We investigate the dynamic mechanisms of generation of subthreshold and phase resonance in two-dimensional linear and linearized biophysical (conductance-based) models, and we extend our analysis to account for the effect of simple, but not necessarily weak, types of nonlinearities. Subthreshold resonance refers to the ability of neurons to exhibit a peak in their voltage amplitude response to oscillatory input currents at a preferred non-zero (resonant) frequency. Phase-resonance refers to the ability of neurons to exhibit a zero-phase (or zero-phase-shift) response to oscillatory input currents at a non-zero (phase-resonant) frequency. We adapt the classical phase-plane analysis approach to account for the dynamic effects of oscillatory inputs and develop a tool, the envelope-plane diagrams, that captures the role that conductances and time scales play in amplifying the voltage response at the resonant frequency band as compared to smaller and larger frequencies. We use envelope-plane diagrams in our analysis. We explain why the resonance phenomena do not necessarily arise from the presence of imaginary eigenvalues at rest, but rather they emerge from the interplay of the intrinsic and input time scales. We further explain why an increase in the time-scale separation causes an amplification of the voltage response in addition to shifting the resonant and phase-resonant frequencies. This is of fundamental importance for neural models since neurons typically exhibit a strong separation of time scales. We extend this approach to explain the effects of nonlinearities on both resonance and phase-resonance. We demonstrate that nonlinearities in the voltage equation cause amplifications of the voltage response and shifts in the resonant and phase-resonant frequencies that are not predicted by the corresponding linearized model. The differences between the nonlinear response and the linear prediction increase with increasing levels of the time scale separation between the voltage and the gating variable, and they almost disappear when both equations evolve at comparable rates. In contrast, voltage responses are almost insensitive to nonlinearities located in the gating variable equation. The method we develop provides a framework for the investigation of the preferred frequency responses in three-dimensional and nonlinear neuronal models as well as simple models of coupled neurons.

中文翻译：

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二维神经模型中振荡输入的频率偏好响应：亚阈值幅度和相位共振的几何方法。

我们研究了二维线性和线性化生物物理（基于电导）模型中产生亚阈值和相位共振的动态机制，并扩展了我们的分析以解释简单但不一定弱的非线性类型的影响。亚阈值共振是指神经元在优选的非零（共振）频率下对振荡输入电流的电压幅度响应表现出峰值的能力。相位共振是指神经元在非零（相位共振）频率下对振荡输入电流表现出零相位（或零相移）响应的能力。我们采用经典的相平面分析方法来解释振荡输入的动态影响，并开发了一种工具，包络平面图，与越来越小的频率相比，它捕捉了电导和时间尺度在放大谐振频带电压响应方面的作用。我们在分析中使用包络平面图。我们解释了为什么共振现象不一定是由于静止时虚特征值的存在而产生的，而是它们是从内在时间尺度和输入时间尺度的相互作用中出现的。我们进一步解释了为什么除了移动谐振和相位谐振频率之外，时间尺度分离的增加还会导致电压响应的放大。这对于神经模型至关重要，因为神经元通常表现出强烈的时间尺度分离。我们扩展这种方法来解释非线性对谐振和相位谐振的影响。我们证明了电压方程中的非线性会导致电压响应的放大以及相应线性化模型无法预测的谐振和相位谐振频率的偏移。非线性响应和线性预测之间的差异随着电压和门控变量之间的时间尺度分离水平的增加而增加，并且当两个方程以可比的速率发展时，它们几乎消失。相比之下，电压响应对选通变量方程中的非线性几乎不敏感。我们开发的方法为研究三维和非线性神经元模型以及耦合神经元的简单模型中的首选频率响应提供了框架。