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Continuous approximation of $ M_t/M_t/ 1 $ distributions with application to production
Numerical investigation of a neural field model including dendritic processing
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 |
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.
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. |
[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. |
[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. |
[4] |
P. C. Bressloff,
New mechanism for neural pattern formation, Phys. Rev. Lett., 76 (1996), 4644-4647.
doi: 10.1103/PhysRevLett.76.4644. |
[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. |
[6] |
S. Coombes,
Waves, bumps, and patterns in neural field theories, Biol. Cybernet., 93 (2005), 91-108.
doi: 10.1007/s00422-005-0574-y. |
[7] |
S. Coombes, P. Beim Graben, R. Potthast and J. Wright, Neural Fields: Theory and Applications, Springer, 2014. |
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
P. L. Nunez, Neocortical dynamics and human EEG rhythms, Physics Today, 49 (1996), 57.
doi: 10.1063/1.2807585. |
[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. |
[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. |
[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. |
[19] |
L. N. Trefethen, Spectral Methods in MATLAB, Software, Environments, and Tools, SIAM, vol. 10, 2000.
doi: 10.1137/1.9780898719598. |
[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. |
[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. |
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. |
[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. |
[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. |
[4] |
P. C. Bressloff,
New mechanism for neural pattern formation, Phys. Rev. Lett., 76 (1996), 4644-4647.
doi: 10.1103/PhysRevLett.76.4644. |
[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. |
[6] |
S. Coombes,
Waves, bumps, and patterns in neural field theories, Biol. Cybernet., 93 (2005), 91-108.
doi: 10.1007/s00422-005-0574-y. |
[7] |
S. Coombes, P. Beim Graben, R. Potthast and J. Wright, Neural Fields: Theory and Applications, Springer, 2014. |
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
P. L. Nunez, Neocortical dynamics and human EEG rhythms, Physics Today, 49 (1996), 57.
doi: 10.1063/1.2807585. |
[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. |
[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. |
[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. |
[19] |
L. N. Trefethen, Spectral Methods in MATLAB, Software, Environments, and Tools, SIAM, vol. 10, 2000.
doi: 10.1137/1.9780898719598. |
[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. |
[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. |




Algorithm 1: IMEX time stepper in matrix form (8), nonlinear term computed with pseudospectral evaluation (21) |
Input: Initial condition Output: An approximate solution 1 begin 2 Compute grid vectors 3 Compute synaptic vectors 4 Compute synaptic vectors 5 Compute quadrature weights 6 Compute sparse $ LU = A \in \mathbb{R}^{n_\xi \times n_\xi}. $ 7 for 8 Set 9 Compute the external input at time 10 Set 11 Set 12 Solve for 13 end 14 end |
Algorithm 1: IMEX time stepper in matrix form (8), nonlinear term computed with pseudospectral evaluation (21) |
Input: Initial condition Output: An approximate solution 1 begin 2 Compute grid vectors 3 Compute synaptic vectors 4 Compute synaptic vectors 5 Compute quadrature weights 6 Compute sparse $ LU = A \in \mathbb{R}^{n_\xi \times n_\xi}. $ 7 for 8 Set 9 Compute the external input at time 10 Set 11 Set 12 Solve for 13 end 14 end |
Algorithm 2: IMEX time stepper in vector form (7), nonlinear term evaluated with quadrature formula (12). |
Input: Initial condition Output: An approximate solution 1 begin 2 Compute grid vectors 3 Compute synaptic vector 4 Compute synaptic vectors 5 Compute quadrature weights 6 Compute sparse $ 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 8 Set 9 Compute the external input at time 10 Compute the nonlinear term 11 Solve for 12 end 13 end |
Algorithm 2: IMEX time stepper in vector form (7), nonlinear term evaluated with quadrature formula (12). |
Input: Initial condition Output: An approximate solution 1 begin 2 Compute grid vectors 3 Compute synaptic vector 4 Compute synaptic vectors 5 Compute quadrature weights 6 Compute sparse $ 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 8 Set 9 Compute the external input at time 10 Compute the nonlinear term 11 Solve for 12 end 13 end |
Algorithm 1 | Algorithm 2 | ||
Lines | Flops | Lines | Flops |
2 | 2 | ||
3 | 3 | ||
4 | 4 | ||
5 | 5 | ||
6 | 6 | ||
2-6 | 2-6 | ||
8 | 8 | ||
9 | 9 | ||
10 | 10 | ||
11 | 11 | ||
12 | |||
8-12 | 8-11 |
Algorithm 1 | Algorithm 2 | ||
Lines | Flops | Lines | Flops |
2 | 2 | ||
3 | 3 | ||
4 | 4 | ||
5 | 5 | ||
6 | 6 | ||
2-6 | 2-6 | ||
8 | 8 | ||
9 | 9 | ||
10 | 10 | ||
11 | 11 | ||
12 | |||
8-12 | 8-11 |
Floating Point Numbers | Algorithm 1 | Algorithm 2 |
Total |
Floating Point Numbers | Algorithm 1 | Algorithm 2 |
Total |
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