Studies in the mathematical theory of excitation

The general linear two-factor nerve-excitation theory of the type of Rashevsky and Hill is discussed and normal forms are derived. It is shown that in some cases these equations are not reducible to the Rashevsky form. Most notable is the case in which the solutions are damped periodic functions. It is shown that in this case one or more—in some cases infinitely many—discharges are predictable, following the application of a constant stimulusS. The number of discharges increases withS, but the frequency is a constant, characteristic of the fiber and independent ofS.     

A note on the mathematical biophysics of central excitation and inhibition

Some relations between the temporally macroscopic theory of central excitation and inhibition and the temporally microscopic theory of nervous nets are suggested.     

A mathematical synthesis of niche and neutral theories in community ecology

The debate between niche-based and neutral community theories centers around the question of which forces shape predominantly ecological communities. Niche theory attributes a central role to niche differences between species, which generate a difference between the strength of intra- and interspecific interactions. Neutral theory attributes a central role to migration processes and demographic stochasticity. One possibility to bridge these two theories is to combine them in a common mathematical framework. Here we propose a mathematical model that integrates the two perspectives. From a niche-based perspective, our model can be interpreted as a Lotka-Volterra model with symmetric interactions in which we introduce immigration and demographic stochasticity. From a neutral perspective, it can be interpreted as Hubbell's local community model in which we introduce a difference between intra- and interspecific interactions. We investigate the stationary species abundance distribution and other community properties as functions of the interaction coefficient, the immigration rate and the strength of demographic stochasticity.     

Equivalence of the theories of nervous excitation of Hill,Monnier, and Rashevsky

New and direct proofs of the equivalence of the three theories of nervous excitation are developed, and by the introduction of theHauptnutzzeit as the unit of time and a special unit for the measurement of the excitatory state, a normal form for Rashevsky's equations is obtained which exhibits clearly the relations among the various independent parameters.     

On the mathematical modelling of pain

In this review a case is presented for the use of mathematical modelling in the study of pain. The philosophy of mathematical modelling is outlined and a recommendation is made for the use of modern nonlinear techniques and computational neuroscience in the modelling of pain. Classic and more recent examples of modelling in neurobiology in general and pain in particular, at three different levels—molecular, cellular and neural networks—are described and evaluated. Directions for further progress are indicated, particularly in plasticity and in modelling brain mechanisms. Major advantages of mathematical modelling are that it can handle extremely complex theories and it is non-invasive, and so is particularly valuable in the investigation of chronic pain. Special issue dedicated to Dr. Herman Bachelard     

On mathematical characterization of mutations

The aim of the present paper is to give for the mutations from genetics a mathematical treatment similar to that utilized for the perturbation in communication theory. The transformations induced by mutations both on codons and on amino acids are investigated.     

On the mathematical basis of the variance-mean power relationship

The mathematical basis of a widely-known variance-mean power relationship of ecological populations was examined. It is shown that the log variance (S ²)—log mean, (m) plot is virtually delimited by two lines logS ²=logn+2 logm and logS ²=logm, thus increasing the chance that a linear regression line can be successfully fitted, without a profoundly behavioural background. This makes difficult the task of interpreting a successful fit of the power law regression and its parameterb in a biologically meaningful manner. In comparison with the power law regression, Iwao'sm ^*-m regression is structurally less constrained, i.e. has a wider spatial region in which data points can scatter. This suggests that a comparison between the two methods in terms of how good a fit is achieved for a particular data set is largely meaningless, since the power law regression may inherently produce a better fit due to its constrained spatial entity. Furthermore, it could be argued that a successful fit in Iwao's method, when found, is less taxed with mathematical arterfacts and perhaps more clearly linked to some biological mechanisms underlying spatial dispersion of populations.     

On a mathematical theory of embolism

A theory of embolism based on an optimization model of blood flow is proposed and used to explain the topographic distribution of emboli in arterial trees.     

On structural theories of basal metabolic rate

Because cells and organisms interface with the environment through surfaces, their design should be governed by surface laws. Yet, basal metabolic rate is not proportional to the 0·67-power of body mass (surface law) but to the 0·75-power of body mass. From the many theories that have derived a surface law, Teissier's dimensional analysis theory was probably the neatest. However, the surface law has been empirically invalidated. Moreover, Teissier assumed that times in the prototype animal and a similar one with different size are in the same ratio as their linear sizes. This is incorrect, however, because heart rates, being inverses of times, should be proportional to the 1/3-power of body mass—but are proportional to the 1/4-power of body mass, which is consistent with a 0·75-power law of basal metabolic rate. McMahon's recent attempt to explain the deviation of the empirical law from a surface law based entirely on structural considerations, is critically examined. It does not appear that purely structural considerations could explain the deviation between the empirical 0·75-law of basal metabolic rate and the surface law.     

On a mathematical model describing the aggregation of amoebae

In this paper an extension of a mathematical model of Keller and Segel (1970) describing the aggregation of amoebae is presented. In their paper (Keller and Segel, 1970) they showed that the onset of the aggregation could be viewed as a spatial instability. Their instability condition involved diffusion constants of the cyclic AMP and of the amoebae as well as a constant describing the chemotactic behavior of the amoebae. In our case we consider a temporal instability that depends only on the kinetics of cyclic AMP production, degradation and transport through the cell wall. Our model then explains the oscillatory behavior of the cyclic AMP in well-stirred suspensions of amoebae. In addition we discuss existence and non-existence of nonuniform steady states of the nonlinear parabolic system involved.     

On the treatment of airline travelers in mathematical models

The global spread of infectious diseases is facilitated by the ability of infected humans to travel thousands of miles in short time spans, rapidly transporting pathogens to distant locations. Mathematical models of the actual and potential spread of specific pathogens can assist public health planning in the case of such an event. Models should generally be parsimonious, but must consider all potentially important components of the system to the greatest extent possible. We demonstrate and discuss important assumptions relative to the parameterization and structural treatment of airline travel in mathematical models. Among other findings, we show that the most common structural treatment of travelers leads to underestimation of the speed of spread and that connecting travel is critical to a realistic spread pattern. Models involving travelers can be improved significantly by relatively simple structural changes but also may require further attention to details of parameterization.     

On the theories of gene regulation and differentiation in eukaryotes

The interrelationships among recent theories on the regulation of gene activity and differentiation in higher organisms are reviewed. Interpretations within these theories of the various components of chromosomes are re-evaluated and a unified conceptual framework of hierarchical genetic control mechanisms in eukaryotes is presented.     

Evolution of spiral and scroll waves of excitation in a mathematical model of ischaemic border zone

Abnormal electrical activity from the boundaries of ischemic cardiac tissue is recognized as one of the major causes in generation of ischemia-reperfusion arrhythmias. Here we present theoretical analysis of the waves of electrical activity that can rise on the boundary of cardiac cell network upon its recovery from ischaemia-like conditions. The main factors included in our analysis are macroscopic gradients of the cell-to-cell coupling and cell excitability and microscopic heterogeneity of individual cells. The interplay between these factors allows one to explain how spirals form, drift together with the moving boundary, get transiently pinned to local inhomogeneities, and finally penetrate into the bulk of the well-coupled tissue where they reach macroscopic scale. The asymptotic theory of the drift of spiral and scroll waves based on response functions provides explanation of the drifts involved in this mechanism, with the exception of effects due to the discreteness of cardiac tissue. In particular, this asymptotic theory allows an extrapolation of 2D events into 3D, which has shown that cells within the border zone can give rise to 3D analogues of spirals, the scroll waves. When and if such scroll waves escape into a better coupled tissue, they are likely to collapse due to the positive filament tension. However, our simulations have shown that such collapse of newly generated scrolls is not inevitable and that under certain conditions filament tension becomes negative, leading to scroll filaments to expand and multiply leading to a fibrillation-like state within small areas of cardiac tissue.     

On the stretching of DNA in the reptation theories of gel electrophoresis

The reptation dynamics of DNA chains in gel electrophoresis is discussed. The differences between the Lumpkin, Déjardin, and Zimm (LDZ), and the Slater and Noolandi (SN), biased reptation models are pointed out and studied. We show that new assumptions are necessary in order to study large-field electrophoresis, and we present preliminary results of a computer simulation of DNA gel electrophoresis.