Showing papers in "Bellman Prize in Mathematical Biosciences in 1971"
TL;DR: It is demonstrated that, in order for the learning process to be stable, pattern storage must be accomplished principally by weakening synaptic weights rather than by strengthening them.
Abstract: A comprehensive theory of cerebellar function is presented, which ties together the known anatomy and physiology of the cerebellum into a pattern-recognition data processing system. The cerebellum is postulated to be functionally and structurally equivalent to a modification of the classical Perceptron pattern-classification device. It is suggested that the mossy fiber → granule cell → Golgi cell input network performs an expansion recoding that enhances the pattern-discrimination capacity and learning speed of the cerebellar Purkinje response cells. Parallel fiber synapses of the dendritic spines of Purkinje cells, basket cells, and stellate cells are all postulated to be specifically variable in response to climbing fiber activity. It is argued that this variability is the mechanism of pattern storage. It is demonstrated that, in order for the learning process to be stable, pattern storage must be accomplished principally by weakening synaptic weights rather than by strengthening them.
2,449 citations
TL;DR: It is shown that in a simple mathematical model the criteria for isolated single-level stability and for total web stability are not identical, but they tend to be similar, so that usually stability at one level goes with stability (or instability) of the whole.
Abstract: First, we consider a simple mathematical model for a many-predator-many-prey system, and show it to be in general less stable, and never more stable, than the analogous one-predator-one-prey community. This result would seem to caution against any simple belief that increasing population stability is a mathematical consequence of increasing multispecies complexity. Second, we take up the question of the relation between stability in any one trophic level by itself and stability of the total trophic web. We find that in a simple mathematical model the criteria for isolated single-level stability and for total web stability are not identical, but they tend to be similar, so that usually stability (or instability) at one level goes with stability (or instability) of the whole. It is possible, however, to construct examples where either stability at one level occurs in an overall unstable system, or alternatively instability at one level goes with total web stability. This model points the way to a synthesis of the diverse views that have been expressed on this subject.
202 citations
TL;DR: The deterministic and stochastic versions of a generalization of Bailey's simple epidemic model are considered in this paper, and the asymptotic values of the expected number of infected as calculated by the two theories do not necessarily agree.
Abstract: The deterministic and stochastic versions of a generalization of Bailey's simple epidemic model are considered. It is shown that the asymptotic values of the expected number of infected as calculated by the two theories do not necessarily agree. Nevertheless, when the mean time to epidemic extinction is calculated from the stochastic theory, it is shown that when the deterministic mean tends to zero, the mean time to extinction remains bounded, whereas when the deterministic mean tends to a finite limit, the mean time to extinction tends to infinity as the total population number grows without bound.
153 citations
TL;DR: Equations governing the size and age structure of a large population composed of two sexes are developed and applications to a population in stable growth and in transient growth are discussed.
Abstract: Equations governing the size and age structure of a large population composed of two sexes are developed. It is assumed that the “significant” environment of the population —that which governs its mortality, nuptiality, and natality patterns and rates— remains constant. Two different social arrangements respecting reproduction are considered. It is assumed first that reproduction is by random mating; it is then assumed that reproduction is via strict monogamous marriage. The equations describing these two cases are coupled and nonlinear. In the case of the random mating model, the problem can be reduced to a single nonlinear integral equation. In the case of the monogamous marriage model, a system of partial differential-integral equations results. Applications of these equations to a population in stable growth and in transient growth are discussed.
120 citations
TL;DR: In this paper, the authors investigated economic optimal policies for the management of natural resource stocks subject to nonconcave reproduction curves and showed that any optimal policy leads either to a sustained yield or to complete destruction of the resource.
Abstract: Economically optimal policies are investigated for the management of natural resource stocks subject to nonconcave reproduction curves. It is shown that any optimal policy leads either to a sustained yield or to complete destruction of the resource. Criteria are established according to which the optimal policy is destructive. The simplifying assumption is made that net revenue (utility) is proportional to the amount of harvest.
111 citations
TL;DR: In this paper, a statistical model that views the microbial population as distributed according to its mass has been analyzed in detail, and solutions were obtained for growth in a well-stirred vessel for batch and continuous systems by the method of weighted residuals.
Abstract: The solution of integropartial differential equations resulting from statistical formulations of the behavior of cell populations has been investigated. In particular, a statistical model that views the microbial population as distributed according to its mass has been analyzed in detail. Solutions were obtained for growth in a well-stirred vessel for batch and continuous systems by the method of weighted residuals. The trial solutions were expressed as linear combinations of basic functions from a set complete in L 2 [ 0 , ∞) and the coefficients of the expansion determined by weighting the residual with functions also from a set complete in L 2 [ 0 , ∞). The solutions obtained by the method of weighted residuals were found to be sufficiently accurate in most cases. By using them as initial approximations to a successive approximation scheme, increased accuracy could be obtained within a short number of iterations. Various choices of basic and weighting functions were considered. The method used in this work may be pertinent to other particle fission processes.
74 citations
TL;DR: In this paper, a technique for solving convolution-type integral equations, previously investigated by the author for the inverse problem of radiography, is extended to biological identification problems and the properties of solutions to convolution type integral equations are formulated in terms of the eigenvalues of a circulant matrix.
Abstract: A technique for solving convolution-type integral equations, previously investigated by the author for the inverse problem of radiography, is extended in this article to biological identification problems. The properties of solutions to convolution-type integral equations are formulated in terms of the eigenvalues of a circulant matrix, the eigenvalues being obtained by diagonalization by the discrete Fourier transform. Statistical properties of the solution, the bias and covariance matrix, are derived, and a discrete Fourier transform technique for computing the covariance matrix is also derived. A brief criticism of biological identification problems in the published literature is also given.
67 citations
TL;DR: In this paper, a review of the central mathematical problem of compartment analysis is presented: What is the set of coefficient matrices appearing in the compartment equations that are compatible with the part of the solution that is observed?
Abstract: A review is presented of the central mathematical problem of compartment analysis: What is the set of coefficient matrices appearing in the compartment equations that are compatible with the part of the solution that is observed? The matrix elements are shown to depend on a generally underdetermined set of nonlinear algebraic equations. Some general properties of the coefficient matrix for an n -compartment system are deduced. The matrix may be made “minimal” by the imposition of null constraints on the matrix elements. The compatibility of such arbitrary constraints with the algebraic equations is investigated. The algebraic simplicity of two-compartment and three-compartment systems permits a complete characterization of the set of minimal matrices in parametric form. Properties of the graphs representing some minimal matrices of a four-compartment system are obtained.
61 citations
TL;DR: In this article, the power spectral densities of the electrocardiograms (ECGs) of seven patients in a coronary care unit were examined and the underlying periodic phenomenon is characterized by only one frequency, the heart rate, and that invariably by the fifteenth multiple of this fundamental frequency almost all the power is accounted for.
Abstract: The power spectral densities of the electrocardiograms (ECGs) of seven patients in a coronary care unit were examined. It was discovered that the underlying periodic phenomenon is characterized by only one frequency, the heart rate, and that invariably by the fifteenth multiple of this fundamental frequency almost all the power is accounted for. Periodic approximations to the ECG based on 7, 10, and 15 harmonics were reconstructed; the fifteenth multiple of the fundamental frequency gave an excellent fit to the actual ECG for each patient. The ECGs were also reconstructed based on the frequencies corresponding to peaks greater than a designated fraction of the peak at the fundamental frequency; 104 elements were required to produce reconstruction comparable to the periodic approximations based on 15 harmonics. In addition, the power spectral densities of the individual waves of the PQRST cycle were determined and analyzed. This method provides an accurate description of the characteristic ECG and the practicality of its application to ECG monitoring for arrhythmia detection is being investigated.
61 citations
TL;DR: Some novel types of control processes arising in the study of optimal drug administration are considered and are treated by dynamic programming techniques.
Abstract: Some novel types of control processes arising in the study of optimal drug administration are considered. These are treated by dynamic programming techniques.
48 citations
TL;DR: In this article, the authors considered a deterministic epidemic model that accounts for certain threshold phenomena occuring in the spread of infection, including a major modification of the earlier model that allows individuals recovered from the infection to eventually become susceptible again to the infection.
Abstract: In an earlier article the authors considered a deterministic epidemic model that accounts for certain threshold phenomena occuring in the spread of infection. The present article includes a major modification of the earlier model that allows individuals recovered from the infection to eventually become susceptible again to the infection. A set of five integral equations is derived from the basic assumptions concerning the spread of the infection. The basic theorem asserts existence, uniqueness, and continuous dependence of a solution of these equations. The proof of these results is accomplished by a contraction mapping argument that also gives a simple proof of the earlier result. A minor modification of the earlier model also is included, which brings into the scope of this study some of the older epidemic models. Some numerical computation carried out for this problem is also described.
TL;DR: In this article, an electric analogue of the saccular aneurysm and its parent vessel is analyzed, and an approximate solution in the form of a Fourier series and graphical analysis is developed, which suggests the danger of either a high pulse rate or increased pressure in leading to critical jump phenomena and increased turbulence of flow within the anouysm.
Abstract: Various physical models of saccular aneurysms of the circle of Willis provide evidence of a nonlinear pressure-volume relationship within the aneurysm. When an electric analogue of the model aneurysm and its parent vessel is analyzed, a form of the Duffing equation results, namely: V +λv− βv 2 +γv 3 =A cos ωt+B where V = velocity of blood flow in the aneurysm, and (Acosωt + B) represents the pulse rate and pressure. An approximate solution in the form of a Fourier series and graphical analysis is developed, which suggests the danger of either a high pulse rate or increased pressure in leading to critical jump phenomena and increased turbulence of flow within the aneurysm.
TL;DR: In this article, the Frobenius theory of nonnegative matrices is applied to derive conditions under which a growth matrix converges to the stationary age distribution, and sufficient conditions for the primitivity of growth matrices are derived from a result in graph theory.
Abstract: Necessary and sufficient conditions for the primitivity of growth matrices are derived from a result in graph theory. The Frobenius theory of nonnegative matrices is applied to derive conditions under which a growth matrix converges to the stationary age distribution.
TL;DR: In this article, the authors consider the compartmental analysis in pharmacokinetics when uncertainties in our knowledge of the rate constants are taken into account and determine the joint probability distribution of the drug concentrations.
Abstract: We consider the compartmental analysis in pharmacokinetics when uncertainties in our knowledge of the rate constants are taken into account. The rate constants are considered as random variables in this article and the joint probability distribution of the drug concentrations are determined. The significance of this stochastic approach is illustrated by simple compartmental models.
TL;DR: In this paper, it was shown that under standard conditions X n m n →V > 0 with probability one, where {Xn} is a Galton-Watson process with immigration, and offspring mean m > 1.
Abstract: In [6] it was shown that under standard conditions X n m n →V > 0 with probability one, where {Xn} is a Galton-Watson process with immigration, and offspring mean m > 1. The authors obtain convergence-rate results under additional conditions for this asymptotic behavior in the form of analogues of the central limit theorem and the law of the iterated logarithm, and similar results for the estimator X n + r X n , which for fixed r ⩾ 1 approaches mr with probability one.
TL;DR: In this paper, a stochastic model is developed to study the effect of density and distribution of prey on total accumulated catch by the predator, which is applicable to search by a predator for prey when the latter are randomly located on a plane with given density, clustered with variable cluster size, or schooling and schools can be represented by disks of variable size.
Abstract: A stochastic model is developed to study the effect of density and distribution of prey on total accumulated catch by the predator. The model is applicable to search by a predator for prey when the latter are randomly located on a plane with given density, clustered with variable cluster size, or schooling and schools can be represented by disks of variable size. Allowance is made for possible escapement of prey. Capturing or delay time for each sighting or catch made can be taken as variable and dependent on the number of prey caught. The radius of detection of prey by the predator may also be fixed or variable. Examples are given for randomly distributed, clustered, and schooling prey.
TL;DR: A statistical method is presented that allows an experimenter to estimate the mean time a cell spends in each of the four phases G 1, S, G 2, and M from the frequencies of these phases in a large population of dividing cells.
Abstract: In this article a model of the cell cycle based on the theory of multitype age-dependent branching processes is developed. The article begins with a brief description of the G 1 , S , G 2 , and M phases of the cell cycle, and in Section 2 a detailed account of the application of the theory of multitype age-dependent branching processes to cell cycle analysis is given. By letting each phase of the cell cycle correspond to a type in the general theory of multitype age-dependent branching processes, it is possible to derive formulas for the frequencies of cells in each of the four phases in a large population of dividing cells. In section 3 it is shown how those formulas for the theoretical frequencies of cells in each of the four phases G 1 , S , G 2 , and M may be used to estimate the mean time a cell spends in each phase, providing we have estimates of the frequencies of cells in each phase in a large population of dividing cells and the total mean generation time of a cell. Finally, a numerical example is given illustrating how the formulas developed in the article may be used in cell cycle analysis. The principal novelty of the results presented in this article center around a statistical method that allows an experimenter to estimate, in a precise way, the mean time a cell spends in each of the four phases G 1 , S , G 2 , and M from the frequencies of these phases in a large population of dividing cells.
TL;DR: This analysis shows that the multiunit multipath characteristics apparently provides adequate signal-to-noise ratio in the summated excitatory postsynaptic potential of the α-motoneuron and is essential for fidelity of information transmission.
Abstract: Information is transmitted in the peripheral nervous system using many similar but not identical channels acting more or less in parallel. This multiunit multipath characteristic of neural communication is utilized, for example, by the monosynaptic spinal reflex (MSR) of the neuromuscular system. In the present article, we investigate the functions performed by this characteristics in the transfer of information in the afferent pathway of the MSR. The factors considered are the multiplicity of elements and neural paths; the distributions of spindle stretch thresholds and afferent conduction speeds; spatial and temporal summation at the motoneuron; and the noise introduced by the sensory encoding process. This analysis shows that the multiunit multipath characteristics apparently provides adequate signal-to-noise ratio in the summated excitatory postsynaptic potential of the α-motoneuron and is essential for fidelity of information transmission.
TL;DR: A complete theoretical system of double inhibitions has been evolved where the inhibitors may be purely or partially competitive, noncompetitive, or purely uncompetitive with the substrate.
Abstract: A complete theoretical system of double inhibitions has been evolved where the inhibitors may be purely or partially competitive, noncompetitive, or purely uncompetitive with the substrate. A method is presented for the determination of interaction constants (a) for inhibitors on (i) the free enzyme, or (ii) the enzyme–substrate complex; and (b) in some cases for the substrate and one of the inhibitors on the enzyme complex formed with the other inhibitor. This method permits determination of the type of a pure inhibition in the presence of an unseparable inhibitor in the enzyme preparation. The different possible mechanisms of action of two inhibitors on the same enzyme are discussed and the method of their determination is described.
TL;DR: In this article, the authors discuss some aspects of the asymptotic behavior of linear differential-difference equations arising in two-compartment models where time lags exist, and present a method for the determination of rate constants and delays on the basis of observations over time.
Abstract: We discuss some aspects of the asymptotic behavior of solutions of linear differential-difference equations arising in two-compartment models where time lags exist, and present a method for the determination of rate constants and delays on the basis of observations over time.
TL;DR: The reformulation of the Hodgkin-Huxley equations to include the effect of series inductance that has recently been suggested by Lieberstein is examined and is found to be about five orders of magnitude too small to influence the conduction velocity of an action potential.
Abstract: The reformulation of the Hodgkin-Huxley equations to include the effect of series inductance that has recently been suggested by Lieberstein is examined. An axon model that includes series inductance is employed to obtain an analytic expression for the conduction velocity of an action potential. A recently developed exact transmission line equivalent circuit is then used to estimate the series inductance for a squid axon. It is found to be about five orders of magnitude too small to influence the conduction velocity.
TL;DR: The motivation for the work is a search for formulation of the general theory of neural nets that could be tied to the theory of triadic intensional relations, as suggested by Warren S. McCulloch.
Abstract: The paper presents the fundamentals of the theory of neural nets with loops based on functional matrices, which are a generalization of the state transition matrices for a net. Problems of analysis and synthesis; given a net, find its functional matrix and vice versa, are treated for probabilistic and deterministic nets. Questions about universal nets, oscillations, and stability are studied for deterministic nets. Reduction of probabilistic nets with loops is considered. It is shown that any probabilistic net with loops can be duplicated by a deterministic net with loops plus a probabilistic loop-free encoder. The motivation for the work is a search for formulation of the general theory of neural nets that could be tied to the theory of triadic intensional relations, as suggested by Warren S. McCulloch.
TL;DR: In this paper, a mathematical model that will accurately reproduce the discrete waves of an electrocardiograph signal is presented. But the model is not suitable for real or observed ECG signals and has many practical applications.
Abstract: It is desirable to construct a mathematical model that will accurately reproduce the discrete waves of an electrocardiograph signal. The usual cycle for one heartbeat includes a P wave (atrial depolarization), QRS waves (ventricular depolarization), and T wave (ventricular repolarization). The model presented simulates real or observed ECG signals and has many practical applications.
TL;DR: In this article, an equation similar to the Goldman equation is derived for the steady-state diffusion of univalent ions across a membrane with arbitrary potential profile and in the presence of electrogenic pumps.
Abstract: An equation similar to the Goldman equation is derived for the steady-state diffusion of univalent ions across a membrane with arbitrary potential profile and in the presence of electrogenic pumps. The presence of electrogenic pumps adds a term proportional to the net pump flux to both numerator and denominator in the Goldman equation. For arbitrary potential profiles the permeabilities of the positive ions are all divided by one potential-dependent factor and the permeabilities of the negative ions are divided by another such factor. Both factors are determined by the potential function and tend to vary inversely. If the symmetric part of the potential function is zero, both factors are equal to 1.0. Hence in the general form of the Goldman equation the relative contributions of the positive and negative ions are weighted by factors that are easily calculated if the potential function is known.
TL;DR: In this article, the authors present a procedure for developing an information science approach to biomathematics and present five assertions that should be in agreement with the three requirements of Section 4 and that exemplify their attitude for developing biomatahematics from the standpoint of information science.
Abstract: Our general views on a procedure for developing an information science approach to biomathematics are presented. Sections 2 and 3 analyze in detail two important fields of mathematics, mathematical analysis and automaton theory, emphasizing that their lack of an information science aspect makes them unsatisfactory for formulating biomathematics. In Section 4 we give three requirements that biomathematics should satisfy in order to provide a mathematical foundation for biological existences and biological phenomena. Section 5 presents five assertions that should be in agreement with the three requirements of Section 4 and that exemplify our attitude for developing biomathematics from the standpoint of information science. Section 6 reviews some works on biological control systems analysis from the standpoint of Section 5.
TL;DR: Quantitative morphological and functional bases are given for an analog computer model of the frog cerebellar cortex based on actual neuronal counts and takes into consideration the three main types of neurons in the cerebellary cortex.
Abstract: Quantitative morphological and functional bases are given for an analog computer model of the frog cerebellar cortex. The model is based on actual neuronal counts and takes into consideration the three main types of neurons in the cerebellar cortex (Purkinje cells, granule cells, and stellate cells) and the two cerebellar afferent systems (mossy and climbing fibers). Physiologically the responses of mossy fibers and Purkinje cells are defined as observed following physiological stimulation of the vestibular system by means of rotative angular acceleration.
TL;DR: In this paper, the authors apply information theory to the multivariate data analysis problem, where given a dependent variable Y and any independent variable X, they associate with X a number that would indicate the importance of X with respect to Y.
Abstract: In this article we apply information theory to the multivariate data analysis problem. Given a dependent variable Y and any independent variable X, we can associate with X a number that would indicate the importance of X with respect to Y. This procedure enables us to select variables based on their importance. Two sets of data (upper abdominal pain and character recognition) are analyzed by this approach. The results indicate that this approach is feasible.
TL;DR: In this article, a method for evaluating the parameters in a differential equation from an experimental solution in the case where the parameters appear as multipliers is described, which gives satisfactory results for precise data but loses accuracy when the data are affected by small random errors.
Abstract: A method is described for evaluating the parameters in a differential equation from an experimental solution in the case where the parameters appear as multipliers. By formal integrations a set of linear equations is obtained with coefficients that can be calculated by numerical integrations. The method gives satisfactory results for precise data but loses accuracy when the data are affected by small random errors.
TL;DR: A new perspective on three papers coauthored by Warren McCulloch emphasizes the paradigmatic role to be played in brain theory by action-oriented studies of distributed computation in somatotopically organized neural networks.
Abstract: A new perspective on three papers coauthored by Warren McCulloch emphasizes the paradigmatic role to be played in brain theory by action-oriented studies of distributed computation in somatotopically organized neural networks.
TL;DR: The problem of a current source of finite extent is solved approximately for the case where the source is located on the plasmalemma itself, and it is shown that dendrites and plasmodesmata give rise to no physiologically significant angular variations in plas malemma potential.
Abstract: It is well known that a considerable angular variation in the plasmalemma potential can be caused by a point source of current located within a cell, but no computationally convenient expression for this variation has been found, except for the case when the point source is located on the plasmalemma itself: such an expression is derived here. In addition, the problem of a current source of finite extent is solved approximately for the case where the source is located on the plasmalemma, and it is shown that dendrites and plasmodesmata give rise to no physiologically significant angular variations in plasmalemma potential.