On the fisher and the cubic-polynomial equations for the propagation of species properties

For precise boundary conditions of biological relevance, it is proved that the steadily propagating plane-wave solution to the Fisher equation requires the unique (eigenvalue) velocity of advance 2(Df)^1/2, whereD is the diffusivity of the mutant species andf is the frequency of selection in favor of the mutant. This rigorous result shows that a so-called “wrong equation”, i.e. one which differs from Fisher's by a term that is seemingly inconsequential for certain initial conditions, cannot be employed readily to obtain approximate solutions to Fisher's, for the two equations will often have qualitatively different manifolds of exact solutions. It is noted that the Fisher equation itself may be inappropriate in certain biological contexts owing to the manifest instability of the lowerconcentration uniform equilibrium state (UES). Depicting the persistence of a mutantdeficient spatial pocket, an exact steady-state solution to the Fisher equation is presented. As an alternative and perhaps more faithful model equation for the propagation of certain species properties through a homogeneous population, we consider a reaction-diffusion equation that features a cubic-polynomial rate expression in the species concentration, with two stable UES and one intermediate unstable UES. This equation admits a remarkably simple exact analytical solution to the steadily propagating plane-wave eigenvalue problem. In the latter solution, the sign of the eigenvelocity is such that the wave propagates to yield the “preferred” stable UES (namely, the one further removed from the unstable intermediate UES) at all spatial points ast→∞. The cubic-polynomial equation also admits an exact steady-state solution for a mutant-deficient or mutant-isolated spatial pocket. Finally, the perpetuating growth of a mutant population from an arbitrary localized initial distribution, a mathematical problem analogous to that for ignition in laminar flame theory, is studied by applying differential inequality analysis, and rigorous sufficient conditions for extinction are derived here.     

Physics,biology, and sociology: A reappraisal

To the extent that all biological phenomena are perceivable only through their physical manifestations, it may be justified to assume that all biological phenomena will be eventually represented in terms of physics; perhaps not of present day physics, but of some “extended” form of it. However, even if this should be correct, it must be kept in mind that representing individual biological phenomena in terms of physics is not the same as deducing from known physical laws the necessity of biological phenomena. Drawing an analogy from pure mathematics, it is possible that while every biological phenomenon may be represented in terms of physics, yet biological statements represent a class of “undecidable” statements within the framework of physics. Such a conjecture is reinforced by the history of physics itself and illustrated on several examples. The 19th century physicists tried in vain todeduce electromagnetic phenomena from mechanical ones. A similar situation may exist in regard to biological and social sciences. Quite generally, the possibility of representing a class B phenomena in terms of class A phenomena does not imply that the phenomena of class B can be deduced from those of class A. The consequences of the above on the relation between physics, biology, and sociology are studied. A tentative postulational formulation of basic biological principles are given and some consequences are discussed. It is pointed out that not only can the study of biological phenomena throw light on some physical phenomena, but that the study of social phenomena may be of value for the understanding of the structures and functions of living organisms. The possibility of a sort of “socionics” is indicated.     

Protention and retention in biological systems

This article proposes an abstract mathematical frame for describing some features of cognitive and biological time. We focus here on the so called “extended present” as a result of protentional and retentional activities (memory and anticipation). Memory, as retention, is treated in some physical theories (relaxation phenomena, which will inspire our approach), while protention (or anticipation) seems outside the scope of physics. We then suggest a simple functional representation of biological protention. This allows us to introduce the abstract notion of “biological inertia”.     

Continuous Models for Cell Migration in Tissues and Applications to Cell Sorting via Differential Chemotaxis

Chemotaxis, the guided migration of cells in response to chemical gradients, is vital to a wide variety of biological processes, including patterning of the slime mold Dictyostelium, embryonic morphogenesis, wound healing, and tumor invasion. Continuous models of chemotaxis have been developed to describe many such systems, yet few have considered the movements within a heterogeneous tissue composed of multiple subpopulations. In this paper, a partial differential equation (PDE) model is developed to describe a tissue formed from two distinct chemotactic populations. For a “crowded” (negligible extracellular space) tissue, it is demonstrated that the model reduces to a simpler one-species system while for an “uncrowded” tissue, it captures both movement of the entire tissue (via cells attaching to/migrating within an extracellular substrate) and the within-tissue rearrangements of the separate cellular subpopulations. The model is applied to explore the sorting of a heterogeneous tissue, where it is shown that differential-chemotaxis not only generates classical sorting patterns previously seen via differential-adhesion, but also demonstrates new classes of behavior. These new phenomena include temporal dynamics consisting of a traveling wave composed of spatially sorted subpopulations reminiscent of Dictyostelium slugs.     

Individual-based models for stage structured populations: formulation of “no regression” development equations

Individual-based models describe the growth dynamics of a population by performing numerical simulations of the life histories of its individuals. The life of an individual is determined by the basic processes of development, reproduction and mortality. In this paper the model equations for the development process are stochastic difference equations with discrete time and describe the time evolution of the status of an individual, in terms of a physiological age. We address the formulation of development models, when “regression” effects (defined as negative development) on the status of an individual are forbidden; this is a natural assumption when the physiological age is defined in terms of an abstract non-decreasing indicator measuring the maturity or the percentage of development. Different stochastic models of the development process are presented, and their behaviours are analyzed by varying the stochasticity level, which takes into account the degree of intraspecific variability. Moreover, remarks on the choice of the time step are reported.     

The probabilistic approach to the effects of radiations and variability of sensitivity

The calculation of the size of the “sensitive volume” or “control center” in biological effects of radiations is discussed from the viewpoint of the probabilistic theory of these phenomena based on the concept of random “effective events”. On the bases of that theory, the resistivity of a microorganism to radiation is defined as its “mean life” under a radiation of one roentgen per minute. This mean is calculated for processes with and without recovery. The case of variable sensitivity, as it occurs for instance during mitosis, is discussed in detail. Methods are given to calculate this variability from survival curves or similar experimental data. The theory is applied to experiments of A. Zuppinger on irradiation ofAscaris eggs with X-rays.     

Solution of large compartmental models using numerical transform inversion

Compartmental models of biological or physical systems are often described by a system of “stiff” differential equations. In this paper an algorithm for solving a system with linear coefficients is presented that employs numerical inversion of the Laplace transform of the model equations. The inversion algorithms and Gear's backward differentiation method are compared for two stiff test problems and a differential system governing a 27-compartment model of bile acid transport and metabolism. The inversion algorithm is reliable, requires modest computation time on a desktop computer and provides better accuracy than Gear's method, especially for the extremely stiff example.     

A reconsideration of “Races” and their impact on the origins of the Chalcolithic in the Levant using available anthropological and archaeological data

Populations of the Chalcolithic Levant as defined by archaeological excavations has in many cases reinforced the traditional scheme that a number of “races” are present. This scheme is usually based not only on differential cultural traditions as identified by archeologists, but also on the available skeletal evidence as discussed by physical anthropologists. Recently this view has been challenged and it has been suggested that the metrical and anatomical range of variability as identified within Chalcolithic populations can be subsumed into a single population or “racial” range. This paper examines both the available biological and archaeological evidence from the Chalcolithic Levant and concludes that there is no strong archaeological or biological evidence to support a multiple “racial” origin for the Chalcolithic of the Levant.     

Theory of transport in linear biological systems: II. Multiflux problems

If in a multiflux system theith flux is given by the integral equation, , the corresponding equation in the Laplace transforms is Γ _i = Σ _j W _ij Γ _j +M _i -the entire system having the matrix formulaion, [I−W]Γ=M. The general solution of this equation and its physical interpretation are discussed. Explicit solutions are given for the general mammillary and catenary systems and for some capillary exchange problems. The theory is applied to the integrated from of the fundamental continuity equation to give equations for total quantity of material in the various “compartments.” If the compartments are uniformly mixed, the integral equation treatment is shown to be mathematically equivalent to the usual differential equation formulation.     

A reinterpretation of the mathematical biophysics of the central nervous system in the light of neurophysiological findings

The fundamental equations for the interaction between neurons used in mathematical biophysics seem at first incompatible with the actual neurophysiological findings on the synaptic transmission. It is shown, however, that those equations may be readily interpreted in terms of accepted neurophysiological views. What has been termed “synapse” in mathematical biophysics must be regarded as a complicated network of internuncial neurons. It is shown that, under rather conditions, the number of those interneurons willstatistically vary with time according to the differential equation postulated for the excitatory and inhibitory factors. The latter are thus interpreted as the number of excitatory and inhibitory interneurons.     

A note on the equations of conditioned reflex

An alternative method is suggested for integrating a certain differential equation associated with a conditioning process, where the stimulus is presented in the form of a “square wave,” i.e., is of constant intensity during an interval of time followed by no stimulus during the next interval, etc. A solution is also given where the stimulus is a rectified sine wave.     

Forms of output distribution between two individuals motivated by a satisfaction function

Motivations of two individuals governed by a satisfaction function are assumed to determine their respective “efforts”, which result in the production of “output”, i.e., objects of satisfaction. In previous papers the sharing of output was prescribed in advance. In the present article, however, the sharing formula itself is determined to a certain extent by the satisfaction function. The rate of remuneration per unit of output for each individual is taken to be proportional to the derivative of the satisfaction of the other individual with respect to the effort of the first. The formulation of this condition leads to a partial differential equation whose solutions determine the sharing formula. Sharing determined in this way is referred to as sharing according to the Condition of Mutual Need (C.M.N.). Satisfaction resulting from five different situations are the computed and compared: (1) an individual producing and consuming alone; (2) two individuals sharing equally and neither taking the “initiative” to determine the optimum output; (3) sharing determined by C.M.N. with optimum output determined as in (2); (4) equal sharing but with one individual taking “initiative” in determining optimal output; and (5) sharing determined by C.M.N. and optiml output by the “initiative” of one individual. further considerations concern conditions imposed on the arbitrary function occurring in the solution of the above-mentioned partial differential equation.     

Dimensional analysis in mathematical biology I. General discussion

Dimensional analysis is discussed from the viewpoint of its basic group properties and shown to be an algebraic Abelian group that is useful for analysis of physical measurements. The application of the method to various types of equations and the formulation of previously unclassified dimensions are discussed. Functional dimensional analysis is applied to the problems of cell size and biomass proliferation; future applications are also noted. A number of dimensionless terms have been formulated for cellular physiochemical phenomena. They apparently represent the first systematic study of biological dimensionless numbers recorded in the literature. A dimensionless proliferation law is suggested. A brief analysis of the physical dimensionality associated with information measures is carried out. Entropy and “information” are shown to be completely different in their dimensional meaning; other informational measures of possible interest in biology are proposed. The dimensional coding and computor analysis of biomathematical equations is suggested.     

Weak convergence of a sequence of stochastic difference equations to a stochastic ordinary differential equation

We consider a sequence of discrete parameter stochastic processes defined by solutions to stochastic difference equations. A condition is given that this sequence converges weakly to a continuous parameter process defined by solutions to a stochastic ordinary differential equation. Applying this result, two limit theorems related to population biology are proved. Random parameters in stochastic difference equations are autocorrelated stationary Gaussian processes in the first case. They are jump-type Markov processes in the second case. We discuss a problem of continuous time approximations for discrete time models in random environments.     

Tuned solutions in dynamic neural fields as building blocks for extended EEG models

The most prominent functional property of cortical neurons in sensory areas are their tuned receptive fields which provide specific responses of the neurons to external stimuli. Tuned neural firing indeed reflects the most basic and best worked out level of cognitive representations. Tuning properties can be dynamic on a short time-scale of fractions of a second. Such dynamic effects have been modeled by localised solutions (also called “bumps” or “peaks”) in dynamic neural fields. In the present work we develop an approximation method to reduce the dynamics of localised activation peaks in systems of n coupled nonlinear d-dimensional neural fields with transmission delays to a small set of delay differential equations for the peak amplitudes and widths only. The method considerably simplifies the analysis of peaked solutions as demonstrated for a two-dimensional example model of neural feature selectivity in the brain. The reduced equations describe the effective interaction between pools of local neurons of several (n) classes that participate in shaping the dynamic receptive field responses. To lowest order they resemble neural mass models as they often form the base of EEG-models. Thereby they provide a link between functional small-scale receptive field models and more coarse-grained EEG-models. More specifically, they connect the dynamics in feature-selective cortical microcircuits to the more abstract local elements used in coarse-grained models. However, beside amplitudes the reduced equations also reflect the sharpness of tuning of the activity in a d-dimensional feature space in response to localised stimuli.     

Multiple relational equilibria: Polymorphism,metamorphosis and other possibly similar phenomena

In a preceding paper (Rashevsky, 1969. “Outline of a Unified Approach to Physics, Biology and Sociology.”Bulletin of Mathematical Biophysics,31, 159–198) certain isomorphisms between biological and social systems on the one hand and physical systems on the other were studied. The notion or relational forces, of which ordinary physical forces are a particular case, was introduced. In the present paper an attempt is made to establish analogies between stable equilibria in physical systems, equilibria due to physical forces, and stable equilibria in biological and social systems which are due to purely relational forces. The notion of relational forces causing multiple equilibria similar to multiple equilibria in some physical systems is studied, and it is outlined how this notion may possibly help the understanding of such phenomena as polymorphism, metamorphosis and the existence of rudimentary organs or rudimentary functions.     

Oscillating reactions in open systems

The probable existence of oscillating chemical reactions has been attracting some interest in recent years for their possible role in explaining certain biological phenomena. Perhaps the simplest model of oscillating reactions is that of Lotka (1910), which consists of a chain of autocatalytic reactions. Two “reactor systems” in which such a chain of reactions could take place are considered in this work and are called homogeneous and compartmental models, respectively. The differential equations governing the temporal behavior of the reacting species are solved on an analog computer, and the conditions under which sustained oscillations occur are obtained and discussed. Comparisons of the solution obtained in the two models are discussed.     

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.     

Chain processes and their biophysical applications: Part I. General theory

A mathematical theory applicable to the biological effects of radiations as chain processes is developed. The theory may be interpreted substantially as a “hit theory” involving the concepts of “sensitive volume” or “target area”. The variability of the sensitivity of the organism to the radiation and its capacity of recovery between single hits is taken into account. It is shown that in a continuous irradiation of a biological aggregate in which the effect of each single hit cannot be observed, recovery and variation of sensitivity are formally equivalent to each other so that a discrimination between these two phenomena is possible only by discontinuous irradiation or by using different radiation intensities. Methods for the calculation of the “number of hits” and for the determination of the kinetics of the processes from “survival curves” or similar experimental data are given. The relation between the recovery and the Bunsen-Roscoe law is discussed. The case in which the injury of the organism is dependent on the destruction of more than one “sensitive volume” is also considered.     

Analytical conditions for the conservative form of the ecological equations

The analytical conditions by which a Volterra's general system describingn interacting species can be put in the “conservative” form have been examined. The cases forn=2, 3, 4 have been analyzed in detail and a general condition for any value ofn is deduced. The analytical and biological constraints following by this approach are compared to the conclusions drawn by Leigh on the ground of purely biological considerations.