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The publisher's final edited version of this article is available at Phys Rev ResAtrial fibrillation (AF) is the most common cardiac arrhytmia, characterized by the chaotic motion of electrical wavefronts in the atria. In clinical practice, AF is classified under two primary categories: paroxysmal AF, short intermittent episodes separated by periods of normal electrical activity; and persistent AF, longer uninterrupted episodes of chaotic electrical activity. However, the precise reasons why AF in a given patient is paroxysmal or persistent is poorly understood. Recently, we have introduced the percolation-based Christensen-Manani-Peters (CMP) model of AF which naturally exhibits both paroxysmal and persistent AF, but precisely how these differences emerge in the model is unclear. In this paper, we dissect the CMP model to identify the cause of these different AF classifications. Starting from a mean-field model where we describe AF as a simple birth-death process, we add layers of complexity to the model and show that persistent AF arises from reentrant circuits which exhibit an asymmetry in their probability of activation relative to deactivation. As a result, different simulations generated at identical model parameters can exhibit fibrillatory episodes spanning several orders of magnitude from a few seconds to months. These findings demonstrate that diverse, complex fibrillatory dynamics can emerge from very simple dynamics in models of AF.
Atrial fibrillation (AF) is the most common cardiac arrhythmia with a growing prevalence worldwide [1]. It is characterized by the rapid, irregular beating of the atria, caused by the chaotic motion of electrical wavefronts. This lack of coordinated contraction may allow blood to clot, making AF the leading cause of ischaemic stroke in people over 75 years of age [2].
Despite over 100 years of extensive research, the mechanisms underlying the initiation and maintenance of AF are still poorly understood [3–8]. There are numerous controversies and conflicts in AF research, primary of which is the question of whether AF is driven and sustained by local (spatially fixed) sources of new fibrillatory waves, or whether AF is self-sustaining from the interaction and fragmentation of multiple meandering electrical wavelets in the atria [4,6,8]. Although this dispute is yet to be resolved, recent evidence appears to strengthen the case for local drivers as the primary mechanism of AF [5,9–18].
Questions concerning the underlying mechanism of AF are of particular importance because they inform potential treatment strategies. Historically, treatment for AF has focused on mitigating potential symptoms and lowering the risk of stroke through the use of rate control, and anti-arrhythmic drugs [19]. However, these treatments do not “cure” AF. Surgical ablation strategies have been developed to destroy, or isolate, the regions of atrial muscle thought to be responsible for initiating and sustaining AF [8]. If local drivers are responsible for AF, then ablating the focus of these drivers may terminate and prevent AF. If meandering wavelets underlie AF, then ablation strategies which minimise the space wavelets can move into may be preferable. Although the leading ablation strategy, pulmonary vein isolation [20], has a success rate of around 60%, ablation still fails in a large subset of patients and AF reoccurs in many patients who were initially free of AF after surgery.
One of the key factors determining the likelihood of ablation success is the fraction of time a patient spends in AF [8]. Clinically, AF is defined as paroxysmal if episodes are short and self-terminating. Conversely, long, uninterrupted AF episodes are referred to as persistent. In general, patients are much more likely to be free of AF after ablative treatment if AF is paroxysmal. The success rate is around 60% for paroxysmal AF while it is 40% for persistent AF after a three year follow-up [21]. Recurrence rates are also significantly higher for persistent AF after an initially successful treatment. However, why a patient exhibits paroxysmal or persistent AF is unclear. In many cases paroxysmal AF will develop into persistent AF, but reversion to paroxysmal AF after years of persistent AF has also been observed [22]. Additionally, of the patients who initially exhibit paroxysmal AF, many develop persistent AF rapidly (after a few months), but others do not progress at all over several years [23].
The progression of AF from paroxysmal to persistent is often associated with the idea that “AF begets AF,” most notably in the goat model [24], but also with some evidence in human AF [25]. During AF, the atria undergo electrophysiological and structural changes which promote the progression of AF. Among these changes, the accumulation of fibrosis is a key factor in determining a patient’s susceptibility to AF [26–28]. Fibrosis is also critical for the formation of reentrant circuits that drive AF [14,16,27]. The emergence of a reentrant circuit begins when the regular propagation of electrical wavefronts is disrupted by unidirectional blocks. These blocks leave an opening for the conduction to reenter back from adjacent muscle fibers [14,29]. When the atria accumulate fibrosis, the distribution of gap junctions between fibers becomes highly anisotropic, that is, adjacent fibers become less and less coupled. In this scenario, the reentering conduction is less likely to be obstructed by refractory atrial muscle cells (myocites), finding the appropriate conditions for initiating a spatially stable circuital conduction (i.e., a reentrant circuit) which drives AF. However, the relationship between the absolute fibrosis burden in the atria and the persistence of AF is not clear—two patients with an equivalent fibrosis burden may have drastically different heart rhythms (e.g., sinus rhythm versus paroxysmal AF versus persistent AF) [23].
In this paper, our aim is to better understand the relationship between AF persistence and the atrial microstructure using computational modeling. Computer models are a well established tool in cardiac electrophysiology, allowing for a range of experimental investigations that are not possible in a clinical, or laboratory setting. There are a wide variety of model types pitched at different scales and levels of complexity [30]. Highly detailed, biophysical models focus on precisely modeling the exchange of ions across cardiomyocyte gap junctions to study the propagation of action potentials across topologically realistic cardiac tissue. However, the resolution of these models is often not ideal and they typically assume continuous cardiac tissue. Conversely, simplified discrete models focus on understanding the microstructure of cardiac tissue and how this effects the propagation of electrical wavefronts. The former are typically preferable when studying what effect a prospective drug might have on AF [30,31], whereas the latter are most often used to study the effect of discontinuous tissue that might arise from the accumulation of fibrosis [29,32,33]. The latter also have the benefit that their simplicity allows for much larger simulations suited to statistical analysis [18,34,35], both in the duration of individual simulations and the resolution of phase spaces which can be generated.
Previously, we have introduced the Christensen-Manani-Peters (CMP) model of AF, a simple percolation-based model that investigates how the formation of reentrant circuits is dependent on the decoupling of neighboring muscle fibers, through the action of fibrosis or otherwise [29]. The model is not a fully realistic representation of the atria and it does not consider the precise evolution and propagation of action potentials across the atrial tissue. However, the model effectively demonstrates from basic principles how reentrant circuits can form if fibrosis accumulates in sufficient quantities in a given local area. Additionally, adaptations of the CMP model to 3D [18] and to a realistic atrial topology based on a sheep heart [36,37] have been successful at explaining a number of key clinical results and have generated a number of new hypotheses. This includes the distribution of reentrant circuits in the atria, notably in the pulmonary vein sleeves and the atrial appendages, the appearance of reentrant circuits as both reentrant and focal sources, and the increased probability of ablation failure as AF becomes more persistent. Machine learning has been applied to the model to test prospective methods for automated reentrant circuit detection from electrogram data [34], and other models inspired by the CMP approach have been used to study the heart rhythm of patients following a heart transplant [33].
Consistent with clinical knowledge, the CMP model has shown that two tissues with the same total fibrosis burden may exhibit very different forms of AF [38]—at the same level of coupling, different simulations may exhibit sinus rhythm, paroxysmal AF, persistent AF, or persistent AF before reverting to paroxysmal AF, see Sec. IV. This is because the formation of reentrant circuits appears to be dependent on the local distribution of fibrosis, not the total fibrosis burden across the atria [38]—this is in line with other computational studies on the effect of fibrosis on AF persistence [39]. Despite these intriguing results, it is so far unclear how the variability in AF persistence arises from the specific processes taking place at the microscopic scale in the CMP model. Hence, the aim of this paper is to dissect the CMP model into its constituent parts to understand which parts of the model microstructure are responsible for the progression from paroxysmal to persistent AF.
A detailed overview of the CMP model will be given in Sec. II, however, the key constituent elements include the lattice representing the atrial tissue, nodes representing individual muscle cells (or a block of cells), locations susceptible to unidirectional conduction block (where the propagating signal has a small probability of extinguishing), and lattice bonds representing the electrical connections between neighboring nodes. The reentrant circuits that form in the CMP model are spatially stable, but temporally intermittent—they can turn on and off as a result of local conduction blocks. This has similarities to the self-regenerating renewal process proposed by others to explain cardiac fibrillation, where fibrillation is driven by the continuous birth and death of temporally intermittent drivers [40].
To dissect the CMP model, we first remove all spatial elements of the model. We do this by deriving a mean-field (MF) model where AF is described by a set of particles, representing critical structures which, when active, correspond to reentrant circuits, evolving as a simple birth-death process. Our results indicate that the MF model significantly underestimates the probability of inducing AF relative to the CMP model, and that the MF model does not explain the emergence of persistent AF.
At a second level of abstraction, we reintroduce the spatial components of the model, but carefully control the reentrant circuits that form by inhibiting the interaction of multiple successive conduction blocks (within the same activation cycle). Like the MF model, this controlled version of the CMP model (cCMP) also underestimates the probability of inducing AF and the time in AF. However, the spatial elements of the cCMP model do not appear to make a difference to the absolute time spent in AF relative to the MF model. Only very small differences in the time in AF are observed between the MF and cCMP models, explained by small differences in the duration of individual fibrillatory events.
Finally, we show that the difference in the probability of inducing AF and the persistence of AF between the cCMP and CMP models can be explained by a series of complex reentrant circuits that exhibit an assymetry between the probability of activating and deactivating. These circuits have a special property that they require fewer successive conduction blocks to initiate fibrillation than are needed to terminate fibrillation. We also demonstrate that in some cases several of these structures are coupled together such that the termination of one reentrant circuit immediately activates a dormant neighboring structure. These mechanisms result in a spectrum of individual fibrillatory events spanning several orders of magnitude from seconds to months.
In the remainder of the paper, we outline the CMP model and review key results including previous work on the persistence of AF. Subsequently, we introduce the MF model and the cCMP model and explain why both these models underestimate the time spent in AF and the persistence of AF relative to the original CMP model. Finally, we put the CMP model and our results into a wider context and discuss their potential clinical impact, the limitations of our approach, and outline proposals for future work.
The atrial muscle consists of tubiform cells (myocytes) of length Δx′ ≈ 100 μm and diameter Δy′ = Δz′ ≈ 20 μm [41,42]. Myocytes are mainly connected longitudinally, composing discrete fibers that sporadically connect transversally. The Christensen-Manani-Peters (CMP) model condenses this branching network of anisotropic cells into an L × L square lattice of nodes [29]. A node represents a single (or multiple) atrial cell(s). Nodes are longitudinally connected to their neighbours with probability ν‖ = 1 and transversally with probability 0 ⩽ ν⊥ ⩽ 1. This creates long arrangements of nodes, mimicking the protracted, interlaced fibers in the atrium. This simplified representation of the myocardial architecture captures the anisotropic distribution of gap junctions [41]. Furthermore, it reproduces the dynamics of electrical impulses which mainly propagate longitudinally (along single muscle fibers) rather than transversally (across multiple fibers) [34]. A cylindrical topology is obtained by applying open boundary conditions longitudinally and periodic boundary conditions transversally.
Nodes follow a well defined electrical cycle characterized by three different states: resting (a node that can be excited), excited, or refractory (after exciting, the node cannot be excited for the next τ time steps). This course mimics the membrane potential of real myocardial cells. At a given time t, an excited node prompts the neighboring resting nodes to become excited at time t + 1. An excited node at time t enters into a refractory state at time t + 1. The duration of the refractory period is τ time steps; see Fig. 1 .
(a) Propagation of the wave of excitation across a small region of the CMP lattice. Nodes are connected longitudinally with probability ν‖ = 1 and transversally with probability 0 ⩽ ν⊥ ⩽ 1. Excited nodes (white squares) continue the propagation of the wave-front by activating their neighboring resting nodes (black squares) before entering into a refractory state (grayscale squares with yellow borders) for the next τ time steps. Depending on the architecture of the region, the excitation can proceed forward, backward and across fibers. (b) The full progression of a node through the three states of the electrical cycle: resting (black), excited (white), and refractory (grayscale with yellow borders).
In the CMP model, a fraction δ of nodes are susceptible to conduction block. These nodes are identified at the beginning of a simulation and are fixed in space. The probability that nodes that are susceptible to conduction block fail to excite is arbitrarily set to ϵ = 0.05; the effect of varying this parameter is discussed in Sec. II C. This probability of failure refers to the probability that a node susceptible to conduction block will not excite when prompted to do so by a neighboring active node. This leaves us with a very simple framework in which the fraction of transversal connections, ν⊥, and the fraction of nodes that are susceptible to conduction block, δ, serve as control parameters. For simplicity, we set δ = 0.01 and examine how the behavior of the system varies with ν⊥. The effect of changing δ is demonstrated in Sec. IV and has been investigated in Ref. [43].
The pacemaker (sinus node) is placed on the left side of the two-dimensional (2D) sheet and nodes lying on this edge regularly excite every T time steps. The excitation propagates as a planar wavefront, mimicking the coordinated contraction of the real atrial muscle. The parameters of the CMP model reflect clinical observations of real human atrial tissues [14,41,42,45–47]. Clinical measurements are translated into model parameters, followed by a coarse-graining procedure leading to a square lattice of size L = 200 nodes, pacemaker period of T = 220 time steps, and refractory period of a node of τ = 50 time steps. A single time step in the model corresponds to approximately 3 ms such that T = 660 ms and τ = 150 ms. This refractory period is relatively short and corresponds to what may be seen clinically during burst pacing. The dynamics of the model are maintained under changes of τ, but the transition from sinus rhythm to fibrillation takes place at a different point in the coupling phase space. The longer (shorter) the refractory period, τ, the smaller (larger) the coupling value, ν⊥, needs to be to induce AF [43].
The CMP model reveals that reentrant circuits may emerge due to a combination of the electrical signal propagating on the branching structure of a heart muscle network, the three-state dynamics of nodes, and the occurrence of nodes susceptible to unidirectional conduction block. These latter nodes may fail to excite in response to an excited neighbor with small probability ϵ, stopping the regular propagation of the wavefront [29]. The wave of excitation proceeds forwards in the adjacent fiber until it reaches a transversal connection, leaking back through the fiber in which conduction has been previously blocked. For reentrant circuits to emerge, the segment between the reentry point and the node that has previously failed to excite must be long enough to prevent the backward propagating wave from being stopped by unresponsive refractory nodes. This happens when the probability of transverse connections decreases, for example, due to fibrosis. In the CMP model the formation of reentrant circuits triggers AF. These activities survive until the circuital motion of the wavefront is annihilated by a subsequent conduction block occurring within the path of the circuit (i.e., self-termination) or by other waves spreading from the neighboring regions; see Fig. 2 . For full activation maps see Ref. [29]; snapshots are shown in Appendix D with accompanying videos.
The formation of a reentrant circuit in the CMP model. The node that is susceptible to conduction block is marked by a red square. (a) An incoming planar wavefront (green arrows) reaches the susceptible node. (b) The node fails to excite (red cross), blocking the progression of the wavefront in the lower fiber. The wavefront advances in the upper fiber, reaching the node with a transversal connection to the lower fiber. (c) At this point, the wavefront spreads both longitudinally and transversally, initiating a retrograde propagation through the lower fiber. (d) If the path denoted by the black segment includes at least τ/2 nodes, then the reentering wavefront will not encounter refractory nodes while propagating backward in the lower fiber. This establishes a structural (i.e., spatially stable) reentrant circuit in the region surrounded by the blue rectangular box. When the conduction blocking node fails to fire again, the reentrant circuit is terminated. The full evolution of this critical structure is shown in the Supplemental Material with an accompanying video [44].
Note, in the CMP model nodes are coupled with probability ν⊥ across the whole tissue. However, in the real atrium only a small patch of fibrosis may be necessary to decouple fibers and induce a reentrant circuit. Such small patches of fibrosis may be too small to see using current MRI technologies [48], inhibiting effective treatment.
The CMP model allows us to analytically compute the risk of developing AF with respect to the fraction of transversal connections ν⊥, as shown in Ref. [29]. The risk is defined as the likelihood that the L × L grid has at least one region that can host a simple reentrant circuit. The probability of having at least one transversal link on a given node is
p ν ⊥ = 1 − ( 1 − ν ⊥ ) 2 .Let ℓ be the distance (in number of nodes) between a node that is susceptible to conduction block and the first node to the right which has at least one transversal connection. By making use of Eq. (1), we find that the probability of ℓ being equal to k nodes is
ℙ ( ℓ = k ) = ( 1 − p ν ⊥ ) k p ν ⊥ .A given region cannot sustain a reentrant circuit if ℓ is strictly smaller than τ/2; see Fig. 2 . The likelihood of this event can be calculated by summing over the probabilities of ℓ from 0 to τ/2 – 1,
ℙ ( ℓ < τ / 2 ) = ∑ j = 0 τ / 2 − 1 ( 1 − p ν ⊥ ) j p ν ⊥ = 1 − ( 1 − ν ⊥ ) τ .Because the average number of nodes that are susceptible to conduction block is δL 2 , the risk, R, of having at least one region that can host a reentrant circuit is the complementary of the probability that the segments departing from these nodes are shorter than τ/2,
R = 1 − [ ℙ ( ℓ < τ / 2 ) ] δ L 2 = 1 − [ 1 − ( 1 − ν ⊥ ) τ ] δ L 2 .Equations (1)–(4) have been derived in Ref. [29]. Equation (4) provides a simple analytical tool to estimate the risk of developing AF in the CMP model. The result indicates that the risk of AF increases as the tissue becomes more decoupled/fibrotic, in agreement with the current clinical understanding [49]. Likewise, the theory predicts that the risk of fibrillation increases as the size of the atrial tissue increases, in agreement with clinical practice where left atrial volume is used as a predictor of the risk of developing AF [50]. The theoretical analysis presented here has additional value in that we can predict how the model will change if the rules or parameters are changed, allowing for a comparison with similar computational models of AF. This is discussed in detail in Sec. V.
This theoretical result builds on the assumption that reentrant circuits form from the failure of a single conduction blocking node. However, this assumption does not account for all instances in which AF is triggered in the model. For instance, the probability of triggering a reentrant circuit varies across the lattice depending on the architecture of the hosting region; see Fig. 3 . Notably, some reentrant circuits may only activate if two nodes susceptible to conduction block fail successively (i.e., in a single activation cycle).
Critical structures in the CMP model. The black segment on top of each structure represents the minimum distance (in number of nodes) between the relevant conduction blocking node (red squared border) and the first regular node to the right which has at least one transversal connection for the structure to sustain a reentrant circuit. The wavefront direction is indicated by the green arrows. (a, b) Simple critical structures are triggered by a single block of the incoming planar wavefront originating from sinus rhythm. These structures might include multiple nodes that are susceptible to conduction block, increasing the probability of self-termination. (c–f) The activation of complex critical structures requires a sequence of conduction blocks of the planar wave front or waves of excitation not originating from sinus rhythm. The probability of triggering these regions is much smaller than in panels (a, b). (c) The presence of at least one transversal connection departing from the conduction blocking node makes the activation more difficult as this node must fail to excite twice before prompting a reentrant circuit. (d) This structure cannot be triggered from sinus rhythm but it can be triggered by a single block of a wave of excitation originating from elsewhere. (e, f) The activation of these structures requires multiple blocks of the planar wavefront to occur in different nodes. Examples of the evolution of each structure are shown in the Supplemental Material with accompanying videos [44].
These details indicate that the CMP theory represents an ideal case for AF driven by simple reentrant circuits only. The theory assumes that if a simple circuit exists, the tissue spends 100% of the time in AF. Therefore, the theory curve sets a limit on the maximum time the model can spend in AF due to simple circuits only.
Local regions that are capable of hosting reentrant circuits are called critical structures; see Fig. 2 . A critical structure is active (inactive) when it hosts (does not host) a reentrant circuit. In the CMP model, critical structures are classified according to the complexity of their activation and deactivation mechanisms. Structures which can activate and terminate from the failure of a single conduction blocking node from sinus rhythm are referred to as simple. This includes cases where a critical structure contains multiple conduction blocking nodes, but only one must fail to allow for the formation of a reentrant circuit. All other configurations in which the planar wavefront from sinus rhythm requires multiple conduction blocks to fail to form a reentrant circuit are referred to as complex. The latter class includes critical structures that are only triggered by waves of excitation not originating from sinus rhythm (proceeding from right to left on the lattice); see Fig. 3(d) .
For large values of ν⊥, the model is in sinus rhythm indefinitely. The high number of transversal connections excludes the presence of regions that are critical for AF initiation and preservation as there are no sections of length ⩾τ/2 without a transverse connection. When ν⊥ decreases, for example due to increasing fibrosis [51], we observe a more pronounced branching structure of the lattice which favours the spontaneous emergence of structures that can host reentrant circuits. This increases the risk of developing AF.
When ν⊥ is sufficiently small, increasing δ extends the time the system spends in AF. This occurs because a larger fraction of nodes are susceptible to conduction block and this increases the number of regions that can host a reentrant circuit. However, the sensitivity of the system to the fraction of conduction blocking nodes, δ, rapidly vanishes as ν⊥ increases, suggesting that weak branching prevents the formation of critical structures independent of the fraction of nodes that are susceptible to conduction block [43]. The probability that a conduction blocking node fails to excite, ϵ, does not significantly influence the relationships between ν⊥ and the fraction of time the system spends in AF [43]. This implies that ϵ is mainly used to set the timescale of the model. More precisely, for simple reentrant circuits, we note that ϵ does not appear in the derivation of the risk of AF in Eq. (4). This is because ϵ effects both the probability that a simple reentrant circuit activates and deactivates. If ϵ is reduced, then it will, on average, take longer for a simple reentrant circuit to activate. However, once active, that reentrant circuit will take longer to de-activate than the equivalent circuit with a larger value of ϵ. That means that ϵ determines the duration of paroxysmal AF episodes and the time between paroxysmal AF episodes, but has a minimal effect on the overall risk of AF in the CMP model. Likewise, ϵ has no effect on the period of any simple reentrant circuits formed. However, if circuits exist with an asymmetry between the probability of activation and deactication, ϵ may play a role in the duration of individual fibrillatory events.
The length of the refractory period, τ, sets the minimum distance between the conduction blocking node and the first regular node to the right which has at least one transversal connection for the structure to sustain a reentrant circuit, see Figs. 2 and and3. 3 . Given a fixed value of δ, lowering τ increases the number of regions that can host reentrant circuits, increasing the time the system spends in AF.
In the CMP model, the system is defined to have entered AF if the number of active nodes per time step a(t) exceeds 1.1 × L (220) nodes for T consecutive time steps,
p CMP AF ( t ) = < 1 if min ( [ a ( t − T ) , … , a ( t ) ] ) ⩾ 220 , 0 otherwise ,where t can take integer values in the range t = T. S and S is the duration of the experiment (in time steps) [52]. We use Eq. (5) to study how the probability of inducing AF varies with the amount of coupling ν⊥ and compare this statistic with its theoretical estimations; see Eq. (4). Note, Eq. (5) gives a working definition of AF in the CMP model and was derived by inspection in previous work [29,38,43].
The definition used here is not unique and is not robust against changes in the pacing frequency T. The definition is designed to measure whether nodes in the model are activated more frequently than would be expected in sinus rhythm. This is based on the principle that if nodes are being activated at a rate higher than the pacing rate, then there must a source of fibrillatory wavefronts other than the sinus node. A superior method would be to measure the average activation frequency of nodes relative to the pacing frequency explicitly, rather than the number of active nodes, since this would be more robust against changes in T. However, to be consistent with previous work we use the existing definition in the current paper. We stress that for fixed T, the two methods give almost identical results. Both methods compare well with a clinical definition of AF where AF is diagnosed from ECG or electrogram recordings; see Appendix C. We do not generate electrograms as standard in the CMP model since this significantly increases the computational burden of the simulations. Additionally, we do not explicitly distinguish between AF and atrial tachycardia (AT) in the CMP model; see Appendix E for further details.
The probability of inducing AF in simulations of the CMP model is systematically higher than in the CMP theory, see Fig. 4 . These findings are somewhat surprising since the CMP theory assumes the most favourable conditions for the emergence of AF from simple reentrant circuits only. We assert that this excess could be explained by the fact that reentrant circuits in the CMP model might have multiple mechanistic origins that are not accounted for in the CMP theory. Furthermore, the CMP theory assumes that reentrant circuits are triggered by single unidirectional conduction blocks, that is, AF is exclusively driven by simple critical structures; see Fig. 3 .
Phase diagram of the probability of inducing AF as a function of the fraction of transversal connections ν⊥. The violet line represents the theoretical risk curve; see Eq. (4). For each value of ν⊥, we perform 200 simulations of the CMP model and compute the average probability of inducing AF (black square); see Eq. (5). The duration of each simulation is S = 10 6 time steps. For both the model and the theory, we observe that the system never (always) develops AF for ν⊥ ⪆ 0.2 (ν⊥ ⪅ 0.1). Within this interval, the probability of developing AF rapidly increases as ν⊥ is lowered. For any value of ν⊥ between 0.1 and 0.2, the probability of inducing AF in the CMP model (black) is always higher than in the CMP theory (violet).
To better understand the discrepancy between theory and experiment, we look at the trace of the number of active nodes in the model. AF is paroxysmal when this statistic exhibits large fluctuations which prevent it from stabilizing above the AF threshold, i.e., the number of active nodes frequently falls below 220 nodes with only short periods of high frequency activity. AF is persistent when the number of active nodes consistently exceeds the AF threshold for extended periods of time. If AF in the model has a unique mechanistic origin, then we would expect tissues at the same level of coupling to exhibit statistically similar behaviors in the number of the active nodes over time. However, we find that this is not the case—there is significant heterogeneity among systems characterized by the same parameters, e.g., the amount of uncoupling, or the fraction of conduction blocking nodes.
In Fig. 5 all tissues are generated using the same parameters, with ν⊥ = 0.11. Tissue (a) remains in sinus rhythm indefinitely. Tissue (b) remains mostly in sinus rhythm, with rare fibrillatory events on the order of 10 3 time steps in the model. In real time, these events are on the order of 1 s. It is plausible that clinically, such short events may be interpreted as an ectopic beat rather than AF. From tissue (c), through to tissue (g), we observe a spectrum of AF persistence. This includes short frequent events in tissue (c), rare intermediate events in tissue (d), frequent intermediate events in tissue (e), a combination of short and intermediate events in tissue (f), and long events with brief interruptions in tissue (g). Only in tissue (h) do we see a permanent transition from short paroxysmal AF, to persistent AF. The event shown in tissue (h) is on the order of 30 min when converting to real time. Repeating those simulations where persistent AF appears to last until the end of the simulation, these simulations are extended to 10 9 time steps without the simulation reverting to sinus rhythm. In real time, these events are on the order of 1 month. For practical reasons, we have not investigated events on timescales longer than 10 9 time steps. Note, that for visual clarity, the example chosen in Fig. 5(h) is driven by a single dominant driver which may be defined as AT rather than AF. However, in most cases, persistent activity is maintained for long time periods with the presence of multiple competing drivers, see Appendix D for an example.
The number of excited nodes per time step a(t) (black thin line) and its moving average 〈a(t)〉 calculated over T = 220 successive time steps (red solid line) in eight different simulations of the CMP model. All simulations are generated with identical model parameters. The coupling value is set at ν⊥ = 0.11. The system is in AF when the number of excited nodes per time step exceed 220 (blue dashed line) for at least T time steps. The figure demonstrates the broad spectrum of AF persistence that naturally emerges in the CMP model, from (a) sinus rhythm, through (b–g) various forms of paroxysmal AF, to (h) fully persistent AF. The figures exhibit a range of different event times, and asymmetries between the period of time in and out of AF. Subfigures (b, e, h) are dominated by short, intermediate, and long AF events respectively. Subfigures (f, g) exhibit an interplay between short and intermediate, and intermediate and long event times respectively. These figures demonstrate that complex behavior can emerge at the model macrostructure from specific details at the model microstructure, independent of the parameters of the model.
The variability in the persistence of AF in the CMP model has been studied previously in Ref. [38]. The authors focused on the relationship between the amount of uncoupling in the lattice (i.e., ν⊥) and the features of the developed AF in 32 independent experiments. In agreement with clinical observations [23,53], they report high degrees of heterogeneity in the progression to persistent AF and in the amount of uncoupling required for AF to emerge. Similarly to Fig. 5 , they observe very different AF patterns across systems characterized by the same amount of uncoupling, asserting that the emergence of reentrant circuits is subject to the local distribution of transversal connections, not the global amount of coupling, i.e., ν⊥. However, the authors do not satisfactorily explain how and why different AF patterns emerge from the microstructure of the CMP model.
The findings presented in Figs. 4 and and5 5 provide two important pieces of evidence against the assumption that AF is exclusively driven by simple reentrant circuits. First, they show that the probability of inducing AF is systematically higher in the CMP model than in the CMP theory; see Fig. 4 . Second, they reveal different activation patterns do not appear consistent with simple structures activating and deactivating with fixed rates. Individual events exhibit a spectrum of lifetimes before reverting to sinus rhythm, from seconds to months. This motivates us to assess whether different mechanistic origins of AF are effectively present in the CMP model and how they eventually relate with the progression to persistent AF from paroxysmal AF.
In the following sections, we take up these challenges by removing layers of complexity from the CMP model. This allows us to derive simpler frameworks in which we can examine whether reentrant circuits have different mechanistic origins and how the features of these activation processes influence the development of AF.
In Sec. III, we start with the simplest approach by removing all the spatial elements of the CMP model. This is done by condensing the CMP model into a mean-field (MF) model in which complex critical structures and interactions between reentrant circuits (i.e., wave collisions) are neglected. This simple framework allows us to study AF under the assumption that fibrillation is exclusively driven by independently activated simple reentrant circuits. We show that the MF model systematically underestimates the probability of inducing AF and the persistence of AF.
In Sec. IV we dissect this discrepancy by reintroducing the spatial elements of the CMP model while carefully controlling the placement of nodes susceptible to conduction block. This prevents the formation of complex critical structures. The main advantage of this controlled CMP model (cCMP) over the simpler MF model is that it allows us to quantify how different activation mechanisms contribute to AF emergence and maintenance. Like the MF model, the cCMP model underestimates the probability of inducing AF and the persistence of AF with respect to the CMP model. However, the cCMP model does not increase the time in AF relative to what is found in the MF model with the exception of very small fluctuations explained by differences in individual event times.
Finally, we confirm that the difference in the probability of inducing AF and the persistence of AF between the CMP and cCMP models stems from the contribution of complex reentrant circuits which exhibit an asymmetry between the probability of activating and deactivating a reentrant circuit. These complex structures may only require a single failure from a conduction blocking node to initiate, but multiple failures to terminate, resulting in long individual event times. Additionally, these structures may be coupled as part of a larger critical structure such that the termination of a reentrant circuit anchored to a specific substructure immediately initiates a new reentrant circuit in a coupled substructure. We demonstrate these mechanisms explicitly and show that as the probability that a node is susceptible to conduction block is lowered, the spatial density of conduction blocking nodes falls to the extent that multiple failing nodes are not required for the termination of a reentrant circuit. As a result, the time the CMP and cCMP models spend in AF collapse onto a single curve. This demonstrates that an increase in the local density of conduction blocking nodes is highly proarrhythmic.
In the CMP model, critical structures activate and deactivate to sustain AF. Initially, the system is in sinus rhythm as planar waves of excitation released from the sinus node (pacemaker) propagate on the lattice. The motion of the planar waves is disrupted now and then by conduction blocks occurring across the grid. At some point in time, a conduction block forms the initial reentrant circuit. This reentrant circuit cannot maintain AF indefinitely because it will either self-terminate or be terminated by waves spreading from the surrounding regions. However, its circuital motion intensifies the model activity, generating disorganized, high-frequency activation wavefronts that spread across the lattice. When the system enters this state, nonplanar waves of excitation spreading from the active reentrant circuit reach dormant critical structures at a much higher frequency than the pacemaker waves. This initiates a chain of asynchronous activations and deactivations of different critical structures located across the lattice, protracting the current AF episode until the complete disappearance of reentrant circuits brings the system back to sinus rhythm.
In the CMP model, it is unclear whether these interactions between simple critical structures are the only drivers of AF. In particular, the results discussed in Figs. 4 and and5 5 motivate us to examine whether other activation mechanisms drive AF and how differences between paroxysmal and persistent AF emerge. The simplest approach to this problem is to derive a framework in which fibrillation is solely driven by independently activated simple reentrant circuits and to compare AF-related statistics against the CMP model.
To do so, we translate the features of the CMP lattice into a simple mean-field (MF) model of AF in which N particles independently turn on and off. For a one-to-one comparison, the number of particles, N, is directly observed from the number of simple critical structures present in the CMP lattice at a given level of coupling. The fact that simple critical structures are characterized by a few well defined architectural features allows us to systematically inspect the grid and detect each region falling into this category.
In the MF model, the system is represented by a simple Markov chain. At a given time t, the state of the chain is the number of active particles Na(t), such that t : Na(t) = N>. When Na(t) = 0, the system is defined as being in sinus rhythm where any existing critical structure has a chance to be triggered every T time steps (pacemaker frequency). However, Na(t) ⩾ 1 is defined as the MF model exhibiting AF. In this case, the length of active reentrant circuits sets the frequency (in time steps) at which inactive critical structures can be triggered. For the sake of simplicity, we assume that particles have the same length 〈ℓ〉 corresponding to the average length (in number of nodes) of the simple critical structures tracked across the CMP lattice. At any time step, inactive particles activate with rate p and active particles deactivate with rate q; see Fig. 6 .
The CMP model is condensed into a mean-field (MF) model of AF. (a) Simple critical structures are mapped into particles which can take two distinct states: active (i.e., hosting a reentrant circuit, blue path) or inactive. We enforce the following assumptions: (i) the location (spatial positioning) of a particle is irrelevant, (ii) all particles activate with rates ϵ/T when the system is in sinus rhythm and ϵ/〈ℓ〉 when the system is in AF, (iii) all particles deactivate with rate ϵ/〈ℓ〉, (iv) all particles have the same length 〈ℓ〉, and (v) particles can change their states at any time step. (b) Simple critical structures (black filled rectangles) found in the CMP lattice are condensed into particles (black filled circles). (c) The evolution of the MF system is driven by N independent particles that activate and deactivate with probability p and q, respectively, depending on the current state of the particle and the system.
Activation rates change depending on the state of the system, mimicking the fact that the presence of at least one reentrant circuit significantly increases the frequency at which dormant critical structures can be triggered. It follows that p and q are given by
