Bifurcations of sleep patterns due to homeostatic and circadian variation in a sleep-wake flip-flop model

TL;DR

This study investigates how variations in homeostatic and circadian influences cause bifurcations in sleep patterns using a mathematical sleep-wake flip-flop model, revealing complex bifurcation sequences and effects of circadian profile steepness.

Contribution

It introduces a comprehensive bifurcation analysis of a sleep-wake flip-flop model considering circadian and homeostatic variations, employing novel mathematical techniques for piecewise-smooth systems.

Findings

Sleep patterns exhibit period adding sequences as homeostatic time constants decrease.

Circadian profile steepness affects the number and stability of sleep solutions.

Bifurcations include saddle-node and border collision types, influenced by parameter values.

Abstract

Differential equation-based physiological models of sleep-wake networks describe sleep-wake regulation by simulating the activity of wake- and sleep-promoting neuronal populations and the modulation of these populations by homeostatic and circadian ($\sim24$ h) drives. Here, we consider a sleep-wake flip-flop network model consisting of mutually inhibitory interactions between wake- and sleep-promoting neuronal populations. Motivated by changes in sleep behavior during early childhood as babies transition from napping to non-napping behavior, we vary homeostatic and circadian modulation and analyze effects on resulting sleep-wake patterns. To identify the types and sequences of bifurcations leading to changes in stable sleep-wake patterns in this piecewise-smooth model, we employ multiple mathematical methods, including fast-slow decomposition and numerical computation of circle maps. We find that the average daily number of sleeps exhibits a period adding sequence as the homeostatic time constants are reduced, and that the temporal circadian profile influences the number of observed solutions in the sequence. These solutions emerge through sequences of saddle-node and border collision bifurcations, where the particular sequence depends on parameter values. When the temporal circadian profile is steep, as is observed with long day lengths, some sleep patterns are lost and bistability of other patterns may occur. We analyze a limiting case of the temporal circadian waveform, a circadian hard switch model, to understand this loss of solutions. Generally, our holistic analysis approach provides an alternative analysis method for model systems that defy conventional numerical bifurcation analysis techniques.