Institute of Mathematics, Humboldt University of Berlin, Unter den Linden 6, 10099 Berlin, Germany.
Philos Trans A Math Phys Eng Sci. 2013 Aug 19;371(1999):20120470. doi: 10.1098/rsta.2012.0470. Print 2013 Sep 28.
Rings of delay-coupled neurons possess a striking capability to produce various stable spiking patterns. In order to reveal the mechanisms of their appearance, we present a bifurcation analysis of the Hodgkin-Huxley (HH) system with delayed feedback as well as a closed loop of HH neurons. We consider mainly the effects of external currents and communication delays. It is shown that typically periodic patterns of different spatial form (wavenumber) appear via Hopf bifurcations as the external current or time delay changes. The Hopf bifurcations are shown to occur in relatively narrow regions of the external current values, which are independent of the delays. Additional patterns, which have the same wavenumbers as the existing ones, appear via saddle-node bifurcations of limit cycles. The obtained bifurcation diagrams are evidence for the important role of communication delays for the emergence of multiple coexistent spiking patterns. The effects of a short-cut, which destroys the rotational symmetry of the ring, are also briefly discussed.
延迟耦合神经元环具有产生各种稳定脉冲模式的惊人能力。为了揭示它们出现的机制，我们对具有延迟反馈和 HH 神经元闭环的 Hodgkin-Huxley（HH）系统进行了分岔分析。我们主要考虑外部电流和通信延迟的影响。结果表明，随着外部电流或延迟时间的变化，通常会通过 Hopf 分岔出现不同空间形式（波数）的周期性模式。Hopf 分岔发生在外部电流值的相对较窄区域内，与延迟无关。通过极限环的鞍结分岔出现具有与现有波数相同的其他模式。所得分岔图证明了通信延迟对于出现多种共存脉冲模式的重要作用。还简要讨论了破坏环的旋转对称性的捷径的影响。
