The state-dependent impulsive control for a general predator-prey model

In this paper, a general predator-prey model with state-dependent impulse is considered. Based on the geometric analysis and Poincaré map or successor function, we construct three typical types of Bendixson domains to obtain some sufficient conditions for the existence of order-1 periodic solutions. At the same time, the existing domains are discussed with respect to the system parameters. Moreover, the Analogue of Poincaré Criterion is used to obtain the asymptotic stability of the periodic solutions. Finally, to illustrate the results, an example is presented and some numerical simulations are carried out.

Stratifications and foliations in phase portraits of gene network models

Periodic processes of gene network functioning are described with good precision by periodic trajectories (limit cycles) of multidimensional systems of kinetic-type differential equations. In the literature, such systems are often called dynamical, they are composed according to schemes of positive and negative feedback between components of these networks. The variables in these equations describe concentrations of these components as functions of time. In the preparation of numerical experiments with such mathematical models, it is useful to start with studies of qualitative behavior of ensembles of trajectories of the corresponding dynamical systems, in particular, to estimate the highest likelihood domain of the initial data, to solve inverse problems of parameter identification, to list the equilibrium points and their characteristics, to localize cycles in the phase portraits, to construct stratification of the phase portraits to subdomains with different qualities of trajectory behavior, etc. Such an à priori geometric analysis of the dynamical systems is quite analogous to the basic section "Investigation of functions and plot of their graphs" of Calculus, where the methods of qualitative studies of shapes of curves determined by equations are exposed. In the present paper, we construct ensembles of trajectories in phase portraits of some dynamical systems. These ensembles are 2-dimensional surfaces invariant with respect to shifts along the trajectories. This is analogous to classical construction in analytic mechanics, i. e. the level surfaces of motion integrals (energy, kinetic moment, etc.). Such surfaces compose foliations in phase portraits of dynamical systems of Hamiltonian mechanics. In contrast with this classical mechanical case, the foliations considered in this paper have singularities: all their leaves have a non-empty intersection, they contain limit cycles on their boundaries. Description of the phase portraits of these systems at the level of their stratifications, and that of ensembles of trajectories allows one to construct more realistic gene network models on the basis of methods of statistical physics and the theory of stochastic differential equations.

Entrainment within hierarchical circadian oscillator networks

Circadian rhythms are endogenous oscillations, widely found across biological species, that have the capability of entraining to the 24-h light-dark cycle. Circadian systems often consist of both central oscillators that receive direct light-dark input and peripheral oscillators that receive input from the central oscillators. In this paper, we address questions related to what governs the time to and pattern of entrainment of these hierarchical circadian systems after an abrupt switch in the light-dark phasing. For a network consisting of a single central oscillator coupled to a chain of N feed-forward peripheral oscillators, we introduce a systematic way to derive an N-dimensional entrainment map whose fixed points correspond to entrained solutions. Using the map, we explain that the direction of reentrainment can involve fairly complicated phase advancing and delaying behavior as well as reentrainment times that depend sensitively on the nature of the perturbation. We also study the dynamics of a hierarchical system in which the peripheral oscillators are mutually coupled. We study how reentrainment times vary as a function of the degree to which the oscillators are desynchronized at the time of the change in light-dark phasing. We show that desynchronizing the peripheral oscillators can, in some circumstances, speed up their ultimate reentrainment following perturbations.

A Scalar Poincaré Map for Anti-phase Bursting in Coupled Inhibitory Neurons With Synaptic Depression

Short-term synaptic plasticity is found in many areas of the central nervous system. In the inhibitory half-center central pattern generators involved in locomotion, synaptic depression is believed to act as a burst termination mechanism, allowing networks to generate anti-phase bursting patterns of varying periods. To better understand burst generation in these central pattern generators, we study a minimal network of two neurons coupled through depressing synapses. Depending on the strength of the synaptic conductance between the two neurons, this network can produce symmetric n : n anti-phase bursts, where neurons fire n spikes in alternation, with the period of such solutions increasing with the strength of the synaptic conductance. Relying on the timescale disparity in the model, we reduce the eight-dimensional network equations to a fully-explicit scalar Poincaré burst map. This map tracks the state of synaptic depression from one burst to the next and captures the complex bursting dynamics of the network. Fixed points of this map are associated with stable burst solutions of the full network model, and are created through fold bifurcations of maps. We derive conditions that predict the bifurcations between n : n and (n + 1) : (n + 1) solutions, producing a full bifurcation diagram of the burst cycle period. Predictions of the Poincaré map fit excellently with numerical simulations of the full network model and allow the study of parameter sensitivity for rhythm generation.

Cardiac and Autonomic Dysfunctions Assessed Through Recurrence Quantitative Analysis of Electrocardiogram Signals and an Application to the 6-Hydroxydopamine Parkinson's Disease Animal Model

A classic method to evaluate autonomic dysfunction is through the evaluation of heart rate variability (HRV). HRV provides a series of coefficients, such as Standard Deviation of n-n intervals (SDNN) and Root Mean Square of Successive Differences (RMSSD), which have well-established physiological associations. However, using only electrocardiogram (ECG) signals, it is difficult to identify proper autonomic activity, and the standard techniques are not sensitive and robust enough to distinguish pure autonomic modulation in heart dynamics from cardiac dysfunctions. In this proof-of-concept study we propose the use of Poincaré mapping and Recurrence Quantification Analysis (RQA) to identify and characterize stochasticity and chaoticity dynamics in ECG recordings. By applying these non-linear techniques in the ECG signals recorded from a set of Parkinson’s disease (PD) animal model 6-hydroxydopamine (6-OHDA), we showed that they present less variability in long time epochs and more stochasticity in short-time epochs, in their autonomic dynamics, when compared with those of the sham group. These results suggest that PD animal models present more “rigid heart rate” associated with “trembling ECG” and bradycardia, which are direct expressions of Parkinsonian symptoms. We also compared the RQA factors calculated from the ECG of animal models using four computational ECG signals under different noise and autonomic modulatory conditions, emulating the main ECG features of atrial fibrillation and QT-long syndrome.

A nonlinear dynamical approach to analysis of emotions using EEG signals based on the Poincaré map function and recurrence plots

Dynamic variations of electroencephalogram (EEG) contain significant information in the study of human emotional states. Transient time methods are well suited to evaluate short-term dynamic changes in brain activity. Human affective states, however, can be more appropriately analyzed using chaotic dynamical techniques, in which temporal variations are considered over longer durations. In this study, we have applied two different recurrence-based chaotic schemes, namely the Poincaré map function and recurrence plots (RPs), to analyze the long-term dynamics of EEG signals associated with state space (SS) trajectory of the time series. Both approaches determine the system dynamics based on the Poincaré recurrence theorem as well as the trajectory divergence producing two-dimensional (2D) characteristic plots. The performance of the methods is compared with regard to their ability to distinguish between levels of valence, arousal, dominance and liking using EEG data from the "dataset for emotion analysis using physiological" database. The differences between the levels of emotional feelings were investigated based on the analysis of variance (ANOVA) test and Spearman's statistics. The results obtained from the RP features distinguish between the emotional ratings with a higher level of statistical significance as compared with those produced by the Poincaré map function. The scheme based on RPs was particularly advantageous in identifying the levels of dominance. Out of the 32 EEG electrodes examined, the RP-based approach distinguished the dominance levels in 23 electrodes, while the approach based on the Poincaré map function was only able to discriminate dominance levels in five electrodes. Furthermore, based on nonlinear analysis, significant correlations were observed over a wider area of the cortex for all affective states as compared with that reported based on the analysis of EEG power bands.

Effect of spontaneous movement on respiration in preterm infants

NEW FINDINGS

What is the central question of this study? The respiratory centres in the brainstem that control respiration receive inputs from various sources, including proprioceptors in muscles and joints and suprapontine centres, which all affect limb movements. What is the effect of spontaneous movement on respiration in preterm infants? What is the main finding and its importance? Apnoeic events tend to be preceded by movements. These activity bursts can cause respiratory instability that leads to an apnoeic event. These findings show promise that infant movements might serve as potential predictors of life-threatening apnoeic episodes, but more research is required.

ABSTRACT

A common condition in preterm infants (<37 weeks' gestational age) is apnoea resulting from immaturity and instability of the respiratory system. As apnoeas are implicated in several acute and long-term complications, prediction of apnoeas may preempt their onset and subsequent complications. This study tests the hypothesis that infant movements are a predictive marker for apnoeic episodes and examines the relation between movement and respiration. Movement was detected using a wavelet algorithm applied to the photoplethysmographic signal. Respiratory activity was measured in nine infants using respiratory inductance plethysmography; in an additional eight infants, respiration and partial pressure of airway carbon dioxide ( P C O 2 ) were measured by a nasal cannula with side-stream capnometry. In the first cohort, the distribution of movements before and after the onset of 370 apnoeic events was compared. Results showed that apnoeic events were associated with longer movement duration occurring before apnoea onsets compared to after. In the second cohort, respiration was analysed in relation to movement, comparing standard deviation of inter-breath intervals (IBI) before and after apnoeas. Poincaré maps of the respiratory activity quantified variability of airway P C O 2 in phase space. Movement significantly increased the variability of IBI and P C O 2 . Moreover, destabilization of respiration was dependent on the duration of movement. These findings support that bodily movements of the infants precede respiratory instability. Further research is warranted to explore the predictive value of movement for life-threatening events, useful for clinical management and risk stratification.

Analysis of a mathematical model with nonlinear susceptibles-guided interventions

In this paper, we considered a mathematical model describing the nonlinear susceptibles-guided vaccination and isolation strategies, incorporating the continuously saturated treatment. In this strategy, we find that the disease-free periodic solution can always exist, and consequently the control reproduction number can be defined through analyzing the stability of the disease-free periodic solution. Also, we discussed the existence and stability of the positive order-1 periodic solution from two points of view. Initially, we investigated the transcritical and pitchfork bifurcation of the Poincaré map with respect to key parameters, and proved the existence of a stable or an unstable positive order-1 periodic solution near the disease-free periodic solution. For another aspect, by studying the properties of the Poincaré map, we verified the existence of the positive order-1 periodic solution in a large range of the control parameters, especially, we verified the co-existence of finite or infinite countable different positive order-1 periodic solutions. Furthermore, numerical simulations show that the unstable order-1 periodic solution can co-exist with the stable order-1, or order-2, or order-3 periodic solution. The finding implies that the nonlinear susceptibles-triggered feedback control strategy can induce much rich dynamics, which suggests us to carefully choose key parameters to ensure the stability of the disease-free periodic solution, indicating that infectious diseases die out.

Enhanced disinfection of bacterial populations by nutrient and antibiotic challenge timing

Several difficult to treat illnesses like tuberculosis, chronic pneumonia, and inner ear infections are caused by tolerant bacteria enmeshed in a biofilm. Bacterial tolerance can be genotypic (resistance-i.e. MRSA), phenotypic (non-heritable) or environmental (e.g. nutrient gradients). Persister formation is a phenotypic expression and this phenotype is highly tolerant of disinfection. Constant dosing is typically ineffective and to generate an effective treatment protocol, we need to understand the dynamics of persister cells. In this study, we investigate how manipulating the application of antibiotics and addition of nutrients enhances the disinfection of a bacterial population in batch culture. Eliminating persister bacteria is considered as a challenge for the food industry or wastewater treatment, since the failure may result in food contamination and disease transmission. Previous studies focused on the antimicrobial agent as a control variable to eliminate the bacterial population. In addition to antibiotic, we consider the significance of the nutrient in eradicating the susceptible and persister cells since the disinfection of susceptible population depends on the nutrient intake. We present a mathematical model that captures the dynamics between susceptible and persister bacteria with antibiotic and nutrient control variables. We investigate the optimal dose-withdrawal of antibiotic timing in several cases including constant nutrient in time, dynamic nutrient in time and piecewise constant nutrient in time.

Effects of pesticide dose on Holling II predator-prey model with feedback control

We establish a Holling II predator-prey system with pesticide dose response non-linear pulses and then study the global dynamics of the model. First, we construct the Poincaré map in the phase set and discuss its main properties. Second, threshold conditions for the existence and stability of boundary periodic solution and order-[Formula: see text] periodic solutions have been provided. The results show that the pesticide dose increases when the period of control increases, while it will decrease as threshold increases. Sensitivity analyses reveal that critical condition for the stability of boundary periodic solution is very sensitive to control parameters and pesticide doses. The bifurcation analysis reveals that the proposed model exists complex dynamics. Compared to the model with fixed moments, it demonstrates that the density of pest population not only can be controlled below the threshold but also can avoid some negative effects due to pesticide application, confirming the importance of biological control.

Control Synergies for Rapid Stabilization and Enlarged Region of Attraction for a Model of Hopping

Inspired by biological control synergies, wherein fixed groups of muscles are activated in a coordinated fashion to perform tasks in a stable way, we present an analogous control approach for the stabilization of legged robots and apply it to a model of running. Our approach is based on the step-to-step notion of stability, also known as orbital stability, using an orbital control Lyapunov function. We map both the robot state at a suitably chosen Poincaré section (an instant in the locomotion cycle such as the mid-flight phase) and control actions (e.g., foot placement angle, thrust force, braking force) at the current step, to the robot state at the Poincaré section at the next step. This map is used to find the control action that leads to a steady state (nominal) gait. Next, we define a quadratic Lyapunov function at the Poincaré section. For a range of initial conditions, we find control actions that would minimize an energy metric while ensuring that the Lyapunov function decays exponentially fast between successive steps. For the model of running, we find that the optimization reveals three distinct control synergies depending on the initial conditions: (1) foot placement angle is used when total energy is the same as that of the steady state (nominal) gait; (2) foot placement angle and thrust force are used when total energy is less than the nominal; and (3) foot placement angle and braking force are used when total energy is more than the nominal.

A Verified ODE Solver and the Lorenz Attractor

A rigorous numerical algorithm, formally verified with Isabelle/HOL, is used to certify the computations that Tucker used to prove chaos for the Lorenz attractor. The verification is based on a formalization of a diverse variety of mathematics and algorithms. Formalized mathematics include ordinary differential equations and Poincaré maps. Algorithms include low level approximation schemes based on Runge–Kutta methods and affine arithmetic. On a high level, reachability analysis is guided by static hybridization and adaptive step-size control and splitting. The algorithms are systematically refined towards an implementation that can be executed on Tucker’s original input data.

Phase-space topography characterization of nonlinear ultrasound waveforms

Fundamental understanding of ultrasound interaction with material discontinuities having closed interfaces has many engineering applications such as nondestructive evaluation of defects like kissing bonds and cracks in critical structural and mechanical components. In this paper, to analyze the acoustic field nonlinearities due to defects with closed interfaces, the use of a common technique in nonlinear physics, based on a phase-space topography construction of ultrasound waveform, is proposed. The central idea is to complement the "time" and "frequency" domain analyses with the "phase-space" domain analysis of nonlinear ultrasound waveforms. A nonlinear time series method known as pseudo phase-space topography construction is used to construct equivalent phase-space portrait of measured ultrasound waveforms. Several nonlinear models are considered to numerically simulate nonlinear ultrasound waveforms. The phase-space response of the simulated waveforms is shown to provide different topographic information, while the frequency domain shows similar spectral behavior. Thus, model classification can be substantially enhanced in the phase-space domain. Experimental results on high strength aluminum samples show that the phase-space transformation provides a unique detection and classification capabilities. The Poincaré map of the phase-space domain is also used to better understand the nonlinear behavior of ultrasound waveforms. It is shown that the analysis of ultrasound nonlinearities is more convenient and informative in the phase-space domain than in the frequency domain.

Reentrainment of the circadian pacemaker during jet lag: East-west asymmetry and the effects of north-south travel

The normal alignment of circadian rhythms with the 24-h light-dark cycle is disrupted after rapid travel between home and destination time zones, leading to sleep problems, indigestion, and other symptoms collectively known as jet lag. Using mathematical and computational analysis, we study the process of reentrainment to the light-dark cycle of the destination time zone in a model of the human circadian pacemaker. We calculate the reentrainment time for travel between any two points on the globe at any time of the day and year. We construct one-dimensional entrainment maps to explain several properties of jet lag, such as why most people experience worse jet lag after traveling east than west. We show that this east-west asymmetry depends on the endogenous period of the traveler's circadian clock as well as daylength. Thus the critical factor is not simply whether the endogenous period is greater than or less than 24 h as is commonly assumed. We show that the unstable fixed point of an entrainment map determines whether a traveler reentrains through phase advances or phase delays, providing an understanding of the threshold that separates orthodromic and antidromic modes of reentrainment. Contrary to the conventional wisdom that jet lag only occurs after east-west travel across multiple time zones, we predict that the change in daylength encountered during north-south travel can cause jet lag even when no time zones are crossed. Our techniques could be used to provide advice to travelers on how to minimize jet lag on trips involving multiple destinations and a combination of transmeridian and translatitudinal travel.

Stable bipedal walking with a swing-leg protraction strategy

In bipedal locomotion, swing-leg protraction and retraction refer to the forward and backward motion, respectively, of the swing-leg before touchdown. Past studies have shown that swing-leg retraction strategy can lead to stable walking. We show that swing-leg protraction can also lead to stable walking. We use a simple 2D model of passive dynamic walking but with the addition of an actuator between the legs. We use the actuator to do full correction of the disturbance in a single step (a one-step dead-beat control). Specifically, for a given limit cycle we perturb the velocity at mid-stance. Then, we determine the foot placement strategy that allows the walker to return to the limit cycle in a single step. For a given limit cycle, we find that there is swing-leg protraction at shallow slopes and swing-leg retraction at steep slopes. As the limit cycle speed increases, the swing-leg protraction region increases. On close examination, we observe that the choice of swing-leg strategy is based on two opposing effects that determine the time from mid-stance to touchdown: the walker speed at mid-stance and the adjustment in the step length for one-step dead-beat control. When the walker speed dominates, the swing-leg retracts but when the step length dominates, the swing-leg protracts. This result suggests that swing-leg strategy for stable walking depends on the model parameters, the terrain, and the stability measure used for control. This novel finding has a clear implication in the development of controllers for robots, exoskeletons, and prosthetics and to understand stability in human gaits.

"Glucose-at-a-Glance": New Method to Visualize the Dynamics of Continuous Glucose Monitoring Data

The standard continuous glucose monitoring (CGM) output provides multiple graphical and numerical summaries. A useful adjunct would be a visualization tool that facilitates immediate assessment of both long- and short-term variability. We developed an algorithm based on the mathematical method of delay maps to display CGM signals in which the glucose value at time t_i is plotted against its value at time t_i+1. The data points are then color-coded based on their frequency of occurrence (density). Examples of this new visualization tool, along with the accompanying time series, are presented for selected patients with type 2 diabetes and non-diabetic controls over the age of 70 years. The method reveals differences in the structure of the glucose variability between subjects with a similar range of glucose values. We also observe that patients with comparable hemoglobin A1c (HbA1c) values may have very different delay maps, consistent with marked differences in the dynamics of glucose control. These differences are not accounted by the amplitude of the fluctuations. Furthermore, the delay maps allow for rapid recognition of hypo- and hyperglycemic periods over the full duration of monitoring or any subinterval. The glucose-at-a-glance visualization tool, based on colorized delay maps, provides a way to quickly assess the complex data acquired by CGM systems. This method yields dynamical information not contained in single summary statistics, such as HbA1c values, and may also serve as the basis for developing novel metrics of glycemic control.