scholarly journals On the Simulation of Nonlinear Bidimensional Spiking Neuron Models

2011 ◽  
Vol 23 (7) ◽  
pp. 1704-1742 ◽  
Author(s):  
Jonathan Touboul
Keyword(s):  
Differential Equation ◽  
Membrane Potential ◽  
Cortical Neurons ◽  
Fixed Time ◽  
Computational Time ◽  
Integration Step ◽  
Large Network ◽  
Time Step ◽  
Integration Schemes ◽  
Adaptation Variable

Bidimensional spiking models are garnering a lot of attention for their simplicity and their ability to reproduce various spiking patterns of cortical neurons and are used particularly for large network simulations. These models describe the dynamics of the membrane potential by a nonlinear differential equation that blows up in finite time, coupled to a second equation for adaptation. Spikes are emitted when the membrane potential blows up or reaches a cutoff θ. The precise simulation of the spike times and of the adaptation variable is critical, for it governs the spike pattern produced and is hard to compute accurately because of the exploding nature of the system at the spike times. We thoroughly study the precision of fixed time-step integration schemes for this type of model and demonstrate that these methods produce systematic errors that are unbounded, as the cutoff value is increased, in the evaluation of the two crucial quantities: the spike time and the value of the adaptation variable at this time. Precise evaluation of these quantities therefore involves very small time steps and long simulation times. In order to achieve a fixed absolute precision in a reasonable computational time, we propose here a new algorithm to simulate these systems based on a variable integration step method that either integrates the original ordinary differential equation or the equation of the orbits in the phase plane, and compare this algorithm with fixed time-step Euler scheme and other more accurate simulation algorithms.

scholarly journals A Sequential Approach to Numerical Simulations of Solidification with Domain and Time Decomposition

2019 ◽  
Vol 9 (10) ◽  
pp. 1972 ◽  
Author(s):  
Elzbieta Gawronska
Keyword(s):  
Parallel Computing ◽  
Computational Cost ◽  
Fixed Time ◽  
Simulation Software ◽  
Computational Method ◽  
Computational Time ◽  
Time Step ◽  
Sequential Approach ◽  
Time Partitioning ◽  
Physical Phenomena

Progress in computational methods has been stimulated by the widespread availability of cheap computational power leading to the improved precision and efficiency of simulation software. Simulation tools become indispensable tools for engineers who are interested in attacking increasingly larger problems or are interested in searching larger phase space of process and system variables to find the optimal design. In this paper, we show and introduce a new approach to a computational method that involves mixed time stepping scheme and allows to decrease computational cost. Implementation of our algorithm does not require a parallel computing environment. Our strategy splits domains of a dynamically changing physical phenomena and allows to adjust the numerical model to various sub-domains. We are the first (to our best knowledge) to show that it is possible to use a mixed time partitioning method with various combination of schemes during binary alloys solidification. In particular, we use a fixed time step in one domain, and look for much larger time steps in other domains, while maintaining high accuracy. Our method is independent of a number of domains considered, comparing to traditional methods where only two domains were considered. Mixed time partitioning methods are of high importance here, because of natural separation of domain types. Typically all important physical phenomena occur in the casting and are of high computational cost, while in the mold domains less dynamic processes are observed and consequently larger time step can be chosen. Finally, we performed series of numerical experiments and demonstrate that our approach allows reducing computational time by more than three times without losing the significant precision of results and without parallel computing.

scholarly journals Importance of the Cutoff Value in the Quadratic Adaptive Integrate-and-Fire Model

2009 ◽  
Vol 21 (8) ◽  
pp. 2114-2122 ◽  
Author(s):  
Jonathan Touboul
Keyword(s):  
Membrane Potential ◽  
Cortical Neurons ◽  
Large Scale ◽  
Quadratic Model ◽  
Fixed Amount ◽  
Cutoff Value ◽  
Fire Model ◽  
Integrate And Fire ◽  
Large Scale Simulations ◽  
Adaptation Variable

The quadratic adaptive integrate-and-fire model (Izhikevich, 2003 , 2007 ) is able to reproduce various firing patterns of cortical neurons and is widely used in large-scale simulations of neural networks. This model describes the dynamics of the membrane potential by a differential equation that is quadratic in the voltage, coupled to a second equation for adaptation. Integration is stopped during the rise phase of a spike at a voltage cutoff value Vc or when it blows up. Subsequently the membrane potential is reset, and the adaptation variable is increased by a fixed amount. We show in this note that in the absence of a cutoff value, not only the voltage but also the adaptation variable diverges in finite time during spike generation in the quadratic model. The divergence of the adaptation variable makes the system very sensitive to the cutoff: changing Vc can dramatically alter the spike patterns. Furthermore, from a computational viewpoint, the divergence of the adaptation variable implies that the time steps for numerical simulation need to be small and adaptive. However, divergence of the adaptation variable does not occur for the quartic model (Touboul, 2008 ) and the adaptive exponential integrate-and-fire model (Brette & Gerstner, 2005 ). Hence, these models are robust to changes in the cutoff value.

scholarly journals COMPOSITION METHODS FOR THE SIMULATION OF ARRAYS OF CHUA'S CIRCUITS

1999 ◽  
Vol 09 (04) ◽  
pp. 723-733
Author(s):  
YVES MOREAU ◽  
JOOS VANDEWALLE
Keyword(s):  
Differential Equation ◽  
Differential Geometry ◽  
Linear Part ◽  
Fixed Time ◽  
Time Step ◽  
Nonlinear Part ◽  
Composition Methods ◽  
Algebra Theory ◽  
Simple Integration ◽  
Integration Rules

Composition methods are methods for the integration of ordinary differential equations arising from differential geometry, or more precisely, Lie algebra theory. We apply them here to the simulation of arrays of Chua's circuits. In these methods, we split the vector field of the array of Chua's circuits into its linear part and its nonlinear part. We then solve the elementary differential equation for each part separately — which is easy since the equations for the nonlinear part are all decoupled — and recombine these contributions into a sequence of compositions. This splitting gives rise to simple integration rules for arrays of Chua's circuits, which we compare to more classical approaches: fixed time-step explicit Euler and adaptive fourth-order Runge–Kutta.

scholarly journals Acceleration of Runge-Kutta integration schemes

2004 ◽  
Vol 2004 (2) ◽  
pp. 307-314 ◽  
Author(s):  
Phailaung Phohomsiri ◽  
Firdaus E. Udwadia
Keyword(s):  
Computational Cost ◽  
Fixed Time ◽  
Lower Number ◽  
Integration Scheme ◽  
Time Step ◽  
Third Order ◽  
Numerical Examples ◽  
Integration Schemes ◽  
Local Accuracy ◽  
Runge Kutta

A simple accelerated third-order Runge-Kutta-type, fixed time step, integration scheme that uses just two function evaluations per step is developed. Because of the lower number of function evaluations, the scheme proposed herein has a lower computational cost than the standard third-order Runge-Kutta scheme while maintaining the same order of local accuracy. Numerical examples illustrating the computational efficiency and accuracy are presented and the actual speedup when the accelerated algorithm is implemented is also provided.

scholarly journals Evaluation of oceanic and atmospheric trajectory schemes in the TRACMASS trajectory model v6.0

2016 ◽  
Author(s):  
Kristofer Döös ◽  
Bror Jönsson ◽  
Joakim Kjellsson
Keyword(s):  
Differential Equation ◽  
General Circulation ◽  
Computational Cost ◽  
Circulation Model ◽  
General Circulation Models ◽  
Velocity Fields ◽  
Time Dependent ◽  
Computational Time ◽  
Time Step ◽  
Large Ensembles

Abstract. Two different trajectory schemes for oceanic and atmospheric general circulation models are compared in two different experiments. The theories of the two trajectory schemes are presented showing the differential equations they solve and why they are mass conserving. One scheme assumes that the velocity fields are stationary for a limited period of time and solves the trajectory path from a differential equation only as a function of space, i.e. "stepwise stationary". The second scheme uses a continuous linear interpolation of the fields in time and solves the trajectory path from a differential equation as a function of both space and time, i.e. "time-dependent". A special case of the "stepwise-stationary" scheme, when velocities are assumed constant between GCM outputs, is also considered, named "fixed GCM time step". The trajectory schemes are tested "off-line", i.e. using the already integrated and stored velocity fields from a GCM. The first comparison of the schemes uses trajectories calculated using the velocity fields from an eddy-resolving ocean general circulation model in the Agulhas region. The second comparison uses trajectories calculated using the wind fields from an atmospheric reanalysis. The study shows that using the "time-dependent" scheme over the "stepwise-stationary" scheme greatly improves accuracy with only a small increase in computational time. It is also found that with decreasing time steps the "stepwise-stationary" scheme becomes more accurate but at increased computational cost. The "time-dependent" scheme is therefore preferred over the "stepwise-stationary" scheme. However, when averaging over large ensembles of trajectories the two schemes are comparable, as intrinsic variability dominates over numerical errors. The "fixed GCM time step" is found to be less accurate than the "stepwise-stationary" scheme, even when considering averages over large ensembles.

scholarly journals Evaluation of oceanic and atmospheric trajectory schemes in the TRACMASS trajectory model v6.0

2017 ◽  
Vol 10 (4) ◽  
pp. 1733-1749 ◽  
Author(s):  
Kristofer Döös ◽  
Bror Jönsson ◽  
Joakim Kjellsson
Keyword(s):  
Differential Equation ◽  
General Circulation Model ◽  
General Circulation ◽  
Computational Cost ◽  
Circulation Model ◽  
Velocity Fields ◽  
Time Dependent ◽  
Computational Time ◽  
Time Step ◽  
Large Ensembles

Abstract. Three different trajectory schemes for oceanic and atmospheric general circulation models are compared in two different experiments. The theories of the trajectory schemes are presented showing the differential equations they solve and why they are mass conserving. One scheme assumes that the velocity fields are stationary for set intervals of time between saved model outputs and solves the trajectory path from a differential equation only as a function of space, i.e. stepwise stationary. The second scheme is a special case of the stepwise-stationary scheme, where velocities are assumed constant between general circulation model (GCM) outputs; it uses hence a fixed GCM time step. The third scheme uses a continuous linear interpolation of the fields in time and solves the trajectory path from a differential equation as a function of both space and time, i.e. a time-dependent scheme. The trajectory schemes are tested offline, i.e. using the already integrated and stored velocity fields from a GCM. The first comparison of the schemes uses trajectories calculated using the velocity fields from a high-resolution ocean general circulation model in the Agulhas region. The second comparison uses trajectories calculated using the wind fields from an atmospheric reanalysis. The study shows that using the time-dependent scheme over the stepwise-stationary scheme greatly improves accuracy with only a small increase in computational time. It is also found that with decreasing time steps the stepwise-stationary scheme becomes increasingly more accurate but at increased computational cost. The time-dependent scheme is therefore preferred over the stepwise-stationary scheme. However, when averaging over large ensembles of trajectories, the two schemes are comparable, as intrinsic variability dominates over numerical errors. The fixed GCM time step scheme is found to be less accurate than the stepwise-stationary scheme, even when considering averages over large ensembles.

Is Hydraulics Over-Used or Under-Used in Storm Drainage?

1994 ◽  
Vol 29 (1-2) ◽  
pp. 53-61
Author(s):  
Ben Chie Yen
Keyword(s):  
Unsteady Flow ◽  
Computational Time ◽  
Time Step ◽  
Flow Routing ◽  
Kinematic Wave Equation ◽  
Storm Drainage ◽  
Saint Venant Equations ◽  
Unsteady Flow Routing ◽  
Selection Of

Urban drainage models utilize hydraulics of different levels. Developing or selecting a model appropriate to a particular project is not an easy task. Not knowing the hydraulic principles and numerical techniques used in an existing model, users often misuse and abuse the model. Hydraulically, the use of the Saint-Venant equations is not always necessary. In many cases the kinematic wave equation is inadequate because of the backwater effect, whereas in designing sewers, often Manning's formula is adequate. The flow travel time provides a guide in selecting the computational time step At, which in turn, together with flow unsteadiness, helps in the selection of steady or unsteady flow routing. Often the noninertia model is the appropriate model for unsteady flow routing, whereas delivery curves are very useful for stepwise steady nonuniform flow routing and for determination of channel capacity.

scholarly journals On the Numerical Simulation of HPDEs Using θ-Weighted Scheme and the Galerkin Method

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 78
Author(s):  
Haifa Bin Jebreen ◽  
Fairouz Tchier
Keyword(s):  
Differential Equation ◽  
Galerkin Method ◽  
Linear Ordinary Differential Equation ◽  
Time Interval ◽  
Hyperbolic Partial Differential Equation ◽  
Time Step ◽  
Numerical Examples ◽  
One Dimensional ◽  
The Stability ◽  
The Galerkin Method

Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the θ-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis.

scholarly journals Investigation of Air Pocket Behavior in Pipelines Using Rigid Column Model and Contributions of Time Integration Schemes

Water ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 785
Author(s):  
Arman Rokhzadi ◽  
Musandji Fuamba
Keyword(s):  
Transient Flow ◽  
Time Integration ◽  
Driving Pressure ◽  
Implicit Time ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Time Step ◽  
Column Model ◽  
Integration Schemes ◽  
Backward Euler

This paper studies the air pressurization problem caused by a partially pressurized transient flow in a reservoir-pipe system. The purpose of this study is to analyze the performance of the rigid column model in predicting the attenuation of the air pressure distribution. In this regard, an analytic formula for the amplitude and frequency will be derived, in which the influential parameters, particularly, the driving pressure and the air and water lengths, on the damping can be seen. The direct effect of the driving pressure and inverse effect of the product of the air and water lengths on the damping will be numerically examined. In addition, these numerical observations will be examined by solving different test cases and by comparing to available experimental data to show that the rigid column model is able to predict the damping. However, due to simplified assumptions associated with the rigid column model, the energy dissipation, as well as the damping, is underestimated. In this regard, using the backward Euler implicit time integration scheme, instead of the classical fourth order explicit Runge–Kutta scheme, will be proposed so that the numerical dissipation of the backward Euler implicit scheme represents the physical dissipation. In addition, a formula will be derived to calculate the appropriate time step size, by which the dissipation of the heat transfer can be compensated.

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