Excitable Media Research Articles

Influence of conduction heterogeneities on transient spatiotemporal chaos in cardiac excitable media

Life-threatening cardiac arrhythmias such as ventricular fibrillation are often based on chaotic spiral or scroll wave dynamics which can be self-terminating. In this work, we investigate the influence of conduction heterogeneities on the duration of such chaotic transients in generic models of excitable cardiac media. We observe that low and medium densities of heterogeneities extend the average transient lifetime, while at high densities very long transients, potentially persistent chaos, and periodic attractors occur. Published by the American Physical Society 2024

The Ithildin library for efficient numerical solution of anisotropic reaction-diffusion problems in excitable media

Ithildin is an open-source library and framework for efficient parallelized simulations of excitable media, written in the C++ programming language. It uses parallelization on multiple CPU processors via the message passing interface (MPI). We demonstrate the library’s versatility through a series of simulations in the context of the monodomain description of cardiac electrophysiology, including the S1S2 protocol, spiral break-up, and spiral waves in ventricular geometry. Our work demonstrates the power of Ithildin as a tool for studying complex wave patterns in cardiac tissue and its potential to inform future experimental and theoretical studies. We publish our full code with this paper in the name of open science.

Experimental studies of spiral wave teleportation in a light sensitive Belousov-Zhabotinsky system.

Cardiac arrythmias are a form of heart disease that contributes toward making heart disease a significant cause of death globally. Irregular rhythms associated with cardiac arrythmias are thought to arise due to singularities in the heart tissue that generate reentrant waves in the underlying excitable medium. A normal approach to removing such singularities is to apply a high voltage electric shock, which effectively resets the phase of the cardiac cells. A concern with the use of this defibrillation technique is that the high-energy shock can cause lasting damage to the heart tissue. Various theoretical works have investigated lower-energy alternatives to defibrillation. In this work, we demonstrate the effectiveness of a low-energy defibrillation method in an experimental 2D Belousov-Zhabotinsky (BZ) system. When implemented as a 2D spatial reaction, the BZ reaction serves as an effective analog of general excitable media and supports regular and reentrant wave activity. The defibrillation technique employed involves targeted low-energy perturbations that can be used to "teleport" and/or annihilate singularities present in the excitable BZ medium.

Dreaming of electrical waves: Generative modeling of cardiac excitation waves using diffusion models.

Electrical waves in the heart form rotating spiral or scroll waves during life-threatening arrhythmias, such as atrial or ventricular fibrillation. The wave dynamics are typically modeled using coupled partial differential equations, which describe reaction-diffusion dynamics in excitable media. More recently, data-driven generative modeling has emerged as an alternative to generate spatio-temporal patterns in physical and biological systems. Here, we explore denoising diffusion probabilistic models for the generative modeling of electrical wave patterns in cardiac tissue. We trained diffusion models with simulated electrical wave patterns to be able to generate such wave patterns in unconditional and conditional generation tasks. For instance, we explored the diffusion-based (i) parameter-specific generation, (ii) evolution, and (iii) inpainting of spiral wave dynamics, including reconstructing three-dimensional scroll wave dynamics from superficial two-dimensional measurements. Furthermore, we generated arbitrarily shaped bi-ventricular geometries and simultaneously initiated scroll wave patterns inside these geometries using diffusion. We characterized and compared the diffusion-generated solutions to solutions obtained with corresponding biophysical models and found that diffusion models learn to replicate spiral and scroll wave dynamics so well that they could be used for data-driven modeling of excitation waves in cardiac tissue. For instance, an ensemble of diffusion-generated spiral wave dynamics exhibits similar self-termination statistics as the corresponding ensemble simulated with a biophysical model. However, we also found that diffusion models produce artifacts if training data are lacking, e.g., during self-termination, and "hallucinate" wave patterns when insufficiently constrained.

Amplitude equations for wave bifurcations in reaction–diffusion systems

A wave bifurcation is the counterpart to a Turing instability in reaction–diffusion systems, but where the critical wavenumber corresponds to a pure imaginary pair rather than a zero temporal eigenvalue. Such bifurcations require at least three components and give rise to patterns that are periodic in both space and time. Depending on boundary conditions, these patterns can comprise either rotating or standing waves. Restricting to systems in one spatial dimension, complete formulae are derived for the evaluation of the coefficients of the weakly nonlinear normal form of the bifurcation up to order five, including those that determine the criticality of both rotating and standing waves. The formulae apply to arbitrary n-component systems ( n⩾3 ) and their evaluation is implemented in software which is made available as supplementary material. The theory is illustrated on two different versions of three-component reaction–diffusion models of excitable media that were previously shown to feature super- and subcritical wave instabilities and on a five-component model of two-layer chemical reaction. In each case, two-parameter bifurcation diagrams are produced to illustrate the connection between complex dispersion relations and different types of Hopf, Turing, and wave bifurcations, including the existence of several codimension-two bifurcations.

Wave propagation in a light-temperature neural network under adaptive local energy balance.

External electric and mechanical stimuli can induce shape deformation in excitable media because of its intrinsic flexible property. When the signals propagation in the media is described by a neural network, creation of heterogeneity or defect is considered as the effect of shape deformation due to accumulation or release of energy in the media. In this paper, a temperature-light sensitive neuron model is developed from a nonlinear circuit composed of a phototube and a thermistor, and the physical energy is kept in capacitive and inductive terms. Furthermore, the Hamilton energy for this function neuron is obtained in theoretical way. A regular neural network is built on a square array by activating electric synapse between adjacent neurons, and a few of neurons in local area is excited by noisy disturbance, which induces local energy diversity, and continuous coupling enables energy propagation and diffusion. Initially, the Hamilton energy function for a temperature-light sensitive neuron can be obtained. Then, the finite neurons are applied noise to obtain energy diversity to explore the energy spread between neurons in the network. For keeping local energy balance, one intrinsic parameter is regulated adaptively until energy diversity in this local area is decreased greatly. Regular pattern formation indicates that local energy balance creates heterogeneity or defects and a few of neurons show continuous parameter shift for keeping energy balance in a local area, which supports gradient energy distribution for propagating waves in the network.

Chaos control in cardiac dynamics: terminating chaotic states with local minima pacing.

Current treatments of cardiac arrhythmias like ventricular fibrillation involve the application of a high-energy electric shock, that induces significant electrical currents in the myocardium and therefore involves severe side effects like possible tissue damage and post-traumatic stress. Using numerical simulations on four different models of 2D excitable media, this study demonstrates that low energy pulses applied shortly after local minima in the mean value of the transmembrane potential provide high success rates. We evaluate the performance of this approach for ten initial conditions of each model, ten spatially different stimuli, and different shock amplitudes. The investigated models of 2D excitable media cover a broad range of dominant frequencies and number of phase singularities, which demonstrates, that our findings are not limited to a specific kind of model or parameterization of it. Thus, we propose a method that incorporates the dynamics of the underlying system, even during pacing, and solely relies on a scalar observable, which is easily measurable in numerical simulations.

Observation and modeling of an unusual spatiotemporal pattern in bioacoustic chorusing

This paper describes an unusual underwater biological chorus recorded in the Southern California Bight and presents a numerical modeling approach that replicates aspects of the chorus. During one experiment, the evolution of the directionality of the chorusing region over time is suggestive of “The Wave”, the human waves performed by fans periodically standing and sitting in sports stadia around the world; here, the region of chorusing periodically propagated upcoast over 20 km of coastline at nearly 1.5 km/s. The chorus occurs predominantly at night in spring and summer, mostly in very shallow waters near the coast. It increases the underwater sound field levels within chorusing regions by 20–30 dB in the 50 Hz to 1 kHz frequency band. The chorus is composed of three parts; 1) a “sunset chorus” which is a 20-to-30 min continuous roar around sunset with received spectral levels up to 100 dB re 1 μPa2/Hz, 2) a “sunrise chorus” of lower level than the sunset chorus, and 3) an all-night-long cycling chorus. The cycling portion is made up of 15-to-20 s periods of higher received spectral levels (up to 90 dB re 1 μPa2/Hz within the chorusing region) each followed by a 10-to-20 s lull in which the spectral levels drop by 4–10 dB. This alternating pattern repeats every 30–40 s throughout the night. The numerical modeling approach is based on the physics of excitable media. A cellular automaton is used to model the two-dimensional spatial grid occupied by the calling animals (units), with each unit being either in the “resting”, “excitable”, or “active (calling)" state at each time step. Transition from resting to excitable and from active back to resting occurs automatically after a fixed period of time in the present state, whereas the probability of transitioning from excitable to active is determined not only by the elapsed time since entering the excitable state, but also by the received sound level at the unit location, creating a non-linear acoustics-based coupling between units. With appropriate inputs, many determined from measurements of the chorus properties and the individual animal calls themselves, simulations with the model can replicate the cycling levels in the night-long chorus, the continuous din of the sunset chorus, and (once properly initialized) the periodic upcoast evolution of the chorusing region (“The Wave”). When noise from a transiting ship is included in the simulations, the spatiotemporal characteristics of the chorus change appreciably, in ways similar to changes observed during the experiment.

Self-Improvising Memory: A Perspective on Memories as Agential, Dynamically Reinterpreting Cognitive Glue.

Many studies on memory emphasize the material substrate and mechanisms by which data can be stored and reliably read out. Here, I focus on complementary aspects: the need for agents to dynamically reinterpret and modify memories to suit their ever-changing selves and environment. Using examples from developmental biology, evolution, and synthetic bioengineering, in addition to neuroscience, I propose that a perspective on memory as preserving salience, not fidelity, is applicable to many phenomena on scales from cells to societies. Continuous commitment to creative, adaptive confabulation, from the molecular to the behavioral levels, is the answer to the persistence paradox as it applies to individuals and whole lineages. I also speculate that a substrate-independent, processual view of life and mind suggests that memories, as patterns in the excitable medium of cognitive systems, could be seen as active agents in the sense-making process. I explore a view of life as a diverse set of embodied perspectives-nested agents who interpret each other's and their own past messages and actions as best as they can (polycomputation). This synthesis suggests unifying symmetries across scales and disciplines, which is of relevance to research programs in Diverse Intelligence and the engineering of novel embodied minds.

Weak synaptic connections may facilitate spiral wave formation under source-sink interactions

This study explores the interaction between two distinct sites, termed the source and the sink, to analyze the possibility of spiral wave formation. To this aim, a grid of memristive FitzHugh–Nagumo elements is designed to simulate biological excitable media, such as the myocardium. The source, characterized by high excitation levels with a gradual increase in the recovery variable, is primed to generate an excitatory wavefront. In contrast, the sink remains quiescent in excitation yet elevated in the recovery state, thus tending to absorb excitation as it is temporarily unexcitable but soon recoverable. Existing literature on spiral wave formation primarily focuses on rotor formation under source–sink adjacency. This work, on the other hand, examines the potential for re-entrant behavior under spatial separation between the source and the sink. Within the context of eight-nearest neighbor coupling, the results indicate that the time delay for the excitation wave to reach the refractory state of the sink, due to the increased distance, may still facilitate re-entry if the intensity of connections is sufficiently weak. However, beyond a certain threshold of the source–sink distance under weak connections, wave breakage may occur without resulting in re-entry. For the birth of a spiral wave rotor, the tip of the excitatory wavefront must converge with its refractory back, specifically at the outermost contour in the final stage of refractoriness, referred to as the wavetail. The formation of a spiral rotor cannot arise from a wavefront encountering any site earlier in the stage of refractoriness.

Optical soliton solutions in a distinctive class of nonlinear Schrödinger’s equation with cubic, quintic, septic, and nonic nonlinearities

The Biswas–Mollivic equation is a special type of nonlinear Schrödinger equation, which explains the spatio-temporal behaviour of excitable media. In this paper, we investigate the optical soliton solutions of the Biswas–Mollivic equation with cubic–quintic–septic–nonic nonlinearities using the generalized Riccati equation mapping method. This method is efficient and provides new perspectives. It also provides novel insights into the dynamics of excitable media. Our findings add to a better understanding of the complex spatio-temporal patterns that develop in excitable media and have potential applications in the design of new technologies for controlling and manipulating pattern formation. To depict optical soliton solutions graphically, we use the MATLAB software.

Representing stimulus motion with waves in adaptive neural fields.

Traveling waves of neural activity emerge in cortical networks both spontaneously and in response to stimuli. The spatiotemporal structure of waves can indicate the information they encode and the physiological processes that sustain them. Here, we investigate the stimulus-response relationships of traveling waves emerging in adaptive neural fields as a model of visual motion processing. Neural field equations model the activity of cortical tissue as a continuum excitable medium, and adaptive processes provide negative feedback, generating localized activity patterns. Synaptic connectivity in our model is described by an integral kernel that weakens dynamically due to activity-dependent synaptic depression, leading to marginally stable traveling fronts (with attenuated backs) or pulses of a fixed speed. Our analysis quantifies how weak stimuli shift the relative position of these waves over time, characterized by a wave response function we obtain perturbatively. Persistent and continuously visible stimuli model moving visual objects. Intermittent flashes that hop across visual space can produce the experience of smooth apparent visual motion. Entrainment of waves to both kinds of moving stimuli are well characterized by our theory and numerical simulations, providing a mechanistic description of the perception of visual motion.

Ultrafast Response Organic Photoswitch Materials and Their Application in Volumetric 3D Display.

Volumetric three-dimensional (3D) display technology based on static screens is a crucial branch of 3D displays. The essential component in volumetric 3D displays is selectively excitable display media that can generate voxels at any position. Here, we synthesized a series of organic photoswitch materials to meet the specific requirements of 3D display mediums. In these photoswitch solutions, voxels are activated ultrafast within tens of picoseconds at the intersection of two control lasers and faded rapidly within tens of milliseconds when switching light is turned off. An experimental volumetric 3D display system utilizing an organic photoswitch solution as a screen is demonstrated. The system not only achieves a dynamic 3D display but also enables 360° viewing. The volumetric 3D display system can display true 3D images without auxiliary glasses and is expected to be applicable in fields as diverse as virtual reality, medical imaging, architectural design, and military visualization.

Spiking at the edge: Excitability at interfaces in reaction–diffusion systems

Excitable media, ranging from bioelectric tissues and chemical oscillators to forest fires and competing populations, are nonlinear, spatially extended systems capable of spiking. Most investigations of excitable media consider situations where the amplifying and suppressing forces necessary for spiking coexist at every point in space. In this case, spikes arise due to local bistabilities, which require a fine-tuned ratio between local amplification and suppression strengths. But, in nature and engineered systems, these forces can be segregated in space, forming structures like interfaces and boundaries. Here, we show how boundaries can generate and protect spiking when the reacting components can spread out: Even arbitrarily weak diffusion can cause spiking at the edge between two non-excitable media. This edge spiking arises due to a global bistability, which can occur even if amplification and suppression strengths do not allow spiking when mixed. We analytically derive a spiking phase diagram that depends on two parameters: i) the ratio between the system size and the characteristic diffusive length-scale and ii) the ratio between the amplification and suppression strengths. Our analysis explains recent experimental observations of action potentials at the interface between two non-excitable bioelectric tissues. Beyond electrophysiology, we highlight how edge spiking emerges in predator-prey dynamics and in oscillating chemical reactions. Our findings provide a theoretical blueprint for a class of interfacial excitations in reaction-diffusion systems, with potential implications for spatially controlled chemical reactions, nonlinear waveguides and neuromorphic computation, as well as spiking instabilities, such as cardiac arrhythmias, that naturally occur in heterogeneous biological media.

Bistable spiral wave dynamics in electrically excitable media.

We show that a positive feedback loop between sodium current inactivation and wave-front ramp-up speed causes a saddle-node bifurcation to result in bistable planar and spiral waves in electrically excitable media, in which both slow and fast waves are triggered by different stimulation protocols. Moreover, the two types of spiral wave conduction may interact to give rise to more complex spiral wave dynamics. The transitions between different spiral wave behaviors via saddle-node bifurcation can be a candidate mechanism for transitions widely seen in cardiac arrhythmias and neural diseases.

Spiral waves in fractal dimensions and their elimination in λ − ω systems with less damaging intervention

The theory of spiral waves in excitable media is considered an important topic in several dynamical systems. They have been observed in various physicochemical and biological systems such as the Belousov-Zhabotinsky chemical reactions, lethal arrhythmia in the heart, amoebas Dictyostelium discoideium, disinhibited mammalian neocortex retinal spreading depression, and seizures in the brain. In this study, spiral waves in an excitable system are analytically studied in fractal dimensions based on the theory of integration and differentiation for a non-integer dimensional space. We consider the generic oscillatory model λ−ω system characterized by a rotational symmetry where spiral waves are expected to occur. It was observed that for lower fractal dimensions with respect to unity, spiral waves may be suppressed in an excitable media without destroying the propagation of normal waves. This may be considered as an alternative approach to experimental and numerical approaches found in the literature. Their eliminations are important since they forbid the transition to a chaotic state in excitable media. Our approach could contribute to an improved therapy of clinical conditions such as atrial fibrillation and suppressing arrhythmias in cardiac models.

Chirality and curvature determine the meandering of spirals in multilayer excitable media

We study the emergence of meandering spiral waves in a multilayer structure where two spirals, originating independently, evolve in space–time. The FitzHugh-Nagumo model, enhanced with electromagnetic induction effects, defines the nodal dynamics. The layers are chemically coupled, and the drift of spirals is influenced by interlayer flux coupling. An external magnetic flux force is also necessary to destabilize and unpin spiral rotors. The effects of unique characteristics of spiral patterns, like chirality and tip curvature, are assessed. We find that spirals often drift within a bounded meandering area; however, determining the drift directions and overall meandering regions is complex. The forced spiral generally drifts a greater distance, except for identical co-rotating spirals, which drift synchronously. Spirals with opposite chirality exhibit asynchronous drift and wavefront asymmetry, even with identical tip curvature. Regardless of chirality, non-identical spirals are geometrically aligned and amplified before drifting. Stimulating the more loosely curved spiral ensures both spirals drift.

Scroll Waves and Filaments in Excitable Media of Higher Spatial Dimension.

Excitable media are ubiquitous in nature, and in such systems the local excitation tends to self-organize in traveling waves, or in rotating spiral-shaped patterns in two or three spatial dimensions. Examples include waves during a pandemic or electrical scroll waves in the heart. Here we show that such phenomena can be extended to a space of four or more dimensions and propose that connections of excitable elements in a network setting can be regarded as additional spatial dimensions. Numerical simulations are performed in four dimensions using the FitzHugh-Nagumo model, showing that the vortices rotate around a two-dimensional surface which we define as the superfilament. Evolution equations are derived for general superfilaments of codimension two in an N-dimensional space, and their equilibrium configurations are proven to be minimal surfaces. We suggest that biological excitable systems, such as the heart or brain which have nonlocal connections can be regarded, at least partially, as multidimensional excitable media and discuss further possible studies in this direction.

Effect of excitability on partially pinned scroll waves in excitable chemical media.

We present an investigation of excitability effects on the dynamics of scroll waves partially pinned to inert cylindrical obstacles in three-dimensional Belousov-Zhabotinsky excitable media. We also report on corresponding numerical simulations with the Oregonator model. The excitability varies according to the concentration of sulfuric acid [H_{2}SO_{4}] in the Belousov-Zhabotinsky (BZ) reaction and the parameter ɛ^{-1} in the Oregonator model. Initially, the freely rotating scroll segment rotates faster than the pinned one. The difference in the frequency of the two parts results in a transition from a straight pinned scroll wave to a twisted one, which helically wraps around the entire obstacle. The wave frequency in the whole volume is equal to that of the freely rotating scroll wave. When the excitability is increased, the time for the transition to the twisted wave structure decreases while the average speed s of the development increases. After the transition, the twisted wave remains stable. In media with higher excitability, the helical pitch is shorter but the twist rate ω is higher. Analysis presented in this study together with our previous findings of the effect of the cylindrical obstacle diameter on the wave dynamics results in common features: The average speed s and the twist rate ω of both studies fit well to functions of the difference in the initial frequency Δf of the freely rotating and untwisted pinned waves. We also demonstrate the robustness of the partially pinned scroll waves against perturbations from spontaneous waves emerging during the wave generation in the BZ medium with high [H_{2}SO_{4}]. Even though the scroll wave is partly disturbed at the beginning of the experiment, the spontaneous waves are gradually suppressed and the typical wave structure is finally developed.

Inter-layer propagation of spiral waves: Effects of time-varying defect blocks and magnetic flows

The variation of magnetic flux with distance and time is relatively common in nature. This paper investigates the inter-layer propagation of spiral waves in bi-layer Hindmarsh-Rose networks under gradient magnetic field and defect blocks motion, with the following conclusions: (1) Magnetic flux slows down the propagation of waves. (2) For defect blocks moving linearly or away from each other, spiral waves appear in the network, while the magnetic flux suppresses the spiral wave formation. (3) For defect blocks close to or merging with each other, the magnetic flow can only convert spiral waves in the network into “eye-like waves”. These findings imply that the propagation of spiral waves is sensitive to the action site of magnetic flux, which may be important for scar tissue (e.g., cardiac tissue with poor postoperative recovery). Moreover, three statistical parameters defined to detect waveform evolution in excitable media show good performance.