Augmented discounting: interaction between ageing and time-preference behaviour

Discounting occurs when an immediate benefit is systematically valued more highly than a delayed benefit of the same magnitude. It is manifested in physiological and behavioural strategies of organisms. This study brings together life-history theory and time-preference theory within a single modelling framework. We consider an animal encountering reproductive opportunities as a random process. Under an external hazard, optimal life-history strategy typically prioritizes immediate reproduction at the cost of declining fertility and increasing mortality with age. Given such ageing, an immediate reproductive reward should be preferred to a delayed reward because of both the risk of death and declining fertility. By this analysis, ageing is both a consequence of discounting by the body and a cause of behavioural discounting. A series of models is developed, making different assumptions about external hazards and biological ageing. With realistic ageing assumptions (increasing mortality and an accelerating rate of fertility decline) the time-preference rate increases in old age. Under an uncertain external hazard rate, young adults should also have relatively high time-preference rates because their (Bayesian) estimate of the external hazard is high. Middle-aged animals may therefore be the most long term in their outlook.     

Asymptotically robust variance estimation for person‐time incidence rates

Person‐time incidence rates are frequently used in medical research. However, standard estimation theory for this measure of event occurrence is based on the assumption of independent and identically distributed (iid) exponential event times, which implies that the hazard function remains constant over time. Under this assumption and assuming independent censoring, observed person‐time incidence rate is the maximum‐likelihood estimator of the constant hazard, and asymptotic variance of the log rate can be estimated consistently by the inverse of the number of events. However, in many practical applications, the assumption of constant hazard is not very plausible. In the present paper, an average rate parameter is defined as the ratio of expected event count to the expected total time at risk. This rate parameter is equal to the hazard function under constant hazard. For inference about the average rate parameter, an asymptotically robust variance estimator of the log rate is proposed. Given some very general conditions, the robust variance estimator is consistent under arbitrary iid event times, and is also consistent or asymptotically conservative when event times are independent but nonidentically distributed. In contrast, the standard maximum‐likelihood estimator may become anticonservative under nonconstant hazard, producing confidence intervals with less‐than‐nominal asymptotic coverage. These results are derived analytically and illustrated with simulations. The two estimators are also compared in five datasets from oncology studies.     

Scale-free neurodegeneration: cellular heterogeneity and the stretched exponential kinetics of cell death

Neurodegenerative disorders are an insidious group of diseases characterized by severe physical and cognitive effects that often have devastating consequences for the lives of affected individuals and their families. One feature common to a significant proportion of these diseases is that affected neurons commit to undergoing an active form of degeneration known as programmed cell death, or apoptosis. Although intense effort over the past several years has resulted is a remarkable increase in our understanding of the molecular events involved in neurodegeneration, our knowledge regarding the cellular and tissue properties that determine the temporal patterns of neuronal attrition is limited. We recently demonstrated that neurodegenerative kinetics in various diseases fit well to exponential decay functions, and proposed a universal one-hit switch mechanism in which mutant and injured neurons exist in a viable state characterized by an increased but constant risk of initiating apoptosis (Nature, 406, p. 195). Here we show that a heavy-tailed stretched exponential function is better able to account for neurodegenerative kinetic data. Moreover, normalization of all available data according to their corresponding best-fit stretched exponential parameters suggest that the generalized model is consistent with a universal mechanism of neuronal cell death that is greatly improved over the constant risk model. In contrast to the original model in which all cells exhibit an identical risk of initiating apoptosis, the stretched exponential model is consistent with each neuron experiencing a constant risk that is different from that experienced by other cells in the degenerating population, perhaps due to spatial differences in the cellular microenvironment. Intriguingly, the predicted distribution of risk across the cell population can be fit by a power-law function, further suggesting that scale-free properties of degenerating neuronal tissues might act as potent regulators of the kinetics of cell death in neural tissue.     

The kinetics of proteinase K digestion of linear prion polymers.

Transmissible spongiform encephalopathies such as scrapie are caused by a protein-only infectious agent, known as a prion. It is not clear how a protein can be capable of replicating itself, and the mechanism remains controversial. One influential model hypothesizes that prions are nucleated, macroscopically linear polymers. We investigated the theoretical kinetics of this model and derived predictions which could be used to test the model. In the model, the polymerization and depolymerization rates are independent polymer size. This leads to an exponential size distribution at equilibrium. In agreement with a prediction stemming from this size distribution, the average size of PrP-res polymers was proportional to the square root of the concentration of PrP-res in a published study of in vitro conversion. Prion digestion by proteinase K (PK) is predicted to be biphasic. The second phase of digestion should be virtually independent of the PK concentration and should depend on the initial size distribution of prion polymers. For initially equilibrated polymers with an exponential size distribution, phase two digestion is exponential at a predicted rate. This rate varies in a defined way with the concentration used for equilibration and with other parameters which affect the average polymer size.     

Dissociable reward and timing signals in human midbrain and ventral striatum

Reward prediction error (RPE) signals are central to current models of reward-learning. Temporal difference (TD) learning models posit that these signals should be modulated by predictions, not only of magnitude but also timing of reward. Here we show that BOLD activity in the VTA conforms to such TD predictions: responses to unexpected rewards are modulated by a temporal hazard function and activity between a predictive stimulus and reward is depressed in proportion to predicted reward. By contrast, BOLD activity in ventral striatum (VS) does not reflect a TD RPE, but instead encodes a signal on the variable relevant for behavior, here timing but not magnitude of reward. The results have important implications for dopaminergic models of cortico-striatal learning and suggest a modification of the conventional view that VS BOLD necessarily reflects inputs from dopaminergic VTA neurons signaling an RPE.     

Identifying Change Points in a Covariate Effect on Time-to-Event Analysis with Reduced Isotonic Regression

Isotonic regression is a useful tool to investigate the relationship between a quantitative covariate and a time-to-event outcome. The resulting non-parametric model is a monotonic step function of a covariate X and the steps can be viewed as change points in the underlying hazard function. However, when there are too many steps, over-fitting can occur and further reduction is desirable. We propose a reduced isotonic regression approach to allow combination of small neighboring steps that are not statistically significantly different. In this approach, a second stage, the reduction stage, is integrated into the usual monotonic step building algorithm by comparing the adjacent steps using appropriate statistical testing. This is achieved through a modified dynamic programming algorithm. We implemented the approach with the simple exponential distribution and then its extension, the Weibull distribution. Simulation studies are used to investigate the properties of the resulting isotonic functions. We apply this methodology to the Diabetes Control and Complication Trial (DCCT) data set to identify potential change points in the association between HbA1c and the risk of severe hypoglycemia.     

Conformational fluctuations and protein reactivity. Determination of the rate-constant spectrum and consequences in elementary biochemical processes

When a protein's active site happens to be strongly coupled with the protein structure, the rate constant of the reaction may eventually be modulated by the conformational fluctuations. Evidence for this effect has long been provided by extensive flash photolysis investigations of liganded hemoproteins and more recently of the non-heme respiratory protein hemerythrin in hydro-organic solvents. Within a given protein conformational substate, an elementary reaction step is characterized by one single free energy barrier and by a first-order rate constant, k, which changes with temperature according to an Arrhenius law. At physiological temperature and low viscosity, ultrafast conformational relaxation causes efficient averaging of the reaction rates and the protein displays exponential kinetics with an average rate constant (k). Under sufficiently general conditions, it can be shown that (k) also follows a simple Arrhenius law with 'effective' values of the pre-exponential factor Aeff and activation enthalpy Heff. It is found that Aeff strongly depends on the overall shape of the rate constant distribution and that Heff actually corresponds to the lower limit of the enthalpy of activation, i.e. the value associated with the highest possible reaction rate. The underlying distribution of rate constants can be reconstructed from a set of experiments in which the kinetics depart from an exponential, i.e. at low temperature and high viscosity. The most probable distribution of exponentials consistent with the observed kinetics of the geminate recombinations of oxygen with photodissociated hemerythrin has been determined by using a new approach, known as the maximum entropy method. The results are consistent with a single pre-exponential value and a distributed enthalpy spectrum. As expected, Heff does not coincide either with the most probable nor with the average value of the enthalpy. The most salient findings are that the probability for any protein molecule to have an enthalpy of activation equal to the effective value Heff vanishes and that Aeff differs by nearly three orders of magnitude from the true value A0. Biochemical reaction rates are actually average values, since protein reactions are measured under physiological conditions, where conformational relaxation is always fast. Our understanding of the significance of Aeff and Heff is therefore entirely dependent on the knowledge of the distribution function of the rate constants. In particular, enthalpy and entropy terms of similar reactions performed by different proteins cannot be compared as long as the distribution of the rate constants remains unknown.     

On the Fisher Information Matrix in Type II Censored Data from the Exponentiated Exponential Family

In this note, we express, in a general setting, the Fisher information matrix under Type II censoring in terms of the hazard function and then obtain the Fisher information matrix under Type II censoring as a single integral for the exponentiated exponential family, which can be easily evaluated. The Fisher information under Type II censoring can also be used to characterize the exponential distribution among the exponentiated exponential family.     

We'll Meet Again: Revealing Distributional and Temporal Patterns of Social Contact

What are the dynamics and regularities underlying social contact, and how can contact with the people in one''s social network be predicted? In order to characterize distributional and temporal patterns underlying contact probability, we asked 40 participants to keep a diary of their social contacts for 100 consecutive days. Using a memory framework previously used to study environmental regularities, we predicted that the probability of future contact would follow in systematic ways from the frequency, recency, and spacing of previous contact. The distribution of contact probability across the members of a person''s social network was highly skewed, following an exponential function. As predicted, it emerged that future contact scaled linearly with frequency of past contact, proportionally to a power function with recency of past contact, and differentially according to the spacing of past contact. These relations emerged across different contact media and irrespective of whether the participant initiated or received contact. We discuss how the identification of these regularities might inspire more realistic analyses of behavior in social networks (e.g., attitude formation, cooperation).     

A generalized Gompertz-Rayleigh model as a survival distribution

A generalized Gompertz-Rayleigh distribution is proposed as a potential survival distribution for the case of modeling dual components of risk acting on a population. The hazard function is expressed as the sum of two risks. The early risk component allows for either a high initial death rate that can decrease rapidly or an initial death rate that increases to a maximum before decaying. The late risk component is generally relatively constant with the possibility of an increase after a long followup period. An example is provided based on data collected on 569 patients who were operated on for the simultaneous replacement of aortic and mitral heart valves at the University of Alabama in Birmingham Medical Center.     

The differing effects of interference on individuals of different rank

A widely studied ecological model considers the distribution of individuals of different competitive abilities between a number of resources, such that each opti-mizes its reward rate. Recently this model has been criticised for making the predic-tion that the reward rate of an individual can increase under some circumstances with increasing competitor number. The originators of the model have challenged the suggestion that this prediction is biologically implausible and challenged empiricists to test this prediction. Here it is shown that a previous study on hares is consistent with the controversial prediction of the original model, but not with that of an alternative formulation.     

Log-normal distribution of the trace element data results from a mixture of stocahstic input and deterministic internal dynamics

In previous article, we showed a log-normal distribution of boron and lithium in human urine. This type of distribution is common in both biological and nonbiological applications. It can be observed when the effects of many independent variables are combined, each of which having any underlying distribution. Although elemental excretion depends on many variables, the one-compartment open model following a first-order process can be used to explain the elimination of elements. The rate of excretion is proportional to the amount present of any given element; that is, the same percentage of an existing element is eliminated per unit time, and the element concentration is represented by a deterministic negative power function of time in the elimination time-course. Sampling is of a stochastic nature, so the dataset of time variables in the elimination phase when the sample was obtained is expected to show Normal distribution. The time variable appears as an exponent of the power function, so a concentration histogram is that of an exponential transformation of Normally distributed time. This is the reason why the element concentration shows a log-normal distribution. The distribution is determined not by the element concentration itself, but by the time variable that defines the pharmacokinetic equation.     

New Better than Used and Other Concepts for a Class of Life Distributions

It is well-known that the inequalities used in the definition of the New Better than Used (N. B. U.) and the New Better than Used in Expectation (N.B.U.E.) concepts, see BARLOW and PROSCHAN (1965, 1975) become equalities if, and only if, the life length of an organism follows an exponential distribution. It is proved in the present paper that these inequalities also reduce to equalities for the class of life distributions that have the “setting the clock back to zero” property. Simple examples of these distributions include the exponential, the linear hazard exponential and the Gompertz distributions. The General Krane distributions (Krane 1963) belong to this class, as well as a recent model introduced by CHIANG and CONFORTI (1989) of a survival distribution in which the hazard rate is a function of the accumulated effect of an individual's continuous exposure to the toxic material in the environment and his biological reaction to the toxin absorbed. As a simple application of the result proved in the paper, the life expectancy of an organism at age γ₀ involved in the N.B.U.E. concept is evaluated for the Gompertzian growth process and for the Chiang and Conforti model.     

Humans can adopt optimal discounting strategy under real-time constraints

Critical to our many daily choices between larger delayed rewards, and smaller more immediate rewards, are the shape and the steepness of the function that discounts rewards with time. Although research in artificial intelligence favors exponential discounting in uncertain environments, studies with humans and animals have consistently shown hyperbolic discounting. We investigated how humans perform in a reward decision task with temporal constraints, in which each choice affects the time remaining for later trials, and in which the delays vary at each trial. We demonstrated that most of our subjects adopted exponential discounting in this experiment. Further, we confirmed analytically that exponential discounting, with a decay rate comparable to that used by our subjects, maximized the total reward gain in our task. Our results suggest that the particular shape and steepness of temporal discounting is determined by the task that the subject is facing, and question the notion of hyperbolic reward discounting as a universal principle.     

Piecewise exponential survival curves with smooth transitions

Several models of a population survival curve composed of two piecewise exponential distributions are developed. In one formulation the hazard rate changes at a point that is an unobservable random variable that varies between individuals. The population hazard function may decrease with age even when all individuals' hazards are increasing. In a second formulation, the population hazard function is modeled directly. Several models are fit to the survival history of a cohort of 5751 highly inbred male Drosophila melanogaster and the British coal mining disaster data.     

On the interpretation and application of mean times to extinction

As a metric of population viability, conservation biologists routinely predict the mean time to extinction (MTE). Interpretation of MTE depends on the underlying distribution of times to extinction (DTE). Despite claims to the contrary, all information regarding extinction risk can be obtained from this single statistic, the MTE, provided the DTE is exponential. We discuss the proper interpretation of MTE and illustrate how to calculate any population viability statistic when only the MTE is known and the DTE is assumed to be exponential. We also discuss the restrictive assumptions underlying the exponential DTE and the conditions under which alternative models for the DTE are preferable to the conventional (exponential) model. Despite superficial similarities between the exponential and alternative DTEs, several key differences can lead to substantially different interpretations of the MTE.     

Colonisation of freshwater habitats by the European eel Anguilla anguilla

1. The spatial distribution of European eels in 18 U.K. rivers was related to distance from tidal limit using a negative exponential model. This function accounted for between 19 and 90% of the variation in eel density where quantitative data was available. For semiquantitative data the negative exponential function was a significant predictor of eel densities in only six out of 10 cases, although all rivers showed a consistent decline in abundance with distance upstream from the tidal limit. 2. The spatial distribution of different age groups of European eel in River Severn showed an initial rapid dispersion into freshwater followed by a much slower dispersion rate. Movement of the population upstream by a wave‐form migration process does not occur in this system. Instead colonisation of freshwaters can be seen as a two‐phase dispersion. Phase‐1 is a rapid dispersion upstream driven by density at the point source. Phase‐2 commences once the eels become yellow eels and is equivalent to random diffusion of particles. 3. These processes have important implications for the penetration of freshwaters with reduced numbers of eel larvae arriving on the coast of Europe and North America. Eel abundance will decrease more in freshwaters in an upstream direction whilst it may remain stable or decrease to a lesser extent in estuaries. They are also able to explain the demography of eels migrating upstream over weirs and the observations of varying sex ratios within catchments. We conclude that a dispersion model dependent on age, temperature, difficulty of migration, habitat quality and density of eels should be an important part of freshwater eel management.     

Linking exponential components to kinetic states in Markov models for single-channel gating

Discrete state Markov models have proven useful for describing the gating of single ion channels. Such models predict that the dwell-time distributions of open and closed interval durations are described by mixtures of exponential components, with the number of exponential components equal to the number of states in the kinetic gating mechanism. Although the exponential components are readily calculated (Colquhoun and Hawkes, 1982, Phil. Trans. R. Soc. Lond. B. 300:1-59), there is little practical understanding of the relationship between components and states, as every rate constant in the gating mechanism contributes to each exponential component. We now resolve this problem for simple models. As a tutorial we first illustrate how the dwell-time distribution of all closed intervals arises from the sum of constituent distributions, each arising from a specific gating sequence. The contribution of constituent distributions to the exponential components is then determined, giving the relationship between components and states. Finally, the relationship between components and states is quantified by defining and calculating the linkage of components to states. The relationship between components and states is found to be both intuitive and paradoxical, depending on the ratios of the state lifetimes. Nevertheless, both the intuitive and paradoxical observations can be described within a consistent framework. The approach used here allows the exponential components to be interpreted in terms of underlying states for all possible values of the rate constants, something not previously possible.     

The effects of reward quality on risk-sensitivity in Rattus norvegicus

Risk-sensitive foraging theory (RSFT) was developed to explain a choice between a variable (risk-prone) or constant (risk-averse) option. In the RSFT literature, qualitative shifts in risk-sensitivity have been explained by fluctuations in daily caloric energy budget (DEB). The DEB rule describes foragers’ choices as being based on fitness and rate of gain. If the DEB rule is correct, rewards that differ in caloric returns should cause differences in foragers’ sensitivity to risk. However, few studies have explored the influence of reward quality on risk-sensitivity in mammals. The present study was designed to examine the effects of reward quality on risk-sensitivity when reward magnitude, delay to reward, body mass, and response effort were controlled. Results from the current study demonstrated that subjects rewarded with a high calorie reward (i.e., sugar) made significantly fewer choices for a variable option than subjects rewarded with a lower calorie reward (i.e., grain). These results are consistent with the predictions of the DEB rule, and add to the RSFT literature where reward quality was manipulated by describing difference in risk-sensitivity in mammals. Suggestions for future research include an examination of risk-sensitivity where flavor and caloric return are manipulated.