SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 953-980, January 2021.
Experimental characterization of neuronal dynamics involves recording both of spontaneous activity patterns and of responses to transient and sustained inputs. While much theoretical attention has been devoted to the spontaneous activity of neurons, less is known about the dynamic mechanisms shaping their responses to transient inputs, although these bear significant physiological relevance. Here, we study responses to transient inputs in a widely used class of neuron models (nonlinear adaptive hybrid models) well-known to reproduce a number of biologically realistic behaviors. We focus on responses to transient inputs that have been previously associated with Type III neurons, arguably the least studied category in Hodgkin's classification, which are those neurons that never exhibit continuous firing in response to sustained excitatory currents. The two phenomena that we study are postinhibitory facilitation, in which an otherwise subthreshold excitatory input can induce a spike if it is applied with proper timing after an inhibitory pulse, and slope detection, in which a neuron spikes to a transient input only when the input's rate of change is in a specific, bounded range. Using dynamical systems theory, we analyze the origin of these phenomena in nonlinear hybrid models. We provide a geometric characterization of dynamical structures associated with postinhibitory facilitation in the system and an analytical study of slope detection for tent inputs. While the necessary and sufficient conditions for these behaviors are easily satisfied in neurons with Type III excitability, our proofs are quite general and valid for neurons that do not exhibit Type III excitability as well. This study therefore provides a framework for the mathematical analysis of these responses to transient inputs associated with Type III neurons in other systems and for advancing our understanding of these systems' computational properties.

中文翻译：

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混合非线性神经元模型中对瞬态输入的III型响应

SIAM应用动力系统杂志，第20卷，第2期，第953-980页，2021年1月。
神经元动力学的实验表征涉及记录自发活动模式以及对瞬时和持续输入的响应。尽管在神经元的自发活动上投入了大量理论关注，但对影响其对瞬态输入的反应的动力学机制知之甚少，尽管这些机制具有显着的生理意义。在这里，我们研究了广泛使用的一类神经元模型（非线性自适应混合模型）中对瞬态输入的响应，该模型众所周知，可以重现许多生物学上的真实行为。我们专注于对以前与III型神经元相关的瞬态输入的响应，这可能是霍奇金分类中研究最少的类别，那些永不响应持续的兴奋性电流而持续放电的神经元。我们研究的两种现象是抑制后促进，在抑制脉冲之后的适当时机应用，否则阈值以下的兴奋性输入会诱发尖峰；以及斜率检测，其中仅当输入的神经元在瞬态输入时，神经元才会尖峰到瞬态输入。变化率在特定的范围内。使用动力学系统理论，我们分析了非线性混合模型中这些现象的起源。我们提供了系统中与抑制后便利相关的动力学结构的几何特征，以及对帐篷输入的坡度检测的分析研究。虽然在具有III型兴奋性的神经元中很容易满足这些行为的必要条件和充分条件，我们的证明是相当笼统的，对于不具有III型兴奋性的神经元也有效。因此，本研究为在其他系统中对与III型神经元相关的瞬态输入的这些响应进行数学分析，以及增进我们对这些系统的计算特性的理解提供了一个框架。