Analysis of a Scenario for Chaotic Quantal Slowing Down of Inspiration
© C. Baesens, R.S. MacKay; licensee Springer 2013
Received: 13 May 2013
Accepted: 24 August 2013
Published: 16 September 2013
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.
Here, we investigate the possibility of 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.
Alternative explanations have been proposed in [4, 5]. The first proposes a model with two physiologically plausible lumped oscillators; on exposure to opiates, they propose that the coupling becomes stochastic in time (one has to read the supporting information to  to find this), but we can generate the observed behaviour with deterministic chaos, which we consider more satisfactory than invoking stochastic coupling. The second simulates a large network of neurons (81 for expiration, 81 for inspiration), with random coupling, which may well be more realistic than just two, but their parameters are such that most oscillators in the same group fire together; indeed they call their model a “dual oscillator”, and it seems to us more simple to gain conceptual understanding by considering just two oscillators and to see whether there are dynamically plausible perturbations which could produce the observed effects.
1.1 Normal Situation
Following , we assume that the several hundred neurons involved in generating breathing rhythm can be reduced to a system of two coupled oscillators. We assume the instantaneous state can be described by a pair of phases (angles on a circle). When θ passes through a certain value , a signal for expiration is generated; when ϕ passes through a certain value , a signal for inspiration is generated. Compare “limit cycle” models, e.g. .
Under normal operation, the dynamics has an attracting 2-torus, containing an attracting periodic orbit of type (one revolution in each of θ and ϕ per period), which crosses and once per period, giving alternating inspiration and expiration.
We consider both cases. The Poincaré case is the relevant one if the expiratory oscillator is sufficiently strong that it is affected little by the inspiratory oscillator; the observed quantisation of inspiratory period may be taken to support this. On the other hand, there must in normal operation be some mutual inhibition to prevent attempted simultaneous inspiration and expiration. It could be achieved by Poincaré flow if the attracting periodic orbit avoids the intersection of the two threshold circles as in Fig. 2(a), but a stronger way would be a Cherry flow with near the repelling equilibrium, as in Fig. 2(b).
For the Cherry flow of Fig. 2(b), an appropriate surface of section is again the set . The return map then has the form of an increasing function with a jump at the image of the intersection of the stable manifold of the saddle with Σ as shown in Fig. 3(b) and (c). We choose to put the origin of Σ at the intersection with the stable manifold of the saddle so the jump corresponds to . Since Σ is a circle, the point represents the same point as , and we obtain the whole circle by taking the interval . Although the return map is discontinuous, we consider the jump to be upward so as to make the resulting map have degree one.
The slopes of the return map for a Cherry flow at the ends of the discontinuity depend on the “exponent” of the saddle, where the eigenvalues are labelled and . If , the slopes are zero; if , the slopes are infinite. More precisely, near the discontinuity (which we have agreed to put at 0), for x small positive and negative, respectively, some constants , (we ignore the special case , which typically involves a logarithmic correction). These are illustrated in Fig. 3(b) and (c).
1.2 Effect of Opiate Exposure
The effect of exposure to opiates is described in the experimental literature as suppressing the activity of the inspiratory oscillator, thus slowing it down in some way.
If there remains an attracting two-torus, then the winding ratio could cease to be locked to , but all trajectories would still have a common winding ratio. In particular, there would not be trajectories carrying out pseudo-random sequences of rotations. The sequence of numbers of expirations between successive inspirations would be a “rotation sequence”. This has a recursive definition. Firstly, there is an integer such that the numbers of expirations between successive inspirations are all either or . Secondly, either the or the occur in singletons. Thirdly, the sequence of numbers of the other between these singletons forms a rotation sequence again. For example, is part of a rotation sequence, whereas is not. Rotation sequences are very special sequences and the results of Fig. 1 speak strongly against such sequences since it shows variation from 2 to 5 expirations between successive inspirations, contradicting the rule.
Such folded attractors with strong stable foliation can easily produce deterministic chaos of the form of pseudo-random sequences of rotations, as we shall now describe.
2 Analysis of the Scenario
The result of our proposal for the effect of opiate exposure is that θ continues to increase regularly in time, but ϕ may behave in a more complicated manner. Its evolution is most simply studied by considering the return map to the surface of section Σ. This map f is the composition of an increasing degree-one map, either continuous or having an upward jump discontinuity (as in Fig. 3), followed by a continuous degree-one circle map having one decreasing interval (resulting from the perturbation with a fold) that we denote .
The slopes of f at 0 and 2π are determined by the exponent of the saddle. If , the slopes are 0; if , they are infinite, except possibly in the special case that one end of I is a or b, when the outcome depends on how compares with the exponent of the fold (which is generically 2).
A key feature of an orbit of a degree-one circle map is its “rotation number”. This is the average rate (if it exists) at which the iterates , , rotate around the circle. In our context, it gives the average ratio of numbers of inspirations to expirations. For a continuous monotone degree-one map of the circle, Poincaré proved that the limit exists and every orbit has the same rotation number. If monotonicity is broken, however, there is the possibility of more than one rotation number and even of orbits for which the average does not exist. More importantly, in this situation, there is the possibility of orbits with pseudo-random sequences of rotations per iteration. We will formalise this behaviour later in this section under the term “rotational chaos”.
For continuous bimodal circle maps, there are some nice results, such as that the set of possible rotation numbers is either a single point or a closed interval, and that existence of two periodic orbits with different rotation numbers implies existence of an invariant subset with pseudo-random sequences of rotations between those of the two periodic orbits . Another result is based on the concept of a badly ordered invariant set. An invariant set for a degree one circle map is called badly ordered if it contains three points in clockwise order whose images are not in clockwise order. Any finite badly ordered invariant set implies similar rotational chaos . We will give an example of a bimodal map for which there is rotational chaos with rotation interval .
Turning now to the case of discontinuous maps, maps of the form of region + were studied by [11–14]. As long as the jump discontinuity is upward, there is a unique rotation number, but for many of those with downward jump, the set of rotation numbers was shown to be an interval rather than a single point, points for which the rotation number does not exist were found and rotational chaos was found. We will give an example of a map in region + with rotational chaos, but the rotational chaos is attracting only when the saddle is repelling; otherwise, the rotational chaos is repelling and most trajectories go to a periodic orbit.
More promising is the region −+. Here, we make examples with attracting rotational chaos even in the case of an attracting saddle.
We will give the examples first and then some general treatment.
2.1 Continuous Bimodal Examples
Consider continuous degree-one circle maps with a single decreasing interval.
Definition Say a 1D map f maps an interval A over an interval B if f is continuous on A and there is a subinterval of A such that (cf. ).
This definition implies that the image of A covers B and possibly more, but is more precise, allowing application of the intermediate value theorem to deduce that if f maps A over B, B over C, etc., then there is a point such that , , etc.
The arcs A, B, C are mapped over each other as indicated in the transition graph of Fig. 6(b). Here, we have introduced notation + to show when trajectories pass 2π, thus making one revolution; it can be taken as indicating inspiration. The beauty of the notion of “mapping over” is that every path in the graph is taken by some trajectory. In particular, pseudo-random paths in the graph give trajectories generating any sequence of inspiration or not between successive expirations. We call it “rotational chaos”.
The downside of examples such as this is that the set exhibiting rotational chaos might not be attracting; most trajectories might do something less interesting, like converging to the attracting fixed point in the interior of arc A, which corresponds to alternating inspiration and expiration.
The condition that the critical points be pre-periodic is strong, but similar chaotic properties with an absolutely continuous probability distribution can be derived if the orbits of the critical points do not come back too close too soon in a suitable sense. The theory of this is even more technical, beginning with the Collet and Eckmann condition (see  for a survey), but the condition holds for a set of parameters of nearly full measure near the cases with pre-periodic critical points.
2.2 Discontinuous Type + Examples
Now we turn to discontinuous circle maps.
The example is very particular in that and (just as we made the orbits of critical points of bimodal maps land on unstable periodic orbits). If these conditions are not met exactly, the dynamics can be more complicated to describe but may often imply similar pseudo-random behaviour. A complete theory of the dynamics has been given if the slope of f is everywhere greater than 1 (or under a weaker topologically expanding condition) [11–14], but we will content ourselves with presentation of examples.
2.3 Discontinuous Type −+ Examples
Although drawn for the case of attracting saddle, the example works equally well with repelling saddle.
2.4 Rotational Chaos
To make precise statements about the behaviour exhibited by the examples, we introduce a definition.
Definition A circle map exhibits rotational chaos if it has an invariant subset with a semi-conjugacy to a topological Markov chain whose “translation co-cycle” is non-trivial.
We have to explain the various terms in this definition.
A topological Markov chain (TMC) is a discrete-time dynamical system based on a finite directed graph with edge set Γ, whose state space is the set of doubly infinite paths endowed with the product topology, and whose map is the shift , . It is irreducible if for every two nodes x, y in the graph there is a path from x to y.
A map is semi-conjugate to if there is a continuous surjection such that .
A co-cycle for a map is a continuous function such that for all , . It is enough to specify for all x, but traditional to define it as above because the interest is in how grows with n.
We endow any TMC arising from a degree-1 circle map f with a translation co-cycle, defined by making a choice S of lifts to ℝ of the subsets representing the vertices of Γ, and a choice of lift of f (map such that , where is ) and letting be the integer such that for x represented by γ, then . If k depends on only we say it is elementary.
A co-cycle k for f is non-trivial if there are no and continuous function b such that .
A simple cycle for a TMC is a closed path in the graph, which visits no vertex more than once.
Theorem An elementary co-cycle on an irreducible TMC is non-trivial iff there are two simple cycles with differing average.
Proof If there are two cycles (simple or not) with differing averages s, t, let N be the least common multiple of their periods. Then for one and Nt for the other, whereas if the cocycle were trivial it would be Nc for both.
In the other direction, if all simple cycles have the same average, call it c, then the same holds for any periodic cycle because it can be broken into segments between repeated vertices which can be rearranged to make a sum of simple cycles, and the value of the co-cycle is simply the sum because it is elementary. Choose a reference vertex, let there and let at the end of each path from the reference vertex. This defines b uniquely because if two paths end at the same vertex then close them back to the reference vertex by a common path to see that b was uniquely defined. Then , so the co-cycle is trivial. □
Corollary If a degree-one circle map f has a set of intervals which are mapped over each other according to an irreducible graph Γ and the graph contains two simple cycles with differing average translation then f shows rotational chaos.
2.5 Quantisation of the Time Between Inspirations
We have proposed a scenario that naturally leads to pseudo-random sequences of numbers of expirations between successive inspirations, but it remains to explain why the time between inspirations is quantised.
Our scenario consists of two coupled oscillators whose dynamics generates an attractor of the form of a torus with a fold. The attractor may be fractal but we suppose that we can still define phases θ and ϕ on it, representing the phase of the expiratory and inspiratory oscillator, respectively.
The quantisation of the time between inspirations is most simply justified if the flow is a “skew-product”, meaning that the θ dynamics is unaffected by the ϕ dynamics, with for all θ. Then the time from one expiration to the next is independent of what ϕ does, and the time between successive inspirations is closely determined by the number of times the trajectory winds in the θ direction for one revolution in ϕ. The skew-product case would give a Poincaré flow under normal operation.
The quantisation of the inspiration period is less easy to justify in the Cherry case. Indeed, if some of the pseudo-random trajectories pass close to the saddle, then they will be slowed down there and there is no reason for the time between inspirations to be close to a multiple of the average expiration period. Close to quantised behaviour might result, however, if the orbits come close to the saddle only rarely.
Thus, if , which is a mild condition on g.
On the other hand, an advantage of our scenario is that it does not automatically quantise the time between inspirations. Increasing the potassium concentration destroys the quantisation, as remarked in Fig. 1, so one wants a model that retains this possibility, as our does.
We have given a possible dynamical systems explanation of the chaotic quantal slowing down of inspiration observed under the effects of opiates in rats.
In summary, we propose that the effect of opiates is to introduce a fold into the dynamics of two coupled oscillators. This can produce pseudo-random sequences of numbers of expirations between successive inspirations. We also showed it is plausible that the times between successive inspirations remain close to multiples of an average inter-expiration period.
It will be interesting to test this proposal physiologically. In particular, could one construct the return map from experimental results such as those used to make Fig. 1? If the phases are not directly observable, one could perhaps reconstruct something equivalent from the times between inspiration or expiration. We elaborate on this last idea in Appendix A, though it is a project which would deserve a paper in itself.
A simple data analysis technique that could reveal something of the “grammar” of the expiration/inspiration sequences is described in Appendix B.
If φ is a flow on a manifold M of dimension d and is a codimension-1 submanifold transverse to the flow, then the return-time function is a function on the subset of Σ which returns. Denote the return map ; it is also . Then, given , define the map on the subset of Σ which returns at least n times, by . For generic φ, Σ, the map Φ is a embedding of into if , by analogy with Takens’ embedding theorem . Thus, given the sequences of return times for some trajectories on an attractor Λ of φ, one can reconstruct up to diffeomorphism. Furthermore, one can reconstruct the return map f from the shift on the sequences: determines a point on and determines its image.
One could attempt to apply this to the breathing data by taking Σ to correspond to expiration, assume suffices, and hence , assume that every orbit on the attractor returns to Σ infinitely often, and apply the method to the sequence of times between successive expirations, to deduce up to diffeomorphism what and the return map to it look like. In practice, however, this is unlikely to reveal much, as the time between successive expirations is close to constant.
One might propose instead to take Σ to correspond to inspiration, for which the return time is quite variable, but the submanifold corresponding to inspiration is unlikely to be transverse to the flow, as the phenomenon is that some approaches to inspiration fail to cross and have to wait for a later attempt.
If both the expiration and inspiration submanifolds (call them E and I) were transverse to the flow, one could contemplate an extension of the above idea to take the sequence of times between both inspiration and expiration events, adding symbols I or E to indicate which, e.g. . Although this representation is discontinuous at , we are expecting Λ to avoid (simultaneous expiration and inspiration). If Λ does come close to , we could make multiple charts for E and I by eliminating events that follow in too short a time, but would then have to deal with the overlap of charts, which although feasible is messy. In any case, for the present application I is unlikely to be transverse to the flow so this would produce a discontinuous representation of , so we leave the discussion here.
We call the future and the past. Then plot the points in the unit square, as t goes from m to for some m such that is not visible. If all sequences occur, then they will fill out a 2D middle Cantor set. If only a rotation sequence occurs, then is a decreasing function of (if it is periodic then only finitely many points appear). For examples like Fig. 9, the sequences fill out a sub Cantor set, revealing the associated transition graph.
Such plots have been used to study the “pruning front” conjecture .
For an example of use of a related technique to reveal a grammar in sequences of positive integers, see .
We are grateful to Jack Feldman for bringing this problem to our attention in 2005, and to Caroline Sandford for initial work on it as an undergraduate project student at the University of Warwick in 2008 (title: Generators of breathing rhythms) in which she rediscovered results of . We are grateful to a referee for suggesting we should propose a method to reveal grammar of expiration/inspiration sequences.
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