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.     

Mathematical biophysics of automobile driving

In a previous paper (Rashevsky, 1959,Bull. Math. Biophysics,21, 299–308) we derived an approximate expression for the maximal speed of driving in terms of the reaction time of the driver. In the present paper the possible effects of unevennesses of the pavement, of such distracting stimuli as road signs etc. on the reaction time are studied theoretically, using previous developments of the mathematical biophysics of the central nervous system. In this manner expressions are derived which determine the maximal safe speed in terms of road conditions and other distracting stimuli. Effects of those conditions on fatigue are also discussed.     

Some remarks on the mathematical aspects of automobile driving

Some aspects which involve the interaction of the human element and of the machine element in driving are discussed. As an example a simple equation is derived for the maximum safe speed of a car on an empty road. The parameters of the equation are partly of physiological nature, partly of mechanical nature. Another example treats in a similar manner the problem of a car passing another car moving in the same direction.     

A note on the mathematical biology of automobile driving

It is pointed out that the three different stimuli for a corrective turn, namely the distance from the edge of the lane, the rate of approach to the edge, and the angle between the direction of the car and the direction of the lane (Bull. Math. Biophysics,28, 645–654, 1966,29, 181–186, 1967) may act all three simultaneously. It is found that in that case the tracking curve of the car is stable below a critical speed and becomes unstable above it.     

Two remarks on the mathematical biology of automobile driving

In connection with a series of previous papers by this author (Bulletin of Mathematical Biophysics,21, 299–308, 375–385;22, 257–262, 263–267;23, 19–29;24, 319–325) results obtained by A. Crawford (Economics 5, 417–428) on the effects of irrelevant lights on reaction times toward a given light stimulus are discussed. The conclusions from a previous paper of this author (Bulletin of Mathematical Biophysics,23, 19–29) are elaborated.     

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.     

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.     

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.     

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 II

The properties of a neural mechanism previously discussed are given in more general form. It is shown that, under certain conditions, a psychophysical scale value is proportional to a difference between excitation factors at a synapse. More general relations are derived which hold among different variables in the two and three categories of judgment. An attempt is made to apply the results to the method of choices. Certain relations derived are compared with available experimental data.     

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.     

Some considerations on mathematical molecular biophysics

The possibility of a single molecule or of a few molecules controlling the basic metabolic reactions in a cell is discussed from the molecular-kinetic point of view. For such a control by a single molecule to be possible, it is necessary to assume a chain reaction, consisting of at least two steps.     

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.     

A contribution to the mathematical biology of automobile driving: II. Passing as a case of psychophysical discrimination

The decision to pass or not to pass in view of an oncoming car is considered as a case of comparative judgment in which it is to be decided whether the time it will take to pass safely is greater or less than the time it will take to collide with the oncoming car. H. D. Landahl's well-known theory of psychophysical discrimination is used, and it is assumed that the “distracting stimuli” considered previously (Rashevsky, 1959,Bull. Math. Biophysics,21, 375–85) tend to increase the standard deviation of Landahl's fluctuation function. Effects of the “distracting stimuli” on the threshold of the neuroelements in Landahl's circuit are also considered. On this basis an expression is derived which gives the probability of a collision accident in passing as a function of the “distracting stimuli.”     

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.