We investigate firing threshold manifolds in a mathematical model of an excitable neuron. The model analyzed investigates the phenomenon of post-inhibitory rebound spiking due to propofol anesthesia and is adapted from McCarthy et al. (SIAM J. Appl. Dyn. Syst. 11(4):1674-1697, 2012). Propofol modulates the decay time-scale of an inhibitory GABAa synaptic current. Interestingly, this system gives rise to rebound spiking within a specific range of propofol doses. Using techniques from geometric singular perturbation theory, we identify geometric structures, known as canards of folded saddle-type, which form the firing threshold manifolds. We find that the position and orientation of the canard separatrix is propofol dependent. Thus, the speeds of relevant slow synaptic processes are encoded within this geometric structure. We show that this behavior cannot be understood using a static, inhibitory current step protocol, which can provide a single threshold for rebound spiking but cannot explain the observed cessation of spiking for higher propofol doses. We then compare the analyses of dynamic and static synaptic inhibition, showing how the firing threshold manifolds of each relate, and why a current step approach is unable to fully capture the behavior of this model.

中文翻译：

---

可兴奋的神经元，激发阈值流形和鸭翼。

我们研究了可兴奋神经元数学模型中的激发阈值流形。所分析的模型研究了由于丙泊酚麻醉引起的抑制后反弹尖峰现象，并改编自 McCarthy 等人。(SIAM J. Appl. Dyn. Syst. 11(4):1674-1697, 2012)。丙泊酚调节抑制性 GABAa 突触电流的衰减时间尺度。有趣的是，该系统会在特定的丙泊酚剂量范围内引起反弹峰值。使用几何奇异扰动理论中的技术，我们确定了几何结构，称为折叠鞍型鸭翼，形成点火阈值流形。我们发现鸭分离线的位置和方向依赖于丙泊酚。因此，相关的缓慢突触过程的速度被编码在这个几何结构中。我们表明，使用静态、抑制性电流步骤协议无法理解这种行为，该协议可以为反弹尖峰提供单一阈值，但无法解释观察到的较高丙泊酚剂量的尖峰停止。然后，我们比较动态和静态突触抑制的分析，显示每个的触发阈值流形如何相关，以及为什么当前的步骤方法无法完全捕捉该模型的行为。