Analysis of a scenario for chaotic quantal slowing down of inspiration

Abstract

On exposure to opiates, preparations from rat brain stems have been observed to continue to produce regular expiratory signals, but to fail to produce some inspiratory signals. The numbers of expirations between two successive inspirations form an apparently random sequence. Here, we propose an explanation based on the qualitative theory of dynamical systems. A relatively simple scenario for the dynamics of interaction between the generators of expiratory and inspiratory signals produces pseudo-random behaviour of the type observed.

Fig. 1

Sequential plot and histograms of inspiratory period for an en bloc in vitro preparation of rat brain stem before and after treatment with an opiate agonist, showing quantal slowing down of inspiration (reproduced with permission from [2]). The horizontal axis on the sequential plot is “cycle number” (meaning number of inspirations since a start time), the vertical axis is time between successive inspirations in seconds. The horizontal axis for the histograms is inspiration period and the histograms are plotted for ten different experiments. When the potassium concentration was increased in two experiments (the last two histograms, in red), the slowing of the inspiratory period was no longer quantised

Fig. 2

Examples of flows on a two-torus: a a Poincaré flow, b a Cherry flow

Fig. 3

Sketch return maps for the flows of Fig. 2: a Poincaré flow, b Cherry flow with attracting saddle, c Cherry flow with repelling saddle

Fig. 4

Insertion of a fold into the dynamics

Fig. 5

Two-parameter plane showing qualitative form of circle map for perturbed Cherry flow as the ends of the decreasing interval $(a,b)$ move relative to the image I of the discontinuity. For this figure only, we put the origin at the left end of the interval I. Note that $(a,b)$ is restricted to the range $0<a<2\pi$, $a<b<a+2\pi$

Fig. 6

a A bimodal map possessing rotational chaos between rotation numbers $\frac{1}{2}$ and 1; b associated transition graph

Fig. 7

a A bimodal example with the orbits of the critical points landing on an unstable period-2 orbit; b associated transition graph

Fig. 8

a A bimodal example with rotation interval $[\frac{1}{5},\frac{1}{2}]$; b transition graph

Fig. 9

a Example of return map of type +; b associated transition graph

Fig. 10

a Variant of Fig. 9 with attracting instead of repelling saddle; b associated transition graph

Fig. 11

a Example of return map of type −+, b associated transition graph

Fig. 12

The map of Fig. 11 expressed in a new coordinate ξ chosen to make the slope everywhere greater than 1

Fig. 13

a Modification of Fig. 11, b associated transition graph

Fig. 14

Example of type −+ with rotation interval $[\frac{1}{5},\frac{1}{2}]$: a map, b associated transition graph

Fig. 15

An example with periodic orbits of two different rotation numbers but no rotational chaos, because the left interval is invariant with rotation number 0, what does not stay in the right interval falls into the left, and what does stay in the right has rotation number 1

Fig. 16

An example with a badly ordered period-2 orbit but no rotational chaos, because the interval $[0,2\pi ]$ is invariant

Fig. 17

Orbit near a saddle point