Cao Ben, Gu Huaguang, Ma Kaihua
School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, 200092 China.
Cogn Neurodyn. 2022 Aug;16(4):917-940. doi: 10.1007/s11571-021-09744-4. Epub 2021 Nov 17.
The hair bundles of inner hair cells in the auditory nervous exhibit spontaneous oscillations, which is the prerequisite for an important auditory function to enhance the sensitivity of inner ear to weak sounds, otoacoustic emission. In the present paper, the dynamics of spontaneous oscillations and relationships to steady state are acquired in a two-dimensional model with fast variable (displacement of hair bundles) and slow variable . The spontaneous oscillations are derived from negative stiffness modulated by two biological factors ( and ) and are identified to appear in multiple two-dimensional parameter planes. In (, ) plane, comprehensive bifurcations including 4 types of codimension-2 bifurcation and 5 types of codimension-1 bifurcation related to the spontaneous oscillations are acquired. The spontaneous oscillations are surrounded by supercritical and subcritical Hopf bifurcation curves, and outside of the curves are two cases of steady state. Case-1 and Case-2 steady states exhibit Z-shaped (coexistence of ) and N-shaped (coexistence of ) -nullclines, respectively. In (, ) plane, left and right to the spontaneous oscillations are two subcases of Case-1, which exhibit the stable equilibrium point locating on the upper and lower branches of -nullcline, respectively, resembling that of the neuron. Lower to the spontaneous oscillations are 3 subcases of Case-2 from left to right, which manifest stable equilibrium point locating on left, middle, and right branches of -nullcline, respectively, differing from that of the neuron. The phase plane for steady state is divided into four parts by nullclines, which manifest different vector fields. The phase trajectory of transient behavior beginning from a phase point in the four regions to the stable equilibrium point exhibits different dynamics determined by the vector fields, which is the basis to identify dynamical mechanism of complex forced oscillations induced by external signal. The results present comprehensive viewpoint and deep understanding for dynamics of the spontaneous oscillations and steady states of hair bundles, which can be used to well explain the experimental observations and to modulate functions of spontaneous oscillations.
听觉神经中内毛细胞的毛束表现出自发振荡，这是一种重要听觉功能的前提条件，该功能可增强内耳对微弱声音的敏感性，即耳声发射。在本文中，通过一个具有快速变量（毛束位移）和慢速变量的二维模型，获得了自发振荡的动力学及其与稳态的关系。自发振荡源自由两个生物学因素（和）调制的负刚度，并被确定出现在多个二维参数平面中。在（，）平面中，获得了包括4种余维2分岔和5种与自发振荡相关的余维1分岔在内的综合分岔。自发振荡被超临界和亚临界霍普夫分岔曲线所包围，曲线之外是两种稳态情况。情况1和情况2稳态分别表现出Z形（共存）和N形（共存）的零倾线。在（，）平面中，自发振荡的左右两侧分别是情况1的两个子情况，它们分别表现出稳定平衡点位于零倾线的上分支和下分支，类似于神经元的情况。自发振荡下方从左到右是情况2的3个子情况，它们分别表现出稳定平衡点位于零倾线的左、中、右分支，与神经元的情况不同。稳态的相平面被零倾线分为四个部分，表现出不同的向量场。从四个区域中的一个相点开始到稳定平衡点的瞬态行为的相轨迹表现出由向量场决定的不同动力学，这是识别外部信号诱导的复杂强迫振荡动力学机制的基础。这些结果为毛束的自发振荡和稳态动力学提供了全面的观点和深入的理解，可用于很好地解释实验观察结果并调节自发振荡的功能。
