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

在初步工作中，我们采用了随机、非线性、偏微分方程中尺度皮层动力学模型来研究癫痫发作波。我们已经将该模型应用于睡眠皮层，并开发了自动、连续、实时睡眠分期的能力。我们使用局部线性嵌入将患者脑电图数据和模型动力学相关联，在模型简化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改善？或消除?在模型中癫痫发作迅速。为此，我们了解了不确定性或波动对分岔的作用。我们已经检查了该模型，以考虑皮层生理学（无限维模型的数学分支）从正常状态到可以支持癫痫发作的变化的可能途径。该项目的技术目标包括：探索睡眠周期不同阶段（模型和患者）的癫痫发作动态倾向；进一步发展我们的癫痫控制算法，以更紧密地跟踪传感和驱动电极的设计；并通过模型中不同类型的分岔，探讨患者癫痫发作的电生理数据与皮层癫痫易感性之间的关系。全世界有超过5000万人患有癫痫发作。250万美国患者中，20%患有癫痫，药物治疗效果不佳；他们可能会接受手术切除受影响的大脑部分作为最后的手段。我们提高对癫痫认识的工作将有助于找到更好的治疗方法。我们的方法是通过计算机模型以及在医院环境中的患者观察来探索癫痫发作的发生和控制。最终，通过感知和应用大脑受影响部位的电场来控制癫痫发作可能成为可能。合适的方法取决于大脑的状态，电极放置和设计的细节，以及测量如何揭示癫痫发作的方式。这些都是项目的研究重点。这些研究将为大脑动力学的新认识提供基础，并将有助于开发有效治疗途径。

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.