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.

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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 
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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. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/2022016/

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