little is known about the role that Ca 2+ mobilization plays in the early UPR. Work from our group has shown that cytosolic phosphorylation of calnexin (CLNX) controls Ca 2+ uptake into the ER via the sarco-endoplasmic reticulum Ca 2+-ATPase (SERCA) 2b. Methodology/Principal Findings: Here, we demonstrate that calcineurin (CN), a Ca 2+ dependent phosphatase, associates with the (PKR)-like ER kinase (PERK), and promotes PERK auto-phosphorylation. This association, in turn, increases the phosphorylation level of eukaryotic initiation factor-2 a (eIF2-a) and attenuates protein translation. Data supporting these conclusions were obtained from co-immunoprecipitations, pull-down assays, in-vitro kinase assays, siRNA treatments and [ 35 S]-methionine incorporation measurements. The interaction of CN with PERK was facilitated at elevated cytosolic Ca 2+ concentrations and involved the cytosolic domain of PERK. CN levels were rapidly increased by ER stressors, which could be blocked by siRNA treatments for CN-Aa in cultured astrocytes. Downregulation of CN blocked subsequent ER-stress-induced increases in phosphorylated elF2-a. CN knockdown in Xenopus oocytes predisposed them to induction of apoptosis. We also found that CLNX was dephosphorylated by CN when Ca 2+ increased. These data were obtained from [c 32 P]-CLN

Nonlinear gap junctions enable long-distance propagation of pulsating calcium waves in astrocyte networks

A new paradigm has recently emerged in brain science whereby communications between glial cells and neuron-glia interactions should be considered together with neurons and their networks to understand higher brain functions. In particular, astrocytes, the main type of glial cells in the cortex, have been shown to communicate with neurons and with each other. They are thought to form a gap-junction-coupled syncytium supporting cell-cell communication via propagating Ca2+ waves. An identified mode of propagation is based on cytoplasm-to-cytoplasm transport of inositol trisphosphate (IP3) through gap junctions that locally trigger Ca2+ pulses via IP3-dependent Ca2+-induced Ca2+ release. It is, however, currently unknown whether this intracellular route is able to support the propagation of long-distance regenerative Ca2+ waves or is restricted to short-distance signaling. Furthermore, the influence of the intracellular signaling dynamics on intercellular propagation remains to be understood. In this work, we propose a model of the gap-junctional route for intercellular Ca2+ wave propagation in astrocytes showing that: (1) long-distance regenerative signaling requires nonlinear coupling in the gap junctions, and (2) even with nonlinear gap junctions, long-distance regenerative signaling is favored when the internal Ca2+ dynamics implements frequency modulation-encoding oscillations with pulsating dynamics, while amplitude modulation-encoding dynamics tends to restrict the propagation range. As a result, spatially heterogeneous molecular properties and/or weak couplings are shown to give rise to rich spatiotemporal dynamics that support complex propagation behaviors. These results shed new light on the mechanisms implicated in the propagation of Ca2+ waves across astrocytes and precise the conditions under which glial cells may participate in information processing in the brain.Comment: Article: 30 pages, 7 figures. Supplementary Material: 11 pages, 6 figure