Studies on fault detection and diagnosis of planetary gearboxes are quite limited compared with those of fixed-axis gearboxes. Different from fixed-axis gearboxes, planetary gearboxes exhibit unique behaviors, which invalidate fault diagnosis methods that work well for fixed-axis gearboxes. It is a fact that for systems as complex as planetary gearboxes, multiple sensors mounted on different locations provide complementary information on the health condition of the systems. On this basis, a fault detection method based on multi-sensor data fusion is introduced in this paper. In this method, two features developed for planetary gearboxes are used to characterize the gear health conditions, and an adaptive neuro-fuzzy inference system (ANFIS) is utilized to fuse all features from different sensors. In order to demonstrate the effectiveness of the proposed method, experiments are carried out on a planetary gearbox test rig, on which multiple accelerometers are mounted for data collection. The comparisons between the proposed method and the methods based on individual sensors show that the former achieves much higher accuracies in detecting planetary gearbox faults. Most investigations, however, focus on fixed-axis gearboxes in which all gears are designed to rotate around their own fixed centers ,2. CompaDifferent from fixed-axis gearboxes, planetary gearboxes exhibit the following unique behaviors: (a) multiple planet gears produce similar vibrations. These vibrations with different meshing phases couple with each other; as a result, some of the excitations of multiple gear meshes can be cancelled or neutralized ; (b) plaet al. [et al. [To solve the problems of fault detection and diagnosis of planetary gearboxes, a few methods have recently been reported in the literature. Keller and Grabill  proposedet al.  introducet al.  developeet al.  proposedet al.  applied et al.  proposed [et al.  presenteThe above studies, however, only utilized the vibration information from individual sensors. It is a fact that for complex gear transmission systems like planetary gearboxes, multiple sensors provide variously sensitive or complementary characteristic information for fault detection and diagnosis. Thus, how to fuse information from multiple sensors to improve accuracy is a vital issue in fault detection and diagnosis of planetary gearboxes.Aiming at solving the above problem, a method for fault detection of planetary gearboxes based on multi-sensor data fusion is proposed in this paper. In this method, two features are employed to characterize the health conditions of planetary gearboxes, and an adaptive neuro-fuzzy inference system (ANFIS) is used to fuse all features from different sensors. Since the two features are particularly developed for planetary gearboxes, and ANFIS combines the advantages of the adaptive capability of neural networks and the qualitative approach of fuzzy logic, the proposed method is expected to produce satisfactory results in detecting planetary gearboxes faults. The remainder of this paper is organized as follows: Section 2 introduces the method based on multi-sensor data fusion. Section 3 shows a planetary gearbox test rig and experiments conducted for testing the proposed method. Gears having four fault modes were installed in the test rig and vibration signals collected by multiple sensors under different loads and various motor speeds. Section 4 compares the proposed method with the methods based on individual sensors. The experimental results show that the proposed method is superior to the methods based on individual sensors in detecting faults of planetary gearboxes. Conclusions are given in Section 5.2.2.1.As mentioned above, two features are extracted from the vibration signals of planetary gearboxes. They are FRMS and NSDS, which are specially designed for fault detection of planetary gearboxes.2.1.1.s(t) , is the tth data point of the filtered signal S, and T is the total number of data points contained in S. The filtered signal is produced by removing the regular meshing elements from the original vibration signal X. The regular meshing elements include the rotating frequency of the damaged gear and its five order harmonics, the meshing frequency and its three order harmonics, and the modulation sidebands of the meshing frequency and its harmonics. The modulation sidebands are excited when the planets pass through the sensors fixed on the gearbox housing [It is generated by calculating the root mean square of the filtered signal instead of the original signal and is defined as: housing ,16.The rationale of FRMS can be explained as follows: the vibration spectrum of a healthy planetary gearbox, measured by sensors mounted at a fixed location, is unlike that of a fixed-axis gearbox. The planet gears rotate past the sensor location causing the vibration transmission path to change in a periodic manner, thus modulating the amplitude of the measured vibration response. These sidebands are therefore the common elements in the frequency spectra of both healthy and damaged planetary gearboxes instead of the indications of faults.2.1.2.yd(i)  denotes the amplitude of the ith spectrum line of the difference spectrum yd of ym minus yh, I is the total number of spectrum lines contained in yd, and ym and yh represent the frequency spectra of the unknown signal and the healthy one respectively. The use of normalization, that is, dividing the sum of the positive amplitudes in the difference spectra by the sum of the unknown signal spectra, is to attenuate or even cancel the effect of the operation conditions such as various rotating speeds and loads. Therefore, NSDS is a dimensionless feature.It is developed by normalizing the summation of the positive amplitudes of the difference spectrum between the signal measured on a gearbox whose health condition is unknown and the signal measured on a healthy gearbox. It is expressed as:It is believed that the vibration response and the frequency spectra will change once faults occur in a planetary gearbox and the amplitudes in the spectra may increase correspondingly. NSDS aims to discover the increment in the frequency spectra of the faulty planetary gearbox compared to the healthy one.2.2.x and y are the inputs, iA and iB are the fuzzy sets, iz are the outputs within the fuzzy region specified by the fuzzy rules, and ip, iq and ir are the design parameters that are determined during the training process. The ANFIS architecture to implement these two rules has five layers and is shown in ANFIS is an integration system which uses neural networks to optimize the fuzzy inference system. ANFIS maps inputs through input membership functions and associated parameters, and then through output membership functions to outputs. The initial membership functions and rules for the fuzzy inference system can be designed by employing human expertise about the target system to be modeled. Then ANFIS can refine the fuzzy if-then rules and membership functions to describe the input/output behavior of a complex system. To explain the ANFIS architecture, two fuzzy if-then rules based on a first order Sugeno model are considered –19:RuleLayer 1: Input membership functioniAu (x) and uBi−2(y) can be adopted any fuzzy membership function. Usually, the bell shaped membership function is employed and given by:ia, ib and ic are the parameters of the membership functions, deciding the bell shaped functions.This layer performs fuzzification of the inputs and all the nodes in this layer are adaptive. The outputs are the membership grade of the inputs:Layer 2: RuleM, indicating that they perform as simple multipliers. The outputs of this layer represent the fuzzy strengths iω of each rule and can be expressed as:All the nodes in this layer are fixed nodes. They are labeled with Layer 3: NormalizationN, indicating that they play a normalization role to the fuzzy strengths from the previous layer. The normalization factor is calculated as the sum of all weight functions. The outputs of this layer, the so-called normalized fuzzy strengths, can be represented as:All the nodes in this layer are also fixed nodes. They are labeled with Layer 4: Output membership functionip, iq and ir are the parameters of the output membership functions respectively.In this layer, the nodes are adaptive nodes. The outputs of this layer are shown by:Layer 5: OutputS. This node performs the summation of all incoming signals. Hence, the overall output of the model is as follows:In this layer, there is only one single fixed node labeled with Generally, a hybrid learning algorithm of the gradient descent approach and least-squares estimate is utilized to tune the parameters of the membership functions. During the forward pass, the node outputs advance until the output membership function layer, where the consequent parameters are identified by the least-squares estimate. Based on the error signals that propagate backward, the backward pass uses the back propagation gradient descent method to update the premise parameters –19.3.In order to test the effectiveness of the proposed method, an experimental system of a planetary gearbox test rig is established and experiments are carried out on the test rig. A schematic model of the test rig is shown in In a planetary gearbox, sun gear teeth are subject to faults because their multiple meshes with the planet gears increase the potential for damage . Thus, ai.e., the axial and the vertical directions of the tri-axial accelerometer, and the vertical direction of the unidirectional accelerometer), are considered and used for multi-sensor data fusion in this paper.Five experiments are conducted under each of the five gear health conditions. The five health conditions involve normal, the cracked tooth on the sun gear of the first stage, the pitted tooth on the sun gear of the first stage, the chipped tooth on the sun gear of the second stage and the missing tooth on the sun gear of the second stage. In each experiment, a sun gear of different health conditions is installed inside of the test rig, whereas all the other gears are normal. The axial and radial vibrations of the first stage are measured by the tri-axial accelerometer, and the radial vibration of the second stage is captured by the unidirectional accelerometer. The vibration signals of three channels, , thirty data samples are obtained. Therefore, for each of the five gear health conditions, there are 240 data samples collected under two loading conditions and four motor speeds.4.1.In this case, the proposed method is applied to detecting three health conditions of the sun gear in the first stage of the planetary gearbox: normal, a cracked tooth and a pitted tooth. Two hundred and forty (240) data samples are acquired for each health condition, and therefore the whole data set including the three conditions altogether contains 720 samples. Three hundred and sixty samples (360) are selected for training and the remaining 360 samples are used to test. This is a three-class classification problem.First, two features FRMS and NSDS are extracted from each data sample. Correspondingly, six features are obtained for each of data samples from the three channels . Then, the six features are fused via ANFIS. Finally, the health conditions of the sun gear of the first stage are detected. The detection accuracies, that is, the percentage of the number of the data samples correctly detected divided by the whole number of the data samples, are presented in To demonstrate the performance of the proposed method based on multi-sensor data fusion, the method based on each individual sensor/channel is also tested using the same data. We refer to these methods as Method 1, Method 2 and Method 3, corresponding to the three individual sensors/channels. The three methods use only two features extracted from an individual sensor/channel, and also utilize ANFIS as the fusion technique. Their detection results are also shown in Since this case is relatively simple, Methods 1∼3 achieve acceptable training and testing accuracies. They are in the range of 88.61–100% for training and 90–100% for testing. Moreover, both Method 2 and the proposed method obtain the highest training and testing accuracies (100%). Observing their classification errors plotted in 4.2.The difference of Case 2 from Case 1 is that in Case 2, damage is created on the sun gear of the second stage. Similarly, 720 data samples are divided into 360 training and 360 testing samples. They are used to compare the proposed method with Methods 1∼3. The fault detection accuracies are given in As mentioned above, faults to be detected in this case locate on the sun gear of the second stage, in which the sun gear operates under lower rotating speeds than the first stage. As a result, the fault characteristics of the damaged sun gear are not excited enough and accordingly not evident either. Therefore, it is more difficult to detect the sun gear faults of the second stage. The training and testing accuracies of Methods 1∼3 for the sun gear of the second stage in 4.3.The purpose of using this case is to test the performance of the proposed method in detecting both gear health conditions and damage locations . Thus, 1,200 data samples covering five gear health conditions  are applied to testing the proposed method. The 1,200 data samples are split into two sets: 600 samples for training and 600 for testing. In this case, we solve the five-class classification problem. It is observed from To examine the effectiveness of the two proposed features FRMS and NSDS, we make another comparison in this case. Two features SI and SLF, presented in Reference , are useAll results of the three cases prove that it is better to fuse the data from different sensors than to utilize the data from individual ones. The multi-sensor fusion method obtains evident improvements of accuracy and robustness in fault detection of planetary gearboxes.5.Multiple sensors mounted on different locations of planetary gearboxes can provide complementary information for fault detection and diagnosis. Based on this understanding, a method using multi-sensor data fusion is proposed in this paper. In this method, an adaptive neuro-fuzzy inference system (ANFIS) is adopted as the fusion technique and two features specifically designed for planetary gearboxes as the inputs of ANFIS. Three cases of planetary gearbox fault detection are used to test and compare the proposed method and the methods using individual sensors. The experimental results demonstrate that the proposed method is superior to the others in terms of fault detection accuracy.


HDX mass spectrometry is a powerful platform to probe protein structure dynamics during ligand binding, protein folding, enzyme catalysis, and such. HDX mass spectrometry analysis derives the protein structure dynamics based on the mass increase of a protein of which the backbone protons exchanged with solvent deuterium. Coupled with enzyme digestion and MS/MS analysis, HDX mass spectrometry can be used to study the regional dynamics of protein based on the m/z value or percentage of deuterium incorporation for the digested peptides in the HDX experiments. Various software packages have been developed to analyze HDX mass spectrometry data. Despite the progresses, proper and explicit statistical treatment is still lacking in most of the current HDX mass spectrometry software. In order to address this issue, we have developed the HDXanalyzer for the statistical analysis of HDX mass spectrometry data using R, Python, and RPY2.apo and ligand-bound protein allow us to evaluate whether the regions have significant changes in structure dynamics or not. Another option of the software is to calculate the deuterium incorporation rate for the comparison. The two types of statistical analyses are Paired Student’s t-test and the linear combination of the intercept for multiple regression and ANCOVA model. The user interface is implemented with wxpython to facilitate the data visualization in graphs and the statistical analysis output presentation. In order to evaluate the software, a previously published xylanase HDX mass spectrometry analysis dataset is processed and presented. The results from the different statistical analysis methods are compared and shown to be similar. The statistical analysis results are overlaid with the three dimensional structure of the protein to highlight the regional structure dynamics changes in the xylanase enzyme.HDXanalyzer package contains three major modules, the data processing module, the statistical analysis module, and the user interface. RPY2 is employed to enable the connection of these three components, where the data processing module is implemented using Python and the statistical analysis module is implemented with R. RPY2 creates a low-level interface for R and allows the effective integration of statistical module for data processing. The data processing module generates the centroid for the peptides in form of m/z value, and the differences of centroids between the peptides derived from Statistical analysis provides crucial evaluation of whether a protein region is significantly protected or unprotected during the HDX mass spectrometry studies. Although there are several other available software programs to process HDX experimental data, HDXanalyzer is the first software program to offer multiple statistical methods to evaluate the changes in protein structure dynamics based on HDX mass spectrometry analysis. Moreover, the statistical analysis can be carried out for both m/z value and deuterium incorporation rate. In addition, the software package can be used for the data generated from a wide range of mass spectrometry instruments. The information allows one to understand which region of the protein is more stabilized or destabilized upon ligand binding in the solvent exchange reaction [apo form and protein ligand complex with technical replicates. The data processing for HDX mass spectrometry thus requires us to compare a large set of the m/z values or percentages of deuterium incorporation for the same peptides derived from apo protein and ligand bound protein.Protein intrinsic dynamics has been more and more recognized as an important consideration for protein functions . Severalreaction -5,14-22.Various software platforms have been developed to analyze the HDX data. Among them include HX-Express, Deuterator, HD Desktop, DEX, Hydra, TOF2H etc. Most of these HDX data analysis software packages focus on calculating the m/z value from the MS raw data for the deuterated peptide, and then evaluate the m/z value increase according to time. For example, HX-Express is a semi-automated software package which exports deuterium uptake curve and peak width plots based on Microsoft excel application . CompareThe integration of statistical analysis with data processing is challenging. In terms of statistical analysis, several software environment including SAS, SPSS, and R can be employed. Among these packages, R is the open source program and can be easily obtained from internet free of charge. Despite the various advantages of R, the software environment does not have strong user-interface supports and thus requires certain level of expertise. In order to develop user-friendly statistical software for HDX mass spectrometry analysis, we hereby employ the latest RPY2 package to connect the statistical module of R with a data processing module implemented by Python and a user-interface implemented by wxpython. Many programming languages including C, java, Perl, Python can be used for data processing and UI development, and each programming language has its pros and cons. Among these programming languages, Python is chosen as the main developing language for two reasons: First, the existed RPY package allowed the seamless and effective implementation between the data processing module and statistical module in R. Second, various BioPython packages have been developed for the analysis of biological data which allows fast and easy embedding. For these reasons, we have developed the HDX mass spectrometry analysis software HDXanalyzer using Python, R, and RPY2 packages.apo and ligand bound protein. Further statistical analysis is carried out by R, where two statistical methods are used. The Paired Student’s t-test is used to compare either the centroid values of the m/z value or the deuterium incorporation rate to derive point estimation, confidence intervals, and p value to indicate if significant differences in structure dynamics exist or not. In addition, the multiple regression (or ANCOVA) model is also involved for the similar analysis through linear combination of the intercepts. HDXanalyzer thus provided novel solutions toward ultimate quantification and statistical evaluation of the structure dynamics changes in the HDX mass spectrometry experiments. The software package addressed the imminent need of statistical evaluation for the HDX mass spectrometry analysis and can be expanded to other applications for HD exchange studies by other techniques.In this article, we hereby present a novel software package HDXanalyzer for statistical analysis of HDX mass spectrometry data to evaluate the protein structure dynamics changes. The software package includes three major components, the data feeding and processing module, the user interface, and the statistical analysis module. The data processing module is developed in Python to process a batch of excel input files containing the m/z value of the peptides from different experiments. The pre-formatted m/z value of the peptides will be processed to derive the centroid of the peptide peaks or the percentage of deuterium incorporation for the statistical analysis. The data is then processed by the Figure Generator to create graphs to visualize the differential HD exchange rates in apo form, ligand bound form, proteins that has been subjected to different HDX exchange time points). The MS/MS analysis often allows us to correlate the peptide ID with a sequence, which is beyond the scope of HDXanalyzer. HDXanalyzer is mainly dealing with the MS analysis data. The raw MS data was exported for intensity and m/z values in the excel format for each peptide as shown in the online Additional Files available at http://people.tamu.edu/~syuan/hdxanalyzer. The Additional Files contain the sample input data derived from HDX mass spectrometry analysis of xylanase enzyme and the executable file for HDXanalyzer software. The data is in a compressed package available at the aforementioned link. The HDXanalyzer then takes all the pre-formatted excel file containing peptide ID and m/z values of the peptides at different HDX exchange time points as the input. A python package for reading and formatting information from excel files named as xlrd was used to develop a parser to extract and process excel spreedsheets. The parser allows us to extract all m/z values and intensity of the peptides. The peptide ID and the treatment (apo or ligand bound) will also be extracted.As shown in Figure The data processing can derive two types of variables for the statistical analysis, centroid of the peptide in form of m/z value or deuterium incorporation rate. Centroid values of each peptide are derived based on the m/z value of the peaks generated by MS analysis. For deuterium incorporation rate, the weighted average m/z values of each peptide ion isotopic cluster are calculated. Basically, the deuterium incorporation rate is calculated based on the centroid of the peptide m/z value and is in form of percentage. The deuteration level of each peptide is calculated based on Equation 1, and corrections for back-exchange are made based on 70% deuterium recovery and accounting for 80% deuterium content in the ion-exchange buffer. These corrections can be defined by users in the data pre-processing procedure.where m/z (P), m/z (N), and m/z (F) are the centroid values of partially deuterated peptide, nondeuterated peptide, and fully deuterated peptide, respectively .The resulted data are then processed into a table format and loaded to Figure Generator to create visualization of the dynamic status of peptide at different time points. Specifically, the figures are graphs displaying the m/z value or deuterium incorporation rate of a peptide at different exchange time. Gnuplot, an open source GNU plotting tool under UNIX/linux, with counterpart in MSDOS & Windows system, was employed to implement the Figure Generator. The advantages of Gnuplot lies in two aspects including the availability from either GNU projects or internet free of charge, as well as the convenience of automated generating multiple outputs using its corresponding scripting language. Besides the graphic display of the HDX data, statistical analysis is carried out to generate the point estimation for differential m/z value or incorporation rate, the confidence intervals and the p value.Statistical analysis is employed to evaluate if a peptide or a region of the protein has significant changes in structure dynamics or not. Such changes are reflected in the differences of either centroid m/z values or the deuterium incorporation rates during the HDX experiments. The m/z value or deuterium incorporation rate from different peptides can be compared with different statistical models to derive parameter estimation and p value. The parameter estimations allow us to evaluate the levels and variations of the differences in structure dynamics of a protein region, and the p value allows us to determine if the differences are significant or not.apo or ligand binding proteins. Paired Student’s T-test is used instead of the regular T-test because of the time effects in the HDX experiments. More specifically, the m/z values or the percentages of deuterium incorporation for a peptide will increase as the hydrogen deuterium exchange time gets longer. As these two values are time dependent, both will reach a plateau when the exchange time is long enough. For this reason, the Paired t-test is used to avoid the time point effects. Besides the pairwise t-test, the multiple regression model or ANCOVA model are utilized to compare the m/z or incorporation rate differences between peptides from the two types of proteins (apo and ligand bound). The multiple regression model is as shown in Equation 2. The linear combination of the Group effects allows us to compare the differences between apo and ligand-bound proteins. For either model, the point estimation of mean differences, confidence intervals, and p value will be computed.Two types of statistical models are used. First, a Paired Student’s t-test (Pairwise t-test) is used to compare the m/z value or percentage of deuterium incorporation for peptides from Y = TXTimeβ + GXGroupβ + TGXTime*XGroup Equation 2βapo or ligand bound proteins. The combination of the two effects may also influence the dependent variable.where Y is the dependent variable that can be either the m/z value or the deuterium incorporation rates of different peptides. Y is dependent on the effects of time points and different groups from either The integration of statistical analysis, data processing, and visualization is usually challenging. The recent developed RPY allows us to integrate the statistical feature of R and the user interface as well as the data processing features of Python. As an open-source language, R has the unique advantages over other statistical languages for software development. RPY enables us to employ the R for statistical analysis of HDX mass spectrometry data. We have also used RPY2 to provide a low-level interface to R. The Python-based system thus can directly call R function through RPY and the software efficiency and effectiveness are greatly improved.The user interface of HDXanalyzer is developed with Wxpython as shown in Figure HDXanalyzer is implemented as a software package to enable the statistical analysis of HDX mass spectrometry data and to allow the evaluation of protein structure dynamics changes. In order to demonstrate the software application, we first analyze a previously published dataset for the HDX mass spectrometry analysis of xylanase enzyme. The example data is available in the online supplementary document. Furthermore, we also apply HDXanalyzer to analyze two peptides from a recent publication, where the HD exchange for the two peptides were statistically evaluated. We hereby discuss the usage of the software, present the output, compare the different results from different statistical models, and interpret the results.apo set and ligand set. The peptide ID can be corresponding to a certain peptide sequence. The upload function is available from the user interface, where input file can be read and processed to generate m/z centroids and deuterium incorporation rates of the peptides as aforementioned. The data are therefore further analyzed for visualization and statistical analysis.The HDXanalyzer aims to integrate statistical analysis for comparing structure dynamics of protein upon ligand or substrate binding. As discussed in the Implementation section, the software takes a batch of pre-formatted excel files containing m/z values for multiple peptides of different treatment and time points as shown in Supplementary File 1 available online (the HDX_Xylohexaose.rar dataset). The data pre-formatting will allow the software to process a uniform input of HDX mass spec data from different instruments. The sample input file is derived from a xylanase structure dynamics study and the m/z values of the peak area for the peptides are included. Each input excel file will contain several sheets for the data from different peptides and treatments. The spreadsheet contains peptide ID, m/z value, charge state, time points for deuterium treatment, and the ligand name to separate different experimental sets, e.g., apo protein and the ligand-bound protein are marked with different colors in the graph. Each graph in the middle panel is clickable, and the enlarged graph for each peptide is presented in the enlarged figure window in the upper right panel as shown in Figure The data output includes three parts as shown in Figure apo vs. ligand) effects in multiple regression (ANCOVA) model are used to derive parameter and p value estimation. A very important decision for data processing is the choice of time point. The early time points after deuterium exchange, especially for exchange less than 1 minute, may lead to very limited exchange even in apo protein. In such case, the statistical analysis of data from these time points cannot represent the real deuterium incorporation level. Therefore, HDXanalyzer offers the users the option to choose the time points for the analysis and we have focused on the time points after ten minutes in our analysis of the example dataset.As aforementioned, two types of statistical analyses are implemented for HDXanalyzer. Both Paired Student’s T-test and the linear combination of intercept for group  model together with MG and JSY. SL greatly improved the software implementation and developed UI with wxpython. MG provided valuable suggestions for multiple regression model development. UU provided the pre-formatted data for the software development, valuable advises for implementation, and generated the overlaid the statistical results with 3D structure of the xylanase enzyme. WS and YZ provided valuable suggestions for the implementation. JSY and SYD drafted most parts of the article and finalized the draft. SL drafted most of the implementation and UU drafted part of the implementation. JSY, SYD and LL revised the article.The authors declare no competing interests.


Congestion in a wireless sensor network causes an increase in the amount of data loss and delays in data transmission. In this paper, we propose a new congestion control technique  based on an adaptive compression scheme for packet reduction in case of congestion. The compression techniques used in the ACT are Discrete Wavelet Transform (DWT), Adaptive Differential Pulse Code Modulation (ADPCM), and Run-Length Coding (RLC). The ACT first transforms the data from the time domain to the frequency domain, reduces the range of data by using ADPCM, and then reduces the number of packets with the help of RLC before transferring the data to the source node. It introduces the DWT for priority-based congestion control because the DWT classifies the data into four groups with different frequencies. The ACT assigns priorities to these data groups in an inverse proportion to the respective frequencies of the data groups and defines the quantization step size of ADPCM in an inverse proportion to the priorities. RLC generates a smaller number of packets for a data group with a low priority. In the relaying node, the ACT reduces the amount of packets by increasing the quantization step size of ADPCM in case of congestion. Moreover, in order to facilitate the back pressure, the queue is controlled adaptively according to the congestion state. We experimentally demonstrate that the ACT increases the network efficiency and guarantees fairness to sensor nodes, as compared with the existing methods. Moreover, it exhibits a very high ratio of the available data in the sink. Sensor networks have attracted significant attention as important infrastructure for data collection in pervasive computing environments. In this field, wireless sensor networks play a special role in home automation, environmental monitoring, military, health, and other applications –9.Some of these applications require sensor nodes to send data continuously, whereas in some applications, sensor nodes should send data only when a specified event or phenomenon occurs. With fast continuous transmission, the variation of the phenomenon in the environment can be monitored precisely; however, the dissipation of energy, which is a very critical resource, can drastically increase. The energy consumption can be reduced by sending only the information of an event occurrence, but the variation in the phenomenon cannot be monitored in detail. For energy efficiency and detailed monitoring, sensor nodes should control the data transmission interval in an inverse proportion to the variation in the phenomenon. Therefore, the transfer rate could be also varied according to the event occurrence. However, variable transfer rates might cause network congestion due to concentrated packets in case of a concurrent occurrence of multiple events. During congestion, sensor nodes usually drop the overflowed packets; however, packet drops lead to data loss and unnecessary energy dissipation. Therefore, routing protocols should efficiently control the network congestion –17.Conventional congestion control protocols usually use a back-pressure scheme, which reduces congestion by reducing the transfer rate of child nodes of a congested node and by decreasing the packet generation rate of a sensor node that causes heavy traffic. However, packet drops might still occur because the back pressure is propagated slowly due to the collision among sensor nodes in case of serious congestion. Slow propagation of the back pressure may also cause fluctuation in packet flows because network state changes frequently with the back pressure. Moreover, a decrease in the packet generation rate leads to a reduction in the fidelity of an event because heavy traffic is generated by a sensor node that detects the specified event. In order to reduce congestion, the routing protocol should reduce the number of packets in the network; however, simply dropping overflowed packets will reduce the data fidelity and increase the energy dissipation. In this paper, we propose a new congestion control technique  based on an adaptive compression scheme for packet reduction in case of congestion. Compression techniques used in the ACT are Discrete Wavelet Transform (DWT), Adaptive Differential Pulse Code Modulation (ADPCM), and Run-Length Coding (RLC). The ACT first transforms the data from the time domain to the frequency domain, reduces the range of the data with the help of ADPCM, and then reduces the number of packets by means of RLC before transferring the data to the source node. It then introduces the DWT for priority-based congestion control because the DWT classifies the data into four groups with different frequencies. Thereafter, it assigns priorities to these data groups in an inverse proportion to the respective frequencies of the data groups and defines the quantization step size of ADPCM in an inverse proportion to the priorities. RLC generates a less number of packets for a packet with a low priority. In the relaying node, the ACT reduces the number of packets by increasing the quantization step size of ADPCM in case of congestion. Moreover, in order to facilitate the back pressure, the queue is controlled adaptively according to the congestion state. In the next section, we review related studies. Section 3 describes the ACT, and Section 4 provides a performance evaluation. We conclude the paper in Section 5.2.A number of previous studies have addressed the topic of congestion control in wireless sensor networks. Most existing congestion control protocols usually adjusts traffic rate at source nodes or intermediate nodes. This approach is helpful to save network resources and more feasible and efficient. According to the control behavior, there are two general methods for traffic control in WSNs: end-to-end and hop-by hop. The end-to-end control can impose exact rate adjustment at each source node and simplify the design at intermediate nodes; however, it results in slow response and relies highly on the round-trip time (RTT). In contrast, the hop-by-hop congestion control has faster response.CODA    uses effDecentralized, Predictive Congestion Control  (DPCC) f3.3.1.There are mainly two types of congestion in WSNs. The first type is the node-level congestion which occurs due to queue overflow inside the node. Queue overflow might lead to packet drop and this leads to retransmission if required and therefore consumes additional energy. Wireless channels are shared by several nodes using Carrier Sense Multiple Access (CSMA)-like protocols and thus collisions among sensor nodes can occur when multiple sensor nodes try to occupy the channel concurrently. This is the second type of congestion-link-level congestion. Link-level congestion increases the packet service time and decreases the link utilization. Both the node level and the link-level congestions have direct impact on energy efficiency and Quality of Data (QoD). Therefore congestion must be efficiently controlled. Congestion control protocol efficiency depends on how much it can achieve the following objectives: first, energy-efficiency should be improved in order to extend system lifetime. Therefore congestion control protocols need to avoid or reduce packet loss due to buffer overflow, and remain lower control overhead that consumes less energy; second, it is also necessary to support traditional QoS metrics such as packet loss ratio, packet delay, and throughput; third, fairness needs to be guaranteed so that each node can achieve fair throughput .The main purpose of the proposed ACT scheme is to guarantee a high quality of data by reducing dropped packets due to the congestion. First ACT reduces the amount of generated packets on the source node with compression scheme-DWT, ADPCM, and RLC. Second ACT reduces the amount of transmitting rate on the relaying node with compression scheme under the congestion by adjusting the quantization step on the ADPCM. Third ACT assigns the priority based on the result of DWT to guarantee the reconstruction of data with packet loss. Last, for the fast propagation of congestion notification, the queue is operated adaptively according to the congestion state and queue state.3.2.We consider a densely deployed wireless sensor network such that each sensor node has one or more sensing devices. A sensor node frequently transfers the sensed data to a sink node or gateway. Sensor nodes should control the data transmission interval in an inverse proportion to the variation in a phenomenon. A tree-based routing protocol is used to construct the path from sensor nodes to the sink node. The path to the sink node can be changed whenever the link states to the upper nodes are unreliable due to node failure or obstacles. The lower nodes in a routing tree have to compete with other nodes for an upper node in a routing tree. The errors and losses that occur in the air are corrected by an error-correcting code such as FEC or ACK, which are based on a retransmission technique.ad-hoc network that is deployed without any infrastructure. Therefore, a sensor node should not only act as a data source but also as a relaying node. Packets in the sending queue of a sensor node can be classified into a transit packet received from another node and a source packet generated by the sensor node itself. The routing protocol should reduce the insertion rate of not only the transit packet but also the source packet. Reducing the transit traffic is called as the back pressure, and reducing the source traffic is referred to as the up pressure.A sensor network is a type of 3.3.3.3.1.The ACT uses three types of compression techniques, namely, DWT, ADPCM, and RLC. DWT and RLC are usually used in image compression such as JPEG2000, and ADPCM is used in audio recording –40.Wavelet transform exploits both the spatial and frequency correlation of data by dilations and translations of mother wavelet on the input data. It supports the multi resolution analysis of data, i.e., it can be applied to different scales according to the details required, which allows progressive transmission and zooming of the data without the need of extra storage. Another encouraging feature of wavelet transform is its symmetric nature that is both the forward and the inverse transform has the same complexity, building fast compression and decompression routines. The Discrete Wavelet Transform (DWT), which is based on sub-band coding is found to yield a fast computation of the Wavelet Transform. It is easy to implement and reduces the computation time and resources required. A time-scale representation of the digital signal is obtained using digital filtering techniques. The signal to be analyzed is passed through filters with different cutoff frequencies at different scales. The DWT is computed by successive low-pass and high-pass filtering of the discrete time-domain signal. This is called the Mallat algorithm or Mallat-tree decomposition. Its significance is in the manner it connects the continuous-time multi resolution to discrete-time filters. At each level, the high pass filter produces detail information, while the low pass filter associated with scaling function produces coarse approximations. The Wavelet Transform at high frequencies gives good time resolution and poor frequency resolution, while at low frequency the Wavelet Transform gives good frequency resolution and poor time resolution. Its characteristics well suited for data compression include the ability to take into account of very good energy compaction capabilities, robustness under transmission, high compression ratio etc. Wavelet transform divides the information of data into approximation and detail sub-signals. The approximation sub-signal shows the general trend of data values and other three detail sub-signals show the vertical, horizontal and diagonal details or changes in the data.Adaptive DPCM (ADPCM) is a variant of Differential Pulse-Code Modulation (DPCM) that varies the size of the quantization step, to allow further reduction of the required bandwidth for a given signal-to-noise ratio. DPCM or differential pulse-code modulation is a signal encoder that uses the baseline of PCM but adds some functionalities based on the prediction of the samples of the signal. The input can be an analog signal or a digital signal. A quantizer is used to reduce the number of bits needed to store the transformed coefficients by reducing the precision of those values. As it is a many-to-one mapping, it is a lossy process and is the main source of compression in an encoder. Quantization can be performed on each individual coefficient, which is called Scalar Quantization (SQ). Quantization can also be applied on a group of coefficients together known as Vector Quantization (VQ). Both uniform and non-uniform quantizers can be used depending on the problems.An entropy encoder supplementarily compresses the quantized values losslessly to provide a better overall compression. It uses a model to perfectly determine the probabilities for each quantized value and produces an appropriate code based on these probabilities so that the resultant output code stream is smaller than the input stream. The most commonly used entropy encoders are the Huffman encoder and the arithmetic encoder, although for applications requiring fast execution, simple Run Length Coding (RLC) is very effective. It is important to note that a properly designed quantizer and entropy encoder are absolutely necessary along with optimum signal transformation to get the best possible compression.These compression techniques are usually used in compression of images or audio which are continuous data and have similar data values. We use these compression techniques because the sensed data are continuous and have similar values. DWT, ADPCM, and DWT can drastically reduce the number of packets required by the sensed data.3.3.2.TinyOS Tools for Wavelet Decomposition on motes  providesDistributed Wavelet Transform for Wireless Sensor Networks  implemen3.3.3.3.4.The ACT consists of an APC (adaptive packet compression coder) and an ARC (adaptive rate controller), as shown in The ACP applies the compression techniques  to the source packet generated by the application layer; in case of a sink node, it applies the inverse compression techniques . The ARC controls the packet rate by reducing the number of transit packets in the relaying queue with the help of ADPCM and RLE. It is a reduced version of the APC without DWT. One may implement only the APC and then use only ADPCM and RLC for the ARC. If the APC and ARC are implemented separately, ADPCM and RLC component (or procedure) can be shared between the APC and the ARC.3.4.1.3.4.2.The ARC controls the output packet rate by re-encoding the transit packets in the relaying queue. It consists of RLC, IRLC, ADPCM, and IADPCM units. 3.4.3.3.5.The queue in a network layer works in the First Come First Serve (FCFS) mode . If ther3.6.These compression techniques are usually used in compression of images or audio which are continuous data and have similar data values. We use these compression techniques because the sensed data are continuous and have similar values. DWT, ADPCM, and DWT can drastically reduce the number of packets required by the sensed data. Moreover, in case of congestion, the probability of a packet drop can be reduced by applying ADPCM with a big quantization step size, which reduces the amount of output data from RLC and thus the number of required packets. Dropping a packet causes the loss of the entire data contained in the dropped packet and thus reduces the amount of data, while increasing the quantization step size reduces the fineness (fidelity) of the data.To reduce the packet drop and guarantee the fairness, ACT controls the forwarding rate and the generating rate together. When a node detects the congestion situation, the node notifies its child nodes to reduce the out rate. Child nodes compute the out rate for itself and its child node by dividing the available out rate with the whole number of its child including itself and then also notify their child nodes the congestion with proper out rate. Child nodes then apply the computed rate to the ARC and APC.The ACT assigns a priority to a packet based on the result of the DWT. The DWT divides data into four groups with different frequencies. Usually, a high-frequency group has little energy and may be disregarded, while a low-frequency group has considerable energy and should not be disregarded. Therefore, groups with a low frequency should be assigned a high priority in order to prevent a packet drop during data transmission, as shown in 3.7.Transmission of data is one of the most energy expensive tasks that a node undertakes and using data compression to reduce the number of bits sent reduces energy for transmission. However, compression requires the computation, which also consumes the energy. Trading the computation for transmission can save the energy since typically on the order of 3,000 instructions can be executed for the energy cost required to transmit one bit over a distance of 100 m by radio . Moreove4.4.1.In order to evaluate the performance of the proposed ACT, we simulated CODA, CODA with compression at the source node (CODA-Comp), CRRT and the ACT together by using TOSSIM, which is a simulator for TinyOS. TinyOS is an operating system developed for event-based sensor networks at UC Berkeley. TOSSIM provides a simulation environment that simulates a real sensor network with TinyOS. An application in TinyOS consists of components from each network layer and hardware. The application running on TOSSIM can be run on real sensor nodes such as a micaz or a telos. Thus, we can say that our implemented simulation reflects the real world. 4.2.The network efficiency measures the efficiency of packet transmission in a sensor node. If a packet is successfully delivered from a source node to a sink node, the efficiency is perfect (100%). However, if a packet is dropped somewhere in a network during transmission, the transmissions from the source node to the relaying node that dropped the packet become unnecessary. If many packets are dropped due to congestion, the network efficiency decreases. The network efficiency is related directly to energy because unnecessary packet transmissions unnecessarily dissipate a large amount of energy, which is a very critical resource.On the other hand, the ACT exhibits high efficiency at the beginning of simulation, because it reduces the generated packets with the APC. Although the total number of sensed data packets is increased and network congestion becomes serious, the ACT exhibits high network efficiency. Moreover, the network efficiency of CODA decreases continuously, while that of the ACT increases with time. With CODA, the back pressure is propagated slowly, and the packet flows greatly fluctuate in case of congestion. Therefore, the packet overflows increase, which leads to a decrease in the network efficiency. On the other hand, the ACT controls the quantization step size in proportion to the congestion state, and thus, the total number of packets in the network is decreased. Moreover, it propagates the back pressure fast with adaptive queue operation. As a consequence, its network efficiency is increases slowly. ACT also shows highest value at the average network efficiency . The con4.3.Fairness measures the packet reception ratio from each sensor node at the sink node. Fairness decreases if the sink node receives more packets from some sensor nodes than other sensor nodes and increases if the sink node receives packets from every node fairly. In the simulation, the fairness might be very low because sensor nodes selected the sensing interval randomly. If a sensor node selects a long sensing interval, it will generate packets at a low rate, and thus, less number of packets will be transferred to the sink node, while a sensor node with a short sensing interval will generate packets at a high rate. Moreover, a large number of packets from sensor nodes with a short sensing interval may cause congestion, and the packets from such sensor nodes will also be dropped. Therefore, the fairness will considerably decrease in case of heavy congestion.On the other hand, the ACT exhibits higher fairness than the other strategies does because it simultaneously uses the back pressure and up pressure. It attempts to reduce the number of packets in the network with the help of the ARC. Further, sensor nodes with a short sensing interval will generate more packets that will in turn occupy more space in the queues of relaying nodes. Therefore, the ARC of the ACT attempts to reduce the number of these packets in the congested node. Moreover, with the ACT, all child nodes of the congested node increase the packet transmission interval, and thus, the ACT can achieve higher fairness than other schemes can.4.4.QoD is defined as the ratio of the amount of data obtained by the sink node to the amount of data sensed by sensor nodes. The entire sensed data should arrive at the sink node and used by user applications. During congestion, the network should make the best effort to deliver as much data to the sink node as possible. 4.5.Sensor nodes are usually powered by batteries and thus have very limited energy, therefore maximizing the network lifetime is the main objective of the majority of WSN research works. With data compression, the traffic is reduced and energy consumption is also reduced and the network lifetime can be extended. However ACT is composed by many compression techniques which could be expensive in terms of energy and thus energy efficiency should be compared. 5.Recent advances in MEMS and microprocessor and wireless communication technologies have enabled the deployment of large-scale sensor networks, where thousands or even tens of thousands of small sensors are distributed over a vast area in order to collect sensing data. A large amount of data from a large number of sensor nodes can provide very important information to the users; however, a very large amount of data might cause large-scale congestion in a network.Conventional congestion control protocols usually use a back-pressure scheme that reduces the congestion by reducing the transfer rate of the child nodes of the congested node and by decreasing the packet generation rate of the sensor node that causes heavy traffic. However, packet drops may still occur because the back pressure is propagated slowly due to the collision among sensor nodes in case of serious congestion.In this paper, we proposed a new congestion control technique, named ACT, based on an adaptive compression scheme for packet reduction in case of congestion. Compression techniques used in the ACT are DWT, ADPCM, and RLC. ACT first transforms the data from the time domain to the frequency domain, reduces the range of the data with the help of ADPCM, and then reduces the number of packets by using RLC before transferring the data to the source node. It then introduces the DWT for priority-based congestion control because the DWT classifies the data into four groups with different frequencies. Subsequently, it assigns priorities to these data groups in an inverse proportion to the respective frequencies of the data groups and defines the quantization step size of ADPCM in an inverse proportion to the priorities. RLC generates a smaller number of packets for a data group with a low priority. In the relaying node, the ACT reduces the number of packets by increasing the quantization step size of ADPCM in case of congestion. Moreover, in order to facilitate the back pressure, the queue is controlled adaptively according to the congestion state. We experimentally demonstrated that the ACT increases the network efficiency and guarantees fairness to sensor nodes, as compared with the existing methods and that it exhibits very high ratio of the overall available data in the sink.


The response of a neuron to a time-dependent stimulus, as measured in a Peri-Stimulus-Time-Histogram (PSTH), exhibits an intricate temporal structure that reflects potential temporal coding principles. Here we analyze the encoding and decoding of PSTHs for spiking neurons with arbitrary refractoriness and adaptation. As a modeling framework, we use the spike response model, also known as the generalized linear neuron model. Because of refractoriness, the effect of the most recent spike on the spiking probability a few milliseconds later is very strong. The influence of the last spike needs therefore to be described with high precision, while the rest of the neuronal spiking history merely introduces an average self-inhibition or adaptation that depends on the expected number of past spikes but not on the exact spike timings. Based on these insights, we derive a ‘quasi-renewal equation’ which is shown to yield an excellent description of the firing rate of adapting neurons. We explore the domain of validity of the quasi-renewal equation and compare it with other rate equations for populations of spiking neurons. The problem of decoding the stimulus from the population response (or PSTH) is addressed analogously. We find that for small levels of activity and weak adaptation, a simple accumulator of the past activity is sufficient to decode the original input, but when refractory effects become large decoding becomes a non-linear function of the past activity. The results presented here can be applied to the mean-field analysis of coupled neuron networks, but also to arbitrary point processes with negative self-interaction. population has proven more difficult. There is a bothersome gap between the dynamics of single adapting neurons and the dynamics of populations. Moreover, if we ignore the single neurons and describe directly the population dynamics, we are faced with the ambiguity of the adapting neural code. The neural code of adapting populations is ambiguous because it is possible to observe a range of population activities in response to a given instantaneous input. Somehow the ambiguity is resolved by the knowledge of the population history, but how precisely? In this article we use approximation methods to provide mathematical expressions that describe the encoding and decoding of external stimuli in adapting populations. The theory presented here helps to bridge the gap between the dynamics of single neurons and that of populations.How can information be encoded and decoded in populations of adapting neurons? A quantitative answer to this question requires a mathematical expression relating neuronal activity to the external stimulus, and, conversely, stimulus to neuronal activity. Although widely used equations and models exist for the special problem of relating external stimulus to the action potentials of a single neuron, the analogous problem of relating the external stimulus to the activity of a  Encoding and decoding of information with populations of neurons is a fundamental question of computational neuroscience is the correct rule to decode population activity? What elements of the population history are relevant? What are the basic principles?When driven by a step change in the input, the population of neurons coding for this stimulus responds first strongly but then adapts to the stimulus. To cite a few examples, the activity of auditory nerve fibers adapt to pure tones Synapse- and network-specific mechanisms merge with intrinsic neuronal properties to produce an adapting population response. Here we focus on the intrinsic mechanisms, commonly called spike-frequency adaptation. Spike-frequency adaptation appears in practically all neuron types of the nervous system Mean-field methods were used to describe: attractors The results described in the present article are based on two principal insights. The first one is that adaptation reduces the effect of the stimulus primarily as a function of the expected number of spikes in the recent history and only secondarily as a function of the higher moments of the spiking history such as spike-spike correlations. We derive such an expansion of the history moments from the single neuron parameters. The second insight is that the effects of the refractory period are well captured by renewal theory and can be superimposed on the effects of adaptation.The article is organized as follows: after a description of the population dynamics, we derive a mathematical expression that predicts the momentary value of the population activity from current and past values of the input. Then, we verify that the resulting encoding framework accurately describes the response to input steps. We also study the accuracy of the encoding framework in response to fluctuating stimuli and analyze the problem of decoding. Finally, we compare with simpler theories such as renewal theory and a truncated expansion of the past history moments.To keep the discussion transparent, we focus on a population of unconnected neurons. Our results can be generalized to coupled populations using standard theoretical methods i. e. the presynaptic population activity.How does a population of adapting neurons encode a given stimulating current ctivity  . The popMathematically, we consider a set of spike trains in which spikes are represented by Dirac-pulses centered on the spike time Since the population activity represents the instantaneous firing probability, it is different from the conditional firing intensity, ries see :(2)Ideally, one could hope to estimate del SRM; , which iTo see that the function most recent spike, adaptation refers to the effect of multiple spikes. Adaptation is most clearly observed as a successive increase of interspike intervals in response to a step current. In contrast, a renewal model While refractoriness refers to the interspike-interval distribution and therefore to the dependence upon the Conceptually, contributions of multiple spikes must accumulate to generate spike frequency adaptation. In the Spike Response Model, this accumulation is written as a convolution: i.e. it relates to an appropriately scaled version of the experimentally measured spike after-potential. A depolarizing (facilitating) SAP is associated with The effects described by last spike. Our hypothesis corresponds to the approximation i.e. the last spike needs to be treated explicitly, but we may average across earlier spike times. This approximation is not appropriate for intrinsically bursting neurons, but it should apply well to other cell types . According to this hypothesis, and in analogy to the time-dependent renewal theory In a population of neurons, every neuron has a different spiking history defined by its past spike train We truncate the series expansion resulting from We note that by removing the integral of Let us now assess the domain of validity of the QR theory by comparing it with direct simulations of a population of SRM neurons. To describe the single neurons dynamics, we use a set of parameters characteristic of L2–3 pyramidal cells strong step increase in current starts with a very rapid peak of activity. Indeed, almost immediately after the strong stimulus onset, most of the neurons are triggered to emit a spike. Immediately after firing at The response to a step increase in stimulating current is a standard paradigm to assess adaptation in neurons and used here as a qualitative test of our theory. We use three different step amplitudes: weak, medium and strong. The response of a population of, say, 25,000 model neurons to a In contrast to renewal models , we observe in response . An apprThe QR equation describes well both the damped oscillation and the adapting tail of the population activity response to steps . Also, tStep changes in otherwise constant input are useful for qualitative assessment of the theory but quite far from natural stimuli. Keeping the same SAP as in Decoding the population activity requires solving the QR equation  and 8 fo see . Input s see  but also see . Again,  see . The numecreases . Finite ecreases ). Anotheecreases –F. Also, to zero . Neverth to zero .We will consider two recent theories of population activity from the literature. Both can be seen as extensions of rate models such as the Linear-Nonlinear Poisson model where the activity of a homogeneous population is all previous spikes, that is, no special treatment for the most recent one. In this caseTo discuss the relation to existing theories, we recall that the instantaneous rate of our model xpansion  offers aWe compare the prediction of EME1, EME2 and renewal theory with the simulated responses to step inputs . All theaptation .Fluctuating input makes the population respond in peaks of activity separated by periods of quiescence. This effectively reduces the coupling between the spikes and therefore improves the accuracy of EME1. The validity of EME1 for encoding time-dependent stimulus  decreaseDecoding with EME1 is done according to a simple relation:In summary, the EMEs yield theoretical expressions for the time-dependent as well as steady-state population activity. These expressions are valid in the limit of small coupling between the spikes which corresponds to either large interspike intervals or small SAP. Renewal theory on the other hand is valid when the single-neuron dynamics does not adapt and whenever the refractory effects dominate.The input-output function of a neuron population is sometimes described as a linear filter of the input We have derived self-consistent formulas for the population activity of independent adapting neurons. There are two levels of approximation, EME1  is validin vivo neurons of the cortex The QR equation captures almost perfectly the population code for time-dependent input even at the high firing rates observed in retinal ganglion cells We have focused here on the Spike Response Model with escape noise which is an instantiation of a Generalized Linear Model. The escape noise model, defined as the instantaneous firing rate et al.The decoding schemes presented in this paper  and 45 rUsing the results presented here, existing mean-field methods for populations of spiking neurons can readily be adapted to include spike-frequency adaptation. In The scope of the present investigation was restricted to unconnected neurons. In the mean-field approximation, it is straight-forward to extend the results to several populations of connected neurons This section is organized in 3 subsections. Subsection A covers the mathematical steps to derive the main theoretical results , 5 and 7earlier spikes so that one Dirac-delta function gives a non-vanishing term because only the integral over The probability density of a train of   spikes  in frontIn order to single out the effect of the previous spike, we replace We can recognize in A derivation of the renewal equation Applying and to survive from all previous spikes. The QR approximation means that we approximate this decay by averaging over all possible spike trains before First consider the expected value in At the steady state with a constant input All simulations were performed on a desktop computer with 4 cores  using Matlab . The Matlab codes to numerically solve the self-consistent equations are made available on the author's websites. The algorithmic aspects of the numerical methods are discussed now.All temporal units in this code are given in milliseconds. Direct simulation of is added . TypicalFor all simulations, the baseline current was 10 pA (except for time-dependent current where the mean was specified), the baseline excitability was Time-dependent input consisted of an Ornstein-Uhlenbeck process which is computed at every time step as:We consider the QR equation, tion see  A and coThe second vector We can therefore calculate the population activity iteratively at each time bin using Isolating the input tion of   and 42, The structure of EME1 and EME2 allows us to use a nonlinear grid spacing in order to save memory resources. The bins should be small where To perform the numerical integration, we define the vector To compute the second order equation, we first build the correlation vector The first-order expansion  can be uof input .When assessing the accuracy of the encoding or the decoding, we used the correlation coefficient. The correlation coefficient is the variance-normalized covariance between two random variables Figure S1Statistics of decoding performance. (A–B) Correlation coefficient between original filtered input recovered from the activity of a population of A) 20 ms and (B) 2 ms. (C) Mean squared error associated with an instantaneous firing rate (D–E) Fraction of input times at which decoding could be performed corresponding to A and B, respectively. Decoding could not be carried out when the stimulus was outside the dynamic range which corresponds to F) Fraction of times where the activity was non-zero as a function of the population size. Colors show different standard deviation of the original input with values in pA, other parameters idem as (TIF)Click here for additional data file.Figure S2Role of SAP for Renewal theory, EME1 and EME2 for step input. Population activity responses  to the step current input . The neuron population follows spike-response model dynamics with effective SAP A–C) shows exemplar traces for different SAP amplitude and input steps: (A) B) C) D Renewal, E EME1, F, EME2) for various values of the SAP amplitude (TIF)Click here for additional data file.Figure S3Encoding time-dependent stimuli in the population activity with Event-Based Moment Expansion (EME). (A) Population activity responses . The difference between direct simulation and theory is shown in the top panel.The stimulus is an Ornstein-Uhlenbeck process with correlation time constant of 300 ms with STD increasing every 2 seconds  and a mean of 10 pA. (B) Correlation coefficients between direct simulation and EME1 for various STDs and mean (in pA) of the input current. Results of (TIF)Click here for additional data file.Figure S4Decoding the stimulus from the population activity with EME1. (A–D) The original  and decoded stimulus  recovered from the PSTH of 25,000 independent SRM neurons  using E) Correlation coefficient of original and decoded input as a function of input STD, shown for three distinct mean input ((TIF)Click here for additional data file.