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 physio-logical 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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An historical discussion is provided of the intellectual trends that caused nineteenth century interdisciplinary studies of physics and psychobiology by leading scientists such as Helmholtz, Maxwell, and Mach to splinter into separate twentieth-century scientific movements. The nonlinear, nonstationary, and nonlocal nature of behavioral and brain data are emphasized. Three sources of contemporary neural network research-the binary, linear, and continuous-nonlinear models-are noted. The remainder of the article describes results about continuous-nonlinear models: Many models of content-addressable memory are shown to be special cases of the Cohen-Grossberg model and global Liapunov function, including the additive, brain-state-in-a-box, McCulloch-Pitts, Boltzmann machine, Hartline-Ratliff-Millet; shunting, maskingfield, bidirectional associative memory, Volterra-Lotka, Gilpin-Ayala, and Eigen-Schuster models. A Liapunov functional method is described for proving global limit or oscillation theorems for nonlinear competitive systems when their decision schemes are globally consistent or inconsistent, respectively. The former case is illustrated by a model of a globally stable economic market, and the latter case is illustrated by a model of the voting paradox. Key properties of shunting competitive feedback networks are summarized, including the role of sigmoid signalling, automatic gain control, competitive choice and quantization, tunable filtering, total activity normalization, and noise suppression in pattern transformation and memory storage applications. Connections to models of competitive learning, vector quantization, and categorical perception are noted. Adaptive resonance

This article describes the concepts and strategies that have guided the design and implementation of this simulator, with emphasis on those features that are particularly relevant to its most efficient use. 1.1 The problem domain

II. Current Source-Density (CSD) Method................................. 39 A. Physical and physiological basis.................................... 39 B. Experimental CSD studies......................................... 4.2 C. Interpretation of CSDs............................................ 44

neutral, amine, and quaternary local anesthetic compounds. The kinetics of the Na currents are governed by a composite of voltage- and time-dependent gating processes with voltage- and time-dependent block of channels by drug. Conventional measurements of steady-state sodium inactivation by use of 50-ms prepulses show a large negative voltage shift of the inactivation curve with neutral benzocaine and with some ionizable amines like lidocaine and tetracaine, but no shift is seen with quaternary QX-572. However, when the experiment is done with repetitive application of a prepulse-testpulse waveform, a shift with the quaternary cations (applied internally) is seen as well. 1-min hyperpolarizations of lidocaine- or tetracaine-treated fibers restore two to four times as many channels to the conducting pool as 50-ms hyperpolarizations. Raising the external Ca ++ concentration also has a strong unblocking effect. These manipulations do not relieve block in fibers treated with internal quaternary drugs. The results are interpreted in terms of a single receptor in Na channels for the different drug types. Lipid-soluble drug forms are thought to come and go from the receptor via a hydrophobic region of the membrane, while charged and less lipid-soluble forms pass via a hydrophilic region (the inner channel mouth). The hydrophilic pathway is open only when the gates of the channel are open. Any drug form in the channel increases the probability of closing the inactivation gate which, in effect, is equivalent to a negative shift of the voltage dependence of inactivation.

description of the time course of the current which flows through the mem-brane 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 con-centration. Choline was used as an inert cation since replacement of sodium with this ion makes the squid axon completely inexcitable, but does not

ABSTRACT Increasing the hydrogen ion concentration of the bathing medium reversibly depresses the sodium permeability of voltage-clamped frog nerves. The depression depends on membrane voltage: changing from pH 7 to pH 5 causes a 60 % reduction in sodium permeability at +20 mV, but only a 20 % reduction at +180 mV. This voltage-dependent block of sodium channels by hydrogen ions is explained by assuming that hydrogen ions enter the open sodium channel and bind there, preventing sodium ion passage. The voltage dependence arises because the binding site is assumed to lie far enough across the membrane for bound ions to be affected by part of the potential difference across the membrane. Equations are derived for the general case where the blocking ion enters the channel from either side of the membrane. For H+ ion blockage, a simpler model, in which H+ enters the channel only from the bathing medium, is found to be sufficient. The dissociation constant of H + ions from the channel site, 3.9 X 10-6 M (pK, 5.4), is like that of a carboxylic acid. From the voltage dependence of the block, this acid site is about one-quarter of the way

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

Cheetahs and beetles run, dolphins and salmon swim, and bees and birds fly with grace and economy surpassing our technology. Evolution has shaped the breathtaking abilities of animals, leaving us the challenge of reconstructing their targets of control and mechanisms of dexterity. In this review we explore a corner of this fascinating world. We describe mathematical models for legged animal locomotion, focusing on rapidly running insects, and highlighting achievements and challenges that remain. Newtonian body-limb dynamics are most naturally formulated as piecewise-holonomic rigid body mechanical systems, whose constraints change as legs touch down or lift off. Central pattern generators and proprioceptive sensing require models of spiking neurons, and simplified phase oscillator descriptions of ensembles of them. A full neuro-mechanical model of a running animal requires integration of these elements, along with proprioceptive feedback and models of goal-oriented sensing, planning and learning. We outline relevant background material from neurobiology and biomechanics, explain key properties of the hybrid dynamical systems that 1 underlie legged locomotion models, and provide numerous examples of such models, from the simplest, completely soluble ‘peg-leg walker ’ to complex neuro-muscular subsystems that are yet to be assembled into models of behaving animals. 1