Entrainment, Instability, Quasi-periodicity, and Chaos in a Compound Neural Oscillator

We studied the dynamical behavior of a class of compound central pattern generator (CPG) models consisting of a simple neural network oscillator driven by both constant and periodic inputs of varying amplitudes, frequencies, and phases. We focused on a specific oscillator composed of two mutually inhibiting types of neuron (inspiratory and expiratory neurons) that may be considered as a minimal model of the mammalian respiratory rhythm generator. The simulation results demonstrated how a simple CPG model— with a minimum number of neurons and mild nonlinearities— may reproduce a host of complex dynamical behaviors under various periodic inputs. In particular, the network oscillated spontaneously only when both neurons received adequate and proportionate constant excitations. In the presence of a periodic source, the spontaneous rhythm was overriden by an entrained oscillation of varying forms depending on the nature of the source. Stable entrained oscillations were inducible by two types of inputs: (1) anti-phase periodic inputs with alternating agonist-antagonist drives to both neurons and (2) a single periodic drive to only one of the neurons. In-phase inputs, which exert periodic drives of similar magnitude and phase relationships to both neurons, resulted in varying disruptions of the entrained oscillations including magnitude attenuation, harmonic and phase distortions, and quasi-periodic interference. In the absence of significant phasic feedback, chaotic motion developed only when the CPG was driven by multiple periodic inputs. Apneic episodes with repetitive alternation of active (intrinsic oscillation) and inactive (cessation of oscillation) states developed when the network was driven by a moderate periodic input of low frequency. %and amplitudes of intermediate strength, Similar results were demonstrated in other, more complex oscillator models (that is, half-center oscillator and three-phase respiratory network model). These theoretical results may have important implications in elucidating the mechanisms of rhythmogenesis in the mature and developing respiratory CPG as well as other compound CPGs in mammalian and invertebrate nervous systems.     

Separable models in age-dependent population dynamics

A class of population models is considered in which the parameters such as fecundity, mortality and interaction coefficients are assumed to be age-dependent. Conditions for the existence, stability and global attractivity of steady-state and periodic solutions are derived. The dependence of these solutions on the maturation periods is analyzed. These results are applied to specific single and multiple population models. It is shown that periodic solutions cannot occur in a general class of single population age-dependent models. Conditions are derived that determine whether increasing the maturation period has a stabilizing effect. In specific cases, it is shown that any number of switches in stability can occur as the maturation period is increased. An example is given of predator-prey model where each one of these stability switches corresponds to a stable steady state losing its stability via a Hopf bifurcation to a periodic solution and regaining its stability upon further increase of the maturation period.     

The role of feedback in morphological computation with compliant bodies

The generation of robust periodic movements of complex nonlinear robotic systems is inherently difficult, especially, if parts of the robots are compliant. It has previously been proposed that complex nonlinear features of a robot, similarly as in biological organisms, might possibly facilitate its control. This bold hypothesis, commonly referred to as morphological computation, has recently received some theoretical support by Hauser et?al. (Biol Cybern 105:355–370, doi:10.1007/s00422-012-0471-0, 2012). We show in this article that this theoretical support can be extended to cover not only the case of fading memory responses to external signals, but also the essential case of autonomous generation of adaptive periodic patterns, as, e.g., needed for locomotion. The theory predicts that feedback into the morphological computing system is necessary and sufficient for such tasks, for which a fading memory is insufficient. We demonstrate the viability of this theoretical analysis through computer simulations of complex nonlinear mass–spring systems that are trained to generate a large diversity of periodic movements by adapting the weights of a simple linear feedback device. Hence, the results of this article substantially enlarge the theoretically tractable application domain of morphological computation in robotics, and also provide new paradigms for understanding control principles of biological organisms.     

A Within-Host Virus Model with Periodic Multidrug Therapy

This paper is devoted to the investigation of the effects of periodic drug treatment on a standard within-host virus model. We first introduce the basic reproduction ratio for the model, and then show that the infection free equilibrium is globally asymptotically stable, and the disease eventually disappears if $\mathcal{R}_{0} < 1$ , while there exists at least one positive periodic state and the disease persists when $\mathcal{R}_{0}>1$ . We also consider an optimization problem by shifting the phase of these drug efficacy functions. It turns out that shifting the phase can certainly affect the stability of the infection free steady state. A numerical study is performed to illustrate our analytic results.     

Quantitative molecular assessment of chimerism across tissues in marmosets and tamarins

Background

Periodic spacing of A-tracts (short runs of A or T) with the DNA helical period of ~10?C11?bp is characteristic of intrinsically bent DNA. In eukaryotes, the DNA bending is related to chromatin structure and nucleosome positioning. However, the physiological role of strong sequence periodicity detected in many prokaryotic genomes is not clear.

Results

We developed measures of intensity and persistency of DNA curvature-related sequence periodicity and applied them to prokaryotic chromosomes and phages. The results indicate that strong periodic signals present in chromosomes are generally absent in phage genomes. Moreover, chromosomes containing prophages are less likely to possess a persistent periodic signal than chromosomes with no prophages.

Conclusions

Absence of DNA curvature-related sequence periodicity in phages could arise from constraints associated with DNA packaging in the viral capsid. Lack of prophages in chromosomes with persistent periodic signal suggests that the sequence periodicity and concomitant DNA curvature could play a role in protecting the chromosomes from integration of phage DNA.     

有毒物影响和Beddington-DeAngelis型功能性反应的捕食系统的全局吸引性

利用微分方程比较原理,重合度理论中Mawhin’s延拓定理,Lya．punov泛函和Barbalat引理,研究了一类有毒物影响和Beddington—DeAngelis型功能性反应的时滞两种群捕食者-食饵系统．我们得到了该系统一致持久性和其周期系统存在唯一全局渐近稳定的周期解的充分条件．改进了范猛和唐贵坚的相关结果．     

Asymptotic behavior in a deterministic epidemic model

The effects of a periodic contact rate and of carriers are considered for a generalization of Bailey's simple epidemic model. In this model it is assumed that individuals become susceptible again as soon as they recover from the infection so that a fixed population can be divided into a class of infectives and a class of susceptibles which vary with time. If the contact rate is periodic, then the number of infectives as time approaches infinity either tends to zero or is asymptotically periodic depending on whether the total population size is less than or greater than a threshold value. The behavior for large time of the number of infectives is determined for three modifications of the model which involve carriers.     

Seasonal dynamics in an SIR epidemic system

We consider a seasonally forced SIR epidemic model where periodicity occurs in the contact rate. This periodical forcing represents successions of school terms and holidays. The epidemic dynamics are described by a switched system. Numerical studies in such a model have shown the existence of periodic solutions. First, we analytically prove the existence of an invariant domain $D$ containing all periodic (harmonic and subharmonic) orbits. Then, using different scales in time and variables, we rewrite the SIR model as a slow-fast dynamical system and we establish the existence of a macroscopic attractor domain $K$ , included in $D$ , for the switched dynamics. The existence of a unique harmonic solution is also proved for any value of the magnitude of the seasonal forcing term which can be interpreted as an annual infection. Subharmonic solutions can be seen as epidemic outbreaks. Our theoretical results allow us to exhibit quantitative characteristics about epidemics, such as the maximal period between major outbreaks and maximal prevalence.     

Germline mutations in the extracellular domains of the 55 kDa TNF receptor, TNFR1, define a family of dominantly inherited autoinflammatory syndromes

Autosomal dominant periodic fever syndromes are characterized by unexplained episodes of fever and severe localized inflammation. In seven affected families, we found six different missense mutations of the 55 kDa tumor necrosis factor receptor (TNFR1), five of which disrupt conserved extracellular disulfide bonds. Soluble plasma TNFR1 levels in patients were approximately half normal. Leukocytes bearing a C52F mutation showed increased membrane TNFR1 and reduced receptor cleavage following stimulation. We propose that the autoinflammatory phenotype results from impaired downregulation of membrane TNFR1 and diminished shedding of potentially antagonistic soluble receptor. TNFR1-associated periodic syndromes (TRAPS) establish an important class of mutations in TNF receptors. Detailed analysis of one such mutation suggests impaired cytokine receptor clearance as a novel mechanism of disease.     

稳定有界的Logistic方程的最优捕获策略

考虑单种群非自治的Logistic方程的开采问题．在R^ 中都存在均值的意义下,作为周期和概周期函数的推广,首先给出稳定有界函数的概念．然后定义一个新的最终最优收获策略用于处理我们的问题．选择单位时间的最大持久收益的极限均值作为管理目标。同时得到了最佳的种群水平．作为应用,我们以概周期系数的Logistic方程为例,表明我们的结果不仅推广了经典的Clark关于自治的Logistic方程的收获问题,而且推广了范猛和王克的关于周期的Logistic方程的收获问题的结果．     

一类中心型半连续动力系统的周期解

讨论了一类中心型半连续动力系统的阶1周期解和阶2周期解的存在性、个数及其稳定性,给出该中心型系统存在唯一、两个、无穷多个阶1周期解和阶2周期解的条件,并给出了相应理论结果的数值模拟.     

Periodic coexistence of four species competing for three essential resources

We show the existence of a periodic solution in which four species coexist in competition for three essential resources in the standard model of resource competition. By assuming that species i is limited by resource i for each i near the positive equilibrium, and that the matrix of contents of resources in species is a combination of cyclic matrix and a symmetric matrix, we obtain an asymptotically stable periodic solution of three species on three resources via Hopf bifurcation. A simple bifurcation argument is then employed which allows us to add a fourth species. In principle, the argument can be continued to obtain a periodic solution adding one new species at a time so long as asymptotic stability can be assured at each step. Numerical simulations are provided to illustrate our analytical results. The results of this paper suggest that competition can generate coexistence of species in the form of periodic cycles, and that the number of coexisting species can exceed the number of resources in a constant and homogeneous environment.     

Convergence Time towards Periodic Orbits in Discrete Dynamical Systems

We investigate the convergence towards periodic orbits in discrete dynamical systems. We examine the probability that a randomly chosen point converges to a particular neighborhood of a periodic orbit in a fixed number of iterations, and we use linearized equations to examine the evolution near that neighborhood. The underlying idea is that points of stable periodic orbit are associated with intervals. We state and prove a theorem that details what regions of phase space are mapped into these intervals (once they are known) and how many iterations are required to get there. We also construct algorithms that allow our theoretical results to be implemented successfully in practice.     

A Malaria Transmission Model with Temperature-Dependent Incubation Period

Malaria is an infectious disease caused by Plasmodium parasites and is transmitted among humans by female Anopheles mosquitoes. Climate factors have significant impact on both mosquito life cycle and parasite development. To consider the temperature sensitivity of the extrinsic incubation period (EIP) of malaria parasites, we formulate a delay differential equations model with a periodic time delay. We derive the basic reproduction ratio \(R_0\) and establish a threshold type result on the global dynamics in terms of \(R_0\), that is, the unique disease-free periodic solution is globally asymptotically stable if \(R_0<1\); and the model system admits a unique positive periodic solution which is globally asymptotically stable if \(R_0>1\). Numerically, we parameterize the model with data from Maputo Province, Mozambique, and simulate the long-term behavior of solutions. The simulation result is consistent with the obtained analytic result. In addition, we find that using the time-averaged EIP may underestimate the basic reproduction ratio.     

Dynamic reduction with applications to mathematical biology and other areas

In a difference or differential equation one is usually interested in finding solutions having certain properties, either intrinsic properties (e.g. bounded, periodic, almost periodic) or extrinsic properties (e.g. stable, asymptotically stable, globally asymptotically stable). In certain instances it may happen that the dependence of these equations on the state variable is such that one may (1) alter that dependency by replacing part of the state variable by a function from a class having some of the above properties and (2) solve the 'reduced' equation for a solution having the remaining properties and lying in the same class. This then sets up a mapping Τ of the class into itself, thus reducing the original problem to one of finding a fixed point of the mapping. The procedure is applied to obtain a globally asymptotically stable periodic solution for a system of difference equations modeling the interaction of wild and genetically altered mosquitoes in an environment yielding periodic parameters. It is also shown that certain coupled periodic systems of difference equations may be completely decoupled so that the mapping Τ is established by solving a set of scalar equations. Periodic difference equations of extended Ricker type and also rational difference equations with a finite number of delays are also considered by reducing them to equations without delays but with a larger period. Conditions are given guaranteeing the existence and global asymptotic stability of periodic solutions.     

Reproductive numbers for nonautonomous spatially distributed periodic SIS models acting on two time scales

In this work we deal with a general class of spatially distributed periodic SIS epidemic models with two time scales. We let susceptible and infected individuals migrate between patches with periodic time dependent migration rates. The existence of two time scales in the system allows to describe certain features of the asymptotic behavior of its solutions with the help of a less dimensional, aggregated, system. We derive global reproduction numbers governing the general spatially distributed nonautonomous system through the aggregated system. We apply this result when the mass action law and the frequency dependent transmission law are considered. Comparing these global reproductive numbers to their non spatially distributed counterparts yields the following: adequate periodic migration rates allow global persistence or eradication of epidemics where locally, in absence of migrations, the contrary is expected.     

A two-step patterning process increases the robustness of periodic patterning in the fly eye

Complex periodic patterns can self-organize through dynamic interactions between diffusible activators and inhibitors. In the biological context, self-organized patterning is challenged by spatial heterogeneities (‘noise’) inherent to biological systems. How spatial variability impacts the periodic patterning mechanism and how it can be buffered to ensure precise patterning is not well understood. We examine the effect of spatial heterogeneity on the periodic patterning of the fruit fly eye, an organ composed of ～800 miniature eye units (ommatidia) whose periodic arrangement along a hexagonal lattice self-organizes during early stages of fly development. The patterning follows a two-step process, with an initial formation of evenly spaced clusters of ～10 cells followed by a subsequent refinement of each cluster into a single selected cell. Using a probabilistic approach, we calculate the rate of patterning errors resulting from spatial heterogeneities in cell size, position and biosynthetic capacity. Notably, error rates were largely independent of the desired cluster size but followed the distributions of signaling speeds. Pre-formation of large clusters therefore greatly increases the reproducibility of the overall periodic arrangement, suggesting that the two-stage patterning process functions to guard the pattern against errors caused by spatial heterogeneities. Our results emphasize the constraints imposed on self-organized patterning mechanisms by the need to buffer stochastic effects. Author summary Complex periodic patterns are common in nature and are observed in physical, chemical and biological systems. Understanding how these patterns are generated in a precise manner is a key challenge. Biological patterns are especially intriguing, as they are generated in a noisy environment; cell position and cell size, for example, are subject to stochastic variations, as are the strengths of the chemical signals mediating cell-to-cell communication. The need to generate a precise and robust pattern in this ‘noisy’ environment restricts the space of patterning mechanisms that can function in the biological setting. Mathematical modeling is useful in comparing the sensitivity of different mechanisms to such variations, thereby highlighting key aspects of their design.We use mathematical modeling to study the periodic patterning of the fruit fly eye. In this system, a highly ordered lattice of differentiated cells is generated in a two-dimensional cell epithelium. The pattern is first observed by the appearance of evenly spaced clusters of ～10 cells that express specific genes. Each cluster is subsequently refined into a single cell, which initiates the formation and differentiation of a miniature eye unit, the ommatidium. We formulate a mathematical model based on the known molecular properties of the patterning mechanism, and use a probabilistic approach to calculate the errors in cluster formation and refinement resulting from stochastic cell-to-cell variations (‘noise’) in different quantitative parameters. This enables us to define the parameters most influencing noise sensitivity. Notably, we find that this error is roughly independent of the desired cluster size, suggesting that large clusters are beneficial for ensuring the overall reproducibility of the periodic cluster arrangement. For the stage of cluster refinement, we find that rapid communication between cells is critical for reducing error. Our work provides new insights into the constraints imposed on mechanisms generating periodic patterning in a realistic, noisy environment, and in particular, discusses the different considerations in achieving optimal design of the patterning network.     

Responsive neuromodulators based on artificial neural networks used to control seizure-like events in a computational model of epilepsy

Deep brain stimulation (DBS) has been noted for its potential to suppress epileptic seizures. To date, DBS has achieved mixed results as a therapeutic approach to seizure control. Using a computational model, we demonstrate that high-complexity, biologically-inspired responsive neuromodulation is superior to periodic forms of neuromodulation (responsive and non-responsive) such as those implemented in DBS, as well as neuromodulation using random and random repetitive-interval stimulation. We configured radial basis function (RBF) networks to generate outputs modeling interictal time series recorded from rodent hippocampal slices that were perfused with low Mg2?/high K? solution. We then compared the performance of RBF-based interictal modulation, periodic biphasic-pulse modulation, random modulation and random repetitive modulation on a cognitive rhythm generator (CRG) model of spontaneous seizure-like events (SLEs), testing efficacy of SLE control. A statistically significant improvement in SLE mitigation for the RBF interictal modulation case versus the periodic and random cases was observed, suggesting that the use of biologically-inspired neuromodulators may achieve better results for the purpose of electrical control of seizures in a clinical setting.     

Periodic solution of a chemostat model with Beddington-DeAnglis uptake function and impulsive state feedback control

In this paper, a chemostat model with Beddington-DeAnglis uptake function and impulsive state feedback control is considered. We obtain sufficient conditions of the global asymptotical stability of the system without impulsive state feedback control. We also obtain that the system with impulsive state feedback control has periodic solution of order one. Sufficient conditions for existence and stability of periodic solution of order one are given. In some cases, it is possible that the system exists periodic solution of order two. Our results show that the control measure is effective and reliable.