Mathematical modeling of endocrine regulation subject to circadian rhythm

The 2017 Nobel Prize in Physiology or Medicine awarded for discoveries of molecular mechanisms controlling the circadian rhythm has called attention to the challenging area of nonlinear dynamics that deals with synchronization and entrainment of oscillations. Biological circadian clocks keep time in living organisms, orchestrating hormonal cycles and other periodic rhythms. The periodic oscillations of circadian pacemakers are self-sustained; at the same time, they are entrainable by external periodic signals that adjust characteristics of autonomous oscillations. Whereas modeling of biological oscillators is a well-established research topic, mathematical analysis of entrainment, i.e. the nonlinear phenomena imposed by periodic exogenous signals, remains an open problem. Along with sustained periodic rhythms, periodically forced oscillators can exhibit various “irregular” behaviors, such as quasiperiodic or chaotic trajectories.This paper presents an overview of the mathematical models of circadian rhythm with respect to endocrine regulation, as well as biological background. Dynamics of the human endocrine system, comprising numerous glands and hormones operating under neural control, are highly complex. Therefore, only endocrine subsystems (or axes) supporting certain biological functions are usually studied. Low-order dynamical models that capture the essential characteristics and interactions between a few hormones can than be derived. Goodwin’s oscillator often serves as such a model and is widely regarded as a prototypical biological oscillator. A comparative analysis of forced dynamics arising in two versions of Goodwin’s oscillator is provided in the present paper: the classical continuous oscillator and a more recent impulsive one, capturing e.g. pulsatile secretion of hormones due to neural regulation. The main finding of this study is that, while the continuous oscillator is always forced to a periodic solution by a sufficiently large exogenous signal amplitude, the impulsive one commonly exhibits a quasiperiodic or chaotic behavior due to non-smooth dynamics in entrainment.     

针对生物振荡网络分析的周期系统目标敏感性方法研究

生物学实验和模型计算结果表明振荡是一种常见的生物学现象, 越来越多的研究人员关注生物系统内部的振荡现象是如何产生的、引起振荡的关键因素是什么等问题. 敏感性分析定量分析系统行为在模型参数、系统输入或者初始条件发生变化时受影响的程度. 对周期系统进行传统的状态敏感性计算时, 得到的灵敏度指标随着时间的增加而发散, 因而对其进行敏感性分析是一项具有挑战性的工作. 本文针对这类系统, 首先提出基本状态敏感性的概念, 由此进一步推导出一种相敏感性分析方法. 在计算周期灵敏度过程中, 提出了一种基于奇异值分解的的改进算法, 简化了基本状态灵敏度的计算. 本文中的目标敏感性分析方法克服了因累积效应引起的发散问题. 通过对一个生物节律模型和一个复杂的信号转导网络系统模型进行敏感性分析, 可以看到改进的周期灵敏度计算方法得到的结果与已有方法一致, 并且新提出的目标敏感性分析方法及其计算在处理存在反应守恒的复杂生物振荡系统分析时是有效的.     

哺乳动物生物钟的数学建模及研究进展

由于数学模型在整合实验数据和分析基因调控网络的动力学方面的独特优势,近年来数学模型在生物节律研究领域越来越受到人们的重视.哺乳动物昼夜节律是由位于视觉交叉上颌的神经元控制的,其中的每个神经元都含有一个内在的生物钟,关键的问题是具有广泛周期分布的神经元振子之间如何达到相同步.在分子水平上结合数学方法中的网络分析与控制的观点构建生物网络,然后用非线性动力学的相关知识进行理论分析和数值模拟,是研究生命现象的一个有效途径.本文从系统生物学的研究思路,对生物钟的数学建模及其动力学研究做了一个综述,并对其今后的研究热点进行了展望.     

Features of Circadian Rhythms Relevant for the Design of Shift Schedules

The first part of this paper reviews properties of circadian rhythms when free-running in constant conditions, or in situations where zeitgebers are insufficient for entrainment. The second part deals with problems of entrainment by natural and artificial zeitgebers, with phase controlling effects of sleep and the timing of meals, and with the need to differentiate between entraining and masking effects. The third part discusses phase shifts of circadian rhythms, especially the dependence of the rate of re-entrainment on the direction of the shift, and the splitting of the circadian system into parts which are shifted in opposite directions (re-entrainment by partition). The relevance of these features of circadian rhythms for the design of shift-work schedules is discussed.     

Basic study about cooperative movement between a human and an agent with entrainment

The use of an agent in the environment, as in medical treatment, welfare, the construction field, and the home, is examined. It is necessary for the agent to be able to work cooperatively with a human, and that may be a problem. Therefore, this study pays attention to the phenomenon that is called entrainment. This is the phenomenon where the rhythm of a certain person and the partner in the communication is synchronized. The purpose of this study is to perform a basic examination into applying an entrainment in order to realize cooperation in operations between a human and an agent. Since many human rhythms are expressed by a nonlinear oscillator, we simulated limited cycle oscillators, which are one of two nonlinear oscillators that cause interactions between agents, as the basic examination for this achievement. Consequently, entrainment of limited cycle oscillators with rhythms which differ has been checked in certain conditions. As a result, the possibility that cooperative movement could be gained was suggested.     

Bayesian sparse solutions to linear inverse problems with non-stationary noise with Student-t priors

Bayesian approach has become a commonly used method for inverse problems arising in signal and image processing. One of the main advantages of the Bayesian approach is the possibility to propose unsupervised methods where the likelihood and prior model parameters can be estimated jointly with the main unknowns. In this paper, we propose to consider linear inverse problems in which the noise may be non-stationary and where we are looking for a sparse solution. To consider both of these requirements, we propose to use Student-t prior model both for the noise of the forward model and the unknown signal or image. The main interest of the Student-t prior model is its Infinite Gaussian Scale Mixture (IGSM) property. Using the resulted hierarchical prior models we obtain a joint posterior probability distribution of the unknowns of interest (input signal or image) and their associated hidden variables. To be able to propose practical methods, we use either a Joint Maximum A Posteriori (JMAP) estimator or an appropriate Variational Bayesian Approximation (VBA) technique to compute the Posterior Mean (PM) values. The proposed method is applied in many inverse problems such as deconvolution, image restoration and computed tomography. In this paper, we show only some results in signal deconvolution and in periodic components estimation of some biological signals related to circadian clock dynamics for cancer studies.     

Decentralized autonomous control of a quadrupedal locomotion robot using oscillators

This article deals with the design of a control system for a quadrupedal locomotion robot. The proposed control system is composed of a leg motion controller and a gait pattern controller within a hierarchical architecture. The leg controller drives actuators at the joints of the legs using a high-gain local feedback control. It receives the command signal from the gait pattern controller. The gait pattern controller, on the other hand, involves nonlinear oscillators. These oscillators interact with each other through signals from the touch sensors located at the tips of the legs. Various gait patterns emerge through the mutual entrainment of these oscillators. As a result, the system walks stably in a wide velocity range by changing its gait patterns and limiting the increase in energy consumption of the actuators. The performance of the proposed control system is verified by numerical simulations. This work was presented in part at the Fifth International Symposium on Artificial Life and Robotics, Oita, Japan, January 26–28, 2000     

Adaptive Lipschitz Observer Design for a Mammalian Model

This paper deals with observer designs for a proposed mathematical model of circadian rhythms which exist in almost every living organism. A 7th order model for mammalian circadian rhythms which captures the main dynamic features is considered in this paper. A recent result of one‐sided Lipschitz observer design in the literature is applied to this mammalian model to show a possibility of reducing measurements for circadian models in system biology. The mammalian model presented may contain an uncertainty parameter. An adaptive design of the Lipschitz observer is then applied to deal with this case. Besides detailed designs of both observers, detailed analysis is also performed for nonlinear functions in the mammalian model to show that the Lipschitz observers can indeed be applied. Several simulation studies of the proposed observers are carried out with the results shown in this paper.     

A controller perspective on biological gait control: Reflexes and central pattern generators

Biological motion control paradigms can be analyzed from a control theory point of view and translated to control applications. This paper revises and updates important aspects about the neural control of biped gait based on the Central Pattern Generator (CPG) and provides a controller view, including possible artificial implementations of these controllers. A first aspect to consider is the biological sensor-actuation system because the particularities of this system: the mammalian muscle. Many current biped robots are based on symmetrical CPG. However, new evidence suggests that biological CPG is asymmetrical with rhythm generator controlling the flexor half in a kind of feedforward control and the extensor behaving in a feedback fashion controlled by external inputs such as ground contact. A theoretical model of the CPG connections with flexors and extensors is presented.     

Computational model of an adaptive rhythm generator within the olivocerebellar system

The aim of this study is to investigate via computation whether the olivocerebellar system is capable of functioning as an adaptive rhythm generator. For this purpose, a detailed and physiologically realistic computational model of the olivocerebellar system is developed, based on the known intrinsic cell and network topological properties of this brain system. The present network, where individual cells are modelled by leaky integrate-and-fire units, converts the irregular spikes produced by the olivary cells into a precise rhythmic signal at the output. The simulation results reveal that the computational model, which normally does not exhibit any rhythmic activity, could be switched into a new mode in which it functions as a rhythm generator producing pulses within three different frequency ranges corresponding to alpha, beta, or gamma bands, respectively. In either mode of operation, the firing rates of all simulated cell types are observed to match real data. The results of this study therefore support the experimental findings of researchers who argue that a biological clock producing rhythmic pulses within different temporal ranges is located within the cerebellum.     

Multivariate autoregressive models and kernel learning algorithms for classifying driving mental fatigue based on electroencephalographic

Long-term driving is a significant cause of fatigue-related accidents. Driving mental fatigue has major implications for transportation system safety. Monitoring physiological signal while driving can provide the possibility to detect the mental fatigue and give the necessary warning. In this paper an EEG-based fatigue countermeasure algorithm is presented to classify the driving mental fatigue. The features of multichannel electroencephalographic (EEG) signals of frontal, central and occipital are extracted by multivariate autoregressive (MVAR) model. Then kernel principal component analysis (KPCA) and support vector machines (SVM) are employed to identify three-class EEG-based driving mental fatigue. The results show that KPCA–SVM method is able to effectively reduce the dimensionality of the feature vectors, speed up the convergence in the training of SVM and achieve higher recognition accuracy (81.64%) of three driving mental fatigue states in 10 subjects. The KPCA–SVM method could be a potential tool for classification of driving mental fatigue.     

The influence of irregularity of rest and activity on performance: a model based on time since sleep and time of day

Abstract

Changes in performance during a 9 day period of an irregular schedule of work and rest were investigated. Performance varied with the circadian rhythm and the length of the work periods. For the majority of subjects these two factors were additive. The time-of-day variation was well approximated by a sinusoid and the time-since-sleep factor was represented by a cubic trend. The acrophase of the circadian rhythm varied throughout the experiment under the influence of the sleep-wake cycle and the time-of-day cues. A model which included time of day and time since sleep was used to predict variations in performance for work periods starting at different times of the day. It is considered that this model can be applied to current issues in the scheduling of civil airline pilots’ duty periods.     

Flatness-based control of PER protein oscillations in a drosophila model

Rhythms, and specially circadian rhythms governed by an oscillating set of genes called the central clock, and their incidence on therapeutics, have become an important concern in biology and medicine. Restoration of altered rhythms for therapeutic purpose can be viewed as an open-loop control problem. We study in this paper a nonlinear oscillator modeling the central clock mechanism through the synthesis cycle of PER protein for fruit fly Drosophila. Its particularly robust bifurcation structure makes it representative for a wide number of oscillating biological systems. We show how classical flatness-based motion planning techniques allow for fast design of chronomodulated injection schemes achieving circadian rhythm restoration.     

The preview control of a class of linear systems and its application in the fault-tolerant control theory

In the fault-tolerant control theory based on model following control, the desired signal of the control system is the output of a reference system. This paper is concerned with the design of the preview controller for a class of fault systems. A composite vector is introduced by including error vector, fault system state vector and reference system state vector. Then, we derived an augmented system from the known system equation, in which the reference input has equal status with the desired signal in the traditional preview control theory. Therefore, we can use the known theory to design the preview controller for the augmented system, then the preview controller of the original fault system can be obtained by the integration method. This paper strictly discusses the connection between stabilisation and detectability of the augmented system and the corresponding characteristics of the original system. Finally, by applying this theory to a real steam generator water level control system, it is found that the actions of the reference input preview and the fault signal preview can effectively eliminate the effect of the fault signal on the water level of the steam generator. The simulation shows the effectiveness of the controller designed.     

Bio-inspired behaviour-based control

The design and development of conventional controllers for robot platforms are sometimes too complex to achieve due to the fact that they require an exact model of the system and of the operating environment. The ability to pre-account for unknown operating environments is an important task for the controller to be robust. In contrast, biological controllers are model free and are based on simple working principles. Due to natural biological principles these controllers are adaptive and more robust than their conventional counterparts. In this paper, a behaviour-based controller has been developed, inspired by the concept of spinal fields found in frogs and rats. The performance of the controller has been verified on a Khepera robot platform.     

Simple robust output-feedback controller for uncertain nonlinearsystems

A robust output-feedback controller is designed by using an observer plus backstepping procedure to solve the tracking problems of a class of uncertain nonlinear systems. In the design procedure, three techniques are applied: a) a single filter whose dynamic order is the same as the system order is constructed; b) a normalization signal is introduced; c) positive nonlinear functions that are used to generate the normalization signal and to construct nonlinear damping terms can be chosen freely. The designed controller has a very simple structure and can be applied to track much broader classes of reference signals. It can guarantee the global boundness of all closed-loop signals and make the output tracking error arbitrarily small. Technique c) may provide the designer with the possibility to find a better choice of positive nonlinear functions for our controller to achieve similar tracking performance with less control effort     

Output feedback preview control for polytopic uncertain discrete‐time systems with time‐varying delay

This paper considers the preview tracking control problem of polytopic uncertain discrete‐time systems with a time‐varying delay subject to a previewable reference signal. First, a model transformation is employed and a discrete‐time system with a time‐invariant delay and an external disturbance is obtained. The difference operator method can be extended to derive an augmented error system that includes future information on the reference signal. Then, a previewable reference signal is fully utilized through reformulation of the output equation while considering the output feedback. Based on the small gain theorem, a static output feedback controller with preview actions is designed such that the output can asymptotically track the reference signal. Finally, numerical simulation examples also illustrate the superiority of the desired preview controller for the uncertain system in the paper.     

A SPATIOTEMPORAL MODEL OF HUMAN CIRCADIAN RHYTHM IN SMART HOMES

This article presents a spatiotemporal model of human circadian activity rhythm in smart homes. A spatiotemporal model is used to represent human activity in a time-based system. This article proposes a learning and prediction algorithm to analyze temporal characteristics of the resident's activity. The algorithms combined Allen's temporal logic and Gaussian distribution to incrementally learn and predict next activity of the inhabitant. The methods show 88.1% prediction accuracy when tested with a practical smart home data set. Further analysis showed that human activity in smart homes follows Gaussian distribution, which previously had been merely an assumption.     

基于光流的四旋翼直升机鲁棒自主着陆控制

针对小型四旋翼无人机自主着陆问题,提出了一种基于光流的高度估计方法和基于信号补偿的高度鲁棒控制器设计方法.首先从通用光流运动模型出发,采用奇异值分解方法求解无人机线速度和深度的比值,通过对垂直速度和高度比值积分获得高度数据.其次,将考虑地效影响和其它不确定性的高度通道非线性模型分解为标称线性模型和等效扰动两部分,并设计基于信号补偿的高度鲁棒控制器,该控制器由标称控制器和鲁棒补偿器组成,其中标称控制器使得标称闭环系统达到期望的高度跟踪特性,鲁棒补偿器用于抑制等效扰动的影响.最后从理论上证明了该控制器可以保证高度跟踪误差在有限时间内收敛至指定的原点邻域内.四旋翼无人机自主着陆的实验结果验证了所提出的基于光流的高度估计和鲁棒控制方法的有效性.     

An adaptive full order sliding mode controller for mismatched uncertain systems

In this paper, an adaptive full order sliding mode (FOSM) controller is proposed for strict feedback nonlinear systems with mismatched uncertainties. The design objective of the controller is to track a specified trajectory in presence of significant mismatched uncertainties. In the first step the dynamic model for the first state is considered by the desired tracking signal. After the first step the desired dynamic model for each state is defined by the previous one. An adaptive tuning law is developed for the FOSM controller to deal with the bounded system uncertainty. The major advantages offered by this adaptive FOSM controller are that advanced knowledge about the upper bound of the system uncertainties is not a necessary requirement and the proposed method is an effective solution for the chattering elimination from the control signal. The controller is designed considering the full-order sliding surface. System robustness and the stability of the controller are proved by using the Lyapunov technique. A systematic adaptive step by step design method using the full order sliding surface for mismatched nonlinear systems is presented. Simulation results validate the effectiveness of the proposed control law.