On the scaling interpretation of exponents in hyperboloid models of delay and probability discounting

Previously, we (McKerchar et al., 2009) showed that two-parameter hyperboloid models ( [Green and Myerson, 2004] and [Rachlin, 2006]) provide significantly better fits to delay discounting data than simple, one-parameter hyperbolic and exponential models. Here, we extend this effort by comparing fits of the two-parameter hyperboloid models to data from a larger sample of participants (N = 171) who discounted probabilistic as well as delayed rewards. In particular, we examined the effects of amount on the exponents in the two hyperboloid models of delay and probability discounting in order to evaluate key theoretical predictions of the standard psychophysical scaling interpretation of these exponents. Both the Rachlin model and the Green and Myerson model provided very good fits to delay and probability discounting of both small and large amounts at both the group and individual levels (all R²s > .97 at the group level; all median R²s > .92 at the individual level). For delay discounting, the exponent in both models did not vary as a function of delayed amount, consistent with the psychophysical scaling interpretation. For probability discounting, however, the exponent in both models increased as the probabilistic amount increased—a finding inconsistent with the scaling interpretation.     

Percent body fat is related to delay and probability discounting for food in humans

This study describes delay and probability discounting patterns for hypothetical food and money in relation to percent body fat (PBF). Sixty university students completed four computerized discounting tasks in which they were asked to make a series of hypothetical decisions between (a) 10 dollars after one of several different delays (1, 2, 30, 180, and 365 days) or a smaller amount of money available immediately; (b) 10 bites of food after one of several delays (1, 2, 5, 10, and 20 h) or a smaller number of bites available immediately; (c) $10 at one of several probabilities (0.9, 0.75, 0.5, 0.25, 0.1) or a smaller amount of money to be received for sure; and (d) 10 bites of food at one of several probabilities (0.9, 0.75, 0.5, 0.25, 0.1) or a smaller number of bites to be received for sure. Median indifference points for all participants across each task were well described using the hyperbolic discounting function. Results suggest that percent body fat predicted discounting for hypothetical food, but not money, using regression analyses with the entire sample and when comparing individuals in the high and low quartiles for PBF. None of the other dietary variables (body mass index, subjective hunger, and time since last meal or snack) were related to discounting patterns. This suggests that individuals with high PBF may exhibit heightened sensitivities to delay and probability when making decisions about food.     

A comparison of two algorithms in computerized temporal discounting procedures

Two algorithms are commonly applied in computerized temporal discounting procedures (Decreasing Adjustment and Double-Limit Algorithms); however, the degree to which the two algorithms produce similar patterns of discounting is unknown. The present experiment compared the two common algorithms across sign (gains and losses) and magnitude ($10 and $1000) conditions. Twenty participants made choices between larger later and smaller sooner alternatives that were presented by each of the algorithms in separate conditions. Strong correlations were found between the two measures; however, the Decreasing Adjustment Algorithm tended to produce lower indifference points and higher rates of discounting than the Double-Limit Algorithm. Both algorithms found significant magnitude effects. Less consistent results were found when comparing the two algorithms across sign. The present results suggest that researchers should apply caution when making comparisons between outcomes of delay discounting studies that have used the two different algorithms. However, the interpretation of findings from individual studies is probably not strongly affected by the use of different computer algorithms.     

Within-subject comparison of degree of delay discounting using titrating and fixed sequence procedures

Different procedures are often used across experiments to estimate the degree of delay discounting, a common measure of impulsivity. In all procedures, participants indicate their choice between a reward available immediately and one available after a delay. The present experiment determined whether there are differences in the degree of discounting for a hypothetical $100 produced by a procedure that titrates the immediate amount (titrating sequence procedure) versus a procedure that presents a fixed sequence of immediate amounts (fixed sequence procedure) using a within-subject design. The adult human participants showed no significant differences in degree of discounting between procedures as assessed by a hyperboloid model and the Area Under the Curve. Furthermore, the Area Under the Curve values from the two procedures showed a strong positive correlation. These findings suggest there may be no systematic difference between the degree of delay discounting as estimated by the titrating sequence and fixed sequence procedures. Given the apparent similarities in the results, it appears researchers may be justified in basing their choice of which procedure to use on convenience.     

Response bias is unaffected by delay length in a delay discounting paradigm

This study examined the contribution of response bias to measures of delay discounting in Long-Evans rats (n = 8) using the adjusting amount procedure. Under this procedure, we assessed preference for 150 μl of 10% sucrose solution delivered following a delay over a variable-amount alternative delivered immediately. Bias was calculated based on relative preference when reinforcers were delivered immediately from both alternatives. We extended this assessment procedure to examine preference when rewards from both alternatives were equally delayed (2, 4, 8, or 16 s) in addition to assessing a traditional delay discounting function. Relative preference was similar across delays and slightly larger than 150 μl. These results indicate that response bias was stable and suggests a relative aversion for the adjusting alternative, which may be due to the variability in reward size associated with that alternative.     

Functional parameters of delay discounting assessment tasks: order of presentation

Several variants in the methods used to estimate delay discounting (DD) have been associated with within-subject differences in degree of DD. This study compared variants in the order of presentation of reward and delay values during assessment of DD of hypothetical cash. Participants were randomly assigned to one of two groups. For one group, the immediate reward values were presented in Ascending order and for the other, they were presented in Descending order. In addition, all participants completed a DD task with the reward values presented in a Random order. Degree of DD, calculated as area under the curve (AUC), was similar between the Ascending and Descending procedures, and was significantly higher with the Random procedure. Within-subjects AUC were positively correlated. Reaction times within choice trials changed systematically as a function of order of presentation of the immediate rewards, and distance to the indifference point within each delay value. Reaction times appear to parallel the effort involved in making the individual choices. Some procedural variants in the assessment of DD yield differences in behavior during the assessment task that affect the magnitude of the estimated delay discounting. Thus, the absolute magnitude of DD may not be directly comparable between methods.     

Effect of delay and social distance on the perceived value of social interaction

Temporal discounting is a process by which the perceived value of an outcome decreases as delay increases. Social discounting is a process by which the perceived value of an outcome decreases as the social distance between the decision-maker and recipient increases. Both temporal and social discounting are well established for monetary outcomes. However, little is known regarding the effect of delay-to-occurrence or increased social distance on the perceived value of social interactions. The current study demonstrates that the decrease in perceived value for delayed social interactions is well described by the same hyperbola-like function used for monetary outcomes and that the rate of discounting decreases (indicating increased preference for larger, more delayed outcomes) as social distance increases. Conceptualizing the current discounting procedure as entailing both potential gains and losses, suggest that under certain circumstances, higher discount rates may not reflect increased preference.     

Parameters of delay discounting assessment: number of trials, effort, and sequential effects.

Procedural variants in estimating delay discounting (DD) have been shown to yield significant within-subject differences in estimated degree of delay discounting as well as variations in the patterns of choice. The purpose of this study was to evaluate the effect of subject control over the number of trials in a delay discounting task, on degree of delay discounting. Participants were assessed with two computerized DD assessments: the full-length method presented participants with a fixed set of 240 trials, and the abbreviated task, where once participants had shown indifference between the immediate and delayed rewards, the remaining trials for that delay value were omitted. While the full-length and abbreviated methods did not differentially affect patterns of choice or estimated delay discounting, the order of presentation (ascending or descending) of immediate rewards produced differences in each measure: rate of delay discounting was significantly lower when estimated with the descending sequence; a larger proportion of area under the discounting curve was concentrated around the indifference point trial with the descending sequence; and a lower correlation was observed between estimates obtained across methods with the descending sequence.     

The genetic basis of delay discounting and its genetic relationship to alcohol dependence

Delay discounting is steeper for individuals who drink heavily or are alcohol dependent, but the reasons for this are unclear. Given the substantial genetic component for alcohol dependence it is not unreasonable to ask whether discounting and alcohol dependence have a genetic relationship. For there to be a genetic relationship, delay discounting must have a genetic component (heritability). A review of the human and animal literature suggests that this is the case. Other literature examining whether discounting is a correlated phenotype in individuals who are genetically predisposed to drink (family history positive individuals and selected lines of rats and mice) is mixed, suggesting that networks of genes are critical for the relationship to be seen. The identities of the genes in this network are not yet known, but research examining polymorphisms associated with differences in discounting is beginning to address this issue.     

Within-subject differences in degree of delay discounting as a function of order of presentation of hypothetical cash rewards

Procedural variants in estimating delay discounting (DD) have been shown to yield significant differences in estimated degree of DD as well as variations in individual patterns of choice. For example, a recent study found significantly different degrees of DD between groups assessed using either an ascending or descending order of presentation of the immediately available rewards. The purpose of this study was to test for within-subject effects of order of presentation of the immediate rewards in a DD task. In a single session, college students (N = 29) were asked to complete two DD tasks, one with the immediate rewards presented in ascending order and one in descending order. Consistent with previous results, significantly larger mean area under the discounting curve (AUC) was observed when the descending sequence was used compared to the ascending order of presentation; and the correlation between both measurements was moderate. These results suggest that some DD assessment tasks may be sensitive to contextual variables such as order and range of the reward and delay values.     

Delay discounting of saccharin in rhesus monkeys

The value of a reinforcer decreases as the time until its receipt increases, a phenomenon referred to as delay discounting. Although delay discounting of non-drug reinforcers has been studied extensively in a number of species, our knowledge of discounting in non-human primates is limited. In the present study, rhesus monkeys were allowed to choose in discrete trials between 0.05% saccharin delivered in different amounts and with different delays. Indifference points were calculated and discounting functions were established. Discounting functions for saccharin were well described by a hyperbolic function. Moreover, the discounting rates for saccharin in all six monkeys were comparable to those of other non-human animals responding for non-drug reinforcers. Also consistent with other studies of non-human animals, changing the amount of a saccharin reinforcer available after a 10-s delay did not affect its relative subjective value. Discounting functions for saccharin were steeper than we found in a previous study with cocaine, raising the possibility that drugs such as cocaine may be discounted less steeply than non-drug reinforcers.     

Domain independence and stability in young and older adults' discounting of delayed rewards

Individual discounting rates for different types of delayed reward are typically assumed to reflect a single, underlying trait of impulsivity. Recently, we showed that discounting rates are orders of magnitude steeper for directly consumable liquid rewards than for monetary rewards (Jimura et al., 2009), raising the question of whether discounting rates for different types of reward covary at the individual level. Accordingly, the present study examined the relation between discounting of hypothetical money and real liquid rewards in young adults (Experiment 1) and older adults (Experiment 2). At the group level, young adults discounted monetary rewards more steeply than the older adults, but there was no significant age difference with respect to liquid rewards. At the individual level, the rates at which young and older participants discounted each reward type were stable over a two- to fifteen-week interval (rs > 70), but there was no significant correlation between the rates at which they discounted the two reward types. These results suggest that although similar decision-making processes may underlie the discounting of different types of rewards, the rates at which individuals discount money and directly consumable rewards may reflect separate, stable traits, rather than a single trait of impulsivity.     

A parameter sensitivity methodology in the context of HIV delay equation models

A sensitivity methodology for nonlinear delay systems arising in one class of cellular HIV infection models is presented. Theoretical foundations for a typical sensitivity investigation and illustrative computations are given.     

A cardiovascular-respiratory control system model including state delay with application to congestive heart failure in humans

This paper considers a model of the human cardiovascular-respiratory control system with one and two transport delays in the state equations describing the respiratory system. The effectiveness of the control of the ventilation rate is influenced by such transport delays because blood gases must be transported a physical distance from the lungs to the sensory sites where these gases are measured. The short term cardiovascular control system does not involve such transport delays although delays do arise in other contexts such as the baroreflex loop (see [46]) for example. This baroreflex delay is not considered here. The interaction between heart rate, blood pressure, cardiac output, and blood vessel resistance is quite complex and given the limited knowledge available of this interaction, we will model the cardiovascular control mechanism via an optimal control derived from control theory. This control will be stabilizing and is a reasonable approach based on mathematical considerations as well as being further motivated by the observation that many physiologists cite optimization as a potential influence in the evolution of biological systems (see, e.g., Kenner [29] or Swan [62]). In this paper we adapt a model, previously considered (Timischl [63] and Timischl et al. [64]), to include the effects of one and two transport delays. We will first implement an optimal control for the combined cardiovascular-respiratory model with one state space delay. We will then consider the effects of a second delay in the state space by modeling the respiratory control via an empirical formula with delay while the the complex relationships in the cardiovascular control will still be modeled by optimal control. This second transport delay associated with the sensory system of the respiratory control plays an important role in respiratory stability. As an application of this model we will consider congestive heart failure where this transport delay is larger than normal and the transition from the quiet awake state to stage 4 (NREM) sleep. The model can be used to study the interaction between cardiovascular and respiratory function in various situations as well as to consider the influence of optimal function in physiological control system performance.Supported by FWF (Austria) under grant F310 as a subproject of the Special Research Center F003 Optimization and ControlMathematics Subject Classification (2000): 92C30, 49J15     

A comparison of three maximum likelihood models for stage-frequency data

A comparison has been made between the estimates obtained from maximum likelihood estimation of gamma, inverse normal, and normal distribution models for stage-frequency data. Results have been compared for six of sets of test data, and from many sets of simulated data. It is concluded that (1) some estimates may differ substantially between the models, (2) estimates from the correct model have little bias, and estimated standard errors are generally close to theoretical values, (3) there are problems in determining degrees of freedom for chi-squared goodness of fit tests, so that it is best to compare test statistics with simulated distributions, and (4) goodness of fit tests may not discriminate well between the three models.     

Role of stimulus-stimulus pairing in matching-to-sample procedure: cross-species comparison of humans and pigeons

The present experiment examined whether, in a matching-to-sample (MTS) procedure, a relation between two stimuli, a sample and a comparison, could be established as a result of just stimulus–stimulus pairing, even if back up reinforcers were never provided for the conditional relation between the sample and comparison stimuli, but rather only for the comparison stimulus. A procedure called “pseudo matching-to-sample” was used in which, when S1 was presented as a sample stimulus, two comparison stimuli (C1 and C2) were presented, and only responses to C1 were reinforced. Conversely, when S2 was presented, only responses to C3 (and not C4) were reinforced. In other words, organisms experiencing this procedure could discriminate C1 from C2, and C3 from C4, by simple discrimination without regard to the conditional sample stimuli. In order to examine cross-species differences, responding by humans in this procedure was compared to that by pigeons. Although the humans developed a discriminative function for the sample stimuli, that is, the humans’ responding was affected by both the sample stimuli and the reinforcers, responding by the pigeons was affected solely by the reinforcers. These data suggest that, in this procedure, humans (but not pigeons) are able to learn relations among stimuli simply as a result of stimulus–stimulus pairing.     

Identification and comparison of stochastic metabolic/hemodynamic models (sMHM) for the generation of the BOLD signal

This paper extends a previously formulated deterministic metabolic/hemodynamic model for the generation of blood oxygenated level dependent (BOLD) responses to include both physiological and observation stochastic components (sMHM). This adds a degree of flexibility when fitting the model to actual data by accounting for un-modelled activity. We then show how the innovation method can be used to estimate unobserved metabolic/hemodynamic as well as vascular variables of the sMHM, from simulated and actual BOLD data. The proposed estimation method allowed for doing model comparison by calculating the model’s AIC and BIC. This methodology was then used to select between different neurovascular coupling assumptions underlying sMHM. The proposed framework was first validated on simulations and then applied to BOLD data from a motor task experiment. The models under comparison in the analysis of the actual data considered that vascular response was coupled to: (I) inhibition, (II) excitation, (III) both excitation and inhibition. Data was best described by model II, although model III was also supported.     

A slowing effect on visual search by advance information in pigeons (Columba livia): a comparison with humans (Homo sapiens)

A comparative study was conducted to investigate whether the search for a target letter was facilitated when the target and prime (preceding stimulus) letters were identical. Pigeons (Section 2) and human participants (Section 3) were first trained to search for “A” among “Y”s and “E” among “D”s in a condition in which a square shape appeared as the prime (Neutral condition). In subsequent testing, a prime was identical either to the corresponding target (Target-priming condition) or to the distractor (Distractor-priming condition). Humans and pigeons responded differently to the two priming conditions. On early trials, the Target prime facilitated search in humans, reducing reaction times (RTs) to targets. In pigeons, however, RTs were longer with Target primes, suggesting that pre-exposure to target letters may directly inhibit the search for targets in subsequent search displays. Furthermore, pre-exposure to the distractor letters may inhibit the processing of the distractor. On later trials, RTs of humans were faster in both priming conditions than in the Neutral condition, suggesting that expectation of a target facilitated search (“Y” predicted “A” and “D” predicted “E”). In contrast, the pigeons showed no evidence of expectation-based facilitation, with constant slowing effects of the Target prime extending across sessions. Possible mechanisms underlying such a slowing priming effect in pigeons were discussed.     

Implementing vertex dynamics models of cell populations in biology within a consistent computational framework

The dynamic behaviour of epithelial cell sheets plays a central role during development, growth, disease and wound healing. These processes occur as a result of cell adhesion, migration, division, differentiation and death, and involve multiple processes acting at the cellular and molecular level. Computational models offer a useful means by which to investigate and test hypotheses about these processes, and have played a key role in the study of cell–cell interactions. However, the necessarily complex nature of such models means that it is difficult to make accurate comparison between different models, since it is often impossible to distinguish between differences in behaviour that are due to the underlying model assumptions, and those due to differences in the in silico implementation of the model. In this work, an approach is described for the implementation of vertex dynamics models, a discrete approach that represents each cell by a polygon (or polyhedron) whose vertices may move in response to forces. The implementation is undertaken in a consistent manner within a single open source computational framework, Chaste, which comprises fully tested, industrial-grade software that has been developed using an agile approach. This framework allows one to easily change assumptions regarding force generation and cell rearrangement processes within these models. The versatility and generality of this framework is illustrated using a number of biological examples. In each case we provide full details of all technical aspects of our model implementations, and in some cases provide extensions to make the models more generally applicable.     

Toxicokinetic models as predictive tools in biomonitoring metals in zooplankton: a synthesis of ideas

The objective of the present study was to evaluate previously reported compartment models as a predictive tool, in particular, to investigate the potential of nonlinear predictions in case of saturation of the uptake process. Examples supporting this idea are presented for the amphipods Gammarus oceanicus from northern Norway and Themisto abyssorum from the Greenland Sea as well as for the copepod Calanoides acutus from the Weddell Sea. A further objective was to investigate verification of model parameters with independent experimental observations, aided by the statistical analysis of the results. This can be regarded as a first step toward a meta-analysis. While modeling results obtained for T. abyssorum in 1993 underestimated observed uptake of Pb and Cu in independent experiments (1999) in the same species, a good agreement was encountered between models obtained for T. abyssorum in 1993 and experimental data obtained for Themisto libellula (1999). Furthermore, a fairly good agreement was found between modeling results obtained for Pb in Calanus hyperboreus from the Greenland Sea (1993) and experimental data obtained for C. acutus (1999). Therefore, the biological species cannot be regarded as a “natural” unit in biomonitoring studies without further evaluation (viz. calibration as biomonitor), while even samples of animals belonging to different species may serve as such a unit, allowing for a comparison of bioaccumulation on large temporal and spatial scales. Finally, the concept of the sensitivity of zooplankton organisms for biomonitoring was elaborated in detail. These procedures involve analytical and experimental variability and model simulations based on previously derived parameters of compartment models. The minimal increments in exposure concentrations estimated in this way were 0.4–1.2 μg Pb l⁻¹ for T. abyssorum, T. libellula, C. hyperboreus, and C. acutus. Corresponding results for Cu varied to a greater extent: 4–9 μg l⁻¹ for the amphipods and 1–5 μg l⁻¹ for the copepods. A correction of these values is proposed, taking into account the ratios of experimental (kinetic) and field bioconcentration factors; this yields exposure concentrations close to those reported for natural seawater. Guest editors: U. M. Azeiteiro, I. Jenkinson & M. J. Pereira Plankton Studies