Exploiting mathematical models to illuminate electrophysiological variability between individuals

Alterations in T-tubule and dyad structure in heart disease: challenges and opportunities for computational analyses

Compelling recent experimental results make clear that sub-cellular structures are altered in ventricular myocytes during the development of heart failure, in both human samples and diverse experimental models. These alterations can include, but are not limited to, changes in the clusters of sarcoplasmic reticulum (SR) Ca(2+)-release channels, ryanodine receptors, and changes in the average distance between the cell membrane and ryanodine receptor clusters. In this review, we discuss the potential consequences of these structural alterations on the triggering of SR Ca(2+) release during excitation-contraction coupling. In particular, we describe how mathematical models of local SR Ca(2+) release can be used to predict functional changes resulting from diverse modifications that occur in disease states. We review recent studies that have used simulations to understand the consequences of sub-cellular structural changes, and we discuss modifications that will allow for future modelling studies to address unresolved questions. We conclude with a discussion of improvements in both experimental and mathematical modelling techniques that will be required to provide a stronger quantitative understanding of the functional consequences of changes in sub-cellular structure in heart disease.

The effects of temperature on the stability of a neuronal oscillator

The crab Cancer borealis undergoes large daily fluctuations in environmental temperature (8-24°C) and must maintain appropriate neural function in the face of this perturbation. In the pyloric circuit of the crab stomatogastric ganglion, we pharmacologically isolated the pacemaker kernel (the AB and PD neurons) and characterized its behavior in response to temperature ramps from 7°C to 31°C. For moderate temperatures, the pacemaker displayed a frequency-temperature curve statistically indistinguishable from that of the intact circuit, and like the intact circuit maintained a constant duty cycle. At high temperatures (above 23°C), a variety of different behaviors were seen: in some preparations the pacemaker increased in frequency, in some it slowed, and in many preparations the pacemaker stopped oscillating ("crashed"). Furthermore, these crashes seemed to fall into two qualitatively different classes. Additionally, the animal-to-animal variability in frequency increased at high temperatures. We used a series of Morris-Lecar mathematical models to gain insight into these phenomena. The biophysical components of the final model have temperature sensitivities similar to those found in nature, and can crash via two qualitatively different mechanisms that resemble those observed experimentally. The crash type is determined by the precise parameters of the model at the reference temperature, 11°C, which could explain why some preparations seem to crash in one way and some in another. Furthermore, even models with very similar behavior at the reference temperature diverge greatly at high temperatures, resembling the experimental observations.

Benchmarking electrophysiological models of human atrial myocytes

Mathematical modeling of cardiac electrophysiology is an insightful method to investigate the underlying mechanisms responsible for arrhythmias such as atrial fibrillation (AF). In past years, five models of human atrial electrophysiology with different formulations of ionic currents, and consequently diverging properties, have been published. The aim of this work is to give an overview of strengths and weaknesses of these models depending on the purpose and the general requirements of simulations. Therefore, these models were systematically benchmarked with respect to general mathematical properties and their ability to reproduce certain electrophysiological phenomena, such as action potential (AP) alternans. To assess the models' ability to replicate modified properties of human myocytes and tissue in cardiac disease, electrical remodeling in chronic atrial fibrillation (cAF) was chosen as test case. The healthy and remodeled model variants were compared with experimental results in single-cell, 1D and 2D tissue simulations to investigate AP and restitution properties, as well as the initiation of reentrant circuits.

Decoding myocardial Ca²⁺ signals across multiple spatial scales: a role for sensitivity analysis

Numerous studies have employed mathematical modeling to quantitatively understand release of Ca(2+) from the sarcoplasmic reticulum (SR) in the heart. Models have been used to investigate physiologically important phenomena such as triggering of SR Ca(2+) release by Ca(2+) entry across the cell membrane and spontaneous leak of Ca(2+) from the SR in quiescent heart cells. In this review we summarize studies that have modeled myocardial Ca(2+) at different spatial scales: the sub-cellular level, the cellular level, and the multicellular level. We discuss each category of models from the standpoint of parameter sensitivity analysis, a common simulation procedure that can generate quantitative, comprehensive predictions about how changes in conditions influence model output. We propose that this is a useful perspective for conceptualizing models, in part because a sensitivity analysis requires the investigator to define the relevant parameters and model outputs. This procedure therefore helps to illustrate the capabilities and limitations of each model. We further suggest that in future studies, sensitivity analyses will aid in simplifying complex models and in suggesting experiments to differentiate between competing models built with different assumptions. We conclude with a discussion of unresolved questions that are likely to be addressed over the next several years.

A network-oriented perspective on cardiac calcium signaling

The normal contractile, electrical, and energetic function of the heart depends on the synchronization of biological oscillators and signal integrators that make up cellular signaling networks. In this review we interpret experimental data from molecular, cellular, and transgenic models of cardiac signaling behavior in the context of established concepts in cell network architecture and organization. Focusing on the cellular Ca(2+) handling machinery, we describe how the plasticity and adaptability of normal Ca(2+) signaling is dependent on dynamic network configurations that operate across a wide range of functional states. We consider how (mal)adaptive changes in signaling pathways restrict the dynamic range of the network such that it cannot respond appropriately to physiologic stimuli or perturbation. Based on these concepts, a model is proposed in which pathologic abnormalities in cardiac rhythm and contractility (e.g., arrhythmias and heart failure) arise as a consequence of progressive desynchronization and reduction in the dynamic range of the Ca(2+) signaling network. We discuss how a systems-level understanding of the network organization, cellular noise, and chaotic behavior may inform the design of new therapeutic modalities that prevent or reverse the disease-linked unraveling of the Ca(2+) signaling network.

Synchronization of early afterdepolarizations and arrhythmogenesis in heterogeneous cardiac tissue models

Early afterdepolarizations (EADs) are linked to both triggered arrhythmias and reentrant arrhythmias by causing premature ventricular complexes (PVCs), focal excitations, or heterogeneous tissue substrates for reentry formation. However, a critical number of cells that synchronously exhibit EADs are needed to result in arrhythmia triggers and substrates in tissue. In this study, we use mathematical modeling and computer simulations to investigate EAD synchronization and arrhythmia induction in tissue models with random cell-to-cell variations. Our major observations are as follows. Random cell-to-cell variations in action potential duration without EAD presence do not cause large dispersion of refractoriness in well-coupled tissue. In the presence of phase-2 EADs, the cells may synchronously exhibit the same number of EADs or no EADs with a very small dispersion of refractoriness, or synchronize regionally to result in large dispersion of refractoriness. In the presence of phase-3 EADs, regional synchronization leads to propagating EADs, forming PVCs in tissue. Interestingly, even though the uncoupled cells exhibit either no EAD or only a single EAD, when these cells are coupled to form a tissue, more than one PVC can occur. When the PVCs occur at different locations and time, multifocal arrhythmias are triggered, with the foci shifting in space and time in an irregular manner. The focal arrhythmias either spontaneously terminate or degenerate into reentrant arrhythmias due to heterogeneities and spatiotemporal chaotic dynamics of the foci.