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Journal ArticleDOI

A quantitative description of membrane current and its application to conduction and excitation in nerve

28 Aug 1952-The Journal of Physiology (Wiley/Blackwell (10.1111))-Vol. 117, Iss: 4, pp 500-544

TL;DR: This article concludes a series of papers concerned with the flow of electric current through the surface membrane of a giant nerve fibre by putting them into mathematical form and showing that they will account for conduction and excitation in quantitative terms.

AbstractThis 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 Physiol116, 424–448; Hodgkin and Huxley, 1952,J Physiol116, 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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Citations
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TL;DR: This historical survey compactly summarizes relevant work, much of it from the previous millennium, review deep supervised learning, unsupervised learning, reinforcement learning & evolutionary computation, and indirect search for short programs encoding deep and large networks.
Abstract: In recent years, deep artificial neural networks (including recurrent ones) have won numerous contests in pattern recognition and machine learning. This historical survey compactly summarizes relevant work, much of it from the previous millennium. Shallow and Deep Learners are distinguished by the depth of their credit assignment paths, which are chains of possibly learnable, causal links between actions and effects. I review deep supervised learning (also recapitulating the history of backpropagation), unsupervised learning, reinforcement learning & evolutionary computation, and indirect search for short programs encoding deep and large networks.

11,176 citations


Cites background from "A quantitative description of membr..."

  • ...Current GPUs, however, are little ovens, much hungrier for energy than biological brains, whose neurons efficiently communicate by brief spikes (FitzHugh, 1961; Hodgkin & Huxley, 1952; Nagumo, Arimoto, & Yoshizawa, 1962), and often remain quiet....

    [...]

  • ...Current GPUs, however, are little ovens, much hungrier for energy than biological brains, whose neurons efficiently communicate by brief spikes (Hodgkin and Huxley, 1952; FitzHugh, 1961; Nagumo et al., 1962), and often remain quiet....

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TL;DR: The major concepts and results recently achieved in the study of the structure and dynamics of complex networks are reviewed, and the relevant applications of these ideas in many different disciplines are summarized, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering.
Abstract: Coupled biological and chemical systems, neural networks, social interacting species, the Internet and the World Wide Web, are only a few examples of systems composed by a large number of highly interconnected dynamical units. The first approach to capture the global properties of such systems is to model them as graphs whose nodes represent the dynamical units, and whose links stand for the interactions between them. On the one hand, scientists have to cope with structural issues, such as characterizing the topology of a complex wiring architecture, revealing the unifying principles that are at the basis of real networks, and developing models to mimic the growth of a network and reproduce its structural properties. On the other hand, many relevant questions arise when studying complex networks’ dynamics, such as learning how a large ensemble of dynamical systems that interact through a complex wiring topology can behave collectively. We review the major concepts and results recently achieved in the study of the structure and dynamics of complex networks, and summarize the relevant applications of these ideas in many different disciplines, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering. © 2005 Elsevier B.V. All rights reserved.

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Abstract: A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

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TL;DR: Van der Pol's equation for a relaxation oscillator is generalized by the addition of terms to produce a pair of non-linear differential equations with either a stable singular point or a limit cycle, which qualitatively resembles Bonhoeffer's theoretical model for the iron wire model of nerve.
Abstract: Van der Pol's equation for a relaxation oscillator is generalized by the addition of terms to produce a pair of non-linear differential equations with either a stable singular point or a limit cycle. The resulting "BVP model" has two variables of state, representing excitability and refractoriness, and qualitatively resembles Bonhoeffer's theoretical model for the iron wire model of nerve. This BVP model serves as a simple representative of a class of excitable-oscillatory systems including the Hodgkin-Huxley (HH) model of the squid giant axon. The BVP phase plane can be divided into regions corresponding to the physiological states of nerve fiber (resting, active, refractory, enhanced, depressed, etc.) to form a "physiological state diagram," with the help of which many physiological phenomena can be summarized. A properly chosen projection from the 4-dimensional HH phase space onto a plane produces a similar diagram which shows the underlying relationship between the two models. Impulse trains occur in the BVP and HH models for a range of constant applied currents which make the singular point representing the resting state unstable.

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Journal ArticleDOI
02 Dec 1999-Nature
TL;DR: General principles that govern the structure and behaviour of modules may be discovered with help from synthetic sciences such as engineering and computer science, from stronger interactions between experiment and theory in cell biology, and from an appreciation of evolutionary constraints.
Abstract: Cellular functions, such as signal transmission, are carried out by 'modules' made up of many species of interacting molecules Understanding how modules work has depended on combining phenomenological analysis with molecular studies General principles that govern the structure and behaviour of modules may be discovered with help from synthetic sciences such as engineering and computer science, from stronger interactions between experiment and theory in cell biology, and from an appreciation of evolutionary constraints

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References
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Journal ArticleDOI
TL;DR: A theoretical picture has been presented based on the use of the general kinetic equations for ion motion under the influence of diffusion and electrical forces and on a consideration of possible membrane structures that shows qualitative agreement with the rectification properties and very good agreementwith the membrane potential data.
Abstract: Impedance and potential measurements have been made on a number of artificial membranes. Impedance changes were determined as functions of current and of the composition of the environmental solutions. It was shown that rectification is present in asymmetrical systems and that it increases with the membrane potential. The behavior in pairs of solutions of the same salt at different concentrations has formed the basis for the studies although a few experiments with different salts at the same concentrations gave results consistent with the conclusions drawn. A theoretical picture has been presented based on the use of the general kinetic equations for ion motion under the influence of diffusion and electrical forces and on a consideration of possible membrane structures. The equations have been solved for two very simple cases; one based on the assumption of microscopic electroneutrality, and the other on the assumption of a constant electric field. The latter was found to give better results than the former in interpreting the data on potentials and rectification, showing agreement, however, of the right order of magnitude only. Although the indications are that a careful treatment of boundary conditions may result in better agreement with experiment, no attempt has been made to carry this through since the data now available are not sufficiently complete or reproducible. Applications of the second theoretical case to the squid giant axon have been made showing qualitative agreement with the rectification properties and very good agreement with the membrane potential data.

2,611 citations

Journal ArticleDOI
TL;DR: The identity of the ions which carry the various phases of the membrane current is chiefly concerned with sodium ions, since there is much evidence that the rising phase of the action potential is caused by the entry of these ions.
Abstract: In the preceding paper (Hodgkin, Huxley & Katz, 1952) we gave a general description of the time course of the current which flows through the membrane of the squid giant axon when the potential difference across the membrane is suddenly changed from its resting value, and held at the new level by a feed-back circuit ('voltage clamp' procedure). This article is chiefly concerned with the identity of the ions which carry the various phases of the membrane current. One of the most striking features of the records of membrane current obtained under these conditions was that when the membrane potential was lowered from its resting value by an amount between about 10 and 100 mV. the initial current (after completion of the quick pulse through the membrane capacity) was in the inward direction, that is to say, the reverse ofthe direction of the current which the same voltage change would have caused to flow in an ohmic resistance. The inward current was of the right order of magnitude, and occurred over the right range of membrane potentials, to be the current responsible for charging the membrane capacity during the rising phase of an action potential. This suggested that the phase of inward current in the voltage clamp records might be carried by sodium ions, since there is much evidence (reviewed by Hodgkin, 1951) that the rising phase of the action potential is caused by the entry of these ions, moving under the influence of concentration and potential differences. To investigate this possibility, we carried out voltage clamp runs with the axon surrounded by solutions with reduced sodium concentration. Choline was used as an inert cation since replacement of sodium with this ion makes the squid axon completely inexcitable, but does not reduce the resting potential (Hodgkin & Katz, 1949; Hodgkin, Huxley & Katz, 1949).

2,181 citations

Journal ArticleDOI
TL;DR: This paper contains a further account of the electrical properties of the giant axon of Loligo and deals with the 'inactivation' process which gradually reduces sodium permeability after it has undergone the initial rise associated with depolarization.
Abstract: This paper contains a further account of the electrical properties of the giant axon of Loligo. It deals with the 'inactivation' process which gradually reduces sodium permeability after it has undergone the initial rise associated with depolarization. Experiments described previously (Hodgkin & Huxley, 1952a, b) show that the sodium conductance always declines from its initial maximum, but they leave a number of important points unresolved. Thus they give no information about the rate at which repolarization restores the ability of the membrane to respond with its characteristic increase of sodium conductance. Nor do they provide much quantitative evidence about the influence of membrane potential on the process responsible for inactivation. These are the main problems with which this paper is concerned. The experimental method needs no special description, since it was essentially the same as that used previously (Hodgkin, Huiley & Katz, 1952; Hodgkin & Huxley, 1952b).

1,484 citations

Journal ArticleDOI
TL;DR: The importance of ionic movements in excitable tissues has been emphasized by a number of recent experiments which are consistent with the theory that nervous conduction depends on a specific increase in permeability which allows sodium ions to move from the more concentrated solution outside a nerve fibre to the more dilute solution inside it.
Abstract: The importance of ionic movements in excitable tissues has been emphasized by a number of recent experiments. On the one hand, there is the finding that the nervous impulse is associated with an inflow of sodium and an outflow of potassiuim (e.g. Rothenberg, 1950; Keynes & Lewis, 1951). On the other, there are experiments which show that the rate of rise and amplitude of the action potential are determined by the concentration of sodium in the external medium (e.g. Hodgkin & Katz, 1949 a; Huxley & Stiimpffi, 1951). Both groups of experiments are consistent with the theory that nervous conduction depends on a specific increase in permeability which allows sodium ions to move from the more concentrated solution outside a nerve fibre to the more dilute solution inside it. This movement of charge makes the inside of the fibre positive and provides a satisfactory explanation for the rising phase of the spike. Repolarization during the falling phase probably depends on an outflow of potassium ions and may be accelerated by a process which increases the potassium permeability after the action potential has reached its crest (Hodgkin, Huxley & Katz, 1949).

1,480 citations