arXiv subjects

Todd R. Young

Publications and source records attributed to Todd R. Young.

4 recordsLinked to original sources

arXiv2026

Population Structures with Positive Feedback and Asymmetric Division

In bioreactor experiments, budding yeast can manifest stable metabolic oscillations. In some instances these oscillations involve the cell cycle and are associated with a dynamical phenomenon called temporal clustering. Since yeast divide asymmetrically, mother cells may be able to divide sooner than the smaller daughter cells. We show that asymmetric division plus positive feedback can result in stable temporal clustering. In this scenario, the cells self-organize into $p$ clusters of mother cells and $q$ clusters of daughter cells, $p \le q$. In simple numerical simulations of a population model with asymmetric division and positive feedback, we show that $p:q$ periodic arrangements form spontaneously from random populations of cells. Our main result is that these structures can be stable.

q-bio.CB↗

arXiv2015

Two dimensional heteroclinic attractor in the generalized Lotka-Volterra system

We study a simple dynamical model exhibiting sequential dynamics. We show that in this model there exist sets of parameter values for which a cyclic chain of saddle equilibria, $O_k$, $k=1, \ldots, p$, have two dimensional unstable manifolds that contain orbits connecting each $O_k$ to the next two equilibrium points $O_{k+1}$ and $O_{k+2}$ in the chain ($O_{p+1} = O_1$). We show that the union of these equilibria and their unstable manifolds form a $2$-dimensional surface with boundary that is homeomorphic to a cylinder if $p$ is even and a Möbius strip if $p$ is odd. If, further, each equilibrium in the chain satisfies a condition called ``dissipativity," then this surface is asymptotically stable.

math.DS↗

arXiv2011

Dynamics of a 3 cluster cell-cycle system with positive linear feedback

In this technical note we calculate the dynamics of a linear feedback model of progression in the cell cycle in the case that the cells are organized into k=3 clusters. We examine the dynamics in detail for a specific subset of parameters with non-empty interior. There is an interior fixed point of the Poincare' map defined by the system. This fixed point corresponds to a periodic solution with period $T$ in which the three cluster exchange positions after time $T/3$. We call this solution 3=cyclic. In all the parameters studied, the fixed point is either: * isolated and locally unstable, or, * contained in a neutrally stable set of period 3 points. In the later case the edges of the neutrally stable set are unstable. This case exists if either the three clusters are isolated from each other, or, if they interact in a non-essential way. In both cases the orbits of all other interior points are asymptotic to the boundary. Thus 3-cyclic solutions are practically unstable in the sense the arbitrarily small perturbations may lead to loss of stability and eventual merger of clusters. Since the single cluster solution (synchronization) is the only solution that is asymptotically stable, it would seem to be the most likely to be observed in application if the feedback is similar to the form we propose and is positive.

math.DS↗

arXiv2005

The Signed Distance Function: A New Tool for Binary Classification

From a geometric perspective most nonlinear binary classification algorithms, including state of the art versions of Support Vector Machine (SVM) and Radial Basis Function Network (RBFN) classifiers, and are based on the idea of reconstructing indicator functions. We propose instead to use reconstruction of the signed distance function (SDF) as a basis for binary classification. We discuss properties of the signed distance function that can be exploited in classification algorithms. We develop simple versions of such classifiers and test them on several linear and nonlinear problems. On linear tests accuracy of the new algorithm exceeds that of standard SVM methods, with an average of 50% fewer misclassifications. Performance of the new methods also matches or exceeds that of standard methods on several nonlinear problems including classification of benchmark diagnostic micro-array data sets.

cs.LG↗