Dynamic models of behavior: An extension of life history theory

Life history theory and behavioral ecology, two branches in the study of adaptation, have relied extensively on mathematical models, but have tended to employ different types of models, and different currencies of fitness. Recently, a new approach based on dynamic, state variable models has been increasingly applied to the study of behavioral adaptations. In fact, this approach amounts to a unification of life history theory and behavioral ecology, to the extent that the line separating the two fields is virtually obliterated. Dynamic models (usually solved by computer) can yield both general principles and testable, quantitative or qualitative predictions about specific behavioral and life history phenomena.     

An extension of the coevolution theory of the origin of the genetic code

Background  

The coevolution theory of the origin of the genetic code suggests that the genetic code is an imprint of the biosynthetic relationships between amino acids. However, this theory does not seem to attribute a role to the biosynthetic relationships between the earliest amino acids that evolved along the pathways of energetic metabolism. As a result, the coevolution theory is unable to clearly define the very earliest phases of genetic code origin. In order to remove this difficulty, I here suggest an extension of the coevolution theory that attributes a crucial role to the first amino acids that evolved along these biosynthetic pathways and to their biosynthetic relationships, even when defined by the non-amino acid molecules that are their precursors.     

Slow changes and wiener analysis of nonlinear summation in contractions of cat muscles

With regular trains of stimuli at a high frequency, the contribution of each stimulus to the force generated over time declines from the second to about the tenth stimulus, but then begins to increase again. This late increase is referred to as tetanic potentiation in analogy with the post-tetanic potentiation of the twitch after such a period of stimulation. With regular trains of stimuli at a low frequency, a progressive decrease in the essentially unfused twitches (negative staircase) is observed in the slow soleus muscle of the cat, while a progressive increase (positive staircase) is observed for the fast plantaris muscle. The time constant for the approximately exponential changes observed is on the order of 10 s. Random trains of stimuli were applied at intermediate frequencies and analyzed in terms of general methods of analysis for nonlinear systems. Systematic decreases in the magnitude and increases in the time course of the average tension per stimulus were observed with increasing mean rates of stimulation. Similar changes were observed for short intervals between stimuli within a given random train at a constant mean rate. These changes can be described in terms of an early depression and a later facilitation described in the previous papers in this series.     

A theory of stochastic harvesting in stochastic environments

We investigate how model populations respond to stochastic harvesting in a stochastic environment. In particular, we show that the effects of variable harvesting on the variance in population density and yield depend critically on the autocorrelation of environmental noise and on whether the endogenous dynamics of the population display over- or undercompensation to density. These factors interact in complicated ways; harvesting shifts the slope of the renewal function, and the net effect of this shift will depend on the sign and magnitude of the other influences. For example, when environmental noise exhibits a positive autocorrelation, the relative importance of a variable harvest to the variance in density increases with overcompensation but decreases with undercompensation. For a fixed harvesting level, an increasing level of autocorrelation in environmental noise will decrease the relative variation in population density when overcompensation would otherwise occur. These and other intricate interactions have important ramifications for the interpretation of time series data when no prior knowledge of demographic or environmental details exists. These effects are important whenever the harvesting rate is sufficiently high or variable, conditions likely to occur in many systems, whether the harvesting is caused by commercial exploitation or by any other strong agent of density-independent mortality.     

An extension of the theory of resonances of biological cells: I. Effects of viscosity and compressibility

The effect of the compressibility and the viscosity of the medium on the resonance of biological cells is discussed mathematically. The resonant frequency is only slightly altered, but the motion is highly damped.     

An extension of the theory of resonances of biological cells: II. Cross-section in a plane wave

The scattering and absorption cross-sections for the resonant modes of biological cells in plane waves are computed. These are both so many orders of magnitude smaller than the physical cross-section that the scattering or absorption by a suspension of cells in a plane wave should be negligible.     

On the stochastic theory of compartments: II. Multi-compartment systems

A stochastic model is developed for a system of interconnected compartments. The generating function of the random variable of any compartment can be constructed from a flow graph involving the expectations of the random variables of all compartments of the system.     

On the Fokker-Planck equation in the stochastic theory of mortality: II

This is the continuation of part I, which was published in the September, 1963, issue ofThe Bulletin. Section 5 treats the special case in which the left absorbing barrier recedes to −∞, leaving essentially only one barrier at a finite distance Λ (>0) from the origin. The eigenfunctions are now parabolic cylinder functions. The limiting cases Λ→+∞ and Λ→0 are also considered. Though meaningless for practical applications to our problem, they are of interest, mathematically, because the Green’s function for the solution of the Fokker-Planck equation assumes a particularly simple form. In section 6 we study, by means of an example, how the “force of mortality” may vary with time before attaining its final asymptotic value. Section7, still dealing with only one absorbing barrier, shows that our results for “strong homeostasis” are identical with those derived by Chandrasekhar for the escape of particles through a potential barrier in the limiting case of quasi-static flow. Precise conditions are given for the validity of both the quasi-static and the Smoluchowski approximations to the Fokker-Planck equation. Finally, in section 8, a brief mention is made of Gevrey’s method for the solution of parabolic partial differential equations.     

On the Forkker-Planck equation in the stochastic theory of mortality: I

This paper, consisting of two parts, gives all the mathematical details that were omitted in a previous work by G. A. Sacher and E. Trucco (“The Stochastic Theory of Mortality.”Ann. N. Y. Acad. Sci.,96, 985–1007, cited here as ST). We assume that the reader is familiar with ST, where the stochastic theory of mortality, originally proposed by Sacher, is discussed at length. We recall that the basic model presented there refers to an ensemble of particles performing Brownian motion in one dimension, with the added constraint of two absorbing barriers. These two points, collectively, are designated as the “lethal bound.” Part I (section 1 to 4) deals with the special case in which the two absorbing barriers are symmetrically located at a finite distance from the origin. The solution of the Fokker-Planck equation is obtained from the theory of eigenvalue problems. Quite generally, the eigenfunctions functions belong to the family of Kummer's confluent hypergeometric functions, but the symmetry condition imposed here results in considerable simplification and makes it possible to estimate the first few eigenvalues by a graphical procedure. In section 3 we show how perturbation theory can be applied in the limiting case of “weak homeostasis,” and section 4 deals with the opposite extreme of “strong homeostasis.” A rigorous proof is given for the result corresponding to equation (28) of ST (asymptotic or quasi-static approximation for the “force of mortality”). This work was performed under the auspices of the U.S. Atomic Energy Commission.     

An approximate stochastic optimal control framework to simulate nonlinear neuro-musculoskeletal models in the presence of noise

Optimal control simulations have shown that both musculoskeletal dynamics and physiological noise are important determinants of movement. However, due to the limited efficiency of available computational tools, deterministic simulations of movement focus on accurately modelling the musculoskeletal system while neglecting physiological noise, and stochastic simulations account for noise while simplifying the dynamics. We took advantage of recent approaches where stochastic optimal control problems are approximated using deterministic optimal control problems, which can be solved efficiently using direct collocation. We were thus able to extend predictions of stochastic optimal control as a theory of motor coordination to include muscle coordination and movement patterns emerging from non-linear musculoskeletal dynamics. In stochastic optimal control simulations of human standing balance, we demonstrated that the inclusion of muscle dynamics can predict muscle co-contraction as minimal effort strategy that complements sensorimotor feedback control in the presence of sensory noise. In simulations of reaching, we demonstrated that nonlinear multi-segment musculoskeletal dynamics enables complex perturbed and unperturbed reach trajectories under a variety of task conditions to be predicted. In both behaviors, we demonstrated how interactions between task constraint, sensory noise, and the intrinsic properties of muscle influence optimal muscle coordination patterns, including muscle co-contraction, and the resulting movement trajectories. Our approach enables a true minimum effort solution to be identified as task constraints, such as movement accuracy, can be explicitly imposed, rather than being approximated using penalty terms in the cost function. Our approximate stochastic optimal control framework predicts complex features, not captured by previous simulation approaches, providing a generalizable and valuable tool to study how musculoskeletal dynamics and physiological noise may alter neural control of movement in both healthy and pathological movements.     

A model for nonlinear stochastic behavior of the pupil

In binocular fusion, pairs of left and right stimuli yielding the same brightness perception constitute an equibrightness curve in a coordinate system whose ordinate and abscissa correspond to the left and right stimulus strengths. A neural network model is presented to elucidate the characteristics of the curve. According to the model, Fechner's paradox is due to the threshold characteristics of the neuron. If the shapes or movements are radically different between the left and right stimuli, the retinal rivalry is caused. That is, only the left stimulus is perceived at one moment and the right stimulus at another moment. The period of left or right eye dominance alternates randomly from time to time. The distribution of the period is approximate to the gamma distribution. In order to account for this fact, a neural network model is proposed, which consists of a pair of neurons receiving inputs with stochastic fluctuations. The computer simulation was carried out with satisfactory results. The model of retinal rivalry is integrated with that of brightness perception.     

An extension of the theory of resonances of biological cells: III. Relationship of breakdown curves and mechanical Q

It has been shown that the experimental breakdown curves are much too sharp to be accounted for simply by a mechanical resonance of the biological cells. A detailed discussion of the coupling of these modes and their action indicate a method of estimating mechanicalQ from the experimental curves. This value agrees with the theoretical predictions.     

Thermodynamics of polyelectrolyte solutions. An empirical extension of the manning theory to finite salt concentrations

The range of application of the Manning theory of polyelectrolyte solutions (J. Chem. Phys., 51 , 924 (1969)) is extended to finite concentrations of simple electrolyte by the empirical superposition of excess free energies arising from (i) interactions between mobile ions and polyions and (ii) mutual interactions between mobile ions. A comparison of published results with the modified theory shows excellent agreement over a wide range of concentrations. Results for a polyelectrolyte of low charge density suggest that the effective inter-charge spacing may be less than that of the fully extended polyion.     

An extension to the metabolic control theory taking into account correlations between enzyme concentrations.

The classical metabolic control theory [Kacser, H. & Burns, J.A. (1973) Symp. Soc. Exp. Biol.27, 65-104; Heinrich, R. & Rapoport, T. (1974) Eur. J. Biochem.42, 89-95.] does not take into account experimental evidence for correlations between enzyme concentrations in the cell. We investigated the implications of two causes of linear correlations: competition between enzymes, which is a mere physical adaptation of the cell to the limitation of resources and space, and regulatory correlations, which result from the existence of regulatory networks. These correlations generate redistribution of enzyme concentrations when the concentration of an enzyme varies; this may dramatically alter the flux and metabolite concentration curves. In particular, negative correlations cause the flux to have a maximum value for a defined distribution of enzyme concentrations. Redistribution coefficients of enzyme concentrations allowed us to calculate the 'combined response coefficient' that quantifies the response of flux or metabolite concentration to a perturbation of enzyme concentration.     

On the stochastic theory of compartments: I. A single-compartment system

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.     

Gaussian approximations for phylogenetic branch length statistics under stochastic models of biodiversity

Stein's method for Gaussian approximations and derived results are used to study the distribution of two phylogenetic branch length statistics: the total height of cherries and the sum of external branch lengths. The Gaussian approximations are obtained under a particular model of phylogenetic tree recently introduced by Popovic. Under an appropriate normalization the model is shown to behave similarly as the coalescent, and the approximations given here are also valid in this context.