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.     

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.     

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.     

Non-linear excitation theory: Non-accommodative,sub-threshold effects

The standard two factor excitation theories should be called “preexcitation” theories since they apply only to those events occurring just up to excitation. A true phenomenological excitation theory which describes thewhole excitation cycle must involve non-linear equations. The nature of these non-linearities is suggested by B. Katz's subthreshold response data. From this data is constructed a “local phenomenological characteristic” which is analogous to the current-voltage characteristic of a non-linear electrical or mechanical system capable of displaying relaxation oscillations. Excitation by constant currents is shown to occur where the slope of the characteristic changes sign. The variation of the time constant of excitation with degree of response, explained by W. A. H. Rushton in terms of a liminal length, is described here in purely formal terms. The theory as presented explicity treats only those events in the excitation cycle up to and a little beyond excitation; the complete excitation cycle (including recovery and repetition) is mentioned as being amenable to mathematical treatment by an extension of the present theory.     

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 suggestion for a new approach to the mathematical theory of imitative behavior

The theory of imitative behavior as developed hitherto by the author was based on the assumption that each individual has a natural preference for one of the two mutually exclusive behaviors. The endogenous fluctuations in the central nervous system then result in the individual’s exhibiting the two behaviors alternately with a relative frequency determined by the natural preference. Imitation shifts the natural preference towards one or the other of the two mutually exclusive behaviors. In the present approach it is suggested that the relative frequency of the two mutually exclusive behaviors exhibited alternately is determined by maximizing the “satisfaction function” of the individual, that is by hedonistic factors rather than by purely random fluctuations. Corresponding equations are developed. It is shown that in certain cases, even when the imitation effect is absent, a sort of “pseudoimitation” may occur. Another situation leads, in the case of two individuals only, to a complete “division of labor” between them, with respect to the two behaviors. Each one exhibits only one behavior. After that imitation is introduced explicitly by assuming that imitation by one individual or another increases the satisfaction function of the imitating individual. Results thus obtained show similarities to the results of the old theory.     

Response time and threshold of a random net

The response time of a random net is defined as the expected time (measured in the number of synaptic delays) required for the excitation in the net (measured by the fraction of neurons firing per unit time) to reach a certain level. The response time is calculated in terms of the net parameters as a function of the intensity of the outside stimulation. Two principal types of cases are studied, 1) an instantaneous initial stimulation, and 2) continuously applied stimulation. It is shown that for a certain type of net where the required level of excitation is small, the response time-intensity equation reduces to the one derived on the basis of the “one-factor” theory applied to a neural connection. More general assumptions, however, give different types of equations. The concept of the “net threshold” is defined, and its calculation indicated. The net threshold for instantaneous stimulation is, in general, greater than that for continuous stimulation. The results are discussed with reference to existing theories of reaction times.     

Towards a molecular theory of the nerve membrane

Ion transport through the nerve membrane is considered in terms of barrier-limited fluxes calculated from absolute reaction rate theory. Equations are developed to describe the conformational transitions of an enzyme embedded in the membrane to provide a low-energy transport site. The enzyme transitions are controlled by binding and hydrolytic release of an acetylcholine-like molecule, which in turn depends on ion association with a single negative charge on the enzyme. Simulation of the equations gives good agreement with typical experimental voltage clamps and action potentials. A steady-state negative resistance is found in isoosmolar potassium, and the model shows excitation by an acetylcholine pulse under conditions mimicking the postsynaptic membrane. The implications of the model for development of a molecular theory of the nerve membrane are considered.     

Dilation-producing stresses and strains obtained with the aid of density measurements

Stress systems in a deformed medium are in general of two kinds: (i) dilation-producing and (ii) nondilation-producing. Ordinarily the dilation is computed from a knowledge of the stresses, which in turn are obtained by solving the general elastostatic equations of equilibrium, under specified boundary conditions. If, however, interest is centered only in dilation-producing stresses, or in the event that only such stresses exist in the deformed system, and the density changes in the medium may be experimentally determined or postulateda priori as functions of the coordinates, these stresses may be directly obtained by a simplified procedure of solving for a particular integral of the Poisson equation without appeal to boundary conditions.^** In irrotational systems in which conservative body forces exist, there is a direct relationship between the dilation and the distributed body forces. Reference to both purely physical and biophysical systems is made. This investigation was supported by a PHS research grant, RG-5067, from the Public Health Service.     

The dependence of calcium-activated potassium currents on membrane potential.

Previous experiments on cholinergic synapses in chick cochlear hair cells have shown that calcium entering through acetylcholine-activated synaptic channels in turn activates calcium-dependent potassium currents, resulting in synaptic inhibition. In voltage-clamp experiments such currents would be expected to increase with depolarization (as the driving force for potassium entry is increased) and then decrease towards zero as the membrane approaches the calcium equilibrium potential (when calcium entry is suppressed). In the hair cells, however, such currents approached zero at about +20 mV, more than 170 mV negative to the calcium equilibrium potential. Another feature of the synapse is its post-junctional morphology: a uniform 20 nm cleft is formed between the postsynaptic membrane and the outermost membrane of an underlying cisterna. Here we present a model in which synaptic activation results in calcium influx into the subsynaptic cleft and thence into the bulk of the cytoplasm. The model suggests that the voltage dependence of the calcium-activated potassium current can be accounted for by only two basic assumptions: (i) entry of calcium through the activated synaptic channels by simple diffusion; and (ii) activation of the potassium channels by the cooperative action of four calcium ions. In addition, the model suggests that during activation the calcium concentration in the restricted subsynaptic space can reach levels adequate to activate the potassium channels, without requiring additional, more complicated, considerations (for example, secondary calcium release from the cisterna).     

Biophysics of nerve excitation

Experimental studies testifying to the presence of an interrelation between the physiological functions of the organism and physical and chemical processes in nerves are discussed. Changes in some physical and chemical parameters observed both upon elicited rhythmic excitation of nerves and during the spontaneous rhythmic activity of neurons are analyzed. Upon rhythmic excitation, a complex of physical and chemical processes is triggered, and reversible structural and metabolic rearrangements at the subcellular and molecular levels occur that do not take place during the generation of a single action potential. Thus, only in conditions of rhythmic excitation of a nerve, it is possible to reveal those processes that provide excitation of nerves in the organism. The future possibilities of the investigations combining the biophysical and physiological approaches are substantiated. Characteristic changes in physicochemical parameters are observed in nerves during the generation of a series of action potentials of different frequency and duration (“frequency dependence”) under normal physiological conditions, as well as in extreme situations and in nerve pathology. The structural and metabolic rearrangements are directly related to the mode of rhythmic excitation and proceed both in the course of rhythmic excitation and after its termination. Shown also is participation of the basic components of the nervous trunk (axon, Schwann cell, myelin, subcellular organelles) in the realization of rhythmic excitation. In the coordination of all processes involved in rhythmic excitation, the main role is played by the systems of redistribution and transport of intercellular and intracellular calcium. The idea is put forward that myelin of nerve fibers is not only an insulator, but also an “intercellular depot” of calcium and participates in the redistribution of different ions. Thus, the rhythmic excitation is of great importance in the realization of some physiological functions, the adaptation to changing conditions, the liquidation of consequences of paralogical processes, the formation of mechanisms of “memory,” etc.     

Revision of the theory of tracer transport and the convolution model of dynamic contrast enhanced magnetic resonance imaging

Counterexamples are used to motivate the revision of the established theory of tracer transport. Then dynamic contrast enhanced magnetic resonance imaging in particular is conceptualized in terms of a fully distributed convection–diffusion model from which a widely used convolution model is derived using, alternatively, compartmental discretizations or semigroup theory. On this basis, applications and limitations of the convolution model are identified. For instance, it is proved that perfusion and tissue exchange states cannot be identified on the basis of a single convolution equation alone. Yet under certain assumptions, particularly that flux is purely convective at the boundary of a tissue region, physiological parameters such as mean transit time, effective volume fraction, and volumetric flow rate per unit tissue volume can be deduced from the kernel.      

Methadone maintenance treatment and drugs

The mental and physical capabilities of drivers in traffic are often seriously challenged these days. Not only do they need to concentrate on driving, predict connections between various phenomena, take appropriate judgements in current situations and foresee the sequence of measures to be taken, but they are also expected to be emotionally stable, etc. The problem with drugs in traffic is often encountered when assessing the actual safe driving capability of a person in a given moment, for example after a car accident or a police check, or medical check-ups that are required for a driving license. The Road Traffic Safety Law considers methadone a drug. Drug addicts do not meet the health standards required of drivers. This research program deals with the attitude of drivers who are in methadone maintenance treatment programs with respect to the driving ability as well as the effects of methadone use in combination with other drugs on driving. It has been established that drivers undergoing the methadone maintenance program, regularly drive not only under the influence of methadone but also under the influence of marijuana (20%) and heroin (18%) and sometimes under the influence of marijuana (58.6%), heroin (55.7%), and alcohol (48.6%). Certain initiatives have been taken by some therapists to give, under certain circumstances, a clean bill of health to responsible methadone maintenance patients who have an adequate level of responsibility for themselves and their deeds, in order to help them obtain a driving license. Since it has been established that methadone maintenance patients use methadone quite commonly in combination with illegal drugs and/or alcohol, the classification of this type of addicts among possible driving candidates remains disputable. Long term interdisciplinary research is still required to determine the basic principles required to asses and possibly admit this type of drivers to participate in traffic, as well as to determine which professional therapists can participate and evaluate the driving capabilities of these patients.     

A logical calculus of the ideas immanent in nervous activity

Because of the “all-or-none” character of nervous activity, neural events and the relations among them can be treated by means of propositional logic. It is found that the behavior of every net can be described in these terms, with the addition of more complicated logical means for nets containing circles; and that for any logical expression satisfying certain conditions, one can find a net behaving in the fashion it describes. It is shown that many particular choices among possible neurophysiological assumptions are equivalent, in the sense that for every net behaving under one assumption, there exists another net which behaves under the other and gives the same results, although perhaps not in the same time. Various applications of the calculus are discussed.     

A fundamental form for the differential equation of colonial and organism growth

When total cell number is used as the basic parameter of growth, rational equations which describe colonial and organism growth under varying circumstances have been derived from a single differential form. These equations result from making specific, but reasonable assumptions about two additive factors, ϕ and θ which determine community growth. The first factor (ϕ) is assumed to arise from conditions within the growing cell itself, while the second factor (θ) arises from interactions between the growing cells of the community. If it is further assumed that the cells of a community are homogeneous with respect to density and volume, it has been shown that the mathematical expressions commonly used to describe growth data may be rationally derived from the general form. Clerical assistance in the preparation of these materials was furnished by the personnel of Work Projects Administration, Official Project No. 65-1-08-62, Unit A-8.     

On the theory of diffusion of electrolytes

In the theory of diffusion of electrolytes the following assumptions are frequently made: (i) the electrolytic solution is electrically neutral everywhere, (ii) the ionic concentrations and the electric potential all depend on a single Cartesian coordinate as the only space variable. Often the electric potential of the solution is determined on the basis of the Poisson equation alone, disregarding any other relation between this potential and the ionic concentrations. Since the Poisson equation only represents a condition which the potential fulfills, the use of this equation alone may lead to error unless the explicit relation for the potential involving a space integration of ionic concentrations is also taken into account. But if this relation is used the Poisson equation becomes redundant and, more important, assumptions (i) and (ii) appear unacceptable, the former because it leads to a zero electric potential everywhere, the latter because it is mathematically incorrect. The present paper is based on general equations of diffusion of ions, excluding the Poisson equation. These equations form a system of nonlinear integrodifferential quations whose number equals the number of ionic species present in the solution. It appears that when all ions are distributed symmetrically around a point all functions related to the above system of equations can be made dependent on a single space coordinate: the distance from the center of symmetry. Two methods of successive approximations are given for the solution of the equations in the case of spherical symmetry with limitation to the steady state. These methods are then applied to the study of the distribution of ionic concentrations and electrical potentials inside a cell of spherical shape in equilibrium with its surroundings. These methods are rapidly convergent; exact theoretical values of the electric potential are calculable on the boundary of the cell. It appears that the potential at the center of the cell is not more than ∼50% higher than at its boundary and that variation of concentration inside the cell is not very large. For instance, with 100 mV on the boundary the ionic concentration there is about four times higher than at the center. Calculations show that extremely small amounts of electricity are sufficient to account for the electric potentials currently observed. In a cell of 100 micra diameter an average concentration of only 10⁻¹⁴ mole/cm³ of a monovalent ion would be sufficient to give 1 millivolt on the boundary. This concentration is directly proportional to the voltage and inversely proportional to the square of the cell diameter. Most of the numerical results given above are obtained by considering only those ions whose electrical charge is not compensated for by ions of an opposite sign. The total concentrations may be much higher than those quoted. The theory does not take into account possible effects of structural heterogeneities which may exist in the cell, particularly of various phase boundaries. An incidental result shows that the Boltzmann distribution function in the form employed in modern theory of electrolytes is fundamentally a consequence of the mathematical theory of diffusion alone. It is pointed out, however, that Boltzmann distribution is not always compatible with the definition of the electric potential.     

Characterizing abundance–occupancy relationships: there is no artefact

Positive abundance–occupancy relationships (AORs) are among the most general macroecological patterns: locally common species are regionally widespread, locally rare species are regionally restricted. In a recent contribution, Wilson (Global Ecology and Biogeography, 2011, 20 , 193–202) made three claims: (1) that AORs are critically dependent on the method used to calculate average abundance; (2) averaging abundance over occupied sites tends to lead to a very high incidence of negative relationships; (3) this represents a statistical artefact that should be considered in studies of AORs. Here we show that this outcome arises in Wilson's simulations purely due to an arbitrary choice of occupancy models and parameter ranges. The resulting negative relationships are not statistical artefacts, but are easily interpreted in terms of spatial aggregation in abundant species. The fact that empirical evidence fails to support a high prevalence of negative AORs suggests, however, that such parameter combinations arise only rarely in nature. We conclude that simulations that are based on untested assumptions, and that produce patterns unsupported by empirical evidence, have limited use in characterizing AORs, and add little to understanding of the processes driving important relationships between local population size and regional occupancy.     

Metastable spiking networks in the replica-mean-field limit

Characterizing metastable neural dynamics in finite-size spiking networks remains a daunting challenge. We propose to address this challenge in the recently introduced replica-mean-field (RMF) limit. In this limit, networks are made of infinitely many replicas of the finite network of interest, but with randomized interactions across replicas. Such randomization renders certain excitatory networks fully tractable at the cost of neglecting activity correlations, but with explicit dependence on the finite size of the neural constituents. However, metastable dynamics typically unfold in networks with mixed inhibition and excitation. Here, we extend the RMF computational framework to point-process-based neural network models with exponential stochastic intensities, allowing for mixed excitation and inhibition. Within this setting, we show that metastable finite-size networks admit multistable RMF limits, which are fully characterized by stationary firing rates. Technically, these stationary rates are determined as the solutions of a set of delayed differential equations under certain regularity conditions that any physical solutions shall satisfy. We solve this original problem by combining the resolvent formalism and singular-perturbation theory. Importantly, we find that these rates specify probabilistic pseudo-equilibria which accurately capture the neural variability observed in the original finite-size network. We also discuss the emergence of metastability as a stochastic bifurcation, which can be interpreted as a static phase transition in the RMF limits. In turn, we expect to leverage the static picture of RMF limits to infer purely dynamical features of metastable finite-size networks, such as the transition rates between pseudo-equilibria.