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

https://www2.econ.iastate.edu/tesfatsi/DeepLearningInNeuralNetworksOverview.JSchmidhuber2015.pdf

Deep learning in neural networks

http://www.maths.qmul.ac.uk/%7Elatora/report_06.pdf

Complex networks: Structure and dynamics

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.

/pdf/pattern-formation-outside-of-equilibrium-1601b9znum.pdf

Pattern formation outside of equilibrium

Impulses and Physiological States in Theoretical Models of Nerve Membrane

Muscle structure and theories of contraction.

1. The variation of isometric tetanus tension with sarcomere length in single fibres from frog striated muscle has been re-investigated with special precautions to ensure uniformity of sarcomere length within the part of the fibre being studied.

2. In most respects the results of Ramsey & Street (1940) were confirmed, but (a) the peak of the curve was found to consist of a plateau between sarcomere lengths of 2·05 and 2·2 μ, (b) the decline of tension above this plateau is steeper than found by Ramsey & Street, and (c) the decline of tension below the plateau becomes suddenly steeper at a sarcomere length of about 1·67 μ.

3. Many features of this length—tension relation are simply explained on the sliding-filament theory.

4. It is concluded that, in the plateau and at greater lengths, the tension on each thin filament is made up of equal contributions from each bridge which it overlaps on adjacent thick filaments.

5. Internal resistance to shortening is negligible in this range but becomes progressively more important with shortening below the plateau.

https://physoc.onlinelibrary.wiley.com/doi/pdf/10.1113/jphysiol.1966.sp007909

The variation in isometric tension with sarcomere length in vertebrate muscle fibres.

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).

Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo

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).

A. F. Huxley

Papers

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

Muscle structure and theories of contraction.

The variation in isometric tension with sarcomere length in vertebrate muscle fibres.

Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo

Measurement of current-voltage relations in the membrane of the giant axon of Loligo.