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Edited by: Axel Loewe, Karlsruhe Institute of Technology (KIT), Germany

Reviewed by: Nele Vandersickel, Ghent University, Belgium; Cristiana Corsi, University of Bologna, Italy

This article was submitted to Cardiac Electrophysiology, a section of the journal Frontiers in Physiology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

A quantitative framework to summarize and explain the quasi-stationary population dynamics of unstable phase singularities (PS) and wavelets in human atrial fibrillation (AF) is at present lacking. Building on recent evidence showing that the formation and destruction of PS and wavelets in AF can be represented as renewal processes, we sought to establish such a quantitative framework, which could also potentially provide insight into the mechanisms of spontaneous AF termination.

Here, we hypothesized that the observed number of PS or wavelets in AF could be governed by a common set of renewal rate constants λ_{f} (for PS or wavelet formation) and λ_{d} (PS or wavelet destruction), with steady-state population dynamics modeled as an M/M/∞ birth–death process. We further hypothesized that changes to the M/M/∞ birth–death matrix would explain spontaneous AF termination.

AF was studied in in a multimodality, multispecies study in humans, animal experimental models (rats and sheep) and Ramirez-Nattel-Courtemanche model computer simulations. We demonstrated: (i) that λ_{f} and λ_{d} can be combined in a Markov M/M/∞ process to accurately model the observed average number and population distribution of PS and wavelets in all systems at different scales of mapping; and (ii) that slowing of the rate constants λ_{f} and λ_{d} is associated with slower mixing rates of the M/M/∞ birth–death matrix, providing an explanation for spontaneous AF termination.

M/M/∞ birth–death processes provide an accurate quantitative representational architecture to characterize PS and wavelet population dynamics in AF, by providing governing equations to understand the regeneration of PS and wavelets during sustained AF, as well as providing insight into the mechanism of spontaneous AF termination.

Atrial fibrillation (AF), the most common human arrhythmia, is characterized by aperiodic and disorganized electrical activation of the atria (

Here, we aimed to develop such a parsimonious mathematical representational paradigm with the potential to explain the complex dynamics of AF. We postulated that such a representational structure, if established, should have the following properties: (i) it should be able reframe the complexity of AF dynamics in the form of simple governing equations; (ii) these equations should be able to make predictions that can be tested by experimental observation; (iii) the predictions made by the equations should be accurate in as wide a possible range of experimental AF conditions, and be invariant under transformation of scale. Such a quantitative framework could be an important advance in AF dynamics to facilitate reasoning about the underlying pathobiology of AF.

We reasoned that such a quantitative conceptual paradigm could potentially be established by understanding the properties of unstable reentrant circuits in AF. Unstable reentrant circuits, whose pivots are known as phase singularities (

In an earlier study, we demonstrated that the formation and destruction of PS in human and experimental AF could be represented as renewal processes (_{f} and λ_{d} (

Here, we seek to build on the renewal process paradigm by using the governing parameters established in our previous study, λ_{f} and λ_{d}, to develop a model to understand the dynamics of PS and wavelets at the steady state population level. Given the finding that PS formation and destruction occur at a constant rate, we reasoned that this could be achieved using an M/M/∞ birth–death process; a specific type of continuous time Markov chain in which the population dynamics of a system can be conceptualized as a multi-server queue if arrivals (in the AF case PS or wavelets) occur at a constant renewal rate (λ_{f}), and each arrival experiences immediate service and is thereby available for destruction (at a constant rate given by λ_{d}) (

Study overview. _{f}) gives a measure of how fast PS/wavelets are created, while rate constant of destruction (λ_{d}) gives a measure of how fast they are destroyed.

In a multispecies and multimodality investigation, we analyzed human, animal experimental and computer simulated AF, hypothesizing the M/M/∞ framework could provide governing equations to explain the steady state average number and population distribution of PS. Due to the intrinsic topological property of PS that connects these circuits to the free ends of wavelets (_{f} and λ_{d} for PS would be correlated with those for wavelets. Additionally we hypothesized that although λ_{f} and λ_{d} would vary with coarse graining and decreased field of view (as occurs with human catheter based mapping), that the M/M/∞ governing equations themselves would show the property of scale invariance, and continue to apply at each of these different scales of observation. Finally, we sought to use the M/M/∞ framework to understand the persistence and termination of AF, with the hypothesis that episodes of terminating AF would be distinguished by alterations in spectral properties of the eigenvalues of the birth–death matrix. We specifically reasoned that slowing of the mixing rate at which the quasi-stationary PS birth–death steady state is achieved would be associated with the spontaneous termination of AF, by allowing greater opportunity for the PS population distribution to deviate from the quasi-stationary steady state, and thereby facilitating spontaneous termination. Collectively, this study aimed to establish whether M/M/∞ birth–death processes could be used as a quantitative representational framework to understand how AF self-sustains.

This study tests the hypothesis that the steady state number of rotors and wavelets could be modeled as a Markov M/M/∞ birth–death process, characterized by rate constants λ_{f} and λ_{d}, respectively. Human, animal, and computational models of AF were used to test this hypothesis at different scales and using different mapping modalities. Methods are presented in two parts: (i)

Phase singularities formation and destruction was modeled using a birth–death process in computer simulated AF. Computer simulations were carried out on two-dimensional square grids [Courtemanche-Ramirez-Nattel cell model (^{TM} -HD grid, Abbott, IL, United States) recordings were obtained, with the HD-grid a 3–3–3 mm catheter with 16 electrodes. During these recordings, the HD grid catheter was placed on the anterior LA, or posterior LA with basket

Signal processing was performed as previously (

Data acquisition, signal processing, example phase maps and tracking.

Phase singularities are topological defects at the pivots of rotors, and at the free ends of wavelets (

A birth–death process is a continuous-time Markov chain used to represent the number of entities in a dynamical system (_{f}) and destruction (λ_{d}) are quasi-stationary (_{f}), and a similarly stable rate of destruction (with renewal rate constant λ_{d}), that occurs in the scenario that arrivals into the system are immediately available for service. In AF, we reasoned that because as soon as PS and wavelets are formed, they are immediately accessible for potential destruction, we reasoned the M/M/∞ birth–death process could potentially be used to model AF PS population dynamics. A characteristic of M/M/∞ birth–death processes is the presence of a well-defined transition matrix (

For an M/M/∞ birth–death processes, the average number

where λ_{f} and λ_{d} are the rates of PS/wavelet formation and destruction, respectively.

The second equation summarizes the PS and wavelet population distribution, which gives the steady-state probability (_{n}) of having a phase singularity or wavelet population size of

To test the hypothesis that these equations explain and summarize PS and wavelet population dynamics, λ_{f} and λ_{d} were obtained as described previously (

As the size of the mapped area would influence the number of PS and wavelet formation and destruction events captured, we hypothesized that this would in turn affect the calculated λ_{f} and λ_{d}, relevant to the clinical application of the steady state equations. To test this hypothesis of scale dependence, we investigated PS/wavelet population dynamics and renewal rates in: (i) basket catheter recordings of human persistent AF, (ii) HD-grid recordings of human persistent AF and (iii) computer simulated AF. To gain insight into this effect, decimation was also used in AF computer simulations to investigate how the spatial density of the mapped area affects PS and wavelet formation and destruction processes (

An interesting, yet incompletely understood characteristic of AF is its ability to spontaneously terminate into sinus rhythm (

We specifically reasoned that spontaneous AF termination would likely occur when steady state is reached more slowly, thereby providing a greater opportunity for the process to diverge from the quasi-steady distribution and break the cycle of PS and wavelet regeneration. To test this hypothesis, we examined the eigenvalue spectral gap in computer simulated and human AF, comparing cases where AF spontaneously terminated to cases where it sustained. The spectral gap determines the ‘_{f}, λ_{d} were estimated from epochs of spontaneous AF termination, and compared to epochs of sustained AF.

To cross-validate, several sensitivity analyses were conducted. To ensure that study findings were not due to PS/wavelet annotation method, analyses were repeated with a second PS detection method (_{f} and λ_{d}.

To calculate λ_{f} and λ_{d}, data was fitted to exponential distributions using maximum likelihood (^{2}) goodness-of-fit tests were used to assess adequacy of distributional fit. Shapiro–Wilk tests were performed to assess normality for reported variables. To assess statistical significance, ^{2} goodness-of-fit tests. Two-tailed

We tested the ability of the M/M/∞ equations to summarize and explain PS and wavelet dynamics by conducting a multimodality multispecies study comparing the average number and population distribution of PS and wavelets predicted by the M/M/∞ equations. _{f} = 0.27/ms and a PS destruction rate of λ_{d} = 0.05/ms. A worked example using the M/M/∞ equation is shown in _{f} = 0.37/ms and λ_{d} = 0.11/ms, giving an average computed number of 3.36 wavelets. Computed PS and wavelets number corresponded to the observed number shown in phase maps (^{2} values of 0.56 (

Summarizing and explaining PS and wavelet number. ^{2} = 0.99; ^{2} = 0.99; _{2} 0.56, _{2} 3.57, ^{2} = 0.89; ^{2} = 0.99; _{2} 1.66, _{2} 0.042, ^{2} = 0.99; ^{2} = 0.99; _{2} 3.32, _{2} 2.29, ^{2} = 0.99; ^{2} = 0.92; _{2} 1.37, _{2} 2.44,

All cases of human basket AF showed similar results. The M/M/∞ predicted and observed average number of PS and wavelets were linearly correlated (PS: ^{2} = 0.99; ^{2} = 0.99; ^{2} = 0.89; ^{2} = 0.99; ^{2} = 0.99; ^{2} = 0.99; ^{2} = 0.99; ^{2} = 0.92;

To investigate the relationship between PS and wavelet rates of formation and destruction, the correlation between PS/wavelet λ_{f} and λ_{d} was studied (_{f} and λ_{d} for wavelet and PS were correlated (λ_{f}: ^{2} = 0.86; _{d}: ^{2} = 0.60; _{f} and λ_{d} are a common set of rate constants.

The key M/M/∞ findings were cross validated by assessing the correlation between the predicted and observed PS/wavelet number detected using various tracking parameters (

As the steady state M/M/∞ equations assume λ_{f} and λ_{d} are stationary (temporally stable) we systematically validated the stationarity of PS and wavelet renewal rates λ_{f}/λ_{d} in all model systems (_{f} and λ_{d} from random short-duration windows converged to long-term λ_{f}/λ_{d}, implying temporal stability of these rate constants.

_{f} (_{f} also increased in larger grids (_{d} is more consistent (_{f} and λ_{d} was observed. This is likely due to the long-lasting PS located at the boundary, which is only captured by the 500 × 500 grid (_{f} and λ_{d}.

Effect of mapped field of view size. _{f}), but consistent PS destruction (λ_{d}). _{f} also increases, but λ_{d} remains consistent. _{f} increases for basket catheter data, but λ_{d} decreases. _{f} increases when mapped with basket catheter, whilst λ_{d} decreases.

To further understand the effect of scale dependence on PS formation and destruction rates, λ_{f} and λ_{d} were calculated for simultaneous basket catheter (8 × 8 electrodes) and HD-grid (4 × 4 electrodes) recordings taken during human persistent AF.

The mean number of PS [1.32, (95% CI, 1.22, 1.42)] and wavelets [0.21, (95% CI, 0.16, 0.25)] captured by the HD-grid catheter was significantly lower than mean the number of PS [2.87, (95% CI, 2.71, 3.04)] and wavelets [1.51, (95% CI, 1.38, 1.65)] captured by the basket catheter (HD-grid:

Differences in rate constants were also seen between the two mapping catheters. The mean PS λ_{f} for HD-grid recordings was 0.002/ms (95% CI, 0.001, 0.003) and mean PS λ_{d} 0.001/ms (95% CI, 0.001, 0.002). For basket catheter recordings the mean PS λ_{f}was 0.030/ms (95% CI, 0.027, 0.033) and mean PS λ_{d} 0.012/ms (95% CI, 0.010, 0.013) (_{f} and λ_{d} measured by the HD-grid was 0.006/ms (95% CI, 0.005, 0.007) and 0.036/ms (95% CI, 0.032, 0.039), respectively, and for basket catheter recordings mean λ_{f} and λ_{d} was 0.037/ms (95% CI, 0.034, 0.39) and 0.029/ms (95% CI, 0.025, 0.033), respectively (_{f} when compared to the basket catheter (HD-grid: _{d} was slightly higher for PS when measured using the HD-grid versus basket catheter (_{d} (

_{f} and λ_{d} (_{f} for wavelets increased as the mapping density decreases, whilst λ_{d} increased (

Effect of mapping density. _{f} and λ_{d} decreases, and wavelet λ_{f} increases whilst λ_{d} decreases. _{f} and λ_{d} change.

We hypothesized that we could gain insight into the mechanism of AF termination via understanding differences in the M/M/∞ birth–death matrix between AF epochs that spontaneously terminated and those that sustained. AF termination was studied in

Insights into spontaneous termination. _{f} and λ_{d} lowers in cases of spontaneous AF termination, suggesting a slowing in the rate of PS/wavelet regeneration.

The mean PS λ_{f} was 0.026/ms (95% CI, 0.023, 0.030) for spontaneous termination and 0.035/ms (95% CI, 0.031, 0.039) for sustained cases. All epochs of spontaneous human AF termination showed a statistically significant difference in PS λ_{f} when compared to sustained human AF epochs (_{d}, with spontaneous termination corresponding to a mean of 0.009/ms (95% CI, 0.008, 0.010) and 0.013/ms (95% CI, 0.012, 0.014) in sustained human AF. A statistically significant difference was also seen for PS λ_{d} between all terminating versus sustained epochs (

The mean wavelet λ_{f} was 0.032/ms (95% CI, 0.029, 0.035) and λ_{d} 0.022/ms (95% CI, 0.021, 0.024) for epochs showing spontaneous termination. In sustained AF epochs, mean wavelet λ_{f} was 0.040/ms (95% CI, 0.036, 0.044) and λ_{d} 0.027/ms (95% CI, 0.025, 0.030) (_{f} and λ_{d} between sustained and terminating cases was statistically significant (λ_{f}: _{d}:

It can be noted that although PS and wavelet λ_{f} and λ_{d} lowered in cases of AF termination, there was no statistically significant difference between the average number of PS and wavelets in sustained versus terminating episodes of AF (

We further reasoned that rather than the average number itself, it may be deviations from the steady state population distribution that differentiates terminating episodes. When analyzing the eigenvalue spectrum of the Markov birth–death transition matrix in computer simulated AF, the second largest eigenvalue modulus (SLEM) of termination cases [mean 0.9839 (95% CI, 0.9668, 1.000)] was consistently higher than for sustained AF cases [mean 0.9771 (95% CI, 0.9558, 0.9919)] (

Similar results were seen for human AF. The SLEM of termination cases [mean 0.8815 (95% CI, 0.7746, 0.9885)] was consistently higher than for sustained AF cases [mean 0.7826 (95% CI, 0.6279, 0.9373)] (

Despite a century of research, the complex and turbulent dynamics of AF remain incompletely understood. The objective of the current study was to develop an ontologically parsimonious mathematical representation of AF dynamics. We reasoned that because of the intrinsically disaggregated nature of wave propagation in AF across the atrial myocardium, over time PS and wavelet formation and destruction inter-event times would converge to exponential distributions associated with stable rate constants λ_{f} and λ_{d}. By analogy with queueing systems in probability theory, we hypothesized that λ_{f} and λ_{d} would thus be fundamental rate parameters that could be combined in an M/M/∞ birth–death model to summarize and understand AF dynamics.

We investigated this hypothesis in a multispecies, multimodality study of AF and demonstrated that:

λ_{f} and λ_{d} can be combined using an M/M/∞ birth death process to accurately model the quasi-stationary population dynamics of PS and wavelets in AF.

Wavelet and PS λ_{f} and λ_{d} are highly correlated in keeping with the topological connection between these two forms of propagation.

Rate constants λ_{f} and λ_{d} are determined by the field of view of AF mapping, both in terms of the size of the mapped field and the scale effect of coarse graining the system via mapping AF at reduced electrode densities, as is performed in clinical settings. However, the operation of the M/M/∞ birth–death process itself is invariant under scale transformation.

Rate constants λ_{f} and λ_{d} are slowed in AF cases with spontaneous termination, leading to a reduced spectral gap and slower mixing rate of the M/M/∞ birth–death Markov matrix, potentially providing an explanation for spontaneous AF termination.

These findings suggest the M/M/∞ birth–death process can be used to provide a new quantitative representational framework to understand PS and wavelet population dynamics in AF. We replicated this finding in a range of AF models in different species, using different mapping modalities and densities, to provide evidence of generality. It should be noted an exponential distribution of PS lifetimes has been a consistent observation in both AF and VF (_{f} and λ_{d} measured at any particular observational level give rise to dynamics internally consistent with an M/M/∞ birth–death process operating at the same level.

In our previous study, we demonstrated that PS formation and destruction could be represented as renewal processes, characterized by rate constants λ_{f} and λ_{d} (_{f} and λ_{d}, the previous study did not use these rate constants to develop governing mathematical equations to quantify the population dynamics of PS and wavelets, or study how λ_{f} and λ_{d} changes under various conditions such as varying mapped fields of view, and in episodes of sustained versus terminating AF. Collectively, the results of the current study extends our previous work by finding a way to utilize λ_{f} and λ_{d} in an M/M/∞ birth–death process, hence providing a new quantitative representational framework to help further understand AF dynamics.

In considering the results of the current study, a number of important issues are worthy of consideration. The first is whether or not the results could have arisen purely by methodological experimental error in signal acquisition, or processing. Several studies have suggested a potential for error in electrogram-based detection of PS with basket catheters (_{f} and λ_{d}.

However, several factors would suggest that experimental error alone would be insufficient explain the consistent observation of exponential distributions and the accuracy of the birth–death equations. The first is that an exponential distribution of phase singularity and rotor lifetimes is a consistent finding throughout the history of AF research. The Jalifé laboratory was the first to report an exponential shaped distribution of PS lifetimes in the classic cholinergic sheep model of AF, where PS detection was based on optically mapped data (

A second important factor to consider in assessing the M/M/∞ framework is to understand that this representational architecture is to understand its logical derivation from the fundamental properties of AF. The fundamental property that separates AF from other atrial arrhythmias is disaggregated electrical activity in the atrium, in both space and time. The intrinsic turbulence of AF as an arrhythmia would suggest that the formation and destruction of PS should be statistically independent events. The exponential distribution of PS inter-formation times, and PS lifetimes, arises as a consequence of this statistical independence, and is a standard finding of probability theory (

A third factor in understanding the findings of the current paper is the importance of considering the scale of observation. In our study, we have shown that the M/M/∞ representational architecture can apply at different scales of observation, created by decimation and coarse-graining the observation of AF. We have shown that the rate constants λ_{f} and λ_{d} are different at each of these scales, but that the equations still apply in each experimental system. This finding has particular relevance to understanding basket catheter data acquired in human, and something that has not previously been considered as an issue before. It is clear based on simulation studies that the under-sampling of AF caused by catheter-based sampling of atrial activity in fibrillation will clearly lead to differences in the nature of PS that are detected (

The M/M/∞ birth–death process is a scientifically attractive conceptual representation of AF dynamics. The equations of the M/M/∞ process are well characterized in probability theory, providing an extensive theoretical framework. Additionally, the framework is also parsimonious, in that the complexity of AF dynamics is shown to be compactly represented by two simple parameters λ_{f} and λ_{d}. This simplicity likely underlies why the M/M/∞ paradigm appears to accurately model PS and wavelet dynamics in a range of model systems. Although these parameters adjust according to the field of view, density of mapping and AF physiology, this consistent model accuracy suggests the M/M/∞ concept itself has the property of invariance under scale.

A key reason the M/M/∞ conceptual paradigm could be quite powerful is that it potentially allows challenging problems within the field to be reframed in quantitative terms, facilitating reasoning about the underlying biology. One such example discussed here is spontaneous termination. The spontaneous termination of AF is a common clinical occurrence that is currently incompletely understood (

An important point to recognize is that we would consider the M/M/∞ birth death process a model, or quantitative representation for the dynamics of AF. Although this model appears stable insofar as it provides accurate representations of PS and wavelet behavior, we suggest that this model could potentially be considered as the starting point for a new way of investigating and understanding AF. Much in the way that many scientific conceptual paradigms have been improved by explaining target phenomena outside the frame of reference of particular model abstractions, it is likely that deviations from the M/M/∞ model will provide additional mechanistic and clinical insights, allowing the model to be refined and adapted to particular constraints or scenarios beyond those directly examined. Having said this, to date no such counterexamples have been identified. It is possible the equations may emerge to be considered as idealized governing laws or boundary conditions in AF dynamics.

An interesting issue to consider is why these simple rate constants repeatedly arise. At its core, AF can be considered as a form of chaos, with intrinsically disaggregated and turbulent dynamical behavior arising as a consequence of the intrinsic non-linearity of the atrial myocardium. This non-linearity arises because of the inhomogeneity of individual excitability of atrial myocytes along with the network effects of inhomogeneity due to variability of micro-architecture, and substrate fibrosis. This functional and structural inhomogeneity can be considered as the fundamental substrate for disorganized behavior. The key reason the M/M/∞ pattern repeatedly emerges may be due to the intrinsically turbulent and aperiodic nature of electrical wave propagation that is the defining feature of AF.

Our findings may contribute to the understanding of the multiple wavelet and rotor theories. The multiple wavelet theory postulates that AF self-sustains by a random process of wavelets moving around the atrium (

The M/M/∞ birth–death process model also makes a potentially important contribution to the rotor theory. According to this theory, AF is sustained by rapid local re-entrant drivers, with new wavebreaks forming PS at locations of anatomical and functional conduction block (

The study findings may have important mechanistic and clinical implications. Mechanistically, the M/M/∞ birth–death process provides compact governing equations to summarize and predict the PS and wavelet population dynamics, providing a novel conceptual framework to understand the underlying mechanisms sustaining AF. This could be used to understand the effects of different AF disease pathophysiology, local microarchitectural effects, or disease substrates, or the effect of potential treatments. It is likely each of the above would lead to alterations in the M/M/∞ constants as a way of mediating their particular effects. The universality of the M/M/∞ constants provides a novel way for the effects of these changes under different experimental conditions to have their overall effect on fibrillatory dynamics precisely quantified. This new conceptual paradigm could potentially allow similarities and differences between these conditions to be understood much more clearly. Clinically, it is clear the M/M/∞ equations are able to model PS and wavelet formation when measured at the typical electrode densities and fields of view during catheter level observation. This would enable the same approach to be applied to compare AF from different patient populations, or patients with different burdens of disease, provided mapping is consistent. The association of these rate constants with the spontaneous termination of AF would suggest that measurement in clinical settings could provide insight into the likely persistence of AF. This may provide new ways to classify AF patients beyond the binary paroxysmal and persistent paradigm. It is likely that although the rate constants were temporally stable over the periods of experimental observation, that changes could occur in longer term follow-up. Longitudinal assessment could potentially provide new physiological approaches to AF classification. In clinical ablation, real-time measurement of the rate constants could potentially be combined with existing or novel ablation strategies to determine treatment effect. If rate slowing is observed, it could imply that the effect of a particular ablation strategy may be having impact on overall dynamics. An additional intriguing possibility is combining local measurement of these rate constants with strategies for detection of potential drivers, potentially providing a means of localization of key regions sustaining AF within the atrium.

As this study primarily investigates PS and wavelet dynamics in AF, the role of other potentially important adjunctive theories such as endo-epicardial dissociation and focal discharges could not be addressed with the experimental data available to us. Our study is based upon the notion of phase transformation, and we acknowledge that while this is a widely utilized approach, it is not the only approach to understand AF dynamics, and that some have investigators have identified that phase singularity identification may be unable to distinguish scenarios of conduction block and rotational activation (

However, applying the framework presented to such scenarios may be an important area for future work to provide new insights into the AF mechanism. An important consideration in applying the findings of the current study is that we make no claim that this approach is superior to other approaches or conceptual theories of AF, only that this approach may potentially provide a quantitative framework to quantify the dynamics of various pattern phenomena (PS, wavefronts), that are observed during AF.

Future studies are required to address the relationship between M/M/∞ equations and different clinical substrates of AF. Features that could be examined include the relationship between these equations and variant ablation strategies, pulmonary vein behavior, voltage characterization in the atrium, as well as spatial characterization of the rate constants throughout the human atrium (

M/M/∞ birth–death processes provide a novel quantitative representational framework to conceptualize and understand PS and wavelet population dynamics in AF. This conceptual paradigm has been shown to apply in all forms of AF studied, at a variety of different scales and densities of mapping, providing opportunities for clinical application. Finally, the spectral properties of the birth–death matrix potentially identify a new way to understand the process of AF termination, suggesting termination of AF may occur due to differences in the mixing rate of the M/M/∞ birth–death matrix. M/M/∞ birth–death processes may thus potentially provide a powerful representational architecture to gain insight into the pathobiology of AF.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

The studies involving human participants were reviewed and approved by Southern Adelaide Local Health Network Human Research Ethics Committee. The patients/participants provided their written informed consent to participate in this study. The animal study was reviewed and approved by South Australian Health and Medical Research Animal Ethics Committee.

DD contributed to the analysis and manuscript draft. EJ contributed to the analysis. MA contributed to the modeling and editorial input. AL contributed to the data collection. JQ contributed to the data collection. KT contributed to the data collection and editorial input. PK, CM, and SW contributed to the data collection and editorial input. RC and MN contributed to the editorial input and computer code. SN contributed to the editorial input. AM contributed to the data collection and editorial input. AG contributed to the concept, study design, data collection, and editorial supervision. All authors contributed to the article and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The Supplementary Material for this article can be found online at: