Periodic pattern formation in reaction-diffusion systems -an introduction for numerical simulation

The aim of the present review is to provide a comprehensive explanation of Turing reaction–diffusion systems in sufficient detail to allow readers to perform numerical calculations themselves. The reaction–diffusion model is widely studied in the field of mathematical biology, serves as a powerful paradigm model for self-organization and is beginning to be applied to actual experimental systems in developmental biology. Despite the increase in current interest, the model is not well understood among experimental biologists, partly because appropriate introductory texts are lacking. In the present review, we provide a detailed description of the definition of the Turing reaction–diffusion model that is comprehensible without a special mathematical background, then illustrate a method for reproducing numerical calculations with Microsoft Excel. We then show some examples of the patterns generated by the model. Finally, we discuss future prospects for the interdisciplinary field of research involving mathematical approaches in developmental biology

Maps of complex motion selectivity in the superior temporal cortex of the alert macaque monkey: a double-label 2-deoxyglucose study

The superior temporal sulcus (STS) of the macaque monkey contains multiple visual areas. Many neurons within these regions respond selectively to motion direction and to more complex motion patterns, such as expansion, contraction and rotation. Single-unit recording and optical recording studies in MT/MST suggest that cells with similar tuning properties are clustered into columns extending through multiple cortical layers. In this study, we used a double-label 2-deoxyglucose technique in awake, behaving macaque monkeys to clarify this functional organization. This technique allowed us to label, in a single animal, two populations of neurons responding to two different visual stimuli. In one monkey we compared expansion with contraction; in a second monkey we compared expansion with clockwise rotation. Within the STS we found a patchy arrangement of cortical columns with alternating stimulus selectivity: columns of neurons preferring expansion versus contraction were more widely separated than those selective for expansion versus rotation. This mosaic of interdigitating columns on the floor and posterior bank of the STS included area MT and some neighboring regions of cortex, perhaps including area MST

Formation of antiwaves in gap-junction-coupled chains of neurons

Using network models consisting of gap junction coupled Wang-Buszaki neurons, we demonstrate that it is possible to obtain not only synchronous activity between neurons but also a variety of constant phase shifts between 0 and \pi. We call these phase shifts intermediate stable phaselocked states. These phase shifts can produce a large variety of wave-like activity patterns in one-dimensional chains and two-dimensional arrays of neurons, which can be studied by reducing the system of equations to a phase model. The 2\pi periodic coupling functions of these models are characterized by prominent higher order terms in their Fourier expansion, which can be varied by changing model parameters. We study how the relative contribution of the odd and even terms affect what solutions are possible, the basin of attraction of those solutions and their stability. These models may be applicable to the spinal central pattern generators of the dogfish and also to the developing neocortex of the neonatal rat

Phase Response Curves of Coupled Oscillators

Many real oscillators are coupled to other oscillators and the coupling can affect the response of the oscillators to stimuli. We investigate phase response curves (PRCs) of coupled oscillators. The PRCs for two weakly coupled phase-locked oscillators are analytically obtained in terms of the PRC for uncoupled oscillators and the coupling function of the system. Through simulation and analytic methods, the PRCs for globally coupled oscillators are also discussed.Comment: 5 pages 4 figur

Strongly nonlinear dynamics of electrolytes in large ac voltages

We study the response of a model micro-electrochemical cell to a large ac voltage of frequency comparable to the inverse cell relaxation time. To bring out the basic physics, we consider the simplest possible model of a symmetric binary electrolyte confined between parallel-plate blocking electrodes, ignoring any transverse instability or fluid flow. We analyze the resulting one-dimensional problem by matched asymptotic expansions in the limit of thin double layers and extend previous work into the strongly nonlinear regime, which is characterized by two novel features - significant salt depletion in the electrolyte near the electrodes and, at very large voltage, the breakdown of the quasi-equilibrium structure of the double layers. The former leads to the prediction of "ac capacitive desalination", since there is a time-averaged transfer of salt from the bulk to the double layers, via oscillating diffusion layers. The latter is associated with transient diffusion limitation, which drives the formation and collapse of space-charge layers, even in the absence of any net Faradaic current through the cell. We also predict that steric effects of finite ion sizes (going beyond dilute solution theory) act to suppress the strongly nonlinear regime in the limit of concentrated electrolytes, ionic liquids and molten salts. Beyond the model problem, our reduced equations for thin double layers, based on uniformly valid matched asymptotic expansions, provide a useful mathematical framework to describe additional nonlinear responses to large ac voltages, such as Faradaic reactions, electro-osmotic instabilities, and induced-charge electrokinetic phenomena.Comment: 30 pages, 17 eps-figures, RevTe

Optical assembly of nanostructures mediated by surface roughness

Rigorous understanding of the self-assembly of colloidal nanocrystals is crucial to the development of tailored nanostructured materials. Despite extensive studies, a mechanistic understanding of self-assembly under non-equilibrium driven by an external field remains an ongoing challenge. We demonstrate self-assembly by optical tweezers imposing an external attractive field for cubic-phase sodium yttrium fluoride nanocrystals. We show that surface roughness of the nanocrystals is a decisive factor for contact leading to assembly between the nanocrystals, manifested by the roughness-dependent hydrodynamic resistivity. This provides direct evidence that dynamics are equally important to energetics in understanding self-assembly. These results have implications in a wide variety of different fields, such as in understanding the factors that mediate oriented attachment-based crystal growth or in interpreting the structure of binding sites on viruses.Comment: 21 pages, 3 main figures, 8 supplemental figures, 2 supplemental videos. Submitted to Physical Review Letter

Nonlinear electrochemical relaxation around conductors

We analyze the simplest problem of electrochemical relaxation in more than one dimension - the response of an uncharged, ideally polarizable metallic sphere (or cylinder) in a symmetric, binary electrolyte to a uniform electric field. In order to go beyond the circuit approximation for thin double layers, our analysis is based on the Poisson-Nernst-Planck (PNP) equations of dilute solution theory. Unlike most previous studies, however, we focus on the nonlinear regime, where the applied voltage across the conductor is larger than the thermal voltage. In such strong electric fields, the classical model predicts that the double layer adsorbs enough ions to produce bulk concentration gradients and surface conduction. Our analysis begins with a general derivation of surface conservation laws in the thin double-layer limit, which provide effective boundary conditions on the quasi-neutral bulk. We solve the resulting nonlinear partial differential equations numerically for strong fields and also perform a time-dependent asymptotic analysis for weaker fields, where bulk diffusion and surface conduction arise as first-order corrections. We also derive various dimensionless parameters comparing surface to bulk transport processes, which generalize the Bikerman-Dukhin number. Our results have basic relevance for double-layer charging dynamics and nonlinear electrokinetics in the ubiquitous PNP approximation.Comment: 25 pages, 17 figures, 4 table

Possible Magnetic Behavior in Oxygen-deficient {\beta}-PtO2

We studied the electronic properties of beta-platinum dioxide ({\beta}-PtO2), a catalytic material, based on density functional theory. Using the GGA+U method which reproduces the GW band structures and the experimental structural parameters, we found that the creation of an oxygen vacancy will induce local magnetic moment on the neighboring Pt and O atoms. The magnetism originates not only from the unpaired electrons that occupy the vacancy induced gap state, but also from the itinerant valence electrons. Because of antiferromagnetic (AF) coupling and the localized nature of gap states, the total magnetic moment is zero for charge-neutral state (V_o^0) and is ~ 1 \mu B for singly-charged states (V_o^\mu). Calculation of grand potential shows that, the three charge states (V_o^0, V_o^\pm) are of the same stability within a small region, and the negatively charged state (V_o^-) is energetically favored within a wide range of the band gap. On this basis we discussed the implication on catalytic behavior.Comment: 45 pages, 11 figures, 3 table