A suggested chain process for radiation damage

It has been suggested that when an organism is exposed to ionizing radiation the initial damage results from the occurrence of ionization in a so-called sensitive volume due to absorption of radiation quanta. The initial radiation damage is then transmitted or amplified to a level of macroscopic perception. In this paper a mechanism by which this transmission may take place and a finite Markov chain model applicable to this transmission are postulated and discussed. This mechanism is assumed to be the depolymerization of essential chain molecules which are connected to some “central group” associated with the seensitive volume. The depolymerization of the macromolecules following a hit in the sensitive volume is postulated to be determined by a chain mechanism, which acts in a manner inverse to the mechanism controlling the polymerization process. A mathematical study of this problem is made using the theory of Markov chains. The probability of complete degradation of the chain macromolecule, and the probability of recombination of the units to give the intact chain were determined, assuming that the probabilty of successive steps in the degradation increase linearly from the intact state to that of complete breakdown.     

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.     

Energy model for contrast detection: spatiotemporal characteristics of threshold vision

A model for contrast detection of spatiotemporal stimuli is proposed which consists of a spatiotemporal linear filter, an energy device and a threshold device. Assuming the existence of independent intrinsic noise, the probability of stimulus detection was approximated by a Weibull function of the response energy. With this assumption, the stimulus energy is a constant at fixed detection probability. This energy model for contrast detection satisfactorily accounted for the elliptical threshold contours of line pairs at stimulus separations within the range 2–30 min and at stimulus onset asynchronies within the range 20–140 ms. The threshold contour at a large stimulus onset asynchrony (300 ms) was in the form of a rounded square. This finding was explained by assuming that the probability of seeing the line pair was determined by the joint probability that at least one stimulus had been detected. With the energy model, the temporal and spatial autocorrelation functions of the response to a flashed line were evaluated. The autocorrelation functions thus determined were used to predict the temporal contrast sensitivity function to a flickering line stimulus and the spatial contrast sensitivity function to flashed gratings, which were in agreement with the experimental data. The data obtained were fitted adequately by an impulse response approximated by a spatiotemporal Gabor-like function. Received: 08 December 1997 / Accepted in revised form: 26 January 1999     

The probability of obtaining complex kinetic curves for enzyme mechanisms with cubic terms in the pseudo-steady state rate equations

A number of details required for the classification of 3 : 3 double reciprocal plots are provided. It is shown that the ν(S) plot for a 3 : 3 function can have at most four inflexions and at most two inflexions adjacent to a turning point. Using this information, a classification of 3 : 3 ν(S) plots into ten main varieties with several subclasses is reported. The problem of defining the probability with which a given mechanism can give rise to specific curve shape features is considered. Applying this technique, the probability with which four simple enzyme mechanisms can give rise to 3 : 3 curve shapes is computed. It is shown that a 3 : 3 saturation function can have no turning points, at most two inflexions and at most one inflexion in double reciprocal space. The probability with which the available 3 : 3 shapes can arise is also computed. It is concluded that, with realistic values for rate constants, chemically reasonable enzyme mechanisms leading to rate equations of degree n : n can generate most of the kinetic profiles available to a rational function of degree n : n with positive coefficients. The probability of obtaining specific curve shapes is not so characteristic of the particular mechanism for 3:3 rate equations as it is for 2:2 rate equations. The probability of obtaining highly complex curves with several turning points or inflexions is rather lower for the enzyme mechanisms than with general 3 : 3 rational functions. There is a high probability that 3 : 3 mechanisms will generate kinetic curves that are geometrically similar to those possible for degree 2 : 2 but this is not so for binding isotherms. Hence differentiating 3 : 3 from 2 : 2 rate equations from experimental kinetic data is more likely to be successful by non-linear regression to the whole data set than by demonstrating a specific 3 : 3 feature. Binding curves, on the other hand, for three or more sites should give Scatchard plots with inflexions, features not possible with second degree equations which are conic sections in this space.     

Analysis of the response properties of a computationally efficient spike initiator model

 A simple model for neuronal spike initiation is presented. It comprises two linear differential equations and is based on the work of Hill, Rashevsky and Monnier (Rashevsky 1933; Monnier 1934; Hill 1936). Three different versions of the model and the corresponding assumptions are described. The intrinsic noise model used with the deterministic equations is described. The equations are analyzed through the direct solution of relevant equations and the technique of phase plane analysis. The analysis reveals that a subset of the model parameters is responsible for the distinction between spike initiator models which fire a single spike and those that fire repetitively in the presence of a sustained stimulus. Relationships between stimulus intensity and different modes of operation are derived. The effects of the three different versions of the model are compared analytically. Received: 5 August 1993/Accepted in revised form: 24 February 1994     

A contribution to the mathematical biophysics of psychophysical discrimination III

The neural mechanism previously discussed is further generalized. The case is considered in which a random variation is associated with each stimulus. The mechanism is generalized and equations are derived for discriminations between stimuli differing in several modalities. The latter indicates an analysis by the factor method. Suggestions are made in connection with the use of triads and with the problem of a multidimensional psychophysics.     

Complex dynamic behaviour of autonomous microbial food chains

 The dynamic behaviour of food chains under chemostat conditions is studied. The microbial food chain consists of substrate (non-growing resources), bacteria (prey), ciliates (predator) and carnivore (top predator). The governing equations are formulated at the population level. Yet these equations are derived from a dynamic energy budget model formulated at the individual level. The resulting model is an autonomous system of four first-order ordinary differential equations. These food chains resemble those occuring in ecosystems. Then the prey is generally assumed to grow logistically. Therefore the model of these systems is formed by three first-order ordinary differential equations. As with these ecosystems, there is chaotic behaviour of the autonomous microbial food chain under chemostat conditions with biologically relevant parameter values. It appears that the trajectories on the attractors consists of two superimposed oscillatory behaviours, a slow one for predator–top predator and a fast one for the prey–predator on one branch at which the top predator increases slowly. In some regions of the parameter space there are multiple attractors. Received 8 November 1995; received in revised form 7 January 1997     

Outline of a probabilistic approach to animal sociology: III

The probabilities of the emergence of the two kinds of social structure in a 3-bird flock (chain and cycle) are deduced under the assumption of certain biases acting on the social dynamics of the flock. In particular a bias against the reversal of peck order and a bias against encounters of individuals of disparate social rank are considered. Like-wise a distribution of an “inherent” fighting ability is considered which influences the outcomes of encounters. A functional relation is derived between the importance of this ability and the initial probability of a chain structure.     

A comparative investigation of certain difference equations and related differential equations: Implications for model-building

Many mathematical models for physical and biological problems have been and will be built in the form of differential equations or systems of such equations. With the advent of digital computers one has been able to find (approximate) solutions for equations that used to be intractable. Many of the mathematical techniques used in this area amount to replacing the given differential equations by appropriate difference equations, so that extensive research has been done into how to choose appropriate difference equations whose solutions are “good” approximations to the solutions of the given differential equations. The present paper investigates a different, although related problem. For many physical and biological phenomena the “continuum” type of thinking, that is at the basis of any differential equation, is not natural to the phenomenon, but rather constitutes an approximation to a basically discrete situation: in much work of this type the “infinitesimal step lengths” handled in the reasoning which leads up to the differential equation, are not really thought of as infinitesimally small, but as finite; yet, in the last stage of such reasoning, where the differential equation rises from the differentials, these “infinitesimal” step lengths are allowed to go to zero: that is where the above-mentioned approximation comes in. Under this kind of circumstances, it seems more natural tobuild themodel as adiscrete difference equation (recurrence relation) from the start, without going through the painful, doubly approximative process of first, during the modeling stage, finding a differential equation to approximate a basically discrete situation, and then, for numerical computing purposes, approximating that differential equation by a difference scheme. The paper pursues this idea for some simple examples, where the old differential equation, though approximative in principle, had been at least qualitatively successful in describing certain phenomena, and shows that this idea, though plausible and sound in itself, does encounter some difficulties. The reason is that each differential equation, as it is set up in the way familiar to theoretical physicists and biologists, does correspond to a plethora of discrete difference equations, all of which in the limit (as step length→0) yield the same differential equation, but whose solutions, for not too small step length, are often widely different, some of them being quite irregular. The disturbing thing is that all these difference equations seem to adequately represent the same (physical or biological) reasoning as the differential equation in question. So, in order to choose the “right” difference equation, one may need to draw upon more detailed (physical or) biological considerations. All this does not say that one should not prefer discrete models for phenomena that seem to call for them; but only that their pursuit may require additional (physical or) biological refinement and insight. The paper also investigates some mathematical problems related to the fact of many difference equations being associated with one differential equation.     

A mathematical model for prokaryotic protein synthesis

A kinetic model for the synthesis of proteins in prokaryotes is presented and analysed. This model is based on a Markov model for the state of the DNA strand encoding the protein. The states that the DNA strand can occupy are: ready, repressed, or having a mRNA chain of length i in the process of being completed. The case i = 0 corresponds to the RNA polymerase attached, but no nucleotides attached to the chain. The Markov model consists of differential equations for the rates of change of the probabilities. The rate of production of the mRNA molecules is equal to the probability that the chain is assembled to the penultimate nucleotide, times the rate at which that nucleotide is attached. Similarly, the mRNA molecules can also be in different states, including: ready and having an amino acid chain of length j attached. The rate of protein synthesis is the rate at which the chain is completed. A Michaelis-Menten type of analysis is done, assuming that the rate of protein degradation determines the ’slow’ time, and that all the other kinetic rates are ‘fast’. In the self-regulated case, this results in a single ordinary differential equation for the protein concentration.     

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.     

The Information Capacity of Nerve Cells Using a Frequency Code

Approximate equations are derived for the amount of information a nerve cell or group of nerve cells can transmit about a stimulus of a given duration using a frequency code (i.e., assuming the mean frequency of nerve impulses measures the intensity of a maintained stimulus). The equations take into account the variability of successive interspike intervals, and any serial correlations between successive intervals, but do not require detailed assumptions about the mechanism of impulse initiation. The errors involved in using these approximations are evaluated for neurons which discharge either completely regularly, completely at random (Poisson process) or show a particular type of intermediate variability (gamma distribution model). The errors become negligibly small as the stimulus duration or the number of functionally similar nerve cells increases. The conditions for applying these equations to experimental data are discussed. The application of these equations should help considerably in eliminating the enormous discrepancies between some earlier estimates for the information processing capabilities of single nerve cells and systems of nerve cells.     

Contribution to the probabilistic theory of neural nets: IV. Various models for inhibition

Input-output formulas are derived for a neuron upon which converge single axones of two other neurons, which are subjected to a Poisson shower, where a number of different assumptions are made concerning the mechanism of inhibition. In one assumption so-called “bilateral pre-inhibition” is considered. That is to say, both neuronsN ₁ andN ₂ may exciteN ₃, but if the stimulus of one of them follows within a certain interval σ of the other, the second stimulus is not effective. This model is essentially no different from that involving two excitatory neurons acting upon a neuron having a refractory period. Another mechanism considered involves so-called “pre-and-post” inhibition, in which if two stimuli fromN ₁ andN ₂ fall within σ,both are ineffective. This case being mathematically much more involved than the preceding, an approximation method is used for deriving the input-output formula. Previous papers of this series are denoted by I, II, and III in this paper.     

Steady states in random nets: I

A neural net is taken to consist of a semi-infinite chain of neurons with connections distributed according to a certain probability frequency of the lengths of the axones. If an input of excitation is “fed” into the net from an outside source, the statistical properties of the net determine a certain steady state output. The general functional relation between the input and the output is derived as an integral equation. For a certain type of probability distribution of connections, this equation is reducible to a differential equation. The latter can be solved by elementary methods for the output in terms of the input in general and for the input in terms of the output in special cases.     

Contributions to the theory of imitative behavior

The imitation effects in a social group depend both on the size of the group and on the distribution of a certain psychobiological quantity ϕ which measures the tendency of an individual towards a given behavior. The distribution function of ϕ determines the ratio μ of the individuals in the society who adopt a given behavior. When the size of the social group is not too large, the actual distribution of ϕ will deviate from the most probable one, and therefore communities of the same size and having the same parameters may have different values of μ. Approximate equations are developed which give the probability of a given μ for a group of a given size. Possible effects of interactions of communities of different sizes are briefly discussed. A generalization of the theory of imitative behavior to any number of mutually exclusive behaviors is given, and its possible sociological implications are discussed.     

Killing Babies: Hrdy on the Evolution of Infanticide

Sarah Hrdy argues that women (1) possess a reproductive behavioral strategy including infanticide, (2) that this strategy is an adaptation and (3) arose as a response to stresses mothers faced with the agrarian revolution. I argue that while psychopathological and cultural evolutionary accounts for Hrdy's data fail, her suggested psychological architecture for the strategy suggests that the behavior she describes is really only the consequence of the operation of practical reasoning mechanism(s) – and consequently there is no reproductive strategy including infanticide as such, nor could the alleged strategy be sufficiently mosaic to count as an adaptation. What might count as an adaptation is a ‘window’ before bonding that permits practical reasoning about the reproductive value of infants and hence variable maternal investment, and which, contra (3) arose early in hominid history due to a combination of increases in infant dependency and increased human abilities for conditional practical reasoning.     

Mathematical biophysics of the galvanic skin response

Beginning with Rashevsky's equation for the development of the excitatory state in a nerve fiber, an equation for the change in skin resistance upon the presentation of an instantaneous stimulus is derived. The mechanism assumed is in conformity with the existing evidence of neuro-physiology. Certain deductions from the equations are made and experimental problems suggested for testing the theory.     

Contribution to the mathematical theory of contagion and spread of information: I. Spread through a thoroughly mixed population

An equation is derived from the spread of a “state” by contact through a thoroughly mixed population, in which the probability of transmission depends both on the over-all duration of the process and on the time an individual has been in the “state.” Cases in which this probability is a function of only one or the other of the two “times” are worked out. It is shown that in the case of dependence on “private time” alone the asymptotic value of the fraction of the population effected is the same as that derived by the random net approach.     

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.     

A neurobiophysical interpretation of certain aspects of the problem of risks

A situation is considered in which an individual is given an opportunity to risk a certain amount of money or goods in order to gain a larger amount, providing the result of some uncertain event proves favorable. A neural mechanism is introduced in which the probability of success can be ordered according to its value. The response to a situation thus becomes dependent upon the probability. An expression is then derived for the amount that would be risked in terms of the probability and of the amount that would be gained in the event of a successful outcome. Similar expressions are obtained for the case of insurance against loss. Results of questionnaires indicate that individuals can be classified according to their pattern of behavior in the situation considered. The various types can be most easily recognized when a plot is made of the relative amount risked against the probability of success. In a general way, these types can be understood in terms of the equations derived.