A note on mathematical models for the interaction of neural elements

By introducing a plausible model for the initiation of axonal impulses the output is obtained as a function of the input incoming impulses. If the temporal aspects of the excitatory process resulting from the afferent impulses are sufficiently rapid one obtains the discontinuous or microscopic model of McCulloch-Pitts. If these are sufficiently slow a continuous model, such as Rashevsky’s one or two factor theory, is a natural model. But the linear relation between the strength of excitation of one axon and excitatory factor of the next will not in general hold. However, under conditions which are not too restrictive the linear relation with threshold can be considered as satisfactory approximation over a fairly wide range of values. This research was supported in whole or in part by the U. S. Air Force under Contract AF 49(638)-414 monitored by the Air Force Office of Scientific Research.     

The diffusion process approach to one-compartmental stochastic models: A mathematical note

We consider the one-dimension (one-compartment) exponential model using a diffusion process approach. In particular, we summarize the known results in the case where the stochastic component of the model is a Gaussian white noise process with mean zero and variance σ². Finally, we briefly illustrate a number of cases where similar forms of model arise.     

A note on the mathematical biology of automobile driving

It is pointed out that the three different stimuli for a corrective turn, namely the distance from the edge of the lane, the rate of approach to the edge, and the angle between the direction of the car and the direction of the lane (Bull. Math. Biophysics,28, 645–654, 1966,29, 181–186, 1967) may act all three simultaneously. It is found that in that case the tracking curve of the car is stable below a critical speed and becomes unstable above it.     

A note on the mathematical biophysics of ameboid movements

Theoretical considerations lead to the expectation of some regularities in the apparently random changes of shape of an ameba. We should expect quasiperiodic fluctuations of the ratio of the perimeter to the area of the optical cross-section of an ameba, when plotted against time. The quasi-period should be characteristic of the species. Preliminary experiments appear to bear out that prediction.     

A technical note on seasonal growth models

The growth of many organisms is seasonal, with a dependence on variation in temperature, light, and food availability. A growth model proposed by Somers (Fishbyte 6:8?C11, 1988) is one of the most widely used models to describe seasonal growth. We point out that three different formulae (beyond numerous typographical errors) have been used in the literature referring to Somers (Fishbyte 6:8?C11, 1988). These formulae correspond to different curves and yield different parameter estimates with different biological interpretations. These inconsistencies have led to the wrong identification of the period of lowest growth rate (winter point) in some papers of the literature. We urge authors to carefully edit their formulae to assure use of the original definition in Somers (Fishbyte 6:8?C11, 1988).     

A note on persistence about structured population models

In this paper, we report some results on persistence in two structured population models: a chronic- age-structured epidemic model and an age-duration-structured epidemic model. Regarding these models, we observe that the system is uniformly strongly persistent, which means, roughly speaking, that the proportion of infected subpopulation is bounded away from 0 and the bound does not depend on the initial data after a sufficient long time, if the basic reproduction ratio is larger than one. We derive this by adopting Thieme's technique, which requires some conditions about positivity and compactness. Although the compactness condition is rather difficult to show in general infinite-dimensional function spaces, we can apply Fréchet-Kolmogorov L(1)-compactness criteria to our models. The two examples that we study illuminate a useful method to show persistence in structured population models.     

A note on stability of discrete population models

P. Cull (1981) and G. Rosencranz (1983) studied a discrete population model described by the first-order difference equation xt+1 = g(xt) and obtained an important result on the global stability of the equilibrium point means when g(x) has only one extreme point (a maximum) in (0, means). Motivated by work of M. Kot and W. M. Schaffer (1984), a more general case is considered in which g(x) can have more then one maximum point in (0, means), and results on global stability are obtained. These results are applied to develop tests for global stability of the equilibrium point that imply other results in the literature on global stability.     

A note on multidimensional models of neutral mutation

A generalisation of the Ohta-Kimura charge-state model for neutral mutation, in which alleles are represented by points in d-dimensional space, seems appropriate in the light of recent experimental developments. It is noted that existing methods of analysis extend almost trivially to this more general model, and the point is illustrated by calculating the effective number of alleles.     

A note on persistence about structured population models

In this paper, we report some results on persistence in two structured population models: a chronic- age-structured epidemic model and an age-duration-structured epidemic model. Regarding these models, we observe that the system is uniformly strongly persistent, which means, roughly speaking, that the proportion of infected subpopulation is bounded away from 0 and the bound does not depend on the initial data after a sufficient long time, if the basic reproduction ratio is larger than one. We derive this by adopting Thieme's technique, which requires some conditions about positivity and compactness. Although the compactness condition is rather difficult to show in general infinite-dimensional function spaces, we can apply Fréchet–Kolmogorov L ¹-compactness criteria to our models. The two examples that we study illuminate a useful method to show persistence in structured population models.     

A note on generation times in epidemic models

The time between the infection of a primary case and one of its secondary cases is called a generation time. The distribution (and mean) of the generation times is derived for a rather general class of epidemic models. The relation to assumptions on distributions of latency times and infectious times or more generally on random time varying infectiousness, is investigated. Serial times, defined as the times between occurrence of observable events in the progress of an infectious disease (e.g., the onset of clinical symptoms), are also considered.     

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 note on the fahraeuslindqvist effect in power law fluid

An axisymmetric flow of a power law fluid through circular tubes under constant pressure gradient with the flow parameters varying radially is analyzed theoretically. The main finding is that for the Fahraeus-Lindqvist (F-L) effect to occur, it is necessary to have at least one of the parametersK (consistency) andn (index) as a discontinuous function ofr in the absence of wall slip; and with slip condition the parameters could be continuous functions ofr under specific conditions. In both the cases the existence of more than one discontinuity cannot be ruled out. The results obtained are consistent with experimental findings of blood flow through narrow tubes.     

Some equivalent compartmental models

Compartmental models with strongly connected digraphs always have a stable equilibrium solution. Each such model may be reduced by a sequence of matrix operations to a mammilary system with the same equilibrium solution. It may also be reduced by a sequence of operations on the digraph. It may then be further reduced to a model whose digraph is a 2-cycle with the same mean first passage time m. If this m is taken as an indicator of the relative stability, then the latter is independent of the complexity as measured by the number of arcs per vertex. In a number of special cases the index m is related to the eigenvalues of the matrix. It is shown to have a simple relation to other time parameters as well.     

A note on conditional AIC for linear mixed-effects models

The conventional model selection criterion, the Akaike informationcriterion, AIC, has been applied to choose candidate modelsin mixed-effects models by the consideration of marginal likelihood.Vaida & Blanchard (2005) demonstrated that such a marginalAIC and its small sample correction are inappropriate when theresearch focus is on clusters. Correspondingly, these authorssuggested the use of conditional AIC. Their conditional AICis derived under the assumption that the variance-covariancematrix or scaled variance-covariance matrix of random effectsis known. This note provides a general conditional AIC but withoutthese strong assumptions. Simulation studies show that the proposedmethod is promising.