Exponential Decay Terms Research Articles

Analysis of Traveling Wavefront Information and Wavefront Protection for Power Electronic Device-Interfaced AC System

The power electronic device-interfaced ac system becomes an important scene of renewable energy power system. The response of the power-electronic devices confronting faults is restricted by their overcurrent withstand capability, showing weak-feed characteristic. Therefore, the sensitivity of protection can be ensured only when the fault can be located before fault ride through of power-electronic device. In this paper, the expression of fault traveling wavefront is derived first and it is pointed out that the expression contains two terms, which are power frequency sine term and exponential decay term. The index of the exponential decay term is only determined by fault distance, whereas the fault occurrence time, fault type, and fault impedance only influence amplitudes of the two terms. In this way, the fault distance and fault severity can be characterized by index and amplitude coefficients respectively and thereby be decoupled. On this basis, an algorithm is proposed to extract fault distance and fault severity information from wavefront, then a novel protection principle is proposed using fault distance information contained in wavefront. The proposed wavefront protection can fast locate faults and have higher sensitivity and reliability compared with the existing non-unit protections, which is verified by numerous simulations and experimental tests.

Proportional-Integral State Estimator for Quaternion-Valued Neural Networks With Time-Varying Delays.

This brief investigates the problem of state estimation of quaternion-valued neural networks (QVNNs) with time-varying delays. First, by extending the Jensen inequality to quaternion domain, an extended Jensen inequality with quaternion term is derived. Second, a class of proportional-integral state estimator (PISE) with exponential decay term is proposed. Then, by constructing a suitable Lyapunov-Krasovskii functional (LKF), some sufficient conditions are obtained to ensure the existence of the designed PISE and the gain matrices of the designed PISE can be directly computed. Simulations are given to illustrate the advantage of the proposed method.

Fully Distributed Formation Control of General Linear Multiagent Systems Using a Novel Mixed Self- and Event-Triggered Strategy

In this study, the state formation control issue of general linear networked multi-agent systems (MASs) is considered. Combining self-triggered strategy with general event-triggered strategy, a novel fully distributed asynchronous mixed self- and event-triggered control strategy is proposed, which can make the MASs achieve the prescribed state formation structure. The control strategy proposed in this study does not depend on any global network information. Therefore, no matter how large scale of the network is, the control strategy is feasible. Meanwhile, the control strategy uses the sampled state information at triggering instants instead of the real-time state information, which efficiently eliminates continuous communication among agents. Different from the existing studies, the event-triggered detector starts to work if and only if the next self-triggering instant comes in the control strategy proposed in this article. Thus, the control strategy can save more communication resources between sensor and event-triggered detector within the same inter-event interval compared with the previous event-triggered strategies. In addition, the control strategy designs a exponential decay term in the triggering function to rule out the unexpected Zeno behavior and reduce the triggering number. Finally, the numerical simulation result of multi-robot formation is given, which demonstrates the feasibility and performance of the proposed control strategy.

A Knowledge Query Network Model Based on Rasch Model Embedding for Personalized Online Learning.

The vigorous development of online education has produced massive amounts of education data. How to mine and analyze education big data has become an urgent problem in the field of education and big data knowledge engineering. As for the dynamic learning data, knowledge tracing aims to track learners’ knowledge status over time by analyzing the learners’ exercise data, so as to predict their performance in the next time step. Deep learning knowledge tracking performs well, but they mainly model the knowledge components while ignoring the personalized information of questions and learners, and provide limited interpretability in the interaction between learners’ knowledge status and questions. A context-aware attentive knowledge query network (CAKQN) model is proposed in this paper, which combines flexible neural network models with interpretable model components inspired by psychometric theory. We use the Rasch model to regularize the embedding of questions and learners’ interaction tuples, and obtain personalized representations from them. In addition, the long-term short-term memory network and monotonic attention mechanism are used to mine the contextual information of learner interaction sequences and question sequences. It can not only retain the ability to model sequences, but also use the monotonic attention mechanism with exponential decay term to extract the hidden forgetting behavior and other characteristics of learners in the learning process. Finally, the vector dot product is used to simulate the interaction between the learners’ knowledge state and questions to improve the interpretability. A series of experimental results on 4 real-world online learning datasets show that CAKQN has the best performance, and its AUC value is improved by an average of 2.945% compared with the existing optimal model. Furthermore, the CAKQN proposed in this paper can not only track learners’ knowledge status like other models but also model learners’ forgetting behavior. In the future, our research will have high application value in the realization of personalized learning strategies, teaching interventions, and resource recommendations for intelligent online education platforms.

Impact of Program/Erase Cycling Interval on the Transconductance Distribution of NAND Flash Memory Devices

This study comprehensively investigates the impact of the time interval (t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">wait</sub> ) between program/erase (P/E) cycles on the oxide quality of NAND Flash memory devices. It is observed that, at room temperature, P/E cycles with a shorter t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">wait</sub> yield a better oxide quality than those with a longer t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">wait</sub> . The oxide charge (Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> ) evolution by distributed P/E cycles can be well described by an analytical equation through the extension of our previous statistical transconductance reduction (G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m,max</sub> ) method. This equation is characterized by a power of the number of P/E cycles multiplied by an exponential decay term of t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">wait</sub> . The former term is related to the Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> generation, whereas the latter is related to the Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> emission during t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">wait</sub> between cycles. This model allows to evaluate the activation energy for both Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> generation (E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A,G</sub> ) and recovery (E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A,R</sub> ). E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A,G</sub> is revealed to be a function of the logarithmic scale of t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">wait</sub> , decreasing from 100 to 85 meV for t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">wait</sub> varying from 0.1 to 4 s. Moreover, E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A,R</sub> is approximately 0.4 eV, which is consistent with the value of electron thermal emission from traps in SiO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> .

Hybrid Event-Based Leader-Following Consensus of Nonlinear Multiagent Systems With Semi-Markov Jump Parameters

This article aims to systematically address the leader-following consensus issue for continuous-time multiagent systems with semi-Markov jump parameters. A more universal random process, semi-Markov process, is utilized to model the variations of parameters which may result from the complexity of environment. The innovative hybrid event-triggered strategy associated with an improved threshold function is employed for economizing the network bandwidth resources. It presents an exponential decay term in the threshold function for decreasing the transmission frequency of superfluous data packets, ulteriorly reducing triggering times. The dominating core concentrates on the design of a controller interrelated with the hybrid event-triggered condition, which can be implemented to ensure the leader-following consensus of the considered systems. On the strength of time-dependent and piecewise Lyapunov–Krasovskii functional, several criteria, which are capable of ensuring the leader-following consensus of the multiagent systems, are derived. Ultimately, two suitable numerical examples and a meaningful practical one are utilized to demonstrate the validity and potential of the derived results.

Variable-sampling-period dependent global stabilization of delayed memristive neural networks based on refined switching event-triggered control

This paper studies the stabilization problem of delayed memristive neural networks under event-triggered control. A refined switching event-trigger scheme that switches between variable sampling and continuous event-trigger can be designed by introducing an exponential decay term into the threshold function. Compared with the existing mechanisms, the proposed scheme can enlarge the interval between two successively triggered events and therefore can reduce the amount of triggering times. By constructing a time-dependent and piecewise-defined Lyapunov functional, a less-conservative criterion can be derived to ensure global stability of the closed-loop system. Based on matrix decomposition, equivalent conditions in linear matrix inequalities form of the above stability criterion can be established for the co-design of both the trigger matrix and the feedback gain. A numerical example is provided to demonstrate the effectiveness of the theoretical analysis and the advantages of the refined switching event-trigger scheme.

Delayed output feedback sliding mode control for uncertain non‐linear systems

This study proposes an output feedback sliding mode control with an artificial delay. The output information-based sliding surface is outlined. An artificial delay as well as an exponential decay term is introduced in the sliding surface, which reflects these additional terms (artificial delay and exponential decay term) in the reduced-order dynamics. The idea and advantage of utilising the artificial delay in the sliding surface are to stabilise the class of systems, which are not able to stabilise with only output feedback. The employment of exponential decay term in the sliding surface provides a reaching phase free behaviour. It implies that the authors' approach is insensitive to external matched disturbances from the very initial time. With the aid of Lyapunov–Krasovskii functional, local input to state stability for the reduced-order dynamics as well as for the closed-loop system is presented. Finally, the simulation results of the translation oscillator with rotating actuator system are presented to validate the effectiveness of the offered technique.

Threshold-Function-Dependent Quasi-Synchronization of Delayed Memristive Neural Networks via Hybrid Event-Triggered Control

This article addresses the quasi-synchronization problem of delayed memristive neural networks (MNNs) via hybrid event-triggered control. First, a hybrid event-triggering mechanism with a novel threshold function is devised. Therein, an exponential decay term and a non-negative constant term are additionally introduced. It can further extend the time span between two successively triggered events and therefore can reduce the amount of triggering times in comparison with some existing event-triggering mechanisms. Then, by constructing a time-dependent and piecewise Lyapunov functional, a less conservative criterion for quasi-synchronization of drive-response delayed MNNs is formulated in terms of linear matrix inequalities. In addition, an explicit expression of the error bound is provided and the design of the feedback gain is presented for a predetermined error bound. Finally, a numerical example is given to demonstrate the effectiveness of the theoretical analysis and the advantages of the proposed event-triggering scheme.

A waiting-time-based event-triggered scheme for stabilization of complex-valued neural networks

This paper addresses the global stabilization of complex-valued neural networks (CVNNs) via event-triggered control. First, a waiting-time-based event-triggered scheme is designed to reduce the data transmission rate. Therein, an exponential decay term is introduced into the predefined threshold function, which may postpone the triggering instant of the necessary data and therefore reduce the frequency of data transmission. Then, with the help of the input delay approach, a time-dependent piecewise-defined Lyapunov–Krasovskii functional is constructed for closed-loop system to formulate a less conservative stability criterion. In addition, by resorting to matrix transformation, the co-design method for both the feedback gains and the trigger parameters is derived. Finally, a numerical example is given to illustrate the feasibility and superiority of the proposed event-triggered scheme and the obtained theoretical results.

On sampling of scattering phase functions

Abstract Monte Carlo radiative transfer, which has been demonstrated as a successful algorithm for modelling radiation transport through the astrophysical medium, relies on sampling of scattering phase functions. We review several classic sampling algorithms such as the tabulated method and the accept–reject method for sampling the scattering phase function. The tabulated method uses a piecewise constant approximation for the true scattering phase function; we improve its sampling performance on a small scattering angle by using piecewise linear and piecewise log-linear approximations. It has previously been believed that certain complicated analytic phase functions such as the Fournier–Forand phase function cannot be simulated without approximations. We show that the tabulated method combined with the accept–reject method can be applied to sample such complicated scattering phase functions accurately. Furthermore, we introduce the Gibbs sampling method for sampling complicated approximate analytic phase functions. In addition, we propose a new modified Henyey–Greenstein phase function with exponential decay terms for modelling realistic dust scattering. Based on Monte Carlo simulations of radiative transfer through a plane-parallel medium, we also demonstrate that the result simulated with the new phase function can provide a good fit to the result simulated with the realistic dust phase function.

Fundamental solutions to the regularised 13-moment equations: efficient computation of three-dimensional kinetic effects

Fundamental solutions (Green’s functions) are derived for the regularised 13-moment system (R13) of rarefied gas dynamics, for small departures from equilibrium; these solutions show the presence of Knudsen layers, associated with exponential decay terms, that do not feature in the solution of lower-order systems (e.g. the Navier–Stokes–Fourier equations). Incorporation of these new fundamental solutions into a numerical framework based on the method of fundamental solutions (MFS) allows for efficient computation of three-dimensional gas microflows at remarkably low computational cost. The R13-MFS approach accurately recovers analytic solutions for low-speed flow around a stationary sphere and heat transfer from a hot sphere (for which a new analytic solution has been derived), capturing non-equilibrium flow phenomena missing from lower-order solutions. To demonstrate the potential of the new approach, the influence of kinetic effects on the hydrodynamic interaction between approaching solid microparticles is calculated. Finally, a programme of future work based on the initial steps taken in this article is outlined.

Assesment of the response of the meteorological/hydrological parameters on the soil gas radon emission at Hsinchu, northern Taiwan: A prerequisite to identify earthquake precursors

The present study is an attempt to assess and quantify the influence of the meteorological (atmospheric temperature and pressure) and hydrological (rainfall and ground water head-GWH) parameters on the soil gas radon emission at Hsinchu, northern Taiwan. The quasi-periodic variations corresponding to diurnal and semi diurnal periods were estimated and eliminated by decomposing the time series for the period of September 16, 2009 to March 5, 2010 to singular spectrum analysis. The reconstructed non-periodic variations, which reproduce the salient feature of recorded time series, were searched for meteorological/hydrological influences in radon emission. The combined response of barometric pressure and atmosphere temperature are found to be small when compared to the total variability in radon. The influence of rainfall on radon is found to be strongest. At the onset of rainfall, radon shows a step-jump that attains peak with a time lag of 12–15h. This enhancement is attributed to entrapment of soil gas in the top soil cover as increased soil moisture prevents escape of radon into the atmosphere (capping effect). The decay of radon after the recession of rainfall is approximated by double exponential decay terms, one corresponding to the natural decay of radon with half life of 3.84days and second representing slow weakening of capping effect. The third effect related to internal loading due to rise and fall of groundwater modulates the propagation of radon in overlying strata, accounting for the long term variations in radon. The rainfall inflicted changes in radon look strikingly similar to earthquake related precursory or co-seismic perturbations, inferred by long term synotopic observations. It is surmised that unless radon variations are corrected for meteorological/hydrological contamination, some precursory signals are masked on one hand while on the other hand some anomalies are falsely viewed as earthquake precursors.

Interpretation of fast-ion signals during beam modulation experiments

Fast-ion signals produced by a modulated neutral beam are used to infer fast-ion transport. The measured quantity is the divergence of perturbed fast-ion flux from the phase-space volume measured by the diagnostic, . Since velocity-space transport often contributes to this divergence, the phase-space sensitivity of the diagnostic (or ‘weight function’) plays a crucial role in the interpretation of the signal. The source and sink make major contributions to the signal but their effects are accurately modelled by calculations that employ an exponential decay term for the sink. Recommendations for optimal design of a fast-ion transport experiment are given, illustrated by results from DIII-D measurements of fast-ion transport by Alfvén eigenmodes. The signal-to-noise ratio of the diagnostic, systematic uncertainties in the modeling of the source and sink, and the non-linearity of the perturbation all contribute to the error in .

Fluvial seed dispersal of riparian trees: transport and depositional processes

AbstractFluvial seed dispersal considers both the transport and deposition of seeds where channel geomorphic structures, hydrology and seed dispersal traits contribute to transport times and depositional locations. This study examines the influence of stream flow patterns on fluvial seed dispersal of buoyant white alder (Alnus rhombifolia) seeds by applying a one‐dimensional transport model. Conceptually, the model separates the stream into two components: (i) the main channel where the seeds are transported downstream; and (ii) the transient storage zone where seeds are temporarily detained or deposited on the river bank. Transport processes are characterized by an advection–dispersion equation which is coupled to a transient storage model using an exponential decay term. The model parameters: longitudinal dispersion (DL), exchange coefficient (α), main channel area (A) and storage zone (As) are estimated based on field experiments conducted in a confined, bedrock‐gravel bed river with pool‐riffle morphology located in coastal northern California. The riparian zone is inhabited by Alnus rhombifolia that disperse buoyant seeds in mid‐spring coinciding with the end of the wet, Mediterranean season. Artificial seeds, with similar traits of buoyancy and density to alder seeds, were used to quantify transport times and depositional locations. Preferential deposition resulted in stream reaches with larger As, high As/A ratios, and faster exchange coefficients corresponding to divergent stream flow (back‐eddies, re‐circulating flow, flow expansions) caused by geomorphic structures such as the ends of bar/riffle features and bends in the stream. The results demonstrate the importance of transient storage for seed transport and depositional processes. Morphological features that increase a channel's complexity create complex flow structures that detain seeds and provide a greater opportunity for deposition to occur. The model provides a simplification of river hydraulics to represent dispersal dynamics and lends itself to further understanding of hydrochory processes and associated population structure. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Effect of insoluble calcium concentration on endogenous syneresis rate in rennet-coagulated bovine milk

The rennet coagulation of milk has been extensively studied. Mathematical modeling of the gelation process has been performed, mainly for the purpose of predicting the gel point. Rheological profiles of rennet gels during aging (long reaction times) have indicated that the gel stiffness (modulus) attains a maximum and thereafter decreases. We wanted to model this type of behavior and used the Carlson model, which includes terms for the proteolysis of κ-casein hairs (creating active sites) and the crosslinking of these activated sites. To account for the observed decrease in the gel modulus with time, we modified the Carlson model by adding an exponential decay term, which we ascribe to endogenous syneresis. We believe that this decay (i.e., syneresis rate) would likely be influenced by the mobility of bonds within casein micelles (in gels as indicated by the rheological loss tangent parameter). To modify the internal structural bonding of casein micelles, reconstituted skim milk was acidified to pH values 6.4, 6.0, 5.8, 5.6, and 5.4, or EDTA was added to milk at concentrations of 0, 2, 4, and 6mM, and the final pH values of EDTA-treated samples were subsequently adjusted to pH 6.0. These treatments were then used to prepare rennet gel samples that were monitored by dynamic low amplitude oscillatory rheometry. When the modified Carlson model was fitted to the actual experimental storage modulus values of each sample, it fitted the data reasonably well (especially the pH trial data). As the pH values of milk decreased, the modulus values at infinite reaction time (G′∞) increased; however, G′∞ decreased with an increase in the EDTA concentration. In the pH trial, the rate constants for the proteolysis of κ-casein hairs and the crosslinking of these activated sites exhibited a maximum at pH 5.6 and 6.0, respectively. The rate constant for endogenous syneresis increased at pH values <6.0. The rate constant for endogenous syneresis was significantly positively correlated (r≥0.96) with the loss tangent values of gels (indicating greater mobility), probably due to the loss of insoluble calcium phosphate crosslinking within micelles, which was significantly negatively correlated (r≥0.81) with the rate constant for endogenous syneresis. In the EDTA trial, with an increase in the EDTA concentration no maximum was observed in the rate constants related to proteolysis of κ-casein hairs or crosslinking of these activated sites. The rate constant for endogenous syneresis decreased at higher EDTA levels. The different rheological/modeling behavior in the EDTA trials was likely due to the very significant inhibition of rennet gelation induced by the use of EDTA, which also resulted in extremely long reaction times. Our modified Carlson model fit our experimental pH trial data very well, which indicates that the rennet gel system has the potential to synerese from the start; indeed this ability is an innate property of the casein micelle. Endogenous syneresis was enhanced by the loss of insoluble calcium phosphate crosslinking within casein micelles as this increased bond mobility within rennet gels.

Correction Approach for Delta Function Convolution Model Fitting of Fluorescence Decay Data in the Case of a Monoexponential Reference Fluorophore.

A correction is proposed to the Delta function convolution method (DFCM) for fitting a multiexponential decay model to time-resolved fluorescence decay data using a monoexponential reference fluorophore. A theoretical analysis of the discretised DFCM multiexponential decay function shows the presence an extra exponential decay term with the same lifetime as the reference fluorophore that we denote as the residual reference component. This extra decay component arises as a result of the discretised convolution of one of the two terms in the modified model function required by the DFCM. The effect of the residual reference component becomes more pronounced when the fluorescence lifetime of the reference is longer than all of the individual components of the specimen under inspection and when the temporal sampling interval is not negligible compared to the quantity (τR−1 – τ−1)−1, where τR and τ are the fluorescence lifetimes of the reference and the specimen respectively. It is shown that the unwanted residual reference component results in systematic errors when fitting simulated data and that these errors are not present when the proposed correction is applied. The correction is also verified using real data obtained from experiment.

Improved Model of Long-Term Gain Increases in Traveling-Wave Tubes

An improved model to predict the gain increases in traveling-wave tubes (TWTs) during long-term operation is presented. The conventional gain growth model describes the pressure variation in a TWT over its life using an exponential decrease from the initial outgassing level to a constant base pressure. This model often shows an inability to capture the gain change behavior of many tubes during the transition between early life burn-in and long-term operation, leading to a significant underprediction of long-term gain increases. The model is improved here first through the introduction of another pressure-related term associated with desorption of gas from the tube's inner surfaces that exhibits a t <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1/2</sup> time dependence. This new pressure dependence is governed by the behavior of a Langmuirian adsorption isotherm. Second, the exponential pressure decay term is separated into two terms associated with early life and long-term operation with different outgassing time constants. The improved model shows a significantly better matching of long-term TWT gain growth data compared to the conventional model. In addition, the improved model predicts a more physical pressure behavior in the TWT with time.

Innovative Mars entry integrated navigation using modified multiple model adaptive estimation

A variety of anticipated missions to Mars, such as Mars base and sample return, will have to utilize precision navigation and guidance for their hypersonic entry to achieve the required landing accuracy. In order to effectively reduce the adverse impact of initial state errors and parameter uncertainties during Mars atmospheric entry and then improve entry navigation accuracy, an innovative high-precision integrated navigation algorithm for Mars entry is developed based on a modified multiple model adaptive estimation (MMAE) in this paper. First, a six degree-of-freedom (DOF) Mars entry dynamics model is derived based on the angular velocity outputs of a gyro, which can be more complete and accurate description of the state variables of an entry vehicle than the traditional three DOF dynamics models. Second, both the accelerometer outputs and radiometric measurements are adopted as the navigation observations embedded to Kalman filter bank to perform state estimation and suppress the measurement noise. Finally, a new modified multiple model adaptive estimation algorithm with exponential decay terms is proposed to overcome the inherent drawback of classical MMAE and then further improve navigation accuracy. The numerical simulation results show that the integrated navigation algorithm developed in this paper is able to accurately estimate the state variables of an entry vehicle, even in the case that the initial entry state errors and larger parameter uncertainties are taken into account.

Neural population coding of movement direction for path integration

Many insects including desert ants (Cataglyphis fortis) and honeybees can return their home after foraging. Desert ants choose a direct path to return home after a long journey, and it is believed that they use path integration approach. Path integration process needs internal integrator to accumulate their traveled distance and direction to estimate the homing vector. Biologists showed that Cataglyphis fortis uses a step integrator as an odometer [1]. The path integration system of the desert ants has an interesting aspect, and Mueller and Wehner [2] showed that there are errors of path integration in two-leg trajectory experiments and their polar coordinate system model fits well to the experimental data. Homing direction error may be systematic and with these estimation errors, desert ants may find their nest efficiently by combining egocentric information and visual information including landmarks in the environment. By observing the desert ant's estimation of distance travelled, Sommer and Wehner [3] found that desert ants underestimate their travelled distance. These underestimated errors grow as their travel distance increases. They tried to fit these underestimated distances versus travelled distance results with some model functions: linear, power, logarithmic and exponential. Among these model functions, logarithmic and exponential models show good fit with experimental data (Sommer and Wehner, 2004). Also Merkle et al. [4] proposed an egocentric polar coordinate model to explain the leaky integrator of the desert arthropods. Wittmann and Schwegler [5] showed a neural network model for path integration, which has a population coding of neurons for movement direction. Here we introduce a new neuronal model for path integration system. It includes the first-order low-pass filter for a leaky integrator and a population coding of neurons with a circular array of direction neurons. The movement direction is encoded in the population coding, which uses allocentric information from a polarized light. An exponential decay term for travelled distance is now expressed with a low-pass filter, which can be easily implemented with a simple neuron. A circular array structure of neurons, where each unit represents a movement direction, can encode the movement direction as a form of population coding, and the integration over neuron activations includes the accumulated path for each direction. A decoding process over those neuron activations can estimate the homing direction as well as the homing distance. Each neuron unit encompasses a leaky integrator system as often observed in a neuronal process. Our experiments show that this leaky integrator system triggers errors in two-leg trajectory movements of an agent, and our simple model fits well to the experimental data for two-leg trajectory experiments. This systematic error in path integration can help desert ants easily find their home with visual landmark navigation. We assume a circular array of directionally sensitive neurons to explain the path integration system of desert ants, but what type of neural structure is available in a real neuron system of desert ants or what kind of neural interactions in a circular array of neurons is needed for the path integration is an open question. We need further study on these subjects. More sophisticated models and experiments might explain those questions. Figure 1 Diagram for neuronal population coding of movement direction.