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: 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.     

Mathematical biology of automobile driving

Continuing a previous study (Bull. Math. Biophysics, 28, 645–654, 1966), the biophysical mechanism of a corrective turn is investigated for the case where the stimulus for the corrective turn is produced not only by the perception of the nearness of an edge of the lane, but also by the rate of approach of the car towards the edge. In that case it is found that the tracking curve of the car may consist of a series of damped sinusoids and safe driving would be possible at any speed if it were not for the endogenous fluctuation in the driver's central nervous system. If the effect of the rate of approach increases sufficiently rapidly as the distance to the edge of the lane decreases, then a stable undamped oscillating tracking curve is possible. The case is also studied where the driver makes a corrective turn in response to a direct perception of the angle between the direction of the lane and the longitudinal axis of the car.     

Automobile driving as psychophysical discrimination

The driver tries to keep the car in the center of the lane. If the car is too near the left edge, this causes the driver to make a “corrective” right turn. If the car is near the right edge, a “corrective” left turn is made. Therefore, a quantity which decreases with increasing distance Δ _L from the left edge may be considered as a stimulusS _R producing the reactionR _R of turning to the right. A similar situation holds for the distance Δ _R from the right edge. When the car is in the center of the lane, Δ _L = Δ _R andS _R =S _L , the stimuli are equal. We thus have here a situation analogous to the one studied by H. D. Landahl in his theory of psychophysical discrimination. In general a reactionR _R (resp.R _L ) will occur only ifR _R −R _L ≥h ^* (resp.R _L −R _R ≥h ^*) whereh ^* is a threshold. Applying Landahl’s theory to this situation, we find thath ^* determines the distance from the edge, at which a corrective turn is made. This distance is not constant, but a function of the speedv of the car. The requirement that a corrective turn should be madebeforre the car runs off the road leads to an expression for the maximum safe speed. Because of the transcendency of the equations involved, closed solutions cannot be obtained. It is, however, shown that the expression for maximum safe speed, given in a previous paper (Bull. Math. Biophysics,21, 299–308, 1959), is a rough first approximation to the expressions found now.     

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 biology of learning to drive an automobile

In learning to drive, an individual must learn to rapidly make small corrective turns to the right or to the left as the car comes too close correspondingly to the left or to the right edges of the lane. The magnitude of the corrective turn depends on the angle at which the edge is approached. Thus, the individual must learn to produce a quantitatively correct response (corrective turn) to any one of an infinite number of possible stimuli (angles of approach). By making a number of highly oversimplifying assumptions, the problem can be reduced to a learning situation, studied previously by H. D. Landahl (Bull. Math. Biophysic,3, 13–26, 71–77, 1941). This is used not so much to obtain any relation that might be considered practically applicable immediately as toillustrate what kind of relation can be obtained from such considerations. It is shown how the safe speed of a driver depends on his total driving experience (total distance driven) as well as on his psychophysical parameters.     

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.     

A note on the development of organismic sets

It is suggested that the development of organismic sets is governed not by the maximalization of the integral survival value, as suggested previously (Bull. Math. Biophysics,28, 283–308, 1966;29, 139–152, 1967;30, 163–174, 1968), but by maximizing the number of new relations which appear as an organismic set develops.     

Analysis of tracer experiments: IV. The kinetics of generalN compartment systems

The methods of C. W. Sheppard and A. S. Householder (Jour. App. Physcis,22, 510–20, 1951), H. D. Landahl (Bull. Math. Biophysics,16, 151–54, 1954) and H. E. Hart (Bull. Math. Biophysics,17, 87–94, 1955;ibid.,19, 61–72, 1957;ibid.,20, 281–87, 1958) are employed in studying the kinetics of generalN compartment systems. It is shown that the nature of the transfer processes occurring in fluid flow systems and the chemical processes occurring in quadratic systems and in catalyzed quadratic systems can in principle be completely determined for all polynomial dependencies. Systems involving three-body and higher-order interactions can be completely solved, however, only if supplementary information is available. Research supported by the Atomic Energy Commission, Contract AT (30-1)-1551.     

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 relational biology

The principle of biotopological mapping (Rashevsky, 1954,Bull. Math. Biophysics,16, 317–48) is given a generalized formulation, as the principle of relational epimorphism in biology. The connection between this principle and Robert Rosen’s representation of organisms by means of categories (1958,Bull. Math. Biophysics,20, 317–41) is studied. Rosen’s theory of (M,R)-systems, (1958,Bull. Math. Biophysics,20, 245–60) is generalized by dropping the assumption that only terminalM _i components are sending inputs into theR _i components. It is shown that, if the primordial organism is an (M,R)-system, then the higher organisms, obtained by a construction well discussed previously (1958,Bull. Math. Biophysics,20, 71–93), are also (M,R)-systems. Several theorems about such derived (M,R)-systems are demonstrated. It is shown that Rosen’s concept of an organism as a set of mappings throws light on phenomena of synesthesia and also leads to the conclusion that Gestalt phenomena must occur not only in the fields of visual and auditory perception but in perceptions of any modality.     

Physics,biology and sociology: II. Suggestion for a synthesis

In continuation of previous studies (Bull. Math. Biophysics,28, 283–308; 655–661, 1966;29, 139–152, 1967) it is shown that the difference between the “metric” aspects of physics and the “relational” aspects of biological and social sciences disappear by accepting the broader definition of “relation”, such as that given in mathematics and logic. A conceptual superstructure then becomes possible from which all three branches of knowledge may be derived, though none of them can be derived from the others.     

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.”     

Some considerations on relational equilibria

A previous study (Bull. Math. Biophysics,31, 417–427, 1969) on the definitions of stability of equilibria in organismic sets determined byQ relations is continued. An attempt is made to bring this definition into a form as similar as possible to that used in physical systems determined byF-relations. With examples taken from physics, biology and sociology, it is shown that a definition of equilibria forQ-relational systems similar to the definitions used in physics can be obtained, provided the concept of stable or unstable structures of a system determined byQ-relations is considered in a probabilistic manner. This offers an illustration of “fuzzy categories,” a notion introduced by I. Bąianu and M. Marinescu (Bull. Math. Biophysics,30, 625–635, 1968), in their paper on organismic supercategories, which is designed to provide a mathematical formalism for Rashevsky's theory of Organismic Sets (Bull. Math. Biophysics,29, 389–393, 1967;30, 163–174, 1968;31, 159–198, 1969). A suggestion is made for a method of mapping the abstract discrete space ofQ-relations on a continuum of variables ofF-relations. Problems of polymorphism and metamorphosis, both in biological and social organisms, are discussed in the light of the theory.     

Homeostatic control of thyroxin concentration expressed by a set of linear differential equations

The purpose of this work is to express current concepts on the relationship between the rates of secretion of thyroxin and of thyroid stimulating hormone (TSH) by a set of linear differential equations (two attempts have been made previously in this direction; cf. Roston,Bull. Math. Biophysics,21, 271–282, 1959; Danziger and Elmergreen,Bull. Math. Biophysics,16, 15–21, 1954), and to show that the solutions to these equations fulfill two criteria: that they correctly express the previously observed behavior of thyroxin and TSH, and that they allow certain predictions to be made which are amenable to experimental verification or disproval by currently existing techniques. This mathematical model is necessarily only an approximation of reality.     

On relations between sets: III

The theory of relations between sets, proposed and outlined in previous publications (Bull. Math. Biophysics,23, 233–235, 1961;28, 117–124, 1966;28, 309–313, 1966), is tentatively expanded and generalized with a view to biological applications.     

The inconsistencies of present-day mathematical-physical methods pertaining to current generators in volume conductors used in electrocardiography

The present-day practices of electrocardiography and vectorardiography are based upon the theory that the surface potential differences can be assumed to be due to a single dipole inside the body. It is shown in this paper that a dipole cannot account for all the surface potentials due to realistic current generators, and hence the determination of the current generator from surface potential measurements based upon such a theory will lead to inconsistent representations of the heart for one and the same subject. To demonstrate this point two eccentric dipoles of different strengths and locations representing two muscle fibers are taken to be the current generator in a homogeneous spherical conductor. The exact surface potentials are then expressed by means of the “interior sphere theorem” of the authors. With these expressions the magnitude, direction, and location of the resultant dipole are determined by the method of D. Gabor and C. V. Nelson (J. App. Physics,25, 413–16, 1954). The surface potentials due to this resultant dipole are again exactly expressed by means of the “interior sphere theorem” and compared with those due to the eccentric dipoles assumed. It can be seen that the differences can be considerable. It is suggested that the multipole model of the authors (Bull. Math. Biophysics,20, 203–16, 1958) be used as a more accurate and the only unique representation of the heart. This investigation was supported by the National Heart Institute under a research grant H-2263(c).     

On multipole theory in electrocardiography

This paper continues a comparison of the Taylor series and spherical harmonic forms of multipole representations initiated by Yeh (Bull. Math. Biophysics,24, 197–207, 1962). It is shown that while transformations from Taylor series form into spherical harmonic form is always possible, the inverse cannot be accomplished as suggested by Yeh; corrected transformation equations are given. It is also shown that direct measurement of Taylor coefficients, as outlined in Yeh, Martinek, and de Beaumont (Bull. Math. Biophysics,20, 203–216, 1958), is actually not possible. Accordingly, only the spherical harmonic coefficients can be determined by measurement of surface potentials, as in electrocardiography.     

Further studies of some aspects of a neurobiophysical model of schizophrenia

The neurobiophysical model of schizophrenia discussed previously (Bull. Math. Biophysics,26, 167–185, 1964;27, 21–26, 1965) is generalized further, to include catatonic and stuporous states. It is concluded that the development of schizophrenia will proceed through different stages of catatonic and non-catatonic states, depending on parameters which characterize on one hand the general inhibition of the individual, on the other hand what may be called his “stability.” Suggestions for possible clinical verifications of the conclusions are made.     

Seeds and seedlings of two neotropical montane understory shrubs respond differently to anthropogenic edges and treefall gaps

Edges resulting from forest clear-cutting and treefall gaps can affect plant populations and consequently the distribution of species across landscapes. These two types of disturbance might interact to exacerbate or ameliorate “edge effects”, a rarely tested possibility. We focused on the effects of distance from forest edge (0–10, 30–40, 60–70, and 190–200 m) and habitat within forest fragments (treefall gaps and intact forest) on the early stages of development of Palicourea gibbosa and Faramea affinis, two common shrubs of montane forests in southwest Colombia. Seed germination and seedling growth did not change with distance from forest edge. Within forest fragments, however, seed germination and seedling growth were higher in treefall gaps than in intact forest understory for both species. In contrast, seed predation was influenced by distance from forest edge and in P gibbosa it depended on habitat. Seed predation was highest in the forest interior (190–200 m from forest edge) and in P. gibbosa this was true only in treefall gap habitats. These results suggest that animal mediated processes such as post-dispersal seed predation are more likely than physiological processes to be affected by anthropogenic edges. Our results provide some evidence that treefall gaps may interact with “edge effects”, however, they are inconclusive as to whether they exacerbate or ameliorate them. Received: 31 August 1998 / Accepted: 18 February 1999