A new mathematical model for the homeostatic effects of sleep loss on neurobehavioral performance

The two-process model of sleep regulation makes accurate predictions of sleep timing and duration for a variety of experimental sleep deprivation and nap sleep scenarios. Upon extending its application to waking neurobehavioral performance, however, the model fails to predict the effects of chronic sleep restriction. Here we show that the two-process model belongs to a broader class of models formulated in terms of coupled non-homogeneous first-order ordinary differential equations, which have a dynamic repertoire capturing waking neurobehavioral functions across a wide range of wake/sleep schedules. We examine a specific case of this new model class, and demonstrate the existence of a bifurcation: for daily amounts of wakefulness less than a critical threshold, neurobehavioral performance is predicted to converge to an asymptotically stable state of equilibrium; whereas for daily wakefulness extended beyond the critical threshold, neurobehavioral performance is predicted to diverge from an unstable state of equilibrium. Comparison of model simulations to laboratory observations of lapses of attention on a psychomotor vigilance test (PVT), in experiments on the effects of chronic sleep restriction and acute total sleep deprivation, suggests that this bifurcation is an essential feature of performance impairment due to sleep loss. We present three new predictions that may be experimentally verified to validate the model. These predictions, if confirmed, challenge conventional notions about the effects of sleep and sleep loss on neurobehavioral performance. The new model class implicates a biological system analogous to two connected compartments containing interacting compounds with time-varying concentrations as being a key mechanism for the regulation of psychomotor vigilance as a function of sleep loss. We suggest that the adenosinergic neuromodulator/receptor system may provide the underlying neurobiology.     

On mathematical modeling of circadian rhythms, performance, and alertness

Mathematical models of neurobehavioral performance and alertness have both basic science and practical applications. These models can be especially useful in predicting the effect of different sleep-wake schedules on human neurobehavioral objective performance and subjective alertness under many conditions. Several relevant models currently exist in the literature. In principle, the development and refinement of any mathematical model should be based on an explicit modeling methodology, such as the Box modeling paradigm, that formally defines the model structure and calculates the set of parameters. While most mathematical models of neurobehavioral performance and alertness include homeostatic, circadian, and sleep inertia components and their interactions, there may be fundamental differences in the equations included in these models. In part, these may be due to differences in the assumptions of the underlying physiology. Because the choice of model equations can have a dramatic influence on the results, it is necessary to consider these differences in assumptions when examining the results from a model and when comparing results across models. This article presents principles of mathematical modeling and examples of how such procedures can be applied to the development and refinement of mathematical models of neurobehavioral performance and alertness. This article also presents several methods of testing and comparing these models, suggests different uses of the models, and discusses problems with current models.     

Accounting for Partial Sleep Deprivation and Cumulative Sleepiness in the Three‐Process Model of Alertness Regulation

Mathematical models designed to predict alertness or performance have been developed primarily as tools for evaluating work and/or sleep‐wake schedules that deviate from the traditional daytime orientation. In general, these models cope well with the acute changes resulting from an abnormal sleep but have difficulties handling sleep restriction across longer periods. The reason is that the function representing recovery is too steep—usually exponentially so—and with increasing sleep loss, the steepness increases, resulting in too rapid recovery. The present study focused on refining the Three‐Process Model of alertness regulation. We used an experiment with 4 h of sleep/night (nine participants) that included subjective self‐ratings of sleepiness every hour. To evaluate the model at the individual subject level, a set of mixed‐effect regression analyses were performed using subjective sleepiness as the dependent variable. These mixed models estimate a fixed effect (group mean) and a random effect that accounts for heterogeneity between participants in the overall level of sleepiness (i.e., a random intercept). Using this technique, a point was sought on the exponential recovery function that would explain maximum variance in subjective sleepiness by switching to a linear function. The resulting point explaining the highest amount of variance was 12.2 on the 1–21 unit scale. It was concluded that the accumulation of sleep loss effects on subjective sleepiness may be accounted for by making the recovery function linear below a certain point on the otherwise exponential function.     

Uncovering physiologic mechanisms of circadian rhythms and sleep/wake regulation through mathematical modeling

Mathematical models of neurobehavioral function are useful both for understanding the underlying physiology and for predicting the effects of rest-activity-work schedules and interventions on neurobehavioral function. In a symposium titled "Modeling Human Neurobehavioral Performance I: Uncovering Physiologic Mechanisms" at the 2006 Society for Industrial and Applied Mathematics/Society for Mathematical Biology (SIAM/SMB) Conference on the Life Sciences, different approaches to modeling the physiology of human circadian rhythms, sleep, and neurobehavioral performance and their usefulness in understanding the underlying physiology were examined. The topics included key elements of the physiology that should be included in mathematical models, a computational model developed within a cognitive architecture that has begun to include the effects of extended wake on information-processing mechanisms that influence neurobehavioral function, how to deal with interindividual differences in the prediction of neurobehavioral function, the applications of systems biology and control theory to the study of circadian rhythms, and comparisons of these methods in approaching the overarching questions of the underlying physiology and mathematical models of circadian rhythms and neurobehavioral function. A unifying theme was that it is important to have strong collaborative ties between experimental investigators and mathematical modelers, both for the design and conduct of experiments and for continued development of the models.     

Accounting for partial sleep deprivation and cumulative sleepiness in the Three-Process Model of alertness regulation

Mathematical models designed to predict alertness or performance have been developed primarily as tools for evaluating work and/or sleep-wake schedules that deviate from the traditional daytime orientation. In general, these models cope well with the acute changes resulting from an abnormal sleep but have difficulties handling sleep restriction across longer periods. The reason is that the function representing recovery is too steep--usually exponentially so--and with increasing sleep loss, the steepness increases, resulting in too rapid recovery. The present study focused on refining the Three-Process Model of alertness regulation. We used an experiment with 4 h of sleep/night (nine participants) that included subjective self-ratings of sleepiness every hour. To evaluate the model at the individual subject level, a set of mixed-effect regression analyses were performed using subjective sleepiness as the dependent variable. These mixed models estimate a fixed effect (group mean) and a random effect that accounts for heterogeneity between participants in the overall level of sleepiness (i.e., a random intercept). Using this technique, a point was sought on the exponential recovery function that would explain maximum variance in subjective sleepiness by switching to a linear function. The resulting point explaining the highest amount of variance was 12.2 on the 1-21 unit scale. It was concluded that the accumulation of sleep loss effects on subjective sleepiness may be accounted for by making the recovery function linear below a certain point on the otherwise exponential function.     

Developing mathematical models of neurobehavioral performance for the "real world"

Work-related operations requiring extended wake durations, night, or rotating shifts negatively affect worker neurobehavioral performance and health. These types of work schedules are required in many industries, including the military, transportation, and health care. These industries are increasingly using or considering the use of mathematical models of neurobehavioral performance as a means to predict the neurobehavioral deficits due to these operational demands, to develop interventions that decrease these deficits, and to provide additional information to augment existing decision-making processes. Recent advances in mathematical modeling have allowed its application to real-world problems. Developing application-specific expertise is necessary to successfully apply mathematical models, in part because development of new algorithms and methods linking the models to the applications may be required. During a symposium, "Modeling Human Neurobehavioral Performance II: Towards Operational Readiness," at the 2006 SIAM-SMB Conference on the Life Sciences, examples of the process of applying mathematical models, including model construction, model validation, or developing model-based interventions, were presented. The specific applications considered included refining a mathematical model of sleep/wake patterns of airline flight crew, validating a mathematical model using railroad operations data, and adapting a mathematical model to develop appropriate countermeasure recommendations based on known constraints. As mathematical models and their associated analytical methods continue to transition into operational settings, such additional development will be required. However, major progress has been made in using mathematical model outputs to inform those individuals making schedule decisions for their workers.     

An improved method for estimating human circadian phase derived from multichannel ambulatory monitoring and artificial neural networks

Recently, we developed a novel method for estimating human circadian phase with noninvasive ambulatory measurements combined with subject-independent multiple regression models and a curve-fitting approach. With this, we were able to estimate circadian phase under real-life conditions with low subject burden, i.e., without need of constant routine (CR) laboratory conditions, and without measuring standard circadian markers, such as core body temperature (CBT) or pineal hormone melatonin rhythms. The precision of ambulatory-derived estimated circadian phase was within an error of 12?±?41?min (mean?±?SD) in comparison to melatonin phase during a CR protocol. The physiological measures could be reduced to a triple combination: skin temperatures, irradiance in the blue spectral band of ambient light, and motion acceleration. Here, we present a nonlinear regression model approach based on artificial neural networks for a larger data set (25 healthy young males), including both the original data and additional data collected in the same protocol and using the same equipment. Throughout our validation study, subjects wore multichannel ambulatory monitoring devices and went about their daily routine for 1 wk. The devices collected a large number of physiological, behavioral, and environmental variables, including CBT, skin temperatures, cardiovascular and respiratory functions, movement/posture, ambient temperature, spectral composition and intensity of light perceived at eye level, and sleep logs. After the ambulatory phase, study volunteers underwent a 32-h CR protocol in the laboratory for measuring unmasked circadian phase (i.e., "midpoint" of the nighttime melatonin rhythm). To overcome the complex masking effects of many different confounding variables during ambulatory measurements, neural network-based nonlinear regression techniques were applied in combination with the cross-validation approach to subject-independent prediction of circadian phase. The most accurate estimate of circadian phase with a prediction error of -3?±?23?min (mean?±?SD) was achieved using only two types of the measured variables: skin temperatures and irradiance for ambient light in the blue spectral band. Compared to our previous linear multiple regression modeling approach, motion acceleration data can be excluded and prediction accuracy, nevertheless, improved. Neural network regression showed statistically significant improvement of variance of prediction error over traditional approaches in determining circadian phase based on single predictors (CBT, motion acceleration, or sleep logs), even though none of these variables was included as predictor. We, therefore, have identified two sets of noninvasive measures that, combined with the prediction model, can provide researchers and clinicians with a precise measure of internal time, in spite of the masking effects of daily behavior. This method, here validated in healthy young men, requires testing in a clinical or shiftwork population suffering from circadian sleep-wake disorders. (Author correspondence: vitaliy.kolodyazhniy@sbg.ac.at ).     

An Improved Method for Estimating Human Circadian Phase Derived From Multichannel Ambulatory Monitoring and Artificial Neural Networks

Recently, we developed a novel method for estimating human circadian phase with noninvasive ambulatory measurements combined with subject-independent multiple regression models and a curve-fitting approach. With this, we were able to estimate circadian phase under real-life conditions with low subject burden, i.e., without need of constant routine (CR) laboratory conditions, and without measuring standard circadian markers, such as core body temperature (CBT) or pineal hormone melatonin rhythms. The precision of ambulatory-derived estimated circadian phase was within an error of 12?±?41?min (mean?±?SD) in comparison to melatonin phase during a CR protocol. The physiological measures could be reduced to a triple combination: skin temperatures, irradiance in the blue spectral band of ambient light, and motion acceleration. Here, we present a nonlinear regression model approach based on artificial neural networks for a larger data set (25 healthy young males), including both the original data and additional data collected in the same protocol and using the same equipment. Throughout our validation study, subjects wore multichannel ambulatory monitoring devices and went about their daily routine for 1 wk. The devices collected a large number of physiological, behavioral, and environmental variables, including CBT, skin temperatures, cardiovascular and respiratory functions, movement/posture, ambient temperature, spectral composition and intensity of light perceived at eye level, and sleep logs. After the ambulatory phase, study volunteers underwent a 32-h CR protocol in the laboratory for measuring unmasked circadian phase (i.e., “midpoint” of the nighttime melatonin rhythm). To overcome the complex masking effects of many different confounding variables during ambulatory measurements, neural network–based nonlinear regression techniques were applied in combination with the cross-validation approach to subject-independent prediction of circadian phase. The most accurate estimate of circadian phase with a prediction error of ?3?±?23?min (mean?±?SD) was achieved using only two types of the measured variables: skin temperatures and irradiance for ambient light in the blue spectral band. Compared to our previous linear multiple regression modeling approach, motion acceleration data can be excluded and prediction accuracy, nevertheless, improved. Neural network regression showed statistically significant improvement of variance of prediction error over traditional approaches in determining circadian phase based on single predictors (CBT, motion acceleration, or sleep logs), even though none of these variables was included as predictor. We, therefore, have identified two sets of noninvasive measures that, combined with the prediction model, can provide researchers and clinicians with a precise measure of internal time, in spite of the masking effects of daily behavior. This method, here validated in healthy young men, requires testing in a clinical or shiftwork population suffering from circadian sleep-wake disorders. (Author correspondence: vitaliy.kolodyazhniy@sbg.ac.at)     

An application of a mixed-effects location scale model for analysis of Ecological Momentary Assessment (EMA) data

Summary .   For longitudinal data, mixed models include random subject effects to indicate how subjects influence their responses over repeated assessments. The error variance and the variance of the random effects are usually considered to be homogeneous. These variance terms characterize the within-subjects (i.e., error variance) and between-subjects (i.e., random-effects variance) variation in the data. In studies using ecological momentary assessment (EMA), up to 30 or 40 observations are often obtained for each subject, and interest frequently centers around changes in the variances, both within and between subjects. In this article, we focus on an adolescent smoking study using EMA where interest is on characterizing changes in mood variation. We describe how covariates can influence the mood variances, and also extend the standard mixed model by adding a subject-level random effect to the within-subject variance specification. This permits subjects to have influence on the mean, or location, and variability, or (square of the) scale, of their mood responses. Additionally, we allow the location and scale random effects to be correlated. These mixed-effects location scale models have useful applications in many research areas where interest centers on the joint modeling of the mean and variance structure.     

The Problem of Auto-Correlation in Parasitology

Explaining the contribution of host and pathogen factors in driving infection dynamics is a major ambition in parasitology. There is increasing recognition that analyses based on single summary measures of an infection (e.g., peak parasitaemia) do not adequately capture infection dynamics and so, the appropriate use of statistical techniques to analyse dynamics is necessary to understand infections and, ultimately, control parasites. However, the complexities of within-host environments mean that tracking and analysing pathogen dynamics within infections and among hosts poses considerable statistical challenges. Simple statistical models make assumptions that will rarely be satisfied in data collected on host and parasite parameters. In particular, model residuals (unexplained variance in the data) should not be correlated in time or space. Here we demonstrate how failure to account for such correlations can result in incorrect biological inference from statistical analysis. We then show how mixed effects models can be used as a powerful tool to analyse such repeated measures data in the hope that this will encourage better statistical practices in parasitology.     

Predicting the timing and duration of sleep in an operational setting using social factors

In recent years, there has been increasing interest in the use of bio-mathematical models to predict alertness, performance, and/or fatigue in operational settings. Current models use only biological factors to make their estimations, which can be limited in operational settings where social and geo-physical factors also dictate when sleep occurs. The interaction between social and biological factors that help determine the timing and duration of sleep during layover periods have been investigated in order to create and initially validate a mathematical model that may better predict sleep in the field. Participants were 32 male transmeridian airline pilots (17 captains, 10 first officers, and 5 second officers) flying the Sydney-Bangkok-London-Singapore-Sydney (SYD-LHR) pattern. Participants continued their regular schedule while wearing activity monitors and completing sleep and work diaries. The theoretical sleep timing model underpinning this analysis consists of separate formulations for short (<32 h) and long (>32 h) break periods. Longer break periods are split into three distinct phases-recovery (break start until first local night), personal (first local night until last local night), and preparation phases (last local night until break end)-in order to exploit potential differences specific to each. Furthermore, an iterative procedure combining prediction and retrodiction (i.e., using future duty timing information to predict current sleep timing) was developed to optimize predictive ability. Analysis found an interaction between the social and circadian sleep pressures that changed over the break period. Correlation analysis indicated a strong relationship between the actual sleep and new model's predictions (r = 0.7-0.9), a significant improvement when compared to existing models (r = 0.1-0.4). Social and circadian pressures play important roles in regulating sleep for international flight crews. An initial model has been developed in order to regulate sleep in these crews. The initial results have shown promise when applied to small sets of data; however, more rigorous validation must be carried out.     

Multilevel modeling of chronotype and weekdays versus weekends to predict nonrestorative sleep

Nonrestorative sleep, a form of subjective sleep disturbance that has been largely neglected in the literature, is newly accessible to researchers via the validated restorative sleep questionnaire (RSQ). The daily version of the RSQ allows for analysis of within-subjects variation in restorative sleep across repeated samplings, and such day-to-day regularity in sleep variables has been highlighted as an important new direction for research. The present study used a sophisticated statistical approach, multilevel modeling, to examine the contributions of circadian chronotype, calendar day of questionnaire completion (weekends versus weekdays), and their interaction in explaining both interindividual and intraindividual variance in restorative sleep. Analyses were conducted using an archival dataset of college undergraduates who continuously completed daily RSQs over a 14-day sampling period. In the final multilevel model, possessing an evening type predicted lower restorative sleep between subjects, while sampling on weekdays predicted lower restorative sleep within subjects. Furthermore, a cross-level interaction was observed, such that the difference in restorative sleep on weekends versus weekdays was more pronounced among those with greater evening circadian preference. All of the effects were maintained after accounting for the significant influence of gender (women had less restorative sleep than men). These results are theoretically consistent with findings that evening types display stronger disparities in sleep schedules across free and workdays (i.e., social jet lag), and attest to the usefulness of multilevel models for statistically investigating how stable traits interact with factors that vary day to day (e.g., work or school schedules) in influencing sleep outcomes.     

THE INFLUENCE OF CIRCADIAN PHASE AND PRIOR WAKE ON NEUROMUSCULAR FUNCTION

Previous forced desynchrony (FD) studies have shown that neurobehavioral function is affected by circadian phase and duration of prior wakefulness. There is some evidence that neuromuscular function may also be affected by circadian phase and prior wake, but these effects have not been systematically investigated. This study examined the effects of circadian phase and prior wake on two measures of neuromuscular function—postural balance (PB) and maximal grip strength (MGS)—using a 28-h FD protocol. Eleven male participants (mean?±?SD: 22.7?±?2.5 yr) lived in a sound-attenuated, light- and temperature-controlled time-isolation laboratory for 12 days. Following two training days and a baseline day, participants were scheduled to seven 28-h FD days, with the ratio between sleep opportunity and wake spans kept constant (i.e., 9.3?h sleep period and 18.7?h wake period). PB was measured during 1?min of quiet standing on a force platform. MGS of the dominant hand was measured using a dynamometer. These two measures were obtained every 2.5?h during wake. Core body temperature was continuously recorded with rectal thermistors to determine circadian phase. For both measures of neuromuscular function, individual data points were assigned a circadian phase and a level of prior wake. Data were analyzed by repeated-measures analysis of variance (ANOVA) with two within-subjects factors: circadian phase (six phases) and prior wake (seven levels). For MGS, there was a main effect of circadian phase, but no main effect of prior wake. For PB, there were no main effects of circadian phase or prior wake. There were no interactions between circadian phase and prior wake for MGS or PB. The significant effect of circadian phase on muscle strength is in agreement with previous reports in the literature. In terms of prior wake, both MGS and PB remained relatively stable across wake periods, indicating that neuromuscular function may be more robust than neurobehavioral function when the duration of wakefulness is within a normal range (i.e., 18.7?h). (Author correspondence: charli.sargent@unisa.edu.au)     

Predicting the Timing and Duration of Sleep in an Operational Setting Using Social Factors

In recent years, there has been increasing interest in the use of bio‐mathematical models to predict alertness, performance, and/or fatigue in operational settings. Current models use only biological factors to make their estimations, which can be limited in operational settings where social and geo‐physical factors also dictate when sleep occurs. The interaction between social and biological factors that help determine the timing and duration of sleep during layover periods have been investigated in order to create and initially validate a mathematical model that may better predict sleep in the field. Participants were 32 male transmeridian airline pilots (17 captains, 10 first officers, and 5 second officers) flying the Sydney‐Bangkok‐London‐Singapore‐Sydney (SYD‐LHR) pattern. Participants continued their regular schedule while wearing activity monitors and completing sleep and work diaries. The theoretical sleep timing model underpinning this analysis consists of separate formulations for short (<32 h) and long (>32 h) break periods. Longer break periods are split into three distinct phases—recovery (break start until first local night), personal (first local night until last local night), and preparation phases (last local night until break end)—in order to exploit potential differences specific to each. Furthermore, an iterative procedure combining prediction and retrodiction (i.e., using future duty timing information to predict current sleep timing) was developed to optimize predictive ability. Analysis found an interaction between the social and circadian sleep pressures that changed over the break period. Correlation analysis indicated a strong relationship between the actual sleep and new model's predictions (r=0.7–0.9), a significant improvement when compared to existing models (r=0.1–0.4). Social and circadian pressures play important roles in regulating sleep for international flight crews. An initial model has been developed in order to regulate sleep in these crews. The initial results have shown promise when applied to small sets of data; however, more rigorous validation must be carried out.     

Emotionally Biased Cognitive Processes: The Weakest Link Predicts Prospective Changes in Depressive Symptom Severity

Emotional biases in attention, interpretation, and memory are predictive of future depressive symptoms. It remains unknown, however, how these biased cognitive processes interact to predict depressive symptom levels in the long-term. In the present study, we tested the predictive value of two integrative approaches to model relations between multiple biased cognitive processes, namely the additive (i.e., cognitive processes have a cumulative effect) vs. the weakest link (i.e., the dominant pathogenic process is important) model. We also tested whether these integrative models interacted with perceived stress to predict prospective changes in depressive symptom severity. At Time 1, participants completed measures of depressive symptom severity and emotional biases in attention, interpretation, and memory. At Time 2, one year later, participants were reassessed to determine depressive symptom levels and perceived stress. Results revealed that the weakest link model had incremental validity over the additive model in predicting prospective changes in depressive symptoms, though both models explained a significant proportion of variance in the change in depressive symptoms from Time 1 to Time 2. None of the integrative models interacted with perceived stress to predict changes in depressive symptomatology. These findings suggest that the best cognitive marker of the evolution in depressive symptoms is the cognitive process that is dominantly biased toward negative material, which operates independent from experienced stress. This highlights the importance of considering idiographic cognitive profiles with multiple cognitive processes for understanding and modifying effects of cognitive biases in depression.     

The use of structural dynamic models to explain successes and failures of biomanipulation

A development of a structural dynamic model, i.e. a model with current change of the most important parameters according to a goal function, is presented with the aim to explain the structural changes observed in lakes, when the nutrient concentration is increased or decreased. This type of models may be important in lake management as it may be possible qualitatively to predict the success or failure of biomanipulation. The answer to the crucial question: àt which phosphorus level will the success of biomanipulation be most probable?' will probably require the development of model which takes into account site specific processes and properties, i.e., a more complicated model. As goal function is proposed the thermodynamic function, exergy, which is defined as the work content of the system (model) compared with the system at thermodynamic equilibrium. It is shown that the structural dynamic modelling approach has been able to explain the shift from large to small zooplankton species at a certain level of phosphorus concentration, accompanied by a shifts from a dominance of zooplankton, and predatory fish to a system dominated by planktivorous fish and phytoplankton. The shift in zooplankton species cannot be explained by application of catastrophe theoretical models, which have been used to explain the hysteresis reaction. The results show that the shift should be expected at approximately 0.12 mg P l-1 and that a typical hysteresis reaction occurs at this concentration in accordance with the expectations. These results are consistent with many observations but should be interpreted with great caution, as the model is simple and general and don't account for a number of processes which may influence the results significantly in specific lake studies. The structural dynamic approach has previously been used in ten case studies with good agreement with the observations, but more case studies are needed before a general recommendation of the use of this type of models can be given. The results from this study point toward to apply this type of models for lake management where biomanipulation is involved, although it should be recommended to improve the presented general model with introduction of site specific properties for a considered lake study.     

Evaluation of Scat Deposition Transects versus Radio Telemetry for Developing a Species Distribution Model for a Rare Desert Carnivore,the Kit Fox

Development and evaluation of noninvasive methods for monitoring species distribution and abundance is a growing area of ecological research. While noninvasive methods have the advantage of reduced risk of negative factors associated with capture, comparisons to methods using more traditional invasive sampling is lacking. Historically kit foxes (Vulpes macrotis) occupied the desert and semi-arid regions of southwestern North America. Once the most abundant carnivore in the Great Basin Desert of Utah, the species is now considered rare. In recent decades, attempts have been made to model the environmental variables influencing kit fox distribution. Using noninvasive scat deposition surveys for determination of kit fox presence, we modeled resource selection functions to predict kit fox distribution using three popular techniques (Maxent, fixed-effects, and mixed-effects generalized linear models) and compared these with similar models developed from invasive sampling (telemetry locations from radio-collared foxes). Resource selection functions were developed using a combination of landscape variables including elevation, slope, aspect, vegetation height, and soil type. All models were tested against subsequent scat collections as a method of model validation. We demonstrate the importance of comparing multiple model types for development of resource selection functions used to predict a species distribution, and evaluating the importance of environmental variables on species distribution. All models we examined showed a large effect of elevation on kit fox presence, followed by slope and vegetation height. However, the invasive sampling method (i.e., radio-telemetry) appeared to be better at determining resource selection, and therefore may be more robust in predicting kit fox distribution. In contrast, the distribution maps created from the noninvasive sampling (i.e., scat transects) were significantly different than the invasive method, thus scat transects may be appropriate when used in an occupancy framework to predict species distribution. We concluded that while scat deposition transects may be useful for monitoring kit fox abundance and possibly occupancy, they do not appear to be appropriate for determining resource selection. On our study area, scat transects were biased to roadways, while data collected using radio-telemetry was dictated by movements of the kit foxes themselves. We recommend that future studies applying noninvasive scat sampling should consider a more robust random sampling design across the landscape (e.g., random transects or more complete road coverage) that would then provide a more accurate and unbiased depiction of resource selection useful to predict kit fox distribution.     

Personality development in psychotherapy: a synergetic model of state-trait dynamics

Theoretical models of psychotherapy not only try to predict outcome but also intend to explain patterns of change. Studies showed that psychotherapeutic change processes are characterized by nonlinearity, complexity, and discontinuous transitions. By this, theoretical models of psychotherapy should be able to reproduce these dynamic features. Using time series derived from daily measures through internet-based real-time monitoring as empirical reference, we earlier presented a model of psychotherapy which includes five state variables and four trait variables. In mathematical terms, the traits modulate the shape of the functions which define the nonlinear interactions between the variables (states) of the model. The functions are integrated into five coupled nonlinear difference equations. In the present paper, we model how traits (dispositions or competencies of a person) can continuously be altered by new experiences and states (cognition, emotion, behavior). Adding equations that link states to traits, this model not only describes how therapeutic interventions modulate short-term change and fluctuations of psychological states, but also how these can influence traits. Speaking in terms of Synergetics (theory of self-organization in complex systems), the states correspond to the order parameters and the traits to the control parameters of the system. In terms of psychology, trait dynamics is driven by the states—i.e., by the concrete experiences of a client—and creates a process of personality development at a slower time scale than that of the state dynamics (separation of time scales between control and order parameter dynamics).     

Integrating coalescent and ecological niche modeling in comparative phylogeography

Understanding the factors that contribute to the formation of population genetic structure is a central goal of phylogeographic research, but achieving this goal can be complicated by the stochastic variance inherent to genetic processes. Statistical approaches to testing phylogeographic hypotheses accommodate this stochasticity by evaluating competing models of putative historical population structure, often by simulating null distributions of the expected variance. The effectiveness of these tests depends on the biological realism of the models. Information from the fossil record can aid in reconstructing the historical distributions of some taxa. However, for the majority of taxa, which lack sufficient fossils, paleodistributional modeling can provide valuable spatial-geographic data concerning ancestral distributions. Paleodistributional models are generated by projecting ecological niche models, which predict the current distribution of each species, onto a model of past climatic conditions. Here, we generate paleodistributional models describing the suitable habitat during the last glacial maximum for lineages from the mesic forests of the Pacific Northwest of North America, and use these models to generate alternative phylogeographic hypotheses. Coalescent simulations are then used to test these hypotheses to improve our understanding of the historical events that promoted the formation of population genetic structure in this ecosystem. Results from Pacific Northwest mesic forest organisms demonstrate the utility of these combined approaches. Paleodistribution models and population genetic structure are congruent across three amphibian lineages, suggesting that they have responded in a concerted manner to environmental change. Two other species, a willow and a water vole, despite being currently codistributed and having similar population genetic structure, were predicted by the paleodistributional model to have had markedly different distributions during the last glacial maximum. This suggests that congruent phylogeographic patterns can arise from incongruent ancestral distributions. Paleodistributional models introduce a much-needed spatial-geographic perspective to statistical phylogeography. In conjunction with coalescent models of population genetic structure, they have the potential to improve our understanding of the factors that promote population divergence and ultimately produce regional patterns of biodiversity.     

Empirical assessment of the relation between ecological connectivity and land complexity based on information-theoretic metrics

Habitat fragmentation and connectivity loss pose significant threats to biodiversity at both local and landscape levels. Strategies to increase ecological connectivity and preserve strong connectivity are important for dealing with the potential threat of habitat degradation. Various metrics have been used to measure (i.e., quantify) landscape composition and configuration in landscape ecology. However, their relationship with ecological connectivity must be understood to interpret landscape patterns comprehensively. In the present study, correlations between ecological connectivity and land complexity are examined based on information-theory metrics. Two primary questions are explored: (1) to what extent are landscape mosaic measures of entropy correlated with ecological connectivity, with landscape gradient-based measures, and with each other? (2) are landscape gradient-based entropy measures correlated with ecological connectivity more than discrete entropy measures? Results show that all information theoretic metrics are statistically significant (p < 0.05) for modelling ecological connectivity. Among categorically-based indices, the relationship between ECI and joint entropy was the most significant, while a generalized additive model indicated that Boltzmann entropy could predict the ecological connectivity index, explaining ∼60% of the variance. Therefore, configurational entropy can be used for improving ecological connectivity models.