This work addresses the prediction of the reacting flow field in a swirl stabilized gas turbine model combustor using large-eddy simulation. The modeling of the combustion chemistry is based on laminar premixed flamelets and the effect of turbulence-chemistry interaction is considered by a presumed shape probability density function. The prediction capabilities of the presented combustion model for perfectly premixed and partially premixed conditions are demonstrated. The effect of partial premixing for the prediction of the reacting flow field is assessed by comparison of a perfectly premixed and partially premixed simulation. Even though significant mixture fraction fluctuations are observed, only small impact of the non-perfect premixing is found on the flow field and flame dynamics. Subsequently, the effect of heat loss to the walls is assessed assuming perfectly premixing. The adiabatic baseline case is compared to heat loss simulations with adiabatic and non-adiabatic chemistry tabulation. The results highlight the importance of considering the effect of heat loss on the chemical kinetics for an accurate prediction of the flow features. Both heat loss simulations significantly improve the temperature prediction, but the non-adiabatic chemistry tabulation is required to accurately capture the chemical composition in the reacting layers. In order to meet the stringent regulations related to pollutant emissions while increasing the cycle efficiency, lean premixed combustion is widely used in stationary gas turbine engines. For aeronautical applications, additional space and weight limitations require a maximum efficiency combined with a high compactness of the combustor design. Therefore, a common design approach for flame stabilization is the use of an aerodynamic swirler. The azimuthal momentum introduced by the swirling flow increases the mixing of fuel and oxidizer and creates a low pressure region forming an internal recirculation zone (IRZ) that increases the overall flow residence time. Computational fluid dynamics (CFD) is routinely applied during the development process of new engine designs. However, high fidelity modeling of the turbulence and combustion processes is required to increase the prediction capabilities and reliability for such complex flows. With the increase in computing power, the application of large-eddy simulation (LES) becomes available at the industrial level to model complex combustion systems. LES allows to obtain not only more accurate predictions of mean values, but also evaluate the effects of unsteadiness and fluctuations of important quantities in the flow. However, due to the different scales involved in turbulent combustion problems, the resolution of detailed chemical reactions still remains out of scope for many complex combustion applications. Instead, turbulent combustion models based on flamelet approaches can be used to accurately predict detailed chemical reactions at reduced computational cost , 2.sgs) turbulence-chemistry interactions is presented. The model takes chemical kinetics into account by means of an optimized reaction progress variable and an enthalpy variable is utilized to account for heat loss. Mixing is considered in the tabulation using a mixture fraction approach. The model is implemented in a low-Mach number flow solver that is specifically designed for large-scale parallel applications and based on the Finite Element Method (FEM). While the proposed combustion model has already been successfully applied to predict the flow field of a confined premixed jet flame [In the current work, a combustion model based on tabulation of laminar premixed flamelets and a presumed-shape Probability Density Function (PDF) to account for subgrid scale  by Meier et al. . The comThe analysis presented in this work pursues two major objectives. In the first part, the influence of the fuel-air mixing for this gas turbine model combustor is addressed using large-eddy simulation. The description of the reaction chemistry is based on tabulation of laminar premixed flamelets with presumed-shape PDFs for turbulence-chemistry interaction. The tabulation is coupled to the LES using a mixture fraction and reaction progress variable approach. Thereby, the prediction capabilities of the presented combustion model for partially premixed conditions are demonstrated. In the second part, the influence of heat loss to the walls on the reacting flow field prediction is assessed. For this analysis, a perfectly premixed fuel-air mixture is used as the impact of local mixture fraction inhomogeneities is shown to be very small confirming the results reported in the literature. The heat transfer to the combustor walls is considered by application of isothermal conditions. The results for simulations using non-adiabatic chemistry tabulation are compared to a simulation in which the effect of heat loss on the chemical kinetics is neglected. This allows to analyze the effect of heat loss on the combustion chemistry. In the last step, the impact of radiative heat transfer is discussed by application of a radiation model based on the optically thin gas assumption , 22.The remainder of the paper is organized as follows. The modeling approach is described in Section sgs is accounted for by the use of a presumed-shape PDF. The conservation equations of mass, momentum and enthalpy are solved in the low-Mach number limit and combined with transport equations for the scalar variables controlling the combustion chemistry. In this section, the scalar controlling variables are defined first. Subsequently, the set of governing equations for LES is introduced. Finally, the chemistry tabulation for the different conditions investigated in the current work are presented including the treatment of turbulence-chemistry interaction.In the current work, the modeling of the reacting flow field is based on a flamelet approach with tabulated chemistry. Turbulence-chemistry interaction in the The description of the combustion process in the reacting flow is based on the assumption of the flamelet regime and the thermochemical properties of the flame in this region are given from a precomputed flamelet database. The tabulation is parametrized in terms of scalar controlling variables that are used to couple the chemical states with the fluid flow. The controlling variables correspond to the mixture fraction, the reaction progress variable and a normalized enthalpy scalar when accounting for heat loss.Zj, the mixture fraction is defined as:O and F indicate the values at the oxidizer and fuel inlet, respectively. Consequently, the mixture fraction is zero for pure oxidizer and unity in pure fuel flow. In the current work, the assumption of equal diffusivity is utilized. Thereby, the minor effects of differential diffusion are neglected reducing significantly the complexity of the modeling. As demonstrated by Donini et al. [j could be utilized in order to construct the mixture fraction. The mixture fraction definition based on element mass fractions is beneficial because elements are conserved during chemical reaction and, therefore, the mixture fraction does not change due to chemical reactions. This also implies that the mixture fraction is constant along each individual premixed flamelet.The mixture fraction describes the local mass ratio between fuel and oxidizer. Based on the element mass fraction i et al. , the infYc, sometimes also referred to as the unscaled progress variable, is defined as a linear combination of species mass fractions Yk:bk describe the contribution of each species k to the composed mass fraction Yc [c:u and b refer to the unburnt and burnt composition, respectively. Using this definition, the reaction progress variable is zero in the limit of unburnt gasses and unity for the fully burnt mixture. For non-premixed conditions, the progress of reaction Yc depends on the mixture fraction, so the scaling process leads to addition of scalar and cross-scalar dissipation rates in the balance equation of the scaled reaction progress [Yc can be solved instead. The latter is the strategy that will be followed here. Details about the transport equations for perfectly premixed and partially premixed conditions are given in Section The composed species mass fraction tion Yc . For preprogress . In ordei defined as:i consequently varies from unity for adiabatic conditions (h = hmax) to zero for maximum heat loss (h = hmin). Due to the application of the equal diffusivity assumption, the enthalpy is constant along each premixed flamelet. The maximum enthalpy value is determined based on the solution of a freely propagating flame configuration while the minimum enthalpy level for the tabulation is based on a vanishing source term. Consequently, if the enthalpy level is lower than hmin, the reaction chemistry remains frozen [For non-adiabatic conditions, flamelets at different enthalpy levels are required. The flamelets are parametrized in terms of the normalized enthalpy scalar s frozen .Vol is the volume of the cell. No transport equation is required for the enthalpy scalar i as it is linked directly to the conservation equation of the enthalpy h. Using the assumption of unity Lewis number for all species, the set of filtered balance equations is defined as:Sct is the turbulent Schmidt number and a value of 0.7 is used for all simulations presented in this work.The transport equations in the low Mach number regime governing the reactive flow correspond to the conservation of continuity, momentum and enthalpy along with transport equations for the controlling variables describing the mixing and reaction processes. The chemical state is described by the mean and first moment of the mixture fraction and reaction progress respectively. A Favre-filtered description of the governing equations is given to avoid the modeling of terms including density fluctuations . Note tho et al.  with a lpth is the thermodynamic pressure, R is the universal gas constant and an represent the NASA coefficients [The set of equations is closed by the use of the ideal gas law:ficients , which dτ is defined based on Stoke’s assumption and the turbulence contribution τ∗ is determined by the use of the Boussinesq approximation:I is the identity tensor. The turbulent viscosity μt is modeled using the Wall-Adapting Local Eddy-viscosity model (WALE) [The viscous stress tensor l (WALE) .an are read as function of the controlling variables   and FlamSubsequently, the chemistry tabulation is discussed. To account for the different conditions that are investigated, three different thermo-chemical databases are used throughout this work. In the following sections, the tabulation process is described with emphasis on the respective conditions.In the baseline case, the influence of the local mixing is analyzed and compared to the assumption of perfectly premixing under adiabatic conditions. Thereafter, these conditions are relaxed and the influence of heat loss on the reaction chemistry is incorporated for perfectly premixed conditions.For perfectly premixed conditions, the mixture fraction is constant and can therefore be omitted. Additionally, using adiabatic thermal conditions, the enthalpy does not vary and the enthalpy scalar can be left out. Therefore, only a single flamelet at the global equivalence ratio is required for the chemistry tabulation using a freely propagating flame configuration. The state of chemical reactions is completely described by the use of a reaction progress variable .bk describe the contribution of each species k to the composed mass fraction Yc  mechanism is used to determine the weighting coefficients [In the definition of the reaction progress variable, the weighting coefficients ficients , 33 for P(c) is the probability density function of the reaction progress. A β-PDF shape is used in the current work to account for the subgrid scale variations of the reaction progress variable, since it was shown that only moderate levels of fluctuations occur for this case [β-PDF is defined by the filtered reaction progress variable sgs variance of the reaction progress variable an of Eq. The turbulence-chemistry interaction is based on the application of a presumed-shaped PDF that takes into account the statistical effect of turbulence. For perfectly-premixed adiabatic conditions, the chemistry depends only on the reaction progress variable and an ensemble Favre-filtered scalar is defined as:his case . The β-Pβ-function differs greatly in shape for small values of the variance, a cubic subdivision of the grid points is chosen for the variance distributing most of the points between zero and 0.25 [In the turbulent database, the chemistry is tabulated at 100 discrete points in reaction progress space with a linear subdivision. For each of these points, the variance of the reaction progress variable is tabulated at 25 discrete values. Since the and 0.25 .The turbulent database for perfectly premixed, adiabatic conditions is visualized in Fig. f as introduced in Section Wk is the molecular weight of species k. The effect of non-perfect premixing can be assessed based on the source term of the reaction progress variable presented in Fig. f = 0.027 up to 0.082. Around the stoichiometric mixture fraction, fst = 0.055 for the current case, the highest values of the source term are obtained and steep source term gradients indicate small chemical time scales. Contrary, in the limits of the tabulated mixture fraction, the source term tends toward zero, corresponding to the lean and rich flammability limits. Outside the flammability range, the reaction source term is zero and only mixing is considered. This is achieved by linear interpolation of the properties tabulated for the lean or rich limit and the inlet properties of the oxidizer or fuel inlet, respectively. Due to this treatment, a significant reduction of the entries in the database is obtained.To take into account the effect of non-perfect mixing of fuel and oxidizer, premixed flamelets at different equivalence ratio are computed and assembled to cover the entire range of equivalence ratio within the flammability limits. The degree of mixing is parametrized in terms of the mixture fraction entclass1pt{minimaf and the scaled reaction progress variable c. Therefore, the PDF formulation to account for turbulence-chemistry interaction is defined by the joint PDF P. Commonly, f and c are considered statistically independent [P(f)P(c). The presumed β-PDF is then applied separately for both one-dimensional PDFs. Transport equations are solved for the filtered progress of reaction For non-premixed calculations, the chemical evolution represented by the laminar database depends on the the mixture fraction ependent , 36, whiThe integrated variables are stored in a turbulent database at 25 discrete points for i as defined in Section For non-adiabatic conditions, several flamelets at different enthalpy levels are determined by the solution of burner stabilized one-dimensional laminar premixed flame simulations with varying mass flow rate , 37, 38.In Fig. i, turbulent fluctuations in i have only a small effect [δ-PDF is employed for the enthalpy. The main advantage of using the δ-PDF is that it only depends on the mean of the enthalpy scalar and higher moments do not need to be computed. The turbulent database is stored at 100 discrete points for hmin and hmax using a linear subdivision.Statistically independent fields are again assumed in order to apply a factorized PDF for reaction progress variable and enthalpy scalar. In general, only weak dependency is expected for enthalpy and reaction progress as the enthalpy is only changing due to heat losses, which are independent of the reaction progress. Furthermore, heat loss occurs mainly in regions in which the reaction progress already equals unity and the reaction progress fluctuations are zero. Additionally, statistical correlations caused by the geometry of the domain or physically allowed values are eliminated by the normalization of the variables. Due to the almost linear dependency of the species mass fractions and temperature on the enthalpy scalar l effect  and a δ-i) and right-hand side (bi), and control the resulting system solution in parallel. Details of the parallel performance and computational efficiency of the code can be found in [The system of equations described in the previous sections is solved using the multiphysics code Alya. The space discretization is based on the variational multiscale method (VMS) using linear finite elements along with the WALE model  to take found in .The current work is focused on the analysis of the flow field and the influence of heat loss and fuel-air mixing for the PRECCINSTA burner, which is a gas turbine model combustor derived from an industrial design by Turbomeca , 43. TheThe gas turbine model combustor investigated in this paper has been experimentally investigated at the German Aerospace Center (DLR) by Meier et al. . The comϕ = 0.83, while the three dimensional velocity field was measured at ϕ = 0.75 using Laser Doppler Velocimetry (LDV). The air mass flow rate was kept constant for both equivalence ratios and the corresponding flame parameters are summarized in Table The azimuthal momentum introduced by the radial swirler forms an inner and outer recirculation zone. Thereby, the flame is stabilized and anchored at the central bluff body of the burner nozzle. The burner is operated at globally lean equivalence ratio. Depending on the fuel-to-air ratio, different combustion dynamics are observed. For a range of operating conditions, the burner is running stable with a quiet flame. However, strong self-excited thermoacoustic oscillations occur for certain conditions. Both regimes have been investigated in the experiments, but we focus on the stable flame only, as model validation and evaluation of the effects of heat loss and fuel-air mixing is mainly pursued in this study. The measurements of the quiet flame were performed for two operating conditions with slightly different equivalence ratios. The laser Raman scattering measurements of major species concentrations and temperature were performed for h = 6, 10, 15, 20, 30, 40, 60 and 80 mm). In contrast, the LDV measurements of the velocity field are conducted at six different axial locations , which differ from the locations of the Raman measurements. The systematic and statistical measurement uncertainties are below 4% and 2.5% for the temperature, less than 5% and 7% for major species  and not more than 0.5% and 2% for the velocity measurements, respectively [The temperature and major species are measured in vertical planes at eight different heights downstream of the injector  elements and is presented in Fig. Tw = 1000 K is prescribed for all quartz glass walls of the combustion chamber for the isothermal cases.The inlet velocity is specified with a top hat profile at the fuel and air inlet such that the mass flow rates specified in Table In this section the simulation results of the radial swirl burner are presented. In the first part, the influence of non-perfect mixing on the reacting flow field is examined using the modeling approach for partially premixed flames. The second part is focused on the effect of heat loss on the dynamics and chemical structure of the flame.In the experiments, fuel is injected into the swirler vanes and mixes with the air before the mixture enters into the combustion chamber. However, the fuel-air mixture is not homogeneous yet and the combustor is considered to operate under technically premixed conditions. In this section, the influence of the fuel-air mixing on the reacting flow field is assessed and the prediction capabilities of the presented combustion model are demonstrated. Therefore, simulations based on the assumption of perfectly premixing are compared to partially premixed simulations, in which the fuel-air mixing is solved by governing equations. The experimental data serves as a reference.fst = 0.055. The color scale corresponds to the velocity magnitude. The fuel injection in the swirler vanes is arranged as a jet-in-crossflow configuration. Therefore, small stagnation regions indicated by low velocity develop upstream of the fuel jet. The azimuthal momentum induced by the swirler results in a helicoidal motion of the fuel jets that enhance the mixing with air. Thereby, an overall lean mixture is obtained already before entering the combustion chamber.For the perfectly premixed simulations, a homogeneous fuel-air mixture at the global equivalence ratio is injected directly into the combustion chamber. Contrary, for the technically premixed case, the fuel is injected into the air in the swirler vanes. Figure In Fig. ϕ = 0.75 (Flame B) while the assessment of temperature and major species prediction is carried out for a global equivalence ratio of ϕ = 0.83 (Flame A). Based on these two operating conditions, the results of the perfectly premixed and partially premixed simulations are compared next.The comparison of the experimentally measured quantities such as velocity, temperature, mixture fraction or species is based on two different operating conditions. This is because LDV and Raman measurements have been performed for slightly different equivalence ratios .A similar effect is observed for the mass fraction profiles of CO, which are given in Fig. The predicted species mass fractions from the non-premixed simulation are also added for completeness. Compared to the baseline case, the prediction of the major species slightly improves, but due to the neglected heat loss, large deviations are found in the reacting layer.The comparison between the isothermal simulations in which the effect of heat loss on the chemical kinetics is either including or neglecting reveals only negligible differences for the predicted temperature field. However, a clear improvement of the predicted chemical composition is observed for the case in which the chemical kinetics are influenced by the local temperature (Cases 3 and 4).An important parameter to evaluate the influence of heat loss is the analysis of the OH distribution. In the framework of the measurement campaign at DLR, laser-induced fluorescence (LIF) is used to visualize the OH radical concentration in the combustion chamber . In Fig.In this work, the modeling of a swirl stabilized gas turbine model combustor is addressed by means of LES. The combustion chemistry is described by laminar premixed flamelets and presumed shape probability density functions for turbulence chemistry interaction in large-eddy simulation. The prediction capabilities of the combustion model for perfectly premixed and partially premixed conditions are demonstrated and good overall agreement with the experimental data is observed.In a first section, the influence of non-perfect premixing on the reacting flow field is assessed by comparison of perfectly premixed and partially premixed simulation results. Even though the fuel-air mixture is not homogeneous, many previous investigations are based on the assumption of perfectly premixed conditions. The flame index obtained in the partially premixed simulation also confirms that flame propagation dominates the combustion process in this case, and small influence of diffusion occurs on the postflame region. The analysis revealed significant mixture fraction fluctuations with a scattering between 0.03 and 0.07 and RMS peak values of about 15% of the mean value. However, only small impact of these fluctuations on the turbulent reacting flow field is observed. The velocity profiles reveal a correct prediction of the main flow features for both, perfectly and partially premixed conditions. The time-averaged temperature as well as the RMS values are accurately captured by both simulations apart from the ORZ, where a significant overprediction of the temperature is observed due to neglected heat loss.In a second section, the influence of heat loss is discussed. Adiabatic thermal conditions are employed in almost all previous modeling attempts of this test case. The neglected heat loss to the walls causes an overprediction of the mean temperature close to the chamber walls, especially in the ORZ. As a consequence, an underprediction of the temperature fluctuations for this region is found. Isothermal boundary conditions are applied for the quartz glass walls of the combustion chamber and significant improvement of the temperature prediction is observed. The time-averaged temperature in the ORZ is reduced between 300 and 500 K, which is in agreement with the measured values. The RMS prediction in this region is increased by about 150 K and improves the alignment with the reference data. The comparison between heat loss simulations using non-adiabatic chemistry tabulation and with a thermo-chemical database in which the effect of heat loss on the chemical kinetics is neglected shows only negligible influence on the prediction of the flame dynamics. This also explains the good prediction of the flame shape in many of the previously reported studies of this combustor, in which adiabatic conditions were assumed. The application of a radiation model based on the optically thin gas assumption does not reveal a major impact of radiative heat transfer for the current case. The equilibrium temperature is reduced by about 50 K, but the flame dynamics are not affected.Despite small impact is observed on the temperature field and flame dynamics by non-adiabatic simulations with adiabatic kinetics, significant differences are found on the chemical structure of the flame. The application of non-adiabatic chemistry tabulation improves the prediction of the major species in the reacting layers, leading to very good agreement with the measured values. In particular, the prediction of intermediate species like CO is remarkably improved, for which significantly overpredicted values are observed if the heat loss effect on the chemistry is neglected. The comparison of the OH concentrations additionally demonstrates the importance of heat loss modeling to predict the flame shape, which in this case transitions from M- to V-shape by the existence of heat loss.


This paper derives optimal forecast combinations based on stochastic dominance efficiency (SDE) analysis with differential forecast weights for different quantiles of forecast error distribution. For the optimal forecast combination, SDE will minimize the cumulative density functions of the levels of loss at different quantiles of the forecast error distribution by combining different time-series model-based forecasts. Using two exchange rate series on weekly data for the Japanese yen/US dollar and US dollar/Great Britain pound, we find that the optimal forecast combinations with SDE weights perform better than different forecast selection and combination methods for the majority of the cases at different quantiles of the error distribution. However, there are also some very few cases where some other forecast selection and combination model performs equally well at some quantiles of the forecast error distribution. Different forecasting period and quadratic loss function are used to obtain optimal forecast combinations, and results are robust to these choices. The out-of-sample performance of the SDE forecast combinations is also better than that of the other forecast selection and combination models we considered. Since the seminal work of Bates and Granger , combiniIn this context, the central question is to determine the optimal weights used in the calculation of combined forecasts. In combined forecasts, the weights attributed to each model depend on the model’s out-of-sample performance. Over time, the forecast errors used for the calculation of optimal weights change; thus, the weights themselves vary over time. However, in empirical applications, numerous papers Clemen  have fouIn this paper, we will follow an approach for the combination of forecasts based on stochastic dominance (SD) analysis, and we test whether a simple average combination of forecasts would outperform forecast combinations with more elaborate weights. In this context, we will examine whether an equally weighted forecast combination is optimal when we analyze the forecast error distribution. Rather than assigning arbitrary equal weights to each forecast, we use stochastic dominance efficiency (SDE) analysis to propose a weighting scheme that dominates the equally weighted forecast combination.Typically, SD comparisons are conducted in a pair-wise manner. Barrett and Donald  developeThe mainstream forecast combination literature obtains the forecast combination weights through the minimization of the total sum of the squared forecast errors (or the mean squared forecast errors) taking into account all the forecasts over the whole period. For instance, the seminal paper of Granger and Ramanathan  employs In order to better understand the distinction between the two approaches, one relying on minimizing the number of forecast errors above a given threshold and the other minimizing the overall squared forecast errors (or absolute forecast errors) for a given quantile, we provide a brief discussion on how SDE methodology complements the mainstream forecast combinations. Forecasters and investors follow a certain strategy and depending on their risk attitudes they try to minimize their losses or forecast errors. Some might consider to minimize the forecast errors for all possible forecast levels, and as such, they minimize the total sum of (squared) forecast errors . Others might want to try to minimize the forecast errors for a given quantile of forecasts (quantile regression). On the other hand, there may be a forecaster (like an insurance company) who compensates above a given threshold level of loss. In that case, the company in question would offer a guarantee to compensate their customers if their forecast error (loss) is above a given level. Hence, this company would like to minimize the forecast errors (losses) that are above this threshold so that to minimize its compensation levels, something that may not be achieved by minimizing the total sum of squared forecast errors (or the absolute forecast errors for this quantile). The latter methods will minimize the overall loss (or quantile loss), but the number of losses above a given threshold level might not be the lowest as derived by the SDE approach. In that context, the SDE methodology is designed to combine forecasts that minimizes the number of forecast errors above a given threshold, and this is obtained by maximizing the empirical cumulative distance between the loss generated by the equally weighted forecasts and the alternative one for this threshold loss level. Therefore, the SDE method produces a forecast combination that complements the more conventional forecast selection and combination methods and can serve forecasters and investors to obtain better forecast combinations depending on their strategy and policy.We use two exchange rate series given in a weekly frequency for the Japanese yen/US dollar and US dollar/Great Britain pound to derive optimal forecast combinations with the SDE methodology for different forecasting periods  and for different forecast horizons. Overall, we find that the optimal forecast combinations with SDE weights perform better than different forecast selection and combination methods for the majority of the cases. However, there are also some very few cases where some other forecast selection and combination model performs equally well at some parts of the forecast error distribution. For the optimal forecast combination obtained with SDE weights, the best forecasting model  includes different sets of models at different parts of the empirical distribution. On average, autoregressive and self-exciting threshold autoregressive models are the main contributors to the optimal forecast combination for both the Japanese yen/US dollar and US dollar/Great Britain pound exchange rate application, and during and after the 2007/2009 financial crisis.The remainder of the paper includes the following. In Sect. m different forecasting models generated at time t for the period of h is the forecast horizon and T is the final forecasting period. Furthermore, let Let us start with data m different models, i.e., e being a vector of ones. With this alternative weighting scheme, one can obtain a forecast combination, i.e., The equally weighted column vector, i.ii.iii. i.e., For this paper, we follow a loss function that depends on the forecast error, i.e.,  Granger : i.\doci) suggests that there is no loss when there is no error, (ii) suggests that the minimum loss is zero, and finally, (iii) suggests that the loss is determined by its distance to zero error irrespective of its sign. and the quantile regression minimizes the sum of absolute errors (or mean absolute errors) for a given quantile. If one were to minimize the squared forecast errors by looking at the whole distribution (or quantile), these approaches could be appropriate. On the other hand, with the SDE methodology one minimizes the number of forecast errors (or squared forecast errors) above a given threshold error level. In that respect, SDE approach complements the existing forecast selection and/or combination methods when one’s priority is to minimize the number of forecasts above a given threshold. For example, this could be the case, when a company promises to compensate its consumers if their forecasts give errors that are above a threshold error level. Standard approaches would minimize an overall single measure (mean squared forecast error or mean absolute error for a given quantile). However, these objective functions are not designed to minimize the number of errors above a given threshold error level and might produce a higher number of losses above this given threshold. In this respect, SDE offers a complementary approach to forecast combination if the number of losses above a threshold is deemed more important than the overall (or quantile) loss.z given z represents the level of lossIn this paper, we test whether the cumulative distribution function (cdf) of the loss function with the equally weighted forecast combination is stochastically efficient or not. z of the loss distribution, z denotes the loss level, then the inequality in the definition means that the proportion of loss obtained with the forecast combination of z is no lower than the value (mass) of the cdf of the loss with the equally weighted forecast combination, z level is less than the one with the equally weighted forecast combination, z.For any two forecast combinations, we say that the forecast combination z. In other words, z level, and hence, it minimizes the number of forecasts that gives a loss above a given threshold, z.More precisely, to achieve stochastic dominance, we maximize the following objective function:j, hereafter z level . In other words, under the null, we analyze whether the equally weighted forecast combination, ,  z , the function z. In other words, there is an alternative weighting scheme, z level when compared to average forecast combination.The general hypotheses for testing whether the equally weighted forecast combination, j of the distribution of the equally weighted forecast combination We obtain SD at the first and second orders when F, which yields the following:z level  when z level with a given forecast combination when The empirical counterpart of  is simplWe consider the weighted Kolmogorov–Smirnov-type test statisticp-values, where the critical values are obtained using a supremum statistic.To make the result operational, we need to find an appropriate critical value To sum up, for a given quantile of loss distribution, we analyze whether the equally weighted forecast combination is optimal or not. We test whether an alternative combination of forecasts provides a loss distribution up to a given quantile of loss that would dominate such distribution when forecasts are combined in an equally weighted way. If an alternative combination of forecasts dominates the equally weighted combination, then there is an alternative combination which yields a distribution of loss that is the optimal one at that given quantile.In this section, we apply the SDE testing methodology to obtain optimal forecast combinations on Japanese yen/US dollar and US dollar/Great Britain pound exchange rate returns data. We use log first differences of the exchange rate levels. The exchange rate series data are expressed with a weekly frequency for the period between 1975:1 and 2010:52.In our out-of-sample forecasting exercise, we concentrate exclusively on univariate models, and we consider three types of linear univariate models and four types of nonlinear univariate models. The linear models are random walk (RW), autoregressive (AR), and autoregressive moving-average (ARMA) models; the nonlinear ones are logistic smooth transition autoregressive (LSTAR), self-exciting threshold autoregressive (SETAR), Markov-switching autoregressive (MS-AR), and autoregressive neural network (ARNN) models.t for the time t.Let p and q are selected to minimize the Akaike information criterion (AIC) with a maximum lag of 24. After estimating the parameters of Eq. ..\documenc are simultaneously estimated by maximum likelihood methods.The LSTAR model isIn the LSTAR model, the direct forecast can be obtained in the same manner as with ARMA, which is also the case for all of the subsequent nonlinear modelsmentclasspt{minimaWhen and Lane  to generand Lane  to perfoThe two-regime MS-AR model that we consider here is as follows:h-step forecast from the MS-AR model isith element of the column vector Although MS-AR models may encompass complex dynamics, point forecasting is less complicated in comparison with other nonlinear models. The d is the logistic function, which is defined above as ARNN, which is the autoregressive single-hidden-layer feed-forward neural network modelby White . In conth, the models are estimated by running regressions with data that were collected no later than the date T refers to the final date in our data. The first h-horizon forecast is obtained using the coefficient estimates from the initial regression. Next, after moving forward by one period, the procedure is repeated. For each h-step forecast, we calculate To obtain pseudo-out-of-sample forecasts for a given horizon m is chosen as the preferred model.Before proceeding with our application, in this section we offer different set of model selection and combination methods that are employed extensively in the literature. Akaike’s information criterion (AIC) and Bayesian information criterion (BIC) are two of the most commonly used selection criteria that serve to select a forecasting model  from the minimal performing model’s AIC (BIC) level. Hence, defining the difference between the AIC(m) (BIC(m)) and the min(AIC)  as m models. Another commonly used method to combine forecasts is to allocate weights to each model inversely proportional to the estimated forecast error variances . Hence, the optimal SDE forecast combination would also dominate the rest of the possible combinations as these are part of the available choice set. We obtain the best forecast combinations of the model-based forecasts for the Japanese yen/US dollar and the US dollar/Great Britain pound exchange rate forecasts by computing the weighting scheme on each forecast model that offers the optimal forecast combination at different quantiles of the loss distribution.z. In other words, one could obtain the number of forecasts that generate loss that is below a given quantile of the loss distribution with the equally weighted forecast combination. We test different quantiles of the empirical loss distribution of the average forecast combination, that is, we test whether the equally weighted forecast combination is the best forecast combination against the alternative combination at different parts of the empirical distribution. In the next section, we report the optimal forecast combination for different percentiles  of the empirical loss distribution for the two applications for different forecast periods and horizons.In our applications, because the loss distribution  with the equally weighted forecast combination is known, we can obtain the number of forecast combinations that generate loss that are less than each given level of loss, z level is minimized.First, we begin our empirical analysis with the weekly Japanese yen/US dollar exchange rate forecasts for different forecast horizons for the financial crisis period of 2007/2009 . We proceed with testing whether the equally weighted forecast combination of the forecasting models for different horizons is the optimal forecast combination at different levels of loss or there are alternative weights on the forecast models that stochastically dominate the equally weighted forecast combination, h). The second column gives the details of the forecast period, whereas the third column reports the loss levels  with the equally weighted forecast combinations at these particular percentiles. The following columns provide the weights of the underlying forecasting models for the optimal forecast combinations at the 50th, 75th, and 95th percentiles of the loss distribution with the equally weighted forecast combination.Table In one step ahead forecast horizon, i.e., when mentclass2pt{minimWe carried out the same application when we extended the forecast horizon for 6 months (26 weeks) and a year (52 weeks) , and the forecast combination methods .SDE weights obtained in the previous section suggested that when the equally weighted forecast combination is the benchmark, there is always an alternative forecast combination which would constitute a better case at different quantiles of the loss distribution for all forecast horizons. To evaluate SDE weights further, we also obtain median forecast, and forecasts with different model selection and combination methods that are mentioned above.z, at the 50th, 75th, and 95th percentiles with the equally weighted forecast combination (EW), median forecast (Median), forecasts with the best model chosen with AIC, BIC, and PLS, and forecast combinations with Bates and Granger, Granger and Ramanathan, AIC, BIC, and quantile regression weights.To make the results more apparent for each forecast horizon, Table z, at the 50th, 75th, and 95th percentiles of the loss distribution from the equally weighted forecast combination. The optimal forecast combinations with the SDE weights are obtained using the weights from Table In Table We carry out the same analysis when we change the forecast horizons. When We only presented the SDE weights for the best forecast combination at 50th, 75th, and 95th percentiles of the loss distribution. However, Table Overall, for the weekly Japanese yen/US dollar exchange rate forecasts, we find that the best forecast combination with SDE weights mostly outperforms the other forecast selection and combination models, with some few exceptions where some other models perform equally well. We also should note that the objective of the SDE weight allocation is to obtain the lowest number of forecasts that give a loss above a given threshold, not to minimize the overall loss. Hence, we do not produce conventional comparisons of different methods, but we simply compare whether SDE approach dominates other forecast selection and combination methods given the loss level. For example, when In this subsection, we obtain the optimal forecast combination for the foreign exchange rate of US dollar/Great Britain pound forecasts for different time horizons at different quantiles of the loss distribution for the financial crisis period of 2007/2009 . Table The optimal weights obtained for the foreign exchange rate of US dollar/Great Britain pound are very similar to the ones obtained for the Japanese yen/US dollar exchange rate data  and we find that the forecast combinations obtained with the SDE produce the lowest number of forecasts that give a loss above a given threshold in most of the cases analyzed over this period. In this section, we repeat our analysis to obtain optimal forecast combination for the US dollar/Great Britain pound and Japanese yen/US dollar exchange rate forecasts with the SDE methodology with the forecasts obtained for the period between 2010 and 2012 and compare its performance with other forecast selection and combination methods.Panels A and B of Table Similar to the previous section, we provide comparisons of forecast combination obtained with the SDE methodology with the standard forecast selection and combination methods where Panels A and B of Table Tables Overall, the SDE model still produces the optimal forecast combination even after changing the forecast period in most percentiles  where there is always a lower number of forecasts that produce a loss above a given threshold. The only difference between the during and the after the financial crisis periods is that the AR and SETAR models contribute relatively more to the optimal combination after the financial crisis period, while the contributions of ARMA, ARNN, and SETAR models are relatively less after the crisis period when compared to the financial crisis period.We also evaluate the out-of-sample performance of the SDE forecast combination when compared to the out-of-sample performance of all other forecast selection and combination models. To do this, we use forecast combination weights obtained for 2010–2012 period for one-step ahead forecasts for the US dollar/Great Britain pound exchange rates , which is obtained by the following loss function: z), we can consider that 50% of the forecasts give absolute forecast errors that is above this level with the equally weighted forecast combination. If one were to change the loss function to obtain the distribution of the squared forecast errors: Let us expand our discussion on this. SDE approach’s objective is to minimize the number of forecasts that give a loss level above a given loss level. In the previous section, SDE approach finds a weight allocation across the forecast models  with the quadratic loss function where the 50th, 75th, and 95th percentiles of the squared forecast errors for Overall, our findings are robust to the choice of the loss function  as altering the loss function does not alter the order of losses obtained with different forecast selection and combination methods. Changing the loss function will indeed change the optimal forecast combination obtained by the mainstream methods used in the forecasting literature as these methods consider all forecasts and minimize the overall deviation or loss . However, the SDE methodology minimizes the number of forecasts that gives a loss above a given threshold level and changing the loss function do not alter the position of the losses in the distribution and the results are robust to the choice of the loss function.In this paper, we provide SDE properties to combine forecasts by which optimal forecast combinations are obtained at different quantiles of the loss distribution when compared with respect to all possible forecast combinations constructed from a set of time-series model forecasts. The SDE approach differs from the mainstream forecast combination approaches and complements them. In particular, mainstream forecast combination methods minimize the total sum of losses (such as the sum of squared forecast errors or absolute forecast errors), but the SDE methodology obtains the forecast combinations that will minimize the number of forecasts that produce losses above a given threshold rather than the aggregate measure of loss. In that respect, the SDE approach complements the existing forecast selection and/or combination methods when the forecasting priority is to minimize the number of forecasts that produce loss levels above a given threshold. In that respect, the SDE methodology is particularly well suited for the cases when a company (such as an insurance company) promises to compensate its consumers if their losses (forecast errors) are above a threshold error level rather than trying to minimize the overall loss.We applied the SDE methodology to construct the optimal forecast combination for different forecast horizons at different percentiles of the loss distribution for weekly Japanese yen/US dollar and US dollar/Great Britain pound foreign exchange rate forecasts during and after the financial crisis. During the financial crisis period, we find that the optimal forecast combination in different areas of the loss distribution for different forecast horizons differs. However, the main contributor to the optimal forecast combination is the AR model both during and after the financial crisis period. Overall, there is also agreement that the SETAR, LSTAR, ARMA, and ARNN models contribute more to the optimal forecast combination at some parts of the loss distribution during the crisis period. However, after the crisis period, only SETAR  and LSTAR are the models that contribute to the optimal forecast and contributions of ARMA and ARNN to the optimal forecast combination after the crisis period is limited compared to the crisis period.AIC, BIC, and PLS, and forecast combination with equal, Bates and Granger, Granger and Ramanathan, AIC, BIC, and quantile regression weights at different parts of the loss distribution. However, there are also few cases where some other forecast selection and combination model may perform equally well at some parts of the loss distribution. In particular, forecast combination obtained with the quantile regression is the second best way of combining forecast in most of the cases.In summary, for the majority of the cases considered, forecast combinations with SDE weights perform better than median forecasts, forecasts from the model that is chosen with To test the robustness of the SDE weights, we also used the quadratic loss function in our analysis. Both the weights obtained with the SDE and the comparison results with the other methods remained the same when we used the squared forecast error distribution. In particular, the SDE methodology minimizes the number of forecasts that gives a loss above a given threshold level and changing the loss function would not alter the position of the forecast errors in the distribution and as such the results are robust to the choice of the loss function.Finally, we only applied the SDE analysis to two specific data sets with a given number (seven) of time-series models and, as such, our results on the optimality of the forecast combination at different quantiles of loss distribution do not generalize beyond the scope of the applications at hand. However, the SDE methodology can offer a useful way of assessing the optimality of forecast combinations by using information available in the entire forecast error distribution and not merely in the first two moments, as typically assumed in the literature.


The hypothalamic-pituitary-adrenal (HPA) axis orchestrates the physiological response to unpredictable acute stressors. Moreover, the HPA axis exhibits prominent circadian activity and synchronizes peripheral circadian clocks to daily environmental cycles, thereby promoting homeostasis. Persistent disruption of homeostatic glucocorticoid circadian rhythmicity due to chronic stress exposure is correlated with the incidence of various pathological conditions including depression, diabetes and cancer. Allostatic habituation of the HPA axis, such that glucocorticoid levels retain homeostatic levels upon chronic exposure to stress, can therefore confer fitness advantages by preventing the sustained dysregulation of glucocorticoid-responsive signaling pathways. However, such allostatic adaptation results in a physiological cost  that might impair the homeostatic stress-responsive and synchronizing functions of the HPA axis. We use mathematical modeling to characterize specific chronic stress-induced allostatic adaptations in the HPA network. We predict the existence of multiple individualized regulatory strategies enabling the maintenance of homeostatic glucocorticoid rhythms, while allowing for flexible HPA response characteristics. We show that this regulatory variability produces a trade-off between the stress-responsive and time-keeping properties of the HPA axis. Finally, allostatic regulatory adaptations are predicted to cause a time-of-day dependent sensitization of the acute stress response and impair the entrainability of the HPA axis. The HPA axis is a central player in the response to unpredictable environmental stressors2 leading to release of GCs and enabling the host to actively re-establish homeostasis3. GCs exhibit circadian rhythmicity, peaking just before the onset of the active phase4. Glucocorticoid rhythms enable the host to adjust its behavior in anticipation of periodic changes, including daily variability in light, temperature, food availability and environmental stressors6. The maintenance of circadian rhythmicity has important functional implications preserving host homeostasis by synchronizing physiology to the environment7.The hypothalamic-pituitary-adrenal (HPA) axis integrates environmental information and calibrates the internal 8. By differentiating between critical homeostatic variables , which must be tightly controlled for host survival, and the homeostatic mediators  which regulate the former, allostasis attempts to offer a more precise and integrated description of physiological regulation and the stress response9. While homeostasis is conventionally based on negative feedback architectures that restore all physiological variables to their basal states in response to stress-induced deviations12, allostasis describes a labile equilibrium by which the critical homeostatic mediators actively deviate from basal levels to trigger adaptive mechanisms  that enable the host to cope with perceived stressors and subsequently re-establish homeostasis followed by their gradual return to pre-stress levels14.The concept of allostasis was introduced as an extension of the homeostatic framework in order to explain the process by which the host actively maintains physiological stability (homeostasis)16. Habituation of the HPA axis, such that GCs return to or maintain homeostatic levels in response to chronic exposure to the same stressor provides the host with fitness advantages by optimizing energetic resources and preventing the persistent dysregulation of downstream GC-responsive signaling mechanisms19. However, successful habituation to chronic stress requires the activation of further adaptive allostatic regulatory mechanisms, which can alter the homeostatic functioning of related physiological systems when the host is exposed to novel environments or stressors. Habituation to chronic stress does not necessarily imply an attenuation of allostatic load, and failure of such attenuation results in long-term disruptions, further compromising homeostasis.However, chronic deviation of glucocorticoids from their homeostatic levels through prolonged activation results in a physiological cost  and is associated with detrimental outcomes20, it is suggested that monitoring changes in the concentrations of the homeostatic mediators such as GCs can be indicative of the allostatic state of the host. However, in the case of a habituating allostatic adaptation to chronic stress, when little discernible change in the levels of glucocorticoids occurs, it becomes increasingly difficult to characterize the allostatic state of the system. Moreover, the observation of apparently nominal levels of glucocorticoids upon chronic stress habituation might conceal underlying regulatory changes in the HPA axis that could impair its homeostatic functioning. In such cases, a modeling-based approach can be a particularly useful tool for generating and evaluating experimentally verifiable hypotheses21.Considering the practical difficulty in measuring changes in internal physiological state (e.g. change in energetic expenditure and intake) in a variable environmentWe proposed a semi-mechanistic mathematical HPA model predicting functionally relevant adaptations that occur upon habituation to chronic stress. Amongst the primary functions of the HPA axis are: a) its role as the major physiological stress response system; and b) its ability to synchronize (or entrain) critical internal physiological processes to periodically varying external environment cues (zeitgebers). Therefore, in this work we highlight chronic stress-induced allostatic adaptations influencing its stress-responsiveness and entrainment properties, individual differences in susceptibility to allostatic load, and the existence of multiple individualized regulatory strategies enabling the host to conserve a basal circadian phenotype while allowing for flexibility of its stress-responsive and entrainment properties. We demonstrate that this regulatory diversity results in trade-offs between the stress-responsiveness of the HPA axis and its time-keeping ability. An improved understanding of the underlying regulatory architectures leading to allostatic adaptations and functional trade-offs can provide insights into the complex physiological signaling of the HPA axis in health and disease.22. The Goodwin oscillator represents a phenomenological prototypical biological oscillator, initially devised to show the possibility of self-sustained oscillations in a genetic circuit with a simple delayed negative feedback loop architecture. The skeletal structure of the Goodwin oscillator adapted to model the circadian oscillations in the HPA axis was initially devised by Sriram et al. and was modified by our group to account for the entrainment of the HPA axis rhythms by light via the SCN in both nocturnal and diurnal species.The primary mediators of the HPA axis are represented by nonlinear ODEs that form a limit-cycle type Goodwin oscillator23.Briefly, according to the scheme of equations presented here, CRH induces the release of ACTH, which subsequently induces the release of CORT (corticosterone for the purpose of our study) Equation –9. While25, generally interpreted as the number of ligand subunits cooperatively binding to a receptor. Moreover, CRH, ACTH and the glucocorticoids all undergo enzymatic metabolism or degradation28. We further account for glucocorticoid receptor (GR)-mediated pharmacodynamics as adapted from Ramakrishnan et al.29  exposure. The CORT response to the acute stressor is characterized by the difference in area under the curve (AUC) between the respective CORT profiles with and without exposure to the acute stressor for 4 hours from the time of in-silico exposure to the acute stressor. Subsequently, we used a method of symbolic representation developed by Lin 41, such as periodic photic and temperature cues (zeitgebers), is fundamental to host survival. We hypothesized that chronic stress habituation of the HPA axis would alter the flexibility of the system to entrain to externally varying zeitgebers. We aimed to characterize both individual differences in the entrainment properties (differences in properties within each parametric subspace) as well as the differences in entrainment properties occurring as a result of the adaptation to the simulated chronic stress regimen (differences in properties between the three parametric subspaces). A standard metric for evaluating the entrainment characteristics of the endogenous oscillator is to assess the range of entrainer periods for which the endogenous oscillator is phase-locked to the entrainer as a function of the coupling strength between them42. The domain of entrainment, referred to as the Arnold tongue, is indicative of how flexibly the endogenous circadian oscillator is entrained to changes in the period of the environmental zeitgeber44. While flexibility in entrainment characteristics might enable the system to adequately adapt to changes in external zeitgebers, over-responsiveness on the other hand, might compromise the time-keeping properties of the oscillator by making it too sensitive to noisy environmental stimuli. In determining the possible parametric dependence of the entrainment properties, we calculated the Arnold Tongue for nine representative parameter sets on each of the three parametric surfaces. The nine points on each surface were selected in sets of three such that each set of three had approximately the same level of adrenal sensitivity (kp3) and such that they span the parametric subspaces accounting for regulatory variability. Specific details on the determination of the Arnold tongues are provided in the Supplementary Materials and Methods.Appropriate temporal adaptation or entrainment of critical physiological systems, such as the HPA axis, to the daily rhythmic variations in the environmentSince photoperiod is the primary entrainer of internal circadian rhythms, we determined the parametric dependence of the ability of the system to respond to two physiologically relevant perturbations in the light-dark schedule. In the first case, we studied the response of the system to a transient (96-hr) inversion in the light schedule, simulating temporary shift-work. During this protocol, the maximum phase difference between nominal profiles and those exposed to transiently inverted light schedule was measured and was used to characterize the extent of phase sensitivity to shift work. Subsequently, we determined the parametric dependence of the ability of the system to re-synchronize to a permanent, abrupt shift in the light/dark schedule, simulating the response to jet-lag. On the day of the simulated permanent shift, the time of lights ON was delayed by 10-hr to 17:00, while maintaining the same photoperiod of the light/dark cycle subsequent to the day of the shift. Profiles were considered to be re-synchronized if the phase difference between two consecutive CORT peaks was within 3 minutes.45. Greater (stable) Floquet exponents  imply the system is able to robustly restore its homeostatic rhythm (relaxation) after a transient perturbation from its homeostatic oscillatory state. The procedure for the calculation of the Floquet exponents is described in the Supplementary Materials and Methods.In addition to the ability of the HPA axis to flexibly entrain to environmental zeitgebers, it must also remain robust to transient perturbations in its rhythm, and swiftly relax to its nominal rhythmic activity after mounting an adequate response to an acute stressor. For a periodic system, this property is quantified by the Floquet exponent, which is representative of the system’s stability to amplitude perturbations20. However, given the central regulatory importance of glucocorticoids there is, at the same time, an evolutionary advantage for the host to conserve basal glucocorticoid activity within reasonable physiological bounds such that compensatory downstream homeostatic processes necessary for host survival are optimally balanced8. Thus, we aim to show computationally that the physiological need for the system to conserve a primary phenotype, i.e., basal circadian activity, while at the same time accounting for adequate reactiveness, permits the existence of multiple individually “customized” regulatory strategies in order to achieve this goal46.In the context of allostasis, substantial flexibility in glucocorticoid responsiveness enables the host to adequately adapt to acute environmental challenges23. We used a light-schedule identical to the one adopted in this study in order to keep our simulations consistent with experimental conditions. Thus, given an appropriate selection of parameters the light-entrained model of the HPA axis generates CORT circadian profiles that qualitatively match the experimentally obtained CORT circadian profiles. A parametric analysis of our model indicates that homeostatic glucocorticoid rhythms32  the feed-forward adrenal sensitivity  of CORT over a 4 hr period following exposure to the (acute) stress49. We determine that habituation to chronic stress results in time-of-day dependent alterations in the system’s acute stress response. Specifically, upon habituation to chronic stress, the acute stress response appears to become more robust if experienced towards the middle of the inactive phase relative to the nominal homeostatic state  and in fibroblasts that for a given zeitgeber strength the range of entrainment of these oscillators varies inversely with their intrinsic amplitude and the rigidity of the oscillator given by its amplitude relaxation rate (inferred from the Floquet exponent). Thus, an oscillator with a higher amplitude, termed a “strong” oscillator has a narrower range of entrainment and is more rigid (higher Floquet exponent) compared to a “weak”, low amplitude oscillator. Multilinear regression analysis shows that the calculated leading Floquet exponents are strongly influenced by the adrenal sensitivity of the system and decrease with increasing adrenal sensitivity for nominal as well as chronically stressed parametric subspaces  since the value of the regression coefficient (b3), which determines the dependence of the Floquet exponent on adrenal sensitivity increases with increasing levels of chronic stress. The increase in rates of relaxation are correlated with a decrease in the domain of entrainment as mentioned above. Therefore, we suggest that while the simulated allostatic adaptation is associated with increased robustness to transient perturbation (inferred from the increased rates of relaxation), these changes are also indicative of a loss of the flexibility of the system to readjust to changes in environmental stimuli.The ability of rhythmic endogenous systems, such as the HPA axis, to synchronize to environmental, as well as internal zeitgebers is of critical evolutionary and adaptive significanceion Fig.  and S3. zeitgeber. We specifically focus on physiologically relevant perturbations in the light-schedule, since photoperiod is the key environmental cue synchronizing HPA rhythms in our model. In this respect, understanding the system’s response to perturbations in the light schedule can provide improved insight into how regulatory diversity might influence an individual’s tolerance to shiftwork or jet lag43. We first tested the hypothesis that differences in the regulatory constitution result in a differential response of the HPA axis to a transient inversion of the light/dark cycle. Studies suggest that the maximal circadian phase-shift during shift work is an important parameter that might be predictive of tolerance to shift work schedules53. Therefore, we characterized the maximal phase-shift in the corticosterone circadian rhythm following a transient inversion of the light/dark cycle. The phase shifts produced by the model during this protocol are comparable to those that have been observed in the experimental literature; on the order of a few hours during (rapidly rotating) shift work56. Our model predicts that the maximal phase-shift is greater for subjects with greater adrenal sensitivity and hypothalamic negative feedback . However, stress-induced allostatic maintenance of homeostasis, enabled through regulatory diversity, comes at a cost as the system, once adapted, needs to trade adaptation for responsiveness Fig. . This co59. Our results suggest that despite substantial regulatory diversity, the HPA axis must adjust compensatory processes to maintain homeostatic levels of the GCs. We hypothesized that the existence of these diverse regulatory strategies to conserve the circadian phenotype will have significant consequences for the critical operational characteristics of the HPA axis, specifically its stress response and entrainment behavior.The circadian rhythmicity of endogenous glucocorticoids is partially indicative of the master circadian clock, located in the SCN, and has important implications for the homeostatic signaling of many physiological systems61. Moreover, a number of investigators have shown that the range of entrainment of an oscillator is related to its phase response curve (PRC), a commonly used and powerful analytic tool for the characterization of circadian systems. In general, both theoretical and experimental studies show that a PRC with a smaller amplitude is associated with a narrow entrainment range, where the range of entrainer periods that an oscillator can adopt is indicated by the maximum and minimum values of the PRC63.In characterizing the entrainment properties of the system, we determined a) its domain of entrainment (as depicted by the Arnold tongues) as a metric of how flexibly the system is entrained to changes in the environmental zeitgeber (light/dark schedule) and b) the behavior of the system to physiologically relevant perturbations in the light/dark schedule. We conclude that regulatory structures with intrinsically wider domains of entrainment (ease of adaptation) exhibit greater sensitivity to abrupt but transient (acute) perturbations in the environmental zeitgeber. On the other hand, regulatory strategies exhibiting the largest phase disruptions in response to transient photoperiod perturbations are associated with improved abilities to re-synchronize following a permanent inversion of the light/dark schedule. These results are consistent with experimental findings, which have shown that oscillatory systems with narrow entrainment ranges generally respond with smaller phase changes in response to zeitgber stimuliversus the ability to be easily entrained to a rhythmic environmental stimulus. Since, the HPA axis conveys circadian information from the central SCN to peripheral tissues such a trade-off between entrainability and robust time-keeping functions is particularly important to consider65. These dynamics are of relevance towards understanding individual differences in circadian disruption occurring, for instance, during jet lag and shift-work. Recent evidence from adrenalectomy studies suggests that robust GC rhythmicity might buffer the phase of peripheral clocks against extrinsic noise. Moreover, Kiessling et al. used a mouse-model of jet-lag to show that manipulation of the GC rhythm could both quicken as well as slow-down the adaptation of activity cycles to changes in the light-dark schedule57. Experimentally re-synchronization of CORT rhythms after a substantial shift in the light/dark schedule has been reported to occur at about a day per time-zone shift in the schedule7. A substantial proportion of the simulated individuals in our virtual population have re-synchronization times comparable to experimental results despite the strict criterion for re-entrainment (phase difference between consecutive peaks less than 3 minutes). In the context of adaptation to jet lag it might beneficial for the circadian system to quickly resynchronize to a (permanent) change in light schedule, thus preventing prolonged periods of internal desynchronization with the external environment57. On the other hand, studies on shift work have found that individuals exhibiting a high amplitude circadian rhythm and a slow adjustment in the phase of the rhythm during the night-shift schedule were more tolerant to shift-work in a rapidly rotating shift system, reporting medical complaints at a lower frequency in comparison to individuals less resistant to shift-work53. Therefore, our model predicts qualitatively that individuals with robust endogenous glucocorticoid rhythmicity will be more tolerant to rapid shift-work rotations, while those with a wide domain of entrainment will be able to flexibly adapt to changes in zeitgeber characteristics. Interestingly, a recent study showed that in comparison to the mammary carcinoma-susceptible Fisher 344 rat strain, the resistant Copenhagen rat strain exhibited minimal phase disruption, to changes in the light/dark schedule that simulated shift-work66.Our results highlight an important trade-off for all circadian systems between the ability to be a precise time-keeping mechanism 67. We find that individuals with higher adrenal sensitivity exhibit a more robust acute stress response. Moreover, given that the system adjusts compensatory regulatory processes in order to maintain the critical homeostatic circadian phenotype, we hypothesized that this might result in a systematic interdependent variation of the response characteristics. Therefore, we assessed whether the regulatory mechanisms of the HPA axis vary in a manner that couples the stress-responsive and entrainment properties of the HPA axis. Our results suggest that higher adrenal sensitivity and negative feedback makes the HPA easier to entrain. Thus, we predict that individuals with high adrenal sensitivity, and hence a more robust acute stress response, will more fluently adapt to jet lag, but will be more susceptible to circadian desynchrony due rapidly-rotating shift-work. Therefore, our model highlights the existence of a critical trade-off between the two primary physiological functions of the HPA axis: its ability to function as a robust timekeeper, resilient to fluctuations in an environment zeitgeber versus its ability to respond to environmental stressors. It is suggested that the optimal strategy between the stress responsiveness and entrainability on one hand, and robustness of rhythmicity will depend on the differences in individual life history and exposure to variable natural habitats. We hypothesize that the regulatory diversity might be representative of phenotypic plasticity where an individual organism is capable of modulating its phenotypic state in response to variations in its perceived environment so as to minimize transient accumulations of allostatic load68. Thus, adopting a robust phenotype might be beneficial to prevent over-activation in a highly variable natural habitat with frequent exposure to stressors, while adopting a more responsive phenotype might be advantageous in a relatively predictable natural habitat69. Therefore, the existence of diverse regulatory schemes might be reflective of multiple evolutionarily stable survival strategies in a context specific manner46.In addition to its synchronizing purpose, the HPA axis functions as the primary humoral stress response mechanism, with a flexible response to stressful external stimuli being essential for host fitness70. We suggest that the chronic stress-induced decrease in the variability of key regulatory features of the HPA axis results in a loss of flexibility of the system, which in turn renders the host more susceptible to subsequent environmental stressors. Natural variability in regulatory parameters is known to contribute to organismal fitness by facilitating effective adaptation to locally varying environments72. A number of studies suggest that natural allelic variation within the same population contributes to phenotypic variation that is thought to enable adaptive fine-tuning of the phenotype to environmental variation73. Deletion of the neuropeptide Y gene (NPY), which is involved in a protective response to stress, resulted in a decrease in inter-individual variability in clock period, a shortening of the clock period, and a slower adaptive response to abrupt changes in the photoperiod74. It has been hypothesized that maintenance of regulatory plasticity enables flexible adaptability in response to changing environmental conditions, with a loss in plasticity associated with reduced stress resilience and negative outcomes75.Our results indicate that the accumulation of allostatic load, due to chronic stress, results in a substantial increase in the strength of hypothalamic negative feedback and adrenal sensitivity, accompanied by a reduction in regulatory flexibility Fig. . Interes78. This phenotype of chronic stress sensitization might be associated with negative outcomes, due to an increased risk of overexposure to GCs. Moreover, we predict that greater adrenal sensitivity leads to a greater sensitization of the acute stress response, and thus might be more likely to lead to a maladaptive response upon exposure to chronic stress. We hypothesize that this might have important implications for understanding individual differences in susceptibility to chronic stress disorders79.Our model predicts that allostatic adaptations to a habituating chronic stressor result in an alteration of the acute stress response of the system. Interestingly, following habituation, the system exhibits a strong time-of-day dependence of the response to a subsequent acute stressor, with maximal response around the middle of the active phase. Notably, a number of studies have shown that habituation to prior chronic stress results in increased sensitivity upon subsequent exposure novel stressors44 predict this decrease in the domain of entrainment to be accompanied by a general increase in the robustness of oscillation to transient perturbation (as inferred from the increased rates of relaxation or Floquet exponents). From the viewpoint of a faithful timekeeper such an allostatic adaptation might seem beneficial by making the system a more rigid circadian clock. However, on the other hand, this also implies that habituation to chronic stress requires regulatory adaptations, which make it more difficult for the system to synchronize to diurnally varying environmental signals. Such a phenomenon of lack of entrainment of the internal circadian clock to external zeitgebers corresponds to non-24-hour sleep-wake syndrome, which has been observed in both sighted and blind patients81. Furthermore, this has also been observed in patients of schizophrenia82, and in subjects exposed to severe psychological stress83.Upon habituation to the chronic stress regimen we find that for the same level of adrenal sensitivity, the domain of entrainment becomes significantly narrower for increasing levels of chronic stress exposure, and thus the system loses is entrainability characteristics. Theoretical analyses84. Moreover, there is disagreement about whether the ultradian and circadian rhythms of the mediators of the HPA axis are endogenous in nature, with recent evidence suggesting that the presence of circadian glucocorticoid rhythms after disconnection from the SCN being due to the peripheral adrenal clock7. Glucocorticoids have also been shown to have complex feedback effects on both peripheral clocks  and central circadian clocks, thus adding further complexity to the circadian dynamics of glucocorticoids86. Nonetheless, it has been shown that many of the essential properties of glucocorticoid rhythms such as the observed temporal dependence of the stress response, amplitude behavior and entrainment properties can be explained using limit cycle oscillators as model systems88. In this sense, we suggest that many of the features of our simulation results might still be preserved even after separately accounting for these influences. Therefore, while our current model does not explicitly describe the disparate and complex influences of the SCN, the peripheral circadian clock, and multiple time-scales on glucocorticoid dynamics, we envision future modifications of the model that gradually incorporate these features.The mathematical model of the HPA axis we use in this study has a number of limitations. Importantly, in this work we focus on only the circadian rhythms of the HPA axis. However, it is well known that the mediators of the HPA axis additionally exhibit prominent ultradian rhythmicity, with a pulsatile frequency on the order of about one pulse per hourIn summary, our model simulations indicate that complex physiological signaling systems such as the HPA axis can be viewed as having multiple, simultaneously competing functional objectives, which necessitate the existence of trade-offs between them. We predict the existence of individualized regulatory strategies, as represented by the parametric surfaces, that enable the system to conserve a basal circadian phenotype while also allowing for adequate flexibility in the stress-responsive and entrainment properties of the HPA axis. Our model predicts that this regulatory variability inextricably links the sensitivity of the HPA axis to external stressors with its ability to integrate and be entrained by environmental cues such as light and temperature. We hypothesize that individuals with high adrenal sensitivity, and hence greater stress-responsiveness, are also more easily entrained by, and more sensitive to zeitgeber stimuli. Thus, our results predict the existence of a trade-off between the stress-responsiveness of the HPA axis and its ability to be a faithful time-keeping mechanism that is robust to noise in environmental conditions. Moreover, we specifically highlight allostatic regulatory adaptations upon chronic stress habituation that influence the stress-responsiveness and entrainment properties of the HPA axis. Our models suggest that by allostatically habituating to chronic stress in a way that homeostatic glucocorticoid rhythms are maintained, the system is forced to compromise either its entrainment characteristics (adaptation) or its responsiveness to external stimuli, or in many cases both. In doing so, we further determine individual differences in the susceptibility to allostatic load accumulation. Specifically, we predict that individuals with greater adrenal sensitivity, exhibit a greater sensitization of the acute stress response upon allostatic habituation to a chronic stressor, and thus might be more susceptible to the incidence of chronic stress disorders. An improved understanding of the regulatory dynamics of the HPA axis can provide new insights into its functioning in both health and disease.Supplementary Materials and Methods


Several technical points for postoperative lung hernia repair are still not fully elucidated. We present an original technical solution to deal with this complication. In a 68-year-old female, the lung hernia was confirmed 5 months after the partial left-sided chest wall and scapula angle resection with primary Mersilene mesh reconstruction for elastofibroma. The patient refused the proposed surgical correction, being only slightly limited in daily activities. The symptoms persisted under analgetic therapy till the moment when patient's daily activities became critically limited, 22 months after surgery. The repeated chest CT showed a slight increase in hernia size with no signs of tumour recurrence, so that reoperation was planned. After the exposure of the mesh region, a lung protrusion (4 × 3 cm) along the anterolateral edge of the mesh was confirmed. By careful dissection, the mesh was separated from a firmly adherent lung and removed. After adhaesiolysis and complete lung liberation, a wedge resection of the afunctional lung tissue of the lingula was done, just in the region of contact with the mesh. After the chest tube insertion, the chest wall defect was reconstructed by using a Mersilene mesh, and the final chest wall stabilization was done by the fixation of two Synthes plates (DePuy Synthes J&J) over the 5th and 6th ribs. The postoperative course was uneventful. One year after the operation, the patient was in good general condition, without the need for analgesics. To the best of our knowledge, the described technique is the original way of dealing with postoperative lung hernia. We find it efficient as a prevention of potential serious hernia-related complications. Lung herniation is a protrusion of the pleural-covered lung parenchyma beyond the normal boundaries of the pleural cavity , with noWe present a novel technique of the postoperative lung hernia repair, by combining the mesh interposition and Synthes plates. To the best of our knowledge, there is only one paper published so far dealing with lung hernia in a similar way.A 68-year-old female underwent a partial left-sided chest wall resection, with partial removal of the 6th and 7th ribs and of the scapula angle for elastofibroma . The cheAfter the excision of the previous skin scar and the incision of the muscular layer, the mesh region was exposed, showing a lung protrusion (4 × 3 cm) along the anterolateral edge of the mesh . The locAt the last contact with the patient, one year after the operation, the general condition was good, without the need for analgesics.Postoperative lung hernias are usually associated with minimally invasive surgery, where the intercostal incision is longer than a skin incision . It was A mesh disruption in the anterolateral part and the technique of the mesh fixation (interrupted stitches) suggest that the upfront mesh reinforcement ought to have been done at a time of the first surgery. The patient reported episodes of dry cough preceding the hernia symptoms, a probable key factor causing mesh disruption.According to the literature, most of the postoperative lung hernias occur weeks to months after surgery, like in the presented case. The probable explanation is, as previously documented, that signs of lung hernia, in the form of bulging with cough or palpable chest wall defects, may or may not be associated with symptoms, sometimes being even misleading. Treatment for an acute systolic heart failure or urgent intubation and embolisation for hemoptysis have been reported as well , 5. In tFrom the standpoint of surgical technique, two points are to be mentioned. First, having in mind the localization of the defect, was a primary mesh reconstruction necessary during the first surgery and would the upfront stabilization have been better? The primary reconstruction was done during the first surgery because of the need to remove the scapula tip and to achieve stability with the time. The upfront stabilization was not deemed as necessary because the defect was not too big (including only two ribs) to justify the presence of the metallic foreign body. We do not recommend the support by metallic plates as the solution in general at initial surgery but only in case of correction failure. In case of anterolateral chest wall defect, we think that the mesh reinforcement should be taken into consideration as well.Second, was titanium plate stabilization necessary and what would have been the alternatives? The chest wall rigidity restoration by using polytetrafluoroethylene, polypropylene, or composite meshes as well as methyl methacrylate sandwich is well established and sufficiently documented, including our earlier experience , 8. In tThe presented case can be of interest from at least three standpoints: (1) possible cause of the hernia; (2) interval between the hernia occurrence and diagnosis; and (3) hernia repair technique. Similar approach was described by using laminar hooks connected to titanium bars in one patient  and intrNo need for analgesics and good function of the shoulder girdle after the one-year follow-up demonstrate the appropriateness of the described technique.To summarize, we believe that the hernia was caused by the tearing of the insufficiently thick soft tissue in the anterolateral intercostal area, where the mesh-fixating sutures were placed. As the mesh was initially fixated by interrupted sutures, we assume that an additional running suture (as performed during the redo surgery) would be helpful to prevent hernia occurrence. Based on our good experience with single sutures for mesh fixation, we do not suggest the upfront switch to running suture for the first suture layer. Furthermore, we do not suggest the upfront chest wall stabilization for hernia prevention. We suggest it only as a method of correction in case of failure of the initial postoperative hernia repair.