Cross-Correlation in cricket data and RMT

We analyze cross-correlation between runs scored over a time interval in cricket matches of different teams using methods of random matrix theory (RMT). We obtain an ensemble of cross-correlation matrices $C$ from runs scored by eight cricket playing nations for (i) test cricket from 1877 -2014 (ii)one-day internationals from 1971 -2014 and (iii) seven teams participating in the Indian Premier league T20 format (2008-2014) respectively. We find that a majority of the eigenvalues of C fall within the bounds of random matrices having joint probability distribution $P(x_1...,x_n)=C_{N \beta} \, \prod_{j<k}w(x_j)| x_j-x_k |^\beta$ where $w(x)=x^{N\beta a}\exp(-N\beta b x)$ and $\beta$ is the Dyson parameter. The corresponding level density gives Marchenko-Pastur (MP) distribution while fluctuations of every participating team agrees with the universal behavior of Gaussian Unitary Ensemble (GUE). We analyze the components of the deviating eigenvalues and find that the largest eigenvalue corresponds to an influence common to all matches played during these periods.Comment: 12 pages, 6 figure

Learning normal form autoencoders for data-driven discovery of universal, parameter-dependent governing equations

Complex systems manifest a small number of instabilities and bifurcations that are canonical in nature, resulting in universal pattern forming characteristics as a function of some parametric dependence. Such parametric instabilities are mathematically characterized by their universal un-foldings, or normal form dynamics, whereby a parsimonious model can be used to represent the dynamics. Although center manifold theory guarantees the existence of such low-dimensional normal forms, finding them has remained a long standing challenge. In this work, we introduce deep learning autoencoders to discover coordinate transformations that capture the underlying parametric dependence of a dynamical system in terms of its canonical normal form, allowing for a simple representation of the parametric dependence and bifurcation structure. The autoencoder constrains the latent variable to adhere to a given normal form, thus allowing it to learn the appropriate coordinate transformation. We demonstrate the method on a number of example problems, showing that it can capture a diverse set of normal forms associated with Hopf, pitchfork, transcritical and/or saddle node bifurcations. This method shows how normal forms can be leveraged as canonical and universal building blocks in deep learning approaches for model discovery and reduced-order modeling.Comment: 18 pages, 7 figure

Homoclinic saddle to saddle-focus transitions in 4D systems

A saddle to saddle-focus homoclinic transition when the stable leading eigenspace is three-dimensional (called the 3DL bifurcation) is analyzed. Here a pair of complex eigenvalues and a real eigenvalue exchange their position relative to the imaginary axis, giving rise to a 3D stable leading eigenspace at the critical parameter values. This transition is different from the standard Belyakov bifurcation, where a double real eigenvalue splits either into a pair of complex-conjugate eigenvalues or two distinct real eigenvalues. In the wild case, we obtain sets of codimension 1 and 2 bifurcation curves and points that asymptotically approach the 3DL bifurcation point and have a structure that differs from that of the standard Belyakov case. We give an example of this bifurcation in a perturbed Lorenz-Stenflo 4D ordinary differential equation model

Homoclinic saddle to saddle-focus transitions in 4D systems

A saddle to saddle-focus homoclinic transition when the stable leading eigenspace is three-dimensional (called the 3DL bifurcation) is analyzed. Here a pair of complex eigenvalues and a real eigenvalue exchange their position relative to the imaginary axis, giving rise to a 3D stable leading eigenspace at the critical parameter values. This transition is different from the standard Belyakov bifurcation, where a double real eigenvalue splits either into a pair of complex-conjugate eigenvalues or two distinct real eigenvalues. In the wild case, we obtain sets of codimension 1 and 2 bifurcation curves and points that asymptotically approach the 3DL bifurcation point and have a structure that differs from that of the standard Belyakov case. We give an example of this bifurcation in a perturbed Lorenz–Stenflo 4D ordinary differential equation model

Glial Chloride Homeostasis Under Transient Ischemic Stress

High water permeabilities permit rapid adjustments of glial volume upon changes in external and internal osmolarity, and pathologically altered intracellular chloride concentrations ([Cl–]int) and glial cell swelling are often assumed to represent early events in ischemia, infections, or traumatic brain injury. Experimental data for glial [Cl–]int are lacking for most brain regions, under normal as well as under pathological conditions. We measured [Cl–]int in hippocampal and neocortical astrocytes and in hippocampal radial glia-like (RGL) cells in acute murine brain slices using fluorescence lifetime imaging microscopy with the chloride-sensitive dye MQAE at room temperature. We observed substantial heterogeneity in baseline [Cl–]int, ranging from 14.0 ± 2.0 mM in neocortical astrocytes to 28.4 ± 3.0 mM in dentate gyrus astrocytes. Chloride accumulation by the Na+-K+-2Cl– cotransporter (NKCC1) and chloride outward transport (efflux) through K+-Cl– cotransporters (KCC1 and KCC3) or excitatory amino acid transporter (EAAT) anion channels control [Cl–]int to variable extent in distinct brain regions. In hippocampal astrocytes, blocking NKCC1 decreased [Cl–]int, whereas KCC or EAAT anion channel inhibition had little effect. In contrast, neocortical astrocytic or RGL [Cl–]int was very sensitive to block of chloride outward transport, but not to NKCC1 inhibition. Mathematical modeling demonstrated that higher numbers of NKCC1 and KCC transporters can account for lower [Cl–]int in neocortical than in hippocampal astrocytes. Energy depletion mimicking ischemia for up to 10 min did not result in pronounced changes in [Cl–]int in any of the tested glial cell types. However, [Cl–]int changes occurred under ischemic conditions after blocking selected anion transporters. We conclude that stimulated chloride accumulation and chloride efflux compensate for each other and prevent glial swelling under transient energy deprivation