American Institute of Mathematical Sciences

Advanced
  • Previous Article
    An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations
  • JCD Home
  • This Issue
  • Next Article
    Continuous approximation of $ M_t/M_t/ 1 $ distributions with application to production
December  2020, 7(2): 271-290. doi: 10.3934/jcd.2020011

Numerical investigation of a neural field model including dendritic processing

Daniele Avitabile 1,2, , Stephen Coombes 3, and  Pedro M. Lima 4,

1. 

Department of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1081a 1081 HV Amsterdam, The Netherlands
2. 

Mathneuro Team, Inria Sophia Antipolis, 2004 Rue des Lucioles, 06902 Sophia Antipolis, Cedex
3. 

Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, NG7 2RD, UK
4. 

CEMAT, Instituto Superior Tecnico, University of Lisbon, Portugal

Received  October 2019 Published  July 2020

We consider a simple neural field model in which the state variable is dendritic voltage, and in which somas form a continuous one-dimensional layer. This neural field model with dendritic processing is formulated as an integro-differential equation. We introduce a computational method for approximating solutions to this nonlocal model, and use it to perform numerical simulations for neuro-biologically realistic choices of anatomical connectivity and nonlinear firing rate function. For the time discretisation we adopt an Implicit-Explicit (IMEX) scheme; the space discretisation is based on a finite-difference scheme to approximate the diffusion term and uses the trapezoidal rule to approximate integrals describing the nonlocal interactions in the model. We prove that the scheme is of first-order in time and second order in space, and can be efficiently implemented if the factorisation of a small, banded matrix is precomputed. By way of validation we compare the outputs of a numerical realisation to theoretical predictions for the onset of a Turing pattern, and to the speed and shape of a travelling front for a specific choice of Heaviside firing rate. We find that theory and numerical simulations are in excellent agreement.

Keywords: Neural fields, integral equations, cable equation, finite difference scheme, travelling waves.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Daniele Avitabile, Stephen Coombes, Pedro M. Lima. Numerical investigation of a neural field model including dendritic processing. Journal of Computational Dynamics, 2020, 7 (2) : 271-290. doi: 10.3934/jcd.2020011
References:
[1]

U. M. Ascher, S. J. Ruuth and B. T. R. Wetton, Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32 (1995), 797-823.  doi: 10.1137/0732037.  Google Scholar

[2]

D. Avitabile, S. Coombes and P. M. Lima, danieleavitabile/neural-field-with-dendrites: Ancillary codes to "Numerical Investigation of a Neural Field Model Including Dendritic Processing", 2020. doi: 10.5281/zenodo.3731920.  Google Scholar

[3]

I. Bojak, T. F. Oostendorp, A. T. Reid and R. Kötter, Connecting mean field models of neural activity to EEG and fMRI data, Brain Topography, 23 (2010), 139-149.  doi: 10.1007/s10548-010-0140-3.  Google Scholar

[4]

P. C. Bressloff, New mechanism for neural pattern formation, Phys. Rev. Lett., 76 (1996), 4644-4647.  doi: 10.1103/PhysRevLett.76.4644.  Google Scholar

[5]

P. C. Bressloff and S. Coombes, Physics of the extended neuron, International Journal of Modern Physics B, 11 (1997), 2343-2392.  doi: 10.1142/S0217979297001209.  Google Scholar

[6]

S. Coombes, Waves, bumps, and patterns in neural field theories, Biol. Cybernet., 93 (2005), 91-108.  doi: 10.1007/s00422-005-0574-y.  Google Scholar

[7]

S. Coombes, P. Beim Graben, R. Potthast and J. Wright, Neural Fields: Theory and Applications, Springer, 2014.  Google Scholar

[8]

S. Coombes, H. Schmidt and I. Bojak, Interface dynamics in planar neural field models, J. Math. Neurosci., 2 (2012), 9, 27 pp. doi: 10.1186/2190-8567-2-9.  Google Scholar

[9]

S. M. Crook, G. B. Ermentrout, M. C. Vanier and J. M. Bower, The role of axonal delay in the synchronization of networks of coupled cortical oscillators, Journal of Computational Neuroscience, 4 (1997), 161-172.   Google Scholar

[10]

S. Heitmann, M. J. Aburn and M. Breakspear, The Brain dynamics toolbox for matlab, Neurocomputing, 315 (2018), 82-88.  doi: 10.1016/j.neucom.2018.06.026.  Google Scholar

[11]

A. Hutt and N. Rougier, Numerical simulation scheme of one-and two dimensional neural fields involving space-dependent delays, Neural Fields, Springer, Heidelberg, (2014), 175–185.  Google Scholar

[12]

P. M. Lima and E. Buckwar, Numerical solution of the neural field equation in the two-dimensional case, SIAM J. Sci. Comput., 37 (2015), B962–B979. doi: 10.1137/15M1022562.  Google Scholar

[13]

E. J. Nichols and A. Hutt, Neural field simulator: Two-dimensional spatio-temporal dynamics involving finite transmission speed, Front. Neuroinform., 9 (2015), 25. doi: 10.3389/fninf.2015.00025.  Google Scholar

[14]

P. L. Nunez, Neocortical dynamics and human EEG rhythms, Physics Today, 49 (1996), 57. doi: 10.1063/1.2807585.  Google Scholar

[15]

K. H. Pettersen and G. T. Einevoll, Amplitude variability and extracellular low-pass filtering of neuronal spikes, Biophysical Journal, 94 (2008), 784-802.  doi: 10.1529/biophysj.107.111179.  Google Scholar

[16]

J. Rankin, D. Avitabile, J. Baladron, G. Faye and D. J. B. Lloyd, Continuation of localized coherent structures in nonlocal neural field equations, SIAM J. Sci. Comput., 36 (2014), B70–B93. doi: 10.1137/130918721.  Google Scholar

[17]

J. Ross, M. Margetts, I. Bojak, R. Nicks, D. Avitabile and S. Coombes, A brain-wave equation incorporating axo-dendritic connectivity, Physical Review E, 101 (2020), 022411. Google Scholar

[18]

P. Sanz-Leon, P. A. Robinson, S. A. Knock, P. M. Drysdale, R. G. Abeysuriya, F. K. Fung, C. J. Rennie and X. Zhao, NFTsim: Theory and simulation of multiscale neural field dynamics, PLoS Computational Biology, 14 (2018), e1006387. doi: 10.1371/journal.pcbi.1006387.  Google Scholar

[19]

L. N. Trefethen, Spectral Methods in MATLAB, Software, Environments, and Tools, SIAM, vol. 10, 2000. doi: 10.1137/1.9780898719598.  Google Scholar

[20]

J. M. Varah, A lower bound for the smallest singular value of a matrix, Linear Algebra and Applications, 11 (1975), 3-5.  doi: 10.1016/0024-3795(75)90112-3.  Google Scholar

[21]

S. Visser, R. Nicks, O. Faugeras and S. Coombes, Standing and travelling waves in a spherical brain model: The Nunez model revisited, Physica D, 349 (2017), 27-45.  doi: 10.1016/j.physd.2017.02.017.  Google Scholar

show all references

References:
[1]

U. M. Ascher, S. J. Ruuth and B. T. R. Wetton, Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32 (1995), 797-823.  doi: 10.1137/0732037.  Google Scholar

[2]

D. Avitabile, S. Coombes and P. M. Lima, danieleavitabile/neural-field-with-dendrites: Ancillary codes to "Numerical Investigation of a Neural Field Model Including Dendritic Processing", 2020. doi: 10.5281/zenodo.3731920.  Google Scholar

[3]

I. Bojak, T. F. Oostendorp, A. T. Reid and R. Kötter, Connecting mean field models of neural activity to EEG and fMRI data, Brain Topography, 23 (2010), 139-149.  doi: 10.1007/s10548-010-0140-3.  Google Scholar

[4]

P. C. Bressloff, New mechanism for neural pattern formation, Phys. Rev. Lett., 76 (1996), 4644-4647.  doi: 10.1103/PhysRevLett.76.4644.  Google Scholar

[5]

P. C. Bressloff and S. Coombes, Physics of the extended neuron, International Journal of Modern Physics B, 11 (1997), 2343-2392.  doi: 10.1142/S0217979297001209.  Google Scholar

[6]

S. Coombes, Waves, bumps, and patterns in neural field theories, Biol. Cybernet., 93 (2005), 91-108.  doi: 10.1007/s00422-005-0574-y.  Google Scholar

[7]

S. Coombes, P. Beim Graben, R. Potthast and J. Wright, Neural Fields: Theory and Applications, Springer, 2014.  Google Scholar

[8]

S. Coombes, H. Schmidt and I. Bojak, Interface dynamics in planar neural field models, J. Math. Neurosci., 2 (2012), 9, 27 pp. doi: 10.1186/2190-8567-2-9.  Google Scholar

[9]

S. M. Crook, G. B. Ermentrout, M. C. Vanier and J. M. Bower, The role of axonal delay in the synchronization of networks of coupled cortical oscillators, Journal of Computational Neuroscience, 4 (1997), 161-172.   Google Scholar

[10]

S. Heitmann, M. J. Aburn and M. Breakspear, The Brain dynamics toolbox for matlab, Neurocomputing, 315 (2018), 82-88.  doi: 10.1016/j.neucom.2018.06.026.  Google Scholar

[11]

A. Hutt and N. Rougier, Numerical simulation scheme of one-and two dimensional neural fields involving space-dependent delays, Neural Fields, Springer, Heidelberg, (2014), 175–185.  Google Scholar

[12]

P. M. Lima and E. Buckwar, Numerical solution of the neural field equation in the two-dimensional case, SIAM J. Sci. Comput., 37 (2015), B962–B979. doi: 10.1137/15M1022562.  Google Scholar

[13]

E. J. Nichols and A. Hutt, Neural field simulator: Two-dimensional spatio-temporal dynamics involving finite transmission speed, Front. Neuroinform., 9 (2015), 25. doi: 10.3389/fninf.2015.00025.  Google Scholar

[14]

P. L. Nunez, Neocortical dynamics and human EEG rhythms, Physics Today, 49 (1996), 57. doi: 10.1063/1.2807585.  Google Scholar

[15]

K. H. Pettersen and G. T. Einevoll, Amplitude variability and extracellular low-pass filtering of neuronal spikes, Biophysical Journal, 94 (2008), 784-802.  doi: 10.1529/biophysj.107.111179.  Google Scholar

[16]

J. Rankin, D. Avitabile, J. Baladron, G. Faye and D. J. B. Lloyd, Continuation of localized coherent structures in nonlocal neural field equations, SIAM J. Sci. Comput., 36 (2014), B70–B93. doi: 10.1137/130918721.  Google Scholar

[17]

J. Ross, M. Margetts, I. Bojak, R. Nicks, D. Avitabile and S. Coombes, A brain-wave equation incorporating axo-dendritic connectivity, Physical Review E, 101 (2020), 022411. Google Scholar

[18]

P. Sanz-Leon, P. A. Robinson, S. A. Knock, P. M. Drysdale, R. G. Abeysuriya, F. K. Fung, C. J. Rennie and X. Zhao, NFTsim: Theory and simulation of multiscale neural field dynamics, PLoS Computational Biology, 14 (2018), e1006387. doi: 10.1371/journal.pcbi.1006387.  Google Scholar

[19]

L. N. Trefethen, Spectral Methods in MATLAB, Software, Environments, and Tools, SIAM, vol. 10, 2000. doi: 10.1137/1.9780898719598.  Google Scholar

[20]

J. M. Varah, A lower bound for the smallest singular value of a matrix, Linear Algebra and Applications, 11 (1975), 3-5.  doi: 10.1016/0024-3795(75)90112-3.  Google Scholar

[21]

S. Visser, R. Nicks, O. Faugeras and S. Coombes, Standing and travelling waves in a spherical brain model: The Nunez model revisited, Physica D, 349 (2017), 27-45.  doi: 10.1016/j.physd.2017.02.017.  Google Scholar

Figure 1.  Schematic of the neural field model. (a) Dendrites are represented as unbranched fibres (red), orthogonal to a continuum of somas (somatic layer, in grey). (b) Model (1) is for a 1D somatic layer, with coordinate $ x \in \mathbb{R} $, and fiber coordinate $ \xi \in \mathbb{R} $. Input currents are generated in a small neighbourhood of the somatic layer, at $ \xi = 0 $ and are delivered to a contact point, in a small neighbourhood of $ \xi = \xi_0 $. The strength of interaction depends on the distance between sources and contact points, measured along the somatic layer, hence the inputs that are generated at $ A $ and transmitted to $ B $, $ C $, and $ D $ depend on $ |x_B - x_A| $, $ |x_C-x_A| $, and $ |x_D - x_A| $, respectively (see (2))
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Coherent structure observed in time simulation of (3), (22), (23). (a): Pseudocolor plot of $ V(x,\xi,t) $ at several time points, showing two counter-propagating waves. (b): Solution at $ \xi = 0 $, showing the wave profile. Parameters: $ \xi_0 = 1 $, $ \varepsilon = 0.005 $, $ \nu = 0.4 $, $ \gamma = 1 $, $ \beta = 1000 $, $ \theta = 0.01 $, $ \kappa = 3 $, $ L_x = 24 \pi $, $ L_\xi = 3 $, $ n_x = 2^{10} $, $ n_\xi = 2^{12} $, $ \tau = 0.05 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  (a) Travelling wave speed versus firing rate threshold, computed analytically via (24), and numerically via the time-stepper. (b)-(d) Convergence of the computed speed to the analytical speed at $ \theta = 0.01 $, as a function of the the kernel support parameter $ \varepsilon $, the steepness of the sigmoid $ \beta $, and the time-stepping parameter $ \tau $, respectively. Parameters as in (2)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Numerical simulation of Turing bifurcation for the model with kernel and firing rate function given in (28). (a): Plots of $ \hat w(p) - w_* $ for $ \beta = 28 $ and $ \beta = 30 $, from which we deduce that a Turing bifurcation occurs for an intermediate value of $ \beta $. We expect perturbations of the trivial state to decay exponentially if $ \beta = 28 $ and to increase exponentially if $ \beta = 30 $, as confirmed in panels (b)-(d). (b): Maximum absolute value of the voltage at $ \xi = 0 $, as function of time, when the trivial steady state is perturbed. (c), (d): Pseudocolor plot of $ V $ when $ \beta = 28 $ and $ \beta = 30 $, respectively. Parameters: $ \xi_0 = 1 $, $ \varepsilon = 0.005 $, $ \nu = 6 $, $ c = 1 $, $ a_1 = 1 $, $ b_1 = 1 $, $ a_2 = 1/4 $, $ b_2 = 1/2 $, $ n_x = 2^9 $, $ L_x = 10\pi $, $ n_\xi = 2^{11} $, $ L_\xi = 2.5 \pi $, $ \tau = 0.01 $
Figure Options

Download full-size image

Download as PowerPoint slide

Table201 
Algorithm 1: IMEX time stepper in matrix form (8), nonlinear term computed with pseudospectral evaluation (21)
Input: Initial condition $ V^0 \in \mathbb{R}^{n_\xi \times n_x} $, time step $ \tau $, number of steps $ n_t $.
Output: An approximate solution $ (V^n)_{n=1}^{n_t} \subset \mathbb{R}^{n_\xi \times n_x} $
1 begin
2      Compute grid vectors $ \xi \in \mathbb{R}^{n_\xi \times 1} $, $ x \in \mathbb{R}^{1 \times n_x} $.
3       Compute synaptic vectors $ w, \hat w = \mathcal{F}_{n_x}[w] \in \mathbb{R}^{1 \times n_x} $.
4       Compute synaptic vectors $ \alpha, \alpha' \in \mathbb{R}^{n_\xi \times 1} $.
5       Compute quadrature weights $ \sigma \in \mathbb{R}^{n_\xi \times 1} $.
6       Compute sparse $ LU $-factorisation of $ A $,
$ LU = A \in \mathbb{R}^{n_\xi \times n_\xi}. $
7       for $ n = 1,\ldots,n_t $ do
8              Set $ V = V^{n-1} \in \mathbb{R}^{n_\xi \times n_x} $.
9               Compute the external input at time $ t_{n-1} $ and store it in $ G \in \mathbb{R}^{n_\xi \times n_x} $.
10               Set $ z = \mathcal{F}_{n_x}\big[(\alpha' \odot \sigma)^T S(V)\big] \in \mathbb{R}^{1 \times n_x} $.
11                Set $ N = h_x \alpha \mathcal{F}^{-1}_{n_x} [ \hat w \odot z] \in \mathbb{R}^{n_\xi \times n_x} $.
12                Solve for $ V^{n} $ the linear problem $ (LU)V^n = V + \tau(N+G) $.
13      end
14 end
Table Options

Download as excel

Algorithm 1: IMEX time stepper in matrix form (8), nonlinear term computed with pseudospectral evaluation (21)
Input: Initial condition $ V^0 \in \mathbb{R}^{n_\xi \times n_x} $, time step $ \tau $, number of steps $ n_t $.
Output: An approximate solution $ (V^n)_{n=1}^{n_t} \subset \mathbb{R}^{n_\xi \times n_x} $
1 begin
2      Compute grid vectors $ \xi \in \mathbb{R}^{n_\xi \times 1} $, $ x \in \mathbb{R}^{1 \times n_x} $.
3       Compute synaptic vectors $ w, \hat w = \mathcal{F}_{n_x}[w] \in \mathbb{R}^{1 \times n_x} $.
4       Compute synaptic vectors $ \alpha, \alpha' \in \mathbb{R}^{n_\xi \times 1} $.
5       Compute quadrature weights $ \sigma \in \mathbb{R}^{n_\xi \times 1} $.
6       Compute sparse $ LU $-factorisation of $ A $,
$ LU = A \in \mathbb{R}^{n_\xi \times n_\xi}. $
7       for $ n = 1,\ldots,n_t $ do
8              Set $ V = V^{n-1} \in \mathbb{R}^{n_\xi \times n_x} $.
9               Compute the external input at time $ t_{n-1} $ and store it in $ G \in \mathbb{R}^{n_\xi \times n_x} $.
10               Set $ z = \mathcal{F}_{n_x}\big[(\alpha' \odot \sigma)^T S(V)\big] \in \mathbb{R}^{1 \times n_x} $.
11                Set $ N = h_x \alpha \mathcal{F}^{-1}_{n_x} [ \hat w \odot z] \in \mathbb{R}^{n_\xi \times n_x} $.
12                Solve for $ V^{n} $ the linear problem $ (LU)V^n = V + \tau(N+G) $.
13      end
14 end
Table202 
Algorithm 2: IMEX time stepper in vector form (7), nonlinear term evaluated with quadrature formula (12).
Input: Initial condition $ U^0 \in \mathbb{R}^{n_\xi n_x} $, time step $ \tau $, number of steps $ n_t $.
Output: An approximate solution $ (U^n)_{n=1}^{n_t} \subset \mathbb{R}^{n_\xi n_x} $
1 begin
2      Compute grid vectors $ \xi \in \mathbb{R}^{n_\xi} $, $ x \in \mathbb{R}^{n_x} $.
3       Compute synaptic vector $ w \in \mathbb{R}^{1 \times n_x} $.
4       Compute synaptic vectors $ \alpha, \alpha' \in \mathbb{R}^{n_\xi \times 1} $.
5       Compute quadrature weights $ \rho \in \mathbb{R}^{n_x} $, $ \sigma \in \mathbb{R}^{n_\xi} $.
6      Compute sparse $ LU $-factorisation of A,
$ LU = \big( (1+\tau \gamma) I_{n_x n_\xi} - \tau \nu I_{n_x} \otimes D_{\xi \xi} \big) \in \mathbb{R}^{n_\xi n_x \times n_\xi n_x }. $
7      for $ n = 1,\ldots,n_t $ do
8              Set $ Z = U^{n-1} \in \mathbb{R}^{n_\xi n_x} $.
9              Compute the external input at time $ t_{n-1} $ and store it in $ G \in \mathbb{R}^{n_\xi n_x} $.
10              Compute the nonlinear term $ N $ using (12).
11              Solve for $ U^{n} $ the linear problem $ (LU)U^n = Z + \tau(N+G) $.
12      end
13 end
Table Options

Download as excel

Algorithm 2: IMEX time stepper in vector form (7), nonlinear term evaluated with quadrature formula (12).
Input: Initial condition $ U^0 \in \mathbb{R}^{n_\xi n_x} $, time step $ \tau $, number of steps $ n_t $.
Output: An approximate solution $ (U^n)_{n=1}^{n_t} \subset \mathbb{R}^{n_\xi n_x} $
1 begin
2      Compute grid vectors $ \xi \in \mathbb{R}^{n_\xi} $, $ x \in \mathbb{R}^{n_x} $.
3       Compute synaptic vector $ w \in \mathbb{R}^{1 \times n_x} $.
4       Compute synaptic vectors $ \alpha, \alpha' \in \mathbb{R}^{n_\xi \times 1} $.
5       Compute quadrature weights $ \rho \in \mathbb{R}^{n_x} $, $ \sigma \in \mathbb{R}^{n_\xi} $.
6      Compute sparse $ LU $-factorisation of A,
$ LU = \big( (1+\tau \gamma) I_{n_x n_\xi} - \tau \nu I_{n_x} \otimes D_{\xi \xi} \big) \in \mathbb{R}^{n_\xi n_x \times n_\xi n_x }. $
7      for $ n = 1,\ldots,n_t $ do
8              Set $ Z = U^{n-1} \in \mathbb{R}^{n_\xi n_x} $.
9              Compute the external input at time $ t_{n-1} $ and store it in $ G \in \mathbb{R}^{n_\xi n_x} $.
10              Compute the nonlinear term $ N $ using (12).
11              Solve for $ U^{n} $ the linear problem $ (LU)U^n = Z + \tau(N+G) $.
12      end
13 end
Table 1.  Flop count for the initialisation step (lines 2-6) and for one time step (lines 8-12) in Algorithms 1, 2
Algorithm 1 Algorithm 2
Lines Flops Lines Flops
2 $ n_\xi + n_x $ 2 $ n_\xi + n_x $
3 $ 2n_x $ 3 $ n_x $
4 $ 2n_\xi $ 4 $ 2n_\xi $
5 $ n_\xi $ 5 $ n_\xi + n_x $
6 $ 2n_\xi-1 $ 6 $ 2n_\xi n_x -1 $
2-6 $ O(n_\xi) + O(n_x) $ 2-6 $ O(n_\xi n_x) $
8 $ n_\xi n_x $ 8 $ n_\xi n_x $
9 $ n_\xi n_x $ 9 $ n_\xi n_x $
10 $ 3 n_\xi n_x + O(n_x \log n_x) + n_\xi - n_x $ 10 $ 2n^2_\xi n_x^2 - n^2_\xi n_x $
11 $ n_\xi n_x + O(n_x \log n_x) + 2n_x $ 11 $ 5n_\xi n_x -4 $
12 $ 5 n_\xi n_x - 4 n_x $ $ $
8-12 $ O(n_\xi n_x) + O(n_x \log n_x) $ 8-11 $ O(n^2_\xi n^2_x) $
Table Options

Download as excel

Algorithm 1 Algorithm 2
Lines Flops Lines Flops
2 $ n_\xi + n_x $ 2 $ n_\xi + n_x $
3 $ 2n_x $ 3 $ n_x $
4 $ 2n_\xi $ 4 $ 2n_\xi $
5 $ n_\xi $ 5 $ n_\xi + n_x $
6 $ 2n_\xi-1 $ 6 $ 2n_\xi n_x -1 $
2-6 $ O(n_\xi) + O(n_x) $ 2-6 $ O(n_\xi n_x) $
8 $ n_\xi n_x $ 8 $ n_\xi n_x $
9 $ n_\xi n_x $ 9 $ n_\xi n_x $
10 $ 3 n_\xi n_x + O(n_x \log n_x) + n_\xi - n_x $ 10 $ 2n^2_\xi n_x^2 - n^2_\xi n_x $
11 $ n_\xi n_x + O(n_x \log n_x) + 2n_x $ 11 $ 5n_\xi n_x -4 $
12 $ 5 n_\xi n_x - 4 n_x $ $ $
8-12 $ O(n_\xi n_x) + O(n_x \log n_x) $ 8-11 $ O(n^2_\xi n^2_x) $
Table 2.  Space requirements, measured in Floating Point Numbers, for Algorithms 1 and 2. Arrays $ d_1, \ldots, d_3 $, store diagonals of the $ LU $-factorisation in the respective algorithms
Floating Point Numbers Algorithm 1 Algorithm 2
$ n_\xi $ $ \xi,\alpha,\alpha',d_1,d_2,d_3,z $ $ \xi, \rho, \alpha, \alpha' $
$ n_x $ $ x,w,\sigma $ $ x, \sigma, w $
$ n_\xi n_x $ $ V,V^n,G,N $ $ U^n,Z,N,G,d_1,d_2,d_3 $
Total $ 4n_x n_\xi + 7n_\xi + 3n_x $ $ 7 n_x n_\xi +2n_\xi + 2n_x $
Table Options

Download as excel

Floating Point Numbers Algorithm 1 Algorithm 2
$ n_\xi $ $ \xi,\alpha,\alpha',d_1,d_2,d_3,z $ $ \xi, \rho, \alpha, \alpha' $
$ n_x $ $ x,w,\sigma $ $ x, \sigma, w $
$ n_\xi n_x $ $ V,V^n,G,N $ $ U^n,Z,N,G,d_1,d_2,d_3 $
Total $ 4n_x n_\xi + 7n_\xi + 3n_x $ $ 7 n_x n_\xi +2n_\xi + 2n_x $
[1]

Tetsuya Ishiwata, Kota Kumazaki. Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field. Conference Publications, 2015, 2015 (special) : 644-651. doi: 10.3934/proc.2015.0644
[2]

Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2025-2039. doi: 10.3934/dcdss.2020402
[3]

Timoteo Carletti. The lagrange inversion formula on non--Archimedean fields, non--analytical form of differential and finite difference equations. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 835-858. doi: 10.3934/dcds.2003.9.835
[4]

Wei Qu, Siu-Long Lei, Seak-Weng Vong. A note on the stability of a second order finite difference scheme for space fractional diffusion equations. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 317-325. doi: 10.3934/naco.2014.4.317
[5]

Matthieu Alfaro, Jérôme Coville, Gaël Raoul. Bistable travelling waves for nonlocal reaction diffusion equations. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1775-1791. doi: 10.3934/dcds.2014.34.1775
[6]

Guoliang Zhang, Shaoqin Zheng, Tao Xiong. A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations. Electronic Research Archive, 2021, 29 (1) : 1819-1839. doi: 10.3934/era.2020093
[7]

Ronald E. Mickens. A nonstandard finite difference scheme for the drift-diffusion system. Conference Publications, 2009, 2009 (Special) : 558-563. doi: 10.3934/proc.2009.2009.558
[8]

Jing Li, Yifu Wang, Jingxue Yin. Non-sharp travelling waves for a dual porous medium equation. Communications on Pure & Applied Analysis, 2016, 15 (2) : 623-636. doi: 10.3934/cpaa.2016.15.623
[9]

Narcisa Apreutesei, Nikolai Bessonov, Vitaly Volpert, Vitali Vougalter. Spatial structures and generalized travelling waves for an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 537-557. doi: 10.3934/dcdsb.2010.13.537
[10]

Narcisa Apreutesei, Arnaud Ducrot, Vitaly Volpert. Travelling waves for integro-differential equations in population dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 541-561. doi: 10.3934/dcdsb.2009.11.541
[11]

Michal Fečkan, Vassilis M. Rothos. Travelling waves of forced discrete nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1129-1145. doi: 10.3934/dcdss.2011.4.1129
[12]

Dmitry Treschev. Travelling waves in FPU lattices. Discrete & Continuous Dynamical Systems, 2004, 11 (4) : 867-880. doi: 10.3934/dcds.2004.11.867
[13]

Hawraa Alsayed, Hussein Fakih, Alain Miranville, Ali Wehbe. Finite difference scheme for 2D parabolic problem modelling electrostatic Micro-Electromechanical Systems. Electronic Research Announcements, 2019, 26: 54-71. doi: 10.3934/era.2019.26.005
[14]

Wen Li, Song Wang. Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme. Journal of Industrial & Management Optimization, 2013, 9 (2) : 365-389. doi: 10.3934/jimo.2013.9.365
[15]

Navnit Jha. Nonpolynomial spline finite difference scheme for nonlinear singuiar boundary value problems with singular perturbation and its mechanization. Conference Publications, 2013, 2013 (special) : 355-363. doi: 10.3934/proc.2013.2013.355
[16]

Houda Hani, Moez Khenissi. Asymptotic behaviors of solutions for finite difference analogue of the Chipot-Weissler equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1421-1445. doi: 10.3934/dcdss.2016057
[17]

Gianluca Frasca-Caccia, Peter E. Hydon. Locally conservative finite difference schemes for the modified KdV equation. Journal of Computational Dynamics, 2019, 6 (2) : 307-323. doi: 10.3934/jcd.2019015
[18]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079
[19]

Stephen Coombes, Helmut Schmidt, Carlo R. Laing, Nils Svanstedt, John A. Wyller. Waves in random neural media. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2951-2970. doi: 10.3934/dcds.2012.32.2951
[20]

Xiaohai Wan, Zhilin Li. Some new finite difference methods for Helmholtz equations on irregular domains or with interfaces. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1155-1174. doi: 10.3934/dcdsb.2012.17.1155

 Impact Factor: 

Tools

Metrics

  • PDF downloads (106)
  • HTML views (261)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences