Sergei S. Pilyugin

Global exponential stability of competitive neural networks with different time scales

The dynamics of cortical cognitive maps developed by self-organization must include the aspects of long and short-term memory. The behavior of such a neural network is characterized by an equation of neural activity as a fast phenomenon and an equation of synaptic modification as a slow part of the neural system. We present a new method of analyzing the dynamics of a biological relevant system with different time scales based on the theory of flow invariance. We are able to show the conditions under which the solutions of such a system are bounded being less restrictive than with the K-monotone theory, singular perturbation theory, or those based on supervised synaptic learning. We prove the existence and the uniqueness of the equilibrium. A strict Lyapunov function for the flow of a competitive neural system with different time scales is given and based on it we are able to prove the global exponential stability of the equilibrium point.

Multiple limit cycles in the chemostat with variable yield

The global asymptotic behavior of solutions of the variable yield model is determined. The model generalizes the classical Monod model and it assumes that the yield is an increasing function of the nutrient concentration. In contrast to the Monod model, it is demonstrated that the variable yield model exhibits sustained oscillations. Moreover, it is shown that the variable yield model may undergo a subcritical Hopf bifurcation and feature at least two distinct limit cycles. Implications for the coexistence of competing populations are discussed.

The rescaling method for quantifying the turnover of cell populations.

The dynamic nature of immune responses requires the development of appropriate experimental and theoretical tools to quantitatively estimate the division and death rates which determine the turnover of immune cells. A number of papers have used experimental data from BrdU and D-glucose labels together with a simple random birth-death model to quantify the turnover of immune cells focusing on HIV/SIV infections [Mohri et al. 279 (1998) 1223-1227, Hellerstein et al. 5 (1999) 83-89, Bonhoeffer et al. 164 (2000) 5049-5054, Mohri et al. 87 (2001) 1277-1287]. We show how uncertainties in the assumptions of the random birth-death model may lead to substantial errors in the parameters estimated. We then show how more accurate estimates can be obtained from the more recent CFSE data which allow to track the number of divisions each cell has undergone. Specifically, we: (i) describe a general stage-structured model of cell division where the probabilities of division and death are functions of time since the previous division; (ii) develop a rescaling method to identify invariant parameters (i.e. the ones that are independent of the specific functions describing division and death); (iii) show how these invariant parameters can be estimated, and (iv) illustrate this technique by applying it to CFSE data taken from the literature.

Competition in the Unstirred Chemostat with Periodic Input and Washout

The model of an unstirred chemostat is generalized to that of a chemostat with time-dependent input/washout rates. The novelty of the new model is that time periodicity appears in the boundary conditions. The asymptotic dynamics of the competition between two microbial populations is determined in terms of the corresponding period map, which is shown to preserve the standard competitive ordering. It is shown that the dynamics of competition is similar to that of a chemostat with constant boundary conditions. Simple criteria for coexistence versus competitive exclusion are presented. 1. Introduction. The chemostat represents a basic model of an open system in microbial ecology. In its simplest form, it consists of three vessels. The rst, called the feed bottle, contains medium with all of the nutrients needed for growth in sur- plus except one, which hereafter is simply called the nutrient. The contents of the feed bottle are pumped at a constant rate into the second vessel, called the culture vessel or bioreactor. The culture vessel is charged with one or more populations of microorganisms. The contents of the culture vessel are pumped into the remaining vessel, called the over∞ow vessel, at a constant rate, keeping the volume of the reactor constant. The organisms compete for the nutrient in a purely exploitative manner. Basic assumptions include that the vessel is well mixed and that all other parame- ters (pH, temperature, etc.) are strictly controlled. The ∞ow rate is assumed to be sucient to preclude wall growth or the accumulation of metabolic products. Let S(t) denote the concentration of the nutrient in the culture vessel and xi(t), i =1 ; 2, denote the concentration of the competitors. LetS 0 denote the concentration of the input nutrient and let D denote the dilution rate (∞ow rate/volume). If growth is assumed to be proportional to consumption then the basic equations take the form

The role of coinfection in multidisease dynamics

We investigate an epidemic model of two diseases. The primary disease is assumed to be a slowly progressing disease, and the density of individuals infected with it is structured by age since infection. Hosts that are already infected with the primary disease can become coinfected with a secondary disease. We show that in addition to the disease-free equilibrium, there exists a unique dominance equilibrium corresponding to each disease. Without coinfection there are no coexistence equilibria; however, with coinfection the number of coexistence equilibria may vary. For some parameter values, there exist two coexistence equilibria. We also observe competitor-mediated oscillatory coexistence. Furthermore, weakly subthreshold (which occur when exactly one of the reproduction numbers is below one) and strongly subthreshold (which occur when both reproduction numbers are below one) coexistence equilibria may exist. Some of those are a result of a two-parameter backward bifurcation. Bistability occurs in several...

Multistrain virus dynamics with mutations: a global analysis

We consider within-host virus models with n >or= 2 strains and allow mutation between the strains. If there is no mutation, a Lyapunov function establishes global stability of the steady state corresponding to the fittest strain. For small perturbations, this steady state persists, perhaps with small concentrations of some or all other strains, depending on the connectivity of the graph describing all possible mutations. Moreover, using a perturbation result due to Smith & Waltman (1999), we show that this steady state also preserves global stability.

The simple chemostat with wall growth

A model of the simple chemostat which allows for growth on the wall (or other marked surface) is presented as three nonlinear ordinary differential equations. The organisms which are attached to the wall do not wash out of the chemostat. This destroys the basic reduction of the chemostat equations to a monotone system, a technique which has been useful in the analysis of many chemostat-like equations. The adherence to and shearing from the wall eliminates the boundary equilibria. For a reasonably general model, the basic properties of invariance, dissipation, and uniform persistence are established. For two important special cases, global asymptotic results are obtained. Finally, a perturbation technique allows the special results to be extended to provide the rest point as a global attractor for nearby growth functions.

Local exponential stability of competitive neural networks with different time scales

Abstract This contribution presents a new method of analyzing the dynamics of a biological relevant neural network with different time scales based on the theory of flow invariance. We are able to show that the resulting stability conditions are less restrictive and more general than with K -monotone theory or singular perturbation theory. The theoretical results are further substantiated by simulation results conducted for analysis and design of these neural networks.

How sticky should a virus be? The impact of virus binding and release on transmission fitness using influenza as an example

Budding viruses face a trade-off: virions need to efficiently attach to and enter uninfected cells while newly generated virions need to efficiently detach from infected cells. The right balance between attachment and detachment—the right amount of stickiness—is needed for maximum fitness. Here, we design and analyse a mathematical model to study in detail the impact of attachment and detachment rates on virus fitness. We apply our model to influenza, where stickiness is determined by a balance of the haemagglutinin (HA) and neuraminidase (NA) proteins. We investigate how drugs, the adaptive immune response and vaccines impact influenza stickiness and fitness. Our model suggests that the location in the ‘stickiness landscape’ of the virus determines how well interventions such as drugs or vaccines are expected to work. We discuss why hypothetical NA enhancer drugs might occasionally perform better than the currently available NA inhibitors in reducing virus fitness. We show that an increased antibody or T-cell-mediated immune response leads to maximum fitness at higher stickiness. We further show that antibody-based vaccines targeting mainly HA or NA, which leads to a shift in stickiness, might reduce virus fitness above what can be achieved by the direct immunological action of the vaccine. Overall, our findings provide potentially useful conceptual insights for future vaccine and drug development and can be applied to other budding viruses beyond influenza.

Immune response to a malaria infection: properties of a mathematical model.

Abstract We establish some properties of a within host mathematical model of malaria proposed by Recker et al. [M. Recker et al., Transient cross-reactive immune responses can orchestrate antigenic variation in malaria, Lett. Nature 429 (2004), pp. 555–558; M. Recker and S. Gupta, Conflicting immune responses can prolong the length of infection in Plasmodium falciparum malaria, Bull. Math. Biol. 68 (2006), pp. 821–835.], which includes the role of the immune system during the infection. The model accounts for the antigenic variation exhibited by the malaria parasite (Plasmodium falciparum). We show that the model can exhibit a wide variety of dynamical behaviours. We provide criteria for global stability, competitive exclusion and persistence. We also demonstrate that the disease equilibrium can be destabilized by non-symmetric cross-reactive responses.