DynSyst_Special_Topics:Mathematical Modeling of Nonlinear Dynamics and Bifurcation Control of Epileptic Seizures

在前期工作中，我们采用了一个随机的、非线性的、中尺度皮层动力学的偏微分方程模型来研究癫痫波。我们已经将该模型应用于睡眠皮质，并开发了自动、连续、实时睡眠分期的能力。我们在使用局部线性嵌入来关联患者脑电数据和模型动力学的模型降阶方面取得了重大进展。我们已经了解了如何应用反馈控制理论的思想来快速改善或消除模型中的癫痫波。要做到这一点，我们已经了解了不确定性或波动对分叉的作用。我们研究了该模型，以考虑从正常状态到支持癫痫发作的皮质生理学变化(无限维模型的数学分支)的可能路径。该项目的技术目标包括：探索睡眠周期不同部分(在模型中和在患者中)的癫痫发作动力学倾向；进一步开发我们的癫痫控制算法，以更紧密地跟踪感觉电极和致动电极的设计；以及探索患者癫痫发作的电生理数据如何通过模型中不同类型的分叉与皮质癫痫易感性相关。全球有超过5000万人患有癫痫发作。在250万如此痛苦的美国人中，20%的人患有癫痫，对药物治疗没有很好的反应；他们可能会接受手术，切除大脑中受影响的部分，作为最后的手段。我们提高对癫痫的认识的工作将有助于我们找到更好的治疗形式的挑战。我们的方法是通过计算机模型以及医院环境中患者的观察来探索癫痫的发生和控制。最终，通过在大脑受影响的部分感应和应用电场，癫痫发作控制可能被证明是可能的。适当的方法将取决于大脑状态、电极放置和设计的细节，以及测量可能如何揭示癫痫发作的发生方式。这些都是项目中的研究重点。这些研究将为新的脑动力学理解提供基础，并将有助于开发一条有效治疗的途径。

In preliminary work, we have adapted a stochastic, nonlinear, partial differential equation model of mesoscale cortical dynamics to study seizure waves. We have applied the model to the sleeping cortex, and developed the capacity for automatic, continuous, real-time sleep staging. We have made a significant advance in model reduction using Locally Linear Embedding to associate patient EEG data and model dynamics. We have developed an understanding of how to apply ideas from feedback control theory to ameliorate?or eliminate?seizure waves rapidly in the model. To do so, we have understood the role of uncertainty or fluctuations on bifurcations. We have examined the model to consider possible pathways to changes in cortical physiology (mathematical bifurcations of the infinite dimensional model) from a normal state to one that can support seizures. Our technical goals for the project include the following: to explore propensity for seizure dynamics at different parts of the sleep cycle (in the model and in patients); to develop further our seizure control algorithms to more closely track designs of sense and actuation electrodes; and to explore the ways in which electrophysiological data of seizure onset in patients is related to cortical seizure susceptibility via different kinds of bifurcations in the model.Over 50M people worldwide suffer from epileptic seizures. Of 2.5M Americans so afflicted, 20% suffer epilepsy that does not respond well to pharmacological treatment; they may undergo surgery to remove the affected part of the brain as a last resort. Our work to improve understanding of epilepsy will aid in the challenge to find a better form of treatment. Our approach is to explore seizure occurrence and control through a computer model, along with patient observations in a hospital setting. Eventually, seizure control may prove to be possible by sensing and application of electrical fields in affected parts of the brain. Appropriate methodologies will depend on brain state, the details of electrode placement and design, and how measurements might reveal they way in which seizure onset occurs. These are all points of study in project. These investigations will provide the foundation for new understanding of brain dynamics and will contribute to the development of a pathway to effective treatment.