Emergence of Complex Behavior
2011; Lippincott Williams & Wilkins; Volume: 4; Issue: 4 Linguagem: Inglês
10.1161/circep.110.961524
ISSN1941-3149
AutoresPeter Spector, Nicole Habel, Burton E. Sobel, Jason H. T. Bates,
Tópico(s)Neural dynamics and brain function
ResumoHomeCirculation: Arrhythmia and ElectrophysiologyVol. 4, No. 4Emergence of Complex Behavior Free AccessResearch ArticlePDF/EPUBAboutView PDFView EPUBSections ToolsAdd to favoritesDownload citationsTrack citationsPermissionsDownload Articles + Supplements ShareShare onFacebookTwitterLinked InMendeleyReddit Jump toSupplemental MaterialFree AccessResearch ArticlePDF/EPUBEmergence of Complex BehaviorAn Interactive Model of Cardiac Excitation Provides a Powerful Tool for Understanding Electric Propagation Peter S. Spector, MD, Nicole Habel, Burton E. Sobel, MD and Jason H.T. Bates, PhD, DSc Peter S. SpectorPeter S. Spector From the Division of Cardiology (P.S.S., N.H., B.E.S.) and Division of Pulmonary/Critical Care Medicine (J.H.T.B.), Department of Medicine, Fletcher Allen Health Care and the University of Vermont, Burlington, VT. , Nicole HabelNicole Habel From the Division of Cardiology (P.S.S., N.H., B.E.S.) and Division of Pulmonary/Critical Care Medicine (J.H.T.B.), Department of Medicine, Fletcher Allen Health Care and the University of Vermont, Burlington, VT. , Burton E. SobelBurton E. Sobel From the Division of Cardiology (P.S.S., N.H., B.E.S.) and Division of Pulmonary/Critical Care Medicine (J.H.T.B.), Department of Medicine, Fletcher Allen Health Care and the University of Vermont, Burlington, VT. and Jason H.T. BatesJason H.T. Bates From the Division of Cardiology (P.S.S., N.H., B.E.S.) and Division of Pulmonary/Critical Care Medicine (J.H.T.B.), Department of Medicine, Fletcher Allen Health Care and the University of Vermont, Burlington, VT. Originally published8 Jun 2011https://doi.org/10.1161/CIRCEP.110.961524Circulation: Arrhythmia and Electrophysiology. 2011;4:586–591Other version(s) of this articleYou are viewing the most recent version of this article. Previous versions: January 1, 2011: Previous Version 1 IntroductionWe have developed a straightforward, physiologically based mathematical in silico model of cardiac electric activity to facilitate understanding of the fundamental principles that determine how excitation propagates through the heart. Despite its simplicity, the model provides a very powerful teaching tool. In fact, its simplicity is integral to the model's utility. The contrast between the minimal set of rules that govern the model's function and the widely varied complex behaviors it can manifest offers insight into the nature of emergent behavior in wave propagation. Emergence in this context refers to the richness of the tissue activation patterns that arise from the aggregate behavior of the simple cells that comprise the tissue. Each cell can be active, inactive, or refractory and interacts only with its immediate neighbors. From these simple building blocks, very elaborate global behaviors emerge.Model UtilityFrom the perspective of the electrophysiology student, the notion of emergent properties can act as a Rosetta stone for deciphering electrophysiological behavior. The spread of electric excitation through the intricate 3D structure of the heart can take widely varied forms, ranging from the orderly propagation seen during sinus rhythm to the marked disorganization seen during ventricular fibrillation. Observation of the diverse and sometimes complex patterns of conduction (eg, unidirectional block, reentry, spiral waves) as well as the responses to pacing maneuvers (eg, entrainment) suggests to the electrophysiology student a nearly infinite array of possibilities, the mastery of which can be daunting. However, with study, it becomes apparent that one need not memorize every possible cardiac behavior. Instead, there are overarching principles of cardiac excitation and propagation1 from which these varied phenomena emerge and through which one can understand and predict rather than memorize electrophysiological behavior. Understanding these fundamental principles is integral to mastering electrophysiology.A framework for interpreting clinical observations predicated on these principles has been formalized in the computer model we have developed. The interactive nature of the model provides a substrate for active learning rather than passive observation. The student can simulate complex conduction, such as unidirectional block and reentry. This requires grappling with identification of the conditions that are fundamental to reentry, thereby providing a durable learning experience. Experimentation with the model facilitates learning and appreciation of the principles responsible for electrophysiological behaviors.In the material to follow, we review the conceptual design of our model. We show several examples of how use of the model demonstrates important electrophysiological principles. What is provided, however, is not intended to be an exhaustive review of the lessons that can be learned from use of the model; rather, it serves as an introduction to the ways in which the model can be instructive.The ModelThe model (along with several sets of prestored parameters) has been provided so that readers can experiment on their own. Additionally, the mathematics have been included for interested readers but are not required for appreciation of the model's utility as a learning tool. Please see the online-only Data Supplement to download the model and accompanying materials; http://circep.ahajournals.org/cgi/content/full/CIRCEP.110.961524/DC1.Model DesignThere are an enormous number of aspects to any complex system that can be described mathematically. How comprehensively one designs a model of such a system depends on the desired use of the model's output. As a guiding principle, a model should be as simple as possible while maintaining sufficient detail to achieve simulation of the desired aspects of the system. The advantage of the simplest model is that it can be used more rapidly and easily than a complex model. There is a deeper conceptual utility conferred by simplicity that is succinctly expressed in a quote attributed to Albert Einstein: "Everything should be made as simple as possible, but not simpler." The phrase "as simple as possible" implies capturing the very essence of a system, excluding all that is incidental or derivative, whereas the phrase "but not simpler" implies leaving out nothing that is fundamental.What We ModeledWe defined a set of rules that govern the behavior of individual "cells" and how they interact with their immediate neighbors.What We Did Not ModelWe did not explicitly model the global behavior of the tissue once it was excited. The spread of electric activity in our model is entirely an emergent property that arises from the programmed behavior of the individual cells and their interactions. We also did not model the details of how cells become excited (ie, the kinetics of ion channels), resorting instead to merely specifying a set of rules that each cell obeys depending on its circumstances. Somewhat surprisingly, the intricate details of how the excitation of a cell is achieved are not critical to the production of a vast array of clinically relevant macroscopic tissue behaviors.The Behavior of the ModelThe Properties of Conduction in TissueFor electric activation to propagate through tissue, current must flow from excited cells to adjacent quiescent, but excitable cells. A group of excited cells is referred to as a source, whereas a group of electrically connected adjacent, but unexcited cells are referred to as a sink. The relative sizes of a source and sink determine the success or failure and the rate of propagation.2 Accordingly, the geometric arrangement of groups of cells and their interconnections influence the source/sink balance. For example, a wedge-shaped accessory pathway can exhibit unidirectional conduction as a consequence of its geometry, as illustrated in Figure 1. Here, a wavefront successfully propagates from the atrium into the wide end of the accessory pathway because the source (comprising the atrial cells surrounding the entrance to the pathway) is larger than the sink (made up of the 3 cells spanning the wide end of the pathway itself). This sink then becomes the source for the narrower region deeper within the pathway, and so on, until the narrowest part of the pathway is reached at the entrance to the ventricle. Now, however, the sink suddenly becomes very large, comprising the ventricular cells fanning out from the end of the pathway. At this point, the single cell in the source at the end of the pathway cannot generate enough current to activate the 3 cells in the sink, and propagation fails. The overall result is conduction block from an accessory pathway to the ventricle. But now consider what happens when activation proceeds in the opposite direction. For the most part, the scenario is the same except in reverse, until the activation is ready to leave the wide end of the pathway and enter the atrium. At this point, there are 3 cells in the source attempting to excite 5 cells in the sink, a more favorable ratio than the previous 1:3 situation, and propagation proceeds. The unidirectional block created by this pathway geometry can provide 1 of the conditions necessary for initiation and maintenance of reentry.3Download figureDownload PowerPointFigure 1. Source/sink relationships and unidirectional block. The geometry of a wedge shaped pathway determines its source/sink relationships creating unidirectional block. A, Atrium-to-ventricle conduction fails: At the ventricular insertion site of the pathway, the small source (1 cell) cannot depolarize the large sink (3 cells). B, ventricle-to-atrium conduction succeeds: At the ventricular insertion site, there is a 3:1 source/sink ratio, and at the atrial insertion site, there is a 3:5 ratio. In both cases, the source/sink ratio is sufficient to allow propagation.Reentry occurs when excitation travels along a path that ends where it began such that propagation repetitively traverses the circuit. Reentry is only possible when the conduction time around the loop is less than (or equal to) the time required for each cell in the circuit to recover excitability (refractory period). If the conduction time is shorter than the refractory period, the wavefront will encounter unexcitable tissue and extinguish. It is customary for electrophysiologists to express these parameters in terms of distances (circuit length, the physical distance over which electric activity must travel; wavelength, the distance from the leading edge to the trailing edge of the wavefront). The model can be used to explore these principles of reentry (Figure 2).Download figureDownload PowerPointFigure 2. Reentry secondary to fixed, anatomic substrate. Circuit length is the physical distance around the path of the reentry loop. Wavelength is the distance from the leading to the trailing edge of the excitation wavefront.With the freedom to manipulate the characteristics of tissue conduction and refractoriness as well as the geometric distribution of healthy and diseased tissue, students can discover for themselves the fundamental conditions required for initiation, maintenance, and termination of reentry. The type of reentry described thus far has a fixed anatomic substrate (ie, there is a physically determined circuit). There is a second type of reentry, however, that has a dynamic, functional substrate. In this latter case, the source/sink relationships determined by the shape of an activation wavefront produce the conditions responsible for reentry (reduced conduction velocity and block). By virtue of its geometry, a flat wavefront has an equal source and sink. A convex wavefront, on the other hand, has a larger sink than source, which causes it to propagate slowly, whereas the opposite is true for a concave wavefront. This situation is illustrated in Figure 3, which shows an initially planar wavefront moving to the right past a vertical linear segment of scar tissue. As the leading edges of the wavefront make their way past the scar, they develop a convex curvature because of the movement of activation into the region to the right of the scar. The concave portions travel the fastest, which eventually causes the wavefront to become planar again once it has moved far enough past the scar. Conduction velocity, and ultimately the success or failure of conduction, are thus functions of wavefront curvature.4Download figureDownload PowerPointFigure 3. The effects of curvature on conduction velocity. As excitation spreads around a linear scar, 2 curved wavefronts coalesce, forming a concave leading edge. The source/sink ratio is greatest at the concave center, which therefore propagates more rapidly than the flatter portions of the wavefront. As a result, the wavefront becomes progressively less concave.An extreme example of the effects of curvature is illustrated in Figure 4, which shows a spiral excitation wave, or rotor. The innermost portions of the rotor maintain a curvature so great that propagation fails (secondary to source/sink mismatch), creating a central area of excitable, but unexcited tissue around which the rotor spins. Manipulation of the model's initial conditions allows users to create multiple rotors that can degenerate into "daughter waves" that in turn can persist, divide, coalesce, or extinguish. Users can then explore the relationships between wavelength and circuit length, confirming for themselves the validity of the mass hypothesis of atrial fibrillation.5Download figureDownload PowerPointFigure 4. Reentry secondary to dynamic, functional substrate. The extreme curvature (center) results in propagation failure and a region of unexcited cells. A curved wavefront propagates around this area of functional block.The Response to Pacing ManeuversMapping provides a means of deducing the underlying activation pattern of the heart, but sometimes even an accurate activation map is insufficient for discerning which areas are responsible for driving an arrhythmia. For example, in the absence of conduction block, all areas of the heart are excited with each beat. However, only some areas of tissue are involved in the maintenance of an arrhythmia (ie, are in the circuit), whereas others are simply passively driven as a result of activity emanating from the arrhythmogenic region (ie, are out of the circuit). Entrainment mapping,6,7 is a clever means of using the response to pacing maneuvers to determine whether a pacing site is in or out of the tachycardia circuit (Figure 5). During entrainment mapping, one paces the heart at a cycle length slightly shorter than the tachycardia cycle length, which allows one to "capture" the circuit (activation occurs as a result of the paced wavefront rather than of the tachycardia wavefront). Pacing is then stopped, and the time from the last paced beat to the subsequent tachycardia beat, the postpacing interval, is measured. After the final paced beat, activation proceeds from the pacing site to the circuit, around the circuit, and back to the pacing site. If the pacing site is in the circuit, then the time from the pacing site to the circuit (and back) is 0. If the conduction velocity around the circuit is the same during tachycardia as it is during pacing, then the time around the circuit is, by definition, equal to the tachycardia cycle length. Thus, if the postpacing interval is equal to the tachycardia cycle length, then the pacing site is in the circuit. If the postpacing interval is greater than the tachycardia cycle length, then either (1) the pacing site is out of the circuit or (2) the conduction velocity around the circuit is slower during pacing than during tachycardia (decremental conduction secondary to functional refractoriness).Download figureDownload PowerPointFigure 5. Entrainment mapping. A reentry circuit is shown wherein the reentrant wavelength is indicated by the red arrow, the excitable gap is indicated by the blue tail, and the passive wavefront is indicated by the yellow arrow. Although every cell is activated with every beat, only the left side of the tissue (1, active area) is "in the circuit," whereas the right side of the tissue (2, passive area) is "out of the circuit." When pacing from inside the circuit (top tracing) the PPI is equal to the TCL. When pacing from outside the circuit (bottom tracing), the PPI is much longer than the TCL. PPI indicates postpacing interval; PCL, pace cycle length; TCL, tachycardia cycle length.Using the model, one can create a reentrant arrhythmia with both passive and active components and then pace at user-defined rates and locations. Examination of the response to pacing can elucidate the concepts integral to understanding entrainment mapping (Figure 5). Entrainment requires an electrically stable reentrant circuit (same cycle length and activation sequence with each beat) and an excitable gap (a period of time during which cells have recovered excitability after 1 excitation and before the arrival of the next wavefront).The preceding are just a few examples of ways in which cardiac electrophysiology can be explored using the model. We invite the reader to experiment with the model on their own. It can be downloaded at http://www.fletcherallen.org/cardiacmodel.Model LimitationsAlthough our computational model exhibits a wide range of usefully realistic behaviors, we must remain cognizant of the fact that it is far from a perfect representation of real cardiac tissue. Indeed, our emphasis has been on simplicity for ease of use and speed of operation, so there are numerous known details of cardiac electrophysiology that are not included. For example, we have not explicitly modeled the behavior of the numerous ion channels that dynamically regulate the flow of ions across cell boundaries. A great deal is known about these channels, which have been incorporated in other models.8–10 Of course, the multiscale nature of biological systems means that we can, to a useful degree, replace the myriad details of these channels and their microscopic behavior with a set of empirical rules. These rules encapsulate most of the mesoscopic consequences of channel physiology, which in turn give rise to the macroscopic electric activity of the model. Nevertheless, our model does not link macroscopic electric activity directly to the microscopic behavior that causes it in a real heart. Another limitation of our model is its geometry; we consider merely a rectangular, 2D sheet of tissue. It is important to remember that the tissue geometry in 3 dimensions can be a critical factor in arrhythmogenesis,11 and even inclusion of a realistic curvature and topology in 2 dimensions has been used to advantage in previous models.10,12 These various limitations of our model reflect the practical tradeoffs we have made in devising a convenient and manageable teaching tool.Historical PerspectiveModeling of ion channels and action potentials got its start with the Hodgkin and Huxley13 model for the behavior of ion channels in the giant axon of the squid. Very powerful models of cardiac cells were subsequently developed that used partial differential equations to describe ion channel behavior.14,15 The accuracy of these models improved as data became available from voltage clamp experiments. These models provided a robust mathematical description of cardiac cell membrane behavior. By linking cells together through resistive gap junctions, such models could provide the building blocks for study of propagation through cardiac tissue.16 Although such model designs provide a relatively detailed accounting of tissue behavior and are well suited to examine the effects of various physiological perturbations (eg, pharmacological interventions or ischemia), they come at a very high computational cost.Different model designs were developed that attempted to preserve much of the functional accuracy of tissue behavior but at reduced computational burden. Models using differential equations that describe dampened oscillators17–19 were developed and manifest many of the complex emergent properties of tissue propagation seen with ion channel-based models but with fewer variables and lower processing requirements.20 An even greater level of design simplification makes use of cellular automata to study propagation in cardiac tissue.21–24 Cellular automaton models (like ours) use (1) discrete cell states (quiescent, excited, refractory), (2) a simple lattice structure (generally square or hexagonal), and (3) a simplified set of rules that govern the way that cells update their state as a function of interaction with the state of their neighbors. The first application of cellular automata to cardiac propagation was conceived by Wiener and Rosenbleuth21 in 1946. Several subsequent modifications of this original cellular automata model have enhanced our understanding of the relationship between various functional parameters (eg, excitability, dispersion, restitution) and complex propagation patterns. Moe et al22 incorporated a partially refractory state and demonstrated the role of dispersion of refractoriness in vortex formation. Fast et al23 expanded the neighborhood beyond the immediately adjacent cells and provided weighted coupling between cells. They and others made various modifications to the cellular automata rules to more accurately mimic tissue properties such as source/sink relationships and restitution.23,24 With these modifications, they beautifully demonstrated emergent behaviors, including conduction velocity dependence on wave curvature, and excitation frequency and the dependence of rotor core trajectory on tissue excitability.Each of these cellular automata models have a purely excitation state-based rule for propagation. The number of neighbors required to excite a cell vary from model to model. Some incorporate a larger neighborhood and a weighting function to determine the connectivity between cells. The present model is a variation on this theme in which cells have an "intrinsic" voltage that varies with time. Cells are electrically connected to their neighbors, and current shifts between cells according to Ohm law. Thus, the intrinsic cell voltage (action potential) is altered through electrotonic interactions with the cell's neighbors during active as well as passive intercellular current flows. To the best of our knowledge, this agent-based model is the first to use such an approach. In so doing, the model allows current to accumulate over several time steps, ultimately reaching threshold for excitation. In this way, a cell's state depends on the state of its neighbors over several time steps. The present model thus exploits the computational efficiency of a cellular automaton for most of its behavior while gaining the precision afforded by differential equations to account for intercellular current flow.ConclusionsThe model described provides a concrete example of how guiding principles offer a means to decipher electrophysiological behavior. Experimentation with the model enables students to interactively use these principles to explore electric propagation through excitable tissue. The model provides several advantages over in vivo laboratory demonstrations, including convenience, speed, flexibility, and independent control of relevant parameters. The speed of the model (related in part to its simplicity) facilitates interaction. Because students can conceive of an experiment, perform it, and observe its results in just a few minutes, the ability to learn iteratively is enhanced compared with the pace of learning from in vivo experimentation. Additionally, as opposed to in vivo experimentation, in silico experiments have the advantage that the initial conditions of 1 experiment are independent of the final conditions of the prior experiment. We hasten to point out that, of course, computer simulations are simply enhanced-thought experiments and cannot be used to replace biological experiments. Rather, scientific discovery benefits from the interaction between experimentation and theoretical work (including computer simulations).This computer model is simply a formalization of our understanding of the principles that govern macroscopic behavior of electric propagation, embodied in a set of rules obeyed by each cell in the model. The appropriateness of these rules is reflected in the extent to which the model behaves realistically under those circumstances that we deem to be of interest. Importantly, even the very minimal set of rules we have used here is adequate to produce a staggering array of complex emergent behavior at the level of the whole tissue. Thus, this model serves not only as a teaching tool for cardiac electrophysiology, but also as an example of how multiscale emergent behavior can arise in complex biological systems.Sources of FundingThis work was supported by National Institutes of Health grant NCRR P20 RR15557.DisclosuresDr Spector receives research grants of more than $10,000 from Medtronic, St Jude, and Biosense Webster and receives more than $10,000 in consulting fees from Medtronic and Biosense Webster. Dr Bates receives research grants of more than $10,000 from Medtronic.FootnotesThe online-only Data Supplement is available at http://circep.ahajournals.org/cgi/content/full/CIRCEP.110.961524/DC1.Correspondence to Peter S. Spector, MD, McClure 1 Cardiology, 111 Colchester Ave, Burlington, VT 05401. E-mail peter.[email protected]eduReferences1. 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Physica D. 1990; 46:392–415.CrossrefGoogle Scholar Previous Back to top Next FiguresReferencesRelatedDetailsCited By Dantas E, Orlande H and Dulikravich G (2022) Thermal ablation effects on rotors that characterize functional re‐entry cardiac arrhythmia, International Journal for Numerical Methods in Biomedical Engineering, 10.1002/cnm.3614 Carrick R, Benson B, Bates O and Spector P (2021) Competitive Drivers of Atrial Fibrillation: The Interplay Between Focal Drivers and Multiwavelet Reentry, Frontiers in Physiology, 10.3389/fphys.2021.633643, 12 Ai W, Patel N, Roop P, Malik A and Trew M Cardiac Electrical Modeling for Closed-Loop Validation of Implantable Devices, IEEE Transactions on Biomedical Engineering, 10.1109/TBME.2019.2917212, 67:2, (536-544) Makowiec D, Wdowczyk J and Struzik Z (2019) Heart Rhythm Insights Into Structural Remodeling in Atrial Tissue: Timed Automata Approach, Frontiers in Physiology, 10.3389/fphys.2018.01859, 9 Yip E, Andalam S, Roop P, Malik A, Trew M, Ai W and Patel N (2018) Towards the Emulation of the Cardiac Conduction System for Pacemaker Validation, ACM Transactions on Cyber-Physical Systems, 10.1145/3134845, 2:4, (1-26), Online publication date: 18-Sep-2018. 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Carrick R, Benson B, Bates J and Spector P (2016) Prospective, Tissue-Specific Optimization of Ablation for Multiwavelet Reentry, Circulation: Arrhythmia and Electrophysiology, 9:3, Online publication date: 1-Mar-2016. Carrick R, Bates O, Benson B, Habel N, Bates J, Spector P and Bondarenko V (2015) Prospectively Quantifying the Propensity for Atrial Fibrillation: A Mechanistic Formulation, PLOS ONE, 10.1371/journal.pone.0118746, 10:3, (e0118746) Bates O, Suki B, Spector P, Bates J and Panfilov A (2015) Structural Defects Lead to Dynamic Entrapment in Cardiac Electrophysiology, PLOS ONE, 10.1371/journal.pone.0119535, 10:3, (e0119535) Benson B, Carrick R, Habel N, Bates O, Bates J, Bielau P and Spector P (2014) Mapping multi-wavelet reentry without isochrones: an electrogram-guided approach to define substrate distribution, Europace, 10.1093/europace/euu254, 16:suppl 4, (iv102-iv109), Online publication date: 1-Nov-2014. 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August 2011Vol 4, Issue 4 Advertisement Article InformationMetrics © 2011 American Heart Association, Inc.https://doi.org/10.1161/CIRCEP.110.961524PMID: 21653806 Manuscript receivedDecember 15, 2010Manuscript acceptedApril 15, 2011Originally publishedJune 8, 2011 Keywordsaction potentialsreentrycomputerselectrophysiologyarrhythmiaPDF download Advertisement SubjectsArrhythmias
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