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1. Brief Announcement: Integrating Temporal Information to Spatial Information in a Neural Circuit

   Wang, Brabeeba ; Lynch, Nancy ( January 2019 , 33rd International Symposium on Distributed Computing (DISC))

   null (Ed.)

   In this paper, we consider networks of deterministic spiking neurons, firing synchronously at discrete times. We consider the problem of translating temporal information into spatial information in such networks, an important task that is carried out by actual brains. Specifically, we define two problems: "First Consecutive Spikes Counting" and "Total Spikes Counting", which model temporal-coding and rate-coding aspects of temporal-to-spatial translation respectively. Assuming an upper bound of T on the length of the temporal input signal, we design two networks that solve two problems, each using O(log T) neurons and terminating in time T+1. We also prove that these bounds are tight. 

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2. Brief Announcement: Integrating Temporal Information to Spatial Information in a Neural Circuit

   Wang, Brabeeba ; Lynch, Nancy ( January 2019 , 33rd International Symposium on Distributed Computing (DISC))

   In this paper, we consider networks of deterministic spiking neurons, firing synchronously at discrete times. We consider the problem of translating temporal information into spatial information in such networks, an important task that is carried out by actual brains. Specifically, we define two problems:“First Consecutive Spikes Counting” and “Total Spikes Counting”, which model temporal-coding and rate-coding aspects of temporal-to-spatial translation respectively. Assuming an upper bound of T on the length of the temporal input signal, we design two networks that solve two problems, each using O(logT) neurons and terminating in time T+ 1. We also prove that these bounds are tight. 

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3. Travelling Waves for Adaptive Grid Discretizations of Reaction Diffusion Systems III: Nonlinear Theory

   https://doi.org/10.1007/s10884-022-10143-4  

   Hupkes, H. J. ; Van Vleck, E. S. ( April 2022 , Journal of Dynamics and Differential Equations)

   In this paper we consider a spatial discretization scheme with an adaptive grid for the Nagumo PDE and establish the existence of travelling waves. In particular, we consider the time dependent spatial mesh adaptation method that aims to equidistribute the arclength of the solution under consideration. We assume that this equidistribution is strictly enforced, which leads to the non-local problem with infinite range interactions that we derived in Hupkes and Van Vleck (J Dyn Differ Eqn, 2021). Using the Fredholm theory developed in Hupkes and Van Vleck (J Dyn Differ Eqn, 2021) we setup a fixed point procedure that enables the travelling PDE waves to be lifted to our spatially discrete setting.

    

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4. High Spatial Order Energy Stable FDTD Methods for Maxwell’s Equations in Nonlinear Optical Media in One Dimension

   Bokil, Vrushali A ; Cheng, Yingda ; Jiang, Yan ; Li, Fengyan ( April 2018 , Journal of scientific computing)

   In this paper, we consider electromagnetic (EM) wave propagation in nonlinear optical media in one spatial dimension. We model the EM wave propagation by the time- dependent Maxwell’s equations coupled with a system of nonlinear ordinary differential equations (ODEs) for the response of the medium to the EM waves. The nonlinearity in the ODEs describes the instantaneous electronic Kerr response and the residual Raman molecular vibrational response. The ODEs also include the single resonance linear Lorentz dispersion. For such model, we will design and analyze fully discrete finite difference time domain (FDTD) methods that have arbitrary (even) order in space and second order in time. It is challenging to achieve provable stability for fully discrete methods, and this depends on the choices of temporal discretizations of the nonlinear terms. In Bokil et al. (J Comput Phys 350:420–452, 2017), we proposed novel modifications of second-order leap-frog and trapezoidal temporal schemes in the context of discontinuous Galerkin methods to discretize the nonlinear terms in this Maxwell model. Here, we continue this work by developing similar time discretizations within the framework of FDTD methods. More specifically, we design fully discrete modified leap-frog FDTD methods which are proved to be stable under appropriate CFL conditions. These method can be viewed as an extension of the Yee-FDTD scheme to this nonlinear Maxwell model. We also design fully discrete trapezoidal FDTD methods which are proved to be unconditionally stable. The performance of the fully discrete FDTD methods are demonstrated through numerical experiments involving kink, antikink waves and third harmonic generation in soliton propagation. 

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5. A spatiotemporal weighted regression model (STWR v1.0) for analyzing local nonstationarity in space and time

   https://doi.org/10.5194/gmd-13-6149-2020  

   Que, Xiang ; Ma, Xiaogang ; Ma, Chao ; Chen, Qiyu ( January 2020 , Geoscientific Model Development)

   null (Ed.)

   Abstract. Local spatiotemporal nonstationarity occurs in various naturaland socioeconomic processes. Many studies have attempted to introduce timeas a new dimension into a geographically weighted regression (GWR) model,but the actual results are sometimes not satisfying or even worse than theoriginal GWR model. The core issue here is a mechanism for weighting the effectsof both temporal variation and spatial variation. In many geographical andtemporal weighted regression (GTWR) models, the concept of time distance hasbeen inappropriately treated as a time interval. Consequently, the combinedeffect of temporal and spatial variation is often inaccurate in theresulting spatiotemporal kernel function. This limitation restricts theconfiguration and performance of spatiotemporal weights in many existingGTWR models. To address this issue, we propose a new spatiotemporal weightedregression (STWR) model and the calibration method for it. A highlight ofSTWR is a new temporal kernel function, wherein the method for temporalweighting is based on the degree of impact from each observed point to aregression point. The degree of impact, in turn, is based on the rate ofvalue variation of the nearby observed point during the time interval. Theupdated spatiotemporal kernel function is based on a weighted combination ofthe temporal kernel with a commonly used spatial kernel (Gaussian orbi-square) by specifying a linear function of spatial bandwidth versus time.Three simulated datasets of spatiotemporal processes were used to test theperformance of GWR, GTWR, and STWR. Results show that STWR significantlyimproves the quality of fit and accuracy. Similar results were obtained byusing real-world data for precipitation hydrogen isotopes (δ2H) in the northeastern United States. The leave-one-out cross-validation(LOOCV) test demonstrates that, compared with GWR, the total predictionerror of STWR is reduced by using recent observed points. Predictionsurfaces of models in this case study show that STWR is more localized thanGWR. Our research validates the ability of STWR to take full advantage ofall the value variation of past observed points. We hope STWR can bringfresh ideas and new capabilities for analyzing and interpreting localspatiotemporal nonstationarity in many disciplines. 

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