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.     

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: 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.     

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.”     

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.     

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 studies on the shape of the tracking curve of an automobile and its dependence on biological parameters of the driver

Following previous studies, differential equations are established which determine the variation of the stimulus towards a corrective turn of the steering wheel and its effect on the excitation of the centers in the brain which results in the production of the corrective turn. The equations are derived under the highly oversimplified assumption that all excitation thresholds are so small that they can be neglected. Under these assumptions it is found that the tracking curve of a car is a sinusoid with negative damping, that is, with an ever increasing amplitude. Driving under these assumptions is imposible since the car will always eventually jump off the road. The possible effects of the threshold as well as stimuli towards corrective turns other than the distance from the edge of the lane are very briefly discussed. In spite of the negative results of the paper, its interest lies in the circumstance that with the complication of the model, we find that driving depends not only on the reaction times as the only “purely biological” parameter, but on three other neurobiophysical constants. In a subsequent paper (Rashevsky, 1967) it is shown how the introduction of one or more purely biological parameters of the driver makes a stable driving regime possible.     

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.     

Evidence for declines in populations of grassland-associated birds in marginal upland areas of Britain

Capsule We report large declines among summer populations between 1968–80 and 2000.

Aim To assess changes in the status of breeding populations of birds in pastoral uplands.

Methods Volunteer observers revisited 13 areas of marginal upland in Britain where Common Birds Census data were collected during 1968–80. This allowed the status of 35 bird species to be examined over about 20 years and to make a comparison between grassland-based and woodland-based species.

Results For 12 species the decline in abundance was significant, particularly among passerines, such as Skylark Alauda arvensis, Wheatear Oenanthe oenanthe, Whinchat Saxicola rubetra, Yellow Wagtail Motacilla flava and Yellowhammer Emberiza citrinella, each of which declined by over 80%. Redshank Tringa totanus, Yellow Wagtail, Dipper Cinclus cinclus, Whinchat, Wheatear, Ring Ouzel Turdus torquatus and Yellowhammer were found on less than half the number of plots on which they were originally recorded. Most declining species were associated with grassland for nesting and foraging, compared to those species that increased (Wood Pigeon Columba palumbus, Pied Wagtail Motacilla alba, Carrion Crow Corvus corone, Jackdaw C. monedula and Goldfinch Carduelis carduelis) that were less specialized in their habitat requirements. Woodland or woodland edge species showed no significant change in status, suggesting that population declines among grassland species were not due to lower observer effort between recording periods.

Conclusion Long-term changes to grassland ecosystems in marginal upland areas of Britain may have influenced the status of bird populations.     

Psychometric function for nectar volume perception of a flower-visiting bat

For foraging pollinators one aspect of floral quality is the volume of nectar available. Thus, nectar-feeding animals should be able to estimate volumes of received nectar. In this study, we determined the psychometric function for nectar volume discrimination of a Neotropical flower-visiting bat Glossophaga soricina. For this, we examined the ability of bats to discriminate between two nectar volumes in a two-alternative forced-choice paradigm. We used a Bayesian inference approach to determine psychometric functions. From the derived psychometric function we assessed the discrimination threshold at a value of 0.69. G. soricina could clearly distinguish between two volumes, when the difference between the two nectar volumes divided by their average exceeded this value. This indicates that bats possess a sense for the perception and discrimination of volumes of nectar that is better developed than in honeybees.     

Analysis from avian visual perspective reveals plumage colour differences among females of capuchino seedeaters (Sporophila)

Females of the closely related capuchino seedeaters are difficult to distinguish from one another based on human visual perception of colouration and morphology. We examined plumage colour differences among females of four species, the tawny‐bellied seedeater Sporophila hypoxantha, the dark‐throated seedeater S. ruficollis, the rufous‐rumped seedeater S. hypochroma, and the chesnut‐bellied seedeater S. cinnamomea. Reflectance values were measured on museum skins, and interspecific differences were analyzed using the Vorobyev‐Osorio avian colour discrimination model. Interspecific distances in the colour space calculated by the model were considerably higher than the threshold for colour discrimination, indicating the presence of colour differences among species that should be detectable by birds. Colour differences between S. hypoxantha and the other three species were the largest. A Discriminant Function Analysis showed that UV‐wavelength was particularly important in species separation. Our results indicate that female plumage of these four species is considerably divergent in colour; these differences being imperceptible to the human eye, thus representing previously unknown morphological evolution in these species.     

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.     

The presence of the enhanced K/Na discrimination trait in diploid Triticum species

Summary A number of accessions of the three species of diploid wheat, Triticum boeoticum, T. monococcum, and T. urartu, were grown in 50 mol m^-3 NaCl+2.5 mol m^-3 CaCl₂. Sodium accumulation in the leaves was low and potassium concentrations remained high. This was not the case in T. durum grown under the same conditions, and indicates the presence in diploid wheats of the enhanced K/Na discrimination character which has previously been found in Aegilops squarrosa and hexaploid wheat. None of the accessions of diploid wheat showed poor K/Na discrimination, which suggests that if the A genome of modern tetraploid wheats was derived from a diploid Triticum species, then the enhanced K/Na discrimination character became altered after the formation of the original allopolyploid. Another possibility is that a diploid wheat that did not have the enhanced K/Na discrimination character was involved in the hybridization event which produced tetraploid wheat, and that this diploid is now extinct or has not yet been discovered.     

Some further comments on the DNA-protein coding problem

A notion of quasi-ergodicity is defined in free monoids, generalizing a notion introduced in a previous paper (Rosen, 1959,Bull. Math. Biophysics,21, 71–95). It is shown that under certain conditions the algebraic properties of quasi-ergodicity are very similar to those derived for the more specialized concept inloc. cit. If it is assumed that the DNA-protein coding processes in nature are of a quasi-ergodic nature, then a condition is specified under which only a finite number of different DNA-protein codes are possible, and an upper bound for the total number of different quasi-ergodic codes is obtained.     

Some possible quantitative aspects of a neurophysiological model of schizophrenias

In continuation of a previous paper (Bull. Math. Biophysics,26, 167–185, 1964) simple equations are derived for the rate of development of schizophrenia as a function of some psychobiological parameters of the individual and of an index which characterizes the frequency of traumatic experiences of the individual. A clue to the understanding of why schizophrenia is more likely to develop at an early adult age is thus provided. Formerly of the Committee on Mathematical Biology, The University of Chicago.     

Probability distributions associated with distinct hits on targets

Some probability distributions connected with distinct hits on targets, using two different firing schemes, are developed. It is assumed that any shot has a probabilityp, not necessarily unity, of hitting the target at which it was aimed. The development uses a well-known expression for the probability that exactlyt ofN possible events occur simultaneously. Some of the formulae developed here include as special cases the probabilities derived separately and by more complicated arguments in papers by N. Rashevsky. (Bull. Math. Biophysics,17, 45–50, 1955) and A. Rapoport (Bull. Math. Biophysics,13, 133–38, 1951).     

On the equilibrium distribution of population in space

Spatial equilibrium distributions of population are derived from the spatial distribution of net rates of reproduction, and from a relationship between migratory flow and gradients of population density and of locational “attractiveness.” Conditions are discussed for which population approaches a uniform spatial density. Under certain conditions a particularly simple statement of the equilibrium conditions is possible in terms of the “potential of population,” a concept introduced by demographers (J. Q. Stewart,Geographical Review,37, 46–85, 1947) to measure the proximity of a point to people. This paper was first written at the Center for Advanced Study in the Behavioral Sciences.     

A note on relations between sets

From the definition of a strong and weakn-ary relation betweenn sets, given in a previous paper (Bulletin of Mathematical Biophysics,27, 477–492), it follows that for a given set ofn sets and givenn-ary relationR between them there can exist only one strong relation, but a large number of weak ones. An expression for the total number of possible weak relations is derived and the notion of the degree of weakness of a relation is introduced and discussed.     

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.