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... The electrical properties of the lens of the eye were resolved by Rae and Eisenberg,172 who explained the substantial differences between measurements made with single microelectrodes and two microelectrodes. The single microelectrode measurements obscured the interesting membrane properties because of three-dimensional effects 123 analyzed first here 173,174 and later by methods of singular perturbation theory. [124][125][126][127][128][129][175][176][177] The misleading three-dimensional effects can be easily removed once theory shows where they come from. ...

... 213 The Atlantic cable provided the first high speed (electrical) communication between Europe and the United States, cutting communication delays from weeks to seconds or minutes. The cable equation was used in biology, where it played a crucial role in our understanding of nerve conduction [214][215][216] and it was not thought necessary to derive it, because it seemed so obvious physically (with a few exceptions, important when cables had large diameters, or recording was done on or near point sources of current [123][124][125][126][127][128][129][173][174][175][176][177][217][218][219][220]. ...

... It is tempting to just write down the physically intuitive field equations for the bidomain model 129,177 -correct, or rather not incorrect, as that approach might be-using the general approach of the structural analysis of electrical properties 79,84,[123][124][125][303][304][305] conjoined to a representation of fluid flow. An abundance of caution and mis-steps using this approach in our several careers, motivated a complete derivation from conservation laws themselves, and indeed a derivation of the underlying fluid flow laws like Starling's equation. ...

Biology is about structure. Structures within structures. Organs within animals, tissues within organs, cells within tissues, and molecules, often proteins within cells. The structures are so complex that they can only be described by numbers. No numbers are of more importance than those that describe proteins. The numbers that describe coordinates of its atoms, often determined by Patterson functions (which are inverse Fourier Transforms of intensities) of crystal diffraction. Without these numbers, structural biology would hardly exist. Without numbers, engineering would not exist. Numbers are surely needed by the engineers who produce the x-rays diffracting from atoms of protein crystals. Devices of engineering have function. They are built to implement equations. Engineering devices use structures to implement equations, when power is supplied at the right places, that produces appropriate flows. Flows are the essence of life. Equilibrium means death in most living systems. Flows in biological structures are hard to analyze because we do not know input output equations in advance. Sometimes we do not know the function of the structures. Flows, forces, and structures of life (like those of engineering) are related by field equations of conservation laws, partial differential equations, constrained by location and properties of structures. Constraints are boundary conditions located on the complicated domain of biological structure. The hierarchy of structures allows a handful of atoms (in proteins and nucleic acids) to control macroscopic function. Dealing with this complexity is simplified if one systematically analyzes structure using the most general field theory known, electricity described by the Maxwell equations, without significant known error. Currents are involved because flows of biology usually involve migration of charges, convection of water and solutes, diffusion of ions that form the plasma of life, and their interactions. Interactions can dominate function. Here I show how a few complex structures can be understood in engineering detail. This approach may be useful in dealing with biological and medical issues in many other cases. In one limited case—the clearance of a toxic waste (potassium ions) from the optic nerve—this approach seems to have succeeded.

... The membrane makes a cell an obstacle to electricity. The membrane is such a high resistance (123)(124)(125)(126)(127)(128)(129) that current flows in a parallel path even if that path is only 10 nm or so of salt solution. It is only when intracellular objects are within a Debye length of so (typically less than a nanometer in biological salt solutions) that current flow in the intracellular solution is significantly impeded, beyond the reduction of cross sectional area. ...

... It is tempting to just write down the physically intuitive field equations for the bidomain model (129, 177)-correct, or rather not incorrect, as that approach might be--using the general approach of the structural analysis of electrical properties ( (79,84,(123)(124)(125)(303)(304)(305) conjoined to a representation of fluid flow. An abundance of caution, and mis-steps using this approach in our several careers, motivated a more formal derivation from conservation laws themselves, and indeed a derivation of the underlying fluid flow laws like Starling's equation (306). ...

... The perturbation structure of the problem shows the mathematical reason the potential of the extracellular phase within the lens does not vary with connexin density. Physically and physiologically it seems obvious that changes in the intracellular domain would have only second order effects on the potential of the extracellular phase within the lens because they are separated by a high resistance membrane that buffers extracellular from the intracellular region (see similar effects in other systems, see qualitative general discussion in (123) and quantitative special cases computed in (125)(126)(127)(128). ...

Biology is about structure. Structures within structures. Organs within animals, tissues within organs, cells within tissues, and molecules, often proteins within cells. The structures are so complex that they can only be described by numbers. No numbers are of more importance than those that describe proteins. The numbers that describe coordinates of its atoms, often determined by Patterson functions (which are inverse Fourier Transforms of intensities) of crystal diffraction. Without these numbers, structural biology would hardly exist. Without numbers, engineering would not exist. Numbers are surely needed by the engineers who produce the x-rays diffracting from atoms of protein crystals. Devices of engineering have function. They are built to implement equations. Engineering devices use structures to implement equations, when power is supplied at the right places, that produces appropriate flows. Flows are the essence of life. Equilibrium means death in most living systems. Flows in biological structures are hard to analyze because we do not know input output equations in advance. Sometimes we do not know their function. Flows, forces, and structures of life (like those of engineering) are related by field equations of conservation laws, partial differential equations, constrained by location and properties of structures. Constraints are boundary conditions located on the complicated domain of biological structure. Dealing with this complexity is simplified if one systematically analyzes structure using the most general field theory known, electricity described by the Maxwell equations, without significant known error. Currents are involved because flows of biology usually involve migration of charges, convection of water and solutes, diffusion of ions that form the plasma of life, and their interactions. Interactions can dominate function. Here I show how a few complex structures can be understood in engineering detail. This approach may be useful in dealing with biological and medical issues in many other cases. In one limited case—the clearance of a toxic waste (potassium ions) from the optic nerve—this approach seems to have succeeded.

... The membrane makes a cell an obstacle to electricity. The membrane is such a high resistance (123)(124)(125)(126)(127)(128)(129) that current flows in a parallel path even if that path is only 10 nm or so of salt solution. It is only when intracellular objects are within a Debye length of so (typically less than a nanometer in biological salt solutions) that current flow in the intracellular solution is significantly impeded, beyond the reduction of cross sectional area. ...

... It is tempting to just write down the physically intuitive field equations for the bidomain model (129, 177)-correct, or rather not incorrect, as that approach might be--using the general approach of the structural analysis of electrical properties ( (79,84,(123)(124)(125)(303)(304)(305) conjoined to a representation of fluid flow. An abundance of caution, and mis-steps using this approach in our several careers, motivated a more formal derivation from conservation laws themselves, and indeed a derivation of the underlying fluid flow laws like Starling's equation (306). ...

... The perturbation structure of the problem shows the mathematical reason the potential of the extracellular phase within the lens does not vary with connexin density. Physically and physiologically it seems obvious that changes in the intracellular domain would have only second order effects on the potential of the extracellular phase within the lens because they are separated by a high resistance membrane that buffers extracellular from the intracellular region (see similar effects in other systems, see qualitative general discussion in (123) and quantitative special cases computed in (125)(126)(127)(128). ...

Biology is about structure. Structures within structures. Organs within animals, tissues within organs, cells within tissues, and molecules, often proteins within cells. The structures are so complex that they can only be described by numbers. No numbers are of more importance than those that describe proteins. The numbers that describe coordinates of its atoms, often determined by Patterson functions (which are inverse Fourier Transforms of intensities) of crystal diffraction. Without these numbers, structural biology would hardly exist. Without numbers, engineering would not exist. Numbers are surely needed by the engineers who produce the x-rays diffracting from atoms of protein crystals. Devices of engineering have function. They are built to implement equations. Engineering devices use structures to implement equations, when power is supplied at the right places, that produces appropriate flows. Flows are the essence of life. Equilibrium means death in most living systems. Flows in biological structures are hard to analyze because we do not know input output equations in advance. Sometimes we do not know their function. Flows, forces, and structures of life (like those of engineering) are related by field equations of conservation laws, partial differential equations, constrained by location and properties of structures. Constraints are boundary conditions located on the complicated domain of biological structure. Dealing with this complexity is simplified if one systematically analyzes structure using the most general field theory known, electricity described by the Maxwell equations, without significant known error. Currents are involved because flows of biology usually involve migration of charges, convection of water and solutes, diffusion of ions that form the plasma of life, and their interactions. Interactions can dominate function. Here I show how a few complex structures can be understood in engineering detail. This approach may be useful in dealing with biological and medical issues in many other cases. In one limited case—the clearance of a toxic waste (potassium ions) from the optic nerve—this approach seems to have succeeded.

... However, neither of these have accounted for the 73 electrodiffusive coupling between the movement of ions and electrical potentials (see 74 Results section titled Loss in accuracy when neglecting electrodiffusive effects on 75 concentration dynamics). Hence, to our knowledge, no morphologically explicit neuron 76 model has so far been developed that ensures biophysically consistent dynamics in ion 77 concentrations and electrical potentials during long-time activity, although useful 78 mathematical framework for constructing such models are available [58][59][60][61][62]. 79 The goal of this work is to propose what we may refer to as "a minimal neuronal 80 model that has it all". By "has it all", we mean that it (1) has a spatial extension, (2) 81 considers both extracellular-and intracellular dynamics, (3) keeps track of all ion 82 concentrations (Na + , K + , Ca 2+ , and Cl − ) in all compartments, (4) keeps track of all 83 electrical potentials (φ m , φ e , and φ i -the latter being the intracellular potential) in all 84 compartments, (5) has differential expression of ion channels in soma versus dendrites, 85 and can fire somatic APs and dendritic calcium spikes, (6) conductor theory (Fig 1A-B). ...

... Previous spatial, electrodiffusive, and 508 biophysically consistent models of spreading depression have targeted the problem at a 509 large-scale tissue-level, using a mean-field approach [30,77,78]. These models were 510 inspired by the bi-domain model [79], which has been successfully applied in simulations 511 of cardiac tissue [80,81]. The bi-domain model is a coarse-grained model, in which the 512 tissue is considered as a bi-phasic continuum consisting of an intracellular and 513 extracellular domain. ...

Most neuronal models are based on the assumption that ion concentrations remain constant during the simulated period, and do not account for possible effects of concentration variations on ionic reversal potentials, or of ionic diffusion on electrical potentials. Here, we present what is, to our knowledge, the first multicompartmental neuron model that accounts for electrodiffusive ion concentration dynamics in a way that ensures a biophysically consistent relationship between ion concentrations, electrical charge, and electrical potentials in both the intra- and extracellular space. The model, which we refer to as the electrodiffusive Pinsky-Rinzel (edPR) model, is an expanded version of the two-compartment Pinsky-Rinzel (PR) model of a hippocampal CA3 neuron, where we have included homeostatic mechanisms and ion-specific leakage currents. Whereas the main dynamical variable in the original PR model is the transmembrane potential, the edPR model in addition keeps track of all ion concentrations (Na ⁺ , K ⁺ , Ca ²⁺ , and Cl ⁻ ), electrical potentials, and the electrical conductivities in the intra- as well as extracellular space. The edPR model reproduces the membrane potential dynamics of the PR model for moderate firing activity, when the homeostatic mechanisms succeed in maintaining ion concentrations close to baseline. For higher activity levels, homeostasis becomes incomplete, and the edPR model diverges from the PR model, as it accounts for changes in neuronal firing properties due to deviations from baseline ion concentrations. Whereas the focus of this work is to present and analyze the edPR model, we envision that it will become useful for the field in two main ways. Firstly, as it relaxes a set of commonly made modeling assumptions, the edPR model can be used to test the validity of these assumptions under various firing conditions, as we show here for a few selected cases. Secondly, the edPR model is a supplement to the PR model and should replace it in simulations of scenarios in which ion concentrations vary over time. As it is applicable to conditions with failed homeostasis, the edPR model opens up for simulating a range of pathological conditions, such as spreading depression or epilepsy.
Author summary
Neurons generate their electrical signals by letting ions pass through their membranes. Despite this fact, most models of neurons apply the simplifying assumption that ion concentrations remain effectively constant during neural activity. This assumption is often quite good, as neurons contain a set of homeostatic mechanisms that make sure that ion concentrations vary quite little under normal circumstances. However, under some conditions, these mechanisms can fail, and ion concentrations can vary quite dramatically. Standard models are thus not able to simulate such conditions. Here, we present what to our knowledge is the first multicompartmental neuron model that in a biophysically consistent way does account for the effects of ion concentration variations. We here use the model to explore under which activity conditions the ion concentration variations become important for predicting the neurodynamics. We expect the model to be of great use for simulating a range of pathological conditions, such as spreading depression or epilepsy, which are associated with large changes in extracellular ion concentrations.

... The current pulse travels through the solution (blood) within the capillary, while simultaneously leaking through the porous capillary wall. This geometry ( Fig. 1) is analyzed using traditional cable theory (Eisenberg and Johnson 1970). The voltage drop across the two electrodes can be used to understand the ionic permeability of the blood vessel. ...

... where a is the radius of the capillary (Crone and Olesen 1982;Eisenberg and Johnson 1970). r i is determined by dividing the resistivity of the blood by the capillary crosssection area (hence /cm). ...

Silicon nanomembranes are ultrathin, highly permeable, optically transparent and biocompatible substrates for the construction of barrier tissue models. Trans-epithelial/endothelial electrical resistance (TEER) is often used as a non-invasive, sensitive and quantitative technique to assess barrier function. The current study characterizes the electrical behavior of devices featuring silicon nanomembranes to facilitate their application in TEER studies. In conventional practice with commercial systems, raw resistance values are multiplied by the area of the membrane supporting cell growth to normalize TEER measurements. We demonstrate that under most circumstances, this multiplication does not ‘normalize’ TEER values as is assumed, and that the assumption is worse if applied to nanomembrane chips with a limited active area. To compare the TEER values from nanomembrane devices to those obtained from conventional polymer track-etched (TE) membranes, we develop finite element models (FEM) of the electrical behavior of the two membrane systems. Using FEM and parallel cell-culture experiments on both types of membranes, we successfully model the evolution of resistance values during the growth of endothelial monolayers. Further, by exploring the relationship between the models we develop a ‘correction’ function, which when applied to nanomembrane TEER, maps to experiments on conventional TE membranes. In summary, our work advances the the utility of silicon nanomembranes as substrates for barrier tissue models by developing an interpretation of TEER values compatible with conventional systems.

... The total current of the Maxwell Current Law (2) helps define the circuit models familiar in the treatment of the nerve action potential [5,6], and the action potential of cardiac and skeletal muscle [7]. The circuits have been derived from the Green's functions of a structural description of complex tissues [80][81][82] thereby linking the biological treatment of electricity and the physical treatment of electrodynamics. Singular perturbation techniques [74], p. 218-238, are helpful in applying the Maxwell Current Law (2) to biological systems of some complexity [80,82,83]. ...

Version 3
Maxwell defined a ‘true’ or ‘total’ current in a way not widely used today. He said that “... true electric current ... is not the same thing as the current of conduction but that the time-variation of the electric displacement must be taken into account in estimating the total movement of electricity”. We show that the true or total current is a universal property of electrodynamics independent of the properties of matter, shown using mathematics without the approximation of a dielectric constant. The resulting Maxwell Current Law is a generalization of the Kirchhoff Law of Conduction Current used in circuit analysis, that also includes the displacement current. Engineers often introduce the displacement current through supplementary circuit elements, called ‘stray capacitances’.
The Maxwell Current Law does not require currents to be confined to circuits. It can be applied to three dimensional systems like the electron transport system of mitochondria and the signaling system of nerve cells. The Maxwell Current Law clarifies the flow of electrons, protons, and ions in mitochondria that generate ATP, the molecule used to store chemical energy throughout life. The currents are globally coupled because mitochondria are short. The Maxwell Current Law approach reinterprets the classical chemiosmotic hypothesis of ATP production. The conduction current of protons in mitochondria is driven by the protonmotive force including its component electrical potential, just as in the classical chemiosmotic hypothesis. The conduction current is, however, just a part of the true current analyzed by Maxwell. Maxwell’s current does not accumulate, in contrast to the conduction current of protons which does accumulate. Details of the accumulation do not have to be considered in the analysis of true current.
The treatment here allows the chemiosmotic hypothesis to take advantage of the knowledge of current flow in the physical and engineering sciences, particularly Kirchhoff and Maxwell Current Laws. That knowledge has been helpful in understanding action potentials in biology, and in technology in general. Knowing the current means knowing an important part of the mechanism of ATP synthesis.

... The total current of the Maxwell Current Law (2) helps define the circuit models familiar in the treatment of the nerve action potential [5,6], and the action potential of cardiac and skeletal muscle [7]. The circuits have been derived from the Green's functions of a structural description of complex tissues [72][73][74] thereby linking the biological treatment of electricity and the physical treatment of electrodynamics. Singular perturbation techniques [68], p. 218-238, are helpful in applying the Maxwell Current Law (2) to biological systems of some complexity [72,74,75]. ...

Maxwell defined a ‘true’ or ‘total’ current in a way not widely used today. He said that “... true electric current ... is not the same thing as the current of conduction but that the time-variation of the electric displacement must be taken into account in estimating the total movement of electricity”. We show that the true or total current is a universal property of electrodynamics independent of the properties of matter, shown using mathematics without the approximation of a dielectric constant. The resulting Maxwell Current Law is a generalization of the Kirchhoff Law of Conduction Current used in circuit analysis, that also includes the displacement current. Engineers often introduce the displacement current through supplementary circuit elements, called ‘stray capacitances’.
The Maxwell Current Law does not require currents to be confined to circuits. It can be applied to three dimensional systems like the electron transport system of mitochondria and the signaling system of nerve cells. The Maxwell Current Law clarifies the flow of electrons, protons, and ions in mitochondria that generate ATP, the molecule used to store chemical energy throughout life. The currents are globally coupled because mitochondria are short. The Maxwell Current Law approach reinterprets the classical chemiosmotic hypothesis of ATP production. The conduction current of protons in mitochondria is driven by the protonmotive force including its component electrical potential, just as in the classical chemiosmotic hypothesis. The conduction current is, however, just a part of the true current analyzed by Maxwell. Maxwell’s current does not accumulate, in contrast to the conduction current of protons which does accumulate. Details of the accumulation do not have to be considered in the analysis of true current.
The treatment here allows the chemiosmotic hypothesis to take advantage of the knowledge of current flow in the physical and engineering sciences, particularly Kirchhoff and Maxwell Current Laws. That knowledge has been helpful in understanding action potentials in biology, and in technology in general. Knowing the current means knowing an important part of the mechanism of ATP synthesis.

... This is a far from trivial problem. Domain-type models are inspired by the bi-domain model [109], which has been used to describe cardiac tissue as a bi-phasic continuum consisting of an intracellular and extracellular domain [110,111]. Similar, tri-domain models (including neurons, ECS, and glia cells) have been used to simulate brain tissue [37][38][39]. ...

Within the computational neuroscience community, there has been a focus on simulating the electrical activity of neurons, while other components of brain tissue, such as glia cells and the extracellular space, are often neglected. Standard models of extracellular potentials are based on a combination of multicompartmental models describing neural electrodynamics and volume conductor theory. Such models cannot be used to simulate the slow components of extracellular potentials, which depend on ion concentration dynamics, and the effect that this has on extracellular diffusion potentials and glial buffering currents. We here present the electrodiffusive neuron-extracellular-glia (edNEG) model, which we believe is the first model to combine compartmental neuron modeling with an electrodiffusive framework for intra- and extracellular ion concentration dynamics in a local piece of neuro-glial brain tissue. The edNEG model (i) keeps track of all intraneuronal, intraglial, and extracellular ion concentrations and electrical potentials, (ii) accounts for action potentials and dendritic calcium spikes in neurons, (iii) contains a neuronal and glial homeostatic machinery that gives physiologically realistic ion concentration dynamics, (iv) accounts for electrodiffusive transmembrane, intracellular, and extracellular ionic movements, and (v) accounts for glial and neuronal swelling caused by osmotic transmembrane pressure gradients. The edNEG model accounts for the concentration-dependent effects on ECS potentials that the standard models neglect. Using the edNEG model, we analyze these effects by splitting the extracellular potential into three components: one due to neural sink/source configurations, one due to glial sink/source configurations, and one due to extracellular diffusive currents. Through a series of simulations, we analyze the roles played by the various components and how they interact in generating the total slow potential. We conclude that the three components are of comparable magnitude and that the stimulus conditions determine which of the components that dominate.

... How voltage and ionic concentrations are distributed and regulated in excitable cells such as neurons, astrocytes, etc.. remains a challenging question, despite decades B D. Holcman david.holcman@ens.fr 1 of experimental and theoretical efforts (Hille 2001;Eisenberg and Johnson 1970;Bezanilla 2008;Koch and Segev 1989;Yuste 2010;Holcman and Yuste 2015;Sylantyev et al. 2013). In particular, the voltage in microdomains such as initial segments, dendrites, dendritic spines, remain difficult to study experimentally due to their small size. ...

The distribution of voltage in sub-micron cellular domains remains poorly understood. In neurons, the voltage results from the difference in ionic concentrations which are continuously maintained by pumps and exchangers. However, it not clear how electro-neutrality could be maintained by an excess of fast moving positive ions that should be counter balanced by slow diffusing negatively charged proteins. Using the theory of electro-diffusion, we study here the voltage distribution in a generic domain, which consists of two concentric disks (resp. ball) in two (resp. three) dimensions, where a negative charge is fixed in the inner domain. When global but not local electro-neutrality is maintained, we solve the Poisson–Nernst–Planck equation both analytically and numerically in dimension 1 (flat) and 2 (cylindrical) and found that the voltage changes considerably on a spatial scale which is much larger than the Debye screening length, which assumes electro-neutrality. The present result suggests that long-range voltage drop changes are expected in neuronal microcompartments, probably relevant to explain the activation of far away voltage-gated channels located on the surface membrane.

... A natural question arises as to how different the behavior of the bidomain model is qualitatively from that of the monodomain model. The bidomain model, initially introduced in [8,24,19], is the standard tissuelevel model of cardiac electrophysiology widely used in simulations (see for instance [4,15,17,6,23]). Well-posedness is studied in [7,3,25,5,11,12]. ...

The bidomain model is the standard model for cardiac electrophysiology. In this paper, we investigate the instability and asymptotic behavior of planar fronts and planar pulses of the bidomain Allen--Cahn equation and the bidomain FitzHugh--Nagumo equation in two spatial dimension. In previous work, it was shown that planar fronts of the bidomain Allen--Cahn equation can become unstable in contrast to the classical Allen--Cahn equation. We find that, after the planar front is destabilized, a rotating zigzag front develops whose shape can be explained by simple geometric arguments using a suitable Frank diagram. We also show that the Hopf bifurcation through which the front becomes unstable can be either supercritical or subcritical, by demonstrating a parameter regime in which a stable planar front and zigzag front can coexist. In our computational studies of the bidomain FitzHugh--Nagumo pulse solution, we show that the pulses can also become unstable much like the bidomain Allen--Cahn fronts. However, unlike the bidomain Allen--Cahn case, the destabilized pulse does not necessarily develop into a zigzag pulse. For certain choice of parameters, the destabilized pulse can disintegrate entirely. These studies are made possible by the development of a numerical scheme that allows for the accurate computation of the bidomain equation in a two dimensional strip domain of infinite extent.

... The cellular compartments can communicate with the ECS via transmembrane ion and water fluxes. This mathematical model extends on the celebrated bidomain model 7,8,43 , and both represent the tissue in a homogenized manner. Homogenized models are coarse-grained, and hence well suited for simulating phenomena on the tissue scale (mm). ...

Mathematical modelling of ionic electrodiffusion and water movement is emerging as a powerful avenue of investigation to provide new physiological insight into brain homeostasis. However, in order to provide solid answers and resolve controversies, the accuracy of the predictions is essential. Ionic electrodiffusion models typically comprise non-trivial systems of non-linear and highly coupled partial and ordinary differential equations that govern phenomena on disparate time scales. Here, we study numerical challenges related to approximating these systems. We consider a homogenized model for electrodiffusion and osmosis in brain tissue and present and evaluate different associated finite element-based splitting schemes in terms of their numerical properties, including accuracy, convergence, and computational efficiency for both idealized scenarios and for the physiologically relevant setting of cortical spreading depression (CSD). We find that the schemes display optimal convergence rates in space for problems with smooth manufactured solutions. However, the physiological CSD setting is challenging: we find that the accurate computation of CSD wave characteristics (wave speed and wave width) requires a very fine spatial and fine temporal resolution.

... However, this 440 model was not based on an electrodiffusive formalism, and did not account for effects of 441 extracellular potentials on neurodynamics and K + transport. Other spatial models of SD [27-29, 40, 88] have been inspired by the coarse-grained bi-domain model [89], which 443 previously has been used to simulate cardiac tissue [90,91]. These models are 444 electrodiffusive, and treat brain tissue as a homogeneous, coarse-grained, continuum, 445 making them computationally efficient to allow for large scale simulations of SD 446 propagation. ...

Computational modeling in neuroscience has largely focused on simulating the electrical activity of neurons, while ignoring other components of brain tissue, such as glial cells and the extracellular space. As such, most existing models can not be used to address pathological conditions, such as spreading depression, which involves dramatic changes in ion concentrations, large extracellular potential gradients, and glial buffering processes. We here present the electrodiffusive neuron-extracellular-glia (edNEG) model, which we believe is the first model to combine multicompartmental neuron modeling with an electrodiffusive framework for intra- and extracellular ion concentration dynamics in a local piece of neuro-glial brain tissue. The edNEG model (i) keeps track of all intraneuronal, intraglial, and extracellular ion concentrations and electrical potentials, (ii) accounts for neuronal somatic action potentials, and dendritic calcium spikes, (iii) contains a neuronal and glial homeostatic machinery that gives physiologically realistic ion concentration dynamics, (iv) accounts for electrodiffusive transmembrane, intracellular, and extracellular ionic movements, and (v) accounts for glial and neuronal swelling caused by osmotic transmembrane pressure gradients. We demonstrate that the edNEG model performs realistically as a local and closed system, i.e., that it maintains a steady state for moderate neural activity, but experiences concentration-dependent effects, such as altered firing patterns and homeostatic
breakdown, when the activity level becomes too intense. Furthermore, we study the role of glia in making the neuron more tolerable to hyperactive firing and in limiting
neuronal swelling. Finally, we discuss how the edNEG model can be integrated with previous spatial continuum models of spreading depression to account for effects of
neuronal morphology, action potential generation, and dendritic Ca 2+ spikes which are currently not included in these models.

... Previous spatial, electrodiffusive, and biophysically consistent models of spreading depression have targeted the problem at a large-scale tissue-level, using a mean-field approach [30, 76,77]. These models were inspired by the bi-domain model [78], which has been successfully applied in simulations of cardiac tissue [79,80]. The bi-domain model is a coarse-grained model, in which the tissue is considered as a bi-phasic continuum consisting of an intracellular and extracellular domain. ...

In most neuronal models, ion concentrations are assumed to be constant, and effects of concentration variations on ionic reversal potentials, or of ionic diffusion on electrical potentials are not accounted for. Here, we present the electrodiffusive Pinsky-Rinzel (edPR) model, which we believe is the first multicompartmental neuron model that accounts for electrodiffusive ion concentration dynamics in a way that ensures a biophysically consistent relationship between ion concentrations, electrical charge, and electrical potentials in both the intra- and extracellular space. The edPR model is an expanded version of the two-compartment Pinsky-Rinzel (PR) model of a hippocampal CA3 neuron. Unlike the PR model, the edPR model includes homeostatic mechanisms and ion-specific leakage currents, and keeps track of all ion concentrations (Na⁺, K⁺, Ca²⁺, and Cl⁻), electrical potentials, and electrical conductivities in the intra- and extracellular space. The edPR model reproduces the membrane potential dynamics of the PR model for moderate firing activity. For higher activity levels, or when homeostatic mechanisms are impaired, the homeostatic mechanisms fail in maintaining ion concentrations close to baseline, and the edPR model diverges from the PR model as it accounts for effects of concentration changes on neuronal firing. We envision that the edPR model will be useful for the field in three main ways. Firstly, as it relaxes commonly made modeling assumptions, the edPR model can be used to test the validity of these assumptions under various firing conditions, as we show here for a few selected cases. Secondly, the edPR model should supplement the PR model when simulating scenarios where ion concentrations are expected to vary over time. Thirdly, being applicable to conditions with failed homeostasis, the edPR model opens up for simulating a range of pathological conditions, such as spreading depression or epilepsy.

... How voltage and ionic concentrations are distributed and regulated in excitable cells such as neurons, astrocytes, etc.. remains a challenging question, despite decades of experimental and theoretical efforts [1][2][3][4][5][6][7]. In particular, the voltage in microdomains such as initial segments, dendrites, dendritic spines, remain difficult to study experimentally due to their small size. ...

The distribution of voltage in sub-micron cellular domains remains poorly understood. In neurons, the voltage results from the difference in ionic concentrations which are continuously maintained by pumps and exchangers. However, it not clear how electro-neutrality could be maintained by an excess of fast moving positive ions that should be counter balanced by slow diffusing negatively charged proteins. Using the theory of electro-diffusion, we study here the voltage distribution in a generic domain, which consists of two concentric disks (resp. ball) in two (resp. three) dimensions, where a negative charge is fixed in the inner domain. When global but not local electro-neutrality is maintained, we solve the Poisson-Nernst-Planck equation both analytically and numerically in dimension 1 (flat) and 2 (cylindrical) and found that the voltage changes considerably on a spatial scale which is much larger than the Debye screening length, which assumes electro-neutrality. The present result suggests that long-range voltage drop changes are expected in neuronal microcompartments, probably relevant to explain the activation of far away voltage-gated channels located on the surface.

... However, cardiac tissue is a syncytium. Analytic solutions to the apparent intracellular impedance for the cable models and two-and three-dimensional cardiac tissue are available ( Eisenberg and Johnson, 1970;Jack et al., 1983;Koch, 2004;Plonsey and Barr, 2007). Additionally, in this paper, we are mainly interested in the extracellular impedance. ...

Theoretical cardiac electrophysiology focuses on the dynamics of the membrane and sarcoplasmic reticulum ion currents; however, passive (e.g., membrane capacitance) and quasi-active (response to small signals) properties of the cardiac sarcolemma, which are quantified by impedance, are also important in determining the behavior of cardiac tissue. Theoretically, impedance varies in the different phases of a cardiac cycle. Our goal in this study was to numerically predict and experimentally validate these phasic changes. We calculated the expected impedance signal using analytic methods (for generic ionic models) and numerical computation (for a rabbit ventricular ionic model). Cardiac impedance is dependent on the phase of the action potential, with multiple deflections caused by a sequential activation and inactivation of various membrane channels. The two main channels shaping the impedance signal are the sodium channel causing a sharp and transient drop at the onset of action potential and the inward rectifying potassium channel causing an increase in impedance during the plateau phase. This dip and dome pattern was confirmed in an ex-vivo rabbit heart model using high-frequency sampling through a monophasic action potential electrode. The hearts were immobilized using a myosin-inhibitor to minimize motion artifacts. We observed phasic impedance changes in three out of four hearts with a dome amplitude of 2 − 4Ω. Measurement of phasic impedance modulation using an extracellular electrode is feasible and provides a non-invasive way to gain insight into the state of cardiac cells and membrane ionic channels. The observed impedance recordings are consistent with the dip and dome pattern predicted analytically.

... In [3], Mori and Peskin constructed a mathematical model which takes into account both geometric effects and ionic concentration dynamics and solved it based on the finite volume method. In the present paper, we adopt the following two-phase problem, known as the 3D cable model, proposed by Eisenberg and Johnson [4]. Although this is a rather simple mathematical model compared with the one proposed in [3], we can construct a straightforward numerical scheme; therefore we expect that this problem enables us to study fundamental phenomena in cellular electrophysiology. ...

A three-dimensional model of cellular electrophysiology, the 3D cable model, is numerically studied. Our numerical scheme is constructed based on the method of fundamental solutions, which is a meshfree numerical solver for homogeneous linear partial differential equations. We numerically show the existence of pulse-like traveling wave solutions for the model.

... The relation of the equivalent circuit to the electrodynamics of the system was studied by solving Poisson's equation by several authors, but the simplification to the usual equivalent circuit only became apparent when series expansions of the solution of the Poisson equation, and the Poisson equation itself were developed [3,12,16,33,[51][52][53][54]. ...

Umfassende Monographie mit ausführlichem Theorieteil und verschiedene Anwendungen der elektrochemischen Impedanzspektroskopie

... The relation of the equivalent circuit to the electrodynamics of the system was studied by solving Poisson's equation by several authors, but the simplification to the usual equivalent circuit only became apparent when series expansions of the solution of the Poisson equation, and the Poisson equation itself were developed [3,12,16,33,[51][52][53][54]. ...

A wide range of materials can be usefully characterized by impedance spectroscopy (IS), namely, electrical and structural ceramics, magnetic ferrites, semiconductors, membranes, polymeric materials, and protective paint films. The measurement techniques used to characterize materials are generally simpler than those used for electrode processes. This chapter first discusses microstructural models describing grains and grain boundaries of differing phase composition, suspensions of one phase within another, and porosity. It then gives examples of the combined use of IS and electron microscopy. The chapter then discusses the models that have been proposed for describing the conductive‐system dispersive responses. It then presents examples of several different applications of IS. Four different devices have been chosen: solid electrolyte chemical sensors (SECSs), secondary batteries, photoelectrochemical devices, and semiconductor‐insulator‐electrolyte sensors. The chapter further reviews the application of IS to the study of corrosion phenomena.

... It is important to note that the continuity of current law has important biological implications in systems more general than a series of chemical reactions. Continuity of current law implies the cable equations (called the telegrapher's equation in the mathematics literature), see derivation from the three dimensional theory in [144][145][146][147] and p. 218-238 of [148]. The cable equation [149] is the foundation of the Hodgkin Huxley model [150][151][152][153][154] of the action potential of nerve and muscle fibers. ...

The law of mass action does not force a series of chemical reactions to have the same current flow everywhere. Interruption of far-away current does not stop current everywhere in a series of chemical reactions (analyzed according to the law of mass action), and so does not obey Maxwell’s equations. An additional constraint and equation is needed to enforce global continuity of current. The additional constraint is introduced in this paper in the special case that the chemical reaction describes spatial movement through narrow channels. In that case, a fully consistent treatment is possible using different models of charge movement. The general case must be dealt with by variational methods that enforce consistency of all the physical laws involved. Violations of current continuity arise away from equilibrium, when current flows, and the law of mass action is applied to a non-equilibrium situation, different from the systems considered when the law was originally derived. Device design in the chemical world is difficult because simple laws are not obeyed in that way. Rate constants of the law of mass action are found experimentally to change from one set of conditions to another. The law of mass action is not robust in most cases and cannot serve the same role that circuit models do in our electrical technology. Robust models and device designs in the chemical world will not be possible until continuity of current is embedded in a generalization of the law of mass action using a consistent variational model of energy and dissipation.

... The relation of the equivalent circuit to the electrodynamics of the system was studied by solving Poisson's equation by several authors, but the simplification to the usual equivalent circuit only became apparent when series expansions of the solution of the Poisson equation, and the Poisson equation itself were developed [3,12,16,33,[51][52][53][54]. ...

Impedance Spectroscopy resolves electrical properties into uncorrelated
variables, as a function of frequency, with exquisite resolution. Separation is
robust and most useful when the system is linear. Impedance spectroscopy
combined with appropriate structural knowledge provides insight into pathways
for current flow, with more success than other methods. Biological applications
of impedance spectroscopy are often not useful since so much of biology is
strongly nonlinear in its essential features, and impedance spectroscopy is
fundamentally a linear analysis. All cells and tissues have cell membranes and
its capacitance is both linear and important to cell function. Measurements
proved straightforward in skeletal muscle, cardiac muscle, and lens of the eye.
In skeletal muscle, measurements provided the best estimates of the predominant
(cell) membrane system that dominates electrical properties. In cardiac muscle,
measurements showed definitively that classical microelectrode voltage clamp
could not control the potential of the predominant membranes, that were in the
tubular system separated from the extracellular space by substantial
distributed resistance. In the lens of the eye, impedance spectroscopy changed
the basis of all recording and interpretation of electrical measurements and
laid the basis for Rae and Mathias extensive later experimental work. Many
tissues are riddled with extracellular space as clefts and tubules, for
example, cardiac muscle, the lens of the eye, most epithelia, and of course
frog muscle. These tissues are best analyzed with a bidomain theory that arose
from the work on electrical structure described here. There has been a great
deal of work since then on the bi-domain and this represents the most important
contribution to biology of the analysis of electrical structure in my view.

... It is important to note that the continuity of current law has important biological implications in systems more general than a series of chemical reactions. Continuity of current law implies the cable equations (called the telegrapher's equation in the mathematics literature), see derivation from the three dimensional theory in [5,55,170,172] and p. 218-238 of [118]. The cable equation [94] is the foundation of the Hodgkin Huxley model [93,96,101,103,104] of the action potential of nerve and muscle fibers. ...

The law of mass action is used widely, nearly universally, in chemistry to describe chemical reactions. The law of mass action does not automatically conserve current, as is clear from the mathematics of a simple case, chosen to illustrate the issues involved. If current is not conserved in a theory, charges accumulate that cannot accumulate in the real world. In the real world, tiny charge accumulation—much less than one per cent—produces forces that change predictions of the theory a great deal. Indeed, in the real world tiny charge accumulation produces forces large enough to destroy biological membranes and thus living systems, forces large enough to ionize atoms, to create a plasma of electrons (like sparks or lightning) and thus make experiments impossible in normal laboratory settings.
The mathematics in this paper shows that the law of mass action violates conservation of current when current flows if the rate constants depend only on the potential (chemical and electrical) in one location and its immediate vicinity. The same difficulty arises when rate models of the Markov type deal with the movement of charge. The implication is that such models cannot deal with current through an open channel, or with the gating properties of channels if the gating mechanism is charged, or with the gating current produced by that mechanism for example.
The essential issue is that rate constants are LOCAL functions of potential (at one place or in a small region) so they cannot know about current flow far away or at boundary conditions. If current is interrupted far away, local chemical reactions obviously change, but rate constant models show no change. Consider for example what happens in a battery when current flow is interrupted far from the battery. The law of mass action does not force a series of chemical reactions to have the same current flow everywhere. Interruption of far-away current does not stop current everywhere in a series of chemical reactions (analyzed with the law of mass action), and so does not obey Maxwell’s equations. An additional constraint and equation is needed to enforce global continuity of current. The additional constraint is introduced in this paper in the special case that the chemical reaction describes spatial movement through narrow channels. In that case, a fully consistent treatment is possible using different models of charge movement. The general case must be dealt with by variational methods that enforce consistency of all the physical laws involved.
The law of mass action does not automatically conserve current, as is clear from the mathematics of a simple case, chosen to illustrate the issues involved. Consider the chemical reaction A ==> B ===>C the currents in the two chemical reactions are not equal. The difference in current is easily computed from the fluxes defined by the law of mass action . The difference can be zero only for special circumstances.
Violations of current continuity arise away from equilibrium, when current flows, and the law of mass action is applied to a non-equilibrium situation, different from the systems considered when the law was originally derived. Non-equilibrium systems are important. Almost all of biology occurs away from equilibrium. Almost all devices of our technology function away from equilibrium.

... . I 25 electrotonic model. It should also be noted that because of the frequency dependence of the space constant, a neurone may be isopotential at low frequencies but not at high frequencies Eisenberg & Johnson, 1970). ...

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.

A classical problem from potential theory (a point source inside a long rigid tube) is revisited. It has an extensive literature but its resolution is not straightforward: standard approaches lead to divergent integrals or require the discarding of infinite constants. We show that the problem can be solved rigorously using classical methods.

Mathematical modelling of ionic electrodiffusion and water movement is emerging as a powerful avenue of investigation to provide a new physiological insight into brain homeostasis. However, in order to provide solid answers and resolve controversies, the accuracy of the predictions is essential. Ionic electrodiffusion models typically comprise non-trivial systems of non-linear and highly coupled partial and ordinary differential equations that govern phenomena on disparate time scales. Here, we study numerical challenges related to approximating these systems. We consider a homogenized model for electrodiffusion and osmosis in brain tissue and present and evaluate different associated finite element-based splitting schemes in terms of their numerical properties, including accuracy, convergence and computational efficiency for both idealized scenarios and for the physiologically relevant setting of cortical spreading depression (CSD). We find that the schemes display optimal convergence rates in space for problems with smooth manufactured solutions. However, the physiological CSD setting is challenging: we find that the accurate computation of CSD wave characteristics (wave speed and wave width) requires a very fine spatial and fine temporal resolution.

Electrical impedance tomography (EIT) is a recently developed technique which produces reconstructed images of the internal distribution of the impedance of an object from measurement with external electrodes. It was assessed for its possible application in imaging intra-cranial disorders non- invasively with the use of scalp electrodes. Cerebral impedance increases of 12-55% were measured by a four electrode method at 50 kHz during global cerebral ischaemia or cortical spreading depression (CSD) in anaesthetized rats. Measured with scalp electrodes in two pairs 2-26 mm apart, impedance increased by 1.8-5.9% during global cerebral ischaemia for 5 or 15 min; the changes correlated in duration with cortical impedance changes, but increased more gradually. Increases of about 2% still were observed when the effects of variations in temperature and local scalp impedance were excluded. A finite element model was used to predict the attenuation of a signal due to cerebral anoxic depolarization by the extracerebral layers. These residual impedance changes were compatible with this, but their cause by other mechanisms related to the method of production of cerebral ischaemia could not be ruled out. An unexpected decrease of 0.8% was observed during CSD with electrodes 0.5mm apart on the scalp. This became undetectable when scalp temperature was kept constant. The model predicted that impedance changes of about 1% could be measured during CSD with scalp electrodes spaced further apart. Images were then collected using a prototype EIT system operating at 51 kHz during the same model of cerebral ischaemia. Test objects in a medium of constant resistivity could be accurately localized, but spatial resolution of intracranial impedance changes was substantially degraded when recorded with scalp electrodes. EIT has the potential for imaging various cerebral physiological or pathological changes, but improvements to the reconstruction algorithm are needed if regional intracerebral changes are to be discriminated during recording with scalp electrodes.

Curricium Vitae CV of Bob (more formally Robert S.) Eisenberg November 8, 2018

Computational modeling of signal propagation in neurons is critical to our understanding of basic principles underlying brain organization and activity. Exploring these models is used to address basic neuroscience questions as well as to gain insights for clinical applications. The seminal Hodgkin Huxley model is a common theoretical framework to study brain activity. It was mainly used to investigate the electrochemical and physical properties of neurons. The influence of neuronal structure on activity patterns was explored, however, the rich dynamics observed in neurons with different morphologies is not yet fully understood. Here, we study signal propagation in fundamental building blocks of neuronal branching trees, unbranched and branched axons. We show how these simple axonal elements can code information on spike trains, and how asymmetric responses can emerge in axonal branching points. This asymmetric phenomenon has been observed experimentally but until now lacked theoretical characterization. Together, our results suggest that axonal morphological parameters are instrumental in activity modulation and information coding. The insights gained from this work lay the ground for better understanding the interplay between function and form in real-world complex systems. It may also supply theoretical basis for the development of novel therapeutic approaches to damaged nervous systems.

The gallbladder was the first tissue in which transepithelial transport was shown to occur in the absence of a sizable spontaneous transepithelial electrical potential difference (Diamond, 1962 b). In the 15 years that have passed since that observation, the experimental evidence accumulated from studies in this and other tissues has increasingly indicated that salt transporting epithelia possess widely different physiological properties, and can no longer be considered or modeled in a uniform fashion. The distinction between “tight” and “leaky” epithelia, first proposed by Frömter and Diamond (1972), although perhaps an over-simplification, is a convenient way of pointing out this physiological heterogeneity.

In formulating cell theory, Mathias Schleiden(1) wrote in 1838: “Every higher organism is an aggregate of fully circumscribed and self-contained unit beings, the cells.” This has been one of the most fruitful tenets in biology which—the theory of evolution and the central dogma of molecular biology aside—has had no equal in influencing biological work and thought. The hypothesis held sway for more than a century. But 20 years ago it underwent a basic change when it was found that the cellular circumscription was commonly not complete; many cells turned out to be interconnected at their junctions.(2,3) The elements in this connection are specialized membrane channels built into the junctional membrane complex, through which a range of mole¬cules can flow from one cell interior to another(4) (Fig. 1). These cell-to-cell channels provide a degree of continuity between cells, without loss of external circumscription. Commonly, a given cell in a tissue is thus coupled to several neighbors, and so whole organs or large organ parts are continuous from within. It is then the connected cell ensemble, not the single cell, that is the functional compartmental unit in respect to the channel-permeant molecules. The cell is the unit only in regard to the macromolecules that are too large for the channels.

Electrical impedance measurements of biologic materials have been performed for at least 75 years. The first systematic studies were done by Hober (Schanne and Ruiz-Ceretti, 1978) in 1910. Later investigators include Fricke (Fricke, 1924, and Fricke, 1925), Oncley (Oncley, 1938 and Oncley, 1942), K.C. Cole (Cole, 1928, and Cole, 1968), and Schwan (Schwan, 1957 and Schwan, 1985). All of these earlier studies utilized cell suspensions and the suspension equation. This equation was first derived by Maxwell (1873) for dielectric spheres in a conducting medium and modified by Fricke (Schanne and Ruiz-Ceretti, 1978) to account for the non-spheroidal geometry of human erythrocytes (HRBC), viz,$$\frac{{{\textstyle{\sigma \over {{\sigma _1} - 1}}}}}{{{\textstyle{\sigma \over {{\sigma _1} + 1}}}}} = \rho '\frac{{{\textstyle{{{\sigma _2}} \over {{\sigma _1} - 1}}}}}{{{\textstyle{{{\sigma _2}} \over {{\sigma _1} + \gamma }}}}}$$ ([1])

In formulating cell theory, Mathias Schleiden(1) wrote in 1838: “Every higher organism is an aggregate of fully circumscribed and self-contained unit beings, the cells.” This has been one of the most fruitful tenets in biology which—the theory of evolution and the central dogma of molecular biology aside—has had no equal in influencing biological work and thought. The hypothesis held sway for more than a century. But 20 years ago it underwent a basic change when it was found that the cellular circumscription was commonly not complete; many cells turned out to be interconnected at their junctions.(2,3) The elements in this connection are specialized membrane channels built into the junctional membrane complex, through which a range of molecules can flow from one cell interior to another(4) (Fig. 1). These cell-to-cell channels provide a degree of continuity between cells, without loss of external circumscription. Commonly, a given cell in a tissue is thus coupled to several neighbors, and so whole organs or large organ parts are continuous from within. It is then the connected cell ensemble, not the single cell, that is the functional compartmental unit in respect to the channel-permeant molecules. The cell is the unit only in regard to the macromolecules that are too large for the channels.

Most of the papers in this book discuss the properties and roles of channels in membranes, and the methods needed to investigate them. Work on channels has evolved (in large measure) from older work on the properties of membranes themselves. Since channels are the major pathways for solute movement, the mechanism of solute movement can best be investigated when channels are embedded in as simple a membrane as possible, attached to as simple an apparatus as possible. Indeed, that is why single channel measurements, as described in several chapters in this book, have created such excitement and are so promising for the future of membrane biophysics.

A technique has previously been described (1–5) by which the impedance of isolated cells can be evaluated over a wide frequency range (DC -20 MHz). The methodology centers about the use of a polycarbonate filter containing well defined and uniform cylindrical pores. A pseudo-epithelium is achieved by the entrapment of cells from a suspension in these pores using hydrostatic pressure. Data analysis by discrete Laplace transformation of the input and output waveforms allows electrochemical kinetic models to be tested via aperiodic equivalent circuit representation. In this study the effect of the state of the cell surface on the impedance parameters is examined. This is achieved by the use of the enzyme neuraminidase on the human red blood cell (HRBC) to reduce the surface charge and by melanocyte stimulating hormone (MSH) to alter the state of differentiation of melanoma cells.

The bidomain model is the standard model describing electrical activity of the heart. Here we study the stability of planar front solutions of the bidomain equation with a bistable nonlinearity (the bidomain Allen-Cahn equation) in two spatial dimensions. In the bidomain Allen-Cahn equation a Fourier multiplier operator whose symbol is a positive homogeneous rational function of degree two (the bidomain operator) takes the place of the Laplacian in the classical Allen-Cahn equation. Stability of the planar front may depend on the direction of propagation given the anisotropic nature of the bidomain operator. We establish various criteria for stability and instability of the planar front in each direction of propagation. Our analysis reveals that planar fronts can be unstable in the bidomain Allen-Cahn equation in striking contrast to the classical or anisotropic Allen-Cahn equations. We identify two types of instabilities, one with respect to long-wavelength perturbations, the other with respect to medium-wavelength perturbations. Interestingly, whether the front is stable or unstable under long-wavelength perturbations does not depend on the bistable nonlinearity and is fully determined by the convexity properties of a suitably defined Frank diagram. On the other hand, stability under intermediate-wavelength perturbations does depend on the choice of bistable nonlinearity. Intermediate-wavelength instabilities can occur even when the Frank diagram is convex, so long as the bidomain operator does not reduce to the Laplacian. We shall also give a remarkable example in which the planar front is unstable in all directions.

Until recently almost all knowledge of the electrophysiological properties of the myometrium and other mammalian smooth muscles was based on descriptive observations of some natural phenomena or on modifications of such phenomena by changes in the ionic environment or produced by hormonal and pharmacological agents. Although much of value has been learned about the resting and action potentials and their relationship to contractions, the information derived from such approaches was limited because of the nature of the methods. In the previous edition of this book, substantial revisions were incorporated into this chapter to include advances then being made with the introduction of certain analytical methods for studying the electrophysiological properties of mammalian smooth muscles. For instance, basic electrical properties of some smooth muscle preparations were being unmasked by the polarization method of Abe and Tomita (1968), and the nature of the ionic currents, their equilibrium potentials, and some aspects of the ionic conductances were being identified by use of the double-sucrose-gap voltage-clamp method (Anderson, 1969; Kao and McCullough, 1975; Inomata and Kao, 1976). All that information, fulfilling a need then existing, was also flawed, because it had to be obtained on multicellular preparations, which presented formidable technical obstacles to an unambiguous understanding of the underlying cellular processes.

The absorption of water, electrolytes, and nutrient substances by the small intestine has been a central concern of gastrointestinal physiology for many years. Beginning as early as the mid-seventeenth century, there is a more or less continuous history of experimental investigation in this field (Parsons 1968). In this area, as in many other areas of experimental science, the first half of this century witnessed a tremendous intensification of effort. This increase in the number of investigations designed to uncover the fundamental mechanisms of salt and water transport resulted in a concomitant proliferation of published results and the elaboration of a variety of ingenious hypotheses (see, e.g., Höber 1946; Schultz and Curran 1968). This activity has increased, in volume and intensity, until the present time and seems likely (given the survival of a reasonably supportive socioeconomic milieu) to continue increasing for many years to come. Despite all this, it is a salutory thought that, as pointed out by Schultz and Curran (1968), modern theories of intestinal absorption can still be described, in general terms, by Heidenhain’s (1894) hypothesis. That is, that the absorption process results from a combination of physicochemical (osmotic) driving forces and the intrinsic driving force (Triebkraft) generated by the absorptive cells.

The gap junction is one form of membrane specialization that appears in the electron microscope as a closely apposed region of plasma membrane separating adjacent cells in contact by a 2- to 5-nm-wide “gap,” from which the name gap junction is derived (Revel and Karnovsky, 1967). In freeze-fracture replicas viewed by electron microscopy, gap junctions are composed of tightly packed clusters of particles (Kreutziger, 1968). Each particle in a gap junction presumably represents the site of a single intramembrane channel that connects with the adjacent cell allowing a direct exchange or “communication” of ions and other small molecules. The functions of these “communicating channels” and the spectrum of activities they regulate are now beginning to be realized. In excitable tissues, gap junctions are synonymous with electrical synapses and are sometimes referred to as a nexus (Dewey and Barr, 1962). Gap junctions allow rapid signaling and the coordination of smooth and cardiac muscle contraction. It is apparent, however, that the role of gap junctions in the nervous system can be more complex and that they may also modify or regulate the behavior of discrete populations of neurons (Getting, 1974; Getting and Willows, 1974; Rayport and Kandel, 1980; Dudek et al., 1983; Marder, 1984; Bennett et al., 1985). The functions of gap junctions in nonexcitable tissues include, but are not limited to, tissue homeostasis (metabolic cooperation, i.e., the sharing of metabolites), coordination of a response to hormones or neurotransmitters, and, in lower invertebrates, adhesion.

The hindgut plays a central role in renal function and osmoregulation in most insects. This organ selectively reabsorbs solutes and water from “primary urine” which is secreted into the gut lumen by Malpighian tubules. The rectum of the desert locust Schistocerca gregaria has been studied in some detail in an attempt to understand the physiology of excretion and to learn more about insect ion transport mechanisms at the cellular level (Hanrahan 1982; Phillips 1981). Sodium, potassium, chloride, water, amino acids, phosphate, and acetate are all reabsorbed from the rectal lumen into the hemocoel by active mechanisms (reviewed by Phillips 1980, 1981). A variety of preparations have been used in these studies, each having particular strengths and weaknesses.

In den Jahren 1968–70, der Zeit nach dem Bericht von SCHILDE (Fortschr. Botan. 30, 44), erlebte die Membranbiologie eine Buchflut, an der die Elektrophysiologie naturgemäß teilhatte. COLE schrieb eine “Spezielle Elektrophysiologie”; PLONSEY betont die physikalischen, LAKSHMINARAYANAIAH die physiologischen Aspekte. PASSOW u. STÄMPFLI sowie LAVALLEE et al. demonstrieren ausgewählte Techniken. Neu aufgelegt wurden die einführenden Werke von BURES et al. und KATZ (in dt. Übers.). Das nach Ablieferung dieses Beitrags erschienene Büchlein von HOPE ist eine vorbildliche Einführung in die elektrischen Aspekte des Ionentransportes der Pflanzenzelle.

Neuronal modelling is the process by which a biological neuron is represented by a mathematical structure that incorporates its biophysical and geometrical characteristics. This structure is referred to as the mathematical model or the model of the neuron. The behavior of this representation may serve a number of purposes: for example, it may be used as the basis for estimating the biophysical parameters of real neurons or it may be used to define the computational and information processing properties of a neuron. Neuronal modelling requires not only an understanding of mathematical and computational techniques, but also an understanding of the what the process of modelling entails. A general treatment of models, however, would necessarily lead to the examination of a number of philosophical questions. Here we simply discuss some aspects of modelling that in our experience have proved to be useful in the construction and application of models. These topics are not usually considered in the neurophysiological modelling literature, but an understanding of the basic assumptions of modelling, and the presumed relation between model and reality is essential for constructive work in computational neuroscience.

This chapter describes the voltage clamping of excitable membranes. The excitability resides in a surface membrane in the form of channels that are embedded in a lipid bilayer with a rather large capacitance. They are either open or closed, and the proportion of the time anyone is open is potential dependent. The excitability properties of nerve and muscle membranes result from the fact that the membrane ionic conductances are voltage dependent. These conductances result from the flow of ions through imperfectly selective membrane channels, and the magnitude of the conductance depends upon the fraction of channels that are open. A very important source of error in practical voltage clamp originates in the capacitances of the voltage electrodes, which, in combination with their resistances, act as low-pass filters. This filtering effect is aggravated by the differential amplifier, which never has a flat frequency response. A longitudinal low-resistance current electrode can also be introduced from the ends and serves the dual purpose of passing current and attaining space clamp conditions.

Cells of organized tissues are interconnected at junctions that provide pathways for direct flow of matter between cell interiors (Loewenstein, 1966, 1979; Furshpan and Potter, 1968). Such permeable junctions are present throughout the phylogenetic scale, from sponges to man. They appear to be membrane appositions that are traversed by clusters of well-insulated aqueous channels, and these channels, during their formation, are resolvable one by one in measurements of electrical conductance (Loewenstein et al., 1978). The channels are widely thought to be contained in the intramembranous particles seen in electron microscope images of freeze-fractured gap junctions (Griepp and Revel, 1977; Makowski et al., 1977).

A fascinating hypothesis on the origin of Metazoa envisions some flagellated Protozoa of the Cambrian period aggregating into small communities where food procurement and mutual protection against the environment could be more efficiently provided (Hyman, 1940). The simplest such colonies may have been the ancestors of modern Porifera, Ctenophora, and Coelenterata.

The input impedance of muscle fibers of the crab was determined with microelectrodes over the frequency range 1 cps to 10 kc/sec. Care was taken to analyze, reduce, and correct for capacitive artifact. One dimensional cable theory was used to determine the properties of the equivalent circuit of the membrane admittance, and the errors introduced by the neglect of the three dimensional spread of current are discussed. In seven fibers the equivalent circuit of an element of the membrane admittance must contain a DC path and two capacitances, each in series with a resistance. In two fibers, the element of membrane admittance could be described by one capacitance in parallel with a resistance. In several fibers there was evidence for a third very large capacitance. The values of the elements of the equivalent circuit depend on which of several equivalent circuits is chosen. The circuit (with a minimum number of elements) that was considered most reasonably consistent with the anatomy of the fiber has two branches in parallel: one branch having a resistance R(e) in series with a capacitance C(e); the other branch having a resistance R(b) in series with a parallel combination of a resistance R(m) and a capacitance C(m). The average circuit values (seven fibers) for this model, treating the fiber as a cylinder of sarcolemma without infoldings or tubular invaginations, are R(e) = 21 ohm cm(2); C(e) = 47 microf/cm(2); R(b) = 10.2 ohm cm(2); R(m) = 173 ohm cm(2); C(m) = 9.0 microf/cm(2). The relation of this equivalent circuit and another with a nonminimum number of circuit elements to the fine structure of crab muscle is discussed. In the above equivalent circuit R(m) and C(m) are attributed to the sarcolemma; R(e) and C(e), to the sarcotubular system; and R(b), to the amorphous material found around crab fibers. Estimates of actual surface area of the sarcolemma and sarcotubular system permit the average circuit values to be expressed in terms of unit membrane area. The values so expressed are consistent with the dielectric properties of predominantly lipid membranes.

THE THEORY OF ELECTROTONUS, WHICH HAS BEEN WELL DEVELOPED FOR SMALL CYLINDERS, IS EXTENDED: the fundamental potential equations for a membrane of arbitrary shape are derived, and solutions are found for cylindrical and spherical geometries. If two purely conductive media are separated by a resistance-capacitance membrane, then Laplace's equation describes the potential in either medium, and two boundary equations relate the transmembrane potential to applied currents and to currents flowing into the membrane from each medium. The core conductor model, on which most previous work on cylindrical electrotonus has been based, gives rise to a one dimensional diffusion equation, the cable equation, for the transmembrane potential in a small cylinder. Under the assumptions of the core conductor model the more general equations developed here are shown to reduce to the cable equation. The two theories agree well in predicting the transmembrane potential in a small cylinder owing to an applied current step, and the extracellular potential for this cylinder is estimated numerically from the general theory. A detailed proof is given for the isopotentiality of a spherical soma membrane.

Four different methods of measuring the resistance of a muscle fiber have been applied to the frog sartorius muscle. The methods, in which the resistance of the microelectrode entered the calculation of the effective resistance of the fiber, resulted in values which were 8 times higher than the resistance values obtained with methods independent of the electrode resistance. A simple cable model of a muscle fiber could not account for the discrepancy in the effective resistance found in these measurements; therefore, an enlarged cable model for a muscle fiber has been proposed, and its biological implications have been discussed. The effective resistance (measured with the two different groups of methods) decreased when the potassium concentration in the bath increased. Using the proposed enlarged cable model for the interpretation of these results, it is shown that not only the membrane resistance but also the myoplasmic resistance decreases with an increasing potassium concentration in the Ringer solution.

The structure of a small strand of rabbit heart muscle fibers (trabecula carnea), 30-80 micro in diameter, has been examined with light and electron microscopy. By establishing a correlation between the appearance of regions of close fiber contact in light and electron microscopy, the extent and distribution of regions of close apposition of fibers has been evaluated in approximately 200 micro length of a strand. The distribution of possible regions of resistive coupling between fibers has been approximated by a model system of cables. The theoretical linear electrical properties of such a system have been analyzed and the implications of the results of this analysis are discussed. Since this preparation is to be used for correlated studies of the electrical, mechanical, and cytochemical properties of cardiac muscle, a comprehensive study of the morphology of this preparation has been made. The muscle fibers in it are distinguished from those of the rabbit papillary muscle, in that they have no triads and have a kind of mitochondrion not found in papillary muscle. No evidence of a transverse tubular system was found, but junctions of cisternae of the sarcoplasmic reticulum and the sarcolemma, peripheral couplings, were present. The electrophysiological implications of the absence of transverse tubules are discussed. The cisternae of the couplings showed periodic tubular extensions toward the sarcolemma. A regularly spaced array of Z line-like material was observed, suggesting a possible mechanism for sarcomere growth.

Current has been passed through the cell membrane of muscle fibres of the isolated rabbit right ventricle with the aid of intracellular double-barrelled microelectrodes. Two types of muscle fibres were distinguished which are called P and V fibres. The relation between the intensity of a hyperpolarising current applied during the rising phase and the maximum amplitude of the action potential was different in these fibres. For P fibres the relation was essentially linear over most of the range of currents used. For V fibres the change in maximum action potential amplitude was either negligible or did not appear until a certain value of hyperpolarising current was reached. This behaviour of V fibres can be understood if a drop in polarisation resistance occurs during the rising phase and is of such short duration that the polarisation resistance has returned to its resting value before the crest of the action potential is reached. P fibres have an estimated mean resting polarisation resistance of (106 +/- 13) K ohms, and a rheobase current strength of (0.08 +/- 0.02) microa. In V fibres the resting polarisation resistance was (47 +/- 29) K ohms and the rheobase current strength (0.47 +/- 0.28) microa.

The theory of electrotonus, which has been well developed for small cylinders, is extended: the fundamental potential equations for a membrane of arbitrary shape are derived, and solutions are found for cylindrical and spherical geometries. If two purely conductive media are separated by a resistance-capacitance membrane, then Laplace's equation describes the potential in either medium, and two boundary equations relate the transmembrane potential to applied currents and to currents flowing into the membrane from each medium. The core conductor model, on which most previous work on cylindrical electrotonus has been based, gives rise to a one dimensional diffusion equation, the cable equation, for the transmembrane potential in a small cylinder. Under the assumptions of the core conductor model the more general equations developed here are shown to reduce to the cable equation. The two theories agree well in predicting the transmembrane potential in a small cylinder owing to an applied current step, and the extracellular potential for this cylinder is estimated numerically from the general theory. A detailed proof is given for the isopotentiality of a spherical soma membrane.

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This article concludes a series of papers concerned with the flow of electric current through the surface membrane of a giant nerve fibre (Hodgkinet al., 1952,J. Physiol.116, 424–448; Hodgkin and Huxley, 1952,J. Physiol.116, 449–566). Its general object is to discuss the results of the preceding papers (Section 1), to put them into mathematical form (Section 2) and to show that they will account for conduction and excitation in quantitative terms (Sections 3–6).

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An instrument is described which enables absolute current measurements (10-12 to 10-9 A) to be made using the Townsend null method. It has proved satisfactory over six months of almost continuous use on free-air chamber and extrapolation chamber work. It is readily adaptable for use in a simple bridge circuit for capacitance measurement.

Electrical potential problems encountered in biology differ from those usually considered in electrical theory first, because
the membranes of tissues satisfy a non-linear relation between current flow and polarization, and second, because the interior
of the tissues are not equipotentials. A Green's function suitable for discussing such problems is defined, and a cylindrical
illustration of such a function is discussed.

This article concludes a series of papers concerned with the flow of electric current through the surface membrane of a giant
nerve fibre (Hodgkinet al., 1952,J. Physiol.
116, 424–448; Hodgkin and Huxley, 1952,J. Physiol.
116, 449–566). Its general object is to discuss the results of the preceding papers (Section 1), to put them into mathematical
form (Section 2) and to show that they will account for conduction and excitation in quantitative terms (Sections 3–6).

1. The membrane potential of isolated muscle fibres in solutions containing tetrodotoxin (TTX) was controlled with a two‐electrode voltage clamp. The striation pattern in the region of the electrodes was observed microscopically.
2. With square steps of depolarization of increasing magnitude, contraction occurs first in the myofibrils just beneath the surface membrane, and then spreads inwards towards the axis of the fibre as the depolarization is increased.
3. From the depolarizations which make the superficial and axial myofibrils contract it is possible to estimate a space constant (λ T ) for electrotonic spread in a transverse tubular network.
4. λ T was found to vary with fibre radius; for a 50 μ fibre it was about 60 μ. λ T was not greatly affected by tetraethylammonium (TEA) chloride (111 m M ), or by sucrose substitution of most of the sodium chloride in the Ringer solution.
5. The ratio of the depolarization threshold for contraction of surface myofibrils and of central myofibrils was smaller for short (3 msec) than for long depolarization.
6. Action potentials, recorded from a sartorius fibre, were used as the command signal for the voltage‐clamped fibre in tetrodotoxin. The central myofibrils of this fibre did not appear to contract unless the imposed ‘action potentials’ were of normal size.
7. The passive electrical characteristics of the transverse tubular system will just allow an action potential, at room temperature, to activate the myofibrils at the centre of a frog muscle fibre. An active potential change would be required to achieve a safety factor appreciably greater than one for this process.

The Hodgkin-Huxley model of the nerve axon describes excitation and propagation of the nerve impulse by means of a nonlinear partial differential equation. This equation relates the conservation of the electric current along the cablelike structure of the axon to the active processes represented by a system of three rate equations for the transport of ions through the nerve membrane. These equations have been integrated numerically with respect to both distance and time for boundary conditions corresponding to a finite length of squid axon stimulated intracellularly at its midpoint. Computations were made for the threshold strength-duration curve and for the repetitive firing of propagated impulses in response to a maintained stimulus. These results are compared with previous solutions for the space-clamped axon. The effect of temperature on the threshold intensity for a short stimulus and for rheobase was determined for a series of values of temperature. Other computations show that a highly unstable subthreshold propagating wave is initiated in principle by a just threshold stimulus; that the stability of the subthreshold wave can be enhanced by reducing the excitability of the axon as with an anesthetic agent, perhaps to the point where it might be observed experimentally; but that with a somewhat greater degree of narcotization, the axon gives only decrementally propagated impulses.

This article concludes a series of papers concerned with the flow of electric current through the surface membrane of a giant nerve fibre (Hodgkinet al., 1952,J. Physiol.116, 424–448; Hodgkin and Huxley, 1952,J. Physiol.116, 449–566). Its general object is to discuss the results of the preceding papers (Section 1), to put them into mathematical form (Section 2) and to whow that they will account for conduction and excitation in quantitative terms (Sections 3–6).

The magnitude and time course of changes in polarisation resistance during the repolarisation phase of action potentials recorded from rabbit ventricular muscle have been measured with intracellular double‐barrelled microelectrodes.
Action potentials from 74 fibres in 7 preparations have been examined. In all fibres, at some time during the repolarisation phase, the polarisation resistance was greater than the resting polarisation resistance. The mean maximum increase above the resting value was 1·63 (± 0·15 S.D.) times.
The polarisation resistance remained constant even with large negative voltage displacements (60 mV). No evidence was found of regenerative repolarisation.

The linear electrical properties of muscle fibres have been examined using intracellular electrodes for a.c. measurements and analyzing observations on the basis of cable theory. The measurements have covered the frequency range 1 c/s to 10 kc/s. Comparison of the theory for the circular cylindrical fibre with that for the ideal, one-dimensional cable indicates that, under the conditions of the experiments, no serious error would be introduced in the analysis by the geometrical idealization. The impedance locus for frog sartorius and crayfish limb muscle fibres deviates over a wide range of frequencies from that expected for a simple model in which the current path between the inside and the outside of the fibre consists only of a resistance and a capacitance in parallel. A good fit of the experimental results on frog fibres is obtained if the inside-outside admittance is considered to contain, in addition to the parallel elements Rm = 3100 Omega cm2 and Cm = 2\cdot 6 mu F/cm2, another path composed of a resistance Re = 330 Omega cm2 in series with a capacitance Ce = 4\cdot 1 mu F/cm2, all referred to unit area of fibre surface. The impedance behaviour of crayfish fibres can be described by a similar model, the corresponding values being Rm = 680 Omega cm2, Cm = 3\cdot 9 mu F/cm2, Re = 35 Omega cm2, Ce = 17 mu F/cm2. The response of frog fibres to a step-function current (with the points of voltage recording and current application close together) has been analyzed in terms of the above two-time-constant model, and it is shown that neglecting the series resistance would have an appreciable effect on the agreement between theory and experiment only at times less than the half-time of rise of the response. The elements Rm and Cm are presumed to represent properties of the surface membrane of the fibre. Re and Ce are thought to arise not at the surface, but to be indicative of a separate current path from the myoplasm through an intracellular system of channels to the exterior. In the case of crayfish fibres, it is possible that Re (when referred to unit volume) would be a measure of the resistivity of the interior of the channels, and Ce the capacitance across the walls of the channels. In the case of frog fibres, it is suggested that the elements Re, Ce arise from the properties of adjacent membranes of the triads in the sarcoplasmic reticulum. The possibility is considered that the potential difference across the capacitance Ce may control the initiation of contraction.

Solutions have been computed for the point polarization of a sheet-like membrane obeying the equations used previously (Noble, 1960, 1962) to reproduce the Purkinje fiber action potential. It was found that, in spite of the gross non-linearity of the membrane current-voltage relations, the relations between total polarizing current and displacement of membrane potential at various distances from the polarizing electrode are remarkably linear. It is therefore concluded that Johnson and Tille's (1960, 1961) results showing linear polarizing current-voltage relations obtained by passing current through the membrane from a microelectrode during the plateau of the rabbit ventricular action potential do not conflict with the Hodgkin-Huxley theory of electrical activity.

Radial Spread of Contraction in Frog Muscle Fibers Conduction of Heat in Solids

- R H Adri~
- Cosra~
- L L Nq
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ADRI~, R. H., CosrA~,nq, L. L. and PEA~, L. D. (1969) Radial Spread of Contraction in Frog Muscle Fibers. J. Physiol. 203. C~aSLAW, H. S. and JAISOER, J. C. (1959) Conduction of Heat in Solids, 2nd edition. Oxford University Press.

Digital computer solutions for excitation and pro-pagation of the nerve impulse Equivalent circuit of crab muscle fibers as determined by impedance measurements with intracellular electrodes Linear electrical properties of striated muscle fibres

- J W Cool~y
- F A Dodge
- G Falk
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On th© Problem of Impulse Conduction in the Atriam

- J W Woodnury
- W E Catty

WOODnURY, J. W. and Catty, W. E. (1961) On th© Problem of Impulse Conduction in the Atriam. In Nervous Inhibition, E. FLonEY, Ed. Pergamon Press Inc., N.Y.

Tables of Modified Quotients of Bessel Functions. Columbia University Press Tables of Laplace Transforms

- M Onoe

ONoE, M. (1958) Tables of Modified Quotients of Bessel Functions. Columbia University Press, N.Y. Konrz~'rs, G. E. and KA~rMAN, H. (1966) Tables of Laplace Transforms. Saunders, Phila-delphia.

Methods of TheoreticalPhysics A general purpose wide range electrometer

- P Morse
- F ~srmach

MoRse, P. and F'~srmAcH, H. (1953) Methods of TheoreticalPhysics. McGraw Hill, N.Y. MURRAY, C. T. (1958) A general purpose wide range electrometer. Australian Atomic Energy Symposium, Section 5: Instrumentation, p. 701 (Canberra).

Cable Theory, in Physical Techniques in

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T^~aLOn, R. E. (1963) Cable Theory, in Physical Techniques in Biological Research W. L. NAS'rUK, Ed. Vol. VI B, p. 219. Academic Press, N.Y. VAN V~m~Ns~G, M. E. (1964) Network Analysis, 2nd edition. Prentice-Hall, Englewood Cliffs, N.J. WEBER, H. (1873) Ueber die stationaren Str6mungen der Elektricit~it in Cylindem. J.fiir die reine und angewandte, Mathematik 76, 1.

The voltage and time dependence of the cardiac membrane conductance Bessel Functions, Part 111Zeroes and Associated Values

- D Noble

NOBLE, D. (1962) The voltage and time dependence of the cardiac membrane conductance. Biophvs. J. 2, 381. eLY,, F. W. J. (1960) Bessel Functions, Part 111Zeroes and Associated Values. Cambridge University Press.

Linear electrical properties of striated muscle fibres

- G Falk
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Tables 0/ Modified Quotients 0/ Bessel Functions

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ONOE, M. (1958) Tables 0/ Modified Quotients 0/ Bessel Functions. Columbia University
Press, N.Y.

Tables 0/ Laplace Trans/orms

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ROBERTS, G. E. and KAUFMAN, H. (1966) Tables 0/ Laplace Trans/orms. Saunders, Philadelphia.

Ueber die stationaren Stromungen der Elektricitat in CyIindern. J. fur die reine und angewandte

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