Institute for Theoretical Biology, Department of Biology, Humboldt-Universität zu Berlin, Philippstrasse 13, Haus 4, 10115 Berlin, Germany and Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany.
Phys Rev E. 2017 May;95(5-1):052203. doi: 10.1103/PhysRevE.95.052203. Epub 2017 May 3.
Prominent changes in neuronal dynamics have previously been attributed to a specific switch in onset bifurcation, the Bogdanov-Takens (BT) point. This study unveils another, relevant and so far underestimated transition point: the saddle-node-loop bifurcation, which can be reached by several parameters, including capacitance, leak conductance, and temperature. This bifurcation turns out to induce even more drastic changes in synchronization than the BT transition. This result arises from a direct effect of the saddle-node-loop bifurcation on the limit cycle and hence spike dynamics. In contrast, the BT bifurcation exerts its immediate influence upon the subthreshold dynamics and hence only indirectly relates to spiking. We specifically demonstrate that the saddle-node-loop bifurcation (i) ubiquitously occurs in planar neuron models with a saddle node on invariant cycle onset bifurcation, and (ii) results in a symmetry breaking of the system's phase-response curve. The latter entails an increase in synchronization range in pulse-coupled oscillators, such as neurons. The derived bifurcation structure is of interest in any system for which a relaxation limit is admissible, such as Josephson junctions and chemical oscillators.
先前，神经元动力学的显著变化归因于起始分岔的特定开关，即 Bogdanov-Takens（BT）点。本研究揭示了另一个相关但迄今为止被低估的过渡点：鞍结环分岔，它可以由几个参数达到，包括电容、漏导和温度。该分岔导致的同步变化比 BT 跃迁更为剧烈。这一结果源于鞍结环分岔对极限环进而对尖峰动力学的直接影响。相比之下，BT 分岔对亚阈值动力学施加直接影响，因此仅间接与尖峰相关。我们特别证明，鞍结环分岔 (i) 在具有鞍结在不变周期起始分岔的平面神经元模型中普遍存在，以及 (ii) 导致系统相位响应曲线的对称性破缺。后者意味着在脉冲耦合振荡器（如神经元）中同步范围增加。该导出的分岔结构对于任何允许弛豫极限的系统都具有意义，例如约瑟夫森结和化学振荡器。
