Nonlinear physics of electrical wave propagation in the heart: a review

The beating of the heart is a synchronized contraction of muscle cells (myocytes) that are triggered by a periodic sequence of electrical waves (action potentials) originating in the sino-atrial node and propagating over the atria and the ventricles. Cardiac arrhythmias like atrial and ventricular fibrillation (AF,VF) or ventricular tachycardia (VT) are caused by disruptions and instabilities of these electrical excitations, that lead to the emergence of rotating waves (VT) and turbulent wave patterns (AF,VF). Numerous simulation and experimental studies during the last 20 years have addressed these topics. In this review we focus on the nonlinear dynamics of wave propagation in the heart with an emphasis on the theory of pulses, spirals and scroll waves and their instabilities in excitable media and their application to cardiac modeling. After an introduction into electrophysiological models for action potential propagation, the modeling and analysis of spatiotemporal alternans, spiral and scroll meandering, spiral breakup and scroll wave instabilities like negative line tension and sproing are reviewed in depth and discussed with emphasis on their impact in cardiac arrhythmias.Peer ReviewedPreprin

The initiation of action potentials and the passive electrical properties of identified snail neurones

Two aspects of neuronal function were investigated: the passive electrical properties of neuronal membranes and the initiation of action potentials. The passive electrical properties of a neurone, together with its morphology, determine the efficiency of synaptic current transfer to the impulse initiation zone. A general analysis was made of the problems of estimating the electrical properties of a neurone from the measured input impedance with the aid of equations for the input impedance. These equations were used to quantify the error resulting from an idealization of the neurones structure. Furthermore, frequency and time domain methods for electrotonic parameter estimation were contrasted and frequency domain methods were shown to be less susceptible to error. Frequency domain methods were applied to the problem of estimating the electrotonic parameters of some identified neurones of the garden snail. The membrane time constants for the group of neurones studied had an average value of 43 ms. The nonlinear properties of snail neurones were characterised by measuring the harmonic content of the voltage response to a sinusoidal current input. The model so deduced accounted for the response of neurones for inputs with peak-to-peak amplitudes up to 2 nA, but the form of the input showed a strong dependence on the DC bias of the input. In the second part of this thesis stochastic and deterministic signals were used to characterise and model the dynamics of spike initiation. Neurones were stimulated with Gaussian white noise current signals. Records of the action potentials evoked together with the input noise allowed measurement of the characteristics of the current trajectories that lead to the initiation of action potentials. These records were analysed in the framework of Wiener's theory of nonlinear systems to obtain a model of the current-to-spike transformation. The models were similar in form to that of a low-pass filler in cascade with a threshold device and predicted 60 to 80% of the observed action potentials. The spiking behaviour evoked by step current inputs was contrasted with that produced by Gaussian white noise and the dynamics of the neurone were shown to depend on the form of the input used

Nonlinear Dynamics, Synchronisation and Chaos in Coupled FHN Cardiac and Neural Cells

Physiological systems are amongst the most challenging systems to investigate from a mathematically based approach. The eld of mathematical biology is a relatively recent one when compared to physics. In this thesis I present an introduction to the physiological aspects needed to gain access to both cardiac and neural systems for a researcher trained in a mathematically based discipline. By using techniques from nonlinear dynamical systems theory I show a number of results that have implications for both neural and cardiac cells. Examining a reduced model of an excitable biological oscillator I show how rich the dynamical behaviour of such systems can be when coupled together. Quantifying the dynamics of coupled cells in terms of synchronisation measures is treated at length. Most notably it is shown that for cells that themselves cannot admit chaotic solutions, communication between cells be it through electrical coupling or synaptic like coupling, can lead to the emergence of chaotic behaviour. I also show that in the presence of emergent chaos one nds great variability in intervals of activity between the constituent cells. This implies that chaos in both cardiac and neural systems can be a direct result of interactions between the constituent cells rather than intrinsic to the cells themselves. Furthermore the ubiquity of chaotic solutions in the coupled systems may be a means of information production and signaling in neural systems

Neuronal computation on complex dendritic morphologies

When we think about neural cells, we immediately recall the wealth of electrical behaviour which, eventually, brings about consciousness. Hidden deep in the frequencies and timings of action potentials, in subthreshold oscillations, and in the cooperation of tens of billions of neurons, are synchronicities and emergent behaviours that result in high-level, system-wide properties such as thought and cognition. However, neurons are even more remarkable for their elaborate morphologies, unique among biological cells. The principal, and most striking, component of neuronal morphologies is the dendritic tree. Despite comprising the vast majority of the surface area and volume of a neuron, dendrites are often neglected in many neuron models, due to their sheer complexity. The vast array of dendritic geometries, combined with heterogeneous properties of the cell membrane, continue to challenge scientists in predicting neuronal input-output relationships, even in the case of subthreshold dendritic currents. In this thesis, we will explore the properties of neuronal dendritic trees, and how they alter and integrate the electrical signals that diffuse along them. After an introduction to neural cell biology and membrane biophysics, we will review Abbott's dendritic path integral in detail, and derive the theoretical convergence of its infinite sum solution. On certain symmetric structures, closed-form solutions will be found; for arbitrary geometries, we will propose algorithms using various heuristics for constructing the solution, and assess their computational convergences on real neuronal morphologies. We will demonstrate how generating terms for the path integral solution in an order that optimises convergence is non-trivial, and how a computationally-significant number of terms is required for reasonable accuracy. We will, however, derive a highly-efficient and accurate algorithm for application to discretised dendritic trees. Finally, a modular method for constructing a solution in the Laplace domain will be developed

Applications of optogenetic tandem-cell units for in vitro study of cardiac electrophysiology

Optogenetics and tandem-cell units are an important tool for studying cardiac electrophysiology, and this work explores a few of the exciting avenues of study they enable. In Chapters 2 and 3, myofibroblasts and fibroblasts are transduced with channelrhodopsin-2 and co-cultured with cardiomyocytes to acutely demonstrate that both are electrically connected enough to cardiomyocytes to produce changes in cardiomyocyte electrophysiology, which has implications for treating conduction slowing after cardiac injury. In Chapter 4, a simple, scalable method to use tandem-cell units to point-pace cells in culture to mature them is developed, which has the potential to make them more useful for in vitro study, drug testing, and tissue engineering. Finally, in Chapter 5, an engineered tissue from decellularized extracellular matrix is developed that represents the next step for the applications in the previous chapters by providing important physiological cues, which should improve their relevance and accuracy

Numerical modelling in transcranial magnetic stimulation

Tese de doutoramento, Engenharia Biomédica e Biofísica, Universidade de Lisboa, Faculdade de Ciências, 2009In this work powerful numerical methods were used to study several problems that still remain unsolved in TMS.The first problem that was studied is related to the difficulties that arise when stimulating sub-cortical deep regions with TMS, due to the fact that the induced field rapidly decays and loses focality with depth. This study's approach to overcome this difficulty was to combine ferromagnetic cores with a coil designed to induce an electric field that decays slowly. The efficacy of this approach was tested by using the FEM to calculate the field induced by this coil / core design in a realistically shaped head model. The results show that the core might make this coil even more suited for deep brain stimulation.The second problem that was tackled is related to the lack of knowledge about the dominant mechanisms through which the induced electric field excites neurons in TMS. In this work the electric field along lines, representing trajectories of actual cortical neurons, was calculated using the FEM. The neurons were embedded in a realistically shaped sulcus model, with a figure-8 coil placed above the model. The electric field was then incorporated into the cable equation. The solution of the latter allowed the determination of the site and threshold of activation of the neurons. The results highlight the importance of axonal terminations and bends and tissue heterogeneities on stimulation of neurons.The third problem that was studied concerns TMS of small animals and the lack of knowledge about the optimal geometry, size and orientation of the used coils. This was studied by using the FEM to calculate the electric field induced in a realistically shaped mouse model by several commercially available coils. The results showed that the smaller coils induced fields with higher magnitude, better focality, and smaller decay than the bigger coils.These results highlight the importance of numerical modelling in TMS, either in coil design, determination of basic neurophysiologic mechanisms or optimization of experimental procedures

Neuronal Signal Modulation By Dendritic Geometry

Neurons are the basic units in nervous systems. They transmit signals along neurites and at synapses in electrical and chemical forms. Neuronal morphology, mainly dendritic geometry, is famous for anatomical diversity, and names of many neuronal types reflect their morphologies directly. Dendritic geometries, as well as distributions of ion channels on cell membranes, contribute significantly to distinct behaviours of electrical signal filtration and integration in different neuronal types (even in the cases of receiving identical inputs in vitro). In this thesis I mainly address the importance of dendritic geometry, by studying its effects on electrical signal modulation at the level of single neurons via mathematical and computational approaches. By ‘geometry’, I consider both branching structures of entire dendritic trees and tapered structures of individual dendritic branches. The mathematical model of dendritic membrane potential dynamics is established by generalising classical cable theory. It forms the theoretical benchmark for this thesis to study neuronal signal modulation on dendritic trees with tapered branches. A novel method to obtain analytical response functions in algebraically compact forms on such dendrites is developed. It permits theoretical analysis and accurate and efficient numerical calculation on a neuron as an electrical circuit. By investigating simplified but representative dendritic geometries, it is found that a tapered dendrite amplifies distal signals in comparison to the non-tapered dendrite. This modulation is almost a local effect, which is merely influenced by global dendritic geometry. Nonetheless, global geometry has a stronger impact on signal amplitudes, and even more on signal phases. In addition, the methodology employed in this thesis is perfectly compatible with other existing methods dealing with neuronal stochasticity and active behaviours. Future works of large-scale neural networks can easily adapt this work to improve computational efficiency, while preserving a large amount of biophysical details

Electrical Coupling Between Micropatterned Cardiomyocytes and Stem Cells

To understand how stem cells functionally couple with native cardiomyocytes is crucial for cell-based therapies to restore the loss of cardiomyocytes that occurs during heart infarction and other cardiac diseases. Due to the complexity of the in vivo environment, our knowledge of cell coupling is heavily dependent on cell-culture models. However, conventional in vitro studies involve undefined cell shapes and random length of cell-cell contacts in addition to the presence of multiple homotypic and heterotypic contacts between interacting cells. Thus, it has not been feasible to study electrical coupling corresponding to isolated specific types of cell contact modes. To address this issue, we used microfabrication techniques to develop different geometrically-defined stem cell-cardiomyocyte contact assays to comparatively and quantitatively study functional stem cell-cardiomyocyte electrical coupling. Through geometric confinements, we will construct a matrix of identical microwells, and each was constructed as a specific microenvironment. Using laser-guided cell micropatterning technique, individual stem cells or cardiomyocytes can be deposited into the microwells to form certain contact mode. In this research, we firstly constructed an in-vivo like cardiac muscle fiber microenvironment, and the electrical conductivity of stem cells was investigated by inserting stem cells as cellular bridges. Then, the electrical coupling between cardiomyocytes and stem cells was studied at single-cell level by constructing contact-promotive/-preventive microenvironments