Mathematical biology of automobile driving IV

Previous studies by this author of the mathematical biology of automobile driving have emphasized only the biological aspects, except for such mechanical factors as the size of the car. Otherwise, the ideal case of an inertialess car was considered. In this paper the first step is made toward introducing the effects of the mass of the car and the side-slip of the tires when the direction of driving is even slightly altered and combining these with the previously studied biological aspects. Some tentative comparisons with available data are made. This work was done at the Mental Health Research Institute, University of Michigan, Ann Arbor, Michigan.     

Mathematical biology of automobile driving: V

The role of some inertial properties of the car, studied previously only for the case when the stimulus for the corrective turn is the perception of the angle between the direction of the car and the direction of a straight lane (Bull. Math. Biophysics,32, 71–78, 1970), is generalized to include such stimuli as the nearness to the edge of the lane and the anticipatory effect for a corrective turn, as well as the combination of all three stimuli. Conditions for stability of driving are deduced and discussed. They now depend on both biological parameters and such parameters as the position of the center of gravity of the car, its mass, and the side slip of the tires. This work was done at the Mental Health Research Institute, University of Michigan, Ann Arbor.     

Mathematical biology of automobile driving: III

In a previous paper (Bull. Math. Biophysics,22, 257–262, 1960), an expression for the probability that a car jumps off a road as a function of the speed and the size of the car was derived mostly from geometric and kinematic considerations, introducing only the reaction time as a biological parameter. In subsequent papers (Bull. Math. Biophysics,29, 181–186, 187–188, 1967) a more detailed study was made of the exact shape of the tracking curve of the car which involved several biological parameters of the driver. In the present paper the results of the previous studies are combined, and a more general equation for the probability of jumping off the road is obtained. This probability, as in the earlier study, increases with the speedv, widths _o and lengthl _o of the car, and decreases with widths of the lane. However, this probability also depends on several parameters which characterize the psychobiological constitution of the driver. Unpublished experiments by Ehrlich, which corroborate the general conclusions, are briefly described.     

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.     

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.     

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.     

Mathematical biology of automobile driving: I. The shape of the tracking curve on an empty straight road

The idea was suggested by the author previously (Bull. Math. Biophysics,22, 257–262, 1962) that the keeping of the car close to the center of the lane is a problem of psychophysical discrimination between two conflicting stimuli, namely a stimulus to turn away from the left, resp. right edge of the lane. This is elaborated in the present paper. The effects of discrimination threshold and of the endogenous fluctuations which result in erroneous judgments are discussed. In order that driving should be possible at all, a relation, derived in this paper, must hold between the threshold of discriminationh, the sensitivity coefficientb of the driver to changes in the distance between the car and the edge of the lane, and the width of the lane. General expressions are derived which characterize the stochastic nature of the tracking curve.     

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.”     

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.     

Mathematical biology of social behavior

When an individual grows up in a society, he learns certain behavior patterns which are “accepted” by that society. He may in general have a tendency toward behavior patterns other than those which are “accepted” by the society. This tendency toward such unaccepted behavior may be due to a process of cerebration which results in doubt as to the “correctness” of the accepted behavior. Thus, on the one hand, the individual learns to follow the accepted rules almost automatically; on the other hand, he may tend to consciously break those rules. Using a neural circuit, suggested by H. D. Landahl in his theory of learning, a neurobiophysical interpretation of the above situation is outlined. Mathematical expressions are derived which describe the social behavior of an individual as a function of his age, social status, and some neurobiophysical parameters.     

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.     

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 models in mammalian cell biology

A report on the Conference on Systems Biology of Mammalian Cells, Dresden, Germany, 22-24 May 2008.     

Mathematical biology of social behavior: III

A society composed of individuals each of whom can perform one of two mutually exclusive activitiesR ₁ andR ₂ is considered. The tendency toward the performance of those activities is measured by the intensities ε₁ and ε₂ of excitation of two corresponding neural centers, which cross-inhibit each other. It follows from the theory developed by H. D. Landahl that an individual with ε₁ − ε₂ = 0, that is one who has no preference for either one of the two activities, will on the average performR ₁ andR ₂ with equal probability. As ε₁ − ε₂ increases, the probabilityP ₁ ofR ₁ increases, tending to 1. As ε₂ − ε₁ increases, the probabilityP ₂ ofR ₂ increases, tending to 1. We haveP ₁+P ₂=1. The effect of imitation is now studied. The total number of individuals in the society which exhibits an activityR ₁ at a given time is considered as constituting a stimulus which increases ε₁. Similarly, the total number of individuals which exhibits activityR ₂ at a given time constitutes a stimulus which increases ε₂. Using the standard equations of the mathematical biophysics of the central nervous system, equations are established which govern the behavior of such a society and the following conclusions are reached. It the quantity ε₁ − ε₂ is distributed in the society in such a way that the distribution function is symmetric with respect to ε₁ − ε₂ = 0, then on the average one-half of the population exhibitsR ₁, the other halfR ₂. This social configuration may, however, be unstable. The slightest accidental excess of individuals exhibiting, for example,R ₁, may bring it into a stable configuration, in which most individuals exhibitR ₁, and only a smaller fraction exhibitR ₂. A slight initial deviation in favor ofR ₂ brings it into a stable configuration, in which most individuals exhibitR ₂. Thus in this case there may be two stable configurations. If the population is in one of those stable configurations, and the distribution function of ε₁ − ε₂ is made asymmetric, favoring the other activity, the population will pass into a stable configuration, in which that other activity is predominant, if the asymmetry of the distribution exceeds a threshold value. By making some drastic simplifications the equations derived here may be reduced to a form which waspostulated by the author previously in his mathematical theory of human relations.     

数学方法与生命科学的发展

数学方法在科学研究中有着非常重要的作用,以哈维血液循环理论的建立,达尔文提出进化论和孟德尔发现分离规律和自由组合规律等具体案例为背景,分析了数学方法在生物学史上对生物学理论发展的推动应用,还对生物学应用教学方法的现状和前景作了介绍。     

Mathematical and computational modeling in biology at multiple scales

A variety of topics are reviewed in the area of mathematical and computational modeling in biology, covering the range of scales from populations of organisms to electrons in atoms. The use of maximum entropy as an inference tool in the fields of biology and drug discovery is discussed. Mathematical and computational methods and models in the areas of epidemiology, cell physiology and cancer are surveyed. The technique of molecular dynamics is covered, with special attention to force fields for protein simulations and methods for the calculation of solvation free energies. The utility of quantum mechanical methods in biophysical and biochemical modeling is explored. The field of computational enzymology is examined.     

Mathematical models for excitable systems in biology and medicine.

The excitable systems play a very important role in Biology and Medicine. Phenomena such as the transmission of impulses between neurons, the cardiac arrhythmia, the aggregation of amoebas, the appearance of organized structures in the cortex of egg cells, all derive from the activity of excitable media. In the first part of this work a general definition of excitable system is given; we then analyze some cases of excitability, distinguishing between electrical and chemical excitability and comparing experimental observations with simulations carried out by appropriate mathematical models. Such models are almost always formulated by partial differential equations of "reaction-diffusion" type and they have the characteristic to describe propagations of electrical waves (neurons, pacemaker cardiac cells, pancreatic b-cells) or chemical and mechanical waves (propagation of Ca++ waves or mechanical waves in the endoplasmic reticulum). The aim is to put in evidence that the biological systems can show not only excitability of electrical type, but also excitability of chemical nature, which can be observed in the first steps of development of egg cells or, for example, in the formation of pigments in vertebrate skin or in clam shells.     

Mathematical models, explanation, laws, and evolutionary biology

It is commonly agreed in the literature on laws of nature that there are at least two necessary conditions for lawhood--that a law must have empirical content and that it must be universal. The main reason offered for the requirement that laws be empirical is as follows: a priori statements are consistent with any imaginable set of observations, so they cannot be informative about the world and therefore they cannot provide explanations. However, we care about laws because we think that laws provide explanations and allow us to make predictions. Thus, if one of the functions of laws is to provide explanations and a priori propositions cannot fulfill this function, they cannot properly be viewed as laws. In this paper, I will aim to show that this argument for the claim that laws must be empirical does not work.