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 biological of social behavior: II

A group of individuals is considered in which each individual has tendencies to exhibit one or another of two mutually exclusive behaviors. Neurobiophysically this may be described in terms of Landahl's reciprocally inhibited parallel reaction chains. The spontaneous excitations ε₁ and ε₂ at the central connections of each chain are a measure of the “natural” tendency of the individual toward one or the other of the two behaviors. According to equations derived by H. D. Landahl, the probability of one or the other behavior is determined by the difference ε₁ − ε₂. A population of individuals is considered in which ε₁ − ε₂ is distributed in some continuous way, and therefore in which the probability of a given behavior is distributed continuously between 0 and 1. The effect of other individuals exhibiting a given behavior is to increase the corresponding ε of the individual. Thus behavior of others affects the probability for a given behavior of each individual. It is shown that the equations describing the behavior of the population on the basis of this neurobiophysical picture reduce in the first approximation to the differential equations which were postulated by the author in his previous work on social behavior.     

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.     

An exact method for the computation of the connectivity of random nets

The problem of finding the “weak connectivity” of a random net is reduced to one involving a Markov process. This provides a mathematically exact treatment of the problem which had previously been treated by an approximation, whose justification was not rigorous. The exact method allows in principle not only the calculation of the “weak connectivity”, but also of the “strong connectivity”, and, in general, the probability that from a randomly selected neuron in the net there exist paths to a specified number of neurons. The computations become exceedingly involved for large nets.     

In vitro separation of a rose chimera

“Fairmount 1 thorny” (“FM1 thorny”) (a Rosa multiflora Thunb ex. J. Murr.) and a thornless sport of “FM1 thorny” (“Fairmount 1” (“FM1”)) were established in vitro to investigate chimeral segregation under various levels of BA and to obtain a pure thornless rose. While the chimeral thornless sport was expected to segregate in vitro and yield both thorny and thornless plantlets, “FM1 thorny” was to yield only thorny plants. “FM1” segregated in vitro into its constituent genotypes and yielded thorny and thornless plantlets, suggesting that “FM1” is chimeral. “FM1 thorny” produced only thorny plants in vitro. These results indicate that the “FM1 thorny” clone was not chimeral (pure thorny) and that the thornless regenerates of “FM1” did not develop via somaclonal variation. There was a significant linear relationship between increasing BA concentration and the percentage of thorny plants. Among a population of 690 tissue culture derived plants from all the BA experiments, 6 plants were classified as pure thornless plants 1 year later.     

A Resource-based Model of Microbial Quiescence

To analyze the ecological features of microbial quiescence, a model is proposed that involves “wake-up” rate and “sleep” rate at which the population transitions from a quiescent to an active state and back, respectively. These rates depend continuously on the resources and turn on and off at resource thresholds which may not coincide. The usual dichotomy is observed: the population is washed out under environmental stress and a single “survival” steady state exists otherwise. Proportional nutrient enrichment is used to explore analytically as well as numerically the nature of the steady state which bifurcates from the washout state.     

Outline of a probabilistic approach to animal sociology: II

Under certain assumptions concerning the probabilities of “mutations,” i.e. changes of structure of bird societies, it is shown that the probability distribution for all possible structures of a society ofN individuals approaches a limit independent of the initial probability distribution. A formula for the limiting distribution is derived.     

Mathematical biophysics of abstraction and logical thinking

Developing further a recent suggestion of H. D. Landahl, equations are derived which give the probability for an individual to judge two stimulus patterns as similar or dissimilar. The possibility of experimental verification of those equations is discussed. Next, a mechanism is described which provided for abstraction by responding only to “essential” components of a stimulus pattern. Equations, verifiable in principle experimentally, are derived. Finally, a mechanism is suggested for logical inferences, and equations are derived which give the probability of making an error in a reasoning consisting of a chain of syllogisms, as well as the probability of being unable to complete the chain of reasoning at all.     

Age changes of the carpal bones in macaques: Application of the fourier analysis on the radiographs

A method based on the Fourier analysis is proposed, which describes and analyzes the contour morphology of carpal bones by separating morphology into factors of shape and size. Here, “size” refers to the average diameter of the contour. The “shape” is expressed byshape factors which are derived from the Fourier series and the “shape” of a monkey is expressed by ashape index which is calculated fromshape factors. The age change in the morphology of the lunate and capitate ofMacaca fuscata fuscata was analyzed by this method. The development of “shape” approximately completes by 3 years of age, whereas increase in “size” begins its spurt at that age as do body weight and anterior trunk length. By applying this method to other macaque species, it was found thatM. mulatta, M. f. yakui andM. cyclopis exhibit similar patterns of growth and development of carpal bones to those ofM. f. fuscata. Patterns found inM. fascicularis differ in that its bones develop faster than in the other macaques with respect to the “shape,” but remain small with respect to the “size.”     

Input output curves of aggregates of “simple counter” neurons

The refractory periods of an aggregate of simple “counter” neurons are assumed distributed according to some probability frequency. The output of the aggregate is computed for rectangular and triangular distributions. In particular, it is shown that the maximum output of an aggregate with any triangular distribution cannot exceed the maximum output of its average neuron by a factor greater than 2 ln 2. This puts an upper bound on the amount of departure from the behavior of the average neuron which an aggregate characterized by a certain type of distribution can show. Next, the aggregate is supposed to be subjected to regularly spaced stimuli. Under these conditions, a single neuron will give a discontinuous output curve. If, however, the refractory periods are distributed according to some frequency, the output curve may be “smoothed out.” A general condition on the distribution is derived which makes the output monotone increasing with the input. The condition is applied to some special cases.     

Atmospheric culturable bacteria associated with meteorological conditions at a summer-time site in the mid-Willamette Valley,Oregon

A set of simultaneously collected quantitative measurements of 12 meteorological and 6 culturable atmospheric bacterial (CAB) variables was made over a grass seed field during several early, mid, and late summer days. The observation site was located between the Cascade and Coastal Mountain Ranges near Corvallis in the Willamette Valley, Oregon. Principal component analysis identified those meteorological variables that had the highest loadings in six eigenvectors that account for 95.9% of variation in the data factors, i.e., temperature @ 6.3 m above ground level (AGL), wind velocity @ 10.0 m AGL, wind velocity difference @ 1.7–10.0 m, barometric pressure, wind direction standard deviation, and wind direction. When these variables were used in a cluster analysis, they formed three statistically distinct meteorological variable clusters with means at ca. “midnight”, ca. “midday”, and ca. “evening.” The highest mean density of CAB (i.e., 153.4 ± 162.5 CFU/m³) was associated with the “midday” meteorological Cluster-1 that had warm, dry “gentle breezes” from the southeast, in the relatively bacteria loaded Willamette Valley air. The lowest mean density of CAB (i.e., 35.5 ± 24.1 CFU/m³) was associated with meteorological Cluster-3 in the late afternoon and “evening” occurring during the hottest and driest part of the day with “fresh breezes” coming from the north northwest in air off the Pacific Ocean. Finally, the last cluster, Cluster-2 occurred about midnight and had an intermediate density of CAB (74.2 ± 76.2 CFU/m³) in “light air” coming from the northwest, perhaps off the Pacific Ocean. The CAB associated with each of the three meteorological clusters was only partially statistically distinct. Partially because the CAB in both the Pacific Ocean derived air masses of the “evening” Cluster-3 and “midnight” Cluster-2 were not statically separable, though both were statistically separable from the midday, Willamette Valley derived Cluster-1. Further indicating their common source, both Pacific Ocean derived air masses had very similar percentages of pigmented bacteria, which were dissimilar to the pigmented bacterial population density in the Willamette Valley air masses. In short, it is speculated that “midnight” atmosphere may simply contain the settling concentrated residual bacterial particles in the abated fresh Pacific Ocean breezes after sundown. It is clearly shown that with the methods employed, it is possible to associate the uniqueness of the quantity, and to a lesser extent the quality, of the CAB population with the atmospheric conditions reported herein. From this project comes speculation that the strategies relating the quasi-conservative bacterial populations associated with distinct but nonconservative air mass properties can help to better understand more of the bacterial dynamics found in such situations. And to a further extent, molecular biological methods could be applied to identify bacterial taxa in specific air masses.     

Contribution to the probabilistic theory of neural nets: I. Randomization of refractory periods and of stimulus intervals

Aggregates of neurons are considered in which the frequency of occurrence of neurons with a specified value of the refractory period follows certain probability distributions. Input-output functions are derived for such aggregates. In particular, if input and output intensities are defined in terms of stimulus frequencies and firing frequencies per neuron respectively, it is shown that a rectangular distribution of refractory periods leads to a logarithmic input-output curve. If input and output are defined in terms of the total number of stimuli and firings in the aggregate, it is shown how the “mobilization” picture leads to the logarithmic input-output curve. By randomizing the intervals between stimuli received by a single neuron and by introducing an inhibitory neuron a very simple “filter net” can be constructed whose output will be sensitive to a particular range of the input, and this range can be made arbitrarily small.     

A statistical approach to the theory of the central nervous system

A “probabilistic” rather than a “deterministic” approach to the theory of neural nets is developed. Neural nets are characterized by certain parameters which give the probability distributions of different kinds of synaptic connections throughout the net. Given a “state” of the net (i.e., the distribution of firing neurons) at a given moment, an equation for the state at the next moment of quantized time is deduced. Certain very special cases involving constant distributions are solved. A necessary condition for a steady state is deduced in terms of an integral equation, in general non-linear.     

Biosemiosis and the Cellular Basis of Mind

The human brain is a complex organ made up of neurons and several other cell types, and whose role is processing information for use in elicitation of behaviors. To accomplish this, the brain requires large amounts of energy, and this energy is obtained by the oxidation of glucose (Glc). However, the question of how the oxidation of Glc by individual neurons in brain results in their collective ability to rapidly generate feats of cognition that allow them to recognize the nature of the universe in which they live and to communicate this information remains unclear. In this article, insights into this process are provided by first considering the brain’ s homeostatic “operating system” for supply of energy to stimulated neurons, and how this system defines the basic unit of brain “structure”. This is followed by consideration of the brain’s “two-cell” neuronal communication mechanism which defines the basic unit of brain “function”. Finally, an analysis of the nature of frequency-encoded “neuronal languages” that enable ensembles of neurons to translate energy derived from the oxidation of Glc into a collective “mind”, the aggregate of all brain processes including those involving perception, thought, insight, foresight, imagination and behavior.     

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.     

Spread of information through a population with socio-structural bias: I. Assumption of transitivity

A previously derived iteration formula for a random net was applied to some data on the spread of information through a population. It was found that if the axon density (the only free parameter in the formula) is determined by the first pair of experimental values, the predicted spread is much more rapid than the observed one. If the successive values of the “apparent axon density” are calculated from the successive experimental values, it is noticed that this quantity at first suffers a sharp drop from an initial high value to its lowest value and then gradually “recovers”. An attempt is made to account for this behavior of the apparent axon density in terms of the “assumption of transitivity”, based on a certain socio-structural bias, namely, that the likely contacts of two individuals who themselves have been in contact are expected to be strongly overlapping. The assumption of transitivity leads to a drop in the apparent axon density from an arbitrary initial value to the vicinity of unity (if the actual axon density is not too small). However, the “recovery” is not accounted for, and thus the predicted spread turns out to beslower than the observed.     

On the spread of information with time and distance

The transmission of some information or behavior pattern is treated as a flow of “particles” which execute random motions over a population of individuals and which may multiply or disappear. Equations are derived for the number density of these “particles” and from this is calculated the number of individuals through which the “particles” have passed. The results are applied to a number of situations such as 1) uniform spatial distribution with multiplication factor decreasing with time because of loss of interest or confusion of the information, 2) multiplication factor constant but the rate of spreal decreasing with multiple hearings, 3) one-dimensional region with a small starting region with or without an absorbing barrier 4) two-dimensional region with absorbing barrier, 5) continous sources of information within a small region in one dimension, 6) uniform spatial distribution in which individuals do not respond to more than one hearing.     

Subpopulation of nestin positive glial precursor cells occur in primary adult human brain cultures

Glial fibrillary acidic protein (GFAP) is an intermediate filament protein considered to be the best astroglial marker. However, the predominant cell population in adult human brain tissue cultures does not express GFAP; these cells have been termed “glia-like” cells. The basic question about histological origin of adult human brain cultures remains unanswered. Some authors showed that “glia-like” cells in adult human brain cultures might be of non-glial origin. We examined primary explant tissue cultures derived from 70 adult human brain biopsies. Within first 5–10 days approximately 5–10% of the small explants became attached. Outgrowing cells were mostly flat cells. These cells formed confluent layer over 3–6 weeks in culture. At confluence the cultures contained 2–5% of microglial cells, 0.1% GFAP-positive astrocytes, less than 0.01% oligodendrocytes and 95–98% GFAP-negative “glia-like” cells. This population of flat “glia-like” cells was positively stained for vimentin, fibronectin, and 20–30% of these cells stained for nestin. Our findings revealed that 1 mM dibutyryl-cAMP addition, in serum free conditions, induced a reversible stellation in 5-10% of the flat “glia-like” cells but did not induce the expression of GFAP or nestin in morphologically changed stellate cells. These results demonstrate that “glia-like” cells in primary adult human brain cultures constitute heterogeneous cell populations albeit with similar morphological features. Two distinct subpopulations have been shown: (i) the one immunostained for nestin; and (ii) the other reactive for dibutyryl-cAMP treatment.     

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.     

Advanced backcross-QTL analysis in spring barley (H. vulgare ssp. spontaneum) comparing a REML versus a Bayesian model in multi-environmental field trials

A common difficulty in mapping quantitative trait loci (QTLs) is that QTL effects may show environment specificity and thus differ across environments. Furthermore, quantitative traits are likely to be influenced by multiple QTLs or genes having different effect sizes. There is currently a need for efficient mapping strategies to account for both multiple QTLs and marker-by-environment interactions. Thus, the objective of our study was to develop a Bayesian multi-locus multi-environmental method of QTL analysis. This strategy is compared to (1) Bayesian multi-locus mapping, where each environment is analysed separately, (2) Restricted Maximum Likelihood (REML) single-locus method using a mixed hierarchical model, and (3) REML forward selection applying a mixed hierarchical model. For this study, we used data on multi-environmental field trials of 301 BC₂DH lines derived from a cross between the spring barley elite cultivar Scarlett and the wild donor ISR42-8 from Israel. The lines were genotyped by 98 SSR markers and measured for the agronomic traits “ears per m2,” “days until heading,” “plant height,” “thousand grain weight,” and “grain yield”. Additionally, a simulation study was performed to verify the QTL results obtained in the spring barley population. In general, the results of Bayesian QTL mapping are in accordance with REML methods. In this study, Bayesian multi-locus multi-environmental analysis is a valuable method that is particularly suitable if lines are cultivated in multi-environmental field trials. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.