A mathematical analysis of carbon dioxide respiration in manChilton, Arthur B.;Stacy, Ralph W.
doi: 10.1007/BF02477818pmid: N/A
Abstract By means of physico-mathematical models, formulas are obtained for the purpose of making analyses of external carbon dioxide respiration in man under conditions of metabolic equilibrium. The methods evolved are flexible, permitting study of a wide variety of physiological assumptions. Sample calculations relate various factors in the process of external respiration, and show good agreement with experimental data. A quantitative description of the effects of various factors on rate of carbon dioxide output and alveolar tension is given. In particular, this theory gives a quantitative prediction of the fluctuation in alveolar carbon dioxide tension over the period of a single respiratory cycle.
The audiogyral illusion and the mechanism of spatial representationMayne, Robert
doi: 10.1007/BF02477820pmid: N/A
Abstract An hypothesis regarding the mechanism of spatial representation in the neural centers is formulated in order to explain the audiogyral illusion. Using this hypothesis and experimental data (Clark and Graybiel, 1949) the time constant of the semicircular canals is calculated. The value obtained in this manner agrees with that previously calculated (Mayne, 1950) using results from experiments on the nystagmic latency of pigeons.
“Ignition” phenomena in random netsRapoport, Anatol
doi: 10.1007/BF02477821pmid: N/A
Abstract The spread of excitation in a “random net” is investigated. It is shown that if the thresholds of individual neurons in the net are equal to unity, a positive steady state of excitation will be reached equal to γ, which previously had been computed as the weak connectivity of the net. If, however, the individual thresholds are greater than unity, either no positive steady state exists, or two such states depending on the magnitude of the axone density. In the latter case the smaller of the two steady states is unstable and hence resembles an “ignition point” of the net. If the initial stimulation (assumed instantaneous) exceeds the “ignition point,” the excitation of the net eventually assumes the greater steady state. Possible connections between this model and the phenomenon of the “preset” response are discussed.
Determination of diffusion and permeability coefficients in muscleOpatowski, I.;Schmidt, George W.
doi: 10.1007/BF02477822pmid: N/A
Abstract A theory is presented for the study of diffusion in heterogeneous tissue-like structures. It is applicable to a common type of measurement in which the change of the amount of substance remaining in the tissue is determined as the substance diffuses from the tissue into an adjacent medium, for instance, Ringer's solution. The main objective of this paper is to obtain a method for the calculation of the diffusion coefficient in the intercellular space and of the permeability coefficients between this space and the cells, based on the type of measurement mentioned above. Although the fundamental ideas upon which the theory is based are applicable to any type of tissue, the formulae derived are limited to the case in which the cells form a flat bundle of parallel fibers. The theory is applied to the experimental results of E. J. Harris and G. P. Burn on diffusion of sodium in the sartorius muscle of the frog. We find that if we know the ratio of the cellular and intercellular volumes of the muscle the ratio of the equilibrium concentrations of sodium outside and inside the cells can be determined. A very simple mathematical analysis of the experimental relation between the amount of substance diffusing out of the muscle and the time of diffusion gives us this ratio. The ratio of the equilibrium sodium concentrations in the case of the sartorius frog muscle is between about 10 and 30, depending on the muscle used. The same mathematical analysis makes it possible to obtain the permeability coefficients of muscle fibers through simple calculations, if their sizes are known. The permeability coefficients for the experimental work mentioned above using sodium are 1.25 to 11.5×10−8 cm/sec for the flow into the fibers and 3.2 to 16×10−7 cm/sec for the flow in the opposite direction. The determination of the diffusion coefficient in the intercellular space is more laborious and yields only an order of magnitude: 10−6 cm2/sec.
Some elementary considerations of neural modelsShimbel, A.
doi: 10.1007/BF02477823pmid: N/A
Abstract The outputs of nervous systems (as expressed in motor activity) are viewed as mathematical transformations on the inputs which enter via the sensory nerves. Simple nerve-ganglion models are exhibited which theoretically account for the arithmetic computations necessary to expedite such transformations.
Input output curves of aggregates of “simple counter” neuronsRapoport, Anatol
doi: 10.1007/BF02477824pmid: N/A
Abstract The refractory periods of an aggregate of simple “counter” neurons are assumed distributed according to some probability frequency. The output of the aggregate is computed for rectangular and triangular distributions. In particular, it is shown that the maximum output of an aggregate with any triangular distribution cannot exceed the maximum output of its average neuron by a factor greater than 2 ln 2. This puts an upper bound on the amount of departure from the behavior of the average neuron which an aggregate characterized by a certain type of distribution can show. Next, the aggregate is supposed to be subjected to regularly spaced stimuli. Under these conditions, a single neuron will give a discontinuous output curve. If, however, the refractory periods are distributed according to some frequency, the output curve may be “smoothed out.” A general condition on the distribution is derived which makes the output monotone increasing with the input. The condition is applied to some special cases.
On the theory of diffusion of electrolytesOpatowski, I.
doi: 10.1007/BF02477825pmid: N/A
Abstract In the theory of diffusion of electrolytes the following assumptions are frequently made: (i) the electrolytic solution is electrically neutral everywhere, (ii) the ionic concentrations and the electric potential all depend on a single Cartesian coordinate as the only space variable. Often the electric potential of the solution is determined on the basis of the Poisson equation alone, disregarding any other relation between this potential and the ionic concentrations. Since the Poisson equation only represents a condition which the potential fulfills, the use of this equation alone may lead to error unless the explicit relation for the potential involving a space integration of ionic concentrations is also taken into account. But if this relation is used the Poisson equation becomes redundant and, more important, assumptions (i) and (ii) appear unacceptable, the former because it leads to a zero electric potential everywhere, the latter because it is mathematically incorrect. The present paper is based on general equations of diffusion of ions, excluding the Poisson equation. These equations form a system of nonlinear integrodifferential quations whose number equals the number of ionic species present in the solution. It appears that when all ions are distributed symmetrically around a point all functions related to the above system of equations can be made dependent on a single space coordinate: the distance from the center of symmetry. Two methods of successive approximations are given for the solution of the equations in the case of spherical symmetry with limitation to the steady state. These methods are then applied to the study of the distribution of ionic concentrations and electrical potentials inside a cell of spherical shape in equilibrium with its surroundings. These methods are rapidly convergent; exact theoretical values of the electric potential are calculable on the boundary of the cell. It appears that the potential at the center of the cell is not more than ∼50% higher than at its boundary and that variation of concentration inside the cell is not very large. For instance, with 100 mV on the boundary the ionic concentration there is about four times higher than at the center. Calculations show that extremely small amounts of electricity are sufficient to account for the electric potentials currently observed. In a cell of 100 micra diameter an average concentration of only 10−14 mole/cm3 of a monovalent ion would be sufficient to give 1 millivolt on the boundary. This concentration is directly proportional to the voltage and inversely proportional to the square of the cell diameter. Most of the numerical results given above are obtained by considering only those ions whose electrical charge is not compensated for by ions of an opposite sign. The total concentrations may be much higher than those quoted. The theory does not take into account possible effects of structural heterogeneities which may exist in the cell, particularly of various phase boundaries. An incidental result shows that the Boltzmann distribution function in the form employed in modern theory of electrolytes is fundamentally a consequence of the mathematical theory of diffusion alone. It is pointed out, however, that Boltzmann distribution is not always compatible with the definition of the electric potential.
Prolegomena to a dynamics of ideologiesRashevsky, N.
doi: 10.1007/bf02477826pmid: N/A
The paper outlines a possible further development of the suggestion made by the author in his recent book. Ideology is defined here as a verbalized, or at least verbalizable, behavior pattern which may be adopted by society. After a brief discussion of possible classifications of ideologies, a study is made of those ideologies which refer to the question of social and ethical interrelations. Different kinds of ideologies may be represented bybehavior matrices, introduced in the author's book. A uniparametric representation of all such matrices is suggested and discussed. Next the previous results on social imitation are extended to the case ofn different behaviors, each of which is determined by a particular value of a continuously varying parameter. It is shown that, depending on some other social parameters, changes from one ideology to another may proceed either quasicontinuously or definitely discontinuously. The paper concludes with some general speculations on the possibility of applying the above results to a mathematical interpretation of history.