In the past few decades, a significant volume of work has been carried out on various aspects of the state estimation problem to estimate an optimum state vector of the power system. This problem has been focused on, in previous studies regarding the computational efficiency and numerical robustness in view to find point estimates for system state parameters. This current investigation, constructed confidence intervals for the unknown state parameters of the system. The research indicates that confidence intervals can yield addition useful information about the estimated parameters.The feasible interval estimates for the system state parameters have been modelled in this study by considering the random uncertainty in the processing measurements. The statistical assumptions of the measurement errors have been utilized to characterize the probabilistic behavior of the estimated parameters in terms of confidence intervals. The Gauss–Newton algorithm has been adopted for maximizing the likelihood function of the processing measurements and obtaining the confidence intervals.The usage of the confidence intervals was demonstrated through Monte Carlo experiments on a real dataset of the 6-bus and IEEE 14-bus power systems for both small and large sample sizes. The confidence intervals were constructed for the test networks for the sample of measurements 18, 28, 44 and 68 based on the redundancy ratio R. The proposed interval estimates outperformed for the sample sizes of 28 in the 6 bus and 68 in the IEEE 14-bus systems, respectively. The poor performance for the constructed interval estimates have been reported even for the large sample sizes in the existence of contaminated measurements.The results of the study show that the method is effective and practically applicable in the state estimation of a power system. The constructed confidence intervals for the system state parameters adequately perform for the lager sample size. However, the existence of the gross errors in the processing measurements had severe effect on the performance of the proposed interval estimates. To achieve safe, effective and economic power delivery, it is mandatory for a control center to precisely estimate the existing state of the system. State estimation is, therefore, routinely practiced within the power industry for estimating the steady state of the system. The purpose of this estimation is to decrease the estimation error with an aim to increase the overall accuracy of the estimates. This estimation procedure is concerned with obtaining the point estimates of the state parameters. The probabilistic nature of the estimated parameters is however, not significantly addressed. The knowledge of the variability of these estimates can be served as additional information; so the reliable measure of the standard errors of these estimates is also of importance. This void forms the essence of the construction of confidence intervals for the state parameters reported in this work.The output results of the state estimation algorithm are point estimates of unknown parameter values which are subsequently used in monitoring and planning the analysis of the system  is a vector of nonlinear mean valued functions in terms of k-dimensional unknown state vector X and ε represents a Gaussian errors vector.In mathematical form the problem can be written asThe details of this model is common in the technical literature on the power systems state estimation (Abur and Exposito E(ε) = 0 (mean vector), and Ω = E(εεT)  with:σi2 is the weight, associated with the accuracy of ith measuring device and assumed to be known and σ2 represents the overall variance of the given measurements. For the construction of the individual confidence intervals, it is further assumed that the system is partially observable however; there is enough redundancy in measurements to estimate the system state parameters. Several least squares based methods for estimating the state parameters of a power system are conventionally used but the WLS approach performs the best in the estimation and identification of bad data , the formulation of the confidence intervals could be straightforward but these intervals are not based on the standard Z, T and Fstatistical tests  = [Zit − hit(X)]T[Zit − hit(X)] and v = σ−2.Under the assumptions listed above, the log likelihood function for the observed sample is given by:X. The concentrated likelihood function for the elements of X is therefore, given by:In a power system state estimation, weights are known as nonnegative quantities which make us able to vary the influence of various measurements to the sum of squares. The maximum likelihood (ML) estimate of the scale parameter ν from Eq.  yields:4Equation  is maximX) but, there is no closed form solution for obtaining the estimate of the parameter vector X. To tackle this nonlinearity in the iterative setting, we have been utilized the technique of profiling out the parameter ν from lc(X). In this way the method of the generalized least squares could be implemented more conveniently to evaluate the value of the optimum state vector X and the corresponding standard errors though a suitable iterative algorithm. The expected maximization algorithm, Fisher Information algorithm, and Newton–Raphson algorithm are the most common techniques for numerically maximizing the likelihood function  around Xr is given by:Let Ignoring the higher order terms in Eq. (7), could yield the iterative solution for the point estimates. In order to find the standard errors of the estimates, we need to find the asymptotic variance covariance matrix for the estimated parameters.X*:In the given setting, we have the following observed Fisher Information matrix at the optimum state v from Eq. (X* from Eq. (7), we obtain the following observed Fisher Information matrix:By using the ML estimate of from Eq.  and the I(X*) = HtT(X*)Ht(X*)and σ2 and different from Eq.  will be resulted by this descend direction. In general, the Gauss–Newton scheme is not essential to be in a descent direction due to the failure of the positive definite status of the Hessian matrix, consequently the iteration process may exhibit non-convergence % confidence interval for the individual state parameter xi can be specified as:ith diagonal element of the matrix given Eq.  is the degree of freedom. The primary assumption of using the linearized confidence intervals (LCIs) is that the measurements are randomly and independently sampled from the measuring devices. The validity of these intervals is supported by the results of the asymptotic theory. The intervals calculated in this way are approximated. Their adequacy depends on the good linear approximation to nonlinear functions which means that the effect of the curvature and the higher terms of the Taylor expansion would be insignificant and to that of the asymptotic result (h(X) for the real and reactive powers are common and well established 100% of such intervals include the true parameter; otherwise, the intervals would be characterized as liberal or conservative if the coverage probability remains below or exceeds the nominal probability (1 - α) respectively. We have explored the performance of the LCIs for the state vector parameters for partially observable systems. It has been seen in Table The coverage probability is the proportion of the samples which are taken with repeated sampling to the total number of samples that contain the true value of the population parameter. It is used as one of criteria for the evaluation of the confidence intervals. Intervals are said to be exact if  of the 6-bus and IEEE 14-bus systems, respectively.The procedure of calculating the desired intervals up to 400 runs can also be seen in Figs. v1 of the 6-bus and IEEE 14-bus networks. Each vertical line that is across the central parallel lines is equivalent to the interval that holds the true value of the parameter. Whereas, the other vertical lines that are below and above the central lines are equivalent to the intervals that do not contain the true parameter The central parallel lines in each graph correspond to the actual values of the estimated parameter Note that, the condition given in Eq.  had beenFigure The results in Table Here, we are assuming that our available data vector did not contain any gross errors as these calculated intervals might not be robust in the presence of bad data. The existence of outlier values may be rigorously impaired the estimation results and needs an adequate caution to be taken. This fact has been studied by many researchers in view of point estimation for the unknown state parameters (Anwar and Mahmood δ is magnitude of gross error in some multiple of standard deviation i.e., δ = kσi and ϑ is a vector of zero elements except one at the ith position for the gross error in the measurement from the ith measuring device. To determine the robustness of the calculated intervals, a bad measurement was interrupted at the real power injection at node one by adding the four times standard deviation of its measuring instrument according to Eq. (Incidentally, the data generating model for the contaminated measurements in accordance with Eq.  can be wIt can be observed that even for the large samples, the performance of the approximate intervals is very poor in the presence of a gross error. This is because of the fact that the inclusion of a single bad value, although only causing a slight, increased in the standard errors from the Fisher matrix, but substantially enlarged the mean standard errors which in return causes the width enlargement for the approximated intervals.At the end of this analysis, the question also arises about the correctness of proposed interval estimates. Thereby, in view to verify further the correctness of proposed method for the construction of individual confidence intervals, we have taken the results of the tightest upper and lower bounds obtained by Al-Othman and Irving  for the Clearly, the validation of the proposed method can be seen by the results presented in Table Figures We have explored the problem in this paper by considering the random uncertainty in the measurements that ultimately originated uncertainty in the estimated parameters. In this way we have found the methodology to find the confidence intervals for the estimated parameters by deploying satisfactorily, the results of the nonlinear estimation theory along with the conventional state estimation algorithm. We have observed from our case studies, that the LCIs yield an adequate performance in the case of larger sample size. However, the lager sample size sample is not only the source of the adequate performance of the LCIs because of the nonlinearity of the estimated model. There may be several other sources of uncertainty that could contribute to the difficulty in calculating the optimum estimates and their asymptotic standard errors. Since, the output from the conventional state estimation algorithm for the point estimates is no more reliable in the presence of gross errors. In the same way, we have also been observed in our study, the poor performance of the LCIs in the presence of the gross error even for the larger sample size. This resulted in, an increase in the widths of the confidence intervals and hence a loss in the precision. Therefore, prior to the construction of the confidence intervals, consideration should be given to analyze the data in order to make sure that it is filtered by gross errors.It is recommended to explore further the robust methods of interval estimation and their impact on the state estimation model that is presented in this study to further refine the computational technique.


We have constructed an experimental setup for gas phase electron diffraction with femtosecond resolution and a high average beam current. While gas electron diffraction has been successful at determining molecular structures, it has been a challenge to reach femtosecond resolution while maintaining sufficient beam current to retrieve structures with high spatial resolution. The main challenges are the Coulomb force that leads to broadening of the electron pulses and the temporal blurring that results from the velocity mismatch between the laser and electron pulses as they traverse the sample. We present here a device that uses pulse compression to overcome the Coulomb broadening and deliver femtosecond electron pulses on a gas target. The velocity mismatch can be compensated using laser pulses with a tilted intensity front to excite the sample. The temporal resolution of the setup was determined with a streak camera to be better than 400 fs for pulses with up to half a million electrons and a kinetic energy of 90 keV. The high charge per pulse, combined with a repetition rate of 5 kHz, results in an average beam current that is between one and two orders of magnitude higher than previously demonstrated. I.Gas electron diffraction has a long history in the determination of the structure of isolated molecules in the gas phase,A major challenge in UGED is to reach femtosecond resolution while maintaining the high electron beam current needed to capture diffraction patterns with a high signal to noise ratio (SNR). The differential scattering cross section drops rapidly with increasing scattering angle; thus, a high beam current is needed to capture the diffraction pattern at the larger angles that are needed to reach high spatial resolution. Working with highly charged electron bunches requires compensating for the Coulomb effect that leads to broadening of the pulse duration. The development of compact electron guns resulted in a resolution of around 200 fs in ultrafast transmission electron diffraction from solid samples.We present here a table-top setup operating at an energy of 90 keV that can reach femtosecond resolution with a high electron beam current. We use a radio frequency (RF) cavity with a longitudinal electric field to compress highly charged electron pulses.In the following, we describe the experimental setup, present measurements of the electron pulse duration, and demonstrate the capability to capture diffraction patterns with a short integration time.II.A.in situ.Figures μm using a small nozzle. The nozzle consists of a 20 mm tube with an inner diameter of 100 μm. A collimating aperture is placed before the sample to remove stray electrons and to match the electron beam diameter to that of the gas jet. The setup includes an optical system to excite the target molecules with laser pulses that are velocity matched to the electrons by the pulse-front tilt technique, which is described briefly in Section 6 electrons per pulse, as expected for a one-photon process.The target molecules are delivered into the vacuum in a gas jet with a diameter of approximately 200 B.A mode locked Ti:Sapphire laser generates the laser seed pulses with a central wavelength of 800 nm and a repetition rate of 75 MHz. The repetition rate is actively stabilized using a frequency counter that provides feedback to an actuated mirror inside the laser cavity to adjust the cavity length. The laser is amplified using chirped pulse amplification to generate 40 fs, 2 mJ pulses at a 5 kHz repetition rate with a central wavelength of 800 nm and 30 nm bandwidth. The laser beam pointing is actively controlled using a feedback loop with two CCD cameras and two actuated mirrors. The laser beam is split into three paths as shown in Figure The laser path to the photocathode consists of a frequency tripler that generates 267 nm UV laser pulses. The frequency tripler uses four crystals in an in-line configuration: a BBO second harmonic generator, a delay crystal to temporally overlap the 400 nm and 800 nm pulses, a half-wave plate that matches the polarization of the two pulses, and finally a second BBO crystal that mixes the two pulses to generate the 267 nm laser pulse. After the last crystal, three harmonic separators are used to filter out the IR and blue components. The UV beam is truncated first by a 400 μJ) is needed to trigger the streak camera. The electron pulse duration measurement is described in Section The second laser path is designed to trigger the streak camera that measures the electron pulse duration. Only a small fraction of the laser pulse energy (approximately 80 c at 90 keV). The laser is demagnified by a factor 13.5, which results in a pulse front that is tilted to match the electron beam at the sample. The tilted pulse front is compressed temporally only at the image plane of the grating; however, we have previously measured that the pulse duration remains below 100 fs for a length of about 1 mm,The last path is designed to excite the target and to compensate for the velocity mismatch between the laser and electron pulses. As is shown in Figure t Figure . The lasμm and the laser beam has a diameter of 300 μm, approximately 90% of the sample will be excited by the laser. The current optical design is optimized for exciting the sample with a laser wavelength of 800 nm. A small change in laser wavelength can be accommodated by changing the angle of incidence of the laser onto the grating and the magnification of the imaging system to deliver the correct tilt angle. For a large change in wavelength, for instance the second or third harmonics of the laser , a different grating will be necessary, but the rest of the optical system will not require major changes.The large incidence angle of the pump laser beam with respect to the electron beam means that the laser beam must be larger than the electron beam to excite the whole sample volume that is probed by the electron beam. For example, if both the electron and gas beams have a diameter of 200 C.The electron pulses are compressed by the longitudinal electric field of the RF cavity.III.We constructed a laser activated streak camera with capability to measure pulse durations below 200 fs Figure . The perFigure 5 electrons and 100 fs for 105 electrons. However, we have observed that the temporal resolution does not improve from 350 fs when reducing the charge from 5 × 105 to 105 electrons, which suggests a large contribution from the timing jitter. The timing jitter could be reduced by adding an additional feedback loop after the microwave amplifier that drives the RF compression cavity. If successful, the temporal resolution would approach the electron pulse duration, which can be less than 100 fs.The main factor limiting the temporal resolution of the current setup is the time of arrival jitter between the laser and electrons at the sample, which is caused by phase jitter in the RF signal. Based on numerical simulations using the General Particle Tracer code, we expect that the electron pulse duration on the target should be 250 fs for 5 × 10IV.3I) and Nitrogen (N2). In order to get accurate interatomic distances from the diffraction patterns, the conversion from detector pixels to the scattering angle must be calibrated. We used the data from CF3I to calibrate the detector and then applied this calibration to extract the bond length from the diffraction pattern of N2.We demonstrate the capability of the setup to capture diffraction patterns with a short integration time using two molecules, Trifluoroiodomethane (CFith atom, and the double summations are over all atoms in the molecule.The intensity of the diffraction pattern on the detector, for a gas of randomly oriented molecules, can be written asith atom, respectively. s, and α is a damping constant that is adjusted to minimize edge effects in the transform.The total scattering intensity can be split into an atomic and a molecular contribution3I on a logarithmic scale. The image shows that there is a significant number of scattered electrons even at the largest diffraction angles captured by the screen. The diffraction was acquired in a single frame with an integration time of 20 s, with a beam current of 70 pA (μm and the backing pressure on the nozzle was 140 Torr. Based on the ratio of scattered to incident electrons and the known atomic scattering factors,3. An image with the gas jet turned off was also acquired and used for background subtraction. The electron beam was purposely placed off-center to increase the s-range of the measurement. A beam stop was used to block the directly transmitted electron beam.Figure 3I,The 3I to extract the N-N bond length. The diffraction pattern was captured with a total integration time of 20 s with a beam current of 70 pA. The backing pressure behind the nozzle was 700 Torr, and the gas jet had a diameter of approximately 200 μm in the interaction region. The density in the interaction region was estimated to be 3. Figure 2, and Figure We have used the calibration obtained with CFIt is important to note that one of the bottlenecks in UGED was that integration times on the order of an hour were needed to achieve a sufficient SNR for structure retrieval. Previous UGED experiments with keV electrons reached a resolution of 850 fs, with an average beam current of 3 pA,V.μm FWHM to match the size of the gas jet. There is a tradeoff between the number of electrons, pulse duration, and the beam divergence, where any parameter can be improved at the expense of the others. The RF compression broadens the energy spread of the electron pulses, so there is an additional tradeoff between the minimum compressed electron pulse duration and the energy spread. Based on numerical simulations, we expect that the energy spread for 105 electrons will be below 1%. An energy spread of a few percent does not significantly affect the quality of the diffraction patterns, but for pulses with more than a million electrons the energy spread may become an issue. Currently, the s-range that can be captured by this setup is limited by the size of the phosphor screen to 6.9 Å−1 with the beam at the center of the screen, which we increased to 8.8 Å−1 by moving the beam off-center, as shown in Figure s-range can be adjusted for each experiment. This flexibility will allow us to fully optimize the system depending on the temporal and spatial scales relevant to a given molecular reaction.A new table-top setup was designed and built for UGED experiments that can reach femtosecond resolution with a high average beam current. The setup uses an RF cavity to compress the electron pulses and a tilted laser pulse to compensate for the group velocity mismatch. We have shown that the device can compress electron pulses with up to half a million electrons; however, we have found the best configuration for UGED experiments to be pulses with


As a reference, the median value of TRE between 0 and 50% phases with rigid registration only was 5.0 mm with one SD of 2.5 mm and the maximum value of 12.0 mm. For the OF algorithm, 81% of voxels were within a difference of 50 HU, and 93% of the voxels were within 100 HU. For the DM algorithm, 69% of voxels were within 50 HU, and 87% within 100 HU. For the DD algorithm, 71% of the voxels were within 50 HU, and 87% within a difference of 100 HU. These data suggest that the three DIR methods perform accurate registrations in the thorax region. The mean TRE for all three DIR methods was less than two voxels suggesting that the registration performed by all methods are equally accurate in the thorax.Deformable image registration (DIR) has been proposed for lung ventilation calculation using 4D CT. Spatial accuracy of DIR can be evaluated using expert landmark correspondences. Additionally, image differences between the deformed and the target images give a degree of accuracy of DIR algorithms for the same image modality registration. DIR of the normal end‐expiration (50%), end‐ inspiration (0%), midexpiration (30%), and midinspiration image (70%) phases of the 4D CT images was used to correlate the voxels between the respiratory phases. Three DIR algorithms, optical flow (OF), diffeomorphic morphons (DM), and diffeomorphic demons (DD) were validated using a 4D thorax model, consisting of a 4D CT image dataset, along with associated landmarks delineated by a radiologist. Image differences between the deformed and the target images were used to evaluate the degree of registration accuracy of the three DIR algorithms. In the validation of the DIR algorithms, the average target registration error (TRE) for normal end‐expiration‐to‐end‐inspiration registration with one standard deviation (SD) for the DIR algorithms was PACS number:89.90 A more recent application of DIR is its use in the calculation of ventilation from four‐dimensional computed tomography (4D CT), which is also our motivation for validating the DIR methods.(–)In image registration, images are aligned by establishing a correspondence between features in both images. We can categorize these algorithms into two categories: rigid registration and deformable image registration (DIR). Rigid registration simply aligns the images, whereas DIR can take into account complex changes such as deformation due to breathing, patient weight loss, or tumor shrinkage. The purpose of DIR is to find a transformation from one image set to another, such that the differences between the deformed and the target image sets are minimized by providing a voxel‐to‐voxel deformation matrix. DIR has been studied since the early 1980s. For many years, neurosciences and neurosurgery have been the driving force for developing various deformation techniques.,2 A An application of DIR as exemplified in this paper is image registration of two images from two different times during a breathing cycle (4D CT). DIR finds a mapping between a voxel in one phase of the image  to a corresponding voxel in the second phase .(–) diffeomorphic demons (DD),(–) and diffeomorphic morphons (DM). A more thorough description of these algorithms is given in the corresponding Methods section.A variety of DIR algorithms have been implemented. While the goal of these algorithms is the same, the algorithms are based on aligning different image features, how they measure similarity, and what type of deformations they allow. In this paper, DIR between different phases  of the breathing cycle was evaluated by three different DIR algorithms, optical flow (OF),–15 d(–) The algorithms investigated in this paper were all evaluated using an anatomical landmark‐based model, point‐validated pixel‐based breathing model (POPI), and the image differences between the deformed and the target lung images. The model consists of anatomical landmarks in the lung chosen by expert radiologists. The intention of the model is to have landmarks that are spread uniformly and correspond to anatomical features such as the carina, calcified nodules, division branch of the pulmonary artery, and apical pulmonary vein of the upper lobe and various bifurcations of smaller structures that could be uniquely identified in each image set by four different medical experts. A more detailed procedure for obtaining the landmarks is given by Sarrut et al.For all DIR algorithms, it is highly desirable to provide an estimate of the accuracy of their registration for the desired application.–26 TThe POPI model is a useful tool that has been made available to researchers, which can be used to estimate the deformation accuracy in the lung. This paper evaluates the accuracy of OF, DD, and DM DIR algorithms that are used for registration of the 4D CT image sets for the application of accurate DIR for the lung. and DD OF assumes that Houndsfield Unit (HU) values remain constant for the corresponding voxels. HU of a voxel at time t is equal to the HU at a later time t+δt, that is:I is the HU value at time t. Applying Taylor series expansion of the terms on the right‐hand side of Eq. v→=∂x→∂t is the velocity and it consists of the displacement between a point in the first image, I, and the corresponding point in the second image I divided by the time between the two images, δt. A velocity constraint, which requires that nearby points move in a similar manner to each point, is introduced in order to obtain a single solution from Eq. OF finds the voxel correspondence by computing a displacement field describing the apparent motion represented by the two image sets by matching the intensity gradients.–15 O∇¯T; however, when the difference is less than zero, M moves against ∇¯T. The demons stop exerting force when the images overlap completely. The optical flow equation was used to calculate the force applied by the demons, but is renormalized because of the effects of small image intensity gradients.Maxwell's demon was adapted by Thirion17 to a  that this method resembles an optimization of a second order gradient descent of the sum of the squared intensity differences (SSD). This indicates that demons would function well in single modality registration, where SSD gradient descent is also appropriate.It was demonstrated by Pennec et al.16 tha A diffeomorphism is a map between manifolds, which is differentiable, and its inverse is differentiable, as well. Organs can be compressed and deformed and undergo invertible spatial transformation.Demons algorithm is adapted to include the gradients of the moving and target images. This optimization makes demons more efficient, and is the method that is implemented for our demons registrations. Additionally, to create physical displacement fields, the demons algorithm is updated to make it diffeomorphic.18 A dThe deformation field produced by DD is smoothed by a Gaussian filter, and iteratively used to transform the moving image, and register on to the static image. The displacement vectors during deformation are calculated independently for every voxel. Due to the image noise, the displacement field may not be smooth, so regularization is applied to get a smoother deformation field. The difference between this algorithm and OF and DD is that the computation of the field is based on the local phase difference rather than the intensity difference between images. The phase difference between periodic signals with the same frequency allows the estimation of the spatial difference between these signals. The morphon iteratively deforms a moving image into a target image by morphing the moving image. The process can be divided into three parts: estimation of displacement, accumulation of the deformation field, and deformation.The DM algorithm is based on matching of edges and lines.25 TheThe estimation of displacement has the aim to find the ways and indications on how to deform the moving image into the target image. Estimation of displacement is based on quadrature phase difference. The accumulation of the deformation field uses the estimate of the displacement to update the deformation field. Finally, the deformation morphs the moving image to the target image according to the accumulated deformation field. These steps are done iteratively, as long as the displacement estimates indicate further morphing to be done.,25 Edges between bright and dark areas have one phase, dark lines have one phase, and lines on dark background have a phase as well as bright patches. The transition as we move from one phase to another is continuous. Therefore, the difference in local phase between the moving and target images is a good measure of how much the moving image has to move to fit the target image. The local displacement is a function of the local phase along its associated direction. To estimate the local displacement a least square estimate is used.Quadrature phase difference is a method used to estimate local displacement between two images. The advantage of this method over other methods, such as the ones based on gradient and polynomial expansion, is its invariance to image intensity and weak gradients.19 EdgThe moving image is deformed based on the accumulated field and then it is compared to the target image in order to estimate a displacement field for the current iteration being performed. The updated field is formed by combining the accumulated field and the displacement field from the current iteration. After acquiring the update field as well as the certainty from the field, a weighted accumulation is used to determine the accumulated displacement field. Additionally, estimation of the displacement vectors is combined with a measure of the estimation confidence, which results in a certainty map. In order to prevent folding, the morphons algorithm is optimized to become diffeomorphic. As in the DD algorithm, the deformation field produced by the DM is smoothed by a Gaussian filter, and iteratively used to transform the moving image and register to the static image.All three DIR algorithms are implemented in an iterative and multiscale scheme. A lower resolution deformation is first approximated, then used for the next level of deformations with a higher resolution of the images to be registered. This process continues until the deformation has been approximated at the highest resolution.C++ with four scales of resolution, and 98 iterations at each scale, which was determined to be the optimal number of iterations for OF. DD and DM were implemented using MATLAB . The settings were the same as the ones described in Janssens et al. that is using eight scales, with a minimum of 10 and a maximum of 20 iterations at each scale, and a Gaussian smoothing with a standard deviation σ=2. The iterative process was stopped if the changes measured in terms of the sum of the squared differences (SSD) were less than 0.01%.OF is implemented using ns et al.25 tha The landmarks are anatomically homologous points that were manually delineated by radiologists at all 10 phases of the original 4D CT image sets. They are based on anatomical features that correspond to various locations in the lung such as the carina, calcified nodules, culmen‐lingula junction, division branch of pulmonary artery, and apical pulmonary vein of the upper lobe. The CT image sets consisted of 512×512×141 voxels with voxel dimensions of 0.97×0.97×2 mm2 (2 mm slice thickness). The phases used for the evaluation of the DIR algorithms were normal end inspiration (0% phase), midexpiration , normal end expiration (50% phase), and the 70% phase which is midinspiration or the phase that lies between the end expiration and end inspiration. For all the methods, the original datasets were used for this validation without any changes to the images.All three DIR algorithms were evaluated using the POPI model. The POPI model consists of a typical clinical 4D CT image dataset of a lung with the images binned into 10 phases along with 41 associated anatomical landmarks. The 4D CT images were acquired using a Phillips Brilliance Big Bore CT , and an air bellows belt with a pressure transducer was used to convert the pressure waveform into respiratory phase information, which was used for the 4D image phase tagging.,30 TUsing deformable registration, one phase is registered to the second phase . This generates deformation vector fields (DVFs) that approximate the anatomical displacements voxel‐by‐voxel. The vector fields point from the target positions to the new displaced positions of the anatomical features. For this study the 0%, 30%, and 70% phases were each registered to the 50% phase.TRE=Xa ‐ Xb, where Xa is the anatomical position of the landmark and Xb is the calculated position of the landmark). This displacement was used as the metric to assess the accuracy of the registration algorithms. For comparison purposes, the difference in position between landmarks with rigid registration (RIG) only was calculated. That difference is defined as the position of the landmarks in the inspiration phase minus the position of the landmarks in the expiration phase.The resulting deformation fields were used to calculate the new location of the landmarks on the end‐expiration phase. Then, the positions of the calculated landmarks were compared to the inspiration phase landmarks, as determined by the radiologists. TRE was then assessed by calculating the displacement between the calculated and the anatomically determined landmarks for each of the radiologist identified points  was drawn to include only the voxels inside the lung and exclude the rest of the image. For comparison, the differences between the moving and the target images were calculated using RIG only.Only 39 of 41 landmarks in the POPI model were used. Two of the landmarks were intentionally not used for the evaluation because one of them was outside the area of interest and the other one corresponded to a 4D CT artifact.Table Figures Figure Image difference is another validation method for the DIR algorithms to determine the degree of accuracy of deformable registration. Figure  Janssens et al. used metallic fiducial markers implanted inside a homogeneous phantom to evaluate deformable image registration with morphons and demons. There are several issues with using phantoms for evaluating DIR algorithms. Phantoms in general are not deformed in the same manner as a normal patient's anatomy; they also deform reproducibly, thus producing superior 4D CT image sets without the variability of actual patient breathing. The deformations of the phantom may also be more predictable and smoothly varying. Phantoms are a good method for testing DIR, but results may not be directly related to 4D CT imaging of patients. Additionally, if phantoms are deformed using mathematical models, then the CT artifacts, noise, and quantum effects may be exactly the same in the image pairs, which would not affect the registration quality. However, for registrations between different phases of 4D CT images, noise and image artifacts are not the same across images and, therefore, degrade the quality of registration.Various groups have used different models to validate image registration algorithms.,,, Ja reported similar results to ours when validating DM and DD using the POPI model. They found that the mean target registration error for DM was 0.9±0.5mm and a maximum of 2.8 mm, the mean target registration error for DD was 1.0±0.5mm and a maximum of 2.8 mm. The reason for the differences between our results may be due to the differences in image resolution, as the voxel size for their study was 2×2×2 mm3, or due to the difference in image quality between the two studies.In a different study, Janssens et al.20 rep tested the demons algorithm on three different cases using a model based on real patient data. In the first case, they used CT images from a prostate patient, with voxel sizes 0.94 ×0.94×3.0 mm3, and then mathematically deformed the images in a well‐defined manner. In the second case, they used CT images from a head‐and‐neck patient, with voxel sizes 0.68 ×0.68×3.0 mm3, and then mathematically deformed the images; in the third case they used a deformable pelvis phantom. They found that the mean error for the first case was 0.5±1.5mm, for the second case 0.2±0.6mm, and 0.8±0.5mm for the third case. Although this is a more advanced model than phantom measurements to test the accuracy of the deformation algorithm, this method also lacks patient variability and, consequently, may overestimate the accuracy of the technique. The accuracy is deemed to be good since CT artifacts, noise, and quantum effects were exactly the same in the image pairs, which would not affect the registration quality, while for registrations between phases of 4D CT images this is not the case. The presence of different CT artifacts and noise in the images is a factor that would degrade the quality of the registration. Another source of differences between our results and results from the Wang study may be because the anatomical regions are not the same. Similarly, Zhong et al. validated demons algorithm using a synthesized prostate model and a CT image‐based lung model. They found that the mean displacement error in the prostate was 2.0 mm, while the mean displacement error for the lung registration was 1.3 mm.Wang et al.26 tes evaluated the OF algorithm using a large number of landmarks. They tested image registration with voxel size5×5×5 mm3 and found that the mean 3D registration error for this algorithm was 6.9±0.1mm. This method is similar to the model we used for our validation because it is based on anatomical landmarks. The voxel size used for this study is quite large and may not reflect the clinical situation; the CT voxel size of the POPI model used in our validation was 0.97×0.97×2 mm2. Additionally, Castillo and colleagues tested a landmark‐based DIR algorithm and found that it performed better than the OF algorithm (error of 2.05±0.02mm). As one would expect, using landmarks to evaluate a landmark‐based algorithm may yield too optimistic results. Additionally, Castillo et al. found the error in OF was greater with DIR than RIG‐only registration for four out of the five cases they reported, while our study found the OF DIR was superior to RIG‐only registration.Castillo et al.31 eva Their results show a mean TRE of 1.2 mm with a standard deviation of 0.4 mm, while ours show a mean TRE of 1.4 mm with a standard deviation of 0.7 mm.In all these validation studies, the voxel size will directly determine a lower limit of accuracy that can be reached using DIR, as the anatomical landmarks are defined by a voxel. For example, the slice thickness is the limiting factor on the accuracy of the landmark points in the superior–inferior direction because a point may lie in between two slices, which would make it hard to identify by the radiologist on the two image sets. In addition, the image sets used are from a 4D CT, which may make the identification of landmarks less clear than identifying landmarks on a breath‐hold CT. Therefore, we consider the registration accuracy in this validation study for the three DIR algorithms to be acceptable, as the mean target registration errors are less than the slice thickness. This is comparable to the validation study for demon's algorithm reported by Vandemeulebroucke et al.24 TheThe POPI model is published and used by many groups for validation studies.–,24 Given the difficulties of manual landmark selection, only one set of 4D CT data is available in the model. Thus in this study, one set of data with 39 points was used in the validation.The task of manually selecting landmarks is difficult, tedious, and time‐consuming. In addition, selecting landmarks in the end‐expiration phase is more difficult because of the change of density of the lung.22 GivDIR algorithms are also evaluated by studying the image difference between the deformed image and the target image. The smaller the difference between the deformed and the target images the better the DIR algorithm. HU values in the lung for the CT sets used ranged from ‐922 to near 0. This large range in values can potentially lead to large differences when comparing two images if they are not perfectly aligned. In the case of the differences between the deformed image and the target image, large differences in HU are due to the variations at interfaces within the lung. The interfaces between the blood vessels and other parts of the lung may lead to large differences in HU between image sets. Each algorithm has its own strengths and weaknesses. The DM algorithm is based on matching edges and lines in a voxel, and if an image has an artifact, then that artifact would propagate throughout the process of image deformation. On the other hand, the OF and DD algorithms are intensity‐based registration algorithms, and they may deal differently with an image artifact. Additionally, the DM and DD algorithms have smoothing filters applied to their deformation fields, which may lead to additional differences in image deformation.While the validation results show that all three DIR algorithms perform well, we speculate that how DIR algorithms address the imperfection of the 4D CT images may contribute to some of the inaccuracies observed.33 Eacp=0.37). The mean target registration error for each of the registration algorithms was less than the slice thickness of the 3D CT volumes, which suggests that the deformable registrations done by either of the algorithms are fairly accurate for the thorax region. Additionally, image differences between the target and the deformed images were comparable between the DIR algorithms.We validated all three DIR algorithms for the thorax using the POPI model, which is a landmark‐based model for the lung. We also used image differences to evaluate the accuracy of DIR algorithms. For normal end‐expiration‐to‐end‐inspiration registration, all three algorithms show a maximum target registration error of less than 4 mm, or two voxels, with insignificant differences between them (


Conventional wetting theories on rough surfaces with Wenzel, Cassie-Baxter, and Penetrate modes suggest the possibility of tuning the contact angle by adjusting the surface texture. Despite decades of intensive study, there are still many experimental results that are not well understood because conventional wetting theory, which assumes an infinite droplet size, has been used to explain measurements of finite-sized droplets. Here, we suggest a wetting theory applicable to a wide range of droplet size for the three wetting modes by analyzing the free energy landscape with many local minima originated from the finite size. We find that the conventional theory predicts the contact angle at the global minimum if the droplet size is about 40 times or larger than the characteristic scale of the surface roughness, regardless of wetting modes. Furthermore, we obtain the energy barrier of pinning which can induce the contact angle hysteresis as a function of geometric factors. We validate our theory against experimental results on an anisotropic rough surface. In addition, we discuss the wetting on non-uniformly rough surfaces. Our findings clarify the extent to which the conventional wetting theory is valid and expand the physical understanding of wetting phenomena of small liquid drops on rough surfaces. The contact angle is a material property determined by the surface tensions between substrate, liquid, and vapor345678et al.12et al.However, the conventional wetting theory110111314151617181101112In this work, we present a more generalized wetting theory by considering the entire free energy landscape including the transition configuration between the pinning points in three wetting modes, P, W, and CB, on a surface with rectangular protrusions as well as non-uniformly rough surfaces. First, we confirm that the pinning phenomena can be understood by the existence of multiple local minima of the free energy landscape separated by energy barriers. For all wetting modes , we show that the contact angle at the global minimum recovers the prediction of conventional wetting theory within 2° when the droplet size  becomes at least 40 times larger than the characteristic scale of the surface roughness (periodicity of the texture). Second, we compute the free energy barriers between available pinning points in all wetting modes. For P and CB modes, the energy barrier between local minima tends to decrease as the contact angle becomes further apart from the contact angle at the global minimum energy and eventually becomes zero, which enables us to predict the ultimate bound of advancing and receding angles. Our theory is then applied to explain the measured contact angle on the surface with anisotropic roughness20For the mathematical simplicity, we consider a two-dimensional (2D) finite-sized liquid droplet. As illustrated in e > 90°), and Case II has the lower free energy for the hydrophilic surfaces (θe < 90°) . The overbar indicates the dimensionless variables. In what follows, all length and energy variables are normalized against the radius of the circular droplet R0 . The vertical dotted line is the contact angle determined by the conventional wetting theory on the Wenzel mode, cos θ = r cos θe. The contact angles at the global free energy minima under different surface roughnesses, which are highlighted by the blue circle, green triangle, and red square, do not match perfectly with the prediction from the conventional wetting theory. Thereafter, one can predict the contact angle as a function of the surface roughness factor r by connecting the contact angles at the global free energy minima, as illustrated in e, is also presented in comparison. The contact angles for the CB and P modes can be obtained in a similar way as functions of Based on the free energy expression as a function of θ, we can find the allowable contact angles with free energy minima at a given roughness factorWe then consider how the contact angle changes with the droplet size by varying the dimensionless variable e by comparing the free energies of the three modese < 90°), the contact angle is given by the higher contact angle between the predictions based on the P mode and W mode and on the hydrophobic surface (θe < 90°) by the lower contact angle between the predictions based on the CB mode and W mode. The choice between the W mode and the CB mode is made by comparing the free energy values in C is determined when the free energy variation to fill an additional groove is 0. If the contact angle of the flat surface θe is smaller than θC, the free energy variation to fill another groove becomes negative, spreading occurs, and the Penetrate mode is selected. In our work, the critical contact angle can be expressed with geometric factors as We can predict the most stable wetting mode for the substrates with different Young’s angle θe or θ > θe, respectively. The disappearance of the energy barrier indicates the ultimate bound of advancing and receding angles which are determined by available local free energy minima. The maximum advancing (receding) angle can be obtained from experiments will be the highest (lowest) contact angle at the pinning points with nonzero energy barrier. We speculate that the conventional Wenzel state has very large contact angle hysteresis because of the non-vanishing energy barrier between local minimaWe then confirm the origin of the pinning effect is the occurrence of multiple local minima and calculate the free energy barriers in terms of geometric factors for the three wetting modes. In the W mode, a drastic free energy change occurs when the three-phase contact line is located near the corner of the step because the additional boundary may form or disappear. The A’ and B’ points at the end of the step in et al.e = 114°) along both perpendicular and parallel directions. The width of step and groove were 23 μm and 25.6 μm each. The height of the step was 30 μm. The measurement of the contact angle were conducted with water droplet of volume from 0.59 mm3 to 5.679 mm3. They confirmed that droplets are in the CB mode, and measure the number of pillars filled by or the base line length of the droplet. It was reported that the contact angle at the direction perpendicular to the grooves is likely to be similar to the advancing contact angle310σLV and the baseline length parallel to the groove direction to dimensionalize the formula to compare the energy barriers for different droplet volume. The energy barrier between B state and C state, at each local minimum is plotted in Now, we apply the proposed theory to understand the wetting experiments on the surface with anisotropic roughnessMAX in the number of filled grooves, n. For example, we compute the free energies of the wetting states when e = 120° as functions of contact angle θ with nMAX = 14 and 19, as depicted in e. It is the case for any nMAX ≤ 14 because the free energy curve with a fixed n has a minimum at θe, as depicted by the red dotted curves in MAX = 19 has a higher free energy than the global minimum. Hence, the contact angle prediction becomes identical to the surface with uniform roughness. Our results agree with the previously proposed wetting theory for a non-uniform rough surface2035We then apply our theory to analyse a surface with non-uniform roughness that has a rough center and flat periphery. In other words, we compute the free energy landscape when there exists an upper bound nIn conclusion, we have developed a wetting theory that can predict the wetting angle and wetting mode when the liquid droplet is not much larger than the surface texture scale. Because the conventional theory assumes a much larger size of the droplet compared to the texture scale, there have been limitations in how to analyse experiments that investigate the wetting of small liquid drops. Our theory suggests that conventional theory should be used when the droplet size is about 40 times larger than the characteristic scale of the surface roughness and provides a deeper physical understanding regarding the wetting of smaller liquid droplets on non-uniform rough surfaces.How to cite this article: Kim, D. et al. Wetting theory for small droplets on textured solid surfaces. Sci. Rep. 6, 37813; doi: 10.1038/srep37813 (2016).Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


The uncertain option task has been recently adopted to investigate the neural systems underlying the decision confidence. Latterly single neurons activity has been recorded in lateral intraparietal cortex of monkeys performing an uncertain option task, where the subject is allowed to opt for a small but sure reward instead of making a risky perceptual decision. We propose a multiple choice model implemented in a discrete attractors network. This model is able to reproduce both behavioral and neurophysiological experimental data and therefore provides support to the numerous perspectives that interpret the uncertain option task as a sensory-motor association. The model explains the behavioral and neural data recorded in monkeys as the result of the multistable attractor landscape and produces several testable predictions. One of these predictions may help distinguish our model from a recently proposed continuous attractor model. Recently many studies began to investigate the brain signature of complex cognitive functions such as decision confidence, the feeling of certainty/uncertainty associated with a decision. To this aim, the uncertain option task has been widely adopted in order to assess the confidence in animals. In this study we present a model, detailed at the neuron and synapse level, able to account for the behavior of animals in this task. In addition our model is able to reproduce the neural dynamics found in monkeys brain during this task. However our model is only equipped with a simple multiple choice mechanism and has no mechanism devoted to calculate the confidence. Therefore our study support the idea that the uncertain option task can be solved without relying on confidence assessment (metacognition). The model is based on the idea that the neural dynamics fluctuates around stable equilibrium points (attractors) and associates the landscape of these attractors with the behavior of the monkeys. Finally, our model makes several predictions that could be easily tested in a new experiment. One of these predictions may help distinguish our model from a different one that has been recently proposed. Decision confidence, a simple form of metacognition, has been receiving increasing attention in neuroscience over the last ten years. Many studies began to unravel possible neural correlates of confidence both in rodents , 2 and iWhile some abstract models have already been proposed for the uncertain option task , 13–15, We characterize the dynamics of the model in terms of the underlying multistable attractor landscape and investigate the influence of sensory evidence on the dynamics. In addition we study the modification of the attractor landscape induced by changes in input intensity and link such modification of the attractor landscape with the choice behavior of the network. This analysis shows that multistability, particularly near a bifuraction is needed to reproduce experimental data. Finally, we show three experimental predictions to falsify or verify our model. 1. Only the maximum activity of decision neurons selective for right and left choices is determining the probability of a sure response. 2. The probability of sure target choices separated in correct and error trials presents an X-pattern. 3. The decision times of the model have a bimodal distribution. Interestingly prediction 1 may offer a way to distinguish the continuous attractor network  and our λ). The input to the network is modeled as Poisson spike trains coming from 800 external neurons. The stimulation protocol of the selective pools R and L is the following. First during 500 ms the pools only receives the background activity  to assure that the network is in resting state. After the resting phase the target input produces a strong response in decision pools L and R. The target input is modeled following two exponential decays . The stimulation protocol is shown in We run simulations of 1000 trials for each stimulus condition . The posee e.g.  for detaDetailed description of the neuron and synapse model is given in Tables We use a mean-field reduction  in orderThe mean-field approximation allows reduction of the number of dynamical variables, by describing the average firing rate of each neuronal pool in the limit of an infinitely large number of neurons. In our network several different stationary states are possible depending on model’s parameters: spontaneous state, where the selective pools have low activity, three selective states with one selective pool firing at a high level and the others inhibited, a mixed state with two stimulated pools highly active. With some parameter values the network could sustain just one stable state, or could show bistable, behaviour, or could show multistable behaviour in which the spontaneous state or the mixed state and each of the two decision states are all possible stable states when the inputs to the network are being applied .Note that although the mean-field reduction is useful to explore the attractor’s landscape of the model, it would probably not be adequate to account for data in : in thatpopulation transfer function, which provides the average population rate as a function of the average input current. The set of stationary, firing rates for the different neuronal pools can be found by solving a set of coupled self-consistency equations.Here we detail the mean-field approximation of our model . Under tV(t) is the membrane potential, x labels the populations, τx is the effective membrane time constant, μx is the mean value the membrane potential would have in the absence of spiking and fluctuations, σx measures the magnitude of the fluctuations and η is a Gaussian process with absolute exponentially decaying correlation function with time constant τAMPA. The quantities μx and νext is the external incoming spiking rate, νI is the spiking rate of the inhibitory population, τm = Cm/gm with the values for the excitatory or inhibitory neurons depending on the population considered and the other quantities are given by:p is the number of excitatory populations, rx is the fraction of neurons in the excitatory x population, ωj,x the weight of the connections from population x to population j, νx is the spiking rate of the x excitatory population.The mean-field approximation assumes that the network of integrate-and-fire neurons is in a stationary state. In this formulation the potential of a neuron is calculated as:ϕ is the transduction function of population x, which gives the output rate of a population x in terms of the inputs, which in turn depend on the rates of all of the populations.u) the error function and τrp the refractory period which is considered to be 2 ms for excitatory neurons and 1 ms for inhibitory neurons. To solve the equations defined by xs we integrate numerically The spiking rate of a population as a function of the defined quantities is then given by:For the numerical integration we used an Euler routine with a step size of 0.1 ms.νL, νR and record which attractor the network falls in; we change the initial values of L and R with a bisection algorithm until finding the boundary with a precision of 0.1 Hz.We also calculate the boundary of the attraction basins of fixed points shown in . In ordeλ in order to obtain a qualitative fit of the probability of a correct response in trials without “sure target”. We then adjusted the target input to pool S in order to obtain proportions of “sure” choices comparable to those of the monkeys. The set of parameters used in the simulation is summarized in We adjusted the parameter Δper se, even with very simple inputs, is capable to reproduce the results of the uncertain option task.The stimulation protocol was designed to match the input to LIP and is shown in νL,νR in square bins of side 1 Hz. We calculated for each trial the time averaged firing rate of decision pools in the 50 ms time window preceding the “sure” target onset. Then each bin in the plane is associated with those trials presenting the average firing rate before “sure” target onset determined by the position of the bin. Finally we counted for each bin how many trials ended up in a “sure” decision and divided by the total number of trials associated with that bin. We did this analysis separately for each stimulus duration collapsing together all coherences (Δλ) and also for each coherence collapsing together all durations. The results were very similar across different conditions therefore we decided to pool together all trials from all stimulus conditions in order to show cleaner results.We calculate the probability of choosing the “sure” option conditioned on the value of firing rate of decision populations R and L. In order to calculate this probability we divided the plane ϑ set at 28 Hz. In addition the firing rate needs to be above the threshold in the subsequent 50 ms. Of course this values are not meaningful and other similar values give the same qualitative results. The reaction time is the sum of the DT and a non-decision time that is usually used to account for sensory processing and motor planning. Since we are not comparing the decision times of the model with experimental data we are not considering any non-decision time.We calculate the decision time of the network as the time when the firing rate of a decision pool crosses a threshold, We calculate the reward amount received by the subjects as a function of the probability of correct response and the probability of “sure” target selection. We arbitrarily consider the value of a correct response to be 1 and that of an error to be 0. The value of the “sure” target has been set to 0.8, since in the experimental setup the subjects received for a “sure” response approximately 80% of the reward associated to the high stakes targets. However we note that this value neither is crucial for our results nor need to be fixed based solely on the experimental setup. Indeed, while the value attributed by the subject to the “sure” option will reflect the objective amount of reward associated, it won’t probably be the same since it will include subjective preferences like risk aversion or risk attraction. We calculated the total reward in a block of trials as the sum of the reward received in each trial according the choice made . Reward functions can be complicated and include many variables like reaction times, penalty time associated to errors, etc. . We deciλ and stimulus duration, we separated the trials in fast and slow using the minimum of DT histogram in the range [1001500] ms as the separation point. The restriction on the range of values of DT is necessary to find the dip in the distribution and not an eventual very small tail. We constructed the distribution of the number of error trials in fast and slow trials using bootstrap technique with 10000 samples. We then performed a t-test on these distributions in order to compare their means. The p-value of all tests was smaller than numeric resolution.For each combination of values of ΔS)) decreases both for longer stimulus durations and for higher motion coherences  that triggers the competition. During the trial one of the two pools will prevail on the other and enter into a decision state (high firing rate). Unbalanced inputs (Δλ = λL − λR > 0) can bias the decision, but given the intrinsic stochasticity of the dynamics, an incorrect decision may be taken even when input currents are very different. Our model implements a simple multiple choices decision-making mechanism [Here we describe a multiple choice decision model implemented in a biologically detailed network. We will show that such a network is able to reproduce the experimental results  in accorechanism : if threechanism , we usedAs shown in In addition to psychophysics our model also reproduces the main neurophysiological results. We note that also for the neural activity no further tuning of the parameters was necessary to reproduce experimental data. Forced choice trials show the typical separation of firing rates when the chosen target was inside or outside the response field of the neuron –23. NeurThe reduced separation of firing rates is a key characteristic of LIP neurophysiology in the uncertain option task . The insνR and νL, as depicted in In order to get more insight into the dynamics of the system we can observe the evolution of the decision process in the phase plane of firing rates νR, νL plane that provides useful and clearer information about the decision process.Since pool S has constant activity around 2-3 Hz until the onset of the sure target, we disregard this pool for this analysis and only consider the projection of the system dynamics onto the λ . We represent attractors and their basins under two different stimulation conditions that are given during a trial , hence we omit it here to keep the figure uncluttered; the attractor landscape for the delay condition is shown in λ in νL = νR ≈ 2 Hz, while νS >>0. Also decision attractors for left and right choices exist with similar symmetric conditions.In The traces in the bottom panels show the trial average dynamics of the spiking network for decisions left (correct), right (error) and sure target. The color of traces represents the evolution of time from black , to yellow , passing through red. Traces terminating in the L (R) basin correspond to left (right) choices. Traces terminating in the sure choice attractor basin (gray dashed lines) correspond to sure choices. Events of the trial are represented on the traces by different symbols (as indicated in panel C bottom). Attractors and their basins are depicted also in bottom panels for reference (attractors during sure target period are represented in bottom panels by gray crosses for clarity). When the motion stimulus is turned on, both populations present relatively high and comparable firing rate due to the previous response to the direction targets.λ = 0  attractor  drives the system to choose the sure target.νL and νR). These probability distributions are interesting predictions of our model that could be easily tested in an experimental set up, where the activity of decision neurons selective for the two choices is recoded simultaneously. Such an experiment would be highly valuable in order to falsify this model and to distinguish it from the model in [First, we represent the state of the system showing the probability distribution of firing rates of the full spiking network at the time of sure target onset . Then wemodel in , as explλ, while columns from left to right represent increasing durations of the stimulus. When there is very low evidence  the distribution of νL, νR is quite symmetric about the diagonal and the duration of the stimulus has very low or no effect. In the second row we can observe that a moderate level of coherence already skews the distribution towards the correct decision area . When duration of the stimulus increases, the skewness gets larger. On the bottom row (very high stimulus coherence), for a short duration, the distribution appears markedly skewed towards the decision area and we can observe that for longer stimulus durations the distribution moves even further into the decision region concentrating towards the correct decision attractor.The panels of S|νL, νR) as shown in S|νL, νR) reproduces the shape of the attraction basin of the sure target as calculated with the mean-field approximation  does not change when the stimulus parameters  are changed  does not change with λ may indicate that the state of the system before sure target onset is enough to characterize the sure target choice behaviour and that the complete trajectory of the system is not necessary. The parameters of the stimulus  affect instead the distribution of firing rates in the νL, νR plane .Considering only the state of the system at the moment the sure target is shown is an approximation that neglects the previous history of the trial. However we note that P, a quantity that they call “confidence”, is a sigmoidal function of the absolute value of the difference between the activity of pool L and R. Similarly in our model 1 − P(S) as a function of the absolute value of the difference between the activity of L and R also resembles a sigmoid function (see S) resembles a sigmoid function  an extrapolation of the prediction in [νL, νR plane: σ(|νL − νR|), where σ(a) is the logistic sigmoid function. The different predictions highlight the diverse mechanisms underlying the continuous and the discrete model, proposed here. A new experiment with the same behavioral setting could test these predictions and falsify one of the two (or both) models, if the activity of neurons selective for right and left choices would be recorded simultaneously.Panel C shows (with synthetic data) ction in  on the νThe definition of correct and error choices in forced choice trials is straightforward. In free choice trials the asynchronous stimulation protocol we used induces first a competition between L and R, while S enters into the competition only after the onset of the sure target. Hence we can also separate trials in correct and error “early” choices, until we constrain the analysis to the period before the stimulation of S.Here we characterize the dynamics of the network, before sure target onset, separately for correct and error choices .While early correct trials are distributed mainly in the decision region, early error trials are located near the diagonal. For example, in trials where the network dynamics is evolving towards the right choice but the evidence for the left choice is actually stronger, the system will remain longer near the basin of attraction of the sure choice. If the sure target is not presented (forced choice) these early error trials will likely end up in a wrong choice. However, if the sure target is presented (free choice), those early error trials which are near to the basin of attraction of the S choice will end up in sure choices. This configuration of the system explains the better performance of the model in free choice compared to forced choice trials.λ). We will show that a key ingredient of the model for reproducing the data in [In this section we study the modification of network attractor landscape induced by the input to decision pools L and R , for 4 sample points in different regions of the bifurcation diagram (marked by stars). For each point we adjusted the parameter Δλ in order to get a qualitative match with experimentally measured P(correct). Panels F-I show the P(S) for the same points. As can be seen the model is able to reproduce qualitatively the performance of monkeys both in the bistable and in the multistable region. Right after the pitchfork bifurcation, in the region where only the “mixed” state attractor is stable, the model still reproduce quite well the P(correct), while, if we further increase λ, moving away from the bifurcation, it is not possible to approximate the experimentally observed P(correct) since the model becomes insensitive to stimulus duration. Interestingly the P(S) changes qualitatively when moving through the different regimes of the network. In the bistable region  the P(S) is less sensitive to changes in stimulus coherence (i.e. Δλ), e.g. the frequency of S choices rapidly decreases as a function of stimulus duration for low evidence trials (red curve), while experimental data show a very flat curve. In addition, when the network is stimulated for 500 ms, the P(S) for different values of Δλ show quite small variation, which is also inconsistent with experimental data. When moving towards the pitchfork bifurcation these effects are gradually reduced and a good qualitative fit of experimental data can be observed in the multistable region or right after the bifurcation.Panels A-D show the accuracy, P(λ = 15), for Δλ = 0, the duration of the stimulus induces a strong change in the distribution of firing rates in the phase plane  is very different in these two regions  increases in trials when the “sure” target was shown but not chosen compared to trials without “sure” target  bring us to hypothesize that the P(S) should show an inverted pattern in early correct and early error trials, when plotted as a function of the discriminability (Δλ). In In this section and in the following we provide two further predictions that may be tested in future experiments. The proportion of early error trials is expected to depend on the discriminability of the stimulus. This fact together with the distribution of firing rates in correct and error trials (see sec. λ and the pattern would be X-shaped).This pattern, is analogous to the X-shaped pattern usually associated with confidence , 28, 29 S|outcome). This quantity is unavailable if we look at the psychophysical measurements. The available measures are the probability of correct choices when the sure target was shown but waived, S). Using Bayes theorem and the fact that This prediction arises by applying a decision criterion . For other values of λ the value of the threshold is different but was always chosen according to the same criterion. Independently on the decision rule another problem remains still open when facing with delayed response tasks. Indeed if after accomplishing the decision rule but before the “go” signal the accumulation process keeps on, a change of mind [We describe next the decision times distributions and their relationship to the network attractor landscape. What we refer to as decision time (DT) is the time (from stimulus onset) that the model takes to integrate the stimulus signal and reach a decision. This time offers an insight into the decision process and could be compared to empirical reaction times of a modified version of this task. We consider here that a decision has been reached when the firing rate of one decision selective pool crosses a threshold,  of mind  could oc of mind , could oWe observe that in forced choice trials, the DTs have a bimodal distribution. We note that the two parts of the distributions correspond to two different types of trials. On some trials the network reaches the decision criterion during the stimulus period and these trials result in fast responses. On the contrary when no decision is met during the stimulus period, once the stimulus is extinguished, the decision process relies only on the memory trace of the stimulus (i.e. the self-sustained activity of the network). These type of decisions are of course slower. We can observe that the fast part of the distribution comprises DT values that depend on the duration of the stimulus: in λ = 15Hz, The bimodality of these distributions is induced by the presence of the third “mixed” attractor. This stable point attracts the dynamics of the network during the stimulus driven regime. This happens on trials when the stimulus input is not strong enough to push the system into a decision basin. When the system is in the basin of attraction of the “mixed” state it will remain there until the stimulus extinguishes destabilizing the attractive point; in this case only after stimulus offset the network is eventually going to enter into the basin of attraction of one decision. As a confirmation we observe that when the mixed attractor is not stable, for example in the bistable regime, the DTs distributions are not bimodal , that we didn’t modeled explicitly since they seem to take place outside of the decision-making network. This analysis is interesting per se to understand the model. Moreover this analysis can supply predictions about novel experiments. In fact, it is believed that the subjects have the ability to change, to some extent, their λ in order, for example, to satisfy different sets of experimental instructions .λ fixed when varying the value of λ. This will simulate the scenario of a subject changing its λ due to experimental manipulation, like time pressure, stress on accuracy, stimulus visibility, etc. In this scenario Δλ shouldn’t be adjusted for each value of λ since, once the subject has learned the task, so that the performance is stable, Δλ should only depend on stimulus coherence if all other experimental conditions are constant.In contrast to previous analysis summarized in λ. As expected the P(C) decreases when λ increases since the signal-to-noise ratio decreases. At the same time the P(S) increases as a function of λ.correct) + P(S) ⋅ wS, where wS is the value of the third option. This is a simplified equation for the reward: we are not interested in the actual value of the third option but in its value relative to that of correct choices, so we fix the value of the correct option to one (the units of reward are then arbitrary). λ and for different values of wS. The bottom curve shows the reward for wS = 0.8, which is close to the relative amount of liquid reward used in [λ. The middle curve correspond to wS = 1.1 and we can observe that in this case the reward amount increases until 50 Hz and then decreases again. For higher values of wS (upper curve) the reward amount keeps increasing (or remains high) over the entire range of λ. Consequently, a prediction of the model is that the relative value of the sure target is not simply proportional to the amount of liquid associated to the sure option, rather subjects would assign a higher value to the sure target. This prediction is based on three hypothesis: 1) The value of λ used by the subjects in the experiment is the one that best fit the experimental data, i.e. multistable region λ ≈ 55 Hz. 2) The subjects have the ability to internally control λ in response to some experimental change. 3) The subjects are trying to maximize the proposed mean reward amount. If we consider these hypothesis plausible, than we may say that subjects were assigning a large value to the sure target in order to be more flexible and to remain open to new options offered at a later stage in the trial. However we note that subjects, although flexible, were far from optimality (in terms of liquid reward). For example, they were choosing the sure option only 60% of the trial when their performance was at chance.This means that the subjects have to trade accuracy for flexibility. If the subjects want to make more correct choices they will consider less the third option presented later in the task. Conversely, if the subjects want to remain more flexible to future possible options they have to accept a higher error rate. Therefore we can infer how much subjects valued flexibility in this task. We can hypothesize that subjects were maximizing the mean reward amount. We calculated the mean reward amount over all trials using the equation: P and predictions. The discrete mechanism that we propose induces an attractor landscape, hence a dynamical system, different from that proposed in [λ and were able to link the uncertain option task with the multistable regime. Importantly, Wei and Wang [S)) can be encoded by a sigmoid function of the absolute value of difference between the activity of neural pools corresponding to the two choices (i.e. σ(|RL − RR|), where σ(a) is the sigmoid function). Conversely, in our model the probability of choosing the sure target does not only depend on the distance between L and R pools but on the maximum of the two Click here for additional data file.S2 FigTsure onset for a given combinations of stimulus evidence (Δλ) and stimulus duration as indicated in the figure. It can be observed that, at Δλ = 0, while the system state is quite spread at short durations, it rapidly concentrates around the attractors for longer durations. This in turn produces a large variation in the probability of choosing the sure target. In addition the distributions for different values of Δλ are quite similar and indeed the corresponding probabilities of choosing the sure target are similar. Both this features are not consistent with experimental results of [Each panels shows the distribution of firing rates at sults of .(EPS)Click here for additional data file.S3 FigS|νL, νR) is not affected by this parameter. Color convention is the same of The general form of P((EPS)Click here for additional data file.S4 FigS), that [νL − νR| shows a sigmoid pattern. B: 1 − P(S) as a function of νL + νR also shows a sigmoid pattern meaning that in our model the activity of both νL and νR is important to determine P(S).A: 1 − P(S), that  defines (EPS)Click here for additional data file.S5 Figλ as indicated in legend. The dashed lines are regression lines for each subset of points according to color. Shaded areas around regression lines show the interval of confidence of regression lines. Pearson correlation coefficients between model and data are indicated using the same color code. B: Scatter plot as in A representing the probability of a correct response in forced choice trials, i.e. without sure target. C: Scatter plot as in A representing the probability of a sure target response.A: Scatter plot showing the probability of correct responses in free choice trials, i.e. with sure target. Each color and symbol represent one value of (EPS)Click here for additional data file.S6 Figλ = 7 Hz for all panels. The thresold for the decision is 24 Hz. The color of the histogram represents the relative contribution of correct (green) and error (red) trials to the global distribution. The bimodality induced by the multistable regime  and free choice trials (bottom row), as in e regime  disappea(EPS)Click here for additional data file.S7 Figϑ is indicated above each panel, λ = 55Hz, Δλ = 7 and stimulus duration is 500 ms for all panels.The value of (EPS)Click here for additional data file.S8 Figϑ is indicated above each panel, λ = 55Hz, Δλ = 7 and stimulus duration is 500 ms for all panels.The value of (EPS)Click here for additional data file.S9 Figλ and stimulus duration the histogram of the number of error trials is shown in blue for fast trials and in magent for slow trials. The vertical line represent the number of error trials in network simulations using the same color scheme. The mean of the distribution for slow and fast trials are different for all value of Δλ and stimulus duration .For each value of Δ(EPS)Click here for additional data file.


Dynamical systems can undergo critical transitions where the system suddenly shifts from one stable state to another at a critical threshold called the tipping point. The decrease in recovery rate to equilibrium  as the system approaches the tipping point can be used to identify the proximity to a critical transition. Several measures have been adopted to provide early indications of critical transitions that happen in a variety of complex systems. In this study, we use early warning indicators to predict subcritical Hopf bifurcation occurring in a thermoacoustic system by analyzing the observables from experiments and from a theoretical model. We find that the early warning measures perform as robust indicators in the presence and absence of external noise. Thus, we illustrate the applicability of these indicators in an engineering system depicting critical transitions. Many complex systems such as ecosystems, climate systems, financial markets and neurons in the mammalian cortex exhibit critical transition12345678912et al.18192126Over the years, many early warning measures have been developed to successfully detect critical transitions observed in complex systems. These early warning measures perform by exploiting the phenomenon of critical slowing down observed in dynamical systems. Dynamical systems display slow recovery from perturbations in the vicinity of a tipping point which is known as critical slowing down161214Critical transitions that occur in many engineering systems result in a huge loss of revenue. One such engineering system is the power generating system where combustion of fossil fuels is the source of power28In general, physical systems display complex dynamics which makes it difficult to predict the transitions or even to describe the nature of the transitions observed in such systems. Therefore, controlled laboratory-scale experiments are necessary to verify the robustness of the early warning measures in physical systems. In this study, we employ ingenious laboratory-scale experiments along with a theoretical model to study the effectiveness of the early warning indicators. This is the first attempt to apply these early warning measures to predict subcritical Hopf bifurcation in a prototypical thermoacoustic system. The experimental set-up consists of a horizontal duct with a heat source located inside the duct. The experimental system could undergo a transition to self-sustained pressure oscillations for suitable choice of its control parameters. In addition to experiments, we use a theoretical model derived from the conservation equations of momentum and energy.In this study, we identify early warning signals such as increase in variance and conditional heteroskedasticity close to a subcritical Hopf bifurcation. We find that these measures are robust even when the system under consideration is under the influence of external noise. Many physical systems undergo critical transition from a non-oscillatory state to an oscillatory state, which can be described via a subcritical Hopf bifurcation. Hence, our study on the viability of the early warning measures to predict transition in a physical system gains considerable significance.The experimental set-up consists of a 1 m long duct of 10 cm × 10 cm square cross-section . The ducAn electrically heated wire mesh located in the upstream half of the duct acts as a source of heat for this prototypical system, unlike a flame in practical combustion systems. The voltage drop across the mesh and the current flowing through the mesh are measured to estimate the electrical power supplied to the mesh. We used loudspeakers to provide external fluctuations in experiments when we studied the robustness of early warning measures in the presence of noise.The theoretical model considered in this study captures the feedback between the fluctuating heat release rate and the sound waves (pressure and velocity fluctuations). This feedback could be modified, by varying certain control parameters, resulting in a state shift to large amplitude periodic oscillations. The formulation of the model is detailed in the Methods section. The following set of equations, derived from the conservation equations of momentum and energy, describes the temporal evolution of the systemThe model described in ε represents strength of the additive Gaussian white noise ϕ(t). The noise term ϕ(t) in β. The procedure to obtain β is detailed in the Methods section. In order to simulate the experimental conditions, we maintained the strength of additive Gaussian white noise ε in the model such that the non-dimensional noise intensity β is the same as that in the experimentswhere, K), was increased in a quasi-steady manner and the time series of unsteady pressure pertaining to each value of the parameter was recorded. When K was increased above a threshold value A, the median of the peak acoustic pressure (P) registered an abrupt increase  and for the reverse path (when K is decreased). This difference in threshold values results in the presence of a bistable regime. The presence of the bistable regime indicates that the transition to the oscillatory state observed in the experimental system is subcritical in natureIn K within the range of approximately 380 ± 20 W.In experiments, we have observed a 5% uncertainty in the value of the Hopf point. The uncertainty arises because the experimental conditions such as the mean temperature and the speed of sound are not exactly repeatable. Thus, in a particular experiment, we can expect the onset of oscillations at any K by 2 W in every 20 s and recorded the corresponding time series of acoustic pressure. A plot that depicts the change in K with time is included as et al.We calculated variance, lag-1 autocorrelation and also conditional heteroskedasticity of the time series in search for forewarning signals before the system reaches the state of self-sustained oscillations. Unlike a laboratory experiment where the control parameter is varied in a quasi-static way, the control parameter must be varied continuously as in the case of real systems. This prompted us to perform experiments by varying the control parameter in a continuous manner and capture the time series depicting the continuous transition from non-oscillatory state to oscillatory state. We varied t  =  644 s which marks the occurrence of Hopf bifurcation (at K = 394 W). The effects of critical slowing down, which can be observed just before the occurrence of the subcritical Hopf bifurcation, will aid in detecting the impending transition. Therefore, we used the time series up to t = 640 s to calculate the early warning measures. We can observe a significant increase in the value of variance well before the transition, which clearly serves as an early warning signal  as a function of time, given by k = 1.667 × 10−3t, and obtained the corresponding time series of unsteady pressure. Here also, we observed a transition to a state of limit cycle oscillationset al.We extended our search for effective early warning measures for predicting a subcritical Hopf bifurcation also in the theoretical model given by β = 0.2 in the system using loudspeakersThe robustness of the early warning indicators must be tested in the presence of fluctuations as practical combustion systems work in a turbulent environment. Therefore, we added external noise of non-dimensional intensity k = 0.62) does not change since the noise in the model is additive in naturek reaches the value of Hopf point only at t = 372. We can notice in We further tested the robustness of early warning signals using the model . In the et al.et al.We then proceed to calculate conditional heteroskedasticity which was introduced as an early warning indicator by Seekell C) for conditional heteroskedasticity, applied to the time series, increases as the critical transition is approached  as the critical point is approached, though it shows an increasing trend in the presence of external noise imposed by a loud speaker. The lack of robustness of autocorrelation as a precursor is due to the fact that we are using a single sample path or a single realizationThis is the first experimental study where we establish the effectiveness of critical slowing down indicators in early warning critical transition in a thermoacoustic system. Our findings are highly pertinent as many real systems exhibit critical transitions. By implementing the early warning measures in systems such as the one considered in the present study, we can prevent critical transitions in real systems. We wish to emphasize that the early warning indicators exploit the phenomenon of critical slowing down which happens near any bifurcation. Therefore, any transition which can be viewed as a bifurcation can be predicted using the early warning measures.There are certain limitations to the approach we have adopted in this study. The early warning indicators considered in the current study may fail to detect the regime shifts that occur as a result of rapid change in the control parameterWe use piezoelectric transducers  to record the variation in acoustic pressure. The sensitivity of the pressure transducer is 217.5 mV/kPa and the resolution is 0.2 Pa. The output from the pressure transducer is acquired using a data acquisition system (PCI 6221) through a signal conditioner. The transducer is mounted 325 mm from the end open to the atmosphere. We add external noise to the system through loud speakers (Ahuja AU 60) mounted on the duct 625 mm from the end open to the atmosphereThe model is derived from the conservation equations of momentum and energyγ is the ratio of specific heats of air. The quantities with tilde are dimensional. The variables pressure . In a practical combustion system, this delay can be attributed to factors such as the convective time scale (time taken by the fuel air mixture to travel the distance between the fuel injection point and the flame location), evaporation time scale, mixing time scale and reaction time scale.The location of the heat source inside the duct is denoted by M is the mean flow Mach number and The quantities with tilde are dimensional. where, The variables that govern the heat conduction from the wire are encapsulated into a single non-dimensional parameter denoted by jth natural mode. The expansions for Here c1 and c2 are the damping coefficients32ωj = jπ represents the non-dimensional angular frequency of the jth mode. The non-dimensional frequency ωj is obtained by non-dimensionalizing the angular frequency of the jth natural mode of a duct open at both ends, c0/L, where c0/L is the travel time of the sound waves in the duct. The value of the parameters used for computation are N (number of modes) = 10, dt (step size) = 0.01, ωj = jπ, c1 = 0.1, c2 = 0.06, xf = 0.25 and τ = 0.2.The damping in the system is modeled using the term I) is non-dimensionalized by the amplitude of limit cycle attained at the Hopf point in the absence of external noise (PH).The amplitude of fluctuations or the noise level in the system  is estimated by measuring the rms amplitude of the acoustic pressure when the system is in the non-oscillatory state. The noise amplitude . We used the time series up to the impending critical transition to calculate the early warning indicators. For variance and autocorrelation, we calculated the temporal trend by estimating the nonparametric Kendall rank correlation (τk)τk is a statistical tool used to measure the association between two measured quantities. Lag-1 autocorrelation is determined by the autocorrelation function (ACF) given below:We performed the statistical analyses using the “Early Warning Signals Tool Box” (P(t) is the value of the state variable at time t, and Pm and σ2 are the mean and variance of P(t) within the time frame considered. Variance is calculated as:where, where We used Lagrange multiplier test to calculate conditional heteroskedasticityThe order is selected according to Akaike information criterionα0 and αi represent the regression coefficients. The residuals give the error variance relationship which shows the properties of conditional heteroskedasticity. We conducted chi square test to compare the values of R2 (Lagrange multiplier test statistic) to a where How to cite this article: Gopalakrishnan, E. A. et al. Early warning signals for critical transitions in a thermoacoustic system. Sci. Rep.6, 35310; doi: 10.1038/srep35310 (2016).