The basic building blocks of the electrophysiology of cardiomyocytes are ion channels integrated ... more The basic building blocks of the electrophysiology of cardiomyocytes are ion channels integrated in the cell membranes. Close to the ion channels there are very strong electrical and chemical gradients. However, these gradients extend for only a few nano-meters and are therefore commonly ignored in mathematical models. The full complexity of the dynamics is modelled by the Poisson-Nernst-Planck (PNP) equations but these equations must be solved using temporal and spatial scales of nano-seconds and nano-meters. Here we report solutions of the PNP equations in a fraction of two abuttal cells separated by a tiny extracellular space. We show that when only the potassium channels of the two cells are open, a stationary solution is reached with the well-known Debye layer close to the membranes. When the sodium channels of the left cell are opened, a very strong and brief electrochemical wave emanates from the channels. If the extracellular space is sufficiently small and the number of sod...
Source code for the chapter "Operator splitting and finite difference schemes for solving th... more Source code for the chapter "Operator splitting and finite difference schemes for solving the EMI model" by Karoline Horgmo Jæger, Kristian Gregorius Hustad, Xing Cai, and Aslak Tveito from the book <em>EMI: Cell-based Mathematical Model of Excitable Cells</em> The code can also be found in a GitHub repository at https://github.com/KGHustad/emi-book-2020-splitting-code
The bidomain model is considered to be the gold standard for numerical simulation of the electrop... more The bidomain model is considered to be the gold standard for numerical simulation of the electrophysiology of cardiac tissue. The model provides important insights into the conduction properties of the electrochemical wave traversing the cardiac muscle in every heartbeat. However, in normal resolution, the model represents the average over a large number of cardiomyocytes, and more accurate models based on representations of all individual cells have therefore been introduced in order to gain insight into the conduction properties close to the myocytes. The more accurate model considered here is referred to as the EMI model since both the extracellular space (E), the cell membrane (M) and the intracellular space (I) are explicitly represented in the model. Here, we show that the bidomain model can be derived from the cell-based EMI model and we thus reveal the close relation between the two models, and obtain an indication of the error introduced in the approximation. Also, we prese...
One of the challenges related to modelling calcium dynamics in cardiac cells is the large di eren... more One of the challenges related to modelling calcium dynamics in cardiac cells is the large di erence in the length scales involved. The dyad, where important processes take place, is very small compared to the whole cell. Therefore, resolutions ne enough to capture the details of what happens in the dyad result in huge computational problems for whole-cell simulations, and the exact choice of resolution has a substantial e ect on the problem size. In this thesis, we investigate what grid resolution is necessary to capture the details of what happens in the dyad. We study simple mathematical models of calcium dynamics in the dyad and nd analytical solutions to some of these simple models. Numerical simulations of the models are carried out for di erent resolutions using nite di erence methods in 1D and 2D and a nite volume method in 3D. The accuracy of the numerical simulations is then studied by comparing the numerical solutions to analytical solutions and ne-grid numerical solutions...
Computational modeling has contributed significantly to present understanding of cardiac electrop... more Computational modeling has contributed significantly to present understanding of cardiac electrophysiology including cardiac conduction, excitation-contraction coupling, and the effects and side-effects of drugs. However, the accuracy of in silico analysis of electrochemical wave dynamics in cardiac tissue is limited by the homogenization procedure (spatial averaging) intrinsic to standard continuum models of conduction. Averaged models cannot resolve the intricate dynamics in the vicinity of individual cardiomyocytes simply because the myocytes are not present in these models. Here we demonstrate how recently developed mathematical models based on representing every myocyte can significantly increase the accuracy, and thus the utility of modeling electrophysiological function and dysfunction in collections of coupled cardiomyocytes. The present gold standard of numerical simulation for cardiac electrophysiology is based on the bidomain model. In the bidomain model, the extracellula...
The EMI model represents excitable cells in a more accurate manner than traditional homogenized m... more The EMI model represents excitable cells in a more accurate manner than traditional homogenized models at the price of increased computational complexity. The increased complexity of solving the EMI model stems from a significant increase in the number of computational nodes and from the form of the linear systems that need to be solved. Here, we will show that the latter problem can be solved by careful use of operator splitting of the spatially coupled equations. By using this method, the linear systems can be broken into sub-problems that are of the classical type of linear, elliptic boundary value problems. Therefore, the vast collection of methods for solving linear, elliptic partial differential equations can be used. We demonstrate that this enables us to solve the systems using shared-memory parallel computers. The computing time scales perfectly with the number of physical cells. For a collection of 512 × 256 cells, we solved linear systems with about 2.5×108 unknows. Since...
Excitable cells are of vital importance in biology, and mathematical models have contributed sign... more Excitable cells are of vital importance in biology, and mathematical models have contributed significantly to understand their basic mechanisms. However, classical models of excitable cells are based on severe assumptions that may limit the accuracy of the simulation results. Here, we derive a more detailed approach to modeling that has recently been applied to study the electrical properties of both neurons and cardiomyocytes. The model is derived from first principles and opens up possibilities for studying detailed properties of excitable cells.We refer to the model as the EMI model because both the extracellular space (E), the cell membrane (M) and the intracellular space (I) are explicitly represented in the model, in contrast to classical spatial models of excitable cells. Later chapters of the present text will focus on numerical methods and software for solving the model. Also, in the next chapter, the model will be extended to account for ionic concentrations in the intrace...
We want to be able to perform accurate simulations of a large number of cardiac cells based on ma... more We want to be able to perform accurate simulations of a large number of cardiac cells based on mathematical models where each individual cell is represented in the model. This implies that the computational mesh has to have a typical resolution of a fewµm leading to huge computational challenges. In this paper we use a certain operator splitting of the coupled equations and showthat this leads to systems that can be solved in parallel. This opens up for the possibility of simulating large numbers of coupled cardiac cells.
Mutations are known to cause perturbations in essential functional features of integral membrane ... more Mutations are known to cause perturbations in essential functional features of integral membrane proteins, including ion channels. Even restricted or point mutations can result in substantially changed properties of ion currents. The additive effect of these alterations for a specific ion channel can result in significantly changed properties of the action potential (AP). Both AP shortening and AP prolongation can result from known mutations, and the consequences can be life-threatening. Here, we present a computational method for identifying new drugs utilizing combinations of existing drugs. Based on the knowledge of theoretical effects of existing drugs on individual ion currents, our aim is to compute optimal combinations that can ‘repair’ the mutant AP waveforms so that the baseline AP-properties are restored. More specifically, we compute optimal, combined, drug concentrations such that the waveforms of the transmembrane potential and the cytosolic calcium concentration of the...
Atrial fibrillation (AF) is a common health problem with substantial individual and societal cost... more Atrial fibrillation (AF) is a common health problem with substantial individual and societal costs. The origin of AF has been debated for more than a century, and the precise, biophysical mechanisms that are responsible for the initiation and maintenance of the chaotic electrochemical waves that define AF, remains unclear. It is well accepted that the outlet of the pulmonary veins is the primary anatomical site of AF initiation, and that electrical isolation of these regions remains the most effective treatment for AF. Furthermore, it is well known that certain ion channel or transporter mutations can significantly increase the likelihood of AF. Here, we present a computational model capable of characterizing functionally important features of the microanatomical and electrophysiological substrate that represents the transition from the pulmonary veins (PV) to the left atrium (LA) of the human heart. This model is based on a finite element representation of every myocyte in a segmen...
Short QT (SQT) syndrome is a genetic cardiac disorder characterized by an abbreviated QT interval... more Short QT (SQT) syndrome is a genetic cardiac disorder characterized by an abbreviated QT interval of the patient’s electrocardiogram. The syndrome is associated with increased risk of arrhythmia and sudden cardiac death and can arise from a number of ion channel mutations. Cardiomyocytes derived from induced pluripotent stem cells generated from SQT patients (SQT hiPSC-CMs) provide promising platforms for testing pharmacological treatments directly in human cardiac cells exhibiting mutations specific for the syndrome. However, a difficulty is posed by the relative immaturity of hiPSC-CMs, with the possibility that drug effects observed in SQT hiPSC-CMs could be very different from the corresponding drug effect in vivo. In this paper, we apply a multistep computational procedure for translating measured drug effects from these cells to human QT response. This process first detects drug effects on individual ion channels based on measurements of SQT hiPSC-CMs and then uses these resul...
Using animal cells and tissues as precise measuring devices for developing new drugs presents a l... more Using animal cells and tissues as precise measuring devices for developing new drugs presents a long-standing challenge for the pharmaceutical industry. Despite the very significant resources that continue to be dedicated to animal testing of new compounds, only qualitative results can be obtained. This often results in both false positives and false negatives. Here, we show how the effect of drugs applied to animal ventricular myocytes can be translated, quantitatively, to estimate a number of different effects of the same drug on human cardiomyocytes. We illustrate and validate our methodology by translating, from animal to human, the effect of dofetilide applied to dog cardiomyocytes, the effect of E-4031 applied to zebrafish cardiomyocytes, and, finally, the effect of sotalol applied to rabbit cardiomyocytes. In all cases, the accuracy of our quantitative estimates are demonstrated. Our computations reveal that, in principle, electrophysiological data from testing using animal v...
Cardiomyocytes derived from human induced pluripotent stem cells (hiPSC-CMs) offer a new means to... more Cardiomyocytes derived from human induced pluripotent stem cells (hiPSC-CMs) offer a new means to study and understand the human cardiac action potential, and can give key insight into how compounds may interact with important molecular pathways to destabilize the electrical function of the heart. Important features of the action potential can be readily measured using standard experimental techniques, such as the use of voltage sensitive dyes and fluorescent genetic reporters to estimate transmembrane potentials and cytosolic calcium concentrations. Using previously introduced computational procedures, such measurements can be used to estimate the current density of major ion channels present in hiPSC-CMs, and how compounds may alter their behavior. However, due to the limitations of optical recordings, resolving the sodium current remains difficult from these data. Here we show that if these optical measurements are complemented with observations of the extracellular potential usi...
Cardiomyocytes derived from human induced pluripotent stem cells hold great potential for drug sc... more Cardiomyocytes derived from human induced pluripotent stem cells hold great potential for drug screening applications. However, their usefulness is limited by the relative immaturity of cells’ electro-physiological properties as compared to native cardiomyocytes in the adult human heart. In this work, we extend and improve on methodology to address this limitation, building on previously introduced computational procedures which predict drug effects for mature cells based on changes in optical measurements of action potentials and Ca2+transients made in stem cell derived cardiac microtissues. This methodology quantifies ion channel changes through the inversion of data into a mathematical model, and maps this response to a mature phenotype through the assumption of functional invariance of fundamental intracellular and membrane channels during maturation.Here we utilize an updated action potential model to represent both immature and mature cells, apply an IC50-based model of dose-d...
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2019
Mathematical models describing the dynamics of the cardiac action potential are of great value fo... more Mathematical models describing the dynamics of the cardiac action potential are of great value for understanding how changes to the system can disrupt the normal electrical activity of cells and tissue in the heart. However, to represent speci c data, these models must be parameterized, and adjustment of the maximum conductances of the individual contributing ionic currents is a commonly used method. Here, we present a method for investigating the uniqueness of such resulting parameterizations. Our key question is: Can the maximum conductances of a model be changed without giving any appreciable changes in the action potential? If so, the model parameters are not unique and this poses a major problem in using the models to identify changes in parameters from data, for instance, to evaluate potential drug e ects. We propose a method for evaluating this uniqueness, founded on the singular value decomposition of a matrix consisting of the individual ionic currents. Small singular values of this matrix signify lack of parameter uniqueness and we show that the conclusion from linear analysis of the matrix carries over to provide insight into the uniqueness of the parameters in the nonlinear case. Using numerical experiments, we quantify the identi ability of the maximum conductances of well-known models of the cardiac action potential. Furthermore, we show how the identi ability depends on the time step used in the observation of the currents, how the application of drugs may change identi ability, and, nally, how the stimulation protocol can be used to improve the identi ability of a model.
The conduction of electrical signals through cardiac tissue is essential for maintaining the func... more The conduction of electrical signals through cardiac tissue is essential for maintaining the function of the heart, and conduction abnormalities are known to potentially lead to lifethreatening arrhythmias. The properties of cardiac conduction have therefore been the topic of intense study for decades, but a number of questions related to the mechanisms of conduction still remain unresolved. In this paper, we demonstrate how the so-called EMI model may be used to study some of these open questions. In the EMI model, the extracellular space, the cell membrane, the intracellular space and the cell connections are all represented as separate parts of the computational domain, and the model therefore allows for study of local properties that are hard to represent in the classical homogenized bidomain or monodomain models commonly used to study cardiac conduction. We conclude that a nonuniform sodium channel distribution increases the conduction velocity and decreases the time delays over gap junctions of reduced coupling in the EMI model simulations. We also present a theoretical optimal cell length with respect to conduction velocity and consider the possibility of ephaptic coupling (i.e. cell-to-cell coupling through the extracellular potential) acting as an alternative or supporting mechanism to gap junction coupling. We conclude that for a non-uniform distribution of sodium channels and a sufficiently small intercellular distance, ephaptic coupling can influence the dynamics of the sodium channels and potentially provide cell-to-cell coupling when the gap junction connection is absent.
ObjectiveMechanistic modeling of neurons is an essential component of computational neuroscience ... more ObjectiveMechanistic modeling of neurons is an essential component of computational neuroscience that enables scientists to simulate, explain, and explore neural activity. The conventional approach to simulation of extracellular neural recordings first computes transmembrane currents using the cable equation and then sums their contribution to model the extracellular potential. This two-step approach relies on the assumption that the extracellular space is an infinite and homogeneous conductive medium, while measurements are performed using neural probes. The main purpose of this paper is to assess to what extent the presence of the neural probes of varying shape and size impacts the extracellular field and how to correct for them.ApproachWe apply a detailed modeling framework allowing explicit representation of the neuron and the probe to study the effect of the probes and thereby estimate the effect of ignoring it. We use meshes with simplified neurons and different types of probe...
The basic building blocks of the electrophysiology of cardiomyocytes are ion channels integrated ... more The basic building blocks of the electrophysiology of cardiomyocytes are ion channels integrated in the cell membranes. Close to the ion channels there are very strong electrical and chemical gradients. However, these gradients extend for only a few nano-meters and are therefore commonly ignored in mathematical models. The full complexity of the dynamics is modelled by the Poisson-Nernst-Planck (PNP) equations but these equations must be solved using temporal and spatial scales of nano-seconds and nano-meters. Here we report solutions of the PNP equations in a fraction of two abuttal cells separated by a tiny extracellular space. We show that when only the potassium channels of the two cells are open, a stationary solution is reached with the well-known Debye layer close to the membranes. When the sodium channels of the left cell are opened, a very strong and brief electrochemical wave emanates from the channels. If the extracellular space is sufficiently small and the number of sod...
Source code for the chapter "Operator splitting and finite difference schemes for solving th... more Source code for the chapter "Operator splitting and finite difference schemes for solving the EMI model" by Karoline Horgmo Jæger, Kristian Gregorius Hustad, Xing Cai, and Aslak Tveito from the book <em>EMI: Cell-based Mathematical Model of Excitable Cells</em> The code can also be found in a GitHub repository at https://github.com/KGHustad/emi-book-2020-splitting-code
The bidomain model is considered to be the gold standard for numerical simulation of the electrop... more The bidomain model is considered to be the gold standard for numerical simulation of the electrophysiology of cardiac tissue. The model provides important insights into the conduction properties of the electrochemical wave traversing the cardiac muscle in every heartbeat. However, in normal resolution, the model represents the average over a large number of cardiomyocytes, and more accurate models based on representations of all individual cells have therefore been introduced in order to gain insight into the conduction properties close to the myocytes. The more accurate model considered here is referred to as the EMI model since both the extracellular space (E), the cell membrane (M) and the intracellular space (I) are explicitly represented in the model. Here, we show that the bidomain model can be derived from the cell-based EMI model and we thus reveal the close relation between the two models, and obtain an indication of the error introduced in the approximation. Also, we prese...
One of the challenges related to modelling calcium dynamics in cardiac cells is the large di eren... more One of the challenges related to modelling calcium dynamics in cardiac cells is the large di erence in the length scales involved. The dyad, where important processes take place, is very small compared to the whole cell. Therefore, resolutions ne enough to capture the details of what happens in the dyad result in huge computational problems for whole-cell simulations, and the exact choice of resolution has a substantial e ect on the problem size. In this thesis, we investigate what grid resolution is necessary to capture the details of what happens in the dyad. We study simple mathematical models of calcium dynamics in the dyad and nd analytical solutions to some of these simple models. Numerical simulations of the models are carried out for di erent resolutions using nite di erence methods in 1D and 2D and a nite volume method in 3D. The accuracy of the numerical simulations is then studied by comparing the numerical solutions to analytical solutions and ne-grid numerical solutions...
Computational modeling has contributed significantly to present understanding of cardiac electrop... more Computational modeling has contributed significantly to present understanding of cardiac electrophysiology including cardiac conduction, excitation-contraction coupling, and the effects and side-effects of drugs. However, the accuracy of in silico analysis of electrochemical wave dynamics in cardiac tissue is limited by the homogenization procedure (spatial averaging) intrinsic to standard continuum models of conduction. Averaged models cannot resolve the intricate dynamics in the vicinity of individual cardiomyocytes simply because the myocytes are not present in these models. Here we demonstrate how recently developed mathematical models based on representing every myocyte can significantly increase the accuracy, and thus the utility of modeling electrophysiological function and dysfunction in collections of coupled cardiomyocytes. The present gold standard of numerical simulation for cardiac electrophysiology is based on the bidomain model. In the bidomain model, the extracellula...
The EMI model represents excitable cells in a more accurate manner than traditional homogenized m... more The EMI model represents excitable cells in a more accurate manner than traditional homogenized models at the price of increased computational complexity. The increased complexity of solving the EMI model stems from a significant increase in the number of computational nodes and from the form of the linear systems that need to be solved. Here, we will show that the latter problem can be solved by careful use of operator splitting of the spatially coupled equations. By using this method, the linear systems can be broken into sub-problems that are of the classical type of linear, elliptic boundary value problems. Therefore, the vast collection of methods for solving linear, elliptic partial differential equations can be used. We demonstrate that this enables us to solve the systems using shared-memory parallel computers. The computing time scales perfectly with the number of physical cells. For a collection of 512 × 256 cells, we solved linear systems with about 2.5×108 unknows. Since...
Excitable cells are of vital importance in biology, and mathematical models have contributed sign... more Excitable cells are of vital importance in biology, and mathematical models have contributed significantly to understand their basic mechanisms. However, classical models of excitable cells are based on severe assumptions that may limit the accuracy of the simulation results. Here, we derive a more detailed approach to modeling that has recently been applied to study the electrical properties of both neurons and cardiomyocytes. The model is derived from first principles and opens up possibilities for studying detailed properties of excitable cells.We refer to the model as the EMI model because both the extracellular space (E), the cell membrane (M) and the intracellular space (I) are explicitly represented in the model, in contrast to classical spatial models of excitable cells. Later chapters of the present text will focus on numerical methods and software for solving the model. Also, in the next chapter, the model will be extended to account for ionic concentrations in the intrace...
We want to be able to perform accurate simulations of a large number of cardiac cells based on ma... more We want to be able to perform accurate simulations of a large number of cardiac cells based on mathematical models where each individual cell is represented in the model. This implies that the computational mesh has to have a typical resolution of a fewµm leading to huge computational challenges. In this paper we use a certain operator splitting of the coupled equations and showthat this leads to systems that can be solved in parallel. This opens up for the possibility of simulating large numbers of coupled cardiac cells.
Mutations are known to cause perturbations in essential functional features of integral membrane ... more Mutations are known to cause perturbations in essential functional features of integral membrane proteins, including ion channels. Even restricted or point mutations can result in substantially changed properties of ion currents. The additive effect of these alterations for a specific ion channel can result in significantly changed properties of the action potential (AP). Both AP shortening and AP prolongation can result from known mutations, and the consequences can be life-threatening. Here, we present a computational method for identifying new drugs utilizing combinations of existing drugs. Based on the knowledge of theoretical effects of existing drugs on individual ion currents, our aim is to compute optimal combinations that can ‘repair’ the mutant AP waveforms so that the baseline AP-properties are restored. More specifically, we compute optimal, combined, drug concentrations such that the waveforms of the transmembrane potential and the cytosolic calcium concentration of the...
Atrial fibrillation (AF) is a common health problem with substantial individual and societal cost... more Atrial fibrillation (AF) is a common health problem with substantial individual and societal costs. The origin of AF has been debated for more than a century, and the precise, biophysical mechanisms that are responsible for the initiation and maintenance of the chaotic electrochemical waves that define AF, remains unclear. It is well accepted that the outlet of the pulmonary veins is the primary anatomical site of AF initiation, and that electrical isolation of these regions remains the most effective treatment for AF. Furthermore, it is well known that certain ion channel or transporter mutations can significantly increase the likelihood of AF. Here, we present a computational model capable of characterizing functionally important features of the microanatomical and electrophysiological substrate that represents the transition from the pulmonary veins (PV) to the left atrium (LA) of the human heart. This model is based on a finite element representation of every myocyte in a segmen...
Short QT (SQT) syndrome is a genetic cardiac disorder characterized by an abbreviated QT interval... more Short QT (SQT) syndrome is a genetic cardiac disorder characterized by an abbreviated QT interval of the patient’s electrocardiogram. The syndrome is associated with increased risk of arrhythmia and sudden cardiac death and can arise from a number of ion channel mutations. Cardiomyocytes derived from induced pluripotent stem cells generated from SQT patients (SQT hiPSC-CMs) provide promising platforms for testing pharmacological treatments directly in human cardiac cells exhibiting mutations specific for the syndrome. However, a difficulty is posed by the relative immaturity of hiPSC-CMs, with the possibility that drug effects observed in SQT hiPSC-CMs could be very different from the corresponding drug effect in vivo. In this paper, we apply a multistep computational procedure for translating measured drug effects from these cells to human QT response. This process first detects drug effects on individual ion channels based on measurements of SQT hiPSC-CMs and then uses these resul...
Using animal cells and tissues as precise measuring devices for developing new drugs presents a l... more Using animal cells and tissues as precise measuring devices for developing new drugs presents a long-standing challenge for the pharmaceutical industry. Despite the very significant resources that continue to be dedicated to animal testing of new compounds, only qualitative results can be obtained. This often results in both false positives and false negatives. Here, we show how the effect of drugs applied to animal ventricular myocytes can be translated, quantitatively, to estimate a number of different effects of the same drug on human cardiomyocytes. We illustrate and validate our methodology by translating, from animal to human, the effect of dofetilide applied to dog cardiomyocytes, the effect of E-4031 applied to zebrafish cardiomyocytes, and, finally, the effect of sotalol applied to rabbit cardiomyocytes. In all cases, the accuracy of our quantitative estimates are demonstrated. Our computations reveal that, in principle, electrophysiological data from testing using animal v...
Cardiomyocytes derived from human induced pluripotent stem cells (hiPSC-CMs) offer a new means to... more Cardiomyocytes derived from human induced pluripotent stem cells (hiPSC-CMs) offer a new means to study and understand the human cardiac action potential, and can give key insight into how compounds may interact with important molecular pathways to destabilize the electrical function of the heart. Important features of the action potential can be readily measured using standard experimental techniques, such as the use of voltage sensitive dyes and fluorescent genetic reporters to estimate transmembrane potentials and cytosolic calcium concentrations. Using previously introduced computational procedures, such measurements can be used to estimate the current density of major ion channels present in hiPSC-CMs, and how compounds may alter their behavior. However, due to the limitations of optical recordings, resolving the sodium current remains difficult from these data. Here we show that if these optical measurements are complemented with observations of the extracellular potential usi...
Cardiomyocytes derived from human induced pluripotent stem cells hold great potential for drug sc... more Cardiomyocytes derived from human induced pluripotent stem cells hold great potential for drug screening applications. However, their usefulness is limited by the relative immaturity of cells’ electro-physiological properties as compared to native cardiomyocytes in the adult human heart. In this work, we extend and improve on methodology to address this limitation, building on previously introduced computational procedures which predict drug effects for mature cells based on changes in optical measurements of action potentials and Ca2+transients made in stem cell derived cardiac microtissues. This methodology quantifies ion channel changes through the inversion of data into a mathematical model, and maps this response to a mature phenotype through the assumption of functional invariance of fundamental intracellular and membrane channels during maturation.Here we utilize an updated action potential model to represent both immature and mature cells, apply an IC50-based model of dose-d...
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2019
Mathematical models describing the dynamics of the cardiac action potential are of great value fo... more Mathematical models describing the dynamics of the cardiac action potential are of great value for understanding how changes to the system can disrupt the normal electrical activity of cells and tissue in the heart. However, to represent speci c data, these models must be parameterized, and adjustment of the maximum conductances of the individual contributing ionic currents is a commonly used method. Here, we present a method for investigating the uniqueness of such resulting parameterizations. Our key question is: Can the maximum conductances of a model be changed without giving any appreciable changes in the action potential? If so, the model parameters are not unique and this poses a major problem in using the models to identify changes in parameters from data, for instance, to evaluate potential drug e ects. We propose a method for evaluating this uniqueness, founded on the singular value decomposition of a matrix consisting of the individual ionic currents. Small singular values of this matrix signify lack of parameter uniqueness and we show that the conclusion from linear analysis of the matrix carries over to provide insight into the uniqueness of the parameters in the nonlinear case. Using numerical experiments, we quantify the identi ability of the maximum conductances of well-known models of the cardiac action potential. Furthermore, we show how the identi ability depends on the time step used in the observation of the currents, how the application of drugs may change identi ability, and, nally, how the stimulation protocol can be used to improve the identi ability of a model.
The conduction of electrical signals through cardiac tissue is essential for maintaining the func... more The conduction of electrical signals through cardiac tissue is essential for maintaining the function of the heart, and conduction abnormalities are known to potentially lead to lifethreatening arrhythmias. The properties of cardiac conduction have therefore been the topic of intense study for decades, but a number of questions related to the mechanisms of conduction still remain unresolved. In this paper, we demonstrate how the so-called EMI model may be used to study some of these open questions. In the EMI model, the extracellular space, the cell membrane, the intracellular space and the cell connections are all represented as separate parts of the computational domain, and the model therefore allows for study of local properties that are hard to represent in the classical homogenized bidomain or monodomain models commonly used to study cardiac conduction. We conclude that a nonuniform sodium channel distribution increases the conduction velocity and decreases the time delays over gap junctions of reduced coupling in the EMI model simulations. We also present a theoretical optimal cell length with respect to conduction velocity and consider the possibility of ephaptic coupling (i.e. cell-to-cell coupling through the extracellular potential) acting as an alternative or supporting mechanism to gap junction coupling. We conclude that for a non-uniform distribution of sodium channels and a sufficiently small intercellular distance, ephaptic coupling can influence the dynamics of the sodium channels and potentially provide cell-to-cell coupling when the gap junction connection is absent.
ObjectiveMechanistic modeling of neurons is an essential component of computational neuroscience ... more ObjectiveMechanistic modeling of neurons is an essential component of computational neuroscience that enables scientists to simulate, explain, and explore neural activity. The conventional approach to simulation of extracellular neural recordings first computes transmembrane currents using the cable equation and then sums their contribution to model the extracellular potential. This two-step approach relies on the assumption that the extracellular space is an infinite and homogeneous conductive medium, while measurements are performed using neural probes. The main purpose of this paper is to assess to what extent the presence of the neural probes of varying shape and size impacts the extracellular field and how to correct for them.ApproachWe apply a detailed modeling framework allowing explicit representation of the neuron and the probe to study the effect of the probes and thereby estimate the effect of ignoring it. We use meshes with simplified neurons and different types of probe...
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Papers by Karoline Jæger