Experimental bifurcation analysis of neurons using control-based continuation

生物系统的行为受一系列复杂现象的支配，这些现象本质上是非线性的，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中起着核心作用。以神经元模型为例，研究这些模型是为了揭示细胞爆发行为背后的分叉[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）和实验确定的模型参数值。在生物实验中，目前还没有直接测量分叉等非线性动力学特征的实验方法。基于控制的延拓（CBC）是布里斯托首创的一种非参数方法，它在实验测试过程中直接绘制出非线性物理系统的动态特征，而不依赖于数学模型或特定模型结构的参数估计。CBC将反馈控制与数值连续算法相结合，在线修改施加到系统的输入，以隔离感兴趣的非线性行为。通过这种方式，CBC提供了详细分析这些动态特征的最佳条件，随着实验可控参数的变化，以及检测和跟踪定性不同类型行为（即分叉）之间的边界。CBC的基本原理已经很好地建立起来，并且该方法已被应用于广泛的非生物（即机电）系统。例如，LR最近在一个多自由度结构上证明了该方法，该结构表现出复杂的非线性现象，如模式相互作用、准周期振荡和孤立响应曲线[2-3]。CBC证明能够提取系统的动态特征，如极限点分叉曲线，这是理解其行为（滞后，多稳定性等）的关键。这个博士项目旨在进一步开发CBC，使其可以应用于神经元，并利用实验来验证其爆发动力学。

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.