Further contributions to the theory of cell polarity and self-regulation

In continuation of a previous study a case of self-regulating polarity is investigated, in which a metabolite neither enters nor leaves the cell, being produced in one part of it and consumed in another. Such a cell possesses a larger degree of autonomy and independence of external conditions than the previously discussed one. Application to cellular locomotion is indicated.     

An alternate approach to the mathematical biophysics of perception of combinations of musical tones

In connection with a previous paper on the same subject this different approach to the problem is outlined, which also leads to an expression for pleasantness values of different binary combinations of musical sounds.     

A contribution to the mathematical biophysics of cell growth and shapes: II

A theoretical study of the growing nerve cell filopodium is made using the assumptions of volume constancy, cylindrical shape, and substrate track of an earlier paper, but assuming additionally that a retarding force per unit area proportional to the rate of elongation is also acting. Equations of elongation for two different cases are derived.     

Some suggestions for a possible approach to a mathematical biophysics of mitosis

A tentative suggestion is made to interpret the phenomenon of mitosis as being due to coiling and uncoiling of long chain molecules, which are assumed to constitute the spindle. A possible quantitative approach is briefly outlined.     

Contribution to the mathematical biophysics of automobile driving

Traffic in one direction on a multilane highway is considered, and a general expression for the number of cars which pass a car travelling at a given velocity, as well as the number of cars which the given car passes, is derived for the case when the speeds of different cars are distributed in some arbitrary manner. Closed expressions are derived and discussed for a rectangular distribution. Each passing by another car or of another car is considered as a distracting stimulus which affects the reaction times of the driver. Using previously derived expressions for the safe speed as a function of reaction times, expressions for the safe average speed are derived, in terms of the volume of traffic and of the spread of the distribution of speeds.     

Contributions to the mathematical biophysics of visual aesthetics

An expression for the intensity of central excitation corresponding to the perception of an angle is derived. A general expression for the aesthetic value of certain types of polygonal patterns is derived there-from and compared with experimental values obtained by the rank order method. The theory is found to agree with the experiment within a limit of about 10%.     

Contributions to the mathematical biophysics of the cardiovascular system

The reflection of pressure waves in a fluid enclosed within a tube with an elastic wall is studied for the case of a localized change in diameter of the tube. The concept of impedance is introduced. The relation of the reflection characteristics of the parts of the tube at either side of the change is derived on the basis of the continuity of pressure and mass flow at the site of the change. This relations is used to derive the expression for the ratio of the pressure oscillations measured in front of, and behind, the constriction in terms of the constants of the system. As a result, a method is indicated to locate the coarctation from measurements of the pressures in front of, and behind it.     

Contributions to the mathematical biophysics of branched circulatory systems

A derivation is given of the reflection coefficient of pressure waves in a vessel whose end branches into many smaller vessles. This coefficient depends on the number of these smaller vessels and their sizes relative to the size of the main vessel. Estimations are made of the order of magnitude of the coefficient. Assuming the main vessel to be of the order of size of an artery, it is shown that the reflection coefficient has a value close to one for reflections at branchings into vessels of arteriolar size. It is pointed out that the result may support the idea that the standing waves in the arterial system are due to reflections at the site of the arterioles.     

Further contributions to the mathematical biophysics of automobile driving

The discussions of a previous paper (Bull. Math. Biophysics,21, 299–308, 1959) are generalized by considering that the angular direction error made by the driver, as well as the driver's reaction time are not constant but are randomly distributed. Instead of a critical speed, at which the car will jump off the road, we now find that for every speed there is a probability of the car to jump off the road but that this probability is vanishingly small for sufficiently low speeds, yet increases rapidly for high speeds. Thus a more realistic picture of the process of driving is obtained. When the standard deviation of the distribution functions for the angle and the reaction time are very small, the expression obtained here reduces to the expression obtained previously.     

Further contributions to the mathematical biophysics of visual aesthetics

In previous papers cases were considered in which a visual pattern consists of a relatively small number of relatively strongly excited elements. Those cases are of interest in the theory of visual perception and aesthetic rating of artificial man made patterns. By an extension of the theory so as to take into account the finite thresholds of the inhibiting fibers a theory of aesthetic ratings of pattern consisting of a very large number of elements is outlined. This type of theory is applicable to natural patterns, such as landscapes.     

A contribution to the mathematical biophysics of visual aesthetics

In continuation of previous studies, inequalities between different parameters of the brain are derived which determine whether an individual prefers in general visual patterns consisting of a relatively small number of relatively strongly excited elements, or such patterns which consist of a very large number of relatively weakly excited elements. As has been discussed in a previous publication, the first type of pattern is usually represented by artificial human-made designs, whereas the second type of pattern is formed predominantly in natural landscapes and sceneries. Thus the inequalities established in this paper give us the biophysical conditions which determine an individual's preference for either artificial designs or for landscapes and other natural objects.     

Contributions to the mathematical biophysics of the central nervous system

Various neural mechanisms are considered which deal with point to point correspondence between two sets of neural elements with a smaller number of conducting elements between them; the transmission of nerve impulses in a limited range of intensities; movement of the transmission of excitation along a contour; the reaction to the size of an object independent of its distance; and an interpretation of the effect of a warning stimulus and of stimulus intensity upon reaction time. For the latter cases a comparison of the theoretical equations is made with some of the available experimental data, and a general agrement is found.     

The relation of mathematical biophysics to experimental biology

Zusammenfassung Nach einer allgemeinen Diskussion des Zusammenhanges zwischen theoretischer und experimenteller Forschung, wird in Hinblick auf die vom Verfasser entwickelten physikalisch-mathematischen Grundlagen der Biologie, eine Reihe von Einzelproblemen betrachtet. Es wird an Hand von Kurvenmaterial gezeigt wie weit die mathematisch vorausgesagten Beziehungen mit den experimentellen Befunden übereinstimmen. Folgende Fragen werden besprochen: Zellatmung, Zellgrössen, deren Abhängigkeit von Stoffwechsel, Zellteilung, Protoplasmaströmungen, Nervenerregung, psychophysische Gesetze, Reaktion auf geometrische Gestalten.

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Résumé Après une mise au point générale de la relation entre les sciences théoriques et expérimentales, diverses questions sont discutées, du point de vue des fondations physico-mathématiques de la biologie, développées récemment par l'auteur. On montre par la comparaison des courbes calculées à celles observées, comment les relations prédites mathématiquement se trouvent vérifiées. Les questions suivantes sont discutées: respiration cellulaire, dimensions cellulaires et leur rapport au métabolisme, division cellulaire, mouvements protoplasmiques, excitation nerveuse, les lois psychophysiques, perception des formes géométriques.

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The author is indebted,to MrGale Young for calculations of the curves of figures 4 and 5. He is also indebted to Prof. A. V.Hill, Dr D. Y.Solandt, The University of Chicago Press, the Editor of Growth and the Psychometric Corporation for permission of reproducing different figures used in this article.     

A contribution to the mathematical biophysics of psychophysical discrimination III

The neural mechanism previously discussed is further generalized. The case is considered in which a random variation is associated with each stimulus. The mechanism is generalized and equations are derived for discriminations between stimuli differing in several modalities. The latter indicates an analysis by the factor method. Suggestions are made in connection with the use of triads and with the problem of a multidimensional psychophysics.     

The mathematical biophysics of Nicolas Rashevsky

N. Rashevsky (1899-1972) was one of the pioneers in the application of mathematics to biology. With the slogan: mathematical biophysics : biology :: mathematical physics ; physics, he proposed the creation of a quantitative theoretical biology. Here, we will give a brief biography, and consider Rashevsky's contributions to mathematical biology including neural nets and relational biology. We conclude that Rashevsky was an important figure in the introduction of quantitative models and methods into biology.     

The mathematical biophysics of some mental phenomena

In this paper some aspects of the mathematical biophysics of the central nervous system, which hitherto have not been treated, are discussed. First, a neurobiophysical mechanism for consciousness is suggested. It provides for the possibility of conscious and unconscious rractions. Next, a mechanism of memory, both on the conscious and subconscious level, is suggested. The gradual forgetting of remote events is ascribed to the inhibition of older memory traces by the more recent ones. On the average, an exponential decay of memory with time is thus obtained, although memory for unusually strong experiences follows a somewhat different law of decay. A homeostatic mechanism is then considered which regulates the level of accumulated excitation or inhibition. Such a mechanism, under certain disturbing conditions, will result in periodical fluctuations of the total cortical excitation with periods varying within a very wide range. Finally, a mechanism for foresight and desire of future events is suggested. The latter provides for the possibility of the formation of subconscious reactions and habits, which may be abolished by bringing them into consciousness.     

Studies in the mathematical biophysics of discrimination and conditioning I

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 return alley, and Lashley's jumping problem. For each case equations are derived which are qualitatively correct as far as was determined. Several qualitative deductions are made and these are substantiated by data available. In principle, the theory makes it possible to predict, for any trial, the number of errors at any junction of a maze provided certain experimental conditions are satisfied, and if a sufficient number of experimental values are given to determine the parameters of the system.     

A note on the mathematical biophysics of ameboid movements

Theoretical considerations lead to the expectation of some regularities in the apparently random changes of shape of an ameba. We should expect quasiperiodic fluctuations of the ratio of the perimeter to the area of the optical cross-section of an ameba, when plotted against time. The quasi-period should be characteristic of the species. Preliminary experiments appear to bear out that prediction.     

Contributions to the mathematical biophysics of the central nervous system with special reference to learning

A learning theory based on the lowering of thresholds of neurons under certain conditions is applied to two “random net” models. The first, a so-called “ganglion-brain” is characterized by completely random connections of all afferent tracts except certain ones which form the pathways for unconditioned responses. Certain expressions are derived which measure the learning potentiality of the ganglion— in particular, with respect to the number of responses which can be learned (conditioning potential) and the amount of interference between the learned responses (redundance potential). The second model concerns the progressive refinement of a response. The efficiency of learning in this case is reflected in the eventual specificity of the response which, in turn, depends on the modification of the distribution of thresholds associated with the neurons governing the responses. Expressions are derived relating the initial distribution of thresholds, the relative effectiveness of the various responses, and certain other parameters to the final distribution of thresholds. For a particular choice of the effectiveness distribution of responses the progressive sharpening of the threshold curve (i.e., progressive specificity of response) is demonstrated. Some implications of the model with respect to the evolution of nervous systems are discussed.