We propose and demonstrate a single channel interferometer that can be used to study how vortex beams propagate through a scatterer. The interferometer consists of a multifunctional diffractive optical element (MDOE) synthesized by the spatial random multiplexing of a Fresnel zone plate and a spiral Fresnel zone plate with different focal lengths. The MDOE generates two co-propagating beams, such that only the beam carrying orbital angular momentum is modulated by an annular stack of thin scatterers located at the focal plane of the Fresnel zone plate, while the other beam passes through the centre of the annulus without any modulation. The interference pattern is recorded at the focal plane of the spiral Fresnel zone plate. The scattering of vortex beams through stacks consisting of different number of thin scatterers was studied using the proposed optical setup. Conflicting results have been reported earlier on whether higher or lower charge beams suffer more deterioration. The proposed interferometer provides a relatively simple and compact means of experimentally studying propagation of vortex beams through scattering medium. These techniques can be broadly classified into two types. In the first type, optical signal processing8, correlation techniques13 and phase retrieval algorithms15 have been developed to decode a speckle signature and convert it into useful information. A second direction of research is to develop adaptive aberration correction methods to compensate for the disturbance introduced by the scattering medium17. In the latter approach, structured light beams with scatter-resistant characteristics are used to reduce the aberrations introduced by the scatterer19.Light propagation through scattering and turbid medium produce speckle signatures from which it is often difficult to extract useful information. Different techniques have been developed in order to look through both static as well as dynamic scatterers20 and optical communication21 experiments. Besides, LG beams have also shown scattering-resistant properties in various studies25 and several techniques have been developed to aid probing and communicating through scattering medium using LG beams27. In most of these techniques28, the deterioration of the phase of the beam is studied using an interferometer. With the need for interferometry, the footprint of the experiment increases, as more components and vibration isolation, have to be added to the experiment. There are a few interference-less optical configurations and techniques used to measure the level of deterioration of the LG beam26 but the information is incomplete without measurement of phase.Vortex or orbital angular momentum (OAM) beams have a unique complex amplitude, which makes them attractive for various applications. For example, Laguerre-Gaussian (LG) beams have been used in optical trappingIn this manuscript, we propose a single channel optical configuration based on a multifunctional diffractive optical element (MDOE) for studying the robustness of LG beams of different charges versus level of scattering. The advanced optical configuration does not require a vibration isolation system, beam splitting, etc., and therefore, is extremely compact. Secondly, the interferogram is designed to directly show whether the charge of the beam changed during propagation, in terms of the number of radial intensity lobes. In that sense, the measurement of the deterioration of LG beams through thin scatterers can be considered a pattern recognition problem. The single channel interferometer has been implemented to study and compare the scattering characteristics of LG beams with different topological charges when interacting with single or multiple thin scattering layers.Several groups have reported theoretical and simulation studies of OAM beam propagation in a scattering medium. While the results depend on several factors including the type of scattering media under study, the characteristics of the OAM beam propagated, and the performance parameters measured; more detailed studies are required to resolve the many different results presented so far. The compact interferometer is proposed as an elegant means by which to carry out further studies in this important field.29 and a spiral Fresnel zone plate32 with different focal lengths. The phase of the Fresnel zone plate (FZP) is given by R is the radial coordinate and the focal length f = f1. On the other hand, the phase of the spiral Fresnel zone plate is given by L is the topological charge, f = f2 and θ is the azimuthal angle in the beam’s cross section. The focal length values are selected such that f1 < f2 and the scatterer is positioned at the plane corresponding to f1, as shown in Fig. Mr(T) represents a virtual mask comprising a binary random pattern  synthesized with a transmittivity T, where T is given by N1/(N1 + N2). N1 and N2 are the number of pixels with value 1 and 0 respectively. As the virtual mask is applied directly to the spiral phase through the operation T can also be considered to control the fraction of the incident energy directed into the LG beam. A larger value of N1, would increase the intensity of the LG beam with respect to the reference beam. Therefore, we will henceforth refer to T, as the splitting ratio.The optical configuration of the single channel, dual beam interferometer is shown in Fig. f1, the former beam is focused to a spot, whereas the latter occupies a larger annular region. There is therefore, no overlap between the beams. On the other hand, at the focal plane f2, the two beams spatially overlap resulting in an interference pattern, with intensity given byE1 is a diverging spherical wavefront,Collimated light from a coherent source is incident on the MDOE. This generates two co-propagating beams with complex amplitudes R is the radial distance from the axis. Neglecting complex constants results in the following expression for the spirally varying phase front E2,In Eq. , R is thA deeper look at the formation of the interference pattern reveals many interesting characteristics of the compact interferometer configuration. In any two-beam interference-based phase measurement technique, the interferogram is processed and the phase is retrieved using phase retrieval algorithms. In the proposed technique, the two wavefronts that interfere in the sensor plane are a diverging spherical wavefront and a helical wavefront. The diverging spherical wave has a constant phase for one value of radius, as the azimuthal coordinate is varied from 0 to 2π. On the other hand, the helical wavefront has a linear phase variation along that same path. Therefore, the resulting interference pattern is a direct representation of the relative phase difference between the two waves. Consequently, a maximum value in the interference pattern represents a phase difference of 0 and a minimum value in the interference pattern represents a phase difference of π. The number of peaks along any ring of infinitesimal thickness represents the topological charge of the helical wavefront.f1. The index p is used to denote the number of layers in the stack. The resultant scattering phase therefore, is the modulo-2π phase addition of p scattering layers (that form the stack). The intensity at the plane f2 will be given by In order to study the effect of scattering, a stack of annular scatterers, with phase I1 and I2 in comparison to the autocorrelation of I1. It is well-known that autocorrelation of a function gives the narrowest peak while the deviation from this ideal function can be quantified against the degree of deterioration. From the previous explanations, cross-correlation measures the degree of deterioration of the phase profile with respect to the ideal one and therefore is a direct and simpler method compared to the phase retrieval methods.The degree of deterioration of the vortex beam can be measured by a cross-correlation between 32. Since, binarization produces a zeroth diffraction order when there is a phase error during fabrication, an additional linear phase with an angle α = 0.03 radians was added to the MDOEs such that the required optical signal is moved away from the optical axis. The images of the design files for MDOEs with T = 0.3, 0.5 and 0.7 for topological charges L = 1 to 5 are shown in Fig. L. When T = 0.5, the incident energy is divided equally between both the diffractive functions, whereas for other T values, one function of the MDOE will dominate changing the splitting ratio. The fabrication procedure is presented in Supplementary Section MDOEs were designed as binary elements and therefore, the calculated function entclass1pt{minimaλ = 632.8 nm) is spatially filtered using a pinhole with a diameter of 100 µm and a refractive biconvex lens La (fa = 40 mm). The spatially filtered light is collimated by a second refractive biconvex lens Lb (fb = 100 mm). The collimated light is incident on the MDOE, which generates several orders. Of interest are the spherical converging beam and a spiral-spherical beam, with focal lengths 25 cm and 30 cm respectively, generated in the +1st order. An image sensor (Thorlabs camera with 1024 × 768 pixels of size 4.65 µm) mounted at a distance of 30 cm from the MDOE captures the interference of these two beams at this plane. The interference pattern will be between a spherical reference beam and either a scattered and or an unscattered LG beam, depending on whether or not the annular scatterer stack is introduced at a distance of f1 (25 cm) from the MDOE. A study of the scattering characteristics of the scatterer stack is presented in Supplementary Section The experimental set up is shown in Fig.  La fa = 0 mm. Thef1 and the interference pattern is once again recorded at the plane f2. The images of the recorded interference patterns for the different topological charges and splitting ratios are shown in the Fig. p = 1 and p = 2 respectively.Following this, the stack of scatterers is introduced in the plane m along the azimuthal direction that match the topological charge L. In the experiment, an LG beam of known charge is sent through the scatterer; any change in the charge of the beam would be easily picked from the number of lobes in the interference pattern.The first point to note is that the resulting interference pattern contains a number of intensity lobes T is increased, the relative intensity of the two beams in the plane f1 varied resulting in the ring pattern around the central spot becoming brighter. This is not surprising, as T controls the amplitude of the LG beam, as described earlier. Figure f2, with the scatterer in place. It is clear that the disturbance to the LG beam (that is the beam that sees the scatterer) causes the interference pattern to deviate from the case without the scatterer. The deviation is even more pronounced for the case two scattering layers (p = 2). It is also observed that the deterioration to the interferogram is greater, when the LG beam’s amplitude is weaker (T = 0.3). The experimental results were verified by simulations, presented in Supplementary Section – From Fig. 33. They are compared with the corresponding autocorrelation results of the un-scattered OAM beam. The cross-correlation results for p = 1 and p = 2 for T = 0.5 are shown in Fig. e2 radius of the cross-correlation and autocorrelation plots for topological charges L = 1 to 5 are calculated and normalized with the corresponding autocorrelation radius. This is plotted in Fig. p = 1 case, as the topological charge increases, the cross-correlation result has more deviation from its corresponding auto-correlation result. This says that lower charge OAM modes are less deteriorated by the scattering layer. It would be because we have used a dense scatterer, which is expected to deteriorate the stability of the propagating OAM mode strongly, resulting in fracturing of the OAM mode35. However, in the p = 2 case, the range of topological charge under study seems to be affected in the same way, which is why the deviation is more or less a constant with a slightly increasing trend.The conventional method for the study of scattering characteristics using phase retrieval algorithm is given in Supplementary Sections 28, the study of beam propagation through scatterers involves bulky optical configurations with a minimum of two optical channels and many optical components. In the proposed method, the entire interferometer consists of only one optical element. The proposed single channel interferometer with a MDOE along with a blind cross-correlation method was found to measure the degree of deterioration of the beams carrying OAM. The experimental results match with the results expected with the scatterers used. Clearly, the results would change dramatically according to the type of scattering media being studied. This conclusion is supported by the simulation results presented in the supplementary material. In particular, Supplementary Fig. 36 indicated that the scattering degree of OAM beams with higher topological charges was higher when they were incident on diffuser, which confirms the preliminary study carried out by the compact interferometer. More thorough studies and comparative analysis of different charged beams in scattering medium are needed. We believe that the proposed set-up and MDOE will allow exactly this. The efficiency of the demonstrated MDOE is limited due to the binary random multiplexing method and the efficiency can be increased by fabricating a greyscale version of the elements.We have proposed and demonstrated a compact, single channel interferometer using a MDOE for the study of propagation of beams with orbital angular momentum through scattering layers. The MDOE generates two complex waves, where OAM is carried by only one of the waves and the other wave is used as a reference. The MDOE generated a focal plane suitable for modulating only the vortex, while transferring the reference wave without any modulation. The optical configuration was designed such that in an axial plane, the topological charge can be measured from the number of intensity peaks. In most existing optical configurationsSupplementary Information2


Chlamydomonas reinhardtii with a data-driven mathematical model of HSR. We based our dynamical model mostly on mass action kinetics, with a few nonlinear terms. The model was parametrized and validated by several independent datasets obtained from the literature. We demonstrate that HSR quantitatively and significantly differs if an increase in temperature of the same magnitude occurs abruptly, as often applied under laboratory conditions, or gradually, which would rather be expected under natural conditions. In contrast to rapid temperature increases, under gradual changes only negligible amounts of misfolded proteins accumulate, indicating that the HSR of C. reinhardtii efficiently avoids the accumulation of misfolded proteins under conditions most likely to prevail in nature. The mathematical model we developed is a flexible tool to simulate the HSR to different conditions and complements the current experimental approaches.Global warming exposes plants to severe heat stress, with consequent crop yield reduction. Organisms exposed to high temperature stresses typically protect themselves with a heat shock response (HSR), where accumulation of unfolded proteins initiates the synthesis of heat shock proteins through the heat shock transcription factor HSF1. While the molecular mechanisms are qualitatively well characterized, our quantitative understanding of the underlying dynamics is still very limited. Here, we study the dynamics of HSR in the photosynthetic model organism  By this mathematical approach to the HSR in C. reinhardtii, we show that at the quantitative level, the HSR differs substantially if an increase in temperature of the same magnitude occurs rapidly, as often applied in typical laboratory heat shock experiments, or gradually, as expected in most natural environments.As a consequence of global warming, plants are more and more subject to heat stress, a condition that can severely reduce crop yield ,2. UnderC. reinhardtii is a widely studied, easy to grow photosynthetic model organism with promising industrial applications like biopharmaceuticals, biofuels and hydrogen. Thus motivated, different aspects of the HSR in C. reinhardtii have been experimentally investigated, as reviewed, e.g. by Schroda et al.  + [P#], [SK] + [SK*] and [G] + [HSF *G] + [HSFG], thus the vertical scale represents the fraction over the total, allowing for direct comparisons with relative values from experiments. Quantities in figure 1d,f,g are expressed in arbitrary units.To investigate the typical behaviour of the model, we simulate a heat shock at time b). The concentration of the functional form P suddenly decreases due to the nonlinear ωTP relation and the misfolded form P# increases correspondingly. The latter form induces the transition of the SKs into their active form SK* (figure 1c). SK* phosphorylates the HSF (figure 1d). The activated HSF* activates gene transcription (figure 1e). The amount of free gene G decreases and the active form with phosphorylated HSF bound, HSF*G, increases rapidly. Simultaneously, the gene bound to the inactive form of HSF (HSFG) increases as well, but with a slower dynamics. The activated gene induces mRNA production (figure 1f) for both HSF and HSP. These mRNAs are translated, leading to an increase of the HSF itself (figure 1d) and of the HSP (figure 1g). The increased chaperon level (figure 1g) leads to re-folding of degenerated proteins into their functional form (figure 1b), which eventually leads to a termination of the response. This analysis illustrates that the model is able to realistically describe the HSR.Owing to the temperature increase, the functional proteins become misfolded  of the deviations between model simulations and experimental data. We first performed a Monte Carlo (MC) scan of the parameter space to gain insight into its structure and then a gradient search to find a set of parameters which locally optimizes the objective function.A general challenge in modelling biological systems is the identification of system parameters because often these are not directly accessible experimentally. None of the rate constants listed in electronic supplementary material, table S4 are explicitly known. Still for some of them a reasonable range can be estimated from databases . In partorted in . Conside  after 06.00. The response remains until the evening and during this time the accumulation of P# due to the increase in temperature is counterbalanced by the HSR . When the temperature becomes sufficiently low in the evening the level of unfolded proteins P# decreases considerably, thus there is no need of a HSR until the next day.As we can see from figure 4#], figure 4a). This is remarkable because this value is more than one order of magnitude smaller than that obtained for a stepwise increase in the temperature, figure 1b. As unfolded proteins are undesired by the cell, it is meaningful that the HSR, which for a sudden increase of 20°C is not fast enough and allows for a certain transient accumulation of unfolded proteins, is on the other hand perfectly capable of preventing the accumulation of unfolded proteins during a heat shock like those that occur in nature.Importantly, the accumulation of unfolded proteins remains considerably smaller than 0.25% of the total amount of proteins  to lead to an accumulation of [P#] above the level which triggered the initiation of the HSR. This allows the cell to remain exposed to high temperatures for several hours during the day accumulating far less unfolded proteins than when the temperature increase occurs within minutes instead of hours. This observation allows us to speculate and propose the hypothesis that the HSR might be adapted to naturally occurring, smooth daily temperature variations rather than abrupt temperature changes. The system's parameters could thus be the result of adaptation to variable thermal environments, such that the HSR would be well suited to handle daily temperature fluctuations.The circadian clock of  studied ,34 and iTlow to the final value Thigh, we systematically simulated heat shocks in which the temperature raised from Tlow = 25°C to Thigh = 42°C in a cosinusoidal way during a time τ and subsequently stayed at Thigh (h). We screened for various values of τ as shown in figure 4i. We observed that for an instantaneous increase in temperature, up to increases which require approximately 1 min (similar to those provided to cells during experiments), the maximum value of the degenerated protein concentration does not change, and exhibits a very large value. For τ between approximately 1 and 100 min there is a steep fall, and another plateau is reached for τ larger than approximately 100 min. These are the timescales which a Chlamydomonas cell is more likely to experience for an increase of temperature in natural conditions. The accumulated concentration of HSP is much lower than for shorter times τ, contributing to generate the hypothesis that the HSR dynamics might be adapted to natural temperature variations . In electronic supplementary material, §E, we show that these results are robust against changes in the parameters m and P0 of the Hill kinetics term ωPS by which SK gets activated .To investigate how the maximum concentration of unfolded proteins accumulated during a heat shock depends on the dynamics of the temperature increase from the initial value at Thigh h. We scra–f: an early heat shock lasting for about the first 3 h, and a late heat shock in which the system shows acclimation (a new steady state is reached), compatible with the experimental findings of  is kept very close to zero. On the other hand, for too high temperatures the HSR cannot prevent the accumulation of degenerated proteins and the cell dies. We have finally been able to systematically investigate how the accumulation of HSPs depends on the combination of temperature and duration of the heat shock in electronic supplementary material, §D.3, which we illustrate in electronic supplementary material, figure S9M.One of the main advantages of a mathematical model, inferred and calibrated from experiments, is that it allows situations to be simulated that are difficult to test with experiments and enables computation of quantities that are difficult to measure. To outline the potential of our model, we employed it to simulate systematically further situations. We have been able, as described in electronic supplementary material, §D.1, to show the ability of the system to acclimate to temperatures higher than usual during heat shocks longer than 3 h by shifting to a new steady state. Two distinct phases are clearly visible in electronic supplementary material, figure S9dings of . We have4.C. reinhardtii, a photosynthetic model organism. We have extracted the signalling network structure from various experiments, and experimental data are used for model parameterization.In this work, we have developed a data-driven mathematical model for the HSR in The model is based on a number of simplifying assumptions. Foremost, we assume that the simulated cells are close to a stationary state, in which the total numbers of proteins remain constant. This, however, may be inappropriate if either extreme and prolonged stress conditions apply, or if synchronized cells in periodic diurnal conditions are considered. Especially the latter is important for our predictive study regarding the response to natural daily temperature variations. It is well known that HSPs also underlie a circadian rhythm. For a future study, it would therefore be interesting to link the gene expression dynamics with the output of a model of the circadian clock, in order to study how the results obtained here would be influenced by including dynamic protein concentrations. Such an extended model would moreover provide the possibility to investigate and predict how the HSR depends on the time of the day when it is applied.For model validation, we have tested the model's behaviour with independent experimental results extracted from the literature; the following were not employed for the calibration: feeding experiments, double heat shock experiments and HP expression. By this, we have shown that the model is able to reproduce very well the main features of various experimental datasets. This capability shows that our conclusion about the signalling mechanism is plausible and robust. Even though the model is based on simplifying assumptions, it exhibits very good agreement with experimental observations. Moreover, it allows analysis of the HSR at different signal levels which are not easily accessible in experiments. An example is represented by our analysis of the double heat shock experiment, where it is interesting to note the (simulated) behaviour of e.g. SK* which is usually not measured in experiments .a–f). An interesting observation of the simulations was that the percentage of misfolded proteins does not exceed 0.25% of total proteins. While these numbers are not experimentally validated and should therefore be interpreted with care, it is remarkable that this fraction is a factor of 20 lower than the predicted 5% unfolded proteins, when a sharp temperature increase (of the same amount of degrees) is simulated . Likewise, and possibly even more important, the maximal fraction of SK that is activated amounts to 80% upon a sharp temperature increase (figure 1c), while it reaches only approximately 4% for a smooth temperature variation (g). These results indicate that for the slow temperature increase the system is not really under stress, whereas the sharp temperature increase, usually applied in heat shock experiments, imposes severe stress, activating the vast majority of the available SKs.We then investigated the system's response to a smooth and more natural variation in temperature mimicking a hot day a–f. An iariation g. These τ of temperature increase from the minimal to the maximal value (i). This systematic investigation has shown that for times τ shorter than 1 min (i.e. a sharp increase in temperature) the maximal concentration of unfolded proteins is approximately 20 times higher than for increases that are slower than approximately 100 min. This result supports the notion that the HSR of C. reinhardtii is able to keep the concentration of unfolded proteins at negligible levels if the increase in temperature occurs on a timescale of several minutes to hours, in sharp contrast to the sudden increases usually applied in experiments. This indicates that the HSR of C. reinhardtii is indeed tailored to handle natural temperature variations due to a match of the intrinsic timescale of the HSR system, which could be easily tested in experiments. Moreover, as we have shown with several applications, our model provides a framework to explore further situations of interest and to generate new hypotheses to be tested by experiments.We have finally studied how the maximal concentration of unfolded proteins reached during heat shock depends on the time al value i. This sWhereas models such as the one presented here may help greatly to elucidate molecular mechanisms and enable the interpretation of experimental data in a sound theoretical framework, they are clearly idealized and represent an ‘average’ cell. Thus, it is not straightforward to employ this model if one wishes to make predictions about the adaptive behaviour of a cell population, in particular, over longer timescales. However, in principle it seems possible that ensembles of models such as the one presented here, each with slightly modified parameters to reflect the plasticity found within a natural population, can be used to predict in which direction a population , or even


A version of the upper left panel of Figure 4D was inadvertently used as the upper left panel of Figure 4E and incorrectly identified as Trichrome staining. This error occurred during the preparation of figures and went unnoticed during the review process and during preparation of final publication.The original and revised Figure 4E is shown below, with the correct Trichrome stain image in the upper left panel. This correction does not alter any of the findings or conclusions of the study.The authors regret this error.


Additionally, some other data were miscategorized (‘capture’ versus ‘escape’).The authors reanalyzed all the data, which showed that the overall results and conclusions were unaltered. This was confirmed by a reviewer of the original paper, who was asked to review the corrected version and specifically state whether or not, in their opinion, the conclusions were unchanged.COPE), which state: “Retractions should usually be reserved for publications that are so seriously flawed (for whatever reason) that their findings or conclusions should not be relied upon”.The editorial policies of BiO state that: “Should an error appear in a published article that affects scientific meaning or author credibility but does not affect the overall results and conclusions of the paper, our policy is to publish a Correction…” and that a Retraction should be published when “…a published paper contain[s] one or more significant errors or inaccuracies that change the overall results and conclusions of the paper…”. We follow the guidelines of the Committee on Publication Ethics .body), flight initiation distance calculated using the nearer prey's eye (FIDeye) and flight initiation distance calculated using the prey's center of mass (FIDCM)] in the largest and smallest initial orientations were smaller than those in the other initial orientations. This trend was detected in the regression curve in the original version, but was not detected in the regression curve in the corrected version . Additionally, there was a significant positive linear relationship between initial orientation and FIDeye in the corrected version, but the relationship was not significant in the original version .In both versions, flight initiation distance [flight initiation distance calculated using the closest margin of the prey's body to the predator's snout  was the largest when the initial orientation was away from predators (150-180°) and the values were similar among the other initial orientations (0-150°). This trend was detected in the regression curve, and was statistically significant in the original version. However, this trend was not detected in the regression curve, and was not statistically significant in the corrected version .Predator speed was the smallest at the 0-30° initial orientation, and the second smallest at the 90-120° initial orientation in the original version. The speed was the smallest at the 120-150°, and the second smallest at the 0-30° initial orientation on the corrected version .In both versions, an increase of initial orientation tended to decrease the mean turning rate. This trend was statistically significant in the original version, but was not statistically significant in the corrected version .The authors wish to particularly draw readers' attention to the following changes in trend:The authors apologise to the journal and readers for these errors.


This study was designed to investigate heat accumulation and bone thermal necrosis for various distances between holes and time delays between drillings.2 coolant was also tested in addition to two common cooling modes in bone drilling.The tests were performed at three distances  and three time delays: 0, 5 and 10 s. To examine the efficiency of coolants, COThe main results were the trend of temperature–time graph, maximum temperature at drilling site, temperature distribution on the surface of drilling site, temperature durability and returning time. The effect of lateral drillings on the initial hole was notable in drilling at a distance of 6 mm without cooling. This effect did not disappear even by increasing the time delay up to 10 s. The results obtained for drilling with normal saline coolant were not sufficiently acceptable due to the manual and non-uniform cooling process as well as the relative obstruction of the chips exit path. Generally, drillings with two common cooling modes, even when the distances between holes and time delays between drillings were controlled, did not yield all favorable conditions for preventing bone thermal necrosis.2 coolant eliminates the risk of bone thermal necrosis completely even in cases that the distances between holes in plates or implants are 6 mm and there is no time delay between drillings. These results can be especially useful in emergency orthopedic surgeries and for designing the location of screw holes in implants and plates.Bone drilling using CO Bone drilling is usually required for inserting implants and fixing plates in fractured bones , 2. DespIn the first group of studies, the variables affecting the manner and procedure of bone drilling to reduce thermal necrosis were evaluated. Gholampour et al. examined the effect of drilling depth and direction on thermal necrosis in tibia . The othIn the second group of studies, the effect of the drilling tool type on reduction of bone thermal necrosis was investigated. Hein et al. and Palmisano et al. compared the heat generated during drillings with industrial and orthopedic drill bits , 25. UdiSome studies also measured the temperature changes at the surface of drilling site using a thermography camera considering the importance and notable impact of temperature distribution at the surface of drilling site on occurrence of thermal necrosis. Stumm et al. proposed the thermo-mechanical coefficients and parameters affecting bone thermography . Kim et Based on the literature, comprehensive researches have been performed regarding the optimal conditions and manner of bone drilling for minimizing thermal necrosis. Inserting implants or fixing plates requires usually drilling several holes in bone. Consequently, determining the optimal distances between holes and proper time delay between each drilling and the next one is important. However, none of the previous studies have examined the effect of heat accumulation due to change in these distances and time delays on bone thermal necrosis. The present study investigates these effects and due to the importance of coolants in reducing the risk of bone thermal necrosis during bone drilling, compares the efficacy of internal gas coolants with that of two cooling modes commonly used in operation rooms  with the aim of improving the efficacy of orthopedic surgeries.The femoral diaphysis of 23 grown-ups  was used as test specimens. These specimens were cut in dimensions of 25 mm to make testing easier and each test was repeated seven times. The specimens were stored according to the recommendation of Sedlin & Hirsch and Hillery et al. at − 20 °C in Ringer’s solution and then were tested , 32.First, the effect of distance between the holes on insertion of plates and implants was investigated during drilling with the aim of minimizing bone thermal necrosis. To do this, an initial hole was created in the bone. Then lateral holes were drilled at a distance of 6 mm from the center of the initial hole so that the effect of accumulation of the generated heat due to drilling of lateral holes on the initial hole and its effect on bone thermal necrosis could be investigated. It should be noted that the three values of 6, 12 and 16 mm are the most common distances between the screw holes in a variety of implants and plates. Thus, the above experiments were repeated for distances of 12 and 16 mm between the centers of the initial and lateral holes.The next variable in successive drillings was the time interval between the first drilling and the lateral drillings. Hence, in the second part of the experiment, the lateral holes were drilled without delay after drilling the initial hole. The experiment was then repeated for time delays of 5 and 10 s to assess the effect of time delay between drillings on reduction or raise of heat accumulation due to lateral drillings at the initial hole. It should be noted that all drillings were done at a depth of 8 mm with a rotational speed of 2000 rpm as the most optimal rotational speed for minimizing thermal necrosis , 16.2 gas. CO2 was selected from among the applicable gas coolants for this study as this gas does not have any biocompatibility problem or infection risk according to previous studies and is suitable for using as coolant during bone drilling [2 is also very common in medicine, e.g. in laparoscopy operation. The conventional drilling machines, however, are not able to internally transfer CO2 gas through the body of drill bit directly to the point of drill bit and ultimately to the drilling site. On the other hand, manual drills are commonly used for bone drilling in operating rooms. Therefore, for better and more realistic simulation of operating room conditions, a manual drill with the ability of direct gas transfer to the drilling site was first developed in this study  of all the data was calculated and the results of Table 2 gas was used. The maximum temperature during drilling with CO2 gas as coolant was 45.2 °C.All investigations in this section were performed by changing the spatial distance between the centers of the laterals and initial holes from 6 to 16 mm while drillings were done without time delay. In drilling at a distance of 6 mm, the temperature–time graph of the drilling site of initial hole experienced temporal temperature increases from 54.1 to 61 °C and from 52.7 to 60.5 °C following drilling of lateral holes without cooling . The temperature dropped after completion of each lateral drilling and the graph returned to its main trend. The relevant amounts were, respectively, 13.2% and 9.1% for drilling with a time delay of 5 s  and 6.0% and 3.2% for a time delay of 10 s . The maximum temperature at the drilling site in this cooling mode exceeded the threshold of human bone thermal necrosis (47 °C). According to Fig. 2 gas as coolants caused no particular changes in the trend of the temperature–time graph of the initial hole. The maximum temperature in CO2 cooling mode did not exceed 47 °C.All examinations in this section were performed by changing the time delay between the initial drilling and lateral drillings from zero to 10 s while the distance between the centers of the lateral holes and the initial hole was 6 mm. In the case of drilling without cooling and without delay, the lateral drillings led, respectively, to a temperature increase of 13.3% and 14.8% in the temperature–time graph of the initial hole . Thus, drilling with CO2 coolant can be proper option regarding the maximum temperature at drilling site and the trend of temperature–time graph. The temperature at the surface of drilling site returned under the threshold value of bone thermal necrosis with increasing the distance from 6 to 12 mm in drilling without cooling  was zero , which indicates the durability of the drilling site temperature at a level above the 47 °C, and the returning time (RT), which indicates the time taken for return of the drilling site temperature to initial temperature  and even without any time delay between drillings, eliminates the risk of occurrence of thermal necrosis and potential post-operation complications completely. The results of this study can be useful for controlling the distances between holes and time delays between drillings required for bone fixation in orthopedic surgeries, especially in emergency orthopedic surgeries. The manufacturers of plates and implants can also use these results for designing the location of holes in implants and plates.The results of examinations of the trend of temperature–time graphs, the maximum temperature at the drilling site, the temperature distribution on the surface of the drilling site, TD and RT show that using CO


In the social sciences it is common practice to test specific theoretically motivated research hypotheses using formal statistical procedures. Typically, students in these disciplines are trained in such methods starting at an early stage in their academic tenure. On the other hand, in psychophysical research, where parameter estimates are generally obtained using a maximum-likelihood (ML) criterion and data do not lend themselves well to the least-squares methods taught in introductory courses, it is relatively uncommon to see formal model comparisons performed. Rather, it is common practice to estimate the parameters of interest  and their standard errors individually across the different experimental conditions and to ‘eyeball’ whether the observed pattern of parameter estimates supports or contradicts some proposed hypothesis. We believe that this is at least in part due to a lack of training in the proper methodology as well as a lack of available software to perform such model comparisons when ML estimators are used. We introduce here a relatively new toolbox of Matlab routines called Palamedes which allows users to perform sophisticated model comparisons. In Palamedes, we implement the model-comparison approach to hypothesis testing. This approach allows researchers considerable flexibility in targeting specific research hypotheses. We discuss in a non-technical manner how this method can be used to perform statistical model comparisons when ML estimators are used. With Palamedes we hope to make sophisticated statistical model comparisons available to researchers who may not have the statistical background or the programming skills to perform such model comparisons from scratch. Note that while Palamedes is specifically geared toward psychophysical data, the core ideas behind the model-comparison approach that our paper discusses generalize to any field in which statistical hypotheses are tested. Indeed, students in these disciplines are trained in such methods starting at an early stage in their academic tenure. On the other hand, in psychophysical research, where parameter estimates are generally obtained using a maximum-likelihood (ML) criterion and data do not lend themselves well to least-squares methods, it is relatively uncommon to see formal model comparisons performed. Rather, it is common practice to estimate the parameters of interest  as well as their standard errors individually across the different experimental conditions and to ‘eyeball’ whether the observed pattern of parameter estimates supports or contradicts some proposed hypothesis. Another common strategy is to perform a least-squares method . This ‘model-comparison approach’ is extremely flexible and has been advanced previously by The PF relates some quantitative stimulus characteristic  to psychophysical performance . A common formulation of the PF is given by:x refers to stimulus intensity, ψ refers to a measure of performance , and γ and 1−λ correspond to the lower and upper asymptote, respectively. F is usually some sigmoidal function such as the cumulative normal distribution, Weibull function, or Logistic function. Parameters γ and λ are generally considered to be nuisance parameters in that they do not characterize the sensory mechanism underlying performance. For example, in a ‘yes/no task,’ in which a single stimulus is presented per trial and the observer must decide whether or not it contains the target, γ corresponds to the false alarm rate which characterizes the decision process. On the other hand, in an mAFC (m Alternative Forced Choice) task in which m stimuli are presented per trial and the observer decides which contains the target, γ is determined by the task and is generally assumed to equal 1/m. The parameter λ is commonly referred to as the ‘lapse rate’ in that it corresponds to the probability of a stimulus-independent negative response . The sensory mechanism underlying performance is characterized by function F. Function F has two parameters: α and β. Parameter α determines the location of F, while Parameter β determines the rate of change of F. The interpretation of α and β in terms of the sensory or perceptual process underlying performance depends on the specific task. For example, in an mAFC contrast detection task, α corresponds to the stimulus intensity at which the probability correct detection reaches some criterion value, usually halfway between the lower and upper asymptote of the psychometric function. In this context α is a measure of the detectability of the stimulus and is often referred to as the ‘threshold.’ However, in appearance-based 2-Alternative Forced Choice (2AFC) tasks . In the remainder of this paper we will use the terms location and slope parameter for α and β, respectively. These terms describe the function itself and carry no implications with regards to the characteristics of the underlying sensory or perceptual process. As such, these terms have the distinct advantage of being appropriate to use regardless of the nature of the task.in which C) tasks , §3.3 suWe will introduce the logic behind the model-comparison approach first by way of a simple one-condition hypothetical experiment. We will then extend the example to include a second experimental condition.Figure 1A displays the hypothetical results from such an experiment. Plotted is the proportion of trials on which the observer reported perceiving the lower line to the left of the upper line as a function of the actual Vernier offset. Figure 1B shows four different models of these data. These models differ as to the assumptions they make regarding the perceptual process underlying performance. All models share a number of assumptions also and we will start with these.Imagine an experimental condition in which an observer is to detect a Vernier offset. The task is a 2AFC task in which the observer is to indicate whether the lower of two vertical lines is offset to the left or to the right relative to the upper line. Five different offsets are used and 50 trials are presented at each offset. Figure 1B make the assumptions of independence and stability. Briefly, this means that the probability of a ‘left’ response is fully determined by the physical Vernier offset. An example violation of the assumption of independence occurs when an observer is less likely to respond ‘left’ on trial six because he responded ‘left’ on all of the previous trials. An example violation of the assumption of stability occurs when an observer over the course of the procedure becomes careless and more likely to respond independently of the stimulus. All models in Figure 1B also assume that the true function describing the probability of a ‘left’ response as a function of Vernier offset has the shape of the Logistic function. Finally, all models assume that the probability that an observer responds independently of the stimulus on any given trial (the lapse rate) equals 0.02. While this assumption is certain to be not exactly correct, data obtained in an experiment like this generally contain very little information regarding the value of the lapse parameter and for that reason, freeing it is problematic  when the two lines are physically aligned. Moving between the two rows places similar restrictions on the slope parameter of the functions. In the two models in the top row the slope parameter is a free parameter, whereas the models in the bottom row fix the slope parameter at the somewhat arbitrary value of 1. We refer to models here by specifying how many location parameter values and slope parameter values need to be estimated. For example, we will refer to the model in the top left corner as ‘1 α 1 β.’Even though the models in Figure 1B restricts the value of the location parameter, whereas moving downward restricts the value of the slope parameter. As a result, any model (‘model B’) in Figure 1B that is positioned to the right and/or below another (‘model A’) can never match the observed p(‘left’) better than this model and we say that model B is ‘nested’ under model A. From the four models shown in Figure 1B we can form five pairs of models in which one of the models is nested under the other model. For any such pair, we use the term ‘lesser model’ for the more restrictive model and ‘fuller model’ for the less restrictive model. For each such pair we can determine a statistical ‘p-value’ using a likelihood ratio test  is distributed asymptotically as the χ2 distribution with degrees of freedom equal to the difference in the number of free parameters between the models1. Thus, the likelihood ratio test can be used to perform a classical (‘Fisherian’) NHST to derive a statistical p-value.Given the considerations above, the appropriate model comparison here is that between the models labeled ‘fuller’ and ‘lesser’ in p-value is 0.691. The difference between the fuller and lesser model was the assumption that the location parameter was equal to zero, thus it appears reasonable to conclude that this assumption is valid. However, remember that the lesser model made additional assumptions. These were the assumptions of independence and stability, the assumption that the guess rate and the lapse rate parameters were equal to 0.02 and that the shape of the function was the logistic function. The model comparison performed above is valid only insofar as these assumptions are valid. We can test these assumptions (except the assumptions of independence and stability) by performing a so-called Goodness-of-Fit test.When the model comparison is performed using the likelihood ratio test, the resulting TLR equals 0.158. With 1 degree of freedom (the fuller model has one more free parameter [the location parameter] compared to the lesser model) the 2. Note that the saturated model makes no assumptions at all regarding how the probability of the response ‘left’ varies as a function of stimulus intensity or experimental condition. As such, it allows the probabilities of all five stimulus intensities that were used to take on any value independent of each other. Thus, the saturated model simply corresponds to the observed proportions of ‘left’ responses for the five stimulus intensities. Note that the assumptions of independence and stability are needed in order to assign a single value for p(‘left’) to all trials of a particular stimulus intensity. Note also that all models in Figure 1B, as well as any other model that makes the assumptions of independence, stability and additional (restrictive) assumptions are nested under the saturated model. Thus for all these we can perform a goodness-of-fit test using a likelihood ratio test. The p-value for the goodness-of-fit of our lesser model was 0.815 indicating that the assumptions that the lesser model makes but the saturated model does not  appear to be reasonable.The model comparison to be performed for a Goodness-of-Fit test is that between our lesser model from above and a model that makes only the assumptions of independence and stability. The latter model is called the saturated model. It is the fact that the fuller model in the comparison is the saturated model that makes this test a Goodness-of-Fit testFigure 3A. We can again apply a number of possible models to these data. Figure 3B shows nine models that can be applied to these data. These models differ as to the assumptions they make regarding the perceptual process underlying performance. Again some assumptions are shared by all nine models. All models make the assumptions of independence and stability. All models also assume again that the true function describing the probability of a ‘left’ response as a function of Vernier offset has the shape of the Logistic function. Finally, all models assume again that the probability that an observer responds independently of the stimulus on any given trial (the lapse rate) equals 0.02. As in the models shown in Figure 1B, the nine models in Figure 3B differ only with respect to the assumptions they make regarding the values of the location and slope parameters. Models in the left column make no assumptions regarding the value of either of the location parameters and allow each to take on any value independent of the value of the other. We say that the values are ‘unconstrained.’ Models in the middle column assume that the two location parameters are equal to each other (‘constrained’). In other words, according to these models the value of the location parameter is not affected by the experimental manipulation. However, these models make no assumption as to the specific value of the shared location parameter. Models in the right column further restrict the location parameters: they assume that both are equal to 0. As we did in the one-condition example, we say that the values for the location parameters are ‘fixed.’ Moving between different rows places similar restrictions on the slope parameters of the functions. Models in the top row allow both slopes to take on any value independent of each other. Models in the middle row assume that the slopes are equal in the two conditions, and models in the bottom row assume a specific value for both slopes . We refer to models here by specifying how many location parameter values and slope parameter values need to be estimated. For example, we will refer to the model in the top left corner as ‘2 α 2 β.’Imagine now that the researchers added a second condition to the experiment in which the observer first adapts to a vertical grating before performing the Vernier alignment trials. Of interest to the researchers is whether Vernier acuity is affected by the adaptation. The results of both conditions are shown in Figure 3B increases the restrictions on the values on the location parameters, whereas moving downward increases the restrictions on the slopes. Thus any model (‘model B’) positioned any combination of rightward and downward steps (including only rightward or only downward steps) relative to another (‘model A’) is nested under that model. From the nine models shown in Figure 3B we can find 27 pairs of models in which one of the models is nested under the other model. Again, for any such pair we can perform a model comparison and again that model comparison would test whether the assumptions that the lesser model makes but the fuller model does not are warranted. Which two models should be compared in order to test whether the adaptation affects Vernier acuity? A difference in Vernier acuity between the two conditions would correspond to a difference in the slope parameters. A higher value for the slope would correspond to a higher acuity. Thus, a model that assumes that adaptation does not affect Vernier acuity assumes that the slope parameters in the two conditions are equal. A model that assumes that Vernier acuity is affected by adaptation assumes that the slope parameters are different between the conditions. Thus, we would compare a model that allows different slopes in the two experimental conditions to a model that constrains the slopes to be identical between conditions. The models to be compared should make identical assumptions regarding the location parameters in the two conditions. This is for the same reason as outlined above in the one-condition example: If the models in the comparison differ with regard to the assumptions they make regarding location parameters as well as slopes and we find that the models differ significantly, we would not be able to determine whether the significance should be attributed to an effect on the location parameters, slope parameters or both. What then should the models in the comparison assume about the location parameters? Depending on the specifics of the experiment it might be reasonable here to assume that the location parameters in both conditions equal 0 . Thus, given the specific research question posed in this example and the considerations above, the appropriate model comparison is that between the fuller model ‘0 α 2 β’ and the lesser model ‘0 α 1 β.’ In Figure 3B we have labeled these two models as ‘Fuller’ and ‘Lesser.’ Figure 4 lists the assumptions of both the fuller and the lesser model. The line connecting the models is labeled with the assumption that the lesser model makes but the fuller does not. When this model comparison is performed using the likelihood ratio test the resulting p-value is 0.016 indicating that the slope estimates differ ‘significantly’ between the two experimental conditions. Note that the p-value is accurate only insofar the assumptions that both models make  are met. All but the first two of these assumptions can be tested by performing a Goodness-of-Fit test of the fuller model. The Goodness-of-Fit model comparison results in a p-value equal to 0.704 indicating that the assumptions that the fuller model makes but the saturated model does not  appear to be reasonable.Moving to the right in the model grid of Figure 3B from which 27 different pairs of models can be identified in which one model is nested under the other.To recap, the essence of the model comparison approach to statistical testing is that it conceives of statistical tests of experimental effects as a comparison between two alternative models of the data that differ in the assumptions that they make. The nature of the assumptions of the two models determines which research question is targeted. Contrast this to a cook book approach involving a multitude of distinct tests each targeting a specific experimental effect. Perhaps there would be a ‘location parameter test’ that determines whether the location parameters in different conditions differ significantly. There would then presumably also be a ‘slope test,’ and perhaps even a ‘location and slope test.’ For each of these there might be different versions depending on the assumptions that the test makes. For example, there might be a ‘location test’ that assumes slopes are equal, another ‘location test’ that does not assume that slopes are equal, and a third ‘location test’ that assumes a fixed value for the slope parameters. Note that the difference between the approaches is one of conception only, the presumed ‘location test’ would be formally identical to a model comparison between a model that does not restrict the location parameters to a model that restricts them to be identical. The model comparison approach is of course much more flexible. Even in the simple two-condition experiment, and only considering tests involving the location and slope parameters, we have defined the nine different models in Note that many more model comparisons may be conceived of even in the simple two-condition experiment of our example. For example, maybe we wish to test the effect on slope again but we do not feel comfortable making the assumption that the lapse rate equals 0.02. We then have the option to loosen the assumption regarding the lapse rate that the fuller and lesser model make. We could either estimate a single, common lapse rate for the two conditions if we can assume that the lapse rates are equal between the conditions or we could estimate a lapse rate for each of the conditions individually if we do not want to assume that the lapse rates in the two conditions are equal. We may even be interested in whether the lapse rate is affected by some experimental manipulation e.g., . We woulThe model-comparison approach generalizes to more complex research designs and research questions. As an example, per se. Rather, research questions concern themselves with relationships among parameter values derived under different experimental conditions. Thus, the common strategy to derive point and spread  estimates on the parameters of PFs in individual conditions is a somewhat peculiar and indirect method to address research questions. Moreover, the determination as to whether parameter estimates are significantly different is often performed by eye-balling parameter estimates and their SEs and often follows questionable rules of thumb  as opposed to following a theoretically sound procedure. Finally, unlike the model-comparison approach, the SE eye-balling approach does not allow model comparisons between models that make different assumptions regarding the value of multiple parameters simultaneously.Note that research questions rarely are concerned with the absolute value of any parameter Our software Palamedes was specifically designed to allow users to test research hypotheses using the model-comparison approach. One necessary ingredient to the model comparison approach is the ability to define models in terms of the assumptions that the model makes. These assumptions are formalized by specifying the relationships among parameter values across different experimental conditions. A second necessary ingredient is the ability to compare statistically any two (nested) models that a user might wish to compare. Together these characteristics allow users to perform statistical tests of a virtually unlimited range of research questions using a single general routine.Figure 3B as an example throughout. The file ModelComparisonSingleCondition.m in the PalamedesDemos folder performs the example model comparison that we performed in the Section “A Simple One-Condition Example Demonstrating the Model-Comparison Approach.” The file ModelComparisonTwoConditions.m in the PalamedesDemos folder performs the example model comparison described in the Section “A Two-Condition Example” using all three methods.Following the spirit of the model comparison approach, Palamedes has just a single routine (PAL_PFLR_Model Comparison.m) that performs model comparisons involving fits of PFs to multiple conditions. Users can tailor the model comparison to be performed by defining the fuller and lesser models. Models are specified using the arguments of the routine. Defining the models occurs by specifying the constraints on each of the four parameters  for both the fuller and the lesser model. Palamedes offers three methods in which to specify the constraints. These methods differ with regard to ease of use, but also with regard to flexibility offered. We will briefly discuss the three methods in decreasing order of ease of use (but increasing flexibility). We will use the model-comparison between the models labeled ‘fuller’ and ‘lesser’ in The easiest method by which to specify constraints on parameters is using the verbal labels of ‘unconstrained,’ ‘constrained,’ and ‘fixed.’ Each of the four parameters of a PF  can be independently specified. In order to perform the test described in the Section “A Two-Condition Example” we would set the location parameters, guess rates, and lapse rates of both the fuller and the lesser model to ‘fixed’ , we would set the slopes of the lesser method to ‘constrained,’ and the slopes of the fuller model to ‘unconstrained.’θ1 = 1 × β1 + 1 × β2, the sum of the slope values. If we include a second row [1 −1], this defines a new parameter θ2 = 1 × β1 + (−1) × β2, the difference between the slope values. In order to allow different slopes in each condition, we instruct Palamedes to estimate both the sum and the difference of the slopes by passing it the matrix . If we wish to constrain the slopes to be equal in the conditions, we instruct Palamedes to estimate only one parameter which corresponds to the sum of the slopes by passing the array [1 1]. Note that the new θ parameters exist behind the scenes only, Palamedes will report the estimated parameters in terms of β1 and β2.The second method by which to specify models is by specifying constraints on parameters using ‘model matrices.’ Model matrices serve to reparameterize parameters into new parameters that correspond to ‘effects.’ For example, a model matrix can be used to reparameterize the two slopes of the PFs into two new parameters, one corresponding to the sum of the slope values, the other to the difference between the slope values. Note that any combination of slope values . Instead of estimating the slopes directly, Palamedes estimates the values of the parameters defined by the model matrix. Each row of the matrix defines a parameter as some linear combination of the original parameters by specifying the coefficients with which the parameters should be weighted to create the linear sum. For example, in order to create a parameter corresponding to the sum of the slopes we include a row in the matrix that consists of two 1’s. The new parameter would then be defined as The advantage of using model matrices rather than the verbal labels above in order to specify models is that it allows model specifications that are not possible using the verbal label. For example, it allows one to specify that only the PSE in condition 2 should be estimated while the PSE in condition 1 should be fixed (by passing the model matrix [0 1]). Especially when there are more than two conditions does the use of model matrices afford much greater flexibility compared to the use of the verbal labels. A model comparison that compares a model in which the location parameters from, say, six conditions are ‘unconstrained’ to a model in which the location parameters are ‘constrained’ merely tests whether there are significant differences among the six location parameters, not where these differences may lie . Model matrices allow researchers to target more specific research questions. For example, if the six location parameters arose from a 2 × 3 factorial design, model matrices can be used to test hypotheses associated with the main effects and their interaction  or the General Linear Model (GLM), may have recognized LM e.g., . ElsewheLM e.g., , Box 9.4The third and most flexible method by which to specify model constraints on parameters is by supplying a custom-written function that defines a reparameterization of the parameter in question. For example, reparameterizing the slopes into parameters that correspond to the sum and difference between them  can also be accomplished using this function:function beta = reparameterizeSlopes(theta)beta(1) = theta(1)+theta(2);beta(2) = theta(1)-theta(2);Note that, somewhat counter-intuitively perhaps, the function takes the ‘new parameters’ (the thetas) as input and returns the parameters that directly correspond to one of the PF’s parameters . We would also specify our guesses for values of theta that will be used as the starting point of the iterative fitting procedure and we specify which, if any, of the thetas should be fixed and which should be estimated. Specifying both thetas to be free parameters is equivalent to using the verbal label ‘unconstrained’ or using the model matrix . Specifying that theta(1) should be estimated while fixing theta(2) (at 0), would be equivalent to using the verbal label ‘constrained’ or using the model matrix [1 1].Note that the size of the input and output arguments need not be equal. To give a simple example, we might fix the location parameters at a value of 0 (as in the example model comparison given above) using the function:function alpha = reparameterizeFixed(theta)alpha(1) = theta;alpha(2) = theta;in which theta is a scalar. We would also specify that theta should be fixed and that the value to be used is 0. To give another example, one can use this method to implement the assumption that location parameters in a series of, say, 10 consecutive experimental sessions follow a learning curve defined as an exponential decay function by using the reparameterization:function alphas = reparameterizeLocations(thetas)session = 1:10;∗exp(-thetas(3)∗(session-1));alphas = thetas(1) + thetas(2)If we use this function to specify location parameters, Palamedes will constrain these parameters to follow an exponential decay function with three parameters: thetas(1) will correspond to the lower asymptote of the function, thetas(2) will correspond to the difference between the (modeled) value of the first location parameter and the lower asymptote and thetas(3) will determine the learning rate. theta parameters) allowing one to compare the values of the new parameters directly. For example, one could compare the learning rates of two observers directly by comparing their estimated thetas(3) in the example above.Using this third method of specifying models in Palamedes is clearly the most flexible of the three. Any model that can be specified using model matrices can also be specified using this third method (while the reverse is not the case). This third method also has the advantage that Palamedes will return the fitted model not only in terms of the PF parameters  but also in terms of the new parameters  to specify that the location parameters of the fuller model are also fixed. An example call to PAL_PFLR_ModelComparison.m that mixes all three of the above methods is given in ModelComparisonTwoConditions.m.2 with degrees of freedom equal to the difference in the number of free parameters between the two models to be compared. PAL_PFLR_ModelComparison also returns the appropriate number of degrees of freedom for the test. Thus, the user can utilize standard χ2 tables or, for example, Matlab’s chi2cdf function to determine a p-value. However, since the theoretical χ2 distribution may not be appropriate for low N experiments, PAL_PFLR_ModelComparison can also be used to generate an empirical sampling distribution from which a p-value can be derived. Especially in the case of low N experiments it is advisable to compare the p-value derived from the theoretical χ2 distribution to that derived from Monte Carlo simulations. As an example, we performed our model comparison between the fuller and lesser model discussed in the Section “A Two-Condition Example” again using an empirical sampling distribution based on 10,000 Monte Carlo simulations. Figure 5 shows the empirical sampling distribution and the appropriately scaled χ2 distribution with 1 degree of freedom. It is clear that the two sampling distributions correspond quite closely for this example. Similarly, the p-value derived from the empirical sampling distribution equaled 0.015, which corresponds closely to the p-value derived from the χ2 distribution (p = 0.016). The code that was used to generate the results shown in Figure 5 and produce the figure in included in the PalamedesDemos folder .The function PAL_PFLR_ModelComparison returns the TLR. If the lesser model is correct, the TLR is asymptotically distributed as χp-values and the NHST in science has been the subject of continuous debate . Homoscedasticity is the assumption that the error variance is equal in all conditions. There the assumption allows one to use all data collected in the experiment to estimate a single error term. Note also that in the model-comparison approach we do not need to create and name a special, specific test to test this assumption. We simply use the general model-comparison approach to compare a model that makes the assumption that all slopes are equal to a model that does not make this assumption.In this regard, another significant advantage of the flexibility of the model comparison approach is that it allows one to reduce the number of parameters of a model greatly. Imagine for example an experiment with several conditions. The researchers are interested in the effect of the experimental variable(s) on the location of the PF. They also feel it is safe to assume that the experimental variable does not affect the slope of the PF. The flexibility of the model-comparison approach allows them to test for specific effects on the location parameters while implementing the assumption that slopes are not affected by constraining the estimated slopes to be equal in all conditions. Trials from all conditions then contribute to the estimation of the single, shared slope value. Note that this is similar in many ways to the assumption of homoscedasticity in many least-squares procedures  resulted in a p-value that would suggest an acceptable fit of the lesser model (p = 0.2571). However, the model comparison between the models labeled ‘fuller’ and ‘lesser’ led us to reject one of the assumptions of the lesser model . Thus there is a discrepancy here. On the one hand, we reject the assumption that the slopes are equal when we compare the fuller and lesser model but are unable to do so when we compare the lesser model to the saturated model in a goodness-of-fit test. The problem is that in the latter the assumption that the slopes are equal is one of many assumptions that are tested simultaneously and this has the effect of diluting the effect of a single false assumption, making it harder to detect this false assumption.One should interpret results of model comparisons in which the models differ with respect to a large number of assumptions cautiously. For example, the model comparison between the model labeled ‘lesser’ in The model-comparison approach allows researchers to target very specific research questions thereby preventing diluting of effects. To give just one example, imagine an experiment consisting of four conditions that differ with respect to the duration for which the stimulus is presented. Let’s say that we expect that the location parameter increases with the duration of the stimulus in a linear fashion. We can, of course, compare a model in which the location parameters are constrained to be equal in the four conditions to a model in which they are allowed to differ. However, this would not be the optimal manner in which to test our hypothesis. Since we expect our location parameter to vary in a specific  manner with duration we should compare a fuller model in which we allow the thresholds to vary according to our expectation  to a lesser model in which they are not allowed to vary. A so-called trend analysis allows us to specify a model in which the location parameters are allowed to vary, but only following a linear function of duration. To do this, we would use polynomial contrasts to reparameterize the four location parameters into four new parameters: one corresponds to the intercept term . In effect, this reparametrization allows us to constrain the location parameters to adhere to polynomial functions of differing degree. In this example, the fuller model would use a model matrix that includes an intercept term and a linear trend allowing location parameters to follow a first-degree polynomial : . The lesser model would specify the constraint on the location parameters using a model matrix containing only the intercept term: [1 1 1 1] . The distinct advantage of this approach is that the fuller and lesser model differ with respect to only one parameter (that defining the linear trend) and we say that the comparison is a one degree of freedom comparison. A fuller model that simply would allow any variation among location parameters would have four parameters and a comparison with our one parameter lesser model would be a three degree of freedom comparison. In case the location parameters indeed do vary as a function of duration in a more or less linear fashion, the one degree of freedom test would be much more likely to result in a significant model comparison. An example trend analysis is performed by the demo program PAL_PFLR_FourGroup_Demo.m in the PalamedesDemo folder.p-value is effectively meaningless. Also, if we do decide to perform several model comparisons, we should be concerned with the family-wise type I error rate, even if we selected our models a priori. A simple procedure to prevent inflation of the family-wise type I error rate that is easily implemented is the Bonferroni correction .We have seen that the model comparison approach allows many different model comparisons even in the case of simple experiments. In our discussion of our example two-condition experiment in the Section “A Two-Condition Example” we showed that there are 27 nested model comparisons possible that address the location and slope parameters only. The number of possible model comparisons will grow exponentially when additional parameters  are compared. Researchers are cautioned to exercise restraint when performing model comparisons. Firstly, as our statistics instructors have impressed upon us, one should decide on the hypotheses to be tested before we inspect our data or even conduct our experiment. As we have argued above, the selection models to be compared should be guided primarily by our research question and by whether our data contain sufficient information to allow estimation of parameters that are not considered in the research question. If, on the other hand, we allow our model comparison to be guided by the results of our experiment , the resulting on e.g.,  in whichwww.palamedestoolbox.org or in While this paper describes the model comparison approach as it can be applied using Palamedes to define and test assumptions regarding the effects of independent variables on the parameters of PFs, Palamedes can do much more. A full description of all of Palamedes’ capabilities is well beyond the scope of this article. Instead, we will list here in general terms what other functionality Palamedes provides. A much more elaborate description may be found on Adaptive methods serve to increase the efficiency of psychophysical testing. Palamedes implements the up/down methods e.g., , variousd’ (“d-prime”), for a variety of psychophysical tasks, e.g., Yes/No, mAFC, Same-Different, Match-to-Sample and Oddity. Unique to Palamedes is that it is able to fit any SDT model to a PF of proportion correct versus stimulus intensity, allowing the user to estimate the power function exponent of the transducer function that relates stimulus intensity to d’, as well as determine a goodness-of-fit of the SDT model. Another unique feature of the toolbox is the ability to model the detection of multiple stimuli according to either a probability or additive summation SDT model, as well as fit summation PFs with either model to determine the relative goodness-of-fit  measure s-of-fit .Maximum-Likelihood Difference Scaling (MLDS)  is a rel2, Palamedes also offers the possibility of creating an empirical sampling distribution using Monte Carlo simulations.With regard to the fitting of PFs, Palamedes allows one to fit individual functions with the freedom to fix or estimate any of the four parameters of the PF. Individual functions can be fit using either a maximum-likelihood or Bayesian criterion. Palamedes allows one to estimate the standard errors of the parameter estimates, using bootstrap analysis e.g.,  when a mα 2 β’ and model ‘1 α 2 β’ and between model ‘2 α 2 β’ and model ‘2 α 1 β.’Another R package that can fit PFs using a maximum-likelihood criterion is quickpsy by Another tool to fit PFs is psignifit  written We have provided an introduction to the model-comparison approach. While we have geared our discussion to testing research hypotheses in psychophysical studies, the model-comparison approach we describe generalizes to any field in which models are compared. We have also discussed in general terms how such tests may be implemented using our free Palamedes Toolbox. As described above, Palamedes allows model specification using three methods that vary in their ease of use and their flexibility. This approach makes it possible for researchers without much modeling experience to begin creating relatively simple models of their data. More seasoned researchers can use Palamedes’ flexibility to implement more complex models.3 or in All user-end routines in the Palamedes toolbox provide extensive help on their use (type ‘help’ followed by the name of the function in the Matlab command window). Included in the Palamedes Toolbox are also many demonstration programs that perform a complete analysis and will report the results in the form of text or a figure. Additional specific information on the use of Palamedes may also be found on the Palamedes websiteNP and FK created the Palamedes software and authored the paper.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Experimental results have revealed a drastic decrease of the friction coefficient for velocities close to the maximum seismic one, independently of the material studied. Due to the extreme loading conditions during seismic slip, many competing physical phenomena occur  that are typically thermal in origin and are changing the nature of the material. Here we show that a large set of experimental data for different rocks can be described by such thermally-activated mechanisms, combined with the production of weak phases. By taking into account the energy balance of all processes during fault movement, we present a framework that reconciles the data, and is capable of explaining the frictional behavior of faults, across the full range of slip velocities (10 Decrease of friction during seismic slip is linked to temperature increase and weak phases production inside the fault core. Here the authors propose a mathematical framework which explains the frictional behaviour of all materials reported in literature and precisely captures material weakening during fault slip. In addition to nucleation, the evolution of the friction coefficient—and thus of the fault’s shearing resistance—determines the propagation and arrest of the fault slip and governs the form and budget of energy dissipation during seismic slip2. The latter is essential as it determines the amount of energy dissipated at the fault, which is radiated on the surface through seismic waves and tremors.The knowledge of the friction (shear strength) evolution along a pre-existing fault is of major importance, as it allows extracting many characteristics and features of seismic slip. In particular, the decrease of the friction with increasing velocity or displacement  determines the possible nucleation of earthquakes. If the weakening rate is larger than a critical value determined by the release rate of elastic energy from the surrounding rock mass, this leads to the triggering of a dynamic slip at the origin of earthquakes3. A drastic decrease of the friction has been observed in most cases for velocities closed to the maximum slip velocity independently of the material considered4, however the physical mechanisms accompanying this rapid weakening are different for each rock type. Following microstructural observations and measurements in the sheared samples, several thermally and mechanically activated weakening mechanisms were proposed to understand the experimental results at seismic slip rates5. The common feature of all these weakening mechanisms is a phase transformation—like mineral decomposition6, nanoparticle lubrication7, melting8—during which a change in the nature of the material takes place.During the last 20 years, a large set of experimental works has been devoted to reproducing the extreme conditions of a seismic slip. The development of high velocity shear apparatus allowed the research community to perform experiments at the maximum velocity reached during an earthquake event (1–10 m/s) and, thus, characterize the behavior of a fault over the full range of possible slip rates−9 to 10 m/s). The similarity of microstructures observed in nature and in experiments suggests that energetic frameworks like the one presented could quantitatively link observations across the scales and provide deep, physics-based insight on the physical mechanisms driving seismic slip.In this paper, a thermo-chemo-mechanical model is developed to describe the thermal sensivity of the materials and the production of weak phases during high velocity experiments. It is shown that the model is able to reproduce the friction at steady-state for different materials and a wide range of velocities (10−5 m/s), so that the mechanisms described above are not triggered. The results are shown in Fig. μ decreases as the weak phase fraction increases. This effect of the weak phase fraction can be captured using an exponential law μ = μ0 + Δμeαw−, where μ0 is the friction coefficient of the weak phase, Δμ = μs − μ0 is the difference of the friction coefficient of the strong and weak phases, w is the weak phase fraction and α a weakening coefficient ranging from 0.1 to 15  sandwiched between two blocks or on bare rock samples12. In the latter, a gouge material is formed after only a few millimeters of displacement13 with a thickness of 100–300 μm. Data are gathered based on the nature of the material sheared and the physical mechanisms that are inferred to operate during the experiments14.The derived exponential decrease of the mechanical strength from the experimental data of Fig. The constitutive law for the mechanical behavior is applied to the gouge material, which accommodates all the deformation and is affected by the temperature, the weak phase fraction and the state of stress. The weak phase creation is modeled as an endothermic first order chemical transformation affecting the energy balance equation and respecting the mass balance. The geometry of the model chosen is larger of one or two orders of magnitude than the gouge in order to impose far field boundary conditions for the temperature and the extend of the phase transformation see Fig. . The stew = 1), the temperature is increasing uncontrollably and the friction coefficient drops towards zero (regime V).The resulting steady state response of the model, in terms of friction and velocity, is depicted in Fig. 16. The experimental data are a collection of several independent studies at different experimental conditions. As shown in Supplementary Fig. 19.After identifying the regimes of the steady frictional response of faults, the model is applied to experimental data. Figure One of the interesting features of the model is the reproduction of strengthening observed experimentally at intermediate velocities , there is no general agreement about which phase transition is at the origin of the weakening. Indeed, several mechanisms are usually triggered and it is difficult to isolate the effect of each one of them. In addition, there might be microstructural features and unstable phases produced during high velocity slip that leaves little or none evidence in the microstructure after the experiment. Nevertheless, the model enables to capture accurately the microstructural observations, when available, and uses as input the material parameters listed in the Supplementary Table 20, melting21, dehydration22, or clay type transition3), when available. In all cases, the temperature of the model is lower than the theoretical one  implying that the phase transition is initiated locally at the contact of the grains where the temperature can be higher than the bulk temperature . For silicate rocks, an unique set of parameters has been used. It is a strong assumption as the melting temperature depends on the silica content, which is is different for gabbro, peridotite or granite, and this effect should be taken into account for a finer description of this set of experiments. The melting temperature reported in Fig. 21. The local phase changes are often hard to detect even though essential for the mechanical behavior. This also explains why the evidences of the phase transformations from specific sensors or microstructural observations  are retrieved for higher velocities in experiments than predicted by the model.The temperature predicted by the model when the weak phase begins to appear can be compared to the theoretical activation temperature of the phase transformation  gouges, the increase of temperature with velocity induces a dehydration of the material that can be evidenced by the presence of talc or illite. Yet, dry illite has a higher shear strength than dry smectite28. A significant increase of CO2 has been detected by electrolyte-type sensors near high-velocity experiments and microstructural observations have shown the presence of very fine-grained powders of lime (CaO) and portlandite (hydrated lime). For this material, the weakening has been attributed to the effect of the thermal decomposition of carbonates into nanograins of lime and CO228, but the microphysics of the weakening is not yet fully understood5.There are many evidences of decarbonation during shearing of carbonate rocks at high velocities13. The presence of fluids can trigger other mechanisms like cavitation29, pressure-solution32, or thermal pressurization36. The addition of such mechanisms could in turn either change the steady state response of the system or induce transients which could drive the system far from its steady-state response37. In that case a detailed model based on the results of high velocity dry experiments and saturated low velocity experiments needs to be carried to assess the dominant mechanisms of slip and earthquake nucleation.These results suggest that heating produced during shear and phase transformations of the material inside the gouge may be the dominant mechanisms during slip in dry conditions. The physics-based model presented here enables us to isolate the main physical mechanisms and constrain the parameter’s values. The knowledge of the variations of these parameters with the normal stress and sample size would allow to extrapolate experimental results to conditions in situ. However, in natural conditions faults are rarely dry and are usually saturated with pressurized fluids, a scenario difficult to study experimentally at high velocity39. A critical velocity is required to activate this mechanism that is related to the processes at asperities or grain contacts23. In this paper, we focus on the response of the material for intermediate and high velocities where the thermal and chemical effect are important (other models have been developed for lower velocities and the nucleation of earthquakes40). Therefore, we consider here that for low velocities experiments the strength of the material is determined by the static friction of the materials in contact or the internal static friction of the granular assembly. For velocities larger than the critical one, the shear stress of the system is calculated by solving a thermo-chemo-mechanical model inside the deforming zone. The critical velocity is retrieved as a result from this model and can be approximated by an analytical solution , fugacity sensitivity, effect of asperities of the contacts etc. Such dependencies would require additional evolution laws for the internal state variables Q.The constitutive law for the mechanical behavior is a rigid elastic-viscoplastic law with the most generic form: an Arrhenius-power law dependency10, Xd. The strength of the material depends on both the plastic strain and this weathering index. It is assumed that the two effects are uncoupled10 and a multiplicative structure of the yield stress is postulated:Ty(ϵp) is constant. Moreover, Xd is considered to be a weak phase volume fraction. As shown in Fig. τ0 is the yield strength of the strong phase and α is the weak phase sensitivity coefficient. From this equation, the friction is determined by:σ is the normal stress applied to the layer.The effect of the non-mechanical state variables on the mechanical behavior of the system can be expressed as a single scalar function called the weathering indexVw and Vs, respectively, we may define the volume ratio 45 obtained from the mass balance of the weak phase fraction and the energy balance equations:T is the temperature, cth the thermal diffusivity, ρC the heat capacity of the mixture considered constant here, Jw the diffusion flux of the weak phase, ΔH the enthalpy of the phase change reaction considered endothermic, rF the reaction rate and ρ1 = ρww. F(y) is a function which value is 1 for y ∈  and 0 otherwise. The reaction rate is expressed as first order chemical reaction with an Arrhenius law.ρs and Ms are the density and molar mass of the strong phase. kF and Qc are the preexponential factor and activation energy of the chemical reaction. Using Eqs. (cw) and the steady state of Eqs.  = ϕi2(y) − 1, allowing to enforce a zero Dirichlet boundary conditions implicitly46. N is the number of polynomials used to simulate the solutions. A convergence analysis has been conducted in each case to verify that N is high enough to have a negligible error on the solution. ai and bi are the interpolation coefficients for the temperature and the weak phase fraction respectively. The interpolation points used for the resolution are the Gauss-Lobato points defined by:6.The solutions of this nonlinear system of differential equations are approximated numerically using pseudospectral methods. The temperature and weak phase fraction fields are interpolated in space using Chebyshev polynomials of the first kind:Supplementary InformationPeer Review File