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Non-linear dynamics of human periodic breathing and implications for sleep apnea therapy

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Abstract

A mathematical model of non-obstructive human periodic breathing (Cheyne–Stokes respiration) or central sleep apnea (CSA) is described which focused on explaining recently reported non-linear behavior. Evidence was presented that CHF (chronic heart failure)–CSA and ICSA (idiopathic central sleep apnea) both involved limit cycle oscillations. The validity of applying linear control theory for stabilization must then be re-examined. Critical threshold values and ranges of parameters were predicted which caused a change (bifurcation) from limit cycle periodic breathing to stable breathing. Changes in lung volume were predicted to form a bifurcation during CHF–CSA where stability and instability can involve a lung volume change as small as 0.1 l. CSA therapy based on reducing control loop gain was predicted to be relatively ineffective during stable limit cycle oscillation. The relative ratios of durations of ventilation to apnea (T v/T a) during periodic breathing were primarily determined by peripheral chemoreceptor dynamics during crescendo, de-crescendo, and apnea phases of CSA.

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Abbreviations

a :

Central chemoreceptor activity threshold

b :

Peripheral chemoreceptor activity threshold

c :

Ventilatory apnea threshold

CHF:

Chronic heart failure

CO2 :

Carbon dioxide

CSA:

Central sleep apnea

\(\Delta P_{{{\text{aCO}}_{{\text{2}}} }} \) :

Arterial partial pressure of carbon dioxide reserve

g c :

Central chemoreceptor gain

g p :

Peripheral chemoreceptor gain

g rc :

Peripheral chemoreceptor rate constant

g rc+ :

Peripheral chemoreceptor “on” transient rate constant

g rc− :

Peripheral chemoreceptor “off” transient rate constant

ICSA:

Idiopathic central sleep apnea

\( P_{{{\text{aCO}}_{{\text{2}}} }} \) :

Arteral partial pressure of carbon dioxide

STD:

Short-term depression

STP:

Short-term potentiation

τ c :

Central chemoreceptor time constant

τ p :

Peripheral chemoreceptor time constant

τ pa :

Peripheral chemoreceptor apnea phase time constant

τ pa− :

Peripheral chemoreceptor apnea phase initiation time constant

τ pa+ :

Peripheral chemoreceptor apnea phase recovery time constant

τ p+ :

Peripheral chemoreceptor “on” transient time constant

τ p− :

Peripheral chemoreceptor “off” transient time constant

T a :

Duration of apnea

T v :

Duration of ventilation

V :

Total minute ventilation

V c :

Central minute ventilation

V in :

Peripheral ventilation volitional input

V p :

Peripheral minute ventilation

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Authors and Affiliations

  1. Biomedical Engineering Department, University of Southern California, Los Angeles, CA, 90089-1451, USA

    S. M. Yamashiro

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Yamashiro, S.M. Non-linear dynamics of human periodic breathing and implications for sleep apnea therapy. Med Bio Eng Comput 45, 345–356 (2007). https://doi.org/10.1007/s11517-006-0153-y

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