Obscured Complexity: How External Cycles Simplify the Dynamics of the Endogenous Circadian Oscillator–take the time series of body temperature records as an example

Abstract

Background: Understanding circadian rhythms is crucial in various fields of biological research, as they play a fundamental role in the regulation of diverse biological processes, ranging from gene expression to physiological functions. Objective: This study aims to explore the complexity of circadian rhythm signals from a biological system. Without the permission of using experimental data, the mathematical model is utilized to simulate the intricate dynamics of the body temperature's circadian rhythms and investigate the impact of parameter variation on system behavior. Methods: The Duffing equation is constructed as the mathematical model for simulating circadian rhythms. A thorough discussion justifies the selection of the Duffing equation and establishes the proper parameter range, ensuring chaotic behavior in the system. Four different values of the driving force parameter gamma (0.32, 0.33, 0.34, and 0.35) are chosen to represent specific cases. Fourier analysis is employed to analyze the simulation data, revealing the frequency components present in the circadian rhythm signals. Entropy analysis along the Poincare sections is utilized to measure the system's behavior and aggregation of points. Results: The simulations exhibit distinct characteristics in terms of plain visualization, Fourier analysis, and entropy analysis along the Poincare sections. Under normal work sleep conditions (gamma=0.35), the system demonstrates specific resetting at particular times within a total period. In shift work (gamma=0.34) or long-term constant temperature (gamma=0.33) conditions, some of the resetting behavior diminishes, and the initial phase of the time series changes. When all external driving forces are eliminated (gamma=0.32), the system undergoes multiple resets within a given period. In such circumstances, the biological clock experiences more frequent resets to adapt to the independent operations of each subsystem. Without relying on external environmental cues for regulation, the biological clock relies on frequent resetting to maintain the stability and coordination of the entire system. These findings align with experimental data, although the lack of available experimental data prevents direct comparison in this paper. Conclusion: The simulations reveals variations in resetting behavior and the importance of frequent resets in the absence of external cues. The complexity arising from chaos allows the biological system to adapt and adjust to the intricacies of the external environment. The endogenous clock within the system, despite its inherent complexity, can dynamically optimize its entrainment with external cycles. However, the full complexity of the endogenous clock may be concealed within the system and not readily observable. These findings contribute to a better understanding of the complex dynamics of circadian rhythms. Future research should aim to validate these results through comparisons with experimental data.