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COULD NETWORK SPINAL ANALYSIS HELP US UNDERSTAND THE DYNAMICS OF THE SINO-ATRIAL NODE?
By E. Jonckheere
Department of Electrical Engineering-Systems
University of Southern California
Los Angeles, CA 90089-2563
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This new direction in our research activity was initiated by an e-mail message we received from Dr. A. Hiebert from Vancouver, Canada, informing us that he had observed some strange Heart Rate Variability (HRV) discrepancies between the readings before and after NSA. His message appeared credible enough to prompt us to pay him a visit and, upon reviewing the data in his office, there were no doubts left in our mind that NSA does affect what the HeartMath Institute refers to as the “coherence ratio.”
On an independent but parallel track, thanks to the public relation skills of Dr. Ray Gin, we were brought in contact with Dr. W. Stuppy, a prominent gastroenterologist, whose interest in the heart stems from the sometimes difficult differential diagnosis between GI and cardiac problems. It is the convergence of these two research tracks that has led us to formulate the question as to whether NSA could be a helper in understanding cardiac dynamics.
Our research effort soon focused on the RR interval, also called the inter-beat time, which is controlled by the sino-atrial (SA) node. Indeed, through its connection to the autonomic nervous system, it is probably that single cardiac parameter that responds best to NSA care. It is also the cardiac parameter the most likely to provide a reading on various stress disorders. The latter is the “heart as a sensor” paradigm of Dr. Stuppy.
From my personal point of view, and that of my students, our interest in the RR interval stems from the fact that it is the cardiac time series the most likely to be amenable to dynamical analysis. Our conjecture had indeed been that the systoles are triggered when the state of the attractor of the sino-atrial (SA) node passes through some subset of the state of the attractor. This is the famous Poincaré return problem (not to be confused with the Poincaré section problem). Using the phraseology of dynamical system theory, the inter-beat interval would be referred to as “Poincaré return time.” It is only very recently that mathematicians, especially at the University of Southern California, have managed to derive the statistics of the Poincaré return time. We immediately thought that this would put us in a cutting edge position if we were to approach the inter-beat interval as a Poincaré return time. Pouring through the 600 Megabytes data supplied to us by Dr. Stuppy, our initial investigation came as a disappointment, as the systoles did not appear to be a Poisson arrival process as the Poincaré return time paradigm was predicting. It took us a little bit of research to realize that the histogram of the RR interval matches, for some patients, the so-called k-fold Erlang distribution, which is known to be the statistics of the k-fold Poisson inter-arrival time: The internal dynamics of the SA node is a mixing process, which is known to have a Poisson single return, but it takes k returns to initiate a heartbeat.
From this austere mathematical point of view, the internal clock of the SA node appears faster than the heartbeats and there are about k SA clock ticks within each RR interval. Of course, the big problem is to find a statistical estimate of k and figure out whether it is consistent with the known fact that the SA node is the fastest oscillator in the heart.
Where does NSA come into play? To understand a complicated dynamical system, it is very instructive to study its bifurcations. This is what mathematicians call “global analysis.” It turns out that the early changes observed in the coherence ratio as a result of NSA care have all of the attributes of a bifurcation. During NSA care, the RR interval goes from an erratic behavior to a sinusoidal pattern. The correlation properties of the RR interval transit from short term to long-range dependency, as shown by the following diagram:
Mathematically, our conjecture is that this is a chaotic to quasi-periodic bifurcation. Another change is the decrease in heart rate that accompanies NSA, indicating some arasympathetic
3.2) beats±response. Choosing 95% confidence in a population of 45 cases, the mean heart rate is (73.7 2.8) beats per min. after NSA. But more dramatic is the±per min. Before NSA and it drops to (69.1 change in the heartbeat pattern as seen from the histograms of the “low coherence ratio” and the “high coherence ratio.” Again, taking 95% confidence in the same population, the mean of the low coherence 8.3) % after NSA. The mean of the high coherence±9.8) % before NSA and it drops to (28±ratio is (61.2 11.4) % after NSA. This is illustrated in the±6.9) % to (45.0±ratio, on the other hand, increases from (26.8 following diagrams:
Histogram of High coherence ratio
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coherence ratio on a 1:100 scale
before entrainment
after entrainment
Histogram of Low coherence ratio
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coherence ratio in a 1:100 scale
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after entrainment
As a word of caution, it should be stressed that this is only a preliminary study. In the future, the data might have to be arranged in a better way. For example, the population contains research subjects of many ages, while it is known that HRV decreases with age. A more accurate study should involve a population of patients of the same age group. Another issue that deserves further investigation is the U-shape of the coherence ratio histograms, a departure from the “bell shaped” curve typical of a Gauss distribution. Nevertheless, this preliminary study clearly indicates that the cardiovascular response to NSA appears worth pursuing. The upshot is that, with NSA able to take the heart through a bifurcation in a simple and noninvasive way, a databank of transitions could easily be built, and this in turn could reveal dynamical features that would otherwise be difficult to pin down.
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