Bifurcations in a mathematical model for circadian oscillations of clock genes

Tsumoto, Kunichika; Yoshinaga, Tetsuya; Hitoshi, Iida; Kawakami, Hiroshi; Aihara, Kazuyuki

doi:10.1016/j.jtbi.2005.07.017

Cited by 45 publications

(25 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Such hysteretic behavior has been sometimes observed in biological systems [4]. Many more theoretical studies have shown the possible occurrence of birhythmicity in models of glycolytic oscillations [5], chemical kinetic equations [6], circadian proteins rhythmics [7][8][9], and biochemical reactions [10]. Perhaps the simplest model that exhibits birhythmicity is a variation of the well known van der Pol oscillator proposed by Kaiser [11] to model enzyme reactions.…”

Section: Introductionmentioning

confidence: 99%

Global stability analysis of birhythmicity in a self-sustained oscillator

Yamapi

Филатрелла

Aziz-Alaoui

2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

We analyze global stability properties of birhythmicity in a self-sustained system with random excitations. The model is a multi-limit cycles variation of the van der Pol oscillator introduced to analyze enzymatic substrate reactions in brain waves. We show that the two frequencies are strongly influenced by the nonlinear coefficients α and β. With a random excitation, such as a Gaussian white noise, the attractor's global stability is measured by the mean escape time τ from one limit-cycle. An effective activation energy barrier is obtained by the slope of the linear part of the variation of the escape time τ versus the inverse noise-intensity 1/D. We find that the trapping barriers of the two frequencies can be very different, thus leaving the system on the same attractor for an overwhelming time. However, we also find that the system is nearly symmetric in a narrow range of the parameters. Some models employed to describe natural systems, such as for instance glycolysis reactions and circadian proteins rhythmics, exhibit spontaneous oscillations at two distinct frequencies. The phenomenon is known as birhythmicity, and the underlying dynamical structure is characterized by the coexistence of two stable attractors, each displaying a different frequency. Being the attractors locally stable, the system would however stay at a single frequency, the one selected by the choice of the initial conditions, unless an external source disturbs the evolution and causes a switch to the other attractor. To investigate such process, we have focused on a particular system of biological interest, a modified van der Pol oscillator (that displays birhythmicity), to determine the global stability properties of the attractors under the influence of noise. More specifically, we have characterized the stability of the attractors with the escape times, or the average time that the system requires to switch from an attractor to the other under the influence of random fluctuations. Such analysis reveals that the two attractors can possess very different properties, with very different relative residence times. Even excluding the most asymmetric cases, the system can spend something like 10 years on one attractor for each second spent on the other. We conclude that although a system can be structurally biorhythmic for the contemporary presence of two locally stable attractors at two different frequencies, actual switch from one frequency to the other could be very difficult to observe. A global stability analysis can therefore help to determine the region of the parameter space in which birhythmic behavior will be genuinely observed.

show abstract

Section: Introductionmentioning

confidence: 99%

Global stability analysis of birhythmicity in a self-sustained oscillator

Yamapi

Филатрелла

Aziz-Alaoui

2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

show abstract

“…The fact that the circadian network relies on multiple interlocked feedback loops allows the generation of complex dynamic behaviours such as chaotic oscillations [85,86]. This kind of irregular behaviour is often considered as pathologic.…”

Section: Exploring Complex Dynamicsmentioning

confidence: 99%

Modeling circadian clocks: Roles, advantages, and limitations

Gonze¹

2011

Open Life Sciences

View full text Add to dashboard Cite

Circadian rhythms are endogenous oscillations characterized by a period of about 24h. They constitute the biological rhythms with the longest period known to be generated at the molecular level. The abundance of genetic information and the complexity of the molecular circuitry make circadian clocks a system of choice for theoretical studies. Many mathematical models have been proposed to understand the molecular regulatory mechanisms that underly these circadian oscillations and to account for their dynamic properties (temperature compensation, entrainment by light dark cycles, phase shifts by light pulses, rhythm splitting, robustness to molecular noise, intercellular synchronization). The roles and advantages of modeling are discussed and illustrated using a variety of selected examples. This survey will lead to the proposal of an integrated view of the circadian system in which various aspects (interlocked feedback loops, inter-cellular coupling, and stochasticity) should be considered together to understand the design and the dynamics of circadian clocks. Some limitations of these models are commented and challenges for the future identified.

show abstract

“…Given that the class II model is more easily synchronized than the class I model, the class II model is applied in studies of synchronization phenomena. Famous class II neuron models include the Hodgkin-Huxley model [14] and Bonhoeffer-van der Pol model [15], mathematical neuron models that form the basis of bioinspired oscillatory pattern generation [16][17][18]. Most of the central pattern generators (CPGs) designed for the synchronized locomotion control of multilegged robots [6][7][8] are also constructed by mathematical neuron models.…”

Section: Introductionmentioning

confidence: 99%

Gait Generation of Multilegged Robots by using Hardware Artificial Neural Networks

Saito¹,

Ohara²,

Abe³

et al. 2018

Advanced Applications for Artificial Neural Networks

View full text Add to dashboard Cite

Living organisms can act autonomously because biological neural networks process the environmental information in continuous time. Therefore, living organisms have inspired many applications of autonomous control to small-sized robots. In this chapter, a small-sized robot is controlled by a hardware artificial neural network (ANN) without software programs. Previously, the authors constructed a multilegged walking robot. The link mechanism of the limbs was designed to reduce the number of actuators. The current paper describes the basic characteristics of hardware ANNs that generate the gait for multilegged robots. The pulses emitted by the hardware ANN generate oscillating patterns of electrical activity. The pulse-type hardware ANN model has the basic features of a class II neuron model, which behaves like a resonator. Thus, gait generation by the hardware ANNs mimics the synchronization phenomena in biological neural networks. Consequently, our constructed hardware ANNs can generate multilegged robot gaits without requiring software programs.

show abstract

Bifurcations in a mathematical model for circadian oscillations of clock genes

Cited by 45 publications

References 43 publications

Global stability analysis of birhythmicity in a self-sustained oscillator

Global stability analysis of birhythmicity in a self-sustained oscillator

Modeling circadian clocks: Roles, advantages, and limitations

Gait Generation of Multilegged Robots by using Hardware Artificial Neural Networks

Contact Info

Product

Resources

About