ALBERT

Notes: Abstract In the theory of organismic sets (Bull. Math. Biophysics,31, 159–198, 1969) we considered organisms as sets endowed with certain “activities,” the latter’s resulting in a set of “products.” Those products may be of a material nature, like a hormone secreted by a cell, or of a non-material nature, like a feeling or an attitude. In the present paper aggressiveness and submissiveness are considered as such non-material products of the activities of the brain cells. A general description of aggressiveness and submissiveness is given in terms of organismic sets. Cycles in “peck order” are thus naturally explained.

Notes: Abstract In line with previous studies on organismic sets, the division of all organismic sets intogeneral autotrophic and heterotrophic is introduced. The first produce their food themselves from some external source of energy, which in general may be an energy of any kind. The others use other organismic sets as the source of their food and energy. On earth we know only one kind of generalgeneral autotrophic organismic sets, namely, the autotrophic plants which use solar radiation as their source of energy and for production of their own food. It is shown why autotrophic animals do not exist on earth except as microorganisms like, e.g.,Euglena. A rigorous proof of the previously derived theorem that in an organismic set of ordern〉1 no element can be completely specialized is given. It requires the introduction of new postulates. Finally, in considering the organic world as a whole, the notion of organismic sets ofmixed order is introduced.

Notes: Abstract It appears to be axiomatic that termolecular and higher order reactions occur relatively rarely. The basis for this judgment seems to lie in the supposition that successful 3-Body collisions of 3 interactive species of molecules cannot occur frequently enought to account for chemical or biochemical transformation. In order to provide a more complete mathematical framework than now exists for examining this hypothesis the probability of effective termolecular “δ-collisions” as a function of time is derived. This amounts to adding to the class of reactions for which stochastic models are now available the termolecular reaction. In common with the unimolecular and bimolecular cases this process is seen to satisfy the criterion of consistency-in-the-mean with respect to deterministic formulations. It is planned next to use the termolecular process and the lower order processes in computer-assistedin numero experimental studies aimed at comparing alternative mechanisms of reaction.

Notes: Abstract Small sample properties of the maximum likelihood estimator for the rate constant of a stochastic first order reaction are investigated. The approximate bias and variance of the maximum likelihood estimator are derived and tabulated. If observations of the system are made at timesiτ,i=1, 2, ...,N; τ〉0, the observational spacing τ which minimizes the approximate variance of the maximum likelihood estimator is found. The non-applicability of large sample theory to confidence interval derivation is demonstrated by examination of the relative likelihood. Bartlett’s method is employed to derive approximate confidence limits, and is illustrated by using simulated kinetic runs.

Notes: Abstract The partial differential equation of the random walk problem with persistence of direction and external bias is derived. By persistence of direction or internal bias we mean that the probability a particle will travel in a given direction need not be the same for all directions, but depends solely upon the particle's previous direction of motion. The external bias arises from an anisotropy of the medium or an external force on the particle. The problem is treated by considering that the net displacement of a particle arises from two factors, namely, that neither the probability of the particle traveling in any direction after turning nor the distance the particle travels in a given direction need be the same for all directions. A modified Fokker-Planck equation is first obtained using the assumptions that the particles have a distribution of travel times and speeds and that the average time of travel between turns need not be zero. The fional equation incopporating the assumption of a persistence of direction and an external bias is then derived. Applications to the study of diffusion and to long-chain polymers are then made.

Notes: Abstract 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.

Notes: Abstract A situation is considered in which a fluid containing a substance flows through a vessel at a constant rate, the substance being permeable to the vessel wall. In the region outside the vessel there is supposed to be rapid mixing in the direction perpendicular to the axis of the vessel but no mixing longitudinally. The solution for the spatial distribution at any time is given for the case of an arbitrary initial distribution along the vessel length in the absence of an input. The solution is also given for the case of a single impulsive input, the concentration being initially zero everywhere.

Notes: Abstract The representation of biological systems by means of organismic supercategories, developed in previous papers (Bull. Math. Biophysics,30, 625–636;31, 59–71;32, 539–561), is further discussed. The different approaches to relational biology, developed by Rashevsky, Rosen and by Băianu and Marinescu, are compared with Qualitative Dynamics of Systems which was initiated by Henri Poincaré (1881). On the basis of this comparison some concrete result concerning dynamics of genetic system, development, fertilization, regeneration, analogies, and oncogenesis are derived.

Notes: Abstract Some years ago (Rosen 1958a, b; 1959) we described a class of metaphorical, relational paradigms for cellular activity which we termed (M, R)-systems. A sizable amount of subsequent work, to be itemized below, has been devoted to an exploration of some of the properties of these systems. The main purpose of the present paper is to put this class of paradigms into a general system-theoretic perspective, with a particular view to appraising the relation between the type of system description embodied in the (M, R)-system and other kinds of physical and mathematical descriptions of cellular systems. Thus, the principal aim is to establish the relationships and connections between the global relational formalism embodied in the (M, R)-systems and the empirical descriptions which still represent the bulk of our biological knowledge.

Notes: Abstract After giving a brief review of the theory of organismic sets (Bull. Math. Biophysics,29, 139–152, 1967;31, 159–198, 1969), in which the concept of relational forces, introduced earlier (Bull. Math. Biophysics,28, 283–308, 1966a) plays a fundamental role, the author discusses examples of possible different structures produced by relational forces. For biological organisms the different structures found theoretically are in general agreement with observation. For societies, which are also organismic sets as discussed in the above references, the structures can be described only in an abstract space, the nature of which is discussed. Different isomorphisms between anatomical structures, as described in ordinary Euclidean space, and the sociological structures described in an abstract space are noted, as should be expected from the theory of organismic sets.

Notes: Abstract Organisms orient themselves to a stimulus by two general methods. One method is by directed orientation (taxis); the other is by undirected locomotory reaction (kinesis). An equation, and the methods for finding the necessary parameters of this equation, is derived for the distribution of organisms within a container, with the following limitations: (1) the organisms have no accommodation, (2) they are always active, and (3) the stimulus changes slowly with position. Necessary modifications of the equation are then derived, so that the last two limitations may be eliminated. The equation cannot be solved excatly because of its complexity; hence an approximation method must be used. This method is discussed, an approximate solution is found, and a time constant for equilibrium to be established is derived. Applications tovarious experiments in the literature are then made with fairly satisfactory results. A new interpretation of the theory of klino-kinesis with accommodation is found upon application of the equations developed to experimental work. Further limitations and uses of these equations are then discussed.

Notes: Abstract Certain parameters are defined which roughly characterize the internal structure of networks. A given network structure uniquely determines the values of the parameters, but the reverse is not true. The parameters therefore define certain classes of networks. One of the parameters, thedispersion D(S) gives an indication of the “compactness” of the internal structure. Addition theorems and inequalities are derived relating the dispersions of sub-systems to the dispersion of the complete structure.

Notes: Abstract A mathematical theory is developed which permits the determination of certain parameters of an inhomogenous tissue, such as a nerve trunk without its epineurium. The parameters are the permeability coefficients for entrance into an exit of a substance from the nerve fibers, and the diffusion coefficient of the interstitial material. The experimental data required are the dimensions of the cross-section, the average diameter of the fibers, and the ratio of the cross-sectional are of the fibers to the total cross-section, as well as the time course of the decrease of the fraction of the substance left in the nerve trunk, when the trunk is immersed in a bathing solution containing none of it.

Notes: Abstract A model based on enzyme localization is developed which gives rise to an apparent active transport of a metabolite into or out of cells. The model is applied to three simple situations, using Fick's equation and the Rashevsky approximation. It is shown that the apparent efficiency can be made as large as desired if, for constant reaction, the outer cell region is made sufficiently small, or, for autocatalytic reaction, if the metabolite concentration in the outer region is sufficiently small. The physical limitations imposed by this mechanism are developed for all three situations.

Notes: Abstract 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.

Notes: Abstract The assumption of transitivity treated in part I is modified in various ways to describe an information-diffusion process, in which a certain amount of randomness of contact does occur. In one model a parameter is introduced which is indicative of a tendency to go beyond one's immediate vicinity to spread the information as the vicinity becomes saturated with knowers. In another model the randomness appears in the assumption that new knowers are uniformly distributed among the knowers. Two of the equations thus derived, each with two free parameters are in good agreement with experimental results.

Notes: Abstract Current psychological research into the inference (diagnostic) process is briefly reviewed, using as a vehicle an investigation of the prediction of the probability of success of hypothetical applicants to a graduate program in biology. Brunswik’s lens model and multiple regression analysis are used, as is a Bayesian approach. Four judges’ (biologists’) predictions are analyzed. Some general conclusions about inference, drawn from the current data in psychology, are presented.

Notes: Abstract A mathematical model has been developed to simulate the glucose-insulin interaction following a glucose load such as occurs in an IVGTT. This model differs from earlier models in that the insulin response to glucose loading is a recurring all or none threshold response. The model has been simulated on a digital computer using the digital analog simulation language CSMP.

Notes: Abstract The composite nature of bone dictates the use of a model for bone which is transversely isotropic. We solve the associated sets of partial differential equations governing the dynamic elastic behavoor of a two-layered cylindrical-shaped bone. The solution is analyzed for long, short, and intermediate length waves. The special case of compact bone is treated for long and short wave lengths and a numerical example is worked out to determine the wave speeds (for short wave lengths) given a set of elastic constants, determined by ultrasonic methods, and the bone density, wave frequency, and radius.

Notes: Abstract A stochastic model is developed for a compartment with a single time-dependent input, and generalized to include inputs from several sources. With the number of particles of a given molecular species in the compartment as the random variable, the mean, variance and third central moment of this variable are calculated from its generating function, and compared with previous results. The behavior of the calculated moments is discussed, and the possibility of applying the model to chemical and biological systems is considered.

Notes: Abstract It is shown that from the definition of organismic sets (Rashevsky,Organismic Sets. Some Reflections on the Nature of Life and Society, Holland, Michigan, Mathematical Biology, Inc. and Grosse Pointe, Michigan, J. M. Richards Laboratory) a complete sensory deprivation of an organismic set of ordern=2 should result in malfunctioning of the set. A generalization to higher order sets is suggested.

Notes: Abstract Optimality in branching structure of the vascular tree was studied. Analysis on its physiological roles as the duct system for blood supply to the capillaries predicted that the vascular tree should be constructed with minimum volume under restriction of determinant pressure, flow and location at the origin and the terminals. Mathematical derivations of this conditional extremum problem yielded some equations expressing the relations between the radii of the branches and their branching angles, which provided numerical solutions for branching points of bi- and poli-terminal minimum volume trees. Comparison of the peritoneal vascular tree in a dog with the minimum volume one computed under the same restrictive conditions showed good agreement in their branching structure.

Notes: Abstract In order to determine the kinetics of passage of a substance through an organ containing a tangle of vessels, we study the response of a tube to various inflows (perfusion, brief injection, ...). The introduction of the catabolic terms and of the spatial dependence between bulk concentration and surface concentration allows one to account for the difference of arteriovenous concentrations observed experimentally for many metabolites. The relationships between the physico-chemical parameters of the organ and the operational parameters of the model demonstrate the importance of the transit time through the considered vessels. If one considers the different pathways as independent, the introduction of the transit time distribution for an inert substance enables one to compute the response of the organ analytically or by recurrence, using convolution. The parameters of the model can be obtained by the moments method.

Notes: Abstract The solution of the diffusion equation in the gas phase of the human lung is very difficult because of the structure of the bronchial tree. It is shown by means of physical arguments, how one can reduce the diffusion equation to a simple one-dimensional form. The solution is then obtained by a stochastic simulation, which is easily realized on a digital computer.

Notes: Abstract A new mathematical model of the oscillatory behavior of the respiratory center has been developed based upon published records of neuronal activity during respiration in the pons and medulla. In contrast with a previous model, four, rather than two, networks are assumed to interact in the respiratory center so as to produce the respiratory oscillation. A mathematical description of this interaction, in the form of a set of four first-order, nonlinear, coupled differential equations, is derived; the behavior of the solutions of this system is studied qualitatively, and expressions for the durations of the inspiratory and expiratory phases are obtained in terms of some parameters. It is found that central and chemical influences drive the medullar neurons to a position somewhere between saturation and full cutoff, and the pontine neurons deeply into cutoff. The control of the duration of the different phases by these chemical and central means is discussed. In order to effect a decrease in the magnitude of the various times, the neurons have to be driven towards operating points of higher central facilitation.

Notes: Abstract In vivo control of calcium is analysed under the assumption that hormonal influences via plasma levels of parathormone and calcitonin are of prime (but not absolutely dominating) importance. A brief review concerning the physiological significance of body calcium and the mode of action of these two hormones is presented as an introduction to the basic philosophy of the study. A theoretical quasi-linear lumped-parameter model is developed to describe variations in ionic calcium, parathormone and calcitonin plasma concentrations to specific input stimuli. Formal evaluation of the system response requires the determination of ten constants, together with quantitation of ingested calcium entry into the plasma compartment which isindependent of hormonal influences. Values for various parameters are deduced from published data and experimental procedures are outlined to facilitate determination of the remaining unknowns. It is suggested that the proposed model should prove useful for investigations concerning general hormonal actions on calcium homeostatic mechanisms in both normal and diseased states, with particular reference to calcitonin.

Notes: Abstract Ann species predator-prey chain is analyzed to determine what oscillations occur in population sizes. It is found that only the populations of the first and second species in the chain must necessarily oscillate around the point of equilibrium if they do not come to equilibrium. The other species may or may not oscillate.

Notes: Abstract Sangren and Sheppard developed a mathematical model for first-order processes taking place in the regional circulation, applicable—for example—to tracer studies of potassium transport. It permits calculation of specific activity at any point along a “tube of flow” or in the cuff of tissue surrounding it as a function of time following a spike injection of tracer. In efforts to relate to the exchange a rate curves obtained within vivo counters pointed at the region of interest, we developed a compartment-system model of the process. In investigating the properties of the Sangren and Sheppard model integrated over an entire circulatory bed, as thein vivo counter would see it, we found that when the distribution of transit times of the “tubes of flow” can be approximated by an exponential sum, the solution reduces to that of the compartment system model. This results in an important simplification in the calculation, and insight into the assumptions underlying the two different models. A curve-fitting computer program for the compartment model has been written and applied to double-isotope studies of potassium transport in the hind leg of the dog.

Notes: Abstract Two chemicals,A andB, are allowed to diffuse together and a reaction described by $$A + B\mathop \rightleftharpoons \limits_{K_{ - 1} }^{K_1 } C$$ is allowed to proceed. This system is described mathematically by a system of partial differential equations. A numerical procedure is presented to find the rate constants ofK 1 andK −1. A systematic analysis of the effects of errors is also presented.

Notes: Abstract Equations are developed to describe the energy expenditure of the human heart. As well as the external potential and kinetic energy terms, general consideration is given to other possible avenues of energy consumption. Emphasis is placed upon using mathematical variables which are readily available for experimental verification. The errors involved in assuming that mean values for the physiological parameters give reasonable estimations for the external mechanical performance are examined, and a theoretical estimation for the discrepancy in the kinetic component is presented. Logical extension of the mathematical derivation leads to a determination of cardiac external mechanical efficiency and clearly demonstrates the significance of the ventricular pressure-volume loop in this context. Finally, experimental procedures are suggested to clarify further some of the conclusions reached through the theoretical analysis.

Notes: Abstract The question is discussed as to the reason why some animal societies, such as bees or ants, are sexually differentiated, that is, onlysome of its members are exhibiting reproducing activities. It is indicated that human society may be on its way to such a sexual differentiation which may eventually come.

Notes: Abstract X-ray diffraction patterns obtained experimentally for fibers, together with their chemical structures, can be analyzed theoretically in terms of an integral equation. The partially unknown electron density function can be solved by iteration. This mathematical technique has been applied with success to study the secondary structures of DNA fibers.

Notes: Abstract For an environmental system described by a system of nonlinear first-order differential equations, the problem of achieving specified terminal conditions in a given time with a minimum expenditure of resources is considered. The initial conditions and the minimum value are found numerically in a particular example.

Notes: Abstract A kinetic model of neural systems is introduced and discussed with statistical mechanics techniques. It is assumed that, for a macroscopic description of the model, it suffices to consider only the distribution for the velocity and position of the impulses, and the distribution for the excitation and position of the neurons, at any timet. Making use of Boltzmann's method for the study of a dilute gas, coupled differential equations for the rate of change with time of the distributions have been constructed.

Notes: Abstract Creeping flow of a Newtonian fluid through a rigid permeable tube is considered and the transmural seepage is assumed to obey Darcy's law. Closed-form solutions for the pressure and velocity fields are presented and equations describing the axial variation of the mean cross-sectional pressure, the axial volumetric flow and the transmural fluid flux are derived. Approximate solutions for small seepage rates are given and are applied to the flow in the proximal renal tubule. Probable values for the epithelium permeability and the intraluminal hydrostatic pressure drop are obtained.

Notes: Abstract The paper demonstrates that it is possible to construct memory models where the information inserted is stored in disseminated form, using sequential coding, the changes in the units forming the models being determined by their geometrical connections and by the incoming stream of information. The models are shown to have large storage capacity and their efficiency can be made insensitive to loss of or damage to a large fraction of their units. The satisfactory verification by computer simulation of the analysis and results described in the present paper will be the subject of a future paper.

Notes: Abstract We examine in detail Edward Kerner’s method for linearizing the equations of enzyme kinetics. Our main result is the determination of canonical forms for systems which can be linearized by the method. This is done both in general and in the special cases of two and three dimensions where complete results are obtained. The practical problem of identifying linearizable systems is also considered and computable necessary criteria are presented.

Notes: Abstract A general method for determination of the volume of a space in a non steady state condition, in case diffusion might be significant, is developed. Instantaneous mixing of indicators with native fluid is assumed in this first stage of investigation. Theoretical expressions are obtained for the volume of the space and the diffusion coefficient as a function of time. An analysis of feasibility of the method is also included.

Notes: Abstract The theory of transfer of low-molecular nonelectrolytes across deformable semipermeable membranes of large curvature developed in Part I (Rubinstein, 1974) is used to describe the dynamics of swelling and shrinking of a muscle fiber at the influx and efflux of low-molecular nonelectrolytes. A large set of computations showed that the theory explains the experiments described in the literature.

Notes: Abstract Green's function for heat and matter transport is calculated for an infinite medium in which a convection field v(r,t) makes a contribution to the total heat and matter current. It is given by a uniformly and absolutely convergent series in which every term is calculated from the preceding one merely by integration. The solution procedure is interpreted physically and illustrated by a simple problem in which v(r,t)=const. in space and time. Since the solution contains no intrinsic spatial symmetry, it can serve as a starting point for a theory of heat and mass transport in perfused biological tissue.

Notes: Abstract The hydrodynamics of a microorganism swimming in a channel is investigated. The microorganism is modeled as a two-dimensional sheet swimming at low Reynolds numbers between two rigid walls. The wavelengths of the propulsive waves passing down the sheet are assummed to be very large compared to the channel spacing, but the amplitude of the propulsive waves is arbitrary. Explicit analytical solutions for the propulsive velocity and the rate of energy dissipated in terms of the wave amplitude, channel spacing, wave number, and wave speeds are given.

Notes: Abstract It is shown from the statistical-mechanical overview of Volterra's ecological model how to reckon the fluctuations of collective variables such as the total population of a genus: and that these fluctuations are much decreased (or that the collective populationsteadiness is enhanced) as the speciation is increased. (A niching of species in time, or phase-niching, is entailed here.) Secondly, it is shown how Preston's log-normal distribution describing the species-abundance relationship, as well as a generalization of such distributions, come forth simply and naturally from the statistical-Volterra-dynamics.

Notes: Abstract The impressed field, “Window Field” (WF), due to a half-wave action potential on a muscle fiber, has been calculated on the basis of potential theory. It has been shown that in spite of the small intensity of the field, its integrated action can transfer the energy needed to induce, contraction from the membrane to the interior of the fiber. The energy of polarization has been found to be sufficient to exceed the energy of, thermal agitation on that length of fiber, which can be identified as the length of a sarcomere. The changes of ion concentration, caused by the WF, if calculated on the assumption of the semipermeability of theZ membranes, was found to be equal to the changes necessary to induce contraction of actomyosinin vitro.

Notes: Abstract Some general properties of the solution of the diffusion equation are deduced for the steady-state, spherically symmetric system. On the basis of these developments some results of N. Rashevsky (Bull. Math. Biophysics,11, 15, 1949) are discussed and the results of a previous investigation (Hearon,Bull. Math. Biophysics,12, 135, 1950b) are extended to more general conditions. In particular these extensions apply to the flow of a soluteagainst its concentration gradient, the nonzero gradient of an inert metabolite, and theaccumulation or exclusion of an inert metabolite in a metabolic system.

Notes: Abstract The approximation method of N. Rashevsky is discussed and reviewed. It is shown that in addition to theexplicit assumptions and approximations there is involved the assumption that the rate of metabolism is the same at every point in the cell and that theaverage rate of metabolism is different from zero. An expression is given for the error in the approximate method when the rate of metabolism is any function of the concentration. It is also shown that a solution in theform of that obtained by the approximate method is not possible if the generalized laws of diffusion are assumed to apply.

Notes: Abstract Rate equations for the enzymatic oxidation of succinic acid are derived on the assumption that when a single molecule of substrate combines with an enzyme molecule, it can do so with either one or two sites on the enzyme, and that oxidation occurs only in the second case. In addition it is assumed that the product of the reaction, fumaric acid, combines reversibly with the enzyme. With certain enzyme preparations the data fitted such an equation satisfactorily. In others the rate was that of a first-order reaction, but addition of cytochrome changed it to the former type. It was concluded that the transfer of hydrogen to oxygen was a first-order reaction and dominated the whole rate when enzyme preparations were used which had been washed relatively free of cytochrome. When the limiting factor was succino-dehydrogenase the rates followed the new equation. Criteria for recognizing noncompetitive inhibition are given, and inhibition by di-tertiary butyl peroxide was shown to be of this type.

Notes: Abstract This paper deals with the following question: Which distributions of radiosensitivity in a population can lead to an exponential survival curve? The problem is solved exactly, with statistical fluctuations in dose fully accounted for. It is shown that only an exponential distribution of sensitivities can give rise to an exponential survival curve.

Notes: Abstract An approximation method is introduced which enables a number of diffusion-type problems to be solved in an approximate but simple manner. Many cases require only the solution of a simple first-order differential equation. The method is applied to a number of cases in which the exact solutions are available. A comparison shows that the method is quite satisfactory in these cases. The method is applied to diffusion problems with rate of consumption proportional to concentration or to the square of the concentration. In the latter case, the result obtained is essentially the same as that found by H. G. Landau (1950) after elaborate calculations.

Notes: Abstract It is shown that a slight modification of a model of excitatory phenomena in irritable tissues, which has been treated before, exhibits spontaneous oscillations. The frequency of these oscillations and the time-course of the potential across the model membrane have been determined, together with the dependence of some of their characteristics on some important parameters, particularly (Ca++).

Notes: Abstract It is demonstrated that, if the variations of viscosity throughout a cell are considered, swelling stresses may produce elongation and division. To do this it is necessary to generalize Betti's theorem to cover systems containing viscosity gradients and such a generalization is presented. On the basis of two special assumptions it is shown that most of the results of the diffusion drag theory of cell division may be duplicated by the present theory.

Notes: Abstract The theory of imitative behavior, developed previously, is applied to the case of two social groups which are separated spatially. If the information of each group as to the behavior of the other is complete, the case reduces to that of a single group. When any information is lacking at all, the two groups are independent. If we have two mutually exclusive behaviorsA andB, all four combinationsAA, AB, BA, andBB are possible. If the mutual information gradually increases from zero, then for a certain value of it, the group which is more informed about the behavior of the other will change to that behavior if it did not already exhibit it. If for constant information the size of the group increases, then above a certain threshold value, the larger group imposes its behavior on the smaller.

Notes: Abstract Local regulation of blood flow as determined by capillary diameter and the number of open capillaries in a region is considered. The local changes in capillary diameter and in the number of open capillaries are assumed to be due to concentration changes of a diffusible, nonspecified metabolite. This metabolite is produced in the tissue and carried away by the blood stream. Using these assumptions and applying pertinent data on capillaries, deductions are made concerning: (a) the law of blood flow as a function of temperature and capillary radius for the hyperemia of high temperature, (b) high flow as it depends on metabolism during strenuous exercise of muscle, and (c) a first approximation to the time duration of occlusion hyperemia.

Notes: Abstract The necessary and sufficient condition is given forn integers to be the score structure of a society with a dominance relation. A proof is also given for a theorem showing that there are members who dominate every other member either directly or indirectly through a single intermediate member.

Notes: Abstract On the basis of a previous general formulation (Bull. Math. Biophysics,15, 21–29, 1953a) a discussion is given of the error in the approximation method of N. Rashevsky. This error, inherent in the method when the metabolic rate is different at each point in the cell, is discussed in detail and numerical values are presented for two particular cases: the rate proportional to the concentration and the rate a prescribed function of the spatial coordinates. It is shown that the formulation for the first case also applies to several other cases, that the error is negligible provided the rate is sufficiently small, and that the error is fairly sensitive to the cell size. If the rate depends upon the coordinatesalone a small rate is not sufficient to insure a negligible error. The relations between the exact method, the standard approximate method, an earlier approximate method (Physics,7 260, 1936), and a more recent refinement (Bull. Math. Biophysics,10, 201, 1948) of the standard method are discussed.

Notes: Abstract It is shown on the basis of (1) conservation of mass, (2) positive concentrations, and (3) the principle of detail balancing that periodic reactions cannot occur in a closed system described bylinear differential equations. The matrix,A, of the rate equations must be such that |A|=0,a ij〉0 fori≠j,a ii〈0, andVAV −1=B, whereV is diagonal andB is symmetric. These properties ofA imply that the latent roots are real and non-positive and that neither catalysis nor inhibition can be described bylinear equations. It is further shown that periodic reactions cannot occur in anopen system for which the matrix associated with the chemical reactions has the above properties and in which thesimple law of diffusion is obeyed. The relation of these results to Onsager's reciprocal relations and to previous work on periodic and cyclic chemical reactions is discussed. The utility of certain of these results for the treatment of isotope kinetics is indicated.

Notes: Abstract It is demonstrated that an explanation of the small radius effect or the so-called sigma phenomenon may be obtained by noting that one of the effects of the presence of suspended particles in a flowing fluid is to increase the velocity of flow near the wall over that existing in the absence of particles. This effect may be considered equivalent to relaxing the boundary conditions at the wall. An expression for the viscosity is compared with data and fit is found to be good.

Notes: Abstract The solution for the spatial distribution of ions in a Donnan equilibrium has been given by J. H. Bartlett and R. A. Kromhout (1952). The present note gives an explicit solution for the case in which the length of the region containing the membrane is large; in biological situations this requires only that the length considered should be greater than a few hundred Ångstrom units. The Donnan equilibrium may be considered to be a special case of a situation in which forces other than electrical act upon the ions; in particular, it represents the case in which only one ion is acted upon and the energy difference on the two sides of the membrane is infinite. An expression is given for the difference in energy of theith in terms of the electrical potential and of the ion concentrations. As an illustration, the results are applied to nerve membrane potentials.

Notes: Abstract A mechanism is described which accounts for the active transport of Na+ ions through a membrane. It is assumed that at one side of the membrane the ion combines with a carrier ion, the resulting carrier compound then diffuses through the membrane and decomposes at the other side of the membrane. The free diffusion of the ions is also taken into account. The time rate of accumulation of the ion in question at the latter side of the membrane is calculated in terms of the concentrations of the ion at both sides of the membrane.

Notes: Abstract The recent extension of the approximation method is applied to enable us to arrive at the time course of the concentrations at both sides of a membrane. From the differential equations which govern these, the steady-state solution is obtained in terms of the parameters, which are determined by the thickness of the diffusion layers, the chemical composition and reactions, and the diffusion constant of the membrane.

Notes: Abstract 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.

Notes: Abstract The reflection of pressure waves in a fluid enclosed within a tube with an elastic wall is studied for the case of a localized change in diameter of the tube. The concept of impedance is introduced. The relation of the reflection characteristics of the parts of the tube at either side of the change is derived on the basis of the continuity of pressure and mass flow at the site of the change. This relations is used to derive the expression for the ratio of the pressure oscillations measured in front of, and behind, the constriction in terms of the constants of the system. As a result, a method is indicated to locate the coarctation from measurements of the pressures in front of, and behind it.

Notes: Abstract The question is raised concerning the possible causes of abnormally small standard deviations found in counting samples in which particles are distributed at random (e.g., blood cells, fat globules in milk, etc.). The effect of discarding abnormal samples is discounted inasmuch as small standard deviations occur even when all samples are counted. An approximation method is used to calculate the effect of finite particle size, of known repulsive forces between particles and of convection currents. This calculation shows that neither finite size nor the known repulsive forces are sufficient to account for the observed abnormality of standard deviation, but that convection currents can possibly account for it. The possible presence of long-range repulsive forces cannot, however, be excluded.

Notes: Abstract The effect of finite particle size on the standard deviation in sample counts is computed for the one-dimensional case. To a first order of approximation the correction is found to be identifical with that found by H. de Vries (1953) using a general approximation method.

Notes: Abstract A critical examination of the “classical” theories of photoreception in view of more recent experimental findings yields the result that these theories do not possess the property to describe all the more significant phenomena of photoreception correctly, and to some extent suffer the lack of more general applicability. The basis for a new and presumably more general theory of photoreception based on dynamical aspects is laid out. Emphasis is put on the time course of afferent and efferent excitation in the photoreception model, consisting of a receptor element, an afferent and an efferent neuron of the one-factor Rashevsky-type, and an effector organ.

Notes: Abstract A mathematical model for the development of human society, beginning with the earliest stages of urban cultures, is outlined. In the early stages of history, behavior was characterized largely by adherence to a number of beliefs and prejudices of diffeirent kinds, which were accepted on faith and not subject to critical rational analysis. Due to psychobiological variability a very small number of individuals spontaneously appear at all times who challenge the accepted beliefs and prejudices and do not follow the accepted patterns of social behavior. The effect of these individuals upon the rest of the society, especially upon the younger generation, depends on the facilities with which information spreads in society. In earliest societies, when modern methods of mass communication were unknown, the channels of communication were practically identical with the channels of economic transport. The latter in its turn depended on the nature of the roads, and especially on the presence of waterway, which facilitated transportation. The sizes of the earliest cities and the distances between them were largely determined by relative ease of transporation. Expressions are derived for the average size of the earliest cities and for the average distance between them. The calculated average populations of the earliest cities are of the order of 103; the distance of the order of 102 km. Both are in agreement with some archaeological findings. An expression for the time spaon required for the development from the earliest stages of urban cultures to the present time is derived and shown to depend on the specific shoreline of the country, that is, the length of the shorline divided by the area of the country. It is pointed out that western Europe's specific shoreline, including land bordering both seas and rivers, is ten times as large as the shoreline area of other parts of the world. It is shown that this greater specific shoreline may account quantitatively for the faster social and technological development of western Europe in the last few centuries. The calculated total span of time of development from earliest urban cultures to our days is found to be of the order of magnitude of ten thousand years. It is shown that the model accounts for the existence at the present time of primitive cultures. A number of suggestions is made in regard to other possible applications of mathematics to history.

Notes: Abstract The assumptions latent in the derivation of the integral equation of Branson are rendered explicit and discussed. It is shown that the equation is valid only for systems in which the substance disappears according to a linear rate law.

Notes: Abstract It is shown that the validity of Branson's integral description of metabolizing systems is subject to severe limitations. The validity is insured only in cases where the reaction is of first order, or quasi of first order. In all other cases Branson's equation has to be modified to insure general applicability. The consequences of a different definition of the metabolizing functionF have also been investigated. With the new definitionF describes the pure effect of metabolization. It is found that in this case the integral equation is only capable of describing first-order reactions. With a slight modification of the integral equation it is possible to describe metabolites “with age”, which do not have reactions of definite order, but which satisfy the superposition principle.

Notes: Abstract A problem in probability is stated with included the problem of the distribution of bacterial mutants as a special case. This problem is solved exactly but since the resulting expressions are too complicated for practical use, various approximate expressions for the distribution are considered, especially for the bacterial mutation case.

Notes: Abstract Simple reaction and discrimination reaction, under the influence of imitation, are considered for the situation in which the stimulus or the stimuli vary slowly with time. The result is analogous to hysteresis under certain conditions. The calculations are facilitated by the solution of $$x = \int_{ - \infty }^{a + \beta x} {g\left( \xi \right)d} \xi ,$$ g(ξ) being the normal error function. Values ofx(α, β) are given in a table.

Notes: Abstract On the basis of simple physical considerations the blood flow in a branching circulatory system is studied. The case of two groups of parallel vessels is treated. The vessels of the same group are supposed to be identical. The resistance of each group is determined by the resistance of each vessel in the group and by the number of vessels in the group. From the dependence of the resistance of each vessel on its radius an expression is obtained for the blood flow through each group of vessels in terms of the numbers and sizes of the vessels in each group. The number of open vessels in an organ and the radius of each of those vessels are assumed to depend on the metabolic rate of that organ. The relations so obtained, together with the expression above, are applied to derive the blood flow through an organ as a function of the metabolic rate of that organ. It is indicated that the relations obtained might describe the shifting of blood from one organ to another if the activity of one of them changes. A way is pointed out to treat neural regulation of this phenomenon.

Notes: Abstract A stochastic model of population growth is treated using the Bellman-Harris theory of agedependent stochastic branching processes. The probability distribution for the population size at any time and the expectation are obtained when it is assumed that there is probability (1−σ), 0≤σ〈1, of the organism dividing into two at the end of its lifetime, and probability σ that division will not take place.

Notes: Abstract In a previous paper, in which a possible mathematical approach to history was outlined, it was shown that urbanization plays an important part in the propagation of new ideas. The rate of such propagation influences the rate of historical developments. The present paper deals in more detail with possible mechanisms of formation of earliest cities. Equations are derived which give the limiting size of such cities and their rate of growth. Of particular importance for the spread of new ideas is the spread of information. The latter largely depends on the fraction of individuals who travel between city and country. Expressions for this quantity are derived. An approach is outlined to the mathematical study of the earliest social classes, which may have been formed as a result of military, religious, or economic stratifications.

Notes: Abstract A stochastic model is developed for an enzyme reaction in an open linear system. The proposed model assumes that the open system maintains the concentration of substrate and inhibitor at constant levels and that the product molecules are removed from the system by a first order reaction. Stochastic models for several enzyme reactions occurring in this open system are shown to correspond to special cases of theGI/M/∞ queue. Takács’ (1958) results for this queueing system are used to obtain the stochastic properties of the enzyme systems. A specific model we studied assumed completely competitive inhibition in an open system. The stationary distribution for the number of product molecules in the system is obtained. The enzyme reaction which incorporated the “intermediate chain hypothesis” can also be investigated by the queueing theory approach. It is shown that for this open system, if the model which incorporated the intermediate chain hypothesis has the same deterministic properties as the Michaelis-Menten model, then the latter has greater stochastic variation than the former.

Notes: Abstract The existing methods to solve the problems of pulsatile flow in the cardiovascular system are based on either linear axisymmetric equations or non-linear one-dimensional equations. The solutions thus obtained give only a mediocre comparison with measurements. In this paper, a non-linear axisymmetric theory is proposed. The starting point of the present theory is a third degree polynomial representation of the velocity profile. Integral methods are then applied to obtain the governing equations. To ascertain the accuracy of the theory proposed above, the calculations for a simple case involving pulsatile flow in a long rigid tube were performed. The results are: (a) the average velocities compare very well with exact solutions and (b) the velocity profiles for a given frequency agree very well with exact solutions for flow in small tubes, but tend to differ as tube size is increased.

Notes: Résumé Le but de ce travail est la mise en évidence d’éventuels “patterns” temporels privilégiés de potentiels d’action neuronaux masqués par la superposition d’une activité aléatoire. Dans la première partie, on propose un modèle susceptible de rendre compte de cette activité aléatoire. Dans la seconde, on expose une méthode d’extraction des patterns privilégiés, compatible avec les paramètres du modèle neuronal proposé. Son algorithme fait notamment intervenir l’estimation de la fonction d’expectation. Cette méthode peut, en fait, ment intervenir l’estimation de la fonction d’expectation. Cette méthode peut, en fait, être appliquée à l’étude de séries temporelles d’événements dans des domaines très divers.

Notes: Abstract The velocity of propagation of a disturbance wave in a liquid flowing in a distensible tube is computed. The mathematical model is more general than those used in previous analyses: the tube wall properties are realistic; the convective part of the axial inertia forces is taken into account; radial inertia forces of both the fluid and tube wall are present; viscous stresses are present. Four parameters influencing the velocity of propagation are obtained and discussed. Curves are plotted illustrating the effects of the parameters. Contrary to the results of previous analyses, viscous effects are shown to be appreciable in blood flow. It is also shown that radial inertia effects can be important in laboratory set-ups.

Notes: Abstract A general theory of the drying of frozen tissue is developed and applied to the measurement of the drying rate of frozen guinea pig liver. It is shown that for a given temperature of the subliming ice crystals the mininum drying time of a piece of guinea pig liver is greater than the minimum sublimation time of a piece of ice of the same size and shape by a factor of the order of one thousand. This fact has many implications in the design of freeze-dry apparatus which will be developed in a following paper.

Notes: Abstract The equations governing the time course of the exchange of substances between the blood in the capillaries and the extracellular space are solved for the case of substances which do not penetrate the cells. The equations given relate the time course of the exchange process to the various tissue and circulation parameters such as the specific capillary wall area, the pore area, the inter-capillary distance, the size of the extra-vascular, extra-cellular space, the diffusion coefficient in this space, and the velocity of blood in the capillaries. Some experimental work on capillary exchange is discussed in relation to the theory and estimates are made of the relative importance of the various tissue and circulation parameters in the exchange of substances in different tissues.

Notes: Abstract An effort is made to begin widening the scope of kinetics by merging the concepts and point of view of molecular set theory with the stochastic approach to kinetics, beginning with the simplest unimolecular molecular set transformation. In this spirit the new concept ofmolecular set variable is introduced as the basic unit of kinetics as opposed to simply the traditionalconcentration (or cardinality) unit, connoting that the composition as well as the size of a molecular set are significant dynamic features of a system. The changes in state (or “value”) of the molecular set variable are characterized by a Markovian stochastic process and the relationship between this process and the corresponding unimolecular process for the concentration variable introduced earlier is discussed. The possible role of molecular set theory in terms of the underlying biomathematical structure of relational biology is also considered.

Notes: Abstract The problem of the forms of plants and models of branching are discussed using experimental data on the mistletoe. The number of branches by division, the distribution of divisions with regard to the number of branches per division and to the level of division, the geometrical characters of branches according to the level of division and the host, the stability of model are studied. One gives an interpretation of the model of branching as a model of growth.

Notes: Abstract A quantum model for the general enzymic reaction,E+S ⇌ ES → P, is presented, starting with the assumptions that any chemical substanceS, which may be a substrate for a particularE (S)-enzyme is a microphysical system and any enzymeE-molecule, capable of interacting with anS-substrate is a “measuring system” which will “measure” one or more of theS-observables. According to the above assumptions a stochastic model of the reaction is constructed and a computer simulation of the steady state performed. The results thus obtained predicted fluctuations in the enzymic reaction rate, function of the substrate “perturbation”. On an experimental basis it is demonstrated that the irradiation of an enzymic substrate with low energies results in the inducement of a dose-dependent oscillatory behavior in the corresponding enzymic reaction rate. In the reaction type, the oscillations thus induced in theE-activity by the corresponding substrates are out-of-phase, realizing a biochemical discriminating net. Likewise, in an $$S_1 \xleftarrow{{E_1 }}S\xrightarrow{{E_2 }}S_2$$ reaction type, the oscillations induced by the irradiatedS-substrate in the activities of the respective enzyme, realize a biochemical switching net.

Notes: Abstract As a further attempt to determine the stresses and strains of the individual myocardial fibers, the heart muscle is considered as an orthotropic material. A theory is presented which leads to the expression of the equilibrium conditions for the left ventricle in the form of three simultaneous differential equations. Solution of these equations would give the changes in shape of the left ventricle throughout the cardiac cycle, and, in addition, the stresses and strains of the individual myocardial fibers. It is pointed out, however, that meaningful solutions of the equations cannot be obtained at the present time because of difficulties in experimental determination of certain parameters.

Notes: Abstract This paper compares two previously published neural models for epilepsies (Bull. Math. Biophysics,33, 539–553, 1971;34, 71–78, 1972). The second model is developed in more detail and an attempt is made to bring it more in line with established neurological findings. The question of classification of some epilepsies is briefly discussed.

Notes: Abstract A formal mathematical model is proposed for a spontaneously repetitively firing neuron. It is based on the assumption that an excitatory and inhibitory substance, possibly different from those involved in synaptic transmissions, is formed in the soma of everynormal neuron. Furthermore, the decay of the substances is ascribed to their combination with some other substances, present in healthy individuals. A generalized two factor system of differential equations is used. It is shown that when the normally present substances are absent, possibly due to genetic defects so that the decay constants become zero, the equations lead to undamped sinusoidal solutions of the difference between excitatory and inhibitory factors, thus producing a trulyspontaneous repetitive discharge, in the absence of external currents or other stimulation. It is suggested that convulsants may act by destroying the substances present in healthy individuals. It is further suggested that by administering to epileptics those substances, which are present in normal healthy persons, perhaps by using brain extracts fromhealthy higher animals which sometimes suffer from epilepsy, an actual cure rather than symptomatic treatment by anticonvulsants may be obtained.