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: 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.     

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.     

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 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.     

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.     

Cycling around a Curve: The Effect of Cycling Speed on Steering and Gaze Behavior

Although it is generally accepted that visual information guides steering, it is still unclear whether a curvature matching strategy or a ‘look where you are going’ strategy is used while steering through a curved road. The current experiment investigated to what extent the existing models for curve driving also apply to cycling around a curve, and tested the influence of cycling speed on steering and gaze behavior. Twenty-five participants were asked to cycle through a semicircular lane three consecutive times at three different speeds while staying in the center of the lane. The observed steering behavior suggests that an anticipatory steering strategy was used at curve entrance and a compensatory strategy was used to steer through the actual bend of the curve. A shift of gaze from the center to the inside edge of the lane indicates that at low cycling speed, the ‘look where you are going’ strategy was preferred, while at higher cycling speeds participants seemed to prefer the curvature matching strategy. Authors suggest that visual information from both steering strategies contributes to the steering system and can be used in a flexible way. Based on a familiarization effect, it can be assumed that steering is not only guided by vision but that a short-term learning component should also be taken into account.     

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.     

On the fundamental nature of perception

The process of recognition or isolation of one or several entities from among many possible entities is termed intellego perception. It is shown that not only are many of our everyday percepts of this type, but perception of microscopic events using the methods of quantum mechanics are also intellego in nature. Information theory seems to be a natural language in which to express perceptual activity of this type. It is argued that the biological organism quantifies its sensations using an information theoretical measure. This, in turn, sets the stage for a mathematical theory of sensory perception.     

Adaptive designs with arbitrary dependence structure

Adaptive designs were originally developed for independent and uniformly distributed p‐values. There are trial settings where independence is not satisfied or where it may not be possible to check whether it is satisfied. In these cases, the test statistics and p‐values of each stage may be dependent. Since the probability of a type I error for a fixed adaptive design depends on the true dependence structure between the p‐values of the stages, control of the type I error rate might be endangered if the dependence structure is not taken into account adequately. In this paper, we address the problem of controlling the type I error rate in two‐stage adaptive designs if any dependence structure between the test statistics of the stages is admitted (worst case scenario). For this purpose, we pursue a copula approach to adaptive designs. For two‐stage adaptive designs without futility stop, we derive the probability of a type I error in the worst case, that is for the most adverse dependence structure between the p‐values of the stages. Explicit analytical considerations are performed for the class of inverse normal designs. A comparison with the significance level for independent and uniformly distributed p‐values is performed. For inverse normal designs without futility stop and equally weighted stages, it turns out that correcting for the worst case is too conservative as compared to a simple Bonferroni design.     

Growth rates and phylogeny in primates

There is sufficient evidence to indicate that man grows for a longer period of time than chimpanzee who in turn has a greater duration of growth than rhesus monkey. The problem of this paper was to determine if there was a concomitant decrease in a rate of growth. Using the relative growth rate of Fisher ('21), it appears that for most of their period of growth, the rate of change is the same. Immediately after birth, when we have no data for children, rhesus monkey grows significantly faster than chimpanzee. By a year and a half their rates are the same, and neither species shows a sex difference. From seven years (age of youngest children) until children start through puberty, there is no sex difference in Homo sapiens, and the human rate does not differ from the chimpanzee rate. Because of the resemblance between the primate curve for rate and that for dairy cattle, it is postulated that this curve is more mammalian than primate and that during phylogeny the primates have merely increased the duration of time when growth is possible. Man does show one new feature, the puberal growth spurt, which is not found in the non-human data considered. The implications of these conclusions for primate phylogeny and for growth are discussed.     

Animal movement tools (amt): R package for managing tracking data and conducting habitat selection analyses

Advances in tracking technology have led to an exponential increase in animal location data, greatly enhancing our ability to address interesting questions in movement ecology, but also presenting new challenges related to data management and analysis. Step‐selection functions (SSFs) are commonly used to link environmental covariates to animal location data collected at fine temporal resolution. SSFs are estimated by comparing observed steps connecting successive animal locations to random steps, using a likelihood equivalent of a Cox proportional hazards model. By using common statistical distributions to model step length and turn angle distributions, and including habitat‐ and movement‐related covariates (functions of distances between points, angular deviations), it is possible to make inference regarding habitat selection and movement processes or to control one process while investigating the other. The fitted model can also be used to estimate utilization distributions and mechanistic home ranges. Here, we present the R package amt (animal movement tools) that allows users to fit SSFs to data and to simulate space use of animals from fitted models. The amt package also provides tools for managing telemetry data. Using fisher (Pekania pennanti) data as a case study, we illustrate a four‐step approach to the analysis of animal movement data, consisting of data management, exploratory data analysis, fitting of models, and simulating from fitted models.     

Endoscopically assisted, intraorally approached corrective rhinoplasty.

In the field of facial surgery, operations that require guesswork can result in unexpected complications. One example of such "blind" facial surgery is the lateral osteotomy procedure in corrective rhinoplasty. In most conventional corrective rhinoplasties, the postoperative results of a lateral osteotomy can be controlled by the surgeon's visual perception or manual dexterity; therefore, an experienced surgeon is indispensable in such elaborate operations. Until now, reports have focused on the endoscopic approach through the nasal dorsum or septum through the nostril. However, because of the difficulty in handling the endoscope with osteotomy instruments, it is considered difficult to perform a precise lateral osteotomy procedure using that approach. The authors think the intraoral endoscopic approach should be considered a viable alternative in corrective rhinoplasty.Through small, bilateral gingivobuccal incisions, both the piriform apertures and nasal bones can be easily exposed, and the exact level of the lateral osteotomy can be confirmed directly under the endoscope. The lateral osteotomy is made simply with a reciprocating saw, and symmetrical cutting can be ascertained during the operation. Sometimes, a particular osteotomy level or the proper repositioning of osteotomed segments can be readily evaluated with assistance from the endoscope during the operation. Eleven cases using this procedure were performed over the past 3 years. These endoscopic repairs for a deviated nose were quite helpful for visual confirmation and accurate correction. No complications occurred when using the endoscope with this procedure.     

An evaluation of the rate constants for the adsorption of the defective phage PBS Z to Bacillus subtilis

The adsorption of the defective phage PBS Z of Bacillus subtilis has previously been assumed to proceed in two steps, a reversible adsorption of extended phages followed by contraction of the adsorbed particles (Steensma, 1981a). This model, also used for other phages, explained the biphasic character of the adsorption curve, but a discrepancy was found between the calculated and observed concentrations of adsorbed, extended phages. Computer simulations indicated that this might be caused by inhomogeneity of the phage preparations with respect to their adsorption properties and that in that case other models would also fit the experimental data. Discrimination between the models was not possible on the basis of the available information on PBS Z and it was therefore concluded that the values reported previously for the rate constants (Steensma, 1981a) should be used with caution.     

Identifying alert concentrations using a model-based bootstrap approach

The determination of alert concentrations, where a pre-specified threshold of the response variable is exceeded, is an important goal of concentration–response studies. The traditional approach is based on investigating the measured concentrations and attaining statistical significance of the alert concentration by using a multiple t-test procedure. In this paper, we propose a new model-based method to identify alert concentrations, based on fitting a concentration–response curve and constructing a simultaneous confidence band for the difference of the response of a concentration compared to the control. In order to obtain these confidence bands, we use a bootstrap approach which can be applied to any functional form of the concentration–response curve. This particularly offers the possibility to investigate also those situations where the concentration–response relationship is not monotone and, moreover, to detect alerts at concentrations which were not measured during the study, providing a highly flexible framework for the problem at hand.     

Chlorophyll-fluorescence measurements in bryophytes: evidence for three main types of light-curve response

Diffusion theory predicts that, except in the lower part of the daylight range, carbon dioxide supply will always be limiting for photosynthesis in a unistratose leaf. We have used chlorophyll fluorometry to survey the photosynthetic responses of numerous bryophytes to a range of light intensities employing the ‘light curve’ approach. Initially, as light intensity is increased in a stepwise manner, electron transport rate (ETR) in bryophytes follows a saturation curve closely fitted by a negative exponential function, y?=?A(1?–?e^–kx ), where y?=?ETR, x?=?light intensity (or photosynthetic photon flux density), A is the asymptote (ETR at infinitely high light intensity), k is a rate constant and e is the base of natural logarithms. The initial slope of the response curve, Ak, approximates maximum quantum efficiency (F_v/F_m) which is measured on dark-adapted plant material. However, at higher intensities ETR frequently veers away from the saturation curve owing to the onset of either photoinhibition or the dissipation of the excitation energy by a photoprotective mechanism, probably involving reduction of O₂. In the latter case, the measurement of ETR significantly overestimates the rate of photosynthetic carbon fixation. We describe a simple approach that enables these instances of photoprotection and photoinhibition to be identified and discuss the wider significance of the results to the ecology of individual species.     

Tropical tree species diversity: a test of the Janzen-Connell model

To test the premises and predictions of the Janzen-Connell model (Janzen's spacing mechanism), seeds of the rainforest canopy tree, Brosimum alicastrum, were placed at different distances from the parent tree and their removal observed over 3 weeks. The number and density of naturally occurring seeds at different distances from the parent tree were also estimated. Predation was not greater near the parent tree, except on the very small spatial scale: the proportion of experimental seeds removed was greater 1 m from the trunk than it was 5–25 m from the trunk. Predation was negatively correlated with seed density, not positively as the Janzen-Connell model assumes-presumably due to predator satiation. The density of seeds after predation peaked 5 m from the tree trunk, but this is well within the crown radius of the parent tree. There is a peak in the number of potential recruits at a distance of 10 m from the parent tree, due to the peaked initial distribution of seeds. This peak is caused by the interaction between the seed density curve and the increasing area of an annulus around the parent tree at increasing distances, not by the product of the density curve and the predation curve. However, it is important to realize that it is not the presence of a peak in recruitment away from the parent that is essential to maintaining tropical tree species diversity, but frequency-dependent recruitment induced by poor recruitment near conspecifics. Predator satiation seems to be an important factor in the survival of B. alicastrum seeds, possibly at several spatial scales. The number of seeds produced by the tree is negatively correlated with the loss to predators, and trees that have a fruiting conspecific nearby also suffer lower levels of predation. Seed predation increases as one moves from the forest edge into the interior, creating an edge effect that may have long-term effects on the forest composition and tree species diversity. More studies are needed, for other species, other localities, and larger spatial and temporal scales, on both the Janzen-Connell mechanism and this edge effect.     

Orientation of male Heterodera rostochiensis Woll. and H. schachtii Schm. to their females

A chemical attractive to their males is emitted by females of Heterodera rostochiensis and H. schachtii. The attraction is evident in agar plates when the females have been killed or removed from the agar, and on 3 mm. thick agar it takes more than 15 min. at 20° C. to diffuse sufficiently to be perceptible to males 5 mm. from the females. The perception is gustatory and seems to elicit klinokinetic aggregation behaviour and klinotaxic orientation. The males respond to a decrease in stimulation by stopping, followed by either a reversal and turn, or a swing and turn. Fatigue of the receptors, leading to loss of acuity of perception, occurs at greater concentrations of the attractant and decreases the frequency of turning. Then the intensity of turning increases and the approach to the female is less well orientated. Females of H. schachtii secreted a less labile or more concentrated attractant; thus their males were more frequently less well orientated and removal of the females caused more direct attraction. Male behaviour was efficient enough to ensure mating even of isolated females.     

Simulation of the conditions of wave excitation for the optimization of the lower hybrid current drive in the Globus-M spherical tokamak

At present, the method of current drive by means of lower hybrid waves is not applied to low-aspect-ratio tokamaks, because, in the traditional approach, it would be necessary to use waves with a very high slowing-down factor. However, studies of new transparency regions for waves in a nonuniform magnetized plasma, performed earlier at the Ioffe Physical Technical Institute, Russian Academy of Sciences, made it possible to develop an approach in which slow waves are excited in the poloidal (rather than toroidal) direction. In this approach, moderately slowed-down waves first propagate in the poloidal direction, but then turn in the toroidalal direction and get into the dense plasma. In this work, this approach is further developed using numerical methods. In particular, the influence of the density profile in the edge plasma on the efficiency of wave excitation for given antenna parameters is studied in detail.     

Investigation of Driving Forces for Charge Extraction in Organic Solar Cells: Transient Photocurrent Measurements on Solar Cells Showing S‐Shaped Current–Voltage Characteristics

The role of drift and diffusion as driving forces for charge carrier extraction in flat heterojunction organic solar cells is examined at the example of devices showing intentional S‐shaped current–voltage (J‐V) characteristics. Since these kinks are related to energy barriers causing a redistribution of the electric field and charge carrier density gradients, they are suitable for studying the limits of charge extraction. The dynamics of this redistribution process are experimentally monitored via transient photocurrents, where the current response on square pulses of light is measured in the μs to ms regime. In combination with drift‐diffusion simulation data, we demonstrate a pile‐up of charge carriers at extraction barriers and a high contribution of diffusion to photocurrent in the case of injection barriers. Both types of barrier lead to S‐kinks in the J‐V curve and can be distinguished from each other and from other reasons for S‐kinks (e.g. imbalanced mobilities) by applying the presented approach. Furthermore, it is also helpful to investigate the driving forces for charge extraction in devices without S‐shaped J‐V curve close to open circuit to evaluate whether their electrodes are optimized.