Cheyne-Stokes Period and Circulatory Delay

Cheyne–Stokes respiration is the chemoreflex limit cycle in the delay-dominated heart-failure regime, and its period scales as about twice the circulatory transport delay — the model returns a period-versus-delay slope of 1.99, i.e. T_CSR ≈ 2×τ_circ. A slower circulation lengthens the cycle. Grade [code output].

In the fast-plant regime the chemoreflex cycle obeys T_CSR ≈ 2 × τ_circ; the measured period/delay slope is 1.992 across delays 8–20 s (band [1.7,2.4]). [code output].

The forced number: period ≈ twice the delay

Cheyne–Stokes respiration is the chemoreflex limit cycle in the fast-plant, delay-dominated regime typical of heart failure. In that regime the cycle period is set by the loop transport delay, and the model returns a period-versus-delay slope of 1.992 — essentially T_CSR ≈ 2 × τ_circ.

This factor of two is a [consistency] property of the model, not a forced biological law: a delayed negative feedback oscillates with a period of about twice its loop delay. It is recovered across a range of circulatory delays, not at a single point.

Why heart failure lengthens the cycle

Because the period scales with the transport delay, a slower circulation (a longer lung-to-chemoreceptor delay, as in low cardiac output) lengthens the Cheyne–Stokes cycle. The mechanism predicts the clinical direction directly.

Grade

The limit-cycle period scaling as about twice the delay is a [code output] across the delay sweep; the slope lands at 1.992, within the acceptance band [1.7, 2.4].

At single delays the ratio period/delay is 2.09–2.22 (table); 1.992 is the slope of a fit across the 8–20 s sweep. [observation] Cheyne–Stokes cycle lengths measured in heart failure are of the order of one minute and lengthen with circulation time (Hall et al. 1996, Am. J. Respir. Crit. Care Med. 154:376). The model periods (17.7–41.7 s for 8–20 s delays) are shorter and approach that range only at the longest delay.