Bifurcations and Hyperchaos in Mathematical Models of Sleep
Adriano Scibilia1, Luigi Fortuna2,3
1Institute of Intelligent Industrial Systems and Technologies for Advanced Manufacturing, National Research Council, 20133 Milan, Italy.
Bioengineering (Basel, Switzerland)
|July 28, 2026
Summary
This study uses nonlinear dynamics to model sleep regulation, revealing complex behaviors like hyperchaos. The findings offer a new framework for understanding irregular sleep patterns and developing digital sleep technologies.
Area of Science:
- Neuroscience
- Complex Systems Science
- Computational Biology
Background:
- Sleep regulation involves intricate nonlinear feedback loops, including homeostatic pressure, circadian rhythms, and neuromodulation.
- Existing sleep models often simplify these complex dynamics, necessitating advanced analytical approaches.
Purpose of the Study:
- To present a nonlinear-dynamical interpretation of sleep regulation, emphasizing bifurcation structure and Lyapunov stability.
- To analyze sleep-stage dynamics using advanced mathematical frameworks and computational models.
- To explore the potential of bifurcation theory in understanding irregular sleep transitions and informing future sleep technologies.
Main Methods:
- Summarized representative sleep models and analyzed sleep-stage dynamics literature.
- Combined bifurcation analysis and Lyapunov exponent calculations on sleep-population models.
- Developed and analyzed a four-dimensional slow-fast feedback model incorporating cortical activation, sleep-promoting activity, homeostatic drive, and circadian pacemaker.
Main Results:
- Identified routes from regular oscillations to period-doubling, chaotic bands, and positive Lyapunov regions using baseline diagrams.
- Lyapunov maps of the four-dimensional model revealed parameter regions supporting hyperchaotic dynamics (positive largest and second-largest Lyapunov exponents).
- Demonstrated how chronic sleep debt and insomnia can be modeled as shifts in homeostatic drive, inhibitory gain, and circadian coupling.
Conclusions:
- Bifurcation theory provides a robust framework for interpreting complex and irregular sleep transitions.
- The nonlinear-dynamical approach offers insights into the fragmentation and intermittency observed in sleep.
- This research paves the way for novel sleep-technology and digital twin applications based on a deeper understanding of sleep dynamics.
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