标题： 具有非对称脉冲耦合的相同神经元中的高阶同步 显示英文标题

标题： High-order synchronization in identical neurons with asymmetric pulse coupling

摘要： 高阶（$p/q$）同步现象，由系统中的\textit{两个不同的}频率引起，在强制振荡器（包括神经元）中已被广泛研究，并在耦合振荡器中有所涉及但研究较少。 它们的频率被锁定，使得一个振荡器每有$p$个周期，另一个振荡器就有$q$个周期。 我们在一对具有\textit{相同的}频率但\textit{不对称}耦合的耦合神经元中展示了这一现象。 具体来说，我们关注一个兴奋性(E)-抑制性(I)神经元对，其中即使具有相等的反向突触强度 $(g)$ 和倒数时间常数 $(\alpha)$，这种不对称性也是自然存在的。 我们通过模拟和分析彻底研究了不对称耦合引起的 $p/q$ 频率锁定结构在 $(g,\alpha)$ 参数空间中的情况。 模拟显示了准周期性、魔鬼阶梯、一种新的斐波那契尖峰序列排列，以及可约和不可约 $p/q$ 区域的存在。 我们引入了一种基于事件驱动映射的分析方法，以确定两个神经元在 $p/q$ 频率锁定状态下任何尖峰序列的存在性和稳定性。 具体来说，这种方法成功处理了非光滑分岔，我们可以利用它来获得相同E-I神经元对在任意耦合强度下的解。 与所谓的阿诺德舌不同，这里得到的 $p/q$ 区域并不是无结构的。 相反，它们具有自己的内部分岔结构，具有不同复杂程度的层次。 序列内和序列间的多稳定性，涉及相同$p/q$状态的尖峰序列，已被发现。 此外，通过$p/q$与$p'/q'$的重叠也会产生多稳定性。 可约和不可约$p/q$区域的边界由各种类型的鞍点和非光滑擦碰分岔定义。 显示英文摘要

摘要： The phenomenon of high-order ($p/q$) synchronization, induced by \textit{two different} frequencies in the system, is well-known and studied extensively in forced oscillators including neurons and to a lesser extent in coupled oscillators. Their frequencies are locked such that for every $p$ cycles of one oscillator there are $q$ cycles of the other. We demonstrate this phenomenon in a pair of coupled neurons having \textit{identical} frequencies but \textit{asymmetric} coupling. Specifically, we focus on an excitatory(E)-inhibitory(I) neuron pair where such an asymmetry is naturally present even with equal reciprocal synaptic strengths $(g)$ and inverse time constant $(\alpha)$. We thoroughly investigate the asymmetric coupling-induced $p/q$ frequency locking structure in $(g,\alpha)$ parameter space through simulations and analysis. Simulations display quasiperiodicity, devil staircase, a novel Farey arrangement of spike sequences, and presence of reducible and irreducible $p/q$ regions. We introduce an analytical method, based on event-driven maps, to determine the existence and stability of any spike sequence of the two neurons in a $p/q$ frequency-locked state. Specifically, this method successfully deals with non-smooth bifurcations and we could utilize it to obtain solutions for the case of identical E-I neuron pair under arbitrary coupling strength. In contrast to the so-called Arnold tongues, the $p/q$ regions obtained here are not structure-less. Instead they have their own internal bifurcation structure with varying levels of complexity. Intra-sequence and inter-sequence multistability, involving spike sequences of same $p/q$ state, are found. Additionally, multistability also arises by overlap of $p/q$ with $p'/q'$. The boundaries of both reducible and irreducible $p/q$ regions are defined by saddle node and non-smooth grazing bifurcations of various types.