Causal Scale of Rotors in a Cardiac System
Companion post to:
Causal Scale of Rotors in a Cardiac System
Hiroshi Ashikaga, Francisco Prieto-Castrillo, Mari Kawakatsu, Nima Dehghani
Frontiers in Computational Physics Volume 6 - 2018
DOI: https://doi.org/10.3389/fphy.2018.00030
The Causal Scale of Rotors in Cardiac Excitable Media
Atrial fibrillation is often described through the language of mechanisms.
Some patterns trigger it, some sustain it, and some are thought to organize the
otherwise irregular electrical activity of the atria. Among the most important
of these proposed organizing structures are rotors: the rotating cores of
spiral waves in cardiac tissue.
Rotors are attractive objects for both physiology and physics. They are macroscopic structures, yet they arise from the local interaction of many excitable elements. They have topology, because their centers can be described as phase singularities. They have dynamics, because they drift, interact, appear, disappear, and sometimes fragment. And they have clinical relevance, because if rotors maintain atrial fibrillation, then finding and ablating them should, in principle, eliminate the arrhythmia.
But this mechanistic picture has never been entirely straightforward. Early clinical attempts to map and ablate rotors in atrial fibrillation produced promising results, but subsequent outcomes were mixed. Part of the difficulty is technical: clinical mapping systems have finite spatial resolution, finite temporal resolution, and incomplete access to the atrial surface. But there is also a deeper conceptual problem. When we say that a rotor is a mechanism, at what scale is that statement supposed to be true?
This paper was written around that question.
The central argument is not that rotors are irrelevant. Nor is it that the microscopic behavior of cardiomyocytes alone is the only legitimate causal level. Instead, the paper asks whether the causal status of a rotor is scale-dependent. In other words: a rotor may be a causal mechanism at one spatiotemporal scale, but not at another.
That is a different way of thinking about mechanism in excitable media. It moves the question from:
Are rotors causal mechanisms?
to:
At what spatiotemporal scale do rotors have causal power?
This distinction matters not only for cardiac electrophysiology, but also for broader problems in complex systems, neurophysiology, and biological computation, where macroscopic patterns are routinely invoked as mechanisms for microscopic dynamics.
Rotors as emergent objects in an excitable medium
Cardiac tissue is an excitable medium. Each local element can be activated, recover, and then become excitable again. When these local elements are coupled spatially, they support propagating waves. Under appropriate conditions, those waves can curl into spiral waves. The center of rotation is a rotor, or more precisely, a phase singularity: a point around which the phase winds but at which the phase itself is undefined.
In a continuous excitable medium, a rotor is not simply an object placed on top of the tissue. It is a collective dynamical structure. It exists because of the local rules of excitation, recovery, conduction, refractoriness, curvature, and coupling. At the same time, once it exists, it appears to organize the activity around it. This is the classical tension of emergence: the macrostructure is generated by the microdynamics, but it may also constrain the future evolution of those microdynamics.
This kind of macro–micro relationship appears throughout physiology and neuroscience. Cortical traveling waves, seizure propagation, sleep oscillations, population bursts, and spiral waves in neural tissue all raise similar questions. We observe a mesoscopic or macroscopic pattern and ask whether it is merely a description of many local events, or whether it has causal force as an organized dynamical entity.
The answer cannot be obtained simply by naming the pattern. A spiral wave is not automatically a causal mechanism just because we can see it. A rotor is not automatically the cause of fibrillation simply because it is correlated with fibrillatory activity. The causal question requires a scale-sensitive description of the system.
Why scale matters for causation
In many biological systems, the choice of scale changes the apparent mechanism. At the microscopic scale, one sees individual components and local interactions. At a more macroscopic scale, one sees collective modes, oscillations, waves, attractors, motifs, and spatial patterns. Both descriptions can be valid, but they do not necessarily carry the same causal information.
A fully microscopic description is not always the most causally informative description. This is one of the important ideas behind causal emergence. A macroscopic description can sometimes have more effective causal power than the underlying microscopic description, because the macroscopic scale can suppress irrelevant degrees of freedom, reduce noise, and expose the state-transitions that matter for the system’s future behavior.
This idea is especially natural from the perspective of statistical physics. Renormalization is not merely a tool for throwing away detail. It is a way of asking which degrees of freedom remain relevant under changes of scale. In a complex dynamical system, coarse-graining can reveal a level of description at which the system’s organization becomes more interpretable and, in some cases, more causally informative.
For an excitable medium such as cardiac tissue, this is a concrete question. If we progressively coarse-grain the system in space and time, does the rotor remain a causal organizing structure? Or does its apparent causal role vanish when the description becomes too coarse?
The paper addresses this by combining three ideas:
- a numerical model of cardiac excitation,
- a renormalization-like construction of spatiotemporal scales,
- an information-theoretic measure of causal emergence.
Modeling spiral waves in cardiac tissue
The simulations used a modified FitzHugh–Nagumo model, a simplified but useful model of excitable dynamics. The model represents the local cardiac action potential through a fast excitation variable and a slower recovery variable:
\[\frac{\partial v}{\partial t} = 0.26v(v-0.13)(1-v) - 0.1vr + I_{\mathrm{ex}} + \nabla \cdot (D\nabla v),\] \[\frac{\partial r}{\partial t} = 0.013(v-r).\]Here, $v$ is the transmembrane potential, $r$ is the recovery variable, $I_{\mathrm{ex}}$ is an external current, and $D$ is the diffusion tensor. In this study, the tissue was modeled as a two-dimensional isotropic homogeneous medium. This was an intentional simplification. The goal was not to reproduce every anatomical detail of atrial tissue, but to isolate the causal relationship between rotors and spiral waves without adding confounds from anisotropy, fibrosis, or tissue heterogeneity.
Spiral waves were induced by random sequential point stimulations. Once the spiral waves were generated, the local phase of the signal was estimated using the analytic signal:
\[\xi(t) = v(t) + i v_H(t) = A(t)e^{i\phi(t)},\]where $v_H(t)$ is the Hilbert transform of $v(t)$, $A(t)$ is the instantaneous amplitude, and $\phi(t)$ is the instantaneous phase.
The rotor was then identified as a phase singularity. Around such a point, the phase winds by an integer multiple of $2\pi$. The topological charge was computed as
\[n_t = \frac{1}{2\pi} \oint_c \nabla \phi \cdot d\mathbf{l}.\]A value of $n_t = +1$ corresponds to a counterclockwise rotor, $n_t = -1$ to a clockwise rotor, and $n_t = 0$ to the absence of a rotor. The magnitude $|n_t|$ was used to quantify the average number of rotors over the time series.
This gives a concrete bridge between excitable-medium physiology and topological characterization. The rotor is not defined visually or heuristically; it is localized through phase structure.
Constructing a spatiotemporal renormalization group
The next step was to describe the same cardiac system at multiple spatiotemporal scales.
Each component of the lattice was first binarized: it was assigned a value of 1 when excited and 0 when resting. This binary representation made it possible to define the system state at each time point as a high-dimensional binary configuration.
Then the system was coarse-grained in space and time.
Spatially, the lattice was decimated by a factor of 2. A block of neighboring sites was reduced to a representative site at the next scale. Temporally, the binary time series was downsampled by a factor of 2. Repeating these operations generated a family of system descriptions, ranging from relatively microscopic to highly macroscopic.
The spatial scales were:
\[30 \times 30,\quad 15 \times 15,\quad 8 \times 8,\quad 4 \times 4,\quad 2 \times 2,\quad 1 \times 1.\]The temporal scales were:
\[400 \ \mathrm{Hz},\quad 200 \ \mathrm{Hz},\quad 100 \ \mathrm{Hz},\quad 50 \ \mathrm{Hz},\quad 25 \ \mathrm{Hz},\quad 12 \ \mathrm{Hz}.\]Together, these produced 36 spatiotemporal descriptions of the same underlying class of rotor dynamics.
This construction is important because it treats scale as an explicit variable. The rotor is not assumed to be causal at all resolutions. Instead, the analysis asks how causal structure changes as the system is progressively coarse-grained.
Effective information and causal emergence
To quantify causal structure, the study used effective information.
At each scale, the system can be treated as a process with possible causes $X$ and possible effects $Y$. Effective information measures how much information is generated when a system, starting from an unconstrained repertoire of possible causes, enters a specific state of possible effects.
Formally, effective information is written as
\[EI(X \to Y) = I(X;Y),\]where
\[I(X;Y) = H(X) + H(Y) - H(X,Y),\]or equivalently,
\[I(X;Y) = \sum_{x,y} p(x,y) \log_2 \frac{p(x,y)}{p(x)p(y)}.\]On its own, mutual information is not a causal measure. It is a measure of statistical dependence. The causal interpretation comes from the construction of $X$: the possible causes are treated as an unconstrained repertoire, with a uniform probability distribution. This corresponds to perturbing the system into all possible states with equal probability and then evaluating how much information is generated by the transition to the observed effect state.
Causal emergence is then defined as the difference in effective information between two scales:
\[CE = EI(X_m \to Y_m) - EI(X_n \to Y_n),\]where $m$ and $n$ refer to different spatiotemporal scales.
In this paper, causal emergence was evaluated relative to the most microscopic description. A positive value of $CE$ means that the more macroscopic description has greater effective information than the microscopic one. In that case, the macro-description is not merely a compressed summary; it has greater causal informativeness. A negative value means that the macro-scale description has lost causal information relative to the microscopic scale.
This is the key conceptual move of the paper. Instead of asking whether the rotor is a cause in an absolute sense, the paper asks whether the scale at which the rotor is described has greater or lesser causal information than the microscopic description.
The main result: causation peaks at an intermediate scale
The first major result is that effective information does not simply increase as the description becomes more microscopic, nor does it monotonically increase as the description becomes more macroscopic.
Instead, effective information peaks at an intermediate spatiotemporal scale.
This is the central finding. The most causally informative description of the rotor-bearing cardiac system was neither the most microscopic lattice nor the most extreme coarse-grained description. It was an intermediate scale, where the system retained enough spatial and temporal structure to capture rotor organization, while removing some of the microscopic degrees of freedom that do not contribute strongly to the causal architecture.
In the aggregate simulations, effective information reached a global maximum around spatial scale 4 and temporal scale 4. Beyond that scale, further coarse-graining reduced effective information. This means that the system has a scale of peak causation.
This has an important interpretation. At scales below the peak, the intermediate macro-description can exert downward causal organization relative to the microscopic description. But at scales above the peak, the system has been coarse-grained too aggressively. The macroscopic description no longer preserves the causal structure needed to describe rotor-mediated dynamics.
Thus, causal power is not located at “the micro” or “the macro” in a simple way. It is distributed across scale, and in this system it is maximized at an intermediate level.
This result is consistent with a broader intuition from complex systems: biological function often does not live at the finest possible resolution. Too much microscopic detail can obscure the collective variables that matter. But too much coarse-graining can erase the very organization one hoped to understand.
Rotors are causal, but not universally causal
The second major result concerns the relationship between the number of rotors and causal emergence.
Across 1,000 simulated datasets, the number of rotors ranged from 0 to 7. The study then asked whether systems with more rotors showed greater causal emergence at each spatiotemporal scale.
At scales up to the scale of peak causation, the relationship was positive: more rotors were associated with greater causal emergence. In this regime, rotor dynamics behaved as an emergent causal organization of the system.
But this relationship did not hold at all scales.
At sufficiently coarse spatial scales, causal emergence became negative, and the correlation between rotor number and causal emergence reversed. In other words, when the system was described too macroscopically, increasing the number of rotors no longer corresponded to stronger emergent causal organization. Instead, rotor dynamics became reducible to the underlying microscopic description.
This is the most important conceptual conclusion of the paper:
Rotors can be causal mechanisms for maintaining spiral waves, but they are not universal causal mechanisms across all spatiotemporal scales.
This statement avoids two overly simple interpretations.
The first simplistic interpretation would be:
Rotors are the mechanism of atrial fibrillation.
The second would be:
Rotors are not real mechanisms, because rotor ablation has produced mixed clinical outcomes.
The scale-dependent view is more precise. Rotors may have causal power at some scales, but not at others. Therefore, the success or failure of a rotor based intervention may depend not only on whether a rotor exists, but also on whether the mapping, interpretation, and intervention operate at the scale at which that rotor has causal relevance.
Why spatial coarse-graining mattered more than temporal coarse-graining
Another important result was that spatial coarse-graining had a stronger effect on effective information than temporal coarse-graining.
This makes sense physiologically. Spiral waves and rotors are spatially organized phenomena. Their identity depends on the spatial arrangement of phase, wavefront curvature, and propagation around a singularity. If spatial coarse-graining becomes too severe, the phase singularity is no longer represented in a meaningful way. The rotor has been erased by the measurement scale.
Temporal coarse-graining also matters, because rotor dynamics unfold over time, and insufficient sampling can distort phase and propagation. But in this model, the loss of spatial structure was more damaging to the causal architecture than the loss of temporal resolution.
For cardiac mapping, this distinction is important. A clinical mapping system does not merely observe the heart; it defines the effective scale at which the heart is represented. If the spatial resolution is too coarse, the mapping system may fail to capture the causal scale of the rotor even if a rotor-like dynamical structure exists in the tissue.
The same point applies to neurophysiology. In cortical systems, waves, oscillations, avalanches, and phase singularities can look very different depending on whether they are measured by intracellular recordings, dense microelectrode arrays, ECoG, EEG, MEG, calcium imaging, or widefield optical signals. A collective pattern observed at one scale may not have the same causal interpretation at another.
Mechanism should not be treated as scale-free
A major philosophical implication of the paper is that mechanism should not be treated as a binary, scale-free property.
In many areas of physiology, we often ask whether a structure or process “is the mechanism” of a phenomenon. But this language hides the fact that mechanistic explanations are almost always tied to a level of description.
A rotor is a mechanism only relative to a scale at which it can be represented and at which its presence changes the effective causal architecture of the system. Below that scale, it may dissolve into local excitation and recovery dynamics. Above that scale, it may disappear into a coarse global descriptor that no longer preserves its causal structure.
This is not a weakness of the rotor concept. It is a more precise way of understanding it.
In complex systems, the same physical process can appear as microscopic interaction, mesoscopic organization, or macroscopic order parameter depending on the observational scale. The causal role of that process is not guaranteed to be invariant under coarse-graining. A mechanism may be real, but only within a finite range of scales.
This is also why information theory is useful here. It provides a way to ask whether a given scale of description carries causal information, rather than assuming that higher resolution is always better or that macroscopic patterns are automatically explanatory.
Clinical implications: rotor ablation and the missing variable of scale
The clinical motivation for this work comes from atrial fibrillation therapy.
Successful arrhythmia treatment requires targeting the mechanism that maintains the arrhythmia, not merely the events that trigger it. Pulmonary vein isolation has been successful in many patients because pulmonary veins are important sources of triggers. But persistent atrial fibrillation often involves a more distributed substrate, and recurrence remains common.
Rotor-guided ablation was motivated by the idea that localized sources or rotors maintain fibrillation. If those sources could be identified and eliminated, the arrhythmia might terminate. But clinical results have been inconsistent.
The analysis in this paper suggests one possible reason: clinical mapping and ablation may not explicitly account for the causal scale of the mechanism. A rotor may be causally meaningful only at a particular spatiotemporal scale. If the mapping system observes at the wrong scale, the inferred rotor may be unstable, incomplete, or causally misleading. Conversely, if ablation changes the tissue at a scale that does not correspond to the rotor’s causal scale, the intervention may fail even if the rotor was correctly identified in a visual or geometric sense.
This does not mean that rotor ablation is invalid. It means that the relevant endpoint may not simply be the visual elimination of rotor-like activity. A more principled endpoint might involve changes in the causal architecture of the tissue: whether the intervention reduces the effective information supporting fibrillatory dynamics, disrupts the scale at which pathological organization is maintained, or changes the communication topology of the excitable medium.
This is especially relevant for ablation strategies that segment the atria through linear lesions, posterior wall isolation, or other substrate-modifying approaches. These interventions change the topology of electrical communication. In doing so, they likely alter the causal architecture of the atrium. The question is whether such changes move the system away from the scale regime that supports fibrillation.
A future clinical version of this framework would not merely map activation. It would estimate patient-specific multiscale causal structure from multi-electrode recordings and ask how that structure changes during intervention.
Relevance beyond the heart: waves, oscillations, and causal scales in neural systems
Although the paper is about cardiac rotors, the underlying question is much broader.
Many neurophysiological phenomena have the same conceptual structure. They are emergent patterns in excitable or active media, and they are observed through measurements that impose a particular scale. Examples include cortical traveling waves, seizure waves, sleep slow oscillations, spindle propagation, phase-amplitude organization, and large-scale synchrony.
In each case, one can ask:
- Is the observed macroscopic pattern merely descriptive?
- Does it constrain local activity?
- At what scale does it have maximal causal information?
- Does coarse-graining reveal the relevant collective variable, or erase it?
- Is the measurement scale aligned with the causal scale of the phenomenon?
For example, a cortical traveling wave observed in LFP or ECoG may organize local spiking activity, but that causal relationship may depend on spatial sampling, temporal bandwidth, laminar depth, and the scale of the neural population being analyzed. Similarly, seizure dynamics may involve spirals, sources, sinks, and traveling fronts whose causal relevance changes with measurement scale.
The broader lesson is that physiological mechanisms should be studied as scale-dependent causal architectures, not merely as patterns detected in data.
What this paper does not claim
It is important to be precise about the limitations.
The model used here is a simplified two-dimensional homogeneous and isotropic excitable medium. Real atrial tissue has anisotropy, fiber structure, heterogeneous conduction, fibrosis, curvature, wall thickness, variable refractoriness, and patient-specific anatomy. Those factors will almost certainly affect the causal architecture.
The point of the model was not to reproduce all clinical details of atrial fibrillation. The point was to isolate a fundamental question: even in a clean excitable medium where rotors can be clearly defined, is the causal status of the rotor invariant across scale?
The answer was no.
That negative answer is valuable precisely because it appears in a simplified system. If scale-dependence is already present in an idealized model, it is unlikely to disappear in the more complex biological heart.
The main lesson
The paper argues for a scale-sensitive view of mechanism in excitable media.
Rotors are not dismissed. They are not declared irrelevant. Instead, they are placed within a causal hierarchy. They are emergent structures whose causal power is maximal only over a finite range of spatiotemporal scales.
This changes how one should think about both theory and intervention. The question is not whether the microscopic or macroscopic description is the “true” one. The question is which scale carries the most causal information for the phenomenon of interest.
For cardiac fibrillation, that means that rotor-based mechanisms should be understood through the scale at which they organize spiral-wave dynamics. For clinical electrophysiology, it means that mapping and ablation should eventually be evaluated not only by anatomical or visual criteria, but by how they alter the multiscale causal architecture of the tissue. For complex systems and neurophysiology, it provides a general lesson: emergent patterns may be real mechanisms, but their causal power is not necessarily scale-free.
A rotor is therefore not simply a thing in the heart. It is a scale-dependent causal object in an excitable medium.
That, in the end, is the central message of the paper.