Geodesic
A multiple timescale network model of intracellular calcium concentrations in coupled neurons: Insights from ROM simulations
A. Bandera1, 2 ; S. Fernández-García1, 2 ; M. Gómez-Mármol1 ; A. Vidal3
1 Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Calle Tarfia s/n, Seville 41012, Spain
2 IMUS, Universidad de Sevilla, Calle Tarfia s/n, Seville 41012, Spain
3 Laboratoire de Mathématiques et Modélisation d’Évry (LAMME), Univ Evry, CNRS, Université Paris-Saclay, IBGBI, 23 Bld de France, Evry 91037, France
Mathematical modelling of natural phenomena, Tome 17 (2022), article no. 11 Cet article a éte moissonné depuis la source EDP Sciences

Voir la notice de l'article

In Fernández-García and Vidal [Physica D 401 (2020) 132129], the authors have analyzed the synchronization features between two identical 3D slow-fast oscillators, symmetrically coupled, and built as an extension of the FitzHugh—Nagumo dynamics generating Mixed-Mode Oscillations. The third variable in each oscillator aims at representing the time-varying intracellular calcium concentration in neurons. The global model is therefore six-dimensional with two fast variables and four slow variables with strong symmetry properties. In the present article, we consider an extension of this model in two different directions. First, we consider heterogeneity among cells and analyze the coupling of two oscillators with different values for one parameter which tunes the intrinsic frequency of the output. We therefore identify new patterns of antiphasic synchronization, with non trivial signatures and that exhibit a Devil’s Staircase phenomenon in signature transitions when varying the coupling gain parameter value. Second, we introduce a network of N cells divided into two clusters: the coupling between neurons in each cluster is excitatory, while the coupling between the two clusters is inhibitory. Such system aims at modelling the interactions between neurons tending to synchronization in each of two subpopulations that inhibit each other, like ipsi- and contra-lateral motoneurons assemblies. To perform the numerical simulations in this case when N is large, as an initial step towards the network analysis, we consider Reduced Order Models in order to save computational costs. We present the numerical reduction results in a network of 100 cells. For the sake of validation of the numerical reduction method, we both compare the outputs and CPU times obtained with the original and the reduced models in different cases of network coupling structures.
DOI : 10.1051/mmnp/2022016

A. Bandera 1, 2 ; S. Fernández-García 1, 2 ; M. Gómez-Mármol 1 ; A. Vidal 3

1 Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Calle Tarfia s/n, Seville 41012, Spain
2 IMUS, Universidad de Sevilla, Calle Tarfia s/n, Seville 41012, Spain
3 Laboratoire de Mathématiques et Modélisation d’Évry (LAMME), Univ Evry, CNRS, Université Paris-Saclay, IBGBI, 23 Bld de France, Evry 91037, France 
@article{10_1051_mmnp_2022016,
     author = {A. Bandera and S. Fern\'andez-Garc{\'\i}a and M. G\'omez-M\'armol and A. Vidal},
     title = {A multiple timescale network model of intracellular calcium concentrations in coupled neurons: {Insights} from {ROM} simulations},
     journal = {Mathematical modelling of natural phenomena},
     eid = {11},
     year = {2022},
     volume = {17},
     doi = {10.1051/mmnp/2022016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/mmnp/2022016/}
}
TY  - JOUR
AU  - A. Bandera
AU  - S. Fernández-García
AU  - M. Gómez-Mármol
AU  - A. Vidal
TI  - A multiple timescale network model of intracellular calcium concentrations in coupled neurons: Insights from ROM simulations
JO  - Mathematical modelling of natural phenomena
PY  - 2022
VL  - 17
UR  - http://geodesic.mathdoc.fr/articles/10.1051/mmnp/2022016/
DO  - 10.1051/mmnp/2022016
LA  - en
ID  - 10_1051_mmnp_2022016
ER  - 
%0 Journal Article
%A A. Bandera
%A S. Fernández-García
%A M. Gómez-Mármol
%A A. Vidal
%T A multiple timescale network model of intracellular calcium concentrations in coupled neurons: Insights from ROM simulations
%J Mathematical modelling of natural phenomena
%D 2022
%V 17
%U http://geodesic.mathdoc.fr/articles/10.1051/mmnp/2022016/
%R 10.1051/mmnp/2022016
%G en
%F 10_1051_mmnp_2022016
A. Bandera; S. Fernández-García; M. Gómez-Mármol; A. Vidal. A multiple timescale network model of intracellular calcium concentrations in coupled neurons: Insights from ROM simulations. Mathematical modelling of natural phenomena, Tome 17 (2022), article  no. 11. doi: 10.1051/mmnp/2022016

[1] A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, C. Zhou Synchronization in complex networks Phys. Rep. 2008 93 153

[2] P. Bak Devil’s staircase Phys. Today 1986 38 45

[3] V.N. Belykh, I.V. Belykh, M. Hasler Connection graph stability method for synchronized coupled chaotic systems Physica D 2004 188 206

[4] V.N. Belykh, E.V. Pankratova Chaotic synchronization in ensembles of coupled neurons modeled by the FitzHugh-Rinzel system Radiophys. Quantum Electr. 2006 910 921

[5] J. Bélair, P. Holmes On linearly coupled relaxation oscillations Quart. Appl. Math. 1984 193 219

[6] E. Benoit, J. Callot, F. Diener, M. Diener Chasse au canard (première partie) Collectanea Math. 1981 37 76

[7] M. Brøns, M. Krupa, M. Wechselberger Mixed mode oscillations due to the generalized canard phenomenon Fields Inst. Commun. 2006 39 63

[8] S.A. Campbell and M. Waite, Multistability in coupled FitzHugh-Nagumo oscillators. Science Direct Working Paper (2001) (S1574-0358), 04.

[9] D. Chapelle, A. Gariah, J. Sainte-Marie Galerkin approximation with proper orthogonal decomposition: new error estimates and illustrative examples ESAIM: M2AN 2012 731 757

[10] J. Drover, J. Rubin, J. Su, B. Ermentrout Analysis of a canard mechanism by which excitatory synaptic coupling can synchronize neurons at low firing frequencies SIAM J. Appl. Math. 2004 69 92

[11] B. Ermentrout, M. Pascal, B. Gutkin The effects of spike frequency adaptation and negative feedback on the synchronization of neural oscillators Neural Comput. 2001 1285 1310

[12] B. Ermentrout, M. Wechselberger Canards, clusters, and synchronization in a weakly coupled interneuron model SIAM J. Appl. Dyn. Syst. 2009 253 278

[13] E.K. Ersöz, M. Desroches, M. Krupa, F. Clément Canard-mediated (de) synchronization in coupled phantom bursters SIAM J. Appl. Dyn. Syst. 2016 580 608

[14] E.K. Ersöoz, M. Desroches, M. Krupa Synchronization of weakly coupled canard oscillators Physica D 2017 46 61

[15] F.D.V. Fallani, M. Corazzol, J.R. Sternberg, C. Wyart, M. Chavez Hierarchy of neural organization in the embryonic spinal cord: Granger-causality graph analysis of in vivo calcium imaging data IEEE Trans. Neural Syst. Rehab. Eng. 2014 333 341

[16] S. Fernéndez-García, A. Vidal Symmetric coupling of multiple timescale systems with mixed-mode oscillations and synchronization Physica D 2020 132129

[17] R. Fitzhugh Impulses and physiological states in theoretical models of nerve membrane Biophys. J. 1961 445 466

[18] J. Guckenheimer Singular Hopf bifurcation in systems with two slow variables SIAM J. Appl. Dyn. Syst. 2008 1355 1377

[19] J. Guckenheimer, P. Meerkamp Unfoldings of singular Hopf bifurcation SIAM J. Appl. Dyn. Syst. 2012 1325 1359

[20] A.L. Hodgkin, A.F. Huxley A quantitative description of membrane current and its application to conduction and excitation in nerve J. Physiol. 1952 500 544

[21] E.M. Izhikevich Phase equations for relaxation oscillators SIAM J. Appl. Math. 2000 1789 1804

[22] K. Jahn, J. Grosskreutz, K. Haastert, E. Ziegler, F. Schlesinger, C. Grothe, J. Bufler Temporospatial coupling of networked synaptic activation of AMPA-type glutamate receptor channels and calcium transients in cultured motoneurons Neuroscience 2006 1019 1029

[23] M. Krupa, N. Popoviéc, N. Kopell, H. Rotstein Mixed-mode oscillations in a three time-scale model for the dopaminergic neuron Chaos 2008 015106

[24] M. Krupa, A. Vidal, F. Cléement A network model of the periodic synchronization process in the dynamics of calcium concentration in GnRH neurons J. Math. Neurosci. 2013 1 24

[25] M. Krupa, M. Wechselberger Local analysis near a folded saddle-node singularity J. Differ. Equ. 2010 2841 2888

[26] T. Kostova-Vassilevska, G.M. Oxberry Model reduction of dynamical systems by proper orthogonal decomposition: error bounds and comparison of methods using snapshots from the solution and the time derivatives J. Comput. Appl. Math. 2018 553 573

[27] K. Kunisch, S. Volkwein Galerkin proper orthogonal decomposition methods for parabolic problems Em Numer. Math. 2001 117 148

[28] E. Lee, D. Terman Stable antiphase oscillations in a network of electrically coupled model neurons SIAM J. Appl. Dyn. Syst. 2013 1 27

[29] E. Lee, D. Terman Stability of antiphase oscillations in a network of inhibitory neurons SIAM J. Appl. Dyn. Syst. 2015 448 480

[30] J. Nagumo, S. Arimoto, S. Yoshizawa An active pulse transmission line simulating nerve axon Proc. IRE 1962 2061 2070

[31] M. Rathinam, L.R. Petzold A new look at proper orthogonal decomposition SIAM J. Numer. Anal. 2003 1893 1925

[32] R. Reimbayev, K. Daley, I.V. Belykh When two wrongs make a right: synchronized neuronal bursting from combined electrical and inhibitory coupling Phil. Trans. R Soc. A 2017 20160282

[33] H.G. Rotstein, R. Kuske Localized and asynchronous patterns via canards in coupled calcium oscillators Physica D 2006 46 61

[34] J.E. Rubin, J. Signerska-Rynkowska, J. Touboul, A. Vidal Wild oscillations in a nonlinear neuron model with resets:(II) Mixed-mode oscillations Discrete Continu. Dyn. Syst. B 2017 4003 4039

[35] A. Stefanski Determining thresholds of complete synchronization, and application World Sci. Ser. Nonlinear Sci. A 2009

[36] D.W. Storti, R.H. Rand Dynamics of two strongly coupled relaxation oscillators SIAM J. Appl. Math. 1986 56 67

[37] D. Storti, R.H. Rand A simplified model of coupled relaxation oscillators Int. J. Non-linear Mech. 1987 283 289

[38] C. Stosiek, O. Garaschuk, K. Holthoff, A. Konnerth In vivo two-photon calcium imaging of neuronal networks Proc. Natl. Acad. Sci. 2003 7319 7324

[39] P. Szmolyan, M. Wechselberger Canards in R3 J. Differ. Equ. 2001 419 453

[40] D. Terman, E. Lee, J. Rinzel, T. Bem Stability of anti-phase and in-phase locking by electrical coupling but not fast inhibition alone SIAM J. Appl. Dyn. Syst. 2011 1127 1153

[41] S. Volkwein, Proper Orthogonal Decomposition: Theory and Reduced-Order Modelling. Lecture Notes. University of Konstanz, Konstanz (2013).

[42] D.D. Wang, A. Bordey The astrocyte odyssey Progr. Neurobiol. 2008 342 367

[43] M. Wechselberger Existence and bifurcation of canards in R3 in the case of a folded node SIAM J. Appl. Dyn. Syst. 2005 101 139

[44] C. Wyart, C. Ybert, L. Bourdieu, C. Herr, C. Prinz, D. Chatenay Constrained synaptic connectivity in functional mammalian neuronal networks grown on patterned surfaces J. Neurosci. Methods 2002 123 131

[45] A.M. Zhabotinsky, H.G. Rotstein, I.R. Epstein, N. Kopell A canard mechanism for localization in systems of globally coupled oscillators SIAM J. Appl. Math. 2003 1998 2019

Cité par Sources :