Modeling of adaptive immunity uncovers disease tolerance mechanisms.

When an organism is challenged with a pathogen a cascade of events unfolds. The innate immune system rapidly mounts a preliminary nonspecific defense, while the acquired immune system slowly develops microbe-killing specialists. These responses cause inflammation, and along with the pathogen cause direct and indirect tissue damage, which anti-inflammatory mediators seek to temper. This interplay of systems is credited for maintaining homeostasis but may produce unexpected results such as disease tolerance. Tolerance is characterized by the persistence of pathogen and damage mitigation, where the relevant mechanisms are poorly understood. In this work we develop an ordinary differential equations model of the immune response to infection in order to identify key components in tolerance. Bifurcation analysis uncovers health, immune- and pathogen-mediated death clinical outcomes dependent on pathogen growth rate. We demonstrate that decreasing the inflammatory response to damage and increasing the strength of the immune system gives rise to a region in which limit cycles, or periodic solutions, are the only biological trajectories. We then describe areas of parameter space corresponding to disease tolerance by varying immune cell decay, pathogen removal, and lymphocyte proliferation rates.

Modeling the presence probability of invasive plant species with nonlocal dispersal.

Mathematical models for the spread of invading plant organisms typically utilize population growth and dispersal dynamics to predict the time-evolution of a population distribution. In this paper, we revisit a particular class of deterministic contact models obtained from a stochastic birth process for invasive organisms. These models were introduced by Mollison (J R Stat Soc 39(3):283, 1977). We derive the deterministic integro-differential equation of a more general contact model and show that the quantity of interest may be interpreted not as population size, but rather as the probability of species occurrence. We proceed to show how landscape heterogeneity can be included in the model by utilizing the concept of statistical habitat suitability models which condense diverse ecological data into a single statistic. As ecologists often deal with species presence data rather than population size, we argue that a model for probability of occurrence allows for a realistic determination of initial conditions from data. Finally, we present numerical results of our deterministic model and compare them to simulations of the underlying stochastic process.

Steady state-Hopf mode interactions at the onset of electroconvection in the nematic liquid crystal Phase V.

We report on a new mode interaction found in electroconvection experiments on the nematic liquid crystal mixture Phase V in planar geometry. The mode interaction (codimension two) point occurs at a critical value of the frequency of the driving AC voltage. For frequencies below this value the primary pattern-forming instability at the onset voltage is an oblique stationary instability involving oblique rolls, and above this value it is an oscillatory instability giving rise to normal traveling rolls (oriented perpendicular to and traveling in the director direction). The transition has been confirmed by measuring the roll angle and the dominant frequency of the time series, as both quantities exhibit a discontinuous jump across zero when the AC frequency is varied near threshold. The globally coupled system of Ginzburg-Landau equations that qualitatively describe this mode interaction is constructed, and the resulting normal form, in which slow spatial variations of the mode amplitudes are ignored, is analyzed. This analysis shows that the Ginzburg-Landau system provides the adequate theoretical description for the experimentally observed phenomenon. The experimentally observed patterns at and higher above the onset allow us to narrow down the range of the parameters in the normal form.

Do fires in savannas consume woody biomass? A comment on approaches to modeling savanna dynamics.

Savanna ecosystems have long been fertile ground for mathematical modeling of vegetation structure and the role of resources and disturbance in tree-grass coexistence. In recent years, several authors have presented models that explore how savanna fires suppress the woody community, alter ecosystem dynamics, and promote grass persistence. We argue, however, that the assumption that fires influence savanna dynamics by consuming woody biomass may be wrong because, in reality, fires kill seedlings and saplings that constitute little biomass relative to adult trees. We present a simple alternative that separates the woody community into a subadult (fire-sensitive) class and an adult (fire-resistant) class and explore how this ecologically more realistic, but still simplified, model may provide better simulations of demographic processes and response to fires in savannas.

Optimal chaos control through reinforcement learning.

A general purpose chaos control algorithm based on reinforcement learning is introduced and applied to the stabilization of unstable periodic orbits in various chaotic systems and to the targeting problem. The algorithm does not require any information about the dynamical system nor about the location of periodic orbits. Numerical tests demonstrate good and fast performance under noisy and nonstationary conditions. (c) 1999 American Institute of Physics.

Storing cycles in Hopfield-type networks with pseudoinverse learning rule: admissibility and network topology.

Cyclic patterns of neuronal activity are ubiquitous in animal nervous systems, and partially responsible for generating and controlling rhythmic movements such as locomotion, respiration, swallowing and so on. Clarifying the role of the network connectivities for generating cyclic patterns is fundamental for understanding the generation of rhythmic movements. In this paper, the storage of binary cycles in Hopfield-type and other neural networks is investigated. We call a cycle defined by a binary matrix Σ admissible if a connectivity matrix satisfying the cycle's transition conditions exists, and if so construct it using the pseudoinverse learning rule. Our main focus is on the structural features of admissible cycles and the topology of the corresponding networks. We show that Σ is admissible if and only if its discrete Fourier transform contains exactly r=rank(Σ) nonzero columns. Based on the decomposition of the rows of Σ into disjoint subsets corresponding to loops, where a loop is defined by the set of all cyclic permutations of a row, cycles are classified as simple cycles, and separable or inseparable composite cycles. Simple cycles contain rows from one loop only, and the network topology is a feedforward chain with feedback to one neuron if the loop-vectors in Σ are cyclic permutations of each other. For special cases this topology simplifies to a ring with only one feedback. Composite cycles contain rows from at least two disjoint loops, and the neurons corresponding to the loop-vectors in Σ from the same loop are identified with a cluster. Networks constructed from separable composite cycles decompose into completely isolated clusters. For inseparable composite cycles at least two clusters are connected, and the cluster-connectivity is related to the intersections of the spaces spanned by the loop-vectors of the clusters. Simulations showing successfully retrieved cycles in continuous-time Hopfield-type networks and in networks of spiking neurons exhibiting up-down states are presented.

Quantitative and qualitative characterization of zigzag spatiotemporal chaos in a system of amplitude equations for nematic electroconvection.

It has been suggested by experimentalists that a weakly nonlinear analysis of the recently introduced equations of motion for the nematic electroconvection by M. Treiber and L. Kramer [Phys. Rev. E 58, 1973 (1998)] has the potential to reproduce the dynamics of the zigzag-type extended spatiotemporal chaos and localized solutions observed near onset in experiments [M. Dennin, D. S. Cannell, and G. Ahlers, Phys. Rev. E 57, 638 (1998); J. T. Gleeson (private communication)]. In this paper, we study a complex spatiotemporal pattern, identified as spatiotemporal chaos, that bifurcates at the onset from a spatially uniform solution of a system of globally coupled complex Ginzburg-Landau equations governing the weakly nonlinear evolution of four traveling wave envelopes. The Ginzburg-Landau system can be derived directly from the weak electrolyte model for electroconvection in nematic liquid crystals when the primary instability is a Hopf bifurcation to oblique traveling rolls. The chaotic nature of the pattern and the resemblance to the observed experimental spatiotemporal chaos in the electroconvection of nematic liquid crystals are confirmed through a combination of techniques including the Karhunen-Loeve decomposition, time-series analysis of the amplitudes of the dominant modes, statistical descriptions, and normal form theory, showing good agreement between theory and experiments.