In this paper, a noise enhanced binary hypothesis-testing problem was studied for a variable detector under certain constraints in which the detection probability can be increased and the false-alarm probability can be decreased simultaneously. According to the constraints, three alternative cases are proposed, the first two cases concerned minimization of the false-alarm probability and maximization of the detection probability without deterioration of one by the other, respectively, and the third case was achieved by a randomization of two optimal noise enhanced solutions obtained in the first two limit cases. Furthermore, the noise enhanced solutions that satisfy the three cases were determined whether randomization between different detectors was allowed or not. In addition, the practicality of the third case was proven from the perspective of Bayes risk. Finally, numerous examples and conclusions are presented. Stochastic resonance (SR) is a physical phenomenon where noise plays an active role in enhancing the performance of some nonlinear systems under certain conditions. Since the concept of SR was first put forward by Benzi et al. in 1981 ,7,8,9,10Studies in recent years indicate that the detection performance of a nonlinear detector in a hypothesis testing problem can be improved by adding additive noise to the system input or adjusting the background noise level based on the Bayesian , MinimaxNumerous researches on how to increase the detection probability according to the Neyman–Pearson criterion have been made. In , S. Kay Few researchers focus on how to reduce the false-alarm probability and there is no evidence to indicate that the false-alarm probability cannot be decreased by adding additive noise on the premise of not deteriorating the detection probability. In fact, it is significant to decrease the false-alarm probability without decreasing the detection probability. In , a noiseActually, in many cases, although the structure of the detector cannot be altered, some of its parameters can be adjusted to obtain a better performance. Even in some particular situations, the structure can also be changed. In this paper, we consider the noise enhanced model established in  for a vaThe remainder of this paper is organized as follows. In Notation: Lower-case bold letters denote vectors, A binary hypothesis-testing problem is considered as followsThe new noise modified observation Then the pdf of The noise modified false-alarm probability From (6) and (7), Actually, although the detector cannot be substituted in many cases, some parameters of the detectors can be adjusted to achieve a better detection performance, such as the decision threshold. Even in some particular cases, the structure of the detector can also be altered. Instead of the strictly fixed Under the constraints that It is obvious that the ranges of (i)When (ii)When (iii)A noise enhanced solution obtained as a randomization between two optimal solutions of case (i) and case (ii) with weights Naturally, Remarkably, if For the noise enhanced detection problem, when the detector is fixed, we only need to consider how to find the suitable or optimal additive noise. Nevertheless, when the detector is variable, we also need to consider how to choose a suitable detector. In this section, the noise enhanced detection problem for the variable detector is discussed for the case where the randomization between different detectors is allowed or not.For the case where no randomization between detectors is allowed, only one detector can be applied for each decision, thereby the noise enhanced detection problem for the variable detector can be simplified to that for a fixed detector For any For the case where the randomization between different detectors is allowed, multiple detector and additive noise pairs can be utilized for each decision, thereby the actual noise modified false-alarm and detection probabilities can be expressed asLet Theorem 1.The optimal noise enhanced solution for case (i) (case (ii)) is a randomization of at most two detectors and discrete vector pairs, i.e.,andwith the corresponding probabilities. The corresponding proof is presented in In this section, we will explore and find the optimal enhanced solutions corresponding to cases (i) and (ii), then achieve case (iii) through utilizing the solutions of cases (i) and (ii). From (8), In this subsection, the main goal is to determine the exact values of two detectors and constant vector pairs in the optimal noise enhanced solution for case (i).Define Theorem 2.If Obviously, when In order to determine the exact values of The focus of this subsection is to determine the exact values of the parameters in the optimal noise enhanced solution for case (ii).Define Theorem 3.If According to Theorem 3, when In addition, in order to determine the exact values of According to the analyses in If The corresponding false-alarm and detection probabilities are calculated asFrom the perspective of Bayesian criterion, the noise modified Bayes risk can be expressed in the form of a false-alarm and detection problem such thatWhen The minimization of Bayes risk for a variable detector has also been discussed in . CompareIn this section, numerical detection examples are given to verify the theoretical conclusions presented in the previous sections. A binary hypothesis-testing problem is given byIn this subsection, suppose that The corresponding decision function isWhen we add an additive noise It is obvious from (27) that the detection performance obtained by setting the threshold as From the definition of function  γ. From , we haveThen the pdf of the optimal additive noise corresponding to case (i) and case (ii) for the detector given in (26) can be expressed asAs plotted in Given From In this example, suppose thatIt is obvious that Through a series of analyses and calculations, it is true that On the other hand, The minimum achievable false-alarm probabilities for As illustrated in Remarkably, the minimum false-alarm probability obtained in the “NRD” case is always superior to the original false-alarm probability for any If the randomization between different decision thresholds is not allowed, the detection probability can be increased by adding additive noise when The probabilities of false-alarm and detection for different According to the analyses above, the original From After some calculations, we know that under the constraints that Naturally, case (iii) can also be achieved by randomization of the noise enhanced solution for case (i) and the new solution for case (ii) with the probabilities As shown in Case (iii) can also be achieved by the randomization of the noise enhanced solutions for case (i) and case (ii) with the probabilities As analyzed in 5.1, if the structure of a detector does not change with the decision thresholds, the optimal noise enhanced detection performances for different thresholds are the same, which can be achieved by adding the corresponding optimal noise. In such case, no improvement can be obtained by allowing randomization between different decision thresholds. On the other hand, if different thresholds correspond to different structures as shown in (39) and (40), randomization between different decision thresholds can introduce new noise enhanced solutions to improve the detection performance further under certain conditions.In this study, a noise enhanced binary hypothesis-testing problem for a variable detector was investigated. Specifically, a noise enhanced model that can increase the detection probability and decrease the false-alarm probability simultaneously was formulated for a variable detector. In order to solve the model, three alternative cases were considered, i.e., cases (i), (ii), and (iii). First, the minimization of the false-alarm probability was achieved without decreasing the detection probability in case (i). For the case where the randomization between different detectors is allowed, the optimal noise enhanced solution of case (i) was proven as a randomization of at most two detectors and additive noise pairs. Especially, no improvement could be introduced by allowing the randomization between different detectors under certain conditions. Furthermore, the maximum noise enhanced detection probability was considered in case (ii) without increasing the false-alarm probability, and the corresponding optimal noise enhanced solution was explored regardless of whether the randomization of detectors is allowed. In addition, case (iii) was achieved by the randomization between two optimal noise enhanced solutions of cases (i) and (ii) with the corresponding weights. Remarkably, numerous solutions for increasing the detection probability and decreasing the false-alarm probability simultaneously are provided by adjusting the weights.Moreover, the minimization of the Bayes risk based on the noise enhanced model was discussed for the variable detector. The Bayes risk obtained in case (iii) is between that obtained in case (i) and case (ii). Obviously, the noise enhanced Bayes risks obtained in the three cases are smaller than the original one without additive noise. It is significant to investigate case (i) and case (iii), especially for the situation where the result of case (ii) is not ideal or even not exists. Such as shown in As a future work, the noise enhanced detection problem can be researched according to the maximum likelihood (ML) or the maximum a posteriori probability (MAP) criterion. Also, the theoretical results can be extended to the Rao test, which is a simpler alternative to the generalized likelihood ratio test (GLRT) , and app


Oxford Nanopore sequencing is an important sequencing technology, which reads the nucleotide sequence by detecting the electrical current signal changes when DNA molecule is forced to pass through a biological nanopore. The research on signal simulation of nanopore sequencing is highly desirable for method developments of nanopore sequencing applications. To improve the simulation accuracy, we propose a novel signal simulation method based on Bi-directional Gated Recurrent Units (BiGRU). In this method, the signal processing model based on BiGRU is built to replace the traditional low-pass filter to post-process the ground-truth signal calculated by the input nucleotide sequence and nanopore sequencing pore model. Gaussian noise is then added to the filtered signal to generate the final simulated signal. This method can accurately model the relation between ground-truth signal and real-world sequencing signal through experimental sequencing data. The simulation results reveal that the proposed method utilizing the powerful learning ability of the neural network can generate the simulated signal that is closer to the real-world sequencing signal in the time and frequency domains than the existing simulation method. Nanopore-based sequencing technology is a new generation of sequencing technology, which has been rapidly developed in recent years ,2. It caThe simulation methods of nanopore sequencing data can be divided into two categories: nanopore sequencing read simulation and nanopore sequencing signal simulation. ReadSim, SiLiCO, and NanoSim  are sequhttps://github.com/zpllx/NanosigSim). BiGRU is a type of artificial neural network, which has been widely used in the field of basecalling to model the sequencing raw signal data. Specifically, by building a signal processing model based on BiGRU instead of using a low-pass filter, the ground-truth signal calculated from the input nucleotide sequence and pore model can be filtered more accurately, and then the final simulated signals can be generated by adding Gaussian noise to the filtered signals. The proposed signal processing model has a novel architecture that couples a three-layer BiGRU and a fully connected layer. This enables it to model the relationship between ground-truth signals and real-world sequencing signals accurately. Utilizing BiGRU neural network to learn the time-frequency characteristics of real-world sequencing signals from empirical data, the proposed method shows higher accuracy in the simulation of nanopore signals than DeepSimulator on four biological sequencing samples. The experimental data also verify the accuracy of the proposed signal simulation method regarding the time and frequency domains.To further improve the simulation accuracy of Oxford Nanopore sequencing signals, we propose a signal simulation method based on Bi-directional Gated Recurrent Units (BiGRU). For the convenience of description, we named the proposed method NanosigSim , and the test dataset consists of 4467 Acinetobacter pittii reads, 15,178 Klebsiella pneumoniae reads, 16,742 Serratia marcescens reads, and 11,047 Staphylococcus aureus reads (https://bridges.monash.edu/articles/Raw_fast5s/7676174).We have used the existing datasets sequenced on MinION R9.4 flowcells to train and evaluate the performance of the proposed signal simulation method . SpecifiWe use DeepSimulator1.5 and the proposed BiGRU-based simulation method to generate the simulated signals corresponding to each real sequencing signal in the test dataset and compare the two simulated signals from multiple aspects. For DeepSimulator, the cutoff frequency was set to 950 Hz, the mean value of the Gaussian noise was set to 0 pA, and the standard deviation of the Gaussian noise was set to 2.0. The standard deviation of the Gaussian noise in our simulation method was also set to 2.0 for fair comparison. The noise level of DeepSimulator and our method was the same. In the two signal simulation methods, the added Gaussian noise was random noise not correlated in time. Firstly, the two different filtered signal waveforms, which are output by the signal processing model based on BiGRU and the low-pass filter respectively, were compared. Then we employed dynamic time warping (DTW) algorithm  and contTo verify whether the proposed BiGRU-based signal processing model can efficiently and accurately filter out the useless high-frequency components embedded in the ground-truth signal, the low-pass filter of the DeepSimulator’s signal generator module and the proposed signal processing model were used to process the ground-truth signal independently. The comparison results of the filtered signal waveforms are shown in The final simulated signals can be generated by adding Gaussian noise to the filtered signals. As shown in In addition to comparing the DTW distance between the simulated signals and the real sequencing signals, we also used CWT algorithm to perform time–frequency analysis on the three signals. The CWT spectrum of each signal is depicted in An important indicator for evaluating the quality of simulated signals is to compare whether the simulated reads generated by the simulated signals have similar error characteristics to the real sequencing reads. We used Guppy, the newest official basecaller, to basecall the two simulated signals and the real sequencing signals, respectively. Then, edlib software  was adopTo compare the error characteristics of the two simulated reads and the four real reads in detail, we further calculated the proportions of the three types of errors in the six reads. As shown in For the Klebsiella pneumoniae sample, the two simulated reads generated by DeepSimulator and NanosigSim are both quite different from the real sequencing reads. To further explore the scalability of NanosigSim, we randomly selected 4000 sequencing reads from the Klebsiella pneumoniae sample to train the BiGRU-based signal processing model, and the remaining sequencing reads were used as the test dataset to verify the performance of the custom model for Klebsiella pneumoniae. As shown in In this paper, we propose a novel signal simulation method of nanopore sequencing based on BiGRU. To improve the simulation accuracy of existing methods, we build a signal processing model based on BiGRU to process the ground-truth signal calculated by the input nucleotide sequence and nanopore sequencing pore model. Compared with low-pass filter, the proposed model can effectively filter out unwanted high-frequency components embedded in the ground-truth signal and retain the useful high-frequency components by using the efficient self-learning ability of the BiGRU neural network. The simulation results show that the proposed method can provide high-quality simulated data for the methods and tools developed for the applications of nanopore sequencing. Although the proposed method can effectively improve the simulation accuracy, it inevitably requires more computational resources. In the future, further structure optimization of the proposed signal processing model is required to reduce the complexity of the signal simulation method.


The effect of entropy at low noises is investigated in five-strategy logit-rule-driven spatial evolutionary potential games exhibiting two-fold or three-fold degenerate ground states. The non-zero elements of the payoff matrix define two subsystems which are equivalent to an Ising or a three-state Potts model depending on whether the players are constrained to use only the first two or the last three strategies. Due to the equivalence of these models to spin systems, we can use the concepts and methods of statistical physics when studying the phase transitions. We argue that the greater entropy content of the Ising phase plays an important role in its stabilization when the magnitude of the Potts component is equal to or slightly greater than the strength of the Ising component. If the noise is increased in these systems, then the presence of the higher entropy state can cause a kind of social dilemma in which the players’ average income is reduced in the stable Ising phase following a first-order phase transition. The large number of payoff parameters causes difficulties in systematic investigations and in the classification of possible behaviors as well. These difficulties can be reduced by recognizing that two games are equivalent if their payoff matrices can be transformed into each other by exchanging some strategy labels.Multi-player evolutionary games are simple models that can be employed to study a wide range of systems across several diverse disciplines ,6,7,8,9.f matrix  if the pn strategies, generalized coordination games are composed of two-strategy coordination subgames between all possible strategy pairs [The linear decomposition of payoff matrices ,12,13,14volution ,25,26,27gy pairs . When thgy pairs ,29,30, sgy pairs  or the Agy pairs ) that exgy pairs ,34,35. ISo far, we have tacitly considered cases where the maximal payoff is unique or degenerate in such a way that equivalent phases possess similar features and symmetries at low noises. What happens when the latter conditions are not satisfied? Which of the possible equilibria will be preferred? For equal average payoffs in the ordered strategy arrangements, it will be shown that these systems select one of those ordered phases that have higher entropy. Furthermore, entropy can have a similar role even when the payoffs of the ordered phases are tuned away from being equal.In this paper we investigate the stabilization effects of entropy in a five-strategy coordination game where the subgames of the first two and the last three strategies are identical to a ferromagnetic Ising and a three-state Potts model, respectively. It is also assumed that the players receive zero income if they choose strategies belonging to different subgames. Using Monte Carlo simulations and cluster variation methods, we show that one of the “Ising phases” is stabilized by its higher entropy at low noises if the payoffs are identical in all five ordered strategy arrangements. Furthermore, this system undergoes an Ising-type phase transition when the noise is increased. Evidently, any small extra payoff in the “Potts phase” can stabilize it in the low noise limit. In the rest of the paper we describe the model in detail and quantitatively study its phase transitions for several payoff values.n available pure strategies to play in all four games. If we represent these strategies with the n Cartesian unit vectors of an n-dimensional Euclidean space, then the income of the player at site x can be calculated asx, the summation runs over this player’s nearest neighbor sites  and last three strategies  are an Ising and a three-state Potts model, respectively. Opposing players whose strategies belong to different subgames both receive zero payoff. We note again that we use pure coordination-type games for which the sum of payoffs is zero in each row and column of the payoff matrix. Consequently, the elements of our Potts payoff matrix are shifted by a uniform irrelevant constant in comparison with the standard diagonal Potts payoff matrix ,37. If th of α/2 . Our anaThe present game is a potential game because its payoff matrix is symmetric  affecting the decision-making abilities of players. This rule exponentially favors higher payoffs, provided that the opponents do not modify their strategies [If the evolution of strategy profile rategies ,29,30,35The main advantage of the application of the logit rule is that the repetition of the above elementary strategy changes will drive potential games into the Boltzmann distribution ,34 when ith strategy (K) the system evolves into a disordered state in which strategies are chosen by almost equal shares of the population, or more precisely, all five strategy frequencies approach K goes to infinity. On the other hand, strategy choices are homogenized in the low-noise limit, and we find the system in one of the competing ordered phases. When The macroscopic behavior of this system can be quantified by the average proportions i at any site. Obviously, these probabilities are normalized, that is, i and strategy j occupying neighboring sites. Both methods evaluate independent equilibrium configuration probabilities as extremal points of a thermodynamic potential Mean-field and pair approximations belong to the family of cluster variation methods ,38,39,40On a square lattice, We used different methods to find the extremal points of the approximated thermodynamic potentials. In the mean-field case a direct numerical approach was employed: At extremal points the partial derivatives of the free energy with respect to one-site probabilities should vanish, that is,We have to keep in mind the nonnegativity and normalization properties of probabilities as well. These equations can be solved numerically. Our results are presented in We also employed Monte Carlo simulations  to numerThe competing Ising and Potts phases were studied using ordered initial states (homogeneously prepared Ising or Potts states) due to the apparent metastability of the ordered strategy arrangements. This way we could reduce the transient time necessary for the slow domain growing process that forms ordered phases if the system is started from a random initial state. However, the application of the latter, slower method was unavoidable when we wished to determine the critical temperature of the first-order transition between the Potts and Ising phases, which turned out to be To illustrate the domain growing process, we have performed some time-consuming simulations starting from random initial strategy distributions near the transition point. ture see a the Potture see b, one ofFirst, we sketch the general features that are already captured by mean-field approximation, and then move on to highlight finer details revealed by pair approximation and Monte Carlo simulations. From the Monte Carlo results, we extract the critical exponents of the system’s order-disorder phase transitions. Finally, we argue that the first-order transitions between the ordered phases are driven by entropy.Equation , stable K is increased. The phase transition between the Potts and the Ising phases is of the first order and it is driven by the higher entropy of the Ising phases.Evidently, the increase of 1.05 see b,c, and In agreement with expectations, Mean-field approaches neglect the correlations between (fixed) neighbors, but their predictions may be applied to well-mixed populations. The common feature of the present games is that the average potential is zero in the disordered phase at the level of the mean-field approximation. Moreover, in the Ising (Potts) phase the Potts (Ising) strategies do not contribute to the average potential. Pair approximation, however, takes into consideration the relevant nearest neighbor correlations and gives more adequate predictions, as shown by comparisons with Monte Carlo data in The predictions of mean-field theory are justified by both pair approximation and Monte Carlo simulations at low temperatures that typically cause point defects in the ordered phases. The latter methods, however, predict a differently structured disordered phase where Ising strategies are chosen slightly more often than Potts strategies due to the correlations favoring the choice of identical neighboring strategies. It is worth mentioning that in the analytic continuation of the disordered solutions the equal frequencies of Potts strategies tend to zero as In general, cluster variation methods overestimate the critical temperatures of continuous order-disorder transitions but give a more accurate approximation of first-order transition points between ordered phases. This expectation is confirmed by comparing the plots of In order to characterize the continuous transitions, we have investigated the critical algebraic behaviors hidden in the quantities  and 1.1 ,44. All  Onsager  for the  Onsager ,16 for aWhy is the Ising state still preferred near the order-disorder transition point when n-strategy elementary coordination game is an extension of the original coordination game [n is greater than the threshold The stabilization effect of entropy can also be observed in elementary coordination games ,45 that ion game ,46,47,48d nth=27 , then thd nth=27 ,50,51, an is increased, critical temperatures should become lower because of the faster increase in the entropy content of the disordered phase.If decoupling of independent game components holds at the mean-field level, then the above-described results should give us an idea about what happens in games made up of an Ising and an n−-strategyIn this paper, we have studied a five-strategy evolutionary potential game that pits an Ising-type and a three-state Potts-type coordination game against each other. This interaction combines the two games as independent subgames of the first two and last three strategies, respectively. The application of the logit strategy update rule has allowed us to utilize various well-known concepts and methods of statistical physics. The systematic investigation of the spatial system is restricted to a model where the players are located at the sites of a square lattice and repeatedly play the five-strategy game against their nearest neighbors. Due to this choice, the observed properties of the stationary states can be compared with universal features characteristic of the original Ising and Potts models.The game is defined by a single parameter Independently of the preferred ordered phases at low noises, the system undergoes a continuous  order-disorder phase transition. The quantitative analysis of the order parameters indicates that in most of the cases these continuous transitions preserve the universal behavior of the corresponding (Ising or Potts) model. Additionally, we have observed non-universal behavior in a narrow range of parameters where the interplay between these two different transitions seems to be relevant. It is worth mentioning that similar non-universal critical phase transitions were also reported in several versions of the Ashkin-Teller model ,52,53,54We have found that the first-order transition from the Potts state to the Ising state is accompanied by a loss of average income for


Electrocorticography (ECoG) has been utilized in many epilepsy cases however, the use of this technique for evaluating electrophysiological changes within tumoral zones is spare. Nonetheless, epileptic activities seem to arise from the neocortex surrounding the gliomas suggesting a link between epileptogenesis and glioma cell infiltration in the peritumoral area. The purpose of this study was to implement novel scale-free measures to assess how cortical physiology is altered by the presence of an invasive brain tumor.Twelve patients undergoing an awake craniotomy for resection of a supratentorial glioma were included. ECoG data over the main tumor and the exposed surroundings was acquired intra-operatively just prior to tumor resection. Six of the patients presented with seizures and had data acquired both in the awake and anesthetic state. The corresponding anatomical location of each electrode in relation to the macroscopically-detectable tumor was recorded using the neuronavigation system based on structural anatomical images obtained pre-operatively. The electrodes were classified into tumoral, healthy or peritumoral based on the macroscopically detectable tumoral tissue from the pre-operative structural MRI.The electrodes overlying the tumoral tissue revealed higher power law exponent (PLE) values across tumoral area compared to the surrounding tissues. The difference between the awake and anesthetic states was significant in the tumoral and healthy tissue (p < 0.05) but not in the peritumoral tissue. The absence of a significant PLE reduction in the peritumoral tissue from the anesthetic to the awake state could be considered as an index of the presence or absence of infiltration of tumor cells into the peritumoral tissue.The current study portrays for the first time distinct power law exponent features in the tumoral tissue, which could provide a potential novel electrophysiological marker in the future. The distinct features seen in the peritumoral tissue of gliomas seem to indicate the area where both the onset of epileptiform activity and the tumor infiltration take place. Although the main presenting symptom of low-grade gliomas are seizures, the pathophysiological mechanisms and related structural-functional abnormalities underlying the epileptogenesis in patients with these tumors remain unclear –3.The recording of electrical changes in the brain by electrodes placed directly on the cerebral cortex, electrocorticography (ECoG), has previously been used for delineation of tumors when the non-invasive imaging techniques were not yet available. Similarly, the electroencephalogram (EEG) had localizing value in cases of brain tumor and epilepsy and the occurrence of delta waves was thought to be a better guide than areas of electrical silence . Subcortin vivo electrophysiological investigation of tumoral tissue in patients undergoing awake craniotomy for tumor resection. Subsequently, we discuss our findings in the context of the available literature.Besides delineating the eloquent areas that need to be preserved during the surgical resection, ECoG use for assessment of electrophysiological changes occurring in the setting of glioma is currently a relatively non-explored domain. It is postulated that the peritumoral neocortex around gliomas represents a pivotal structure both for the genesis of glioma related epilepsy (GRE) and for infiltration by glioma cells, which has significant implications in terms of the treatment recommended to the patients and their oncological prognosis , 9–12. ITwelve patients with a newly diagnosed intracranial glioma, out of which six presenting with multiple seizures, were enrolled after providing informed consent. Patients were eligible for inclusion if they were aged ≥18 years, presented with a supratentorial lesion suggestive of a glioma, and had an indication for surgery as confirmed by an experienced neurosurgery team . Of the patients sequentially admitted for treatment, we excluded patients not eligible for awake surgery, patients with other brain pathologies and those who had undergone previous cranial irradiation or chemotherapy and/or for whom electrophysiological recordings could not be obtained.The patients’ demographics including the presenting symptoms are presented in The ECoG grid was placed over the exposed brain surface as indicated by the needs of the patient. ECoG signals were recorded with a 20, 32 or 48-channel grid of 4.0 mm diameter electrodes with 2.3mm exposure  connected to a BP amplifier . The reference electrode was placed on the contralateral earlobe to avoid any potential increase in coherence due to the reference electrode location.5 min of recording in the awake state were acquired prior to the tumor resection. For the six patients presenting with seizures, the ECoG signal was also acquired in the anesthetic state. The signals were sampled at 1,000 Hz with a hardware filter between 0.01 and 1,000 Hz. The ECoG electrodes for each patient were classified in TT , HT  and PT  according to the radiological defined boundaries as per the conventional structural MRI.http://afni.nimh.nih.gov/afni). Intra-operative photograph was superposed on the segmented cortical surface in order to confirm the electrode location in respect to the tumor with the help of visible sulcal patterns on the cortical surface obtained from the pre-op MRI . Image guided localization of the electrode location were performed as soon as possible after the craniotomy was completed in order to minimize potential brain shift. The electrode location was recorded using the intra-operative Medtronic neuronavigation system Figure 2ECoG signals were manually inspected for quality using EEGLAB for full-length data visualization. Data for electrodes, or channels, were deemed to be of untrustworthy quality if obvious artifacts were present in at least ¼ of the full length of channel data. The entire set of data for these bad channels was removed from further analysis. We calculated the power law exponent (PLE) using an in-house MATLAB script as per previous publications –21 usingWhen assessing scale-free measures, we noticed that the electrodes over the tumoral tissue were found to have a higher PLE value than the electrodes over the macroscopically healthy tissue for the twelve patients Figure 4For the six patients presenting with multiple seizures for which ECoG recordings were obtained in the two anesthetic states , we observed that the electrodes overlying the peritumoral tissue displayed smaller changes between the anesthesia and awake state than the changes observed across the adjacent tissues, respectively over the TT and macroscopically HT Figure 5In addition, given that the glial tumor can displace otherwise healthy non-glioma neural tissue and that this mass effect can cause electrophysiological changes in the intraoperative ECoG, we performed a subgroup analysis according to the presence of mass effect. Two subjects included in this study presented with intracranial mass effect, as identified on the preoperative MRI. The PLE of these subjects  was not found to be different from the PLE of the subjects presenting without a mass effect . Based on these results, the intracranial mass effect was not considered to influence the PLE in this study. It should be noted however, that only two subjects presented with glioma-related mass effect. Also, given the limited time available for intra-operative recordings during the awake procedures, we, unfortunately, could not perform post-resection ECoG recordings.Furthermore, the distance between tumor and the brain surface is also important and to further explore this factor we looked at the influence of the cortical/subcortical tumor location. To study the effect that glioma’s proximity to the cortex has on the PLE results, we compared the PLE of patients with gliomas predominantly involving the cortical surface with the PLE of patients having a more extensive subcortical involvement. The PLE of patients with predominant cortical involvement  was lower than the PLE of patients with both cortical and subcortical glioma . The PLE of patients with both cortical and subcortical glioma was less variant than the PLE of patients with only cortical glioma. This suggests that further analysis should focus on studying the subcortical glioma involvement, and a continuous representation of glioma depth from the cortical surface could be incorporated in the future to study the effect that glioma depth has on the PLE of the surrounding neural tissue.Lastly, the importance of the glioma’s molecular and histological characteristics cannot be overemphasized given the known strong correlation with clinical outcomes. Although we have a small sample of subjects, in order to study the effect that this mechanism could have on the PLE results, all channels for each subject were labeled as low-grade (grade II) or high-grade (grade III & IV) which coincided with an IDH-mutant and respectively an IDH wild-type status. The PLE of low-grade glioma-associated channels  was not found to be different from the PLE of high-grade associated channels . From this analysis, it was concluded that there was not enough evidence to suggest that an unbalanced representation of glioma graded tissue in this study would bias the PLE results presented.In summary, ECoG recordings performed prior to glioma resection allowed us to investigate scale-free related changes in the resting state activity of tumor tissue and its reactivity to anesthesia. We observed an increased PLE in the anesthetic state for all types of tissues  and additionally, a different behavior of the PT when comparing PLE in the two anesthetic states . We thus noticed a lack of reaction to anesthesia in the peritumoral tissue and hypothesized that it is related to a different tissue type which maintain the already high PLE values.Extrapolating our group’s previous work in functional MRI, for the first time to our knowledge, the analysis of the signal from ECoG electrodes in this study concentrated on power law exponent. Power laws have been reported in many distinct scientific, geographical and social phenomena and have been associated with long memory behavior, self-similarity and fractal structures . Power lFurthermore, the tumor’s surrounding tissue showed specific features. A universal definition of the peritumoral area in glioma is lacking and its complex features are not yet elucidated , 31. NonThe origins and mechanisms of GRE are multifactorial  and the Electrocorticography plays an important role in the localization of epileptogenic foci and evaluation of the effects of microsurgical procedures intraoperatively. An accurate estimation of the epileptogenic cortex and its removal requires the estimation of spatial spread of ECoG. Recent studies estimating the spatial spread of ECoG suggested that although brains’ signals capture different features of the neural network, ECoG records a local signal (diameter of ∼3 mm) confirming the precision required for its use in determining the epileptogenic focus . FurtherIn tumoral epilepsy surgery, the first aim is to maximize the extent of tumor resection while minimizing postsurgical morbidity, in order to increase the median survival as well as to preserve the quality of life . Gross-tThe concept of individualized surgery for managing glial tumors is based on the goal of achieving a maximal tumor resection and seizure freedom without inducing new neurological deficits –64. We aIn summary, we present for the first time distinct power law exponent features in the tumoral and peritumoral tissue of gliomas that seems to represent a critical structure both for the onset and propagation of epileptiform activity and for infiltration by glioma cells. Although neuro-oncology and epilepsy surgery have traditionally been separate disciplines, we postulate that the analysis of the mechanisms underlying the tumor related epilepsy might equally yield significant insight into tumorigenesis. We expect that the contributions made from a multidisciplinary approach to unravel the complex pathways underlying the clinical features of gliomas related epilepsy will lead to meaningful progress for the patients affected by these devastating tumors.The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.The studies involving human participants were reviewed and approved by the Huashan Hospital ethics board . The patients/participants provided their written informed consent to participate in this study.Conception and design: GN, J-SW, JL, and DG. Acquisition of data: DG, JL, and J-SW. Analysis and interpretation of data: BL, DG, J-SW, NJ, JL, GN, ET, and AS. Drafting the article: DG. Critically revising the article: J-SW, DG, NJ, ET, AS, and GN. Reviewed submitted version of manuscript: all authors. Approved the final version of the manuscript on behalf of all authors: GN. Statistical analysis: BL, DG, and JL. Administrative/technical/material support: J-SW and GN. Study supervision: J-SW and GN. All authors contributed to the article and approved the submitted version.This study was supported by The National Key Technology R&D Program of China (grant no. 2014BAI04B05 [J-SW]), and Shanghai Shenkang Hospital Development Center (grant no. SHDC12018114 [J-SW]).The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Despite the advances in the field of brain computer interfaces (BCI), the use of the sole electroencephalography (EEG) signal to control walking rehabilitation devices is currently not viable in clinical settings, due to its unreliability. Hybrid interfaces (hHMIs) represent a very recent solution to enhance the performance of single-signal approaches. These are classification approaches that combine multiple human-machine interfaces, normally including at least one BCI with other biosignals, such as the electromyography (EMG). However, their use for the decoding of gait activity is still limited. In this work, we propose and evaluate a hybrid human-machine interface (hHMI) to decode walking phases of both legs from the Bayesian fusion of EEG and EMG signals. The proposed hHMI significantly outperforms its single-signal counterparts, by providing high and stable performance even when the reliability of the muscular activity is compromised temporarily  or permanently . Indeed, the hybrid approach shows a smooth degradation of classification performance after temporary EMG alteration, with more than 75% of accuracy at 30% of EMG amplitude, with respect to the EMG classifier whose performance decreases below 60% of accuracy. Moreover, the fusion of EEG and EMG information helps keeping a stable recognition rate of each gait phase of more than 80% independently on the permanent level of EMG degradation. From our study and findings from the literature, we suggest that the use of hybrid interfaces may be the key to enhance the usability of technologies restoring or assisting the locomotion on a wider population of patients in clinical applications and outside the laboratory environment. Neurological diseases after brain or spinal cord lesions are one of the major causes of locomotion impairments, leading to functional disability and loss of independence in daily-living activities for patients  is certainly one of the most used techniques to control rehabilitation and assistive devices, particularly in robotic applications. EMG has been already successfully used to decode gait activity  fusing the motion information coming from the brain signals with the information coming from the natural muscular activation with the purpose of decoding walking activity. The aim of the fusion method proposed in this study is to enhance the reliability of the human-machine interface by integrating the outputs of an EEG and an EMG classifiers using principles of Bayesian inference  or input devices   and offset  of right and left legs. Finally, their predictions are integrated at the decision-making level using a Bayesian belief fusion method, based on the uncertainty of each classifier in predicting that the subject is in a certain gait phase. We believe that the proposed solution may guarantee more stable performance of the human-machine interface controlling the walking restoration device, by adapting the fusion parameters to the changes of input signals reliability in clinical and daily-living applications.www.seniam.org). Swing and stance events were identified from the signals acquired with four foot-switches positioned under the shoes of each subject, two under the heel and two under the toes. Data from the foot-switches were wirelessly acquired through the EMG system with a sampling frequency of 1,000 Hz. EEG, EMG and foot-switches data were all resampled to 1,024 Hz. This dataset was previously published and was acquired within the MoBI framework presented in Artoni et al.  while walking. During each session, the subject was asked to walk on a treadmill at 2.5 and 3.5 km/h, for 10 min for each walking speed, thus 20 min overall. EEG data were recorded with a signal pre-amplifying active electrode cap  and a 64-channel EEG amplifier  with a sampling rate of 2,048 Hz. EMG data were simultaneously recorded with a wireless EMG system (BTS Free EMG 300) at a sampling rate of 1,000 Hz. EMG electrodes were placed on Tibialis Anterior (TA), Vastus Medialis (VM) and Biceps Femoris (BF) of each leg, according to SENIAM guidelines , a validation set  and a test set . Networks were trained with data of the training set only, while the parameters for the Bayesian belief fusion were obtained from the validation set. Once the Bayesian fusion parameters have been estimated, the networks were re-trained with the ensemble of training and validation set, in order to maximize the data available for classifier's learning. Finally, the overall performance were evaluated on the data of the test set. All the processing procedures applied to the test set are compatible with an online usage of the interface. A representation of the proposed hHMI is schematically shown in Generally, EEG data are contaminated by many external sources  and irregular artifacts  were removed through Artifacts Subspace Reconstruction (ASR)  were rejected by backprojecting the EEG signals to the original domain using only the components related to brain activity  to enhance their reliability  = 100 Hz. A Common Average Reference (CAR) filter is applied to the chunk of EEG data inside each window and the δ-θ band (1-8 Hz) was extracted with a 4th-order zero-lag Butterworth filter. This frequency range was selected since it provided the best performance in a previous study  only on the EEG data used for the estimation of the independent components  of each window. The EMG preprocessing is schematically shown in Gait decoding classifiers were implemented as a deep neural network with recurrent layers, namely Recurrent Neural Network (RNN), whose output is a non-linear function of the current input  and its outputs at the previous time steps  against the double-stance phase (STANCE). On the other hand, the EMG network was trained to solve a 3-class classification problem, detecting the swing phase of the right leg (RIGHT), of the left leg (LEFT), and the double-stance phase (STANCE). An EEG and an EMG network were trained for each subject independently with the data from the training set only. The networks' architectures , shown in In this study, two separated LSTM networks, running in parallel, were implemented to decode gait events from EEG and EMG, respectively, and their predictions are subsequently integrated (see section 2.4). Each network is provided with a sequence input layer with size equal to the number of channels used for classification. Then, channels' signals are processed by one or more LSTM layers to extract information from each time series that is correlated with the gait events to be classified. Channels' information is integrated in the fully-connected layer, with size equal to the number of classes to be identified, and the posterior probability of each class is provided in output by the softmax layer see . The EEGThe reliability of a classifier can be comprehensively described by means of confusion matrix, measuring the errors distribution of each network over the classes:c1, …, cM) the samples truly belong, as defined by the foot-switch signals, and each column to the class which the samples are assigned to. Thus, i − th class that were predicted as the j − th class by the k − th classifier. Given K classifiers predicting among M classes, the general belief measure of correct assignment can be computed for each class i by combining the estimation ek of each classifier as Ruta and Gabrys  is the prior probability of class i and P(ci|ek = cj) is the probability that the true class is i when the k − th classifier outputs class j. In our case, K = 2 since we have one EEG network and one EMG network. However, the two networks do not provide the same number of classes M, as explained in section 2.3. Thus, we adapted (2) to our scenario as follow:where cEEGi and in cEEGj the SWING class is repeated twice since the EEG network provides the same contribution for either the right swing (RIGHT) and left swing (LEFT) events classified by the EMG network. In short, the EEG network detects if the subject is taking a step forward, regardless of which leg is moving, while only the EMG network is in charge of discriminating between the two legs. The class ci with the highest belief Bel(ci) is chosen as the final decision of the classification framework. A schematic representation of the Bayesian belief fusion applied in this work is shown in STANCE class. Nevertheless, the same considerations could be applied when the two networks predict different classes. The conditional probabilities P(cki|ek = ckj) of each classifier were obtained on the validation set and the performance of the fusion method was evaluated on the test set. Since the walking speed is not known to the classifier, we used the same prior probability for all the classes.In s02 and s04 were excluded from the analysis since their EEG networks did not achieve performance above the level of significance . Since the performance and analysis of the EEG network on this dataset, including subjects s02 and s04, have been previously published . As shown in post-hoc, p < 0.05). post-hoc, p < 0.05) and 90%  of attenuation. In particular, the stance detection rate was always above 80% even at the highest levels of EMG alteration. p > 0.05) were found between the performance achieved when the same attenuation is applied to both set (white bars), or when the levels of attenuation differ of +10% (dark gray bars) or −10% (light gray bars).In order to evaluate the performance of the proposed hybrid approach, that integrates the information from the brain signals to support the compromised muscular activity, in the experiments we evaluated the performance of the single modality EMG network alone with respect to the performance obtained by the fusion method in detecting right swing, left swing and double-stance phases. p > 0.05), with more than 80% of correctly classified samples on average in all the conditions for both the EMG network and the hybrid approach. Differently, a significant improvement in the stance recognition rate became apparent with the fusion of EEG and EMG signals , with the proposed hHMI keeping stable performance above 80%, regardless of the SNR level.In this study, we propose a hybrid approach based on the fusion of brain and muscle activity exploiting the principles of Bayesian inference with the purpose of decoding gait events. In particular, we show the benefit of coupling a BCI and a myoelectric interface in order to cope with impairments affecting the reliability of muscular signals, either temporary  or permanent . Indeed, from our experiments, the proposed hHMI achieved significantly better results than the single-signal EMG approach, particularly at higher levels of muscular signals degradation. Nevertheless, it is worth noting that the introduction of the EEG network did not affect the performance of the interface even if no alteration of the EMG signal was applied. Indeed, the EEG network achieved almost 90% of accuracy in discriminating between swing and stance phases, independently from the experimental condition, similarly to what has been obtained in Tortora et al. .The detection of walking patterns with the EEG signal alone is, not surprisingly, less accurate with respect to the results obtained using the EMG signal. Likely, while the location of the EMG channels provides a clear indication of which leg the subject intends to move, the localization of brain activity related to each leg has been always very challenging in BCI, due to their representation over the sensorimotor cortex very close to the deep interhemispheric fissure .The physiological origin of muscle fatigue has been investigated in many studies Enoka,  and bothIn rehabilitation applications, it is expected for the motor functions of the patient to change as the therapy sessions progress, and even inside the same session, hopefully by progressively reducing the movement impairment and making the muscular activity more reliable. In this sense, the analysis of the performance of the hybrid classifier, contrary to the EMG only classifier, and the analysis of the parameters of the Bayesian fusion could provide a good quantitative assessment of motor functions improvement over time, guiding the therapy and the control strategy of the rehabilitation device.While many papers may be found in literature on gait phase classification from EMG ] only based on the performance estimated on the validation set , without considering the confidence the classifier had on that decision . As an example, considering a binary classification problem, the posterior probability  can be considered as a measure of the confidence of the classifier. Thus, future works can investigate the implementation of a probabilistic fusion method to integrate the probability distribution of each classifier over the classes, so as to allow the decision making process to take into consideration both the uncertainty during validation and during online usage. Another limitation of this study is related to the dataset used for the evaluation of the hHMI performance, containing only data from healthy subjects walking over a treadmill, that will be accounted for in future studies by acquiring and testing this approach on real end users . Indeed, muscle weakness following paresis is a complicated phenomenon still under study, with many contributors, like loss of descending corticospinal pathway activation to motoneurons , improving the results we previously achieved with an EEG-based LSTM network (Tortora et al., tortora@dei.unipd.it.The data analyzed in this study is subject to the following licenses/restrictions: data are available from corresponding author upon reasonable request. Requests to access these datasets should be directed to Stefano Tortora, Ethical review and approval was not required for the study on human participants in accordance with the local legislation and institutional requirements. The patients/participants provided their written informed consent to participate in this study.ST and FA analyzed the data and wrote the manuscript. CC and FA collected the data. SM, LT, and EM contributed to the interpretation of the results and writing of the manuscript. All authors discussed the results, reviewed and approved the final version of the manuscript.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


The problem of uncertainty quantification (UQ) for multi-sensor data is one of the main concerns in structural health monitoring (SHM). One important task is multivariate joint probability density function (PDF) modelling. Copula-based statistical inference has attracted significant attention due to the fact that it decouples inferences on the univariate marginal PDF of each random variable and the statistical dependence structure  among the random variables. This paper proposes the Copula-UQ, composing multivariate joint PDF modelling, inference on model class selection and parameter identification, and probabilistic prediction using incomplete information, for multi-sensor data measured from a SHM system. Multivariate joint PDF is modeled based on the univariate marginal PDFs and the copula. Inference is made by combing the idea of the inference functions for margins and the maximum likelihood estimate. Prediction on the PDF of the target variable, using the complete  or incomplete information (due to missing data caused by sensor fault issue) of the predictor variable, are made based on the multivariate joint PDF. One example using simulated data and one example using temperature data of a multi-sensor of a monitored bridge are presented to illustrate the capability of the Copula-UQ in joint PDF modelling and target variable prediction. The problem of uncertainty quantification (UQ) for multi-sensor data has been one of the main concerns in nondestructive testing and structural health monitoring (SHM) over the years ,6,7,8,9.In recent years, copula-based statistical inference has attracted significant attention due to the fact that it decouples the inference on the univariate marginal PDF of each random variable and the statistics dependence structure  among the random variables. In the areas of the SHM and structural assessment, Zhang and Kim  investigAlthough the copula-based statistical inference has been widely applied, there are two limitations in previous works related to the SHM and structural assessment. The first limitation is insufficient types of probabilistic model candidates for univariate marginal PDF modelling. From the parametrization point of view, there are parametric models and nonparametric models for PDF modelling. The former type, assuming that sample data come from a distribution that has a fixed set of parameters, is suitable for data with regular statistical pattern; the latter type, being not specified a priori but being instead adaptively determined from data, is suitable for data with an irregular statistical pattern. In the SHM, it is well known that the statistical regularities of data from multiple sensors can be significantly different from each other. Thus, due to the complexity of real SHM data, only considering one type of probabilistic model in PDF modelling bounds the solution space for UQ, leading to incapability of capturing a statistical pattern of data. However, this important issue was not realized in previous works, so parametric models and nonparametric models were not considered simultaneously. For example, References ,20,21,23This paper proposes the copula-based UQ (Copula-UQ), composing multivariate joint PDF modelling, inference on model class selection and parameter identification, and probabilistic prediction using incomplete information, for multi-sensor data measured from a SHM system. The proposed Copula-UQ contains two stages. The first stage is the copula-based multivariate joint PDF modelling. It is based on the univariate marginal PDFs and the copula. The second stage is copula-based inference and prediction. Inference, including determination of optimal parameters and selection of optimal model classes, is made by combining the idea of the inference functions for margins (IFM) and the maximum likelihood estimate (MLE). Prediction on the PDF of the target variable, using the complete or incomplete information of the predictor variable, are made based on the copula-based multivariate joint PDF.The structure of this paper is outlined as follows. Let For the Parametric modeling methods assume that sample data come from a distribution that has a fixed set of parameters. In this paper, the parametric model class candidates include: Normal distribution, Lognormal distribution, Weibull distribution, Gamma distribution, Gumbel distribution and Uniform distribution.Nonparametric models differ from parametric models in that the model structure is not specified a priori but is instead determined from data. The term nonparametric is not meant to imply that such models completely lack parameters but that the number and nature of the parameters are flexible and not fixed in advance. In this paper, the kernel density estimation method is used. Based on the  adopted :(1)pxd|θLet Applying the probability integral transform to The copula of 2,⋯,UDmD :(8)Cu1m1That is Consider the joint CDF of Sklar’s theorem states that there exists a uch that :(10)Px1,Thus, the joint PDF, In this paper, the multivariate Gaussian copula and the associated copula density function are introduced as follows :(13)Cu1mThis stage is to make inference on ns (IFM) , Θ . For nosolution :(18)θ^dmThe optimal marginal PDFs  the necessary introduction of both parametric and nonparametric models for breaking through the first limitation , (2) the capability of the multivariate joint PDF modelling of the proposed Copula-UQ (as the true joint PDF is known) and (3) the performance of the proposed Equation (23) for prediction on the target variable given complete or incomplete information (due to missing data caused by a sensor fault issue). For the real SHM data example, the performance of the proposed Copula-UQ for prediction on the target variable under complete  or incomplete (fault sensors) information is further validated by considering the following two cases: (1) the test dataset is identical to the training dataset, and (2) the test dataset is different from the training dataset.This example applies the proposed Copula-UQ for multivariate joint PDF modelling and prediction of five-dimensional random variables, cted see . Then, tThus, the analytical form of the joint PDF of High correlation  . TherefoFor further validating the prediction capacity of the proposed Copula-UQ under data missing by sensor fault issue, a new computation is conducted as follows: (1) the monitored dataset was divided into the training dataset  and the test dataset (complement of the training dataset), (2) the marginal PDF along with the copula model was inferred based on the training dataset and (3) the prediction capacity of the trained copula model was validated based on the test dataset with or without data missing by sensor fault issue. ide (S1) . The preThis paper proposed the Copula-UQ for multivariate joint PDF modelling, inference on model class selection and parameter identification, and probabilistic prediction using incomplete information, and presented one example using simulated data and one example using temperature data of a multi-sensor of a monitored bridge. For inference on univariate marginal PDFs, the results show that, in general cases, the optimal univariate marginal PDFs of different dimensions are different, so the introduction of both parametric and nonparametric models is necessary because it provides a large solution space for uncertainty quantification. For prediction on the target variable using the complete  or incomplete information (due to missing data caused by a sensor fault issue) of the predictor variable, the proposed Copula-UQ is capable of obtaining the PDF of the target variable. The proposed methodology can be extended to tackle different multivariate joint PDF modelling problems in SHM with emphasizing the prediction purpose under incomplete information with a sensor fault issue. This important piece of information of the PDF of the target variable can be utilized for uncertainty propagation in further analysis.


The neural response is a noisy random process. The neural response to a sensory stimulus is completely equivalent to a list of spike times in the spike train. In previous studies, decreased neuronal response variability was observed in the cortex’s various areas during motor preparatory in reaching tasks. The reasons for the reduction in Neural Variability (NV) are unclear. It could be influenced by an increased firing rate, or it could result from the intrinsic characteristic of cells during the Reaction Time (RT).A neural response function with an underlying deterministic instantaneous firing rate signal and a random Poisson process spike generator was simulated in this research. Neural stimulation could help us understand the relationships between the complex data structures of cortical activities and their stability in detail during motor intention in arm-reaching tasks.Our measurements indicated a similar pattern of results to the cortex, a sharp reduction of the normalized variance of simulated spike trains across all trials. We also observed a reverse relationship between activity and normalized variance.The present study findings could be applied to neural engineering and brain-machine interfaces for controlling external devices, like the movement of a robot arm. The simulation of neural spike trains in the cerebral cortex was conducted during a reach-to-grasp task performed by a monkey.The definition of neural mechanisms and trial-to-trial variability in the execution of movements by the simulation model of spike trains as neural activity; they were measured by normalized variance as neural variability.The reduction of simulated neural variability, compared to similar behavior in the real variation of cortical activity after target onset or before the movement was observed.The brain activities are nonidentical in repeated trials during the motor preparation and hand movements. This variation of cortical neural responses could result from the internal and external properties of individual neurons or neural populations. The stimulus onset reduces neural cortical variability during the hand movement. If this reduction is coordinated to variability in the sensorimotor neural population, then the observational error movement should have represented low values. To understand the mechanism of reduction, the neural response has been simulated as a random Poisson process spike generator with or without noise. The simulation model demonstrated a decrease in variability of simulated neural responses with noise after the target onset; however, no changes of activity was observed in the absence of noise. The simulation of this mechanism provides a great opportunity for a better understanding of the neurophysiological substrate of the neurodegenerative and mental conditions. Applying neural variability in predicting hand movements control could support brain-machine interfaces for stroke patients’ paralysis and even neuro-feedback therapy. Controlling hand movement could be modeled in a closed-loop neural circuit with added noise and neural variability.1.Cortical cells receive inputs from many sources, and the distribution of post-synaptic potentials is Poisson-like . The braAfter designing a behavioral task and training the monkeys for a specific experiment, the neural data could be recorded. There are several methods for recording neural activity. For instance, the extracellular activity could be recorded with electrodes  using a Thomas Recording system.In the current study, the neural Poisson distribution time-locked to the target onset was simulated. Indeed, the struggle was the quantification of a relationship between the simulated neural Poisson spike process and the measured neural variability. There are several rate coding methods to measure the firing rate of single cells or population activity . The end2.Where ε=ε′=0.01 and k=0.1.The first and second-order moment statistics were applied to estimate the mean and variance parameters of neuronal responses. The NV, i.e. the time function of trial-to-trial variability, is defined as the variance-to-mean ratio of spike density function across trials. This method is proposed by some scholars  with theTo prevent the ratio of NV from becoming zero in the dominator in some cases, the constant value was added. Based on the assumption that neural activity is commonly generated as the Poisson process, the mean values of the firing activity and variance are identical; thus, NV is unity. This value of fraction could be changed depending on the internal and external states of the individual and population neurons.The neural activity of each neuron in contact with the neural network in various areas of the cortex exhibits chaotic patterns; it implies the intrinsic and extrinsic characteristics of that cell. Therefore, to detect a cell’s characteristics for repeated trials (in random responses to the same stimulus), firing rates were simulated  in two types of pure Poisson distribution of firing rates and pure Poisson activities by adding a random noise (noisy Poisson distribution). The aforementioned random noise contains random numbers that follow a Gaussian distribution. The pure Poisson distribution is a set of Poisson random numbers generating random numbers from the Poisson distribution with an average number of occurrences μ over a period. Poisson distribution is an event with a small probability of occurrence and a large number of independent trials taking place. The obtained NV has revealed different behaviors for each group. To use the Poisson spike generator in a simulation, a model neuron is required. The probability of spikes for a Poisson distribution is as Where is the average number of spikes per intervals, e is the Euler’s number, and the parameter is a Poisson random variable that takes non-negative discrete values 0, 1, 2, and so on.3.To determine the relation between NV and firing rates, the mean firing rate of simulated activity was firstly computed. Then, the normalized value of neural response was found through calculating the maximum value of the firing rate. Next, the normalized value of the amplitude of the neural response was divided into three levels, i.e. the maximum value of the response, (0.8%–1%) of maximum value, (0.5%–0.8%) of maximum, and <50%, and finally, the distribution value of NV related to each level was found.There was a reverse relationship between the firing rate and normalized variance. The neural variability was declined by an increased firing rate.Previous studies demonstrated the temporal evolution NV  significantly declined in the frontal areas after the target presentation or before the movement onset. To overcome unreliable statistical results across repeated trials, selected neurons and all directions were pooled. The neural signal and neural variability were averaged across all trials and directions. This strength of NV reduction in the frontal lobe could be a result of increased activity only within specific time windows that depends on the considered area. In our study, the simulated level of neural activity was changing during the time course of behavior . The obtThe simulation in The copy of the movement was compared with electrical stimulation (input of a closed-loop system). The neural variability method could be applied in the brain-machine interfaces to support neurodegenerative diseases, like optic ataxia in human and animal research studies, to obtain a better understanding of the performance of this closed-loop neural circuits in the cortex, a mutual connection between the motor and the parietal cortex.4.No single study was performed on simulated neural variability to characterize the role of the noise in the brain. These study findings suggested that the neural variability, as an internal-external state of noise in the brain, was a function of modulations of trial-to-trial firing rate variability on the single and population level in the parietal, dorsal premotor, and motor cortices. In real data, the across-trial variability of firing rates through the normalized variance was first measured. In all investigated frontal and parietal areas, after the target presentation, a significant decline of NV was observed.The neural variability is a disturbance or noise  in this system and sensory feedback (forward model) to compare the changing status (movement) and the desired motor plan (target position). The closed-loop system in the motor cortex controls the motor plan.To assess the dependency between the activity and the temporal or non-temporal pattern variability across repeated trials, spike trains designed through a random process were simulated; it generated pure Poisson distribution with no change of variability. By adding the noise to signal, a normalized variance was increased across contaminated trials. Moreover, the significant increment of the averaged noisy activity (by supposing time-locked to the stimulus onset) appeared a predominant impact on relative and absolute refractory periods of neuronal responses and a sharp decline of normalized variance on time patterns.5.The neural variability associated with the composition of a motor plan described the task events and the behavioral performance of the animal in a satisfactory way. Moreover, there was a strong correlation between the neural and movement variability where the NV could predict the temporal evolution of movement planning and behaviors in humans. However, the mechanisms underlying the temporal variability of the neural planning in the human motor cortex and its characteristic changes in mental disorders could be related to the neurophysiological substrate of the disorder. By improving this method, as a scanner of the neural firing rate in the cortex, other forms of bio-signals in the other areas of the brain might also have potentials as a predictor of the behavior. These results could be related to the neurophysiological substrate of the disorder. To study this active inhibition notion, active task involvement is necessary to imply an interactive task performance. By improving the therapy with the suggestions mentioned above, other forms of neuro-feedbacks might also have potentials as a treatment for Attention-Deficit Hyperactivity Disorder (ADHD).


The multiaxial fatigue critical plane method can be used to evaluate the extremely-low-cycle fatigue (ELCF) damage of beam-to-column welded joints in steel frameworks subjected to strong seismic activity. In this paper, fatigue damage models using structural detail parameters are studied. Firstly, the fatigue properties obtained from experiments are adopted to assess ELCF life for steel frameworks. In these experiments, two types of welded specimens, namely, plate butt weld (PB) and cruciform load-carrying groove weld (CLG), are designed according to the structural details of steel beam and box column joints, in which both structural details and welded factors are taken into account. Secondly, experiments are performed on three full-scale steel welded beam-to-column joints to determine the contribution of stress and/or strain to damage parameters. Finally, we introduce a modification of the most popular fatigue damage model of Fatemi and Socie (FS), modified by us in a previous study, for damage evaluation, and compare this with Shang and Wang (SW) in order to examine the applicability of the fatigue properties of PB and CLG. This study shows that the modified FS model using the fatigue properties of CLG can predict the crack initiation life and evaluate the damage of beam-to-column welded joints, and can be subsequently used for further investigation of the damage evolution law. Steel beam-to-column welded joints may undergo extremely-low-cycle fatigue (ELCF) during strong seismic activity ,2,3, whiWith the application of the micromechanical approach in fracture predictions of welded joints, researchers have paid attention to investigating the fatigue failure behavior ,25,26,27cie (FS) , which ccie (FS) ,33,34,35ang (SW) , but hasThis paper aims to propose a fatigue damage parameter for ELCF crack initiation life prediction and damage assessment for steel beam-to-column welded joints, in which the I-beam and box-column profile is considered in the joint. Firstly, ELCF tests are carried out by designing two types of welded specimens according to welded structural details for obtaining material parameters in multiaxial fatigue damage modeling. Secondly, reverse cyclic loading tests are performed on three full-scale welded joints to obtain the local stress and strain response of welded joints, and the experimental data are used to form the fatigue damage parameter. In addition, the FS model is modified from our previous study using the nonproportionality factor In this section, ELCF tests of welded structural details and reverse cyclic loading tests of I-beam to box-column welded joints are carried out to obtain the parameters in the multiaxial fatigue damage model.The material and fatigue parameters in the heat affected zone (HAZ) are usually much lower and worse than the corresponding properties of the base materials. Indeed, fatigue failure of welded steel structures may happen at the HAZ. However, according to the damage investigation of the Northridge earthquake in 1994, it was found that about 100 steel frameworks suffered from different degrees of damage at the beam-to-column joints. It was also shown that the cracks originating in the weld zone occupied comprised as much as 90% of the total number, while only about 10% originated from the base metal. There were obvious cracks initiating at the weld seam of the beam-to-column welded joints, and the cracks propagated to the members connected to the joints, causing the failure of the steel frame. It is difficult to determine the location of the HAZ and to take a sample for analysis. Therefore, we focus on the weld zone in this paper.As shown in These fatigue tests were conducted on an INSTRON1342 fatigue test machine, and controlled by uniaxial constant strain amplitude using sinusoidal wave forms, as shown in The experimental results were listed in  as shownThe fatigue properties of the welded structural details, which are correlated by the Manson–Coffin equation expressed by Equation (1), are listed in Three cruciform beam-to-column welded joints were designed in this experiment, in which the inflection point of the beam-to-column connection was selected. The column had box sections with a size of 300 × 300 × 16 × 16 (mm) and had a height of 2 m. The beam had an I-shaped cross-section with a size of 250 × 150 × 8 × 10 (mm) and a beam length of 1.35 m. The weld form between the beam flange and the column flange was a full penetration groove using ER50-6 gas shielded welding wire. Welding process quality inspection was performed using a digital ultrasonic flaw detector; the grade of the weld seam was I. The beam-to-column welded joint specimens and welded structural details are shown in The test device is shown in According to the numerical simulation results of ABAQUS finite element analysis, the cruciform I-beam to box-column welded joint is in yield state when the displacement is greater than 12 mm, which is the noted yield displacement. It can be seen from The bottom flange at the northwest of the second specimen, which is marked with NWB_2, was selected to analyze the strain response. This was validated by a test showing that the weld root on the beam flange plate of the welded connection is in a position where fatigue cracks are likely to initiate. From the experiment, it was observed that under cyclic loading with a constant displacement amplitude, the tiny visible crack in NWB_2 was initiated just after the 7th cycle, i.e., at the time period between the 7th and 8th cycle. Strain data are obtained by placing three strain rosettes at the three coordinate planes in the vicinity of the weld root. Three normal strains and three shear strains of the weld root are then calculated using the elastic mechanics theory. The extracted strain response time history curves of NWB_2, ranging from the beginning of the trial to crack initiation, are drawn in It can be seen from In this section, two fatigue critical plane methods are used. We outline damage analysis steps and fatigue damage parameters ranging from strain response on the critical plane to nonproportional factors, and cumulative damage criteria are introduced.(1)The obtained strain responses are used to determine the candidate material plane of the dangerous position based on the coordinate transformation. Then, the critical damage plane is determined by means of the average weighted method.(2)The cyclic rain flow counting method is applied to count the normal and shear strain response time histories on the critical plane. Then, the stress or strain components required in the SW model and the modified FS model are determined, and the load spectra of variable amplitude fatigue are compiled.(3)The damage parameters and fatigue properties obtained from ELCF tests are substituted into the SW model and modified FS model, respectively. The fatigue life corresponding to each strain amplitude is calculated according to the load spectrum of variable amplitude fatigue.(4)The fatigue damage is evaluated based on the Miner linear damage accumulation criterion, and the applicability of the two damage evaluation models is analyzed by comparing the fatigue damage assessment with the test results.The multiaxial fatigue analysis method is used to analyze the extremely-low-cycle fatigue damage of beam-to-column welded joints; the steps are shown in The critical plane approach has been widely accepted in multiaxial fatigue analysis. It is the primary method for performing strain state decomposition according to six strain components in which the strain tensors on the candidate material plane are calculated, and the location of the fatigue damage critical plane by means of the weighted average method is then determined.The z-axis is chosen parallel to the loading axis at the beam end, and the x-axis coincides with the outside normal of the surface. Then, the strain tensor at the weld root can be expressed asA candidate material plane is characterized by the critical location with the orientation of the unit normal  θ and ϕ , as showThe shear strain component and normal strain component on the candidate material plane can be derived from Equations (4) and (5):The shear and normal strains on the candidate material plane It can be seen from In terms of fatigue damage assessment, it is necessary that the critical damage plane is representative. It is considered that the normal and shear strain on the material plane often have several different peaks during the response time history; the angle of the critical damage plane In light of the above steps, the location of the critical damage plane of the dangerous position at the weld zone of NWB_2 is calculated as Based on the obtained strain responses on the critical damage plane, the double rain flow counting method can be used to compile the load spectrums of variable amplitude fatigue. In practice, the number of cycles, shear strain range The double cyclic rain flow counting results for the critical damage plane at the weld zone of NWB_2 are shown in As shown in In light of the above studies, the fatigue damage parameters for fatigue life prediction are determined by the results of the cyclic rain flow counting. The damage parameters of the SW model and modified FS model are as follows:It is indicated by the tension–torsion multiaxial fatigue test that the amplitude of  loading . FurtherThe multiaxial fatigue damage formula relating to the Manson–Coffin equation is expressed as Equation (10), that is, the SW model:In the case of proportional loading, Equation (10) reduces the equivalent strain approach form. In the case of uniaxial loading, Equation (10) reduces the Manson–Coffin equation to a uniaxial form. Thus, Shang and Wang proposed that Equation (10) could be used as a unified fatigue damage criterion under either proportional loading  or nonproportional loading.In terms of multiaxial fatigue study, researchers have made great progress in the area of damage accumulation and life prediction based on the strain components or the combined strain with stress components on the critical plane. The other popular approach is that proposed by Fatemi and Socie. They introduced a maximum normal stress  loading . The FS relation :(12)εn,mmined by (13)L=σ¯mined by (14)Φ=2AIt is believed that Equation (12) can account for the nonproportionality of both material additional hardening and the strain path. However, it is worth noting that not all metallic materials exhibit nonproportional additional hardening characteristics under nonproportional loading. The value of given by (16)k*=1It can be observed from Equation (16) that the value of The nonproportionality of the strain path cannot be adequately accounted for if determined by Equation (14). This is because the value of  inertia . As showThe strain path of the dangerous position at the weld zone of NWB_2 is shown in It can be seen from In light of the above study, the nonproportional parameters in the modified FS model are shown in Fatigue damage is an irreversible process in which material properties deteriorate continuously under reverse cyclic loading. It is also a process in which damage accumulates gradually. Generally speaking, most of the components are subjected to variable amplitude cyclic loading. Therefore, it is very important for fatigue damage evaluations to study the damage accumulation rule and fatigue failure criteria under variable amplitude cyclic loading.It is known that the Miner linear damage accumulation rule can predict the mean life of engineering structures under random loading. The fatigue damage per cycle, The obtained fatigue damage parameters are respectively substituted into Equations (10) and (15) to calculate the fatigue lives (1)In terms of the same fatigue properties of either CLG or PB, ranging from the 3rd to the 8th cycle, the damage values of the SW model are much smaller than those of the modified FS model. From the comparison between the experiment and damage evaluation, it is indicated that for box-column joints, the SW model is inclined to underestimate the risk. This may be because the nonproportionality of material additional hardening and the strain path both aggravate fatigue damage under nonproportional loading , whereas(2)In terms of the same fatigue damage model, either SW or the modified FS model, the fatigue values calculated by the fatigue properties of CLG are much larger than those calculated by the fatigue properties of PB, which are more coincident with the reverse cyclic loading test results. This may be because the geometric shape and force characteristics of the CLG specimens are more similar to those of box-column joints than those of PB specimens. It is shown that the structural details, i.e., geometric shape and force characteristics, play an important role in damage evaluation; the more similar the structural details, the better the damage estimation.(3)D = 1. The Miner linear damage accumulation rule indicates that a microscopic crack will be initiated at the time when the 7th cycle is coming to an end, while it was previously stated in In terms of the modified FS model using the fatigue properties of CLG, the calculated damage value of the dangerous position at the weld zone of NWB_2 is 1.0321 at the 7th cycle, which is slightly larger than As shown in In summary, for ELCF damage evaluations of box-column welded joints, the method can provide good estimations in cases where the fatigue properties of CLG are adopted as the proposed modified FS model parameters.(1)The failure of beam-to-column welded joints is multiaxial and nonproportional ELCF when subjected to severe seismic excitation.(2)From a comparison between the FS model containing a nonproportional factor and the SW model without this factor, it was shown that the effect of nonproportional additional hardening plays a key role in damage evaluation.(3)The nonproportional factor in the FS model previously proposed by us was again modified and introduced to reflect the effect of nonproportionality. This accounts for both the strain path and additional material hardening. It was found that the fatigue damage model including this proposed factor can successfully evaluate the fatigue damage of beam-to-column welded joints with reasonable accuracy.(4)The fatigue properties of PB and CLG when considering welded structural details were respectively adopted as the damage parameters in the modified FS model and the SW model. From the comparison between experiments and damage estimation, it was shown that the fatigue properties of CLG result in the damage evaluation being more consistent with the test results than that of PB because of the highly similar geometric shape and force characteristics.Based on numerous existing studies, it is difficult to accurately assess the damage of a steel structure during the ELCF fracture process using the macroparameters of the whole structure or a structure component, since local structural details and fatigue crack initiation mechanics are ignored. A damage evaluation methodology based on the structural details of I-beam to box-column welded joints, extended from multiaxial fatigue critical plane theory, is proposed to assess ELCF crack initiation life and damage. Based on experimental studies associated with analytical studies, the key conclusions are as follows:In conclusion, the modified FS model with the fatigue properties of CLG is most in line with the test results, which verifies the applicability of the model. The presented work makes an important contribution to ELCF damage evaluations of steel frameworks.


A better understanding of how the COVID-19 pandemic responds to social distancing efforts is required for the control of future outbreaks and to calibrate partial lock-downs. We present quantitative relationships between key parameters characterizing the COVID-19 epidemiology and social distancing efforts of nine selected European countries. Epidemiological parameters were extracted from the number of daily deaths data, while mitigation efforts are estimated from mobile phone tracking data. The decrease of the basic reproductive number ( An epidemic outbreak can recede either due to reduction of the transmission probability across contacts, or due to a gradual build up of immunity within the population. According to currently available immunological data2 at most locations only a relatively small fraction of the population was infected, typically well below 10%, thus the receding disease mostly reflects changes in social behavior and the associated reduction in disease transmission. Changes in social behavior can include state-mandated control measures as well as voluntary reduction of social interactions. However, especially with the view of potential future outbreaks, it is important to better understand how the timing and extent of social distancing impacted the dynamics of the COVID-19 pandemic.The COVID-19 pandemic started in late 2019 and within a few months it spread around the World infecting 9 million people, out of which half a million succumbed to the disease. As of June 2020, the transmission of the disease is still progressing in many countries, especially in the American continent. While there have been big regional differences in the extent of the pandemic, in most countries of Europe and Asia the initial exponential growth has gradually transitioned into a decaying phase17. A statistical analysis of the number of diagnosed cases, deaths and patients in intensive care units (ICU) in Italy and Spain have indicated that the epidemic started to decrease only after the introduction of strict lock-down action3. This was especially visible in Italy, where the final strict social distancing has been reached in a number of consecutive steps. A statistically more comprehensive analysis of hospitalized and ICU patients in France identified a 77% decrease in the growth rate of these numbers after the introduction of the lock-down4. The comparison of social distancing efforts in China, South Korea, Italy, France, Iran and USA5 revealed that the initial doubling time of identified cases was about 2 days which was prolonged substantially by the various restrictions introduced in these countries. Epidemic models are widely used to estimate how various intervention strategies affect the transmission rate, however, the quantitative relationship between social interventions and epidemic parameters are hardly known7.Previous studies estimated the effect of social distancing on the dynamics of COVID-19 pandemic either by direct data analysis or by modeling methods8. Statistical data on COVID-19-related deaths are considered to be more robust than that of daily cases of new infections. The latter is affected by the number of tests performed as well as by the testing strategy – e.g. its restriction to symptomatic patients – which may be highly variable across countries and often changes during the course of the epidemic. The time course of daily deaths can be considered as a more reliable indirect delayed indicator of daily infections. We thus do not address apparent differences in the case fatality ratio, and restrict our focus to the recorded COVID-19-associated death toll.In this paper, we quantitatively characterise the time course of the COVID-19 pandemic using daily death data from nine selected European countriesOur choice of countries was motivated by the requirements that each of these countries (i) had a relatively large disease-associated death toll  so we can assume that the deterministic component of the epidemic dynamics dominates over random fluctuations. (ii) The selected countries spent a suitably long time in the decaying phase of the epidemic, thus allowing its precise characterisation. Based on these two criteria, we analysed data from the following, socio-economically similar countries, each implementing a somewhat distinct social distancing response: Italy, Spain, France, UK, Germany, Switzerland, Netherlands, Belgium and Sweden.9. Our aim is to quantitatively determine the characteristic features of the progression of the epidemic, such as initial growth rate, timing of the peak, and final decay rate and investigate how these parameters are determined by the timing and strictness of social distancing measures. We demonstrate that the overall death burden of the epidemic can be well explained by both the timing and extent of social distancing, often organized voluntarily well in advance of the state-mandated lock-down, and present quantitative relationships between changes in epidemiological parameters and mobile phone mobility data.To characterise social distancing responses we used mobile phone mobility trend data from Apple Inc.D(t), is presented in Fig. t = 0 reference time point as the date when the daily death rate first exceeded 5 deaths. In each country D(t) indicates the presence of a well defined exponential growth phase, followed by a crossover region and later an exponential decay stage. The initial growth and final decay regions are characterized by fitting the exponential 10 and 12, however our analysis and results do not rely on the value of this parameter or on any modelling assumption regarding the disease dynamics.Daily COVID-19 death data, To characterize the growth and decay phases, the values of mentclass2pt{minimmentclass2pt{minimmentclass2pt{minimD(t) can be due to gradual implementation of social distancing as well as to the case-to-case variability of the time elapsed from infection to death.A third parameter characterizes the transition from the growth to decay phases, i.e., the peak of the epidemic in terms of deaths. We define The total death toll of the disease can be estimated analytically using the three parameters Approximations by Eq.  are compmentclass2pt{minim9. Although the mobility data may have bias in the mobility signal, it may have relatively little direct effect on data reported for the countries studied here. In the 90-day period considered in this study, the data indicates the mobility levels before and after implementing the social distancing. For clarity, to set a unique time-scale for the mobility data for all the countries, we added more data points after The timing and strictness of the often voluntary social distancing is quantified from the average mobility data , unfortunately it is difficult to know in real-time. The country-specific values of As Fig. mentclass2pt{minimD(t). We propose that the increased time between the peak and the time of the social distancing may indicate the presence of subpopulations in which the disease continues to propagate with the initial reproduction number Our analysis thus indicates that social distancing has two effects: it reduces the basic reproduction number of the infection as expected, and shortens the time required for the epidemic to peak. This latter effect is unexpected, as changes in behavior should reduce transmission immediately, which, after a fixed delay – involving manifestation of symptoms and in a fraction of the patients death – should also appear in the daily death toll In this paper we analysed the interdependence of epidemic and mobility data and identified a quantitative relation between parameters of social distancing and key characteristics of the COVID-19 pandemic. Our sample consisted of 9 European countries where suitable data was available at current time. We found that the total death toll does not correlate well with any single parameter such as the timing of the official lock-down or the strictness of social distancing extracted from mobile phone location data. The total death toll, however, could be well estimated by a non-linear combination of three parameters: exponents characterising (i) the initial exponential growth rate (or reproductive number, Since the estimated daily death toll at the peak increases exponentially with the difference between the effective lock-down time (The higher death toll in Belgium cannot be explained by the proposed set of parameters (10.While in the case of Sweden the initial growth rate was the slowest and even the timing (In the case of the UK the timing of the social distancing (14. While mobile phone tracking data thus can characterize the relatively early stages of social distancing efforts, it may be less useful to detect more subtle efforts like wearing masks or staggered working shifts at later stages of the pandemic.The phone mobility data is an indirect measure of the social distancing and disease transmission probabilities. However, it can be compared to more traditional measures in the Netherlands, where a study directly determined the number of daily personal interactions before and after the lock-down. Using questionnaires, the study identified a 71% decrease in interactions on average, which is fairly close to the 65% drop estimated from the phone mobility dataIn conclusion, we demonstrated a quantitative relation between social distancing efforts and statistical parameters describing the current COVID-19 pandemic. We identified an unexpected extension of the exponential growth phase when social distancing efforts are weak, which can substantially increase the death toll of the disease. Future studies are required to extend this analysis to countries with a markedly different socio-economical arrangements.Supplementary Information.