Six interesting works by Alston S. Householder, all found in The Bulletin of Mathematical Biophysics, 1939-19412. Offered in six individual issues of the journal, scarce in their original wrappers. Six issues, $850.00
Householder, Alston. Mathematical biophysics of cellular forms and movements. The Bulletin of Mathematical Biophysics (1941) 3: 27-38, March 01, 1941. Rashevsky's equations for describing the joint variation of cell shape and concentration of a metabolite are discussed. Conditions for the existence of non-spherical equilibria and the location of these are obtained and involve only two parametersa andA. Sufficient (but not necessary) conditions for the stability of these equilibria can also be expressed in terms of these parameters alone. Necessary conditions involve in some cases a third parameterB. Quasi-periodic fluctuations about a stable nonspherical equilibrium may occur, but only in caseB lies on a certain finite range which can be defined in terms of a and A....
Householder, Alston S. A theory of steady-state activity in nerve-fiber networks II: The simple circuit. The Bulletin of Mathematical Biophysics (1941) 3: 105-112, September 01, 1941. It is found that for a simple circuit of neurons, if this contains an odd number of inhibitory fibers, or none at all, or if the product of the activity parameters is less than unity, then the stimulus pattern always determines uniquely the steady-state activity. For circuits not of one of these types, it is possible to classify exclusively and exhaustively all possible activity patterns into three types, here called “odd”, “even”, and “mixed”. For any pattern of odd type and any pattern of even type there always exists a stimulus pattern consistent with both, but in no other way can such an association of activity patterns be made.
Householder, Alston S. A theory of steady-state activity in nerve-fiber networks: I. Definitions and preliminary lemmas. The Bulletin of Mathematical Biophysics (1941) 3: 63-69, June 01, 1941. As an essay towards the determination of the effect of structural relations among nerve fibers upon the character of their activity, preliminary consideration is given to the steady-state activity of some simple neural structures. It is assumed as a first approximation that while acted upon by a constant stimulus, each fiber reaches a steady-state activity whose intensity is a linear function of the applied stimulus. It is shown by way of example that for a simple two-fiber circuit of inhibitory neurons knowledge of the stimuli applied to the separate fibers does not necessarily suffice to determine uniquely the activity that will result. On the other hand, there are deduced certain restrictions on the possible types of activity that may be consistent with a given pattern of applied stimulation.
Householder, Alston S. A note on the horopter. The Bulletin of Mathematical Biophysics (1940) 2: 135-140, September 01, 1940. By assuming the fixity (but not the symmetry) of corresponding points on the two retinae, it is possible to derive the equation of any horopter when one is known. In particular when, as experiment shows, one horopter is linear, then all horopters must be conics. These have the form given by Ogle, but whereas Ogle leaves one parameter undetermined at each fixation, on our assumption the only arbitrary parameter is determined by the position of the linear horopter.
Householder, A. S. A neural mechanism for discrimination: II. Discrimination of weights. The Bulletin of Mathematical Biophysics (1940) 2: 1-13, March 01, 1940. A theoretical central mechanism for the discrimination of intensities as previously developed, together with plausible assumptions concerning the receptors, are employed for the derivation of the discriminable difference between lifted weights as a function of the smaller of these weights. The function so derived depends upon three parameters, one parameter being the weight of the supporting member. Some empirical data are compared with the theoretical predictions, and a few remarks are added to describe the physiological significance of the parameters.
Householder, A. S. Studies in the mathematical theory of excitation. The Bulletin of Mathematical Biophysics (1939) 1: 129-141, September 01, 1939. The general linear two-factor nerve-excitation theory of the type of Rashevsky and Hill is discussed and normal forms are derived. It is shown that in some cases these equations are not reducible to the Rashevsky form.
Other contributors to these six journals include:
Rashevsky. Note on the mathematical biophysics of temporal sequences of stimuli. The Bulletin of Mathematical Biophysics (1941) 3: 89-92, September 01, 1941. By Rashevsky, N. Some general considerations are given regarding the effects of temporal sequences of stimuli in a neuronic network, which consists of a set of parallel chains of excitatory fibers with cross-connections made of inhibitory fibers. It is shown that, in general, the excitation produced by any individual stimulus of the series is a function of the order and duration of the previous stimuli, and that the effect of each stimulus thus depends on the whole temporal pattern considered.
Rashevsky, N. A note on the nature of correlations between different characteristics of organisms. The Bulletin of Mathematical Biophysics (1941) 3: 93-95, September 01, 1941. Different anatomical and physiological characteristics of organisms affect their interreaction with the inorganic world as well as their mutual interreactions. In this way they all may affect indirectly the total rate of reproduction of a species. It is shown that the requirement of a maximum rate of reproduction defines the distribution functions of the different characteristics and through those distribution functions determines statistical correlations between the characteristics.
Coombs, Clyde H. Mathematical biophysics of the galvanic skin response. The Bulletin of Mathematical Biophysics (1941) 3: 97-103, September 01, 1941. Beginning with Rashevsky's equation for the development of the excitatory state in a nerve fiber, an equation for the change in skin resistance upon the presentation of an instantaneous stimulus is derived. The mechanism assumed is in conformity with the existing evidence of neuro-physiology. Certain deductions from the equations are made and experimental problems suggested for testing the theory.
Williamson, Robert R. Electrical charges and potentials in cells resulting from metabolism of electrolytes. The Bulletin of Mathematical Biophysics (1941) 3: 79-87, September 01, 1941. An approximate solution for the relation of the charges and potentials in a spherical cell to the production of electrolytes in the cell and their diffusion resistances is derived. The potential obtained by introducing reasonable values of the constants is of the proper order of magnitude. The equations are applied to a respiratory chain and the relation between oxygen consumption, glycolytic coefficient, and potentials is determined. Available experimental data is compared with the theory...
Landahl, H. D. Studies in the mathematical biophysics of discrimination and conditioning II: Special case: Errors, trials, and number of possible responses. The Bulletin of Mathematical Biophysics (1941) 3: 71-77, June 01, 1941. A special case of a problem discussed in a previous paper is treated in greater detail. An equation in the three variables, errors, trials and number of possible choices, is developed and compared with the results of an experiment performed under conditions closely approximating those required for the development of the equation. The agreement is excellent...
Rashevsky, N. The dynamics of cell constriction during division. The Bulletin of Mathematical Biophysics (1941) 3: 57-62, June 01, 1941. The approximate equation is derived for the rate of constriction of a dividing cell, describing the phenomenon from its early stages. The equation previously derived by G. Young for the case when the constriction has already considerably progressed is obtained as a limiting case.
Rashevsky, N. Some remarks on the movement of chromosomes during cell division. The Bulletin of Mathematical Biophysics (1941) 3: 1-3, March 01, 1941. The possibility of drawing conclusions about the nature of the forces acting upon the chromosomes during division from observation of the rates of their movement in anaphase is pointed out. Some available data are discussed, and shown to agree quantitatively with the assumption that during anaphase the chromosomes are pulled apart by contracting elastic fibers...
Young, Gale. On reinforcement and interference between stimuli. The Bulletin of Mathematical Biophysics (1941) 3: 5-12, March 01, 1941. It is shown that the current “two-factor” theory of nerve excitation can account for sustained inhibition or enhancement by a sequence of stimulus pulses, and for the decrease in the reinforcement period with each successive pulse of the train...
Weinberg, Alvin M. Weber's theory of the kernleiter. The Bulletin of Mathematical Biophysics (1941) 3: 39-55, June 01, 1941 The potential distribution about a kernleiter is determined according to Weber's method. It is shown that the distribution reduces to the solution of a telegrapher's equation when the volume of the external medium is small. The velocity of propagation as a function of the external volume is determined approximately.
Landahl, H.D. Studies in the mathematical biophysics of discrimination and conditioning I. The Bulletin of Mathematical Biophysics (1941) 3: 13-26, March 01, 1941. A mechanism with properties of discrimination and conditioning is discussed mathematically with reference to special cases in the problem of error elimination: elimination of the longer of two paths to a goal, elimination of a blind as well as a..
Rashevsky, N. Physicomathematical aspects of some problems of organic form. The Bulletin of Mathematical Biophysics (1940) 2: 109-121, September 01, 1940. In connection with previous mathematical studies on cell polarity, the possible application of the results obtained before to different embryological phenomena is discussed. Methods for a quantitative mathematical approach to such phenomena as gastrulation, formation of different folds, closing of a half blastula, etc. are outlined.
Rashevsky, N. Contributions to the mathematical biophysics of organic form III. Deformation of shell shaped cellular aggregates. The Bulletin of Mathematical Biophysics (1940) 2: 123-126, September 01, 1940. The average concentrations of a substance diffusing into or from an open spherical shell, in which it is consumed or produced at a constant rate, are calculated by the approximation method. An application of the result to the problem of deformation of such a shell under the influence of diffusion forces is indicated.
Weinberg, Alvin M. On the formal theory of nerve conduction. The Bulletin of Mathematical Biophysics (1940) 2: 127-133, September 01, 1940. A general solution of the formal nerve conduction problem is given. As illustrations of the general method, the capacitative single-factor and the non-capacitative Lapicque problems are solved. Comparisons between velocity formulae for capacitative and non-capacitative models indicate that previously determined non-capacitative velocities are considerably too high.
Peters, H. C. The general fluid circuit theory of active chloride absorption. The Bulletin of Mathematical Biophysics (1940) 2: 141-143, September 01, 1940. The original fluid circuit theory used to explain active intestinal absorption of chloride is modified to include diffusion and secretion of chloride and osmosis. The general differential equation developed is integrated in a particular case. The definition, “effective concentration of chloride in the fluid passing into the intestinal lumen,” leads to simplified general expressions.
Young, Gale. A generalization of Cunningham's extension of Stoke's law for the force on a sphere. The Bulletin of Mathematical Biophysics (1940) 2: 105-108, September 01, 1940. Cuningham's formula for the force on a sphere moving within a larger concentric spherical boundary is extended to cover a general state of motion of the fluid between them.
Rashevsky, N. Contributions to the mathematical biophysics of organic form I. Formation of cavities in cellular aggregates. The Bulletin of Mathematical Biophysics (1940) 2: 27-36, March 01, 1940. The action of diffusion forces in aggregates of metabolising cells is studied mathematically, and it is shown that under definite conditions these forces may lead to the formation of inner cavities inside of the cellular aggregate. Different quantitative relations are derived and the possible bearing of these results on some embryological phenomena is discussed. A possible application of these considerations to the theory of formation of vacuoles is also discussed.
Landahl, H. D. Discrimination between temporally separated stimuli. The Bulletin of Mathematical Biophysics (1940) 2: 37-47, March 01, 1940. A mechanism is discussed which has the property that two temporally separated stimuli presented over a common path may be discriminated as though they had been presented simultaneously over separate paths.
Rashevsky, N. An approach to the mathematical biophysics of biological self-regulation and of cell polarity. The Bulletin of Mathematical Biophysics (1940) 2: 15-25, March 01, 1940. In a cell, in which the permeability to a metabolite is a function of the concentration of that metabolite, situations may occur, in which the diffusion field will exhibit certain assymetric patterns, even though the cell may possess geometrically spherical symmetry. This pattern results in a polarity of the cell. Moreover, the pattern being the result of a dynamic equilibrium, it possesses the property of self-regulation. Dividing the cell in two results in the appearance of a similar patterns in each half-cell.
Young, Gale. Convective diffusion in parallel flow fields. The Bulletin of Mathematical Biophysics (1940) 2: 49-59, March 01, 1940. Laminar motion of two viscous incompressible fluids through each other is treated for two cases: flow along the axis of a circular cylinder, and flow between parallel flat plates. Motion of either fluid entails that of the other. Regarding one fluid as a...
Rashevsky, N. Mathematical biophysics of growth. The Bulletin of Mathematical Biophysics (1939) 1: 119-127, September 01, 1939. The rate of growth of a tissue is studied mathematically in its dependence on the metabolism of the cells. A high glycolytic coefficient, which facilitates cell division, as has been shown before, does in this way also increase indirectly the rate of growth of the tissue. There is however also a possible direct effect of glycolysis on the rate of growth, which is also studied analytically. Equations are derived, giving the total rate of growth of a tissue in its dependence on the glycolytic coefficient.
Reiner, John M.. Diffusion in colloidal media. The Bulletin of Mathematical Biophysics (1939) 1: 143-149, September 01, 1939. When the molecules of a solute diffuse through a medium containing large colloidal particles, which absorb the diffusing molecules, the latter are transported in the diffusion flow not as free molecules, but as absorbtion compounds: solute+colloid. When the colloidal particle is much larger than the molecule of the solute, and has therefore a much smaller mobility, this results in a reduction of the apparent diffusion coefficient for the solute. The biological implications of this are discussed.