Mixing Dynamics of Heart Rate Variability

The theoretical Erlang distribution of the k –fold Poincaré return time of a mixing dynamical system is a very good fit of the experimental RR histograms of normal subjects. From this perspective, a heartbeat is emitted when the state of the attractor has returned k consecutive times to some finite region of the phase space of an abstract dynamical system that generates the RR sequence. The higher frequency, k times that of normal heartbeats, is hypothesized to be related to the synchronization of the array of pacemaker cells in the SA node. For arrhythmia patients, the RR histogram deviates from the Erlang distribution, significantly to the point that it is bimodal. In this case, the distribution can be fitted with the weighted average of an Erlang and another distribution, revealing that the heart in arrhythmia cases operates near the boundary between a mixing attractor and a more complicated one. INTRODUCTION In this chapter, we develop a statistical dynamical systems theory approach to heart dynamics. While there are many variables that can be associated with the cardiovascular system, here we only retain the RR sequence. Recall that the RR interval is the amount of time between consecutive heartbeats (R waves). In a sense, we propose corroborative evidence of a hypothesis emitted in [1] where it is argued that the RR interval is a Poincaré return time. In heart physiology, the sinus node (also called sinoatrial (SA) node) is known as the dominant cardiac pacemaker, which is under the influence of the vagal nerve. The SA node cells have the capability of selfexcitation and they behave like a large population of electrically coupled oscillators (pacemakers) with differing intrinsic frequencies [2], [3]. Hence, no single cell in the SA node serves as pacemaker, and by Michaels et al. hypothesis [2], the pacing rate is not that of the intrinsically fastest pacemaker, but it may actually be generated from a “democratic consensus” dynamical process of contribution of all cells to establish the rhythm. The SA node depolarization events are transmitted to both the antrioventricular (AV) node and the Purkinje fibers. The new impulse from the SA node discharges both the AV node and Purkinje fibers before their own threshold for self-excitation can occur in either of these [4]. Thus, the SA node actually controls the heartbeats and it is virtually always the pacemaker of the normal heart. More specifically, this chapter develops a new heart dynamics paradigm based on two hypotheses that are consistent the clinically observed fact that the RR interval is Erlang distributed for control subjects. A tentative explanation of this phenomenon has already been put forward for atrial fibrillation (AFib) patients [5]. The hypothesis was that the AV node would transform the exponentially-distributed arrivals from the many ectopic sources on the atria to an Erlang-distributed process. However, our data support Erlang distribution on normal subjects! The discrepancy can be explained on the basis that [5] failed to take into consideration the effect of digitalis, on which the patients had been put. Our paradigm, on the other hand, is rooted in recent results [9] of ergodic theory, saying that the k-fold Poincaré return time of a mixing dynamical system is Erlang distributed. The return time rate, k times faster than heart rate, is conjectured to be resulting from the synchronization of the pacemaker cells in the SA node. Under some cardiac risk conditions, the distribution becomes bimodal, which can be dynamically explained within the ergodic theory paradigm that the heart dynamical system operates near the boundary between two attractors: a “universal” mixing attractor and another one that seems to depend on the specific cardiac risks [1]. Our corroborating studies have revealed a correlation between departure from Erlang fit and abnormal heart condition. MIXING DYNAMICS IN THE HEART—A GENTLE INTRODUCTION In order to define mixing heart dynamics, we proceed from much easier physics: Consider a cylindershaped glass, standing vertically (along the z-axis) and full of water. Assume we can number the molecules as N i ,..., 2 , 1 = and assume that at every time t we can register both the positions ( ) ) ( ), ( ), ( ) ( t z t y t x t q i i i i = , N i ,..., 2 , 1 = , and the velocities ( ) ) ( ), ( ), ( ) ( t z t y t x t v i i i i    = , N i ,..., 2 , 1 = , of all molecules. In the preceding, ) (t zi is the coordinate along the vertical axis of the ith molecule, while ( ) ) ( ), ( t y t x i i are its coordinates in some horizontal plane. A “dot” notation denotes the rate of change (time-derivative.) We define the state of the ith molecule as ( ) ) ( ), ( ) ( t v t q t i i i = ω and define the state of the overall system (all molecules together with their interactions) as ( ) ) ( ),..., ( ), ( ) ( 2 1 t t t t N ω ω ω ω = . The positions are constrained to lie in the cylinder; the velocities could in theory be as fast as possible along any direction, except that on the boundary they should be tangent to the walls of the glass. The set of all possible positions and velocities is denoted by Ω , the so-called phase space or sample space of the system. If the water is still, ) (t ω would be constant, except for some microscopic thermal agitation. Here, however, on top of this we will consider larger scale motion created, for example, by agitating the liquid with a spoon. In general, we need a rule that allows us to predict the state of the system at any time in the future, knowing its previous state. Under the spoon agitation motion, the equations of the overall system could be )) ( ( ) ( t T t t ω ω = ∆ + , where t ∆ is the sampling period and ) (⋅ T , the time-shifted operator, is a nonlinear function . Another model, over a very short amount of time, takes the form )) ( ( ) ( t f dt t d ω ω = , where ) (⋅ f is in general a complicated nonlinear function. Regardless of discrete or continuous evolution, such a micro-canonical model can hardly be constructed in such a manner as to accurately anticipate the motion, but its usefulness comes if it successfully reproduces some large-scale measurable quantities, like the water temperature. Via the Boltzmann constant k , the water temperature is defined as the average kinetic energy of the molecules, where the average could be understood either as the timeaverage 2 ) ( 3 2 1 lim 2 t v m k T T ∞ → or the ensemble average ∫ Ω ) ( 2 3 2 2 ω μ d v m k . In the preceding, ) ( ω μ d is the probability of finding the state in a small “box” around Ω ∈ ω . (Precisely, ( ) ⋅ μ is the ergodic measure defined to be invariant along the motion.) It is very much in this spirit that we will conjecture existence of a micro-canonical model of the heart that reproduces some specific features associated with the ECG measurements, more specifically the histogram of the RR interval. Naturally, this dynamical model of the heart is not about water molecules, but about the hydrodynamics of the blood, the ion channels, the conductivity between the SA and AV nodes, etc. Under mild conditions, a motion that has an ergodic measure has Poincaré return, that is, if we isolate a subset A of the water glass, any molecule initially in the subset will eventually return, and it will return infinitely often, in the subset A . The time it takes to return, the so-called Poincaré return time, is of paramount importance in dynamical system theory. If indeed it can be measured, it indicates some qualitative properties of the dynamics, no matter how complicated it is, no matter our degree of confidence in the model. The equality between the time and the ensemble averages mentioned in the simple case of the kinetic energy of water molecules is a fact of ergodic theory. The motion of water molecules (to take a simple case) is said to be ergodic if it has no invariant sets. An invariant subset A is such that, if initially the state is in A , it remains in A during the rest of the motion. For example, if the water is stirred in such a way that every molecule rotates around the axis of symmetry of the cylinder, the movement is certainly not ergodic (and should there be some drops of ink in the glass, the ink and the water will not “mix.”) Unfortunately, ergodicity is inadequate to answer such questions as to whether the motion is chaotic. (It is tempting to define Heart Rate Variability as chaos in the RR sequence; however, this concept is flawed.) There is no universal definition of chaos, but here we shall follow the formal mathematical way and equate chaos to mixing. To define mixing intuitively, put a few drops of ink in the glass and stir it. If the stirring is complicated enough with both vertical and horizontal components, the ink and the water will mix and the water will have homogeneous color. To somewhat formalize the concept, in physics, a dynamical system is said to be mixing if subsets B A, (ink, water, resp.) of the phase space of the system become strongly intertwined as time goes on. Another canonical example is the Cuba libre: suppose that a glass initially contains 20% rum and 80% cola in separate regions, formalized as subsets A and B , resp. After properly stirring the glass, any region of the glass will contain approximately 20% rum. Furthermore, the stirred mixture is in a certain sense inseparable: no matter where one looks, or how small the region one looks at, one will find 80% cola and 20% rum. The formal definition of mixing is in the Appendix. It appears a daunting task to determine whether such a motion mathematically specified as )) ( ( ) ( t T t t ω ω = ∆ + is mixing, let alone to come up with the time shift operator ) (⋅ T from first principles. However, the key point is that the return time, assuming it can be measured in some way, allows us to determine whether or not a system is mixing. The mixing criterion is whether or not the return time τ , viewed as a random variable that can be sampled, follows the Erlang distribution