修饰的FHN神经元环的振荡模式
Oscillation patterns in tori of modified FHN neurons
摘要：
我们分析了电耦合，修改FitzHugh-Nagumo（FHN）振荡器的网络的动力学。网络构建块架构是一个二维正方形阵列形状为一个圆环，单向最近邻耦合在两个方向。线性逼近的起源一个单一的环面，揭示了阵列能够振荡通过一个霍普夫分岔，控制的神经元间的耦合常数。对一个环面的动力学进行群论分析，得到在直积群$\mathbb{Z}_N\times\mathbb{Z}_N\times\mathbb{Z}_2\times\mathbb{S}^1.$的影响下，离散的旋转波在正方形阵列中对角运动。然后，我们研究了由两个耦合环面构成的网络中多频振荡模式的存在性。我们表明，当作用于行波时，这组保持不变，而当它作用于同相振荡时，它们在时间上移动了$ϕ.$。因此，我们证明了一种振荡模式的可能性，其中一个环面产生恒定相移的行波，而第二个环面显示同步同相振荡，频率为行波所示频率的$N-$倍。
英文摘要：
We analyze the dynamics of a network of electrically coupled, modified FitzHugh-Nagumo (FHN) oscillators. The network building-block architecture is a bidimensional squared array shaped as a torus, with unidirectional nearest neighbor coupling in both directions. Linear approximation about the origin of a single torus, reveals that the array is able to oscillate via a Hopf bifurcation, controlled by the interneuronal coupling constants. Group theoretic analysis of the dynamics of one torus leads to discrete rotating waves moving diagonally in the squared array under the influence of the direct product group $\mathbb{Z}_N\times\mathbb{Z}_N\times\mathbb{Z}_2\times\mathbb{S}^1.$ Then, we studied the existence multifrequency patterns of oscillations, in networks formed by two coupled tori. We showed that when acting on the traveling waves, this group leaves them unchanged, while when it acts on the in-phase oscillations, they are shifted in time by $ϕ.$ We therefore proved the possibility of a pattern of oscillations in which one torus produces traveling waves of constant phase shift, while the second torus shows synchronous in-phase oscillations, at $N-$ times the frequency shown by the traveling waves.
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