How the anisotropy of the intracellular and extracellular conductivities influences stimulation of cardiac muscle

A boundary element method of bidomain modeling for predicting cellular responses to electromagnetic fields

Objective.Commonly used cable equation approaches for simulating the effects of electromagnetic fields on excitable cells make several simplifying assumptions that could limit their predictive power. Bidomain or 'whole' finite element methods have been developed to fully couple cells and electric fields for more realistic neuron modeling. Here, we introduce a novel bidomain integral equation designed for determining the full electromagnetic coupling between stimulation devices and the intracellular, membrane, and extracellular regions of neurons.Approach.Our proposed boundary element formulation offers a solution to an integral equation that connects the device, tissue inhomogeneity, and cell membrane-induced E-fields. We solve this integral equation using first-order nodal elements and an unconditionally stable Crank-Nicholson time-stepping scheme. To validate and demonstrate our approach, we simulated cylindrical Hodgkin-Huxley axons and spherical cells in multiple brain stimulation scenarios.Main Results.Comparison studies show that a boundary element approach produces accurate results for both electric and magnetic stimulation. Unlike bidomain finite element methods, the bidomain boundary element method does not require volume meshes containing features at multiple scales. As a result, modeling cells, or tightly packed populations of cells, with microscale features embedded in a macroscale head model, is simplified, and the relative placement of devices and cells can be varied without the need to generate a new mesh.Significance.Device-induced electromagnetic fields are commonly used to modulate brain activity for research and therapeutic applications. Bidomain solvers allow for the full incorporation of realistic cell geometries, device E-fields, and neuron populations. Thus, multi-cell studies of advanced neuronal mechanisms would greatly benefit from the development of fast-bidomain solvers to ensure scalability and the practical execution of neural network simulations with realistic neuron morphologies.

A Boundary Element Method of Bidomain Modeling for Predicting Cellular Responses to Electromagnetic Fields

Objective

Commonly used cable equation-based approaches for determining the effects of electromagnetic fields on excitable cells make several simplifying assumptions that could limit their predictive power. Bidomain or "whole" finite element methods have been developed to fully couple cells and electric fields for more realistic neuron modeling. Here, we introduce a novel bidomain integral equation designed for determining the full electromagnetic coupling between stimulation devices and the intracellular, membrane, and extracellular regions of neurons.

Methods

Our proposed boundary element formulation offers a solution to an integral equation that connects the device, tissue inhomogeneity, and cell membrane-induced E-fields. We solve this integral equation using first-order nodal elements and an unconditionally stable Crank-Nicholson time-stepping scheme. To validate and demonstrate our approach, we simulated cylindrical Hodgkin-Huxley axons and spherical cells in multiple brain stimulation scenarios.

Main Results

Comparison studies show that a boundary element approach produces accurate results for both electric and magnetic stimulation. Unlike bidomain finite element methods, the bidomain boundary element method does not require volume meshes containing features at multiple scales. As a result, modeling cells, or tightly packed populations of cells, with microscale features embedded in a macroscale head model, is made computationally tractable, and the relative placement of devices and cells can be varied without the need to generate a new mesh.

Significance

Device-induced electromagnetic fields are commonly used to modulate brain activity for research and therapeutic applications. Bidomain solvers allow for the full incorporation of realistic cell geometries, device E-fields, and neuron populations. Thus, multi-cell studies of advanced neuronal mechanisms would greatly benefit from the development of fast-bidomain solvers to ensure scalability and the practical execution of neural network simulations with realistic neuron morphologies.

A multi-scale simulation of retinal physiology

We present a detailed physiological model of the (human) retina that includes the biochemistry and electrophysiology of phototransduction, neuronal electrical coupling, and the spherical geometry of the eye. The model is a parabolic-elliptic system of partial differential equations based on the mathematical framework of the bi-domain equations, which we have generalized to account for multiple cell-types. We discretize in space with non-uniform finite differences and step through time with a custom adaptive time-stepper that employs a backward differentiation formula and an inexact Newton method. A refinement study confirms the accuracy and efficiency of our numerical method. Numerical simulations using the model compare favorably with experimental findings, such as desensitization to light stimuli and calcium buffering in photoreceptors. Other numerical simulations suggest an interplay between photoreceptor gap junctions and inner segment, but not outer segment, calcium concentration. Applications of this model and simulation include analysis of retinal calcium imaging experiments, the design of electroretinograms, the design of visual prosthetics, and studies of ephaptic coupling within the retina.

Bidomain modeling of electrical and mechanical properties of cardiac tissue

Throughout the history of cardiac research, there has been a clear need to establish mathematical models to complement experimental studies. In an effort to create a more complete picture of cardiac phenomena, the bidomain model was established in the late 1970s to better understand pacing and defibrillation in the heart. This mathematical model has seen ongoing use in cardiac research, offering mechanistic insight that could not be obtained from experimental pursuits. Introduced from a historical perspective, the origins of the bidomain model are reviewed to provide a foundation for researchers new to the field and those conducting interdisciplinary research. The interplay of theory and experiment with the bidomain model is explored, and the contributions of this model to cardiac biophysics are critically evaluated. Also discussed is the mechanical bidomain model, which is employed to describe mechanotransduction. Current challenges and outstanding questions in the use of the bidomain model are addressed to give a forward-facing perspective of the model in future studies.

Anisotropic Cardiac Conduction

Anisotropy is the property of directional dependence. In cardiac tissue, conduction velocity is anisotropic and its orientation is determined by myocyte direction. Cell shape and size, excitability, myocardial fibrosis, gap junction distribution and function are all considered to contribute to anisotropic conduction. In disease states, anisotropic conduction may be enhanced, and is implicated, in the genesis of pathological arrhythmias. The principal mechanism responsible for enhanced anisotropy in disease remains uncertain. Possible contributors include changes in cellular excitability, changes in gap junction distribution or function and cellular uncoupling through interstitial fibrosis. It has recently been demonstrated that myocyte orientation may be identified using diffusion tensor magnetic resonance imaging in explanted hearts, and multisite pacing protocols have been proposed to estimate myocyte orientation and anisotropic conduction in vivo. These tools have the potential to contribute to the understanding of the role of myocyte disarray and anisotropic conduction in arrhythmic states.

Mechanical bidomain model of cardiac muscle with unequal anisotropy ratios

The properties of cardiac muscle are anisotropic, and the degree of anisotropy may be different in the intracellular and extracellular spaces. In the electrical bidomain model, such "unequal anisotropy ratios" of the conductivity lead to unanticipated behavior. In the mechanical bidomain model, unequal anisotropy ratios of the mechanical moduli might also result in unanticipated behavior. In this study, mathematical modeling based on the mechanical bidomain model is used to calculate the distribution of mechanotransduction in cardiac tissue when it is stretched. This analysis demonstrates that unexpected phenomena arise when the mechanical anisotropy ratios are unequal.

Design and application of a novel gastric pacemaker

Omnipresent bioelectrical events known as slow waves are responsible for coordinating motility in the gastrointestinal tract. Functional motility diseases, such as gastroparesis, are associated with slow wave dysrhythmias. Electrical stimulation is a potential therapy to correct abnormal slow wave patterns. We present the design and application of a new gastric pacemaker. Real-time changes to the stimulation parameters such as period, amplitude and pulse width were applied using a graphical user interface, which communicated with the microcontroller to deliver the stimulus. The new pacemaker allows the voltage, delivered current and resistance between pacing electrodes to be continuously monitored. The pacing device was applied experimentally and was able to modulate and entrain gastric slow wave activity. After the onset of pacing, the direction of slow wave propagation was altered. Furthermore, the mean velocity and amplitude of slow wave activity increased from 4.7±1.5 to 5.4±1.3 mm/s, and from 1.1±1.1 to 1.7±0.9 mV, respectively. A simplified bidomain electrical model was used to simulate the recorded stimulus artifact. The model illustrated a new approach to evaluate if the stimulus has been delivered to the gastric tissue. The new pacing device and model will be used to investigate the mechanisms that allow pacing to entrain slow wave activity.

Highly trabeculated structure of the human endocardium underlies asymmetrical response to low-energy monophasic shocks

Novel low-energy defibrillation therapies are thought to be driven by virtual-electrodes (VEs), due to the interaction of applied monophasic electric shocks with fine-scale anatomical structures within the heart. Significant inter-species differences in the cardiac (micro)-anatomy exist, however, particularly with respect to the degree of endocardial trabeculations, which may underlie important differences in response to low-energy defibrillation protocols. Understanding the interaction of monophasic electric fields with the specific human micro-anatomy is therefore imperative in facilitating the translation and optimisation of these promising experimental therapies to the clinic. In this study, we sought to investigate how electric fields from implanted devices interact with the highly trabeculated human endocardial surface to better understand shock success in order to help optimise future clinical protocols. A bi-ventricular human computational model was constructed from high resolution (350 μm) ex-vivo MR data, including anatomically accurate endocardial structures. Monophasic shocks were applied between a basal right ventricular catheter and an exterior ground. Shocks of varying strengths were applied with both anodal [positive right ventricle (RV) electrode] and cathodal (negative RV electrode) polarities at different states of tissue refractoriness and during induced arrhythmias. Anodal shocks induced isolated positive VEs at the distal side of "detached" trabeculations, which rapidly spread into hyperpolarised tissue on the surrounding endocardial surfaces following the shock. Anodal shocks thus depolarised more tissue 10 ms after the shock than cathodal shocks where the propagation of activation from VEs induced on the proximal side of "detached" trabeculations was prevented due to refractory endocardium. Anodal shocks increased arrhythmia complexity more than cathodal shocks during failed anti-arrhythmia shocks. In conclusion, multiple detached trabeculations in the human ventricle interact with anodal stimuli to induce multiple secondary sources from VEs, facilitating more rapid shock-induced ventricular excitation compared to cathodal shocks. Such a mechanism may help explain inter-species differences in response to shocks and help to develop novel defibrillation strategies.

Intramyocardial injection of hydrogel with high interstitial spread does not impact action potential propagation

Injectable biomaterials have been evaluated as potential new therapies for myocardial infarction (MI) and heart failure. These materials have improved left ventricular (LV) geometry and ejection fraction, yet there remain concerns that biomaterial injection may create a substrate for arrhythmia. Since studies of this risk are lacking, we utilized optical mapping to assess the effects of biomaterial injection and interstitial spread on cardiac electrophysiology. Healthy and infarcted rat hearts were injected with a model poly(ethylene glycol) hydrogel with varying degrees of interstitial spread. Activation maps demonstrated delayed propagation of action potentials across the LV epicardium in the hydrogel-injected group when compared to saline and no-injection groups. However, the degree of the electrophysiological changes depended on the spread characteristics of the hydrogel, such that hearts injected with highly spread hydrogels showed no conduction abnormalities. Conversely, the results of this study indicate that injection of a hydrogel exhibiting minimal interstitial spread may create a substrate for arrhythmia shortly after injection by causing LV activation delays and reducing gap junction density at the site of injection. Thus, this work establishes site of delivery and interstitial spread characteristics as important factors in the future design and use of biomaterial therapies for MI treatment.

STATEMENT OF SIGNIFICANCE

Biomaterials for treating myocardial infarction have become an increasingly popular area of research. Within the past few years, this work has transitioned to some large animals models, and Phase I & II clinical trials. While these materials have preserved/improved cardiac function the effect of these materials on arrhythmogenesis, which is of considerable concern when injecting anything into the heart, has yet to be understood. Our manuscript is therefore a first of its kind in that it directly examines the potential of an injectable material to create a substrate for arrhythmias. This work suggests that site of delivery and distribution in the tissue are important criteria in the design and development of future biomaterial therapies for myocardial infarction treatment.

Screening system for drug-induced arrhythmogenic risk combining a patch clamp and heart simulator

To save time and cost for drug discovery, a paradigm shift in cardiotoxicity testing is required. We introduce a novel screening system for drug-induced arrhythmogenic risk that combines in vitro pharmacological assays and a multiscale heart simulator. For 12 drugs reported to have varying cardiotoxicity risks, dose-inhibition curves were determined for six ion channels using automated patch clamp systems. By manipulating the channel models implemented in a heart simulator consisting of more than 20 million myocyte models, we simulated a standard electrocardiogram (ECG) under various doses of drugs. When the drug concentrations were increased from therapeutic levels, each drug induced a concentration-dependent characteristic type of ventricular arrhythmia, whereas no arrhythmias were observed at any dose with drugs known to be safe. We have shown that our system combining in vitro and in silico technologies can predict drug-induced arrhythmogenic risk reliably and efficiently.

Modelling extracellular electrical stimulation: part 4. Effect of the cellular composition of neural tissue on its spatio-temporal filtering properties

OBJECTIVE

The objective of this paper is to present a concrete application of the cellular composite model for calculating the membrane potential, described in an accompanying paper.

APPROACH

A composite model that is used to determine the membrane potential for both longitudinal and transverse modes of stimulation is demonstrated.

MAIN RESULTS

Two extreme limits of the model, near-field and far-field for an electrode close to or distant from a neuron, respectively, are derived in this paper. Results for typical neural tissue are compared using the composite, near-field and far-field models as well as the standard isotropic volume conductor model. The self-consistency of the composite model, its spatial profile response and the extracellular potential time behaviour are presented. The magnitudes of the longitudinal and transverse components for different values of electrode-neurite separations are compared.

SIGNIFICANCE

The unique features of the composite model and its simplified versions can be used to accurately estimate the spatio-temporal response of neural tissue to extracellular electrical stimulation.

Nonlinear and Stochastic Dynamics in the Heart

In a normal human life span, the heart beats about 2 to 3 billion times. Under diseased conditions, a heart may lose its normal rhythm and degenerate suddenly into much faster and irregular rhythms, called arrhythmias, which may lead to sudden death. The transition from a normal rhythm to an arrhythmia is a transition from regular electrical wave conduction to irregular or turbulent wave conduction in the heart, and thus this medical problem is also a problem of physics and mathematics. In the last century, clinical, experimental, and theoretical studies have shown that dynamical theories play fundamental roles in understanding the mechanisms of the genesis of the normal heart rhythm as well as lethal arrhythmias. In this article, we summarize in detail the nonlinear and stochastic dynamics occurring in the heart and their links to normal cardiac functions and arrhythmias, providing a holistic view through integrating dynamics from the molecular (microscopic) scale, to the organelle (mesoscopic) scale, to the cellular, tissue, and organ (macroscopic) scales. We discuss what existing problems and challenges are waiting to be solved and how multi-scale mathematical modeling and nonlinear dynamics may be helpful for solving these problems.

Effects of macroscopic heterogeneity on propagation in a computationally inexpensive 2D model of the heart

We have developed a computationally inexpensive, two-dimensional, bidomain model of the heart to demonstrate the effect of tissue heterogeneity on propagation of cardiac impulses generated by the sino-atrial node (SAN). The geometry consists of a thin sheet of cardiac tissue with designated areas that represent the SAN and atria. The SAN auto-generates continuous impulses that result in waves of normal propagation throughout the tissue. On the introduction of heterogeneous patches with low tissue conductivities, the rhythm of the waveform becomes irregular. The study suggests that simplified and computationally inexpensive models can be insightful tools to better understand the mechanisms that cause atrial fibrillation (AF) and hence more effective treatment methods.

A Simplified Approach for Simultaneous Measurements of Wavefront Velocity and Curvature in the Heart Using Activation Times

The velocity and curvature of a wave front are important factors governing the propagation of electrical activity through cardiac tissue, particularly during heart arrhythmias of clinical importance such as fibrillation. Presently, no simple computational model exists to determine these values simultaneously. The proposed model uses the arrival times at four or five sites to determine the wave front speed (v), direction (θ), and radius of curvature (ROC) (r₀). If the arrival times are measured, then v, θ, and r₀ can be found from differences in arrival times and the distance between these sites. During isotropic conduction, we found good correlation between measured values of the ROC r₀ and the distance from the unipolar stimulus (r = 0.9043 and p < 0.0001). The conduction velocity (m/s) was correlated (r = 0.998, p < 0.0001) using our method (mean = 0.2403, SD = 0.0533) and an empirical method (mean = 0.2352, SD = 0.0560). The model was applied to a condition of anisotropy and a complex case of reentry with a high voltage extra stimulus. Again, results show good correlation between our simplified approach and established methods for multiple wavefront morphologies. In conclusion, insignificant measurement errors were observed between this simplified approach and an approach that was more computationally demanding. Accuracy was maintained when the requirement that ε (ε = b/r₀, ratio of recording site spacing over wave fronts ROC) was between 0.001 and 0.5. The present simplified model can be applied to a variety of clinical conditions to predict behavior of planar, elliptical, and reentrant wave fronts. It may be used to study the genesis and propagation of rotors in human arrhythmias and could lead to rotor mapping using low density endocardial recording electrodes.

Modeling for cardiac excitation propagation based on the Nernst-Planck equation and homogenization

The bidomain model is a commonly used mathematical model of the electrical properties of the cardiac muscle that takes into account the anisotropy of both the intracellular and extracellular spaces. However, the equations contain self-contradiction such that the update of ion concentrations does not consider intracellular or extracellular ion movements due to the gradient of electric potential and the membrane charge as capacitive currents in spite of the fact that those currents are taken into account in forming Kirchhoff's first law. To overcome this problem, we start with the Nernst-Planck equation, the ionic conservation law, and the electroneutrality condition at the cellular level, and by introducing a homogenization method and assuming uniformity of variables at the microscopic scale, we derive rational bidomain equations at the macroscopic level.

The Mechanical Bidomain Model: A Review

The mechanical bidomain model is a new mathematical description of the elastic behavior of cardiac tissue. Its primary advantage over previous models is that it accounts for forces acting across the cell membrane arising form differences in the displacement of the intracellular and extracellular spaces. In this review, I describe the development of the mechanical bidomain model. I emphasize new predictions of the model, such as the existence of boundary layers at the tissue surface where the membrane forces are large, and pressure differences between the intracellular and extracellular spaces. Although the theoretical analysis is quite mathematical, I highlight the types of experiments that could be used to test the model predictions. Finally, I present open questions about the mechanical bidomain model that may be productive future directions for research.

COMPUTER MODELS OF DEPOLARIZATION ALTERATIONS INDUCED BY MYOCARDIAL ISCHEMIA: THE EFFECT OF SUPERIMPOSED ISCHEMIC INHOMOGENEITIES ON PROPAGATION IN SPACE AND TIME-FREQUENCY DOMAINS

A two-dimensional modified Luo-Rudy model was created to represent a 40 mm by 40 mm slab of myocardial tissue. An inhomogeneity was introduced to simulate acute myocardial ischemia, with components of hyperkalemia, acidosis and anoxia. Simulations were carried out for various degrees of ischemia, to study both the interaction of the propagation front with the inhomogeneity, and the reconstructed signals. The simulations utilized a modified LR model, with a realistic anisotropy of myocardial tissue. Each cluster (.4 mm ×.4 mm) was given bulk electric properties, Rx and Ry (25Ω and 250Ω, respectively). The slab was stimulated and the 2D depolarization pattern was computed by numerical integration. To study ischemia, a circular inhomogeneity with concentric regions (ro=12.8 mm{border zone, BZ} , ri=11.2 mm{extreme zone, EZ} ) regions was introduced in the model. From the 2D simulations and the regional action potentials (AP), unipolar and bipolar lead potentials were reconstructed. Time-frequency decomposition was performed on the lead signals by wavelet analysis. Isochrone and (dV/dt) max maps were obtained to study depolarization. Our results indicate that spatial inhomogeneities yield dramatic spatial dispersion of the wavefront and are the origin of mid-frequency intra-QRS components in cardiac signals. Severe APD shortening and spatial distortion of the isochrone and upstroke maps are also observed.

Mechanical bidomain model of cardiac tissue

Intracellular and extracellular spaces are separately considered in an electrical bidomain model of tissue. We propose a mechanical bidomain model separately considering the intracellular and extracellular spaces, coupled through a linear restoring force proportional to the displacement difference of the two spaces. We consider a mechanically passive model of heart fibers (no tension) with an action potential, and an electrically passive model (no action potential) in tissue with an ischemic boundary. We find the pressure and displacement fields arising from our consideration of a bidomain instead of a monodomain and note interesting characteristics evident only with a bidomain approach.

Cathodal stimulation in the recovery phase of a propagating planar wave in the rabbit heart reveals four stimulation mechanisms

The stimulation of cardiac tissue in the recovery phase has significant importance in relation to reentry induction. In the theoretical experiment proposed by Winfree, termed the 'pinwheel' experiment, a point stimulus (S2) is applied in the wake of a freely propagating planar wave (S1). Reentry induced from this S1-S2 pinwheel protocol has been observed experimentally in heart preparations. However, in these experiments, which focused on activation outcomes, only mapping of extracellular voltages has been conducted. The lack of transmembrane potential (Vm) distribution data makes it impossible to analyse the underlying stimulation mechanisms which precede the reentry induction. In this work we sought to elucidate the stimulation mechanisms throughout the heart cycle using the pinwheel protocol. We examined the cardiac tissue responses during and immediately after cathodal stimulation in the refractory wake of a propagating planar wave. The voltage-sensitive dye di-4-ANEPPS was utilized to measure Vm directly from quasi two-dimensional preparations of cryoablated Langendorff-perfused rabbit hearts. Four stimulation mechanisms were observed that depended on the Vm magnitude during S2 cathodal stimulation. Make stimulation always occurred during diastolic stimulation. When stimulation was at the beginning of the relative refractory period (RRP), transitional make-break stimulation was detected. During the RRP the excitation was due to the break mechanism. While approaching the effective refractory period (ERP), the tissue response is characterized by a damped wave mediated response. These four stimulation mechanisms were observed in all hearts whether the S1 planar wave propagation was parallel or perpendicular to the fibre direction. This study is the first examination of Vm and the stimulation mechanisms throughout the cardiac cycle using the pinwheel protocol, and the results have implications in the development and improvement of pacing protocols for artificial cardiostimulators.

Automated detection of qualitative spatio-temporal features in electrocardiac activation maps

OBJECTIVE

This paper describes a piece of work aiming at the realization of a tool for the automated interpretation of electrocardiac maps. Such maps can capture a number of electrical conduction pathologies, such as arrhytmia, that can be missed by the analysis of traditional electrocardiograms. But, their introduction into the clinical practice is still far away as their interpretation requires skills that belongs to very few experts. Then, an automated interpretation tool would bridge the gap between the established research outcome and clinical practice with a consequent great impact on health care.

METHODS AND MATERIAL

Qualitative spatial reasoning can play a crucial role in the identification of spatio-temporal patterns and salient features that characterize the heart electrical activity. We adopted the spatial aggregation (SA) conceptual framework and an interplay of numerical and qualitative information to extract features from epicardial maps, and to make them available for reasoning tasks.

RESULTS

Our focus is on epicardial activation isochrone maps as they are a synthetic representation of spatio-temporal aspects of the propagation of the electrical excitation. We provide a computational SA-based methodology to extract, from 3D epicardial data gathered over time, (1) the excitation wavefront structure, and (2) the salient features that characterize wavefront propagation and visually correspond to specific geometric objects.

CONCLUSION

The proposed methodology provides a robust and efficient way to identify salient pieces of information in activation time maps. The hierarchical structure of the abstracted geometric objects, crucial in capturing the prominent information, facilitates the definition of general rules necessary to infer the correlation between pathophysiological patterns and wavefront structure and propagation.

Quatrefoil Reentry Caused by Burst Pacing

BACKGROUND

Experiments and clinical studies have shown that high-frequency (burst) pacing can induce reentry and fibrillation without a strong shock. We hypothesize that a train of weak stimuli induces quatrefoil reentry, and investigate the mechanism and threshold for this mode of reentry induction.

METHODS

We apply a train of weak stimuli at different pacing rates to determine the threshold necessary to induce quatrefoil reentry. Numerical calculations are used to simulate cardiac tissue, based on the bidomain model with unequal anisotropy ratios. We consider both anodal and cathodal stimuli.

RESULTS

Quatrefoil reentry is initiated using much smaller currents during burst pacing (0.9 mA) compared to a single premature pulse (8.6 mA). As we varied the pacing rate, we observed reentry at the border between different modes of phase locking, such as between 1:1 and 2:1 responses.

CONCLUSION

Burst pacing can significantly reduce the threshold for reentry. However, the extreme sensitivity of reentry induction to the exact number of stimuli in the pulse train makes the method difficult to use as a consistent, reproducible way to induce reentry.

How to explain why "unequal anisotropy ratios" is important using pictures but no mathematics

"Unequal anisotropy ratios" is an important property of cardiac tissue. Many of the fundamental mechanisms governing how the heart responds to an electrical shock require unequal anisotropy ratios. In this paper, I explain the role of unequal anisotropy ratios using pictures rather than mathematics. My goal is to develop physical insight, so has to understand qualitatively why the condition of unequal anisotropy ratios is so important.

A bidomain model of epiretinal stimulation

A computational bidomain model of epiretinal stimulation is presented, consisting of a continuum description of active retinal tissue in contact with bulk vitreous fluid. Results from two-electrode and four-electrode bipolar stimulation suggest that a biphasic cathodic-anodic stimulus sequence is effective in providing targeted focal activation of retinal tissue. Undesired secondary activations beneath each electrode return may be eliminated by using multiple returns for each stimulus electrode.

Field Stimulation of 2-D Sheets of Excitable Tissue

This paper develops equations for the transmembrane potentials (Vm) that occur in two-dimensional (2-D) sheets of tissue in response to field stimulation from an electrode near but not on the surface of the tissue. Comparison of results with those for one dimension shows that an additional term is present in the 2-D equations that influences the evolution of Vm in the interval between the end of the stimulus and the active propagation that may follow. The results provide an analytical framework for understanding Vm in response to field stimulation in two dimensions, both during the tissue's critical linear phase and thereafter.

New currents in electrical stimulation of excitable tissues

Electric fields can stimulate excitable tissue by a number of mechanisms. A uniform long, straight peripheral axon is activated by the gradient of the electric field that is oriented parallel to the fiber axis. Cortical neurons in the brain are excited when the electric field, which is applied along the axon-dendrite axis, reaches a particular threshold value. Cardiac tissue is thought to be depolarized in a uniform electric field by the curved trajectories of its fiber tracts. The bidomain model provides a coherent conceptual framework for analyzing and understanding these apparently disparate phenomena. Concepts such as the activating function and virtual anode and cathode, as well as anode and cathode break and make stimulation, are presented to help explain these excitation events in a unified manner. This modeling approach can also be used to describe the response of excitable tissues to electric fields that arise from charge redistribution (electrical stimulation) and from time-varying magnetic fields (magnetic stimulation) in a self-consistent manner. It has also proved useful to predict the behavior of excitable tissues, to test hypotheses about possible excitation mechanisms, to design novel electrophysiological experiments, and to interpret their findings.

Quantal and non-quantal current and potential fields around individual sympathetic varicosities on release of ATP

The electrical phenomena that occur at sympathetic varicosities due to the release of ATP include spontaneous and evoked excitatory junction potentials (SEJPs and EJPs; recorded with an intracellular electrode) as well as fast and slow excitatory junctional currents (EJCs; recorded with a loose-patch electrode placed over varicosities). The electrical analysis of these transients is hampered by lack of a detailed theory describing how current and potential fields are generated upon the release of a quantum of ATP. Here, we supply such a theory and develop a computational model for the electrical properties of a smooth muscle syncytium placed within a volume conductor, using a distributed representation for the individual muscle cells. The amplitudes and temporal characteristics of both SEJPs and fast EJCs are predicted by the theory, but those of the slow EJCs are not. It is shown that these slow components cannot arise as a consequence of propagation of fast quantal components from their site of origin in the muscle syncytium to the point of recording. The possibility that slow components arise by a mechanism of transmitter secretion that is different from quantal release is examined. Experiments that involve inserting peptide fragments of soluble N-ethylmaleimide-sensitive fusion attachment protein (alpha-SNAP) into varicosities, a procedure that is known to block quantal release, left the slow component of release unaffected. This work provides an internally consistent description of quantal potential and current fields about the varicosities of sympathetic nerve terminals and provides evidence for a non-quantal form of transmitter release.

How electrode size affects the electric potential distribution in cardiac tissue

We investigate the effect of electrode size on the transmembrane potential distribution in the heart during electrical stimulation. The bidomain model is used to calculate the transmembrane potential in a three-dimensional slab of cardiac tissue. Depolarization is strongest under the electrode edge. Regions of depolarization are adjacent to regions of hyperpolarization. The average ratio of peak depolarization to peak hyperpolarization is a function of electrode radius, but over a broad range is close to three.

Role of virtual electrodes in arrhythmogenesis: pinwheel experiment revisited

INTRODUCTION

Recent experimental evidence demonstrates that a point stimulus generates a nonuniform distribution of transmembrane potential (virtual electrode pattern) consisting of large adjacent areas of depolarization and hyperpolarization. This simulation study focuses on the role of virtual electrodes in reentry induction.

METHODS AND RESULTS

We simulated the electrical behavior of a sheet of myocardium using a two-dimensional bidomain model with straight fibers. Membrane kinetics were represented by the Beeler-Reuter Drouhard-Roberge model. Simulations were conducted for equal and unequal anisotropy ratios. S1 wavefront was planar and propagated parallel or perpendicular to the fibers. S2 unipolar stimulus was cathodal or anodal. With regard to unequal anisotropy, for both cathodal and anodal stimuli, the S2 stimulus negatively polarizes some portion of membrane, deexciting it and opening an excitable pathway in a region of otherwise unexcitable tissue. Reentry is generated by break excitation of this tissue and subsequent propagation through deexcited and recovered areas of myocardium. Figure-of-eight and quatrefoil reentry are observed, with figure-of-eight most common. Figure-of-eight rotation is seen in the direction predicted by the critical point hypothesis. With regard to equal anisotropy, reentry was observed for cathodal stimuli only at strengths > -95 A/m.

CONCLUSION

The key to reentry induction is the close proximity of S2-induced excited and deexcited areas, with adjacent nonexcited areas available for propagation.

Mechanism for polarisation of cardiac tissue at a sealed boundary

If current is flowing in cardiac tissue, and if the myocardial fibres approach a sealed boundary at an angle, then the tissue within a few length constants of the boundary is polarised. This polarisation occurs when the cardiac tissue has different anisotropy ratios in the intracellular and extracellular spaces. This new mechanism of tissue polarisation is demonstrated using a simple, analytical model, and it is shown quantitatively that this polarisation can be nearly as large as that occurring near an electrode.

Patterns of and mechanisms for shock-induced polarization in the heart: a bidomain analysis

This paper examines the combined action of cardiac fiber curvature and transmural fiber rotation in polarizing the myocardium under the conditions of a strong electrical shock. The study utilizes a three-dimensional finite element model and the continuous bidomain representation of cardiac tissue to model steady-state polarization resulting from a defibrillation-strength uniform applied field. Fiber architecture is incorporated in the model via the shape of the heart, an ellipsoid of variable ellipticity index, and via an analytical function, linear or nonlinear, describing the transmural fiber rotation. Analytical estimates and numerical results are provided for the location and shape of the "bulk" polarization (polarization away from the tissue boundaries) as a function of the fiber field, or more specifically, of the conductivity changes in axial and radial direction with respect to the applied electrical field lines. Polarization in the tissue "bulk" is shown to exist only under the condition of unequal anisotropy ratios in the extra- and intracellular spaces. Variations in heart geometry and, thus, fiber curvature, are found to lead to change in location of the zones of significant membrane polarization. The transmural fiber rotation function modulates the transmembrane potential profile in the radial direction. A higher gradient of the transmural transmembrane potential is observed in the presence of fiber rotation as compared to the no rotation case. The analysis presented here is a step forward in understanding the interaction between tissue structure and applied electric field in establishing the pattern of membrane polarization during the initial phase of the defibrillation shock.

A novel mechanism of anode-break stimulation predicted by bidomain modeling

-Anodal stimulation by external pacemakers has been explained on the basis of bidomain models of cardiac tissue. Bidomain models predict that anodal stimuli will hyperpolarize the underlying tissue while adjacent regions become depolarized (virtual cathodes), initiating excitation. We investigated the contribution of active cellular properties to anode-break stimulation. A bidomain model was implemented in which each cell contained realistic ionic currents, including those recruited by hyperpolarization. Simulations reveal that anode-break excitation can originate at the site of stimulation itself and not only from adjacent regions of induced depolarization. The threshold for initiating excitation at the site of stimulation is lower than that for stimulation initiating from adjacent depolarized regions. Thus, incorporation of active cellular properties into a bidomain model predicts a novel mechanism for anode-break stimulation of the heart. The results will improve our understanding of anodal pacing and its risks and benefits in patients.

Electrical stimulation of cardiac tissue by a bipolar electrode in a conductive bath

A three-dimensional (3-D) computer simulation of the electrical stimulation of passive cardiac tissue from a bipolar electrode placed within a conductive bath is presented. Through the bidomain model, the syncytial and anisotropic properties of cardiac tissue are taken into account; tissues with equal anisotropy and no transverse coupling are also considered. The membrane is represented by a capacitor and passive resistor in parallel. Located within an isotropic bath, the bipolar electrode is oriented either perpendicular or parallel to the tissue surface. For anisotropic tissue with a small cathode-tissue separation, the tissue surface is highly depolarized under the cathode with the depolarization persisting a considerable distance from the electrode in the transverse fiber direction. Adjacent to this region in the longitudinal direction, areas of hyperpolarization exist. At large distances from the cathode, the tissue surface is hyperpolarized in all directions when the electrode axis is perpendicular to the tissue. In the parallel case, surface depolarization creates buried regions of hyperpolarization. For the perpendicular configuration, the ratio of the steady-state maximum depolarization to steady-state maximum hyperpolarization, an estimate of the ratio of anodal to cathodal threshold, decreases rapidly with increasing cathode-tissue separation. In the parallel case, the depth of the conductive bath significantly affected the transmembrane potential distribution in the tissue. The use of a 3-D model more realistically simulates real-life electrical stimulation (such as stimulation with an implantable pacemaker) and provides insight into the effect of the volume conductor adjacent to the tissue.

Virtual electrode effects in defibrillation

This modeling study demonstrates that a re-entrant activity in a sheet of myocardium can be extinguished by a defibrillation shock delivered via extracellular point-source electrodes which establish spatially non-uniform applied field. The tissue is represented as a homogeneous bidomain with unequal anisotropy ratios in the cardiac conductivities. Spiral wave re-entry is initiated in the bidomain sheet following an S1-S2 stimulation protocol. The results indicate that the point-source defibrillation shock establishes large-scale changes in transmembrane potential in the tissue (virtual electrodes) that are 'superimposed' over regions of various degrees of membrane refractoriness in the myocardium. The close proximity of large-scale shock-induced regions of alternating membrane polarity is central to the ability of the shock to terminate the spiral wave. The new wavefronts generated following anode/cathode break phenomena restrict the spiral wave and render the tissue too refractory to further maintain the re-entry. In contrast, shocks delivered via line electrodes establish, in close proximity to the electrode, changes in transmembrane potential that are of same-sign polarity. These shocks are incapable of terminating the re-entrant activation.

Sympathetic neuromuscular transmission at a varicosity in a syncytium

The autonomic neuromuscular junction at a varicosity in the vas deferens is defined by the localization of the vesicle-associated protein syntaxin in high concentrations in the axolemma and a high density of P2x1 receptors in a cluster beneath the varicosity. Calcium fluxes have been observed in all individual varicosities of a nerve terminal on the arrival of an impulse even though recordings made from these varicosities of the electrical signs of transmission with loose-patch electrodes over the varicosities show that they have very different probabilities for the secretion of a quantum. The fact that some varicosities seldom release a quantum on the arrival of an impulse is supported by the observation that antibodies against the N-terminus of synaptotagmin, which uniquely label the inside of synaptic vesicles when they undergo exocytosis, fail to do so in some varicosities during nerve stimulation whereas they do in others. It is suggested that the probability for secretion from a varicosity depends on the number of secretosomes that the varicosity possesses, where a secretosome is a complex of syntaxin, synaptotagmin, an N-type calcium channel, and a synaptic vesicle.

Virtual electrode effects in transvenous defibrillation-modulation by structure and interface: evidence from bidomain simulations and optical mapping

INTRODUCTION

Our goal in this combined modeling and experimental study was to gain insight into the transmembrane potential changes in defibrillation conditions, namely, when shocks are delivered by an implantable cardioverter defibrillator (ICD). Two hypotheses concerning the presence and characteristics of virtual electrode effects (VEE) during an ICD shock were tested numerically and experimentally: (H1) anisotropy-dependent VEE are induced over a considerable portion of the "bulk" myocardium; and (H2) surface (epicardial and endocardial) VEE are generated under special tissue bath conditions and are not fully anisotropy determined.

METHODS AND RESULTS

Optical mapping was performed on Langendorff-perfused rabbit hearts (n = 4) stained with di-4-ANEPPS. Monophasic shocks were applied during the plateau phase of an action potential through a 9-mm long distal electrode in the right or left ventricle and a 6-cm proximal electrode positioned 3 cm posteriorly to the heart. We modeled the experiment using an ellipsoidal bidomain heart with transmural fiber rotation, placed in a perfusing bath, and subjected to defibrillation shocks delivered by an electrode configuration as described. Our numerical simulations demonstrated VEE occupying a significant portion of the myocardium in the conditions of unequal anisotropy ratios for the intra- and extracellular domains. Statistically significant differences in epicardial polarization patterns were predicted numerically and confirmed experimentally when the interface conditions varied.

CONCLUSION

The present study concludes that VEE are present in transvenous defibrillation. They are shaped by the combined effect of cardiac tissue characteristics and interface conditions. Because of their size, VEE might contribute significantly to defibrillation outcome.

Mechanism of anode break stimulation in the heart

Anodal stimulation is routinely observed in cardiac tissue, but only recently has a mechanism been proposed. The bidomain cardiac tissue model proposes that virtual cathodes induced at sites distant from the electrode initiate the depolarization. In contrast, none of the existing cardiac action potential models (Luo-Rudy phase I and II, or Oxsoft) predict anodal stimulation at the single-cell level. To determine whether anodal stimulation has a cellular basis, we measured membrane potential and membrane current in mammalian ventricular myocytes by using whole-cell patch clamp. Anode break responses can be readily elicited in single ventricular cells. The basis of this anodal stimulation in single cells is recruitment of the hyperpolarization-activated inward current I(f). The threshold of activation for I(f) is -80 mV in rat cells and -120 mV in guinea pig or canine cells. Persistent I(f) "tail" current upon release of the hyperpolarization drives the transmembrane potential toward the threshold of sodium channels, initiating an action potential. Time-dependent block of the inward rectifier, I(K1), at hyperpolarized potentials decreases membrane conductance and thereby potentiates the ability of I(f) to depolarize the cell on the break of an anodal pulse. Inclusion of I(f), as well as the block and unblock kinetics of I(K1), in the existing Luo-Rudy action potential model faithfully reproduces anode break stimulation. Thus active cellular properties suffice to explain anode break stimulation in cardiac tissue.

Spread of excitation in 3-D models of the anisotropic cardiac tissue. II. Effects of fiber architecture and ventricular geometry

We investigate a three-dimensional macroscopic model of wave-front propagation related to the excitation process in the left ventricular wall represented by an anisotropic bidomain. The whole left ventricle is modeled, whereas, in a previous paper, only a flat slab of myocardial tissue was considered. The direction of cardiac fibers, which affects the anisotropic conductivity of the myocardium, rotates from the epi- to the endocardium. If the ventricular wall is conceived as a set of packed surfaces, the fibers may be tangent to them or more generally may cross them obliquely; the latter case is described by an "imbrication angle." The effect of a simplified Purkinje network also is investigated. The cardiac excitation process, more particularly the depolarization phase, is modeled by a nonlinear elliptic equation, called an eikonal equation, in the activation time. The numerical solution of this equation is obtained by means of the finite element method, which includes an upwind treatment of the Hamiltonian part of the equation. By means of numerical simulations in an idealized model of the left ventricle, we try to establish whether the eikonal approach contains the essential basic elements for predicting the features of the activation patterns experimentally observed. We discuss and compare these results with those obtained in our previous papers for a flat part of myocardium. The general rules governing the spread of excitation after local stimulations, previously delineated for the flat geometry, are extended to the present, more realistic monoventricular model.

Quantal currents and potential in the three-dimensional anisotropic bidomain model of smooth muscle

The potential generated in the smooth muscle of the vas deferens on release of a quantum of transmitter from a varicosity was analyzed using a three-dimensional bidomain continuum model. Current was injected at the origin of the bidomain; this current had the temporal characteristics of the junctional current. The membrane potential, intracellular potential, and extracellular potential, as well as the extracellular current, were then calculated throughout the bidomain at different times. Calculations were performed to show the effect of changing the anisotropy ratios of the intracellular and extracellular conductivities on the spread of current and potential in each of the three dimensions. These results provide a theoretical framework for ascertaining the time course of transmitter interaction at a varicosity following the secretion of a quantum of transmitter.

Electrical conductivity values used with the bidomain model of cardiac tissue

Electrical conductivities in the bidomain model of cardiac tissue are expressed as functions of four parameters. These expressions allow simulations to be performed using nominal, equal, and reciprocal anisotropy without introducing undesired effects, such as length constant variations. Relative values of the bidomain conductivities are estimated to be: sigma iL = 1, sigma iT = 0.1, sigma eL = 1, and sigma eT = 0.4.

Discrete versus syncytial tissue behavior in a model of cardiac stimulation--I: Mathematical formulation

This paper presents a model describing the steady-state response of a two-dimensional (2-D) slice of myocardium to extracellular current injection. The model incorporates a continuous representation of the multicellular, syncytial cardiac tissue based on the bidomain model. The classical bidomain model is modified by introducing periodic conductivities to better represent the electrical properties of the intracellular space. Thus, junctional discontinuity between abutting myocytes is reflected in the macroscopic representation of cardiac tissue behavior. Since a solution to the resulting coupled differential equations governing the intracellular and extracellular potentials in the tissue preparation is not computationally tractable when traditional numerical approaches, such as finite element or finite difference methods are used, spectral techniques are employed to reduce the problem to the solution of a set of algebraic equations for the transform of the bidomain potentials. Further, the solution to the "periodic" bidomain problem in the Fourier space is decomposed into two separate solutions: One for the classical-bidomain potentials where it is assumed that the intracellular conductivity values along and across cells incorporate the average contribution from cytoplasm and junction, and another for the junctional potential component. The decomposition of the total solution allows to approximately solve for the junctional component thus achieving high overall computational efficiency. The results of simulation are presented in an accompanying paper.

Autonomic neuromuscular transmission at a varicosity

This review attempts to clarify the definition of what constitutes an autonomic neuromuscular function formed by a varicosity. Ultrastructural studies of serial sections through varicosities, partly or wholly bare of Schwann cell covering, show that areas of close apposition occur between varicosities and muscle cell membrane that vary between 20 and 150 nm, depending on the muscle considered. Consideration of the diffusion of purine transmitters and their receptor kinetics after secretion in a packet show that the number of purinergic receptor channels opened at a site of 150 nm apposition by a varicosity is about 15% of that at a site of 50 nm apposition. These results, together with the analysis of the stochastic fast component and the deterministic slow components of the rising phase of the EJP suggest that the stochastic fast component is due to varicosities that form especially close appositions (20-50 nm), whereas the deterministic slow component is due to the large number of varicosities at distances up to about 150 nm. Varicosities forming appositions of 20-150 nm with muscle cells several hundred micrometers long possess junctional receptor types distinct from extrajunctional receptors. According to this argument, then, there are two different classes of varicosities: one that gives rise to a relatively large junctional current and another that is responsible for a very small junctional current. Present evidence suggests that two subclasses of varicosities can be discerned amongst the varicosities that generate large junctional currents. One of these subclasses of varicosity possesses relatively few post-junctional receptors compared with the amount of transmitter reaching the receptors from the varicosity, so that the junctional current generated is determined by the size of the receptor population; in this case, the size of the transmitter packages released from these varicosities is unknown and the size of the junctional current is relatively constant. The other subclass of varicosity possesses large receptor patches, sufficient to accommodate the largest amounts of transmitter released from the varicosities: in this case, the size of the transmitter packages is shown to be highly non-uniform. These speculations await confirmation by direct labelling of the receptor patches beneath varicosities, a possibility that is likely to be realized in the near future.

Discrete versus syncytial tissue behavior in a model of cardiac stimulation--II: Results of simulation

The research presented here combines mathematical modeling and computer simulation in developing a new model of the membrane polarization induced in the myocardium by the applied electric field. Employing this new model termed the "period" bidomain model, the steady-state distribution of the transmembrane potential is calculated on a slice of cardiac tissue composed of abutting myocytes and subjected to two point-source extracellular current stimuli. The goal of this study is to examine the relative contribution of cellular discreteness and macroscopic syncytial tissue behavior in the mechanism by which the applied electric field alters the transmembrane potential in cardiac muscle. The results showed the existence of oscillatory changes in the transmembrane potential at cell ends owing to the local resistive inhomogeneities (gap-junctions). This low-magnitude sawtooth component in the transmembrane potential is superimposed over large-scale transmembrane potential excursions associated with the syncytial (collective) fiber behavior. The character of the cardiac response to stimulation is determined primarily by the large-scale syncytial tissue behavior. The sawtooth contributes to the overall tissue response only in regions where the large-scale transmembrane potential component is small.

Spatial changes in transmembrane potential during extracellular electrical shocks in cultured monolayers of neonatal rat ventricular myocytes

This study investigated the role of different types of discontinuities in tissue architecture on the spatial distribution of the transmembrane potential. Specifically, we tested the occurrence of so-called "secondary sources," ie, localized hyperpolarizations and depolarizations during the application of extracellular electrical shocks (EESs). Changes in transmembrane potential relative to action potential amplitude (delta Vm/APA) were measured in patterned cultures of neonatal rat myocytes, stained with voltage-sensitive dye (RH-237), by optical mapping (96-channel photodiode array, 6- to 30-micron resolution) during the application of EES (field strength, 8 to 22 V/cm; duration, 6 ms). Across narrow cell strands (width, 218 +/- 59 [mean +/- SD] microns), EES applied during the relative refractory period produced a linear and symmetrical profile of delta Vm/APA (-65 +/- 23% maximal hyperpolarization versus +64 +/- 15% maximal depolarization). In contrast, the profile of delta Vm/APA was asymmetrical when EESs were applied during the action potential plateau (-95 +/- 32% versus +37 +/- 14%). At high magnification, no secondary sources were observed at the borders between cells. In dense isotropic cell monolayers or in monolayers and strands showing intercellular clefts, secondary sources were frequently observed. Intercellular clefts of the size of one to several myocytes were sufficient to produce secondary sources of the same magnitude as those that elicited action potentials in dense cell strands. There was a close correlation between the location of secondary sources during EES and localized conduction slowing during propagation. Thus, densely packed cultured cell strands behave as an electrical continuum with no secondary sources occurring at cell borders. Small intercellular clefts can create secondary sources of sufficient magnitude to exert a stimulatory effect.

Strength-interval curves for cardiac tissue predicted using the bidomain model

INTRODUCTION

Strength-interval curves are predicted for unipolar anodal and cathodal stimulation of cardiac muscle.

METHODS AND RESULTS

Cardiac tissue is represented by the bidomain model, and the active properties of the membrane are described by the Beeler-Reuter model. Two successive stimuli (S1 and S2) are delivered through a single extracellular electrode. The S2 threshold is determined as a function of the S1-S2 interval, for anodal and cathodal S2 stimuli with 2-, 5-, 10-, and 20-msec durations. Each of the resulting cathodal and anodal strength-interval curves is divided into two parts: one section corresponding to make stimulation (long intervals) and the other section corresponding to break stimulation (short intervals). Generally, the cathodal strength-interval curves are decreasing functions of interval, except for an anomalous section of the 20-msec duration cathodal curve in the interval range from 310 to 318 msec. At short intervals, the anodal strength-interval curve contains a deep dip, which is more prominent for longer S2 durations. The cathodal threshold is less than the anodal threshold for all intervals except those corresponding to the end of the refractory period.

CONCLUSION

The bidomain model predicts complex anodal and cathodal strength-interval curves, with the anodal curve containing a dip (supernormal stimulation). These results resemble the experimental observations of Dekker.

A mathematical model of make and break electrical stimulation of cardiac tissue by a unipolar anode or cathode

Numerical simulations of electrical stimulation of cardiac tissue using a unipolar extracellular electrode were performed. The bidomain model with unequal anisotropy ratios represented the tissue, and the Beeler-Reuter model represented the active membrane properties. Four types of excitation were considered: cathode make (CM), anode make (AM), cathode break (CB), and anode break (AB). The mechanisms of excitation were: for CM, tissue under the cathode was depolarized to threshold; for AM, tissue at a virtual cathode was depolarized to threshold; for CB, a long cathodal pulse produced a steady-state depolarization under the cathode and hyperpolarization at a virtual anode. At the end (break) of the pulse, the depolarization diffused into the hyperpolarized tissue, resulting in excitation. For AB, a long anodal pulse produced a steady-state hyperpolarization under the anode and depolarization at a virtual cathode. At the end (break) of the pulse, the depolarization diffused into the hyperpolarized tissue, resulting in excitation. For AB stimulation, decay of the hyperpolarization faster than that of the depolarization was necessary. The thresholds for rheobase and diastolic CM, AM, CB, and AB stimulation were 0.038, 0.41, 0.49, and 5.3 mA, respectively, for an electrode length of 1 mm and a surface area of 1.5 mm2. Threshold increased as the size of the electrode increased. The strength-duration curves for CM and AM were similar except when the duration was shorter than 0.2 ms, in which case the AM threshold rose more quickly with decreasing duration than did the CM threshold. CM and AM resulted in similar strength-frequency curves. The model agrees qualitatively, but (in some cases) not quantitatively, with experiments.