Abstract

We investigate the effects of reflecting connectivity in a network composed of FitzHugh–Nagumo (FHN) elements linked in a ring topology. Reflecting connectivity is inspired by the linking between the opposite hemispheres in the mammalian brain. To study synchronization phenomena and coexistence of synchronous and asynchronous domains (chimera states) under the influence of this connectivity, we use two versions of the FHN model: Version I where membrane and recovery potentials are interlinked via a rotational matrix (Omelchenko et al. PRL 110:224101, 2013), and version II where only the membrane potentials are linked and not the recovery ones (Shepelev et al., Phys. Lett. A 381:1398, 2017). In both realizations, the reflecting connectivity forces the activity to organise in two connected semirings. Our numerical results give evidence that, for FHN-I and positive (negative) coupling strength, coherent (incoherent) oscillatory regions develop and reside in the junctions between the two semirings. For the FHN-II model which supports multistability, our simulations indicate that for appropriate values of the parameters the two semirings are separated via oscillating elements. The center parts of the two semirings are composed either by elements which are frozen at the FHN fixed points (coexistence of fixed points and oscillatory states), or by coherent oscillatory elements (coexistence of coherent and incoherent nodes). Notable, here, is the case of oscillation death, which has also been reported for other oscillators coupled via one of their variables. These results can be useful for interpreting inhomogeneous and localised coordinated activity observed in the mammalian brain.

Graphic abstract

中文翻译：

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FitzHugh-Nagumo网络中的Chimera州具有反射性的连通性

摘要

我们调查了由FitzHugh-Nagumo（FHN）元素链接成环形拓扑组成的网络中反映连接性的影响。反映性的连通性是受到哺乳动物大脑中相对的半球之间的连接的启发。为了研究在这种连接性的影响下同步现象和同步域和异步域（嵌合体状态）的共存，我们使用了两个版本的FHN模型：版本I，其中膜和恢复电位通过旋转矩阵互连（Omelchenko等人，PRL） 110：224101，2013）和版本II，其中仅膜电位相关，而恢复电位不相关（Shepelev等人，Phys.Lett.A 381：1398，2017）。在这两种实现方式中，反射性的连通性迫使活动组织在两个相连的半环中。我们的数值结果表明，对于FHN-1和正（负）耦合强度，相干（非相干）振荡区域发展并驻留在两个半环之间的交界处。对于支持多重稳定性的FHN-II模型，我们的仿真表明，对于适当的参数值，两个半环通过振荡元素分开。两个半环的中心部分由冻结在FHN固定点上的元素（固定点和振荡状态的共存）或相干振荡元素（相干和不相干节点的共存）组成。这里值得注意的是振荡死亡的情况，对于通过其变量之一耦合的其他振荡器也有报道。这些结果可用于解释在哺乳动物脑中观察到的不均匀和局部的协调活动。

图形摘要