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We study a population model where cells in one part of the cell cycle may affect the progress of cells in another part. If the influence, or feedback, from one part to another is negative, simulations of the model almost always result in multiple temporal clusters formed by groups of cells. We study regions in parameter space where periodic 'k-cyclic' solutions are stable. The regions of stability coincide with sub-triangles on which certain events occur in a fixed order. For boundary sub-triangles with order ' r s 1 ', we prove that the k-cyclic periodic solution is asymptotically stable if the index of the sub-triangle is relatively prime with respect to the number of clusters k and neutrally stable otherwise. For negative linear feedback, we prove that the interior of the parameter set is covered by stable sub-triangles, i.e. a stable k-cyclic solution always exists for some k. We observe numerically that the result also holds for many forms of nonlinear feedback, but may break down in extreme cases. [ABSTRACT FROM AUTHOR]

We investigate the transient dynamics of Block Coordinate Descent algorithms in valleys of the optimization landscape. Iterates converge linearly to a vicinity of the valley oor and then progress in a zig-zag fashion along the direction of the valley oor. When the valley sides are symmetric, the contraction factor to a vicinity of the valley oor appears to be no worse than 1/8, but without symmetry the contraction factor can approach 1. Progress along the direction of the valley oor is proportional to the gradient on the valley oor and inversely proportional to the narrowness" of the valley. We quantify narrowness using the eigenvalues of the Hessian on the valley oor and give explicit formulas for certain cases. Progress also depends on the direction of the valley with respect to the blocks of coordinates. When the valley sides are symmetric, we give an explicit formula for this dependence and use it to show that in higher dimensions nearly all directions give progress similar to the worst case direction. Finally, we observe that when starting the algorithm, the ordering of blocks in the first few steps can be important, but show that a greedy strategy with respect to objective function improvement can be a bad choice. [ABSTRACT FROM AUTHOR]

We study a model of the cell cycles of a large ensemble of cells that was conceived to explain the coupling between metabolism and the cell cycle observed in cell cycle related, metabolic oscillations in yeast continuous culture. An essential feature of the model is that a negative feedback mechanism robustly produces temporal clustering: cohorts of cells proceeding through the cell cycle in phase synchrony, a phenomenon that was observed in the yeast experiments. In this paper we focus on the possibility of temporal clustering in which different cohorts are of unequal sizes. We observe in numerical simulations of the model that the clusters tend very strongly toward having nearly equal numbers of cells. This is consistent with the available experimental data. We study the case of two unequal clusters analytically and conclude that solutions with two unequal clusters are locally asymptotically stable in the clustered subspace. However, local asymptotic stability of unbalanced clusters in the full phase space depends delicately on the parameters of the model. The global dynamics reveal a more robust picture. If clusters become unbalanced, the inuence of the larger cluster can radically shift the basins of attraction of individual clusters, making it more likely for an unclustered cell to join a smaller cluster. We propose that this points to a general principle: systems that form temporal clusters via nonlocal coupling tend to form nearly equal clusters. Finally, we describe an exception. When the number of clusters is a composite, we observe that groups of clusters may decouple with different numbers of cells in each group. [ABSTRACT FROM AUTHOR]

The undamped small-amplitude superimposed horizontal motion of a load supported symmetrically between identical isotropic hyperelastic springs, each subjected to an initial finite uniaxial static stretch, is reviewed in general terms. The small superimposed motion is discussed for the classical incompressible Mooney–Rivlin, James–Guth, and Gent models, as well as two limit classes of compressible Blatz–Ko material models, f = 0 and f = 1. The small-amplitude vibrational frequency is presented for each model, and the effects of limited extensibility are demonstrated for the James–Guth and Gent materials. Unstable equilibrium states are exhibited for the Blatz–Ko compressible foamed rubber material with f = 0 , while the others exhibit infinitesimally stable motion for an essentially arbitrary initial static stretch. It is shown that unstable equilibrium states exist for the general compressible Blatz–Ko model with 0 < f < 1 , and these states are characterized graphically in terms of the static stretch and the material Poisson ratio. The article concludes with a discussion of the Blatz–Ko and Gent–Thomas foamed rubber materials and their relation to the classical molecular-based uni-constant theory of elasticity. [ABSTRACT FROM AUTHOR]

We investigate the possibility that slow metabolic, cell-cycle-related oscillations in yeast and associated temporal clustering of cells within the cell cycle could be due to an interplay between near-critical metabolism and cell cycle checkpoints. We construct a dynamical model of the cell cycles of a large culture of cells that incorporates checkpoint gating and metabolic mode switching that are triggered by resource thresholds. We investigate the model analytically and prove that there exist open sets of parameter values for which the model possesses stable periodic solutions that exhibit metabolic oscillations with cell cycle clustering. Simulations of the model give evidence that such solutions exist for large sets of parameter values. This demonstrates that checkpoint gating coupled with critical resources can be a robust mechanism for producing the phenomena observed in experiments. [ABSTRACT FROM AUTHOR]

As technological improvements continue to infiltrate and impact medical practice, it has become possible to non-invasively collect dense physiological time series data from individual patients in real time. These advances continue to improve physicians' ability to detect and to treat infections early. One important benefit of early detection and treatment of nascent infections is that it leads to earlier resolution. In response to current and anticipated advances in data capture, we introduce the Early Treatment Gain (ETG) as a measure to quantify this benefit. Roughly, we define the gain to be the limiting ratio: ETG = differential change in time of resolution / differential change in treatment time . We study the gain using standard dynamical models and demonstrate its use with time series data from Surgical Intensive Care Unit (SICU) patients facing ventilator associated pneumonia. The main conclusion from the mathematical modelling is that the ETG is always greater than one unless there is an effective immune response, in which case the ETG can be less than one. Using real patient time series data, we observe that the formula derived for a linear model can be applied and that this produces a ETG greater than one. [ABSTRACT FROM AUTHOR]

Combining the results of brain imaging and nonlinear dynamics provides a new hierarchical vision of brain network functionality that is helpful in understanding the relationship of the network to different mental tasks. Using these ideas it is possible to build adequate models for the description and prediction of different cognitive activities in which the number of variables is usually small enough for analysis. The dynamical images of different mental processes depend on their temporal organization and, as a rule, cannot be just simple attractors since cognition is characterized by transient dynamics. The mathematical image for a robust transient is a stable heteroclinic channel consisting of a chain of saddles connected by unstable separatrices. We focus here on hierarchical chunking dynamics that can represent several cognitive activities. Chunking is the dynamical phenomenon that means dividing a long information chain into shorter items. Chunking is known to be important in many processes of perception, learning, memory and cognition. We prove that in the phase space of the model that describes chunking there exists a new mathematical object - heteroclinic sequence of heteroclinic cycles - using the technique of slow-fast approximations. This new object serves as a skeleton of motions reflecting sequential features of hierarchical chunking dynamics and is an adequate image of the chunking processing. [ABSTRACT FROM AUTHOR]

Noninvertible circle maps may have a rotation interval instead of a unique rotation number. One may ask which of the numbers or sets of numbers within this rotation interval may be observed with positive probability in term of Lebesgue measure on the circle. We study this question numerically for families of circle maps. Both the interval and "observed" rotation numbers are computed for large numbers of initial conditions. The numerical evidence suggests that within the rotation interval only a very narrow band or even a unique rotation number is observed. These observed rotation numbers appear to be either locally constant or vary wildly as the parameter is changed. Closer examination reveals that intervals with wild variation contain many subintervals where the observed rotation numbers are locally constant. We discuss the formation of these intervals. We prove that such intervals occur whenever one of the endpoints of the rotation interval changes. We also examine the effects of various types of saddle-node bifurcations on the observed rotation numbers. [ABSTRACT FROM AUTHOR]

We study Hopf-Andronov bifurcations in a class of random differential equations (RDEs) with bounded noise. We observe that when an ordinary differential equation that undergoes a Hopf bifurcation is subjected to bounded noise then the bifurcation that occurs involves a discontinuous change in the Minimal Forward Invariant set.

Ventilator associated pneumonia is a common and costly complication in critically ill and injured surgical patients. The diagnosis of pneumonia remains problematic and non-specific. Using clinical criteria, a diagnosis of pneumonia is typically not made until an infection is well established. Semi-quantitative cultures of endotracheal aspirate and broncho-alveolar lavage are employed to improve the accuracy of diagnosis but are invasive and require time for culture results to become available. We report data that show that an inexpensive, rapid and non-invasive alternative may exist. In particular we show that: 1). Bio-aerosols evolved in the breath of ventilated patients and captured in the hygroscopic condenser humidifier filter of the ventilator circuit contain pathogenic micro-organisms. 2). The number (CFU/ml) and identity (Genus, species) of the pathogens in the aerosol samples can rapidly and inexpensively be determined by PCR. 3). Data from a convenience sample of filters correlate with clinical findings from standard microbiological methods such as broncho-alveolar lavage. The evaluation of the bacterial load evolved in exhaled breath by PCR is amenable to repeated sampling. Since increasing bacterial burden is believed to correlate with the establishment of infection, the use of quantitative PCR may provide a method to rapidly, inexpensively, and effectively detect and diagnose the early onset of pneumonia and identify pathogens involved.

Biologists have long observed periodic-like oxygen consumption oscillations in yeast populations under certain conditions, and several unsatisfactory explanations for this phenomenon have been proposed. These ‘autonomous oscillations’ have often appeared with periods that are nearly integer divisors of the calculated doubling time of the culture. We hypothesize that these oscillations could be caused by a form of cell cycle synchronization that we call clustering. We develop some novel ordinary differential equation models of the cell cycle. For these models, and for random and stochastic perturbations, we give both rigorous proofs and simulations showing that both positive and negative growth rate feedback within the cell cycle are possible agents that can cause clustering of populations within the cell cycle. It occurs for a variety of models and for a broad selection of parameter values. These results suggest that the clustering phenomenon is robust and is likely to be observed in nature. Since there are necessarily an integer number of clusters, clustering would lead to periodic-like behaviour with periods that are nearly integer divisors of the period of the cell cycle. Related experiments have shown conclusively that cell cycle clustering occurs in some oscillating yeast cultures.

In random differential equations with bounded noise minimal forward invariant (MFI) sets play a central role since they support stationary measures. We study the stability and possible bifurcations of MFI sets. In dimensions 1 and 2 we classify all minimal forward invariant sets and their codimension one bifurcations in bounded noise random differential equations.