Experimental bifurcation analysis of neurons using control-based continuation

生物系统的行为受到一系列复杂现象的支配，这些现象本质上是非线性的，并在不同的时间和空间尺度上相互作用。数学建模目前在理解这些非线性系统的动态行为方面发挥着核心作用。例如，研究神经元模型是为了揭示细胞爆裂行为背后的分叉[1]。这种方法的主要警告是，所发现的分叉和其他非线性特征严重依赖于模型假设(即捕获的物理)和通过实验识别的模型参数值。目前，还没有一种实验方法可以直接测量生物实验中的分叉等非线性动力学特征。由布里斯托尔首创的基于控制的延拓(CBC)是一种非参数方法，它在实验测试期间直接绘制出非线性物理系统的动态特征，而不依赖于对数学模型或特定模型结构的参数的估计。将反馈控制和数值连续算法相结合，CBC在线修改系统的输入，以隔离感兴趣的非线性行为。通过这种方式，CBC提供了最好的条件来详细分析这些动态特征，在实验可控参数改变时跟踪它们，并检测和跟踪定性不同类型的行为之间的边界(即分叉)。CBC的基本原理已经确立，该方法已经应用于广泛的非生命(即机电)系统。例如，LR最近在一个多自由度结构上演示了该方法，该结构表现出复杂的非线性现象，如模式相互作用、准周期振荡和孤立的响应曲线[2-3]。CBC被证明能够提取系统的动态特征，例如极限点分叉曲线，这些特征对于理解系统的行为(滞后、多稳定性等)是关键。该博士项目旨在进一步开发CBC，以便将其应用于神经元，并用于实验表征其爆裂动力学。

The behaviour of biological systems is governed by a wide range of complex phenomena that are intrinsically nonlinear and interact on different time and spatial scales. Mathematical modelling currently plays a central role in understanding the dynamic behaviour of these nonlinear systems. Take, for example, neuron models that are studied to unveil the bifurcations underlying the cell's bursting behaviour [1]. The major caveat with this approach is that discovered bifurcations and other nonlinear features critically depend on the model assumptions (i.e. captured physics) and the model parameter values identified experimentally. As of now, there exist no experimental method that can directly measure nonlinear dynamic features such as bifurcations directly in biological experiments. Pioneered at Bristol, control-based continuation (CBC) is a non-parametric method that maps out the dynamic features of a nonlinear physical system directly during experimental tests, without relying on the estimation of the parameters of a mathematical model, or a particular model structure. Combining feedback control with numerical continuation algorithms, CBC modifies, on-line, the input applied to the system in order to isolate the nonlinear behaviour of interest. In this way, CBC offers the best conditions to analyse these dynamic features in detail, to follow them as experimentally-controllable parameters are changed, and to detect and track boundaries between qualitatively different types of behaviour (i.e. bifurcations). The fundamental principles of CBC are well established, and the method has been applied to a wide range of non-living (i.e. electro-mechanical) systems. For instance, the method was recently demonstrated by LR on a multi-degree-of-freedom structure exhibiting complex nonlinear phenomena such as mode interaction, quasi-periodic oscillations and isolated response curves [2-3]. CBC proved able to extract dynamic features of the system such as curves of limit-point bifurcations that are key to the understanding of its behaviour (hysteresis, multi-stability, etc.). This PhD project aims to further develop CBC such that it can be applied to neurons and exploited to experimentally characterise their bursting dynamics.