Symmetric coupling of multiple timescale systems with Mixed-Mode Oscillations and synchronization

A multiple timescale network model of intracellular calcium concentrations in coupled neurons: Insights from ROM simulations

“…Although the results obtained in [16] represent a first step in the theoretical understanding of the panel and repartition of synchronization types according to the coupling strength, two assumptions are too restrictive for dealing with realistic problems: first, the two cells are assumed to be identical while heterogeneity in intrinsic quantitative features of single cell ICC oscillations is a well-known fact [38,44]; second, if we consider each of the 3D systems as a cluster, as mentioned above, we are assuming that the single cells dynamics constituting the cluster are perfectly synchronized. In order to advance in this line towards more realistic problems, in the present article, we consider an extension of the system analyzed in [16] in two different directions:…”

“…A network model has also been built for reproducing episodic synchronization identified in GnRH neuron populations. Afterwards, authors in [16] have proposed a 6D system obtained by symmetrically coupling two such 3D oscillators generating MMOs and performed a theoretical study of the synchronization features in oscillatory ICC patterns of two coupled cells. The coupling type chosen for building this model relies on the underlying dynamics of ICC, on which the synaptic transmission as well as gap-junctions between neurons has little effect (see [22] for instance).…”

“…Glial cells such as astrocytes are involved in the CICR channel through the increase of Inositol trisphosphate (IP3). Since this mechanism is slower than neuronal transmission, the coupling in [16] was chosen on the recovery variables that evolves on a consistent timescale for modeling this mechanism [42]. The choice of a linear coupling, although it consists in a strong simplification of the biological interactions, allows us to obtain a mathematically-tractable model while preserving the qualitative properties of the dynamical interactions between neuron ICC levels.…”

“…Considering both excitatory and inhibitory coupling, the aim of the study was two-folded : (i) identify the conditions for obtaining in-phase and perfectly in-phase synchronization between two coupled cells (for the excitatory case), (ii) analyze the synchronization behaviors of two identical populations of neurons inhibiting each others, considering that one 3D oscillator is also a reduced model for a population of perfectly synchronized neurons, which occurs (following (i)) if the excitatory coupling is strong enough. The analysis performed in [16] is based on geometric singular perturbation theory and takes advantage of the symmetry of the system. We summarize the identified behaviors and some of the results obtained in this case in Section 2, since they are mandatory for understanding the present study.…”

“…We study the synchronization patterns of the system with respect to two parameters: the ratio between the intrinsic frequencies of the oscillators and the coupling strength. Therefore, the system is no longer symmetric and several arguments used for the theoretical analysis in [16] are no longer valid. Thus, we proceed through direct numerical simulations through a fourth order Runge-Kutta solver.…”