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Register a new userEntrainment dynamics of forced hierarchical circadian systems revealed by 2-dimensional maps
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The ability of a circadian system to entrain to the 24-hour light-dark cycle is one of its most important properties. A new tool, called the entrainment map, was recently introduced to study this process for a single oscillator. Here we generalize the map to study the effects of light-dark forcing in a hierarchical system consisting of a central circadian oscillator that drives a peripheral circadian oscillator. We develop techniques to reduced the higher dimensional phase space of the coupled system to derive a generalized 2-D entrainment map. Determining the nature of various fixed points, together with an understanding of their stable and unstable manifolds, leads to conditions for existence and stability of periodic orbits of the circadian system. We use the map to investigate how various properties of solutions depend on parameters and initial conditions including the time to and direction of entrainment. We show that the concepts of phase advance and phase delay need to be carefully assessed when considering hierarchical systems.
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A stochastic averaging technique based on energy-dependent frequency is extended to dynamical systems with triple-well potential driven by colored noise. The key procedure is the derivation of energy-dependent frequency according to the four different motion patterns in triple-well potential. Combined with the stochastic averaging of energy envelope, the analytical stationary probability density (SPD) of tri-stable systems can be obtained. Two cases of strongly nonlinear triple-well potential systems are presented to explore the effects of colored noise and validate the effectiveness of the proposed method. Results show that the proposed method is well verified by numerical simulations, and has significant advantages, such as high accuracy, small limitation and easy application in multi-stable systems, compared with the traditional stochastic averaging method. Colored noise plays a significant constructive role in modulating transition strength, stochastic fluctuation range and symmetry of triple-well potential. While, the additive and multiplicative colored noises display quite different effects on the features of coherence resonance (CR). Choosing a moderate additive noise intensity can induce CR, but the multiplicative colored noise cannot.
The light-based minimum-time circadian entrainment problem for mammals, Neurospora, and Drosophila is studied based on the mathematical models of their circadian gene regulation. These models contain high order nonlinear differential equations. Two model simplification methods are applied to these high-order models: the phase response curves (PRC) and the Principal Orthogonal Decomposition (POD). The variational calculus and a gradient descent algorithm are applied for solving the optimal light input in the high-order models. As the results of the gradient descent algorithm rely heavily on the initial guesses, we use the optimal control of the PRC and the simplified model to initialize the gradient descent algorithm. In this paper, we present: (1) the application of PRC and direct shooting algorithm on high-order nonlinear models; (2) a general process for solving the minimum-time optimal control problem on high-order models; (3) the impacts of minimum-time optimal light on circadian gene transcription and protein synthesis.
We consider a class of parametrically forced Hamiltonian systems with one-and-a-half degrees of freedom and study the stability of the dynamics when the frequency of the forcing is relatively high or low. We show that, provided the frequency of the forcing is sufficiently high, KAM theorem may be applied even when the forcing amplitude is far away from the perturbation regime. A similar result is obtained for sufficiently low frequency forcing, but in that case we need the amplitude of the forcing to be not too large; however we are still able to consider amplitudes of the forcing which are outside of the perturbation regime. Our results are illustrated by means of numerical simulations for the system of a forced cubic oscillator. In addition, we find numerically that the dynamics are stable even when the forcing amplitude is very large (beyond the range of validity of the analytical results), provided the frequency of the forcing is taken correspondingly low.
Given a piecewise $C^{1+beta}$ map of the interval, possibly with critical points and discontinuities, we construct a symbolic model for invariant probability measures with nonuniform expansion that do not approach the critical points and discontinuities exponentially fast almost surely. More specifically, we code the lift of these measures in the natural extension of the map.
We propose a multiscale chemo-mechanical model of cancer tumour development in an epithelial tissue. The model is based on transformation of normal cells into the cancerous state triggered by a local failure of spatial synchronisation of the circadian rhythm. The model includes mechanical interactions and chemical signal exchange between neighbouring cells, as well as division of cells and intercalation, and allows for modification of the respective parameters following transformation into the cancerous state. The numerical simulations reproduce different dephasing patterns - spiral waves and quasistationary clustering, with the latter being conducive to cancer formation. Modification of mechanical properties reproduces distinct behaviour of invasive and localised carcinoma.
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