Roberto Suárez-Ántola - Academia.edu (original) (raw)

Papers by Roberto Suárez-Ántola

Historical aspects ……………………………………………………………... 4 Results related with the isocoric model of van de... more Historical aspects ……………………………………………………………... 4 Results related with the isocoric model of van der Vaart …..…………….....8 A generalization of von Bertalanffy open system model and a study of its anisocoric versions……………………….………………………………………11 Formal kinetic models for open homogeneous reaction-diffusion systems with composition-dependent volume…..………………………………………….34 Heuristic derivation of an equation analogous to the von Bertalanffy growth equation………………………………………………………………………..36 Active droplet growth and stability ………………………………………….39 Discussion and Final Conclusions.……………………………………………53

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A new formula is obtained here to relate the emf induced in a coil of a measuring system, with tw... more A new formula is obtained here to relate the emf induced in a coil of a measuring system, with two fields: the so called impressed electric current density field (due to the action potentials in the membranes of the excitable cells located in the volume conductor of the biological tissues), and a certain stationary field, known as the magnetic lead field. The proposed formula modifies the one due to Robert Plonsey, published in 1972. After carrying out a review of the deduction of the original formula, its modification is deduced in the framework of electromagnetic theory and by analytical methods. The resultant modification emphasizes the weight assigned to the high frequencies in the induced emf. The deduction of the magnetic lead field is made here from a different approach. Some consequences of the modified formula are discussed and suggestions for further work are given. In Appendix A, the phasor field solutions of Maxwell equations corresponding to monochromatic oscillations in a general linear continuous medium are reviewed. Then, a general version of Lorentz reciprocity theorem for linear, nonhomogeneous, anisotropic, and lossy dispersive media is presented as a basis for deducing the modified formula proposed in this work. In Appendix B the restrictions that allow to apply the quasi-stationary approximation to electromagnetic theory in both, the quasi-electro-stationary and the magneto-quasi-stationary approaches are considered.
Key words: Plonsey´s formula of biomagnetism, reciprocity theorem, magnetic field leads, impressed electric currents, inhomogeneous and anisotropic volume conductor, quasi-stationary approximations

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Both in-vitro and in-vivo observations and experiments show that the radial distribution of forme... more Both in-vitro and in-vivo observations and experiments show that the radial distribution of formed elements in blood flowing in arteries is, under certain circumstances, not uniform. The non-uniform distribution of the vast majority component in blood cells, erythrocytes, is related with a spatial variation in blood viscosity and modifications in blood shear rates. The variations both in shear rates and in blood cell concentrations near the endothelium, modifies fluid-to-wall mass transport processes in arteries. The purpose of this article is: (a) Discuss conditions for in vivo appearance of the axial accumulation of red cells, considered as one of the self-fluidization mechanisms of moving blood in the arterial conduits. Discuss the non-uniform distribution of the other formed elements (platelets and leukocytes). (b) Analyze possible consequences and physiological and pathological interrelationships of these phenomena. Some biomechanical aspects of growth, remodeling, and damage in arterial walls during the life cycle are considered. The discussion is framed in the relation between hemodynamics, an optimum principle (Bejan´s constructal law) and the biomechanics of the arterial wall (considered as a local mechanical unit). Both short and long terms modifications in the geometric and mechanical properties of the arterial wall and the mechanisms of vascular control and the geometry of the arterial wall are considered. The self-fluidification of blood at high shear rates and the self-impeding of flow at low shear rates are considered as non-equilibrium phase transition. The association between changes in blood rheology and cardiovascular risk factors, including psychological stress and psychiatric illness are discussed in the framework of psycho-immunoneuroendocrinology. Then the subject is considered from the point of view of evolutionary biology. Aggregation as a mechanism that intensifies the axial accumulation of erythrocytes and reinforces the self-fluidization properties of blood is observed in athletic species but not in sedentary species. In athletic species, such as homo sapiens, the increase in oxygen demand may require a significant increase in the flow that supplies certain tissues. This allows us to give an evolutionary interpretation to the increased velocity profile that produces the disaggregation of erythrocytes and reduces the viscosity of the blood in the arterioles and in the venules.

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Progress in Nuclear Energy, 2019

Asymptotic analytical methods are applied to study some problems related with bifurcations (both ... more Asymptotic analytical methods are applied to study some problems related with bifurcations (both local and global) and stability in three simple mathematical models of nuclear reactors. The first case is a reactor which is subcritical at rest and driven by a distributed external neutron source. Under suitable conditions in the temperature feedback reactivity, the mathematical model predicts a static global bifurcation with three critical states, two stable and one unstable, with the possibility of a runaway. The dynamics of this system is studied, in a framework of slow manifold theory, by methods of restricted nonlinear modal analysis, and the results of a digital simulation are summarized. The model could be modified and extended to study the space time dynamics of sub-critical multiplying systems driven by external neutron sources (neutron beams produced by accelerators). The second case is related with in phase and out of phase xenon oscillations in large thermal reactors. The known subcritical Hopf bifurcation that appears in the context of global mode oscillations is revisited. After applying a nonlinear modal analysis to the mathematical model, a normal form is derived by an averaging method. Approximate analytical formulae for the radii of the unstable limit cycles and for the trajectories of the state variables in a neighborhood of the bifurcation point are obtained. The third case is a non-trivial modification and development of a simple mathematical model intended to describe certain mechanical kinetic effects stemming from a possible coupling of nuclear, thermal and mechanical vibration processes. We show that, under suitable conditions, a dynamic supercritical Hopf bifurcation in the reactor power appears in the framework of our modified model. A normal form is derived by an averaging method. Analytical formulae for the radii of the stable limit cycles and the trajectories of the state variables in a neighborhood of the bifurcation are given.

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In this research note some aspects of an experimental method proposed by Luis María Sánchez and R... more In this research note some aspects of an experimental method proposed by Luis María Sánchez and Roberto Suárez-Antola (Sánchez and Suárez-Antola, 1970) with the purpose of measuring the electro-osmotic parameters of the wall of plant cells are discussed. For the measurement system considered here, it is expected that the joint influence of the transition layers on the value of the measured electroosmotic parameters is not very important. The mechanical instability of the barrier formed by the cell wall fragments suggests that the degree of cohesion between these fragments is quite low. Having had to resort to an osmotic pressure gradient to determine the hydraulic permeability introduces an additional parameter to be determined experimentally: the Stavermann coefficient of reflection.
A pending question is to identify the mechanism by which the barrier disintegrates when a hydrostatic pressure difference is used but remains stable when an osmotic pressure difference is used to produce the same volume flow. As already stressed in the conclusions of the paper that motivates the present discussion (Sánchez and Suárez-Antola, 1970), a possible relation between the scattering of the experimental measurements and the nature of the mechanical instability of the barrier should be investigated also.
Beyond the statistical variations observed in the experimental results, the discussion carried out suggests that the packed wall fragments form a system of interconnected sub-barriers. The substance in the capillary tube can be considered as a porous granular-type medium, the cell wall fragments being the grains, and the interconnected pores being the ionic solution filled regions bordered by the surfaces of the wall fragments.
The measured volume flows suggest that the main contribution to the electro-osmotic parameters of these barriers is due to this network of microchannels and pores, and not to the cell wall fragments.
As consequence, the method that motivates this note should be substantially modified if what is wanted is to characterize the cellular structures themselves from the electro-kinetic point of view.

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Some preliminary experiments were carried out to develop a method designed to separately evaluate... more Some preliminary experiments were carried out to develop a method designed to separately evaluate the contributions of the different structures of the plant cell to the electro-osmotic flow. The classical Dainty apparatus was used, with slight variations, to measure conductance, hydraulic permeability, and the electro-osmotic coefficient of aggregates of fragments of the Saccharomyces cereviciae cell wall in stationary growth phase. From the measured values of the electro-osmotic parameters, the phenomenological coefficients, direct and crossed, were calculated. These coefficients link the volume flow and the electric current intensity with the potential difference between the electrodes and the hydrostatic pressure difference across the permeability barrier. As it was found that the barrier does not withstand (under the conditions in which it was prepared) a hydrostatic pressure difference in steady state large enough to be able to measure flows with sufficient accuracy to estimate the hydraulic permeability and the electric conductance, a sucrose solution added to one of the chambers was used. Under the conditions of the experiment, the Stavermann reflection coefficient of the barrier for sucrose was practically equal to unity. Introduction:

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A framework of formal physical-chemical kinetics with focus in the mathematical modelling of lump... more A framework of formal physical-chemical kinetics with focus in the mathematical modelling of lumped parameters reaction-diffusion systems is presented. Limit steps for mass transport, either located in the systems boundaries or due to bulk diffusion of certain components inside the systems, are employed to give a comparative discussion of the dynamic behavior of several simple models of open systems. Two cases of anisocoric (with variable volume) systems whose volume depends of the system composition are studied. One case is closely related with Perret and Levey’s biophase as a model of a growing population, without scale effects. Considering a biophase as a population of growing and dividing spherical droplets, a probabilistic derivation of the linear relation between area and volume is given. The other case is an anisocoric generalization of a classical open system model due to Ludwig von Bertalanffy, to introduce scale effects in the model. Grounded in slow manifold theory, a heuristic generalization of the derivation of Bertalanffy’s growth equation for open physico-chemical systems with a complex network of chemical reactions and with scale effect is proposed. For growing active droplets with porous matrices, the possibility of a limit step for mass transport due to bulk diffusion through the pore space is considered. The stability threshold to droplet division is studied using a simplified mathematical model, based on the theory of averaged fields on representative volume elements in a two phases medium. Filtration pressure, interfacial tension, viscous-plastic deformation and flow of the polymer matrix, and chemical activity inside the droplet are included. An approximate analytical formula is derived for the droplet’s critical radius as a function of reaction rates, surface tensions of the droplet interfaces and effective transport parameters of the interiors and boundaries of droplets and of the environment. These results are criticized in the light of the progress made up to the present time, when a joint work of physicists and biologists in Dresden, Germany, and in other places and countries, found several simple mechanisms that could explain how droplets might have proliferated, growing and dividing and, perhaps, evolving into the first cell from an early Earth’s primordial soup.
Key words: Mathematical modelling, dynamical systems, bifurcation theory, slow manifolds, singular perturbations, nonequilibrium open systems, formal chemical kinetics, reaction-diffusion systems, aniso-coric models, biophase, von Bertalanffy’s and van der Vaart’s mathematical models, active droplets divi-sion, prebiotic systems, origin of life.

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Progress in Nuclear Energy (Special Issue on Nonlinear Stability Analysis: Nonlinear nuclear reactor stability analysis and special aspects of modelling of complex dynamical systems), 2019

Asymptotic analytical methods are applied to study some problems related with bifurcations (both ... more Asymptotic analytical methods are applied to study some problems related with bifurcations (both local and global) and stability in three simple mathematical models of nuclear reactors.
The first case is a reactor which is subcritical at rest and driven by a distributed external neutron source. Under suitable conditions in the temperature feedback reactivity, the mathematical model predicts a static global bifurcation with three critical states, two stable and one unstable, with the possibility of a runaway. The dynamics of this system is studied, in a framework of slow manifold theory, by methods of restricted nonlinear modal analysis, and the results of a digital simulation are summarized. The model could be modified and extended to study the space time dynamics of sub-critical multiplying systems driven by external neutron sources (neutron beams produced by accelerators).
The second case is related with in phase and out of phase xenon oscillations in large thermal reactors. The known subcritical Hopf bifurcation that appears in the context of global mode oscillations is revisited. After applying a nonlinear modal analysis to the mathematical model, a normal form is derived by an averaging method. Approximate analytical formulae for the radii of the unstable limit cycles and for the trajectories of the state variables in a neighborhood of the bifurcation point are obtained.
The third case is a non-trivial modification and development of a simple mathematical model intended to describe certain mechanical kinetic effects stemming from a possible coupling of nuclear, thermal and mechanical vibration processes. We show that, under suitable conditions, a dynamic supercritical Hopf bifurcation in the reactor power appears in the framework of our modified model. A normal form is derived by an averaging method. Analytical formulae for the radii of the stable limit cycles and the trajectories of the state variables in a neighborhood of the bifurcation are given.
Keywords: nuclear reactor dynamics, reduced order models, restricted nonlinear modal analysis, averaging methods, bifurcation, stability, mathematical modeling, homogenized equivalent reactor

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Dresden Scientific Workshop on Reactor Dynamics and Safety, 2012

One of the goals of nuclear power systems design and operation is to restrict the possible states... more One of the goals of nuclear power systems design and operation is to restrict the possible states of certain critical subsystems to remain inside a certain bounded set of admissible states and state variations. In the framework of an analytic or numerical modeling process of a BWR power plant, this could imply first to find a suitable approximation to the solution manifold of the system of nonlinear partial differential equations describing the stability behavior, and then a classification of the different solution types concerning their relation with the operational safety of the power plant. Inertial manifold theory gives a foundation for the construction and use of reduced order models (ROM's) of reactor dynamics to discover and characterize meaningful bifurcations that may pass unnoticed during digital simulations done with full scale computer codes of the nuclear power plant. The main aspects of approximate inertial manifolds and forms are briefly reviewed in the introduction of the paper.A complete numerical study of reactor dynamics using a realistic ROM currently involves the digital simulation of the behavior of approximately twenty state variables interrelated by a corresponding system of coupled nonlinear ordinary differential equations. The success of hybrid analytical-numerical bifurcation codes to detect interesting behavior, such as global bifurcations in BWR's, may be enhanced by studying suitable simplifications of ROM's, that is ROM's of ROM's. A previous generalization of the classical March-Leuba's model of BWR is briefly reviewed and a nonlinear integral-differential equation in the logarithmic power is derived. The asymptotic method developed by Krilov, Bogoliubov and Mitropolsky (KBM) is applied to obtain approximate equations of evolution for the amplitude and the phase of a manifold of oscillatory solutions jointly with a relation between an offset and the abovementioned amplitude. First, to exemplify the method working with a simpler problem, the KBM tentative solution (ansatz) is applied to construct approximate solutions of, and to study local bifurcations in, a van der Pol equation with continuous and discrete distribution of time delays. Then, the afore-mentioned ansatz is applied to the full nonlinear integral-differential equation of the BWR model. Analytical formulae are derived for the offset, the rate of change in the phase (the instantaneous frequency of oscillation) and the rate of change in the amplitude of oscillation, given as functions of the amplitude and the model parameters (steady state power and coolant flow, temperature and void reactivity coefficients, fuel to coolant heat transfer coefficient and other parameters from neutronics and thermal hydraulics). The obtained analytical formulae are applied to start a semi-analytical, mainly qualitative, approach to bifurcations and stability of the steady states located in different regions of parameters space. This includes a qualitative discussion of the possibility of both, super and subcritical Poincaré-Andronov-Hopf bifurcations, as well as a Bautin's bifurcation scenario. The preliminary qualitative results outlined in this study are consistent with results of recent digital simulations done with a full-scale reduced order model of BWR (PSI-TU Valencia-TU Dresden) and with the results obtained with the application of hybrid approaches to bifurcation theory done with the simplified March-Leuba's model of BWR. Key words: nuclear reactor dynamics, differential-integral equations, reduced order models of boiling water reactors, averaging methods, Hopf supercritical and subcritical bifurcation, Bautin scenario, nuclear reactor stability and control.

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Annals of Nuclear Energy, 2014

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Abstract: A mathematical model is constructed to study the diffuse axial and radial growth of the... more Abstract: A mathematical model is constructed to study the diffuse axial and radial growth of the primary wall of plant cells. Analytical formulas are obtained for Erickson's anisotropy quotient of growth and for the parameters of the Ortega´s augmented equation, as a function of the parameters of the new model. A non-linear constitutive relationship is introduced in the Lockhart model and some aspects of the axial growth model thus generalized are considered.

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The variational approach in the natural sciences is discussed from historical and epistemological... more The variational approach in the natural sciences is discussed from historical and epistemological points of view, stressing its application to relativity theory.

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The relation of the Polya-Aeppli distribution of probabilities (also known as the Poisson-Geometr... more The relation of the Polya-Aeppli distribution of probabilities (also known as the Poisson-Geometric distribution) with random processes in the stationary case is considered in this article. Then the above mentioned distribution is derived, working with the probability generating functions, as limit of the distribution of rare events in a succession of Bernouilli trials with first order Markov dependence (that is, as a limit of Markov chains of rare events).

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Resumen: Tanto las observaciones in-vivo e in-vitro como los experimentos muestran que la distr... more Resumen: Tanto las observaciones in-vivo e in-vitro como los experimentos muestran que la distribución radial de los elementos formes de la sangre que fluye en las arterias es, bajo ciertas circunstancias, no uniforme. Esta distribución no uniforme de las células de la sangre conlleva una variación espacial en la viscosidad y una modificación en las velocidades de cizalla de la sangre.
A su vez, tanto las variaciones en la velocidad de cizalla como los cambios en la concentración de elementos formes en las adyacencias de la pared arterial, modifican los procesos de transporte de masa entre el fluido y el endotelio. El propósito de esta segunda parte del artículo es: (a) Discutir con-diciones para la manifestación in vivo de la acumulación axial de los eritrocitos, considerada como uno de los mecanismos de auto-fluidificación de la sangre en movimiento en los conductos arteriales. (b) Analizar posibles consecuencias e interrelaciones fisiológicas y patológicas de este fenómeno. Se consideran algunos aspectos biomecánicos de los procesos de crecimiento, remodelado, y daño de la pared arterial durante el ciclo de vida.

Palabras clave: Reología de la sangre. Sinergética. Ley de Bejan. Acumulación axial de elementos formes. Procesos de transporte de masa. Arteriosclerosis. Aterosclerosis.

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Contribuciones al estudio de las posibles consecuencias fisiológicas y fisiopatológicas de la acu... more Contribuciones al estudio de las posibles consecuencias fisiológicas y fisiopatológicas de la acumulación axial de los elementos formes de la sangre. Primera parte: modelo matemático. Resumen: Tanto las observaciones in-vivo e in-vitro como los experimentos muestran que la distribución radial de los elementos formes de la sangre que fluye en las arterias es a menudo no uniforme. Los modelos reológicos comunes no tienen en cuenta esta distribución no uniforme de las células de la sangre, y en consecuencia la variación espacial en la viscosidad que esta no uniformi-dad conlleva. Además, las variaciones en las velocidades de cizalla de la sangre modifican los procesos de transporte de masa entre el fluido y la pared en las arterias. El propósito de esta primera parte del artículo es tomar en cuenta esta viscosidad espacialmente no uniforme en la medida en que está relacionada con y afecta a la hemodinámica y los procesos de transporte de masa fluido-pared en las arterias, enfatizando un enfoque analítico aproximado contrastado con simulaciones digitales. En el intervalo de hematocritos globales de tubo considerado, tanto los perfiles de velocidad de cizalla como los parámetros de los procesos de transporte de masa se ven influidos por el modelo macro-reológico utilizado y por el campo de hematocrito local según el modelo-micro-reológico propuesto. Cuando se toma en cuenta las variaciones espaciales de la viscosidad tanto la hemodi-námica local resultante como los procesos de transporte entre la sangre y la pared arterial y la po-tencia de bombeo necesaria para mantener un flujo pulsátil en una arteria pequeña, pueden modifi-carse significativamente. Palabras clave: Reología de la sangre. Acumulación axial de elementos formes. Mecánica de fluidos computacional (CFD). Procesos de transporte de masa.

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Resumen: Al igual que en la primera parte, en esta segunda parte se emplea una ecuación de cable ... more Resumen: Al igual que en la primera parte, en esta segunda parte se emplea una ecuación de cable no lineal homogeneizada para describir la interacción entre un campo eléctrico externo y una fibra nerviosa mielínica. Aplicando nuevamente el concepto de intervalo de influencia del campo sobre el axón, y un modelo simplificado de membrana excitable con dos variables de estado, se aproxima la ecuación a derivadas parciales no lineal del mode-lo original por un sistema de ecuaciones diferenciales ordinarias en amplitudes de modo. Se discuten algunos resultados de simulación digital en el espacio de amplitudes de modo y se identifican una amplitud dominante que permite simplificar el orden del sistema de ecuaciones diferenciales ordinarias. Se estudia la dinámica del umbral en el sistema de orden reducido, empleando los conceptos de conjunto de decaimiento, conjunto amplifi-cador y barrera umbral. Se fundamentan las teorías clásicas de dos factores para la excita-ción. Se obtienen fórmulas analíticas para las curvas intensidad-duración correspondien-tes a la estimulación catódica de cierre y la estimulación anódica de apertura en el caso eléctrico, y para las situaciones análogas en el caso magnético. Se discuten las limitacio-nes del modelo y se sugieren mejoras. Palabras clave: Ecuación del cable homogeneizada, curvas intensidad-duración, umbra-les, cronaxia, reobase, parámetro sigma, estimulación eléctrica funcional, estimulación magnética, fibra nerviosa mielínica, análisis modal no lineal, estimulación catódica de cierre y apertura, estimulación anódica de apertura y cierre. Abstract: A nonlinear and homogenized cable equation is employed here, as it was in the first part of this work, to describe the interaction between an external electric field and a myelin nervous fiber. Applying again the concept of interval of influence of the field over the axon, and a simplified model of excitable membrane with two state variables, the nonlinear partial differential equation of the original model is approximated by a system of nonlinear differential equations in mode amplitudes. Some digital simulation results in the space of mode amplitudes are discussed, and dominant mode amplitude is identified and used to simplify the system of ordinary differential equations. Threshold dynamics is studied in the reduced order system applying the concepts of decaying set, amplifying ser and threshold barrier. A foundation is given to the classical two factor theories of excita-tion. A mathematical model of strength-duration curves for electric and magnetic stimulation is constructed. Analytical formulae for strength-duration curves are obtained for

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Resumen: Se emplea una ecuación de cable no lineal homogeneizada para describir la interacción en... more Resumen: Se emplea una ecuación de cable no lineal homogeneizada para describir la interacción entre un campo eléctrico externo y una fibra nerviosa mielínica. Aplicando el concepto de intervalo de influencia del campo sobre el axón, y un modelo simplificado de membrana excitable, se utiliza un método de Galerkin para aproximar la ecuación a deri-vadas parciales no lineal del modelo original por un sistema de ecuaciones diferenciales ordinarias en las amplitudes de modo resultantes. Se presentan algunos resultados de simulación digital en un espacio de amplitudes de modo, para casos en los que la variable de recuperación se fija en su valor de reposo. Se identifican una amplitud dominante y se utiliza para simplificar el sistema de ecuaciones diferenciales ordinarias. Se estudia la dinámica del umbral en el sistema de orden reducido y se construye un modelo matemáti-co de las curvas intensidad-duración aplicable a la estimulación eléctrica y magnética. Se obtienen fórmulas analíticas para la curva intensidad-duración, en particular para la cro-naxia y la reobase correspondientes a la estimulación catódica de cierre. Se analizan la estimulación anódica de cierre y se sugiere como abordar un análisis del bloqueo anódico por el cátodo. Se ponderan las limitaciones del modelo de axón y de las técnicas de análi-sis utilizadas. Palabras clave: Ecuación del cable homogeneizada, curvas intensidad-duración, umbra-les, cronaxia, reobase, estimulación eléctrica funcional, estimulación magnética, fibra nerviosa mielínica, análisis modal no lineal, bifurcaciones globales, estimulación catódica de cierre, estimulación anódica de cierre. Abstract: A nonlinear and homogenized cable equation is employed to describe the interaction between an external electric field and a myelin nervous fiber. Applying the concept of the interval of influence of the field over the axon, and a simplified model of ex-citable membrane, Galerkin's method is used to approximate the nonlinear partial differential equation of the original model by a system of nonlinear differential equations in the resulting mode amplitudes. Some digital simulation results in the space of mode amplitudes are reported, for the recovery variable fixed at its rest value. Dominant mode amplitude is identified and used to simplify the system of ordinary differential equations. Threshold dynamics is studied in the reduced order system and a mathematical model of strength-duration curves for electric and magnetic stimulation is constructed. Analytical formulae for strength-duration curves, in particular for chronaxie and rheobase, are ob

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Resumen: Esta segunda parte comienza con la reformulación de la teoría de bidominio de la elec-tr... more Resumen: Esta segunda parte comienza con la reformulación de la teoría de bidominio de la elec-trocardiología planteada en el trabajo previo. Se extiende el enfoque teórico-analítico mediante análisis modal no lineal para tener en cuenta la anisotropía desigual y la heterogeneidad del miocar-dio. Se tienen en cuenta las variables de recuperación de las membranas excitables del miocardio empleando un modelo matemático simplificado de la excitación. Se estudia nuevamente, justo hasta el umbral y teniendo en cuenta la anisotropía desigual, la dinámica del sistema electrodo-miocardio estimulado a través de electrodos de marcapaso. Se re-examina el concepto de región de influencia del electrodo de marcapaso sobre el miocardio anisótropo y se tiene en cuenta la función activante generalizada para construir la región de influencia. Luego se plantea un problema mixto, homogé-neo de Dirichlet y no homogéneo de Neumann a partir de las ecuaciones no lineales del modelo de bidominio, de modo de describir el efecto polarizante del electrodo, incluyendo la aparición de ánodos y cátodos virtuales. Al igual que en la primera parte, se emplea un enfoque basado en el análisis modal no lineal para transformar las ecuaciones de campo en un sistema de ecuaciones diferenciales ordinarias no lineales en las amplitudes de modo y el valor umbral de la perturbación que causa inestabilidad se estima empleando una versión modificada del método de Eckhaus. Para calcular las funciones y los valores propios del operador de Sturm-Louville que permite definir las amplitudes de modo se utiliza un espacio virtual (alibi) en el cual el miocardio ventricular puede representarse en forma aproximada mediante un modelo de sincicio eléctrico homogéneo e isótropo. Se identifican patrones espaciales de polarización de las membranas que corresponden a una fami-lia de estados umbral en el espacio de estados construido a partir de las amplitudes de modo, en el marco del modelo generalizado y se obtienen las formas y los tamaños de las regiones liminales (que deben ser despolarizadas por encima del umbral uniforme de membrana para que pueda gene-rarse un potencial de acción propagado) teniendo en cuenta la anisotropía del músculo cardíaco. Se analizan las estimulaciones catódica y anódica de cierre y de apertura. Se deducen fórmulas para las curvas intensidad-duración correspondientes a la estimulación catódica de cierre, incluyendo fórmu-las analíticas para las constantes de tiempo (cronaxias) y reobases, en términos de parámetros geo-métricos y electroquímicos del electrodo, de la distancia entre el electrodo y el miocardio ventricu-lar, de las propiedades eléctricas del conductor de volumen de los tejidos, y de la distribución local de los parámetros electrotónicos y de excitabilidad del músculo cardíaco. Se explican varios resul-tados experimentales, se confirman algunos de los resultados de las simulaciones digitales de la dinámica del umbral basados en la discretización de las ecuaciones de bidominio y se predicen algunos fenómenos nuevos. Palabras clave: Modelos de bidominio del miocardio. Diseño de electrodos de marcapaso. Umbra-les de excitabilidad. Anisotropía desigual. Heterogeneidad. Alibi. Análisis modal no lineal. Familias de estados umbral. Regiones liminales. Ánodos y cátodos virtuales. Estimulación catódica y anódi-ca de cierre y apertura. Curvas intensidad-duración.

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Resumen: En esta primera parte, después de resumir y reformular la teoría de bidominio de la elec... more Resumen: En esta primera parte, después de resumir y reformular la teoría de bidominio de la electrocardiología, se emplea un modelo matemático simplificado de la excitación del miocardio a través de electrodos de marcapaso para estudiar, justo hasta el umbral, la dinámica del sistema electrodo-miocardio. Se introduce una región de influencia del electrodo de marcapaso sobre el miocardio, para poder plantear un problema mixto, ho-mogéneo de Dirichlet y no homogéneo de Neumann a partir de las ecuaciones no lineales del modelo de bidominio, de modo de describir el efecto polarizante del electrodo. Un enfoque basado en el análisis modal no lineal se emplea para transformar las ecuaciones de campo en un sistema de ecuaciones diferenciales ordinarias no lineales en las amplitudes de modo. El valor umbral de la perturbación que causa inestabilidad se estima em-pleando una versión modificada del método de Eckhaus. Se identifican patrones espacia-les de polarización de las membranas que corresponden a una familia de estados umbral en el espacio de estados construido a partir de las amplitudes de modo. Se obtienen las formas y los tamaños de las regiones liminales (que deben ser despolarizadas por enci-ma del umbral uniforme de membrana para que pueda generarse un potencial de acción propagado). Se deducen fórmulas para curvas intensidad-duración (I-D), obtenidas me-diante pulsos catódicos de corriente controlada, incluyendo fórmulas analíticas para las constantes de tiempo (cronaxias) y reobases, en términos de parámetros geométricos y electroquímicos del electrodo, de la distancia entre el electrodo y el miocardio ventricular, de las propiedades eléctricas del conductor de volumen de los tejidos, y de la distribución local de los parámetros electrotónicos y de excitabilidad del músculo cardíaco. Se expli-can varios resultados experimentales (las regiones liminales y el comportamiento de los parámetros de las curvas I-D) y se predicen algunos fenómenos nuevos (relacionados con las regiones liminales, las familias de estados umbral y los parámetros de las curvas I-D). Este enfoque teórico-analítico mediante análisis modal no lineal se extenderá, en la Parte II de este trabajo, para tener en cuenta la anisotropía desigual y la heterogeneidad del miocardio. Palabras clave: Modelos de bidominio del miocardio. Diseño de electrodos de marcapa-so. Umbrales de excitabilidad. Análisis modal no lineal. Familias de estados umbral. Regiones liminales. Estimulación catódica de cierre. Curvas intensidad-duración.

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Progress in Nuclear Energy, 2019

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Progress in Nuclear Energy (Special Issue on Nonlinear Stability Analysis: Nonlinear nuclear reactor stability analysis and special aspects of modelling of complex dynamical systems), 2019

Asymptotic analytical methods are applied to study some problems related with bifurcations (both ... more Asymptotic analytical methods are applied to study some problems related with bifurcations (both local and global) and stability in three simple mathematical models of nuclear reactors.
The first case is a reactor which is subcritical at rest and driven by a distributed external neutron source. Under suitable conditions in the temperature feedback reactivity, the mathematical model predicts a static global bifurcation with three critical states, two stable and one unstable, with the possibility of a runaway. The dynamics of this system is studied, in a framework of slow manifold theory, by methods of restricted nonlinear modal analysis, and the results of a digital simulation are summarized. The model could be modified and extended to study the space time dynamics of sub-critical multiplying systems driven by external neutron sources (neutron beams produced by accelerators).
The second case is related with in phase and out of phase xenon oscillations in large thermal reactors. The known subcritical Hopf bifurcation that appears in the context of global mode oscillations is revisited. After applying a nonlinear modal analysis to the mathematical model, a normal form is derived by an averaging method. Approximate analytical formulae for the radii of the unstable limit cycles and for the trajectories of the state variables in a neighborhood of the bifurcation point are obtained.
The third case is a non-trivial modification and development of a simple mathematical model intended to describe certain mechanical kinetic effects stemming from a possible coupling of nuclear, thermal and mechanical vibration processes. We show that, under suitable conditions, a dynamic supercritical Hopf bifurcation in the reactor power appears in the framework of our modified model. A normal form is derived by an averaging method. Analytical formulae for the radii of the stable limit cycles and the trajectories of the state variables in a neighborhood of the bifurcation are given.
Keywords: nuclear reactor dynamics, reduced order models, restricted nonlinear modal analysis, averaging methods, bifurcation, stability, mathematical modeling, homogenized equivalent reactor

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Dresden Scientific Workshop on Reactor Dynamics and Safety, 2012

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Annals of Nuclear Energy, 2014

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Palabras clave: Reología de la sangre. Sinergética. Ley de Bejan. Acumulación axial de elementos formes. Procesos de transporte de masa. Arteriosclerosis. Aterosclerosis.

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In a previous preprint paper published in researchgate ("A modification of Plonsey´s formula for ... more In a previous preprint paper published in researchgate ("A modification of Plonsey´s formula for biomagnetism") a formula was obtained to relate the emf induced in a coil of a measuring system, with two fields: the so called impressed electric current density field (due to the action potentials in the membranes of the excitable cells located in the volume conductor of the biological tissues), and a certain stationary field, known as the magnetic lead field. The new formula proposed in "A modification of Plonsey´s formula for biomagnetism" corrects and modifies the one due to Robert Plonsey, published in 1972. Here, in this short communication we show that an intermediate formula obtained during the derivation process gives the magnetic flux that passes through one or more measuring coils as a volume integral of the scalar product of the vector field of electric current density impressed on biological tissues with a stationary vector field usually called the magnetic lead field. The latter is a property of the system formed by the measuring instrument, the biological tissues whose activity is measured, and the medium interposed between them, under quasi-stationary conditions.

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The convergence of the integral that gives the proper time as a function of inertial time, when s... more The convergence of the integral that gives the proper time as a function of inertial time, when speed tends to the speed of light fast enough as inertial time tends to infinity, was studied by Suárez and Ferrari in 1985.Here this problem is reconsidered, in the framework of special relativity. Nothing special (besides the rate of growth towards infinity) seems to characterize the behavior of the tangent component of the fields of 3-force along the path of the accelerated clock, in relation with the convergence or divergence of the integral that gives proper time as function of inertial time. However, seen from the viewpoint of the proper acceleration of the clock, a physically meaningful difference appears between the tangential proper acceleration histories that give a finite proper time for an infinite inertial time, and those tangential proper 3-acceleration histories that give an infinite proper time for an infinite inertial time: either a singularity in proper 3-acceleration for a finite value of proper time in one case, or its absence in the other case.

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La idea de que (al menos en las ciencias físicas) el objetivo final y la ley causal son dos form... more La idea de que (al menos en las ciencias físicas) el objetivo final y la ley causal son dos formas de expresar lo mismo, no parece ser todo lo conocida que cabría esperar por científicos y filósofos. De la validez de
los principios de extremo no se desprende de modo alguno que el mundo físico haya sido diseñado, es decir, creado con un propósito, pero tampoco lo excluyen. En este trabajo se analizan, desde un punto de vista físico y filosófico, algunos aspectos de la interrelación entre principios de extremo, ciencia y religión.

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 Abstract— Threshold problems in the electric stimulation of nerve, muscle and myocardium, using... more  Abstract— Threshold problems in the electric stimulation of nerve, muscle and myocardium, using external electrodes, may be studied from a theoretical standpoint using the excitation functional. As shown previously, the linear approximation to the excitation functional and the theory of linear matched filters can be applied to the determination of optimal pulse shapes for biphasic pulses employed in functional electric stimulation and cardiac pacing. Here this approach is extended using Volterra series to obtain higher order nonlinear approximations to the excitation functional. The determination of optimal pulse shapes is reconsidered in this wider framework, now using the methods of the calculus of variations to pose and solve the corresponding nonlinear optimization problems. The pulse shapes thus obtained are compared with those obtained using the linear approximation.

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An analytical approach to threshold problems in functional magnetic stimulation of nerve and skel... more An analytical approach to threshold problems in functional magnetic stimulation of nerve and skeletal muscle fibers was recently proposed, framed in the concept of excitation functional. Three generations of available equipments for magnetic stimulation are briefly considered, stressing the corresponding pulse shape in the stimulation coils. Using the criterion of minimum energy dissipated in biological tissues, an optimal shape for a current pulse in the coil that produces a just threshold depolarization in a nerve or skeletal muscle fiber is found. The method can be further developed and applied to other threshold problems in functional electric stimulation.

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Threshold problems in electric stimulation of nerve and muscle fibers have been studied from a th... more Threshold problems in electric stimulation of nerve and muscle fibers have been studied from a theoretical standpoint using the excitation functional. Here the excitation functional is extended to magnetic stimulation of excitable nerve and muscle fibers. A unified derivation of the functional is done, for (non myelinated) nerve and muscle fibers, by means of the nonlinear cable equation with a Fitzhugh-Nagumo membrane model and a generalized Rattay's activating function. The identification problem of the excitation functional for magnetic stimulation, from strength-duration experimental data, is briefly considered.

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In a generalized sense modal analysis of partial differential equations is a procedure that allow... more In a generalized sense modal analysis of partial differential equations is a procedure that allows: (a) The expansion of the solutions (fields) of partial differential equations in time and space variables as a series of given space functions weighted by time functions known as modal amplitudes. (b) The derivation of evolution equations for the mode amplitudes (a set of ordinary differential equations).
As the space functions are known, by means of modal analysis a complex space- time field dynamics is reduced to the study of the evolution of a representative point in the space of mode amplitudes.
If the original partial differential equation is nonlinear, the ordinary equations for mode amplitudes are nonlinear also: in this case we have nonlinear modal analysis. Applying suitable asymptotic methods, analytical formulae can be derived to characterize the dynamic behavior of a small number of dominant modes, and the understanding thus obtained serves as a guide to the design of in depth digital simulations, using modal equations or directly by discretization of the original nonlinear field equations.

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The mathematical modelling of electric current flow in excitable tissues is briefly reviewed and ... more The mathematical modelling of electric current flow in excitable tissues is briefly reviewed and three fundamental problems of the stimulation by external electrodes are defined and posed, in terms of suitably space-averaged fields. Then, it is given an outline of a new analytical approach to study just-threshold dynamics when certain target elements in excitable tissues are stimulated by external electrodes. Both time and spatial aspects of membrane's non uniform polarization are considered. The concept of " region of influence " of an external field of applied electric current over the target element is introduced as a key concept that allows a nonlinear modal analysis of membrane's polarization in the target element. In the fiber's case we have a one-dimensional interval of influence. In the electric syncytium case, we have a two or three dimensional region of influence. Membrane voltage, activation variables and recovery variables are assumed to remain in their rest values on the boundary of the region of influence until a threshold pattern is achieved, inside the region, due to the perturbation produced by the applied electric field. So, the nonlinear partial differential equations that describe the stimulation of the target element (fiber or electric syncytium) can be studied inside the region of influence, with homogeneous boundary conditions, relative to the constant rest state of the membranes. Then modal analysis allows the representation of the different fields (functions of space and time) like membrane voltage, activation variables, recovery variables, and so on, as linear combinations of known space functions with unknown time dependent mode amplitudes: the modal representation of the fields inside the region of influence. The known space functions belong to the complete set of orthogonal eigenfunctions of a suitable posed Sturm-Liouville problem. Substituting the modal representation of the fields in the nonlinear partial differential equations that describe the stimulation process, a set of nonlinear ordinary differential equations in the mode amplitudes is derived. The excitation dynamics due to a perturbation caused by an imposed external electric field can be studied as a trajectory in the space of mode amplitudes, from the rest state and up to the threshold states beyond which an action potential emerges. In a suitable trunkated mode amplitude space, a threshold manifold, a decaying set and an amplifying set are introduced to study the emergence of an action potential. The threshold value of a perturbation (from the rest state) that causes instability in the space of mode amplitudes is estimated using a modified version of Eckhaus method. It is then outlined a general procedure to derive strength-duration formulae, giving the derivation of Lapicque-Hill's formula, Lapicque-Weiss' formula and a new formula that could be used in the current practice of cardiac pacing. An extension of the Bonhoeffer-van der Pol-Fitzhugh model for an excitable membrane is proposed, and then applied to the modal analysis of the resulting nonlinear cable equations for excitable fibers and excitable electric syncytia. Also, it is pointed how a solution could be given to several old problems of electro stimulation that have remained open until now, using the theoretical framework proposed in this paper. The concept called " family of threshold states " (for each given target element) is proposed and its applications to synaptic excitation and to spontaneous electrical activity of excitable tissues are suggested.

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Colección de temas al estado del arte en magnetismo y superconductividad con aplicaciones de físi... more Colección de temas al estado del arte en magnetismo y superconductividad con aplicaciones de física pura, química, ciencia de materiales, ensayos industriales y biología.

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Fundamentos de Tecnología Nuclear Energética, 2020

Este libro se basa, en buena medida, en un libro previo del autor “La Energía Nuclear” publicado... more Este libro se basa, en buena medida, en un libro previo del autor “La Energía Nuclear” publicado en 2009. Se puede considerar como una introducción a algunos temas básicos sobre reactores y centrales nucleares: física nuclear, física de reactores nucleares, termo-hidráulica, combustibles nucleares, fundamentos de ingeniería mecánica, química y eléctrica de centrales nucleares, nociones sobre instrumentación y control de reactores nucleares, protección radiológica y seguridad nuclear, emplazamiento y licenciamiento de centrales nucleares.
A través del estudio de esos y otros temas, el libro intenta contribuir al proceso de discusión pública de las alternativas energéticas para este siglo. Por este motivo, el último capítulo y un apéndice extenso abarcan temas relacionados con la sociedad, el ambiente, la política y la filosofía que escapan del ámbito propio de la ingeniería nuclear.
Si bien la mayor parte del texto presupone que el lector ha adquirido conocimientos básicos de matemática y física a nivel universitario, se han separado las partes de naturaleza más complicada desde el punto de vista físico, matemático, o de ingeniería, de tal forma que sea posible omitirlas si se desea.
Se presentan muy pocas citas bibliográficas integradas en el texto, a diferencia de lo que es usual en trabajos académicos. Las fuentes principales se presentan al final en una bibliografía comentada.
Así cabe esperar que el texto resulte de lectura más fácil, sea de interés, y pueda tener alguna utilidad para un conjunto numeroso de lectores.
Los capítulos del libro están divididos en secciones. Algunas de ellas se subdividen en subsecciones. Las figuras se numeran y citan según el capítulo al que pertenecen (por ejemplo Fig. 7.2 alude a la segunda figura del séptimo capítulo). Las ecuaciones se numeran y citan según la sección en la que aparecen, con independencia de la subsección, si es que esta existe. Por ejemplo, la ecuación 14.2 [3] alude a la tercera ecuación de la segunda sección del capítulo catorce.
El libro comienza con un capítulo extenso que resume en forma bastante completa los fundamentos de la tecnología orientada a las centrales de conversión nucleoeléctrica. Este capítulo sirve como una introducción al desarrollo más detallado del tema que se encuentra en los capítulos siguientes. No requiere formación específica, a nivel universitario, ni en matemática ni en física, y puede leerse con independencia del resto del libro.
El estudio de los temas de ingeniería nuclear se limita a las tecnologías basadas en la fisión nuclear, puesto que son las únicas que se encuentran disponibles para la generación de potencia eléctrica y pueden considerarse tecnologías maduras.
En contraposición, las tecnologías basadas en la fusión de núcleos livianos, que por varias razones parecería que deberían ser preferidas frente a las tecnologías basadas en la fisión de núcleos pesados, están todavía demasiado atrasadas.
A diferencia de lo que se hace en otros libros similares, aquí no se estudian los aceleradores de partículas. Solo se consideran en el capítulo 16, durante la descripción de los reactores híbridos conducidos por aceleradores.
El fundamento de las armas nucleares de fisión se describe someramente en el capítulo 10, en el marco del problema que plantea la proliferación. Las cuatro generaciones de armas nucleares y su estado de desarrollo se abordan brevemente en el apéndice “Diplomacia y problemas de seguridad relacionados con las armas biológicas, nucleares y cibernéticas: abordaje de algunos aspectos sociales y políticos”.

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This is a second Spanish edition, corrected and slightly augmented, of a book published in 2011. ... more This is a second Spanish edition, corrected and slightly augmented, of a book published in 2011. It is intended for an audience interested in the problems that arise around scientific knowledge in the so-called post-modern era, and at the same time willing to study with some depth the physical mathematical bases of the theory of relativity, as well as some of its applications in astrophysics and cosmology. It is assumed that the reader has basic knowledge of algebra, infinitesimal calculus of one and several variables, and general physics (elements of mechanics, electromagnetism and optics, thermal and quantum physics), such as those that can be acquired during the first two years of university studies of science or engineering. Bearing this in mind, some complementary topics of mathematics and physics are presented with which the reader may not be familiar. On this basis the foundations of the restricted and generalized theories of relativity are developed. The results of experiments and confirmatory observations are considered. Some difficulties of the theory are pointed out and alternative paths that have been or are being explored are mentioned. The reader is not supposed to be familiar with epistemology or other branches of philosophy. Philosophical contents, particularly epistemological ones, are introduced as they are needed, beginning with the first chapter that is completely dedicated to them. Chapters two, three and four describe the development of ideas in Newtonian particle mechanics and classical field theory, including electromagnetism, using Galileo's relativity principle as the guiding thread. In chapter five we examine the crisis in the foundations of nineteenth-century physics that led to the restricted theory of relativity. Chapters six, seven and eight are intended to study the foundations, limitations and some applications of restricted theory. The experiments reported in 2012 that debunk the results reported in 2011 (and mentioned in the first edition of the book) about faster-than-light neutrinos are taken into account. A problem related with the slowing down of accelerated clocks, posed in 1985, is treated with more detail in this second edition of the book, both in chapter eight and in chapter thirteen, where its relation with the “unus mundus” problem is considered. At the end of chapter eight, the epistemological relations between the theories of classical physics are examined after the restricted theory of relativity was accepted. Chapter nine is intended to supplement the knowledge of mathematics. In particular, this chapter presents elements of differential geometry of surfaces and their extension to manifolds in n dimensions, including notions of tensor calculation, parallel transport, geodesics and curvature of manifolds. Chapter ten is intended to expose the foundations of the generalized theory of relativity. The epistemological relations between the theories of classical physics after the generalized theory of relativity was accepted are considered at the end of the chapter. Chapter eleven presents some applications of David Hilbert´s version of Schwartzschild solution of gravitational field equations and summarizes the confirmatory evidence available in relation to generalized theory. In addition, the basics of the calculus of variations are introduced. In chapters twelve and thirteen, astronomical applications and topics of relativistic physical cosmology, accompanied by historical and epistemological considerations, are then developed in some detail. In this second edition of the book the recent observation of an object with the Event Horizon Telescope, considered as the first black hole registered image, is mentioned. Some critical bibliography related with black holes is given. In chapter fourteen, the structure of the restricted theory of relativity, generalized theory and physical cosmology are examined from the epistemological point of view. In chapter fifteen some aspects of the work of scientific research from the psychological and sociological points of view are briefly considered. Emphasis is placed on the topics that allow us to understand avatars in the development of scientific ideas, until they disappear definitively into oblivion or become part of a stabilized and dominant paradigm. In chapter sixteen, the principle of relativity and some of its consequences are discussed from a broader philosophical perspective, emphasizing the self-imposed limitations of science. The detailed bibliographical references and some important complements appear in the form of footnotes. At the end a list of books and web pages is suggested that may be of interest to reaffirm some concepts or to deepen in the subjects of physical-mathematical sciences, cosmology and philosophy presented here. It has been sought, as far as it has been possible, that the development of the subject be self-contained, that the physical and philosophical ideas, as well as the basic hypotheses, be highlighted and explained, and that their consequences naturally arise during the development of the themes. Likewise, an attempt has been made to involve the reader in the construction of the theory, avoiding two extremes equally. One of them consists in making reference to advanced mathematical tools, which usually the reader does not know, but which are not explained to him, and that therefore he is forced to accept or study elsewhere. The other extreme avoids the necessary mathematical foundation by resorting to purely verbal statements and graphic representations, suggestive but often close to a systematized fantasy. To the extent that these objectives are achieved, the book should be useful as an introductory study material and as a basis for accessing more advanced treatments, in books and research articles, some of which include bibliographical references on foot of page

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This work describes the main measures taken to assure the quality of EDXRF Lab results, in order ... more This work describes the main measures taken to assure the quality of EDXRF Lab results, in order to verify the analytical capabilities to develop environmental studies useful for decision makers.

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En este trabajo ciertos problemas de la biología de los tejidos excitables son replanteados como ... more En este trabajo ciertos problemas de la biología de los tejidos excitables son replanteados como problemas físicos y son estudiados mediante los conceptos y los métodos de la física no lineal. Se trata de un trabajo fundamentalmente de biofísica matemática, motivado por un conjunto de problemas originados en el estudio del significado electrofisiológico de los parámetros que se suelen emplear para caracterizar las curvas intensidad umbral versus duración en los tejidos biológicos excitables .
Presentamos un análisis cualitativo de la dinámica del umbral las membranas uniformemente polarizadas y analizamos en profundidad los diferentes conceptos de umbral local de membrana.
Caracterizamos las geometrías del campo de corriente eléctrica generado por un electrodo en el conductor de volumen equivalente a los tejidos por medio de desarrollos multipolares.
Introducimos el concepto de región de influencia del electrodo sobre la fibra o el sincicio funcional blanco de la excitación, para poder efectuar un análisis modal no lineal de las ecuaciones de campo que describen la interacción entre el tejido (fibra o sincicio funcional) y el electrodo estimulador.
Introducimos el conjunto de decaimiento, la pseudo-variedad umbral y el conjunto amplificador en el espacio de configuración formado por las amplitudes de modo. Describimos la acción polarizante del electrodo mediante las proyecciones del factor de forma de la función activante para una fibra, y de la función activante generalizada para un sincicio eléctrico con anisotropía desigual. Analizamos el proceso de excitación en el espacio de configuración empleando la teoría de la estabilidad y las bifurcaciones de los sistemas dinámicos, obteniendo así ejemplos concretos y cuantitativos del concepto básico de familia de estados umbral.
Trabajando con varios modos acoplados, consideramos el bloqueo anódico por el cátodo, la excitación anódica de cierre y la excitación catódica de apertura. Explicamos la excitación anódica de cierre y la excitación catódica de apertura en términos de una desestabilización de modos al superar la correspondiente amplitud umbral (desestabilización “dura” del estado de reposo).
Fundamentamos los modelos clásicos de uno y dos factores para la excitación y esbozamos su generalización en un nuevo modelo de tres factores. La idea original de funcional de excitación resulta fundamental en todo ese desarrollo, así como el uso de los sistemas rompientes de Vogel, la linealización de la dinámica en el conjunto de decaimiento y la aproximación de la pseudo-variedad umbral por una variedad lineal.
Obtenemos diferentes expresiones analíticas para las curvas intensidad umbral versus duración de membranas uniformemente polarizadas (capítulo III) y para el caso tanto de fibras como de sincicios funcionales (capítulos V a VIII) polarizados en forma no uniforme. En estas fórmulas los parámetros globales del umbral utilizados en electrofisiología clínica se expresan en función de los parámetros locales utilizados en electrofisiología básica, de las dimensiones características del tejido (diámetro de la fibra, fracción de membranas y fracciones de volumen del sincicio funcional), de sus parámetros eléctricos (resistividad del citoplasma de la fibra, conductividades del continuo intracelular y del continuo intersticial del sincicio eléctrico), y de las características del campo de corriente generado por el electrodo estimulador en el conductor de volumen equivalente a los tejidos.
Además, introducimos la distinción entre excitabilidad global y excitabilidad local, y entre respuesta eléctrica activa y respuesta eléctrica pasiva.
Construimos un modelo del sistema electrodo-tejidos considerado como dipolo eléctrico (en el sentido de la teoría de circuitos) en el que relacionamos la polarizabilidad de la interfase electrodo-tejidos con su dimensión fractal, interpretando los parámetros de la impedancia de Drake.
Analizamos las curvas intensidad-duración para pulsos estimuladores de voltaje controlado. Discutimos el incremento fisiológico del umbral. Introducimos el "electrodo manzana" como electrodo representativo de los electrodos cóncavos.
Analizamos la propagación de un potencial de acción en un sincicio funcional. Empleamos una adaptación del método de Kompaneyetz y la teoría de perturbaciones singulares para tener en cuenta los procesos de activación y de recuperación en las membranas excitables.
Mediante el empleo del concepto de frente virtual relacionamos la velocidad de propagación con la curvatura media del frente, con los procesos de activación y recuperación, y con las propiedades de heterogeneidad y anisotropía (tal como aparecen expresadas en el tensor de dispersión eléctrica de las membranas), en condiciones de propagación estable y auto-sostenida del potencia1 de acción.
Exploramos las relaciones entre la curvatura media crítica para un frente convexo y las dimensiones críticas de la región de despolarización inicial, a partir de la cual se genera el potencial de acción, conectando de esa forma la posibilidad de propagación estable con la posibilidad de generación del potencial de acción.
Discutimos la generalización de la idea de familias de estados umbral más allá de las condiciones de la estimulación de fibras y sincicios funcionales por electrodos extra-tisulares.
Obtenemos una estimación para las constantes de tiempo de fibras y sincicios funcionales despolarizados por micro-electrodos internos.
Planteamos la aplicación del-concepto de familias de estados umbral para el estudio de la influencia, sobre el proceso de excitación en una fibra o en un sincicio funcional, de una distribución espacial arbitraria del voltaje transmembrana y de las variables de activación, recuperación y adaptación, originadas en una distribución espacio-temporal de acciones sinápticas, en la actividad de grupos de células marcapaso, en los potenciales de campo producidos por la interacción eléctrica entre células vecinas o a través de la acción de mediadores químicos de acción difusa distribuidos en el espacio intersticial, en la actividad de focos ectópicos, etc.
A lo largo de todo el desarrollo y la discusión final relacionamos los resultados teóricos con los datos experimentales o de simulación digital. Generalmente ha habido buen acuerdo entre la teoría y el experimento. No obstante, la confirmación o rectificación de varios de estos resultados teóricos exige una contraparte experimental que hasta el momento parece no estar disponible. Es necesario, entonces, diseñar y llevar a cabo experimentos nuevos.
La idea de la región de influencia del electrodo y el empleo del metódo de Kompaneyetz conlleva ciertas limitaciones que, por un lado, nos condujeron a estudiar la polarización de la zona blanco justo hasta el umbral, y por el otro nos condujeron a considerar la propagación de un potencial de acción plenamente desarrollado. Queda fuera del alcance de nuestro análisis la descripción detallada de la transición entre un potencial de acción incipiente, que emerge de lo que hubiera sido una respuesta local; y un potencial de acción que se propaga ya en forma estable, en su plenitud. Este es un tema fundamental para estudiar en el futuro.
No obstante, a través del presente trabajo se han aclarado varias cuestiones electrofisiológicas que permanecieron confusas durante muchos años. Asimismo se tienen nuevas bases para el desarrollo de una teoría físico-matemática de la estimulación eléctrica de tejidos biológicos -tanto fibras como sincicios funcionales- mediante electrodos externos, posiblemente utilizable en electrofisiología clínica, en terapéutica y en ingeniería biomédica.

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