Quantifying uncertainty in perturbed brain networks: towards a decision support tool for epilepsy surgery

许多自然和人造系统都可以用网络来描述，在网络中，一组节点通过边连接起来，形成一个网络结构。例子包括通信或传输网络，以及生物系统中的网络。在现实世界的应用中，这些网络的节点通常是动态行为的：它们的属性随着时间的推移而变化。了解网络结构如何导致不同的动力学行为是应用非线性动力学中一个尚未解决的基本问题。我们还缺乏对网络扰动(例如移除特定节点)导致网络动态变化的方式的基本了解。以前对这些问题的研究往往集中在特定类型的网络行为上，例如振荡的同步或更复杂的动力学。然而，重要的是将这些研究扩展到包括在本质上不同的状态之间进行零星转换的动力学。这样的系统支撑了“动态疾病”的概念，因此，对具有这些动态的网络的扰动的研究自然属于EPSRC医疗保健技术主题。一个恰当的例子是癫痫；这是一种普遍存在的神经疾病，大脑网络中的节点零星地产生异常活动，导致一个人癫痫发作。我们可以认为，癫痫大脑中大脑网络的动力学经历了从正常功能时期到癫痫发作中异常活动时期的“状态转换”。有很多我们不知道的为什么癫痫会在网络中发生，特别是我们经常不知道如何治疗特定人的癫痫，使癫痫不再发生。最不为人所知的治疗形式之一是手术，即移除大脑网络的节点，希望这将阻止癫痫的发生。为了更好地了解癫痫发作以及手术可能如何减轻癫痫发作，我将研究大脑网络的数学模型，这些网络可以产生类似癫痫发作的状态转换动力学。在这些模型中，可以通过从网络中移除节点并量化状态切换减少到什么程度来模拟手术。为了做到这一点，我需要开发一些方法来为给定的网络选择应该删除哪些节点，以便最有效地降低从一种状态切换到另一种状态的能力。在大型网络中，测试移除每一组可能的节点对其动态性能的影响很快就变得困难起来。因此，我将开发计算方法来有效地估计应该删除的节点集，以便限制网络在状态之间切换的能力。另一个关键问题是，有许多不同的数学模型可以用来生成状态切换动态，这些模型可能会产生关于哪些节点应该被移除的不同预测。为了量化预测中的这种不确定性，我将使用我开发的计算方法来计算不同模型选择下的预测，并量化预测在多大程度上取决于模型的选择。为了测试开发的方法的适用性和对现实世界临床问题的理解，我将把我的方法应用于一组来自接受癫痫手术的患者的数据。我将从这些数据中得出每个人大脑的网络表示，然后使用我的数学工具来预测哪些节点应该被移除，以便使它们免于癫痫发作。这些预测是可以检验的，因为我们知道在患者的手术中哪些结节实际上被切除了，以及这些结节的切除是否导致了癫痫发作的自由。

Many natural and man-made systems can be described in terms of networks, in which a set of nodes is connected by edges to make a network structure. Examples include communication or transport networks, as well as networks in biological systems. Often, in real world applications, the nodes of these networks behave dynamically: their properties change over time. Understanding the ways in which network structure can lead to different dynamic behaviors is a fundamental unsolved problem in applied nonlinear dynamics. We also lack fundamental understanding of the ways in which perturbations to networks, for example the removal of particular nodes, leads to changes in network dynamics. Previous investigations into these problems have often focused on particular kinds of network behavior, such as the synchronisation of oscillations or more complex dynamics. However it is important to extend these studies to include dynamics that undergo sporadic switching between qualitatively different states. Such systems underpin the concept of "dynamic diseases" and therefore studies of perturbations to networks with these dynamics falls naturally into the EPSRC Healthcare Technologies theme. A pertinent example is epilepsy; a prevalent neurological disorder in which nodes in networks of the brain sporadically produce abnormal activity, causing a person to suffer a seizure. We can consider that the dynamics of brain networks in the epileptic brain undergo "state-switching" from periods of healthy functioning to periods in which abnormal activity in seizures occur. There is much we do not know about why seizures occur in networks, and in particular, we often do not know how to treat a particular person's epilepsy, so that seizures no longer occur. One of the least understood forms of treatment is surgery, in which nodes of brain networks are removed, with the hope that this will stop the occurrence of seizures. In order to better understand seizures and how surgery may abate them, I will study mathematical models of brain networks that can generate seizure-like state-switching dynamics. In these models, surgery can be simulated by removing nodes from the network and quantifying to what extent state-switching is reduced. In order to do this, I need to develop ways to choose, for a given network, which nodes should be removed in order to most effectively reduce the ability to switch from one state to another. In large networks, it soon becomes intractable to test the effect that the removal of every possible set of nodes has on its dynamics. I will therefore develop computational approaches to efficiently estimate the set of nodes that should be removed in order to limit the ability of a network to switch between states. Another critical problem is that there are many different mathematical models that can be used to generate state-switching dynamics, and these may yield different predictions for which nodes should be removed. In order to quantify this uncertainty in predictions, I will use the computational methods I develop to calculate predictions under different choices of models, and quantify to what extent predictions depend on the choice of model. To test the applicability of the developed methods and understanding to the real world clinical problem, I will apply my methods to a set of data derived from patients who have undergone epilepsy surgery. I will derive network representations of each persons brain from this data and then use my mathematical tools to predict which nodes should have been removed in order to render them seizure free. These predictions can be tested since we know which nodes were actually removed in the patients' surgery, and whether the removal of those nodes resulted in seizure freedom.