Oscillations and waves in neuronal systems

振荡和其他形式的神经元活动出现在整个中枢神经系统。这些活动模式与睡眠节律、感觉处理、工作记忆和病理节律的产生有关，这些节律在精神分裂症和帕金森病等疾病状态下出现。虽然已经为这些系统提出了许多数学模型，但对这些模型的数学分析很少。这是因为模型是高度非线性的，它们表现出极其复杂的时空动态行为结构，并且往往以不直观的方式依赖于模型中的众多参数。本研究开发了数学和计算工具来分析一类一般的神经元模型。特别是，该研究：(i)有助于解释最近在昆虫触角叶早期感觉加工实验中看到的复杂时空模式的机制；（ii）基于前额皮质内的钙动力学和兴奋-抑制相互作用，开发了一种新的工作记忆模型；（iii）开发新的数学和计算工具，用于研究具有复杂几何形状的空间域中的钙和膜电位波传播。所考虑的问题在整个大脑的许多神经元系统中出现，新的数学工具也将在这些其他系统的研究中非常有用。大脑所做的一切都取决于神经元的放电特性。这包括对运动、学习、记忆、情感和感觉处理的控制。神经元放电模式的变化与记忆和衰老有关，病理模式与精神分裂症和帕金森病等神经系统疾病有关。随着新的复杂实验技术的发展，神经科学家现在开始更好地了解这些放电模式在正常大脑处理中的功能作用，这些模式背后的生物物理机制以及在学习，衰老和疾病期间这些模式变化的原因。然而，越来越清楚的是，数学建模和计算方法在解释大量新数据、测试新假设和理解复杂过程如何相互作用以产生复杂的大脑节律方面至关重要。新的数学工具被开发用于分析在许多大脑系统中出现的详细模型。特别是，工作记忆模型的构建和分析；这与我们在短时间内存储和操作信息以完成复杂任务的能力相对应。工作记忆问题与包括精神分裂症在内的几种神经系统疾病有关，该项目探索了多巴胺等神经递质的变化如何导致与这些神经系统疾病相关的病理节律。

Oscillations and other patterns of neuronal activity arise throughout the central nervous system. These activity patterns have been implicated in the generation of sleep rhythms, sensory processing, working memory and pathological rhythms that arise in diseased states such as schizophrenia and Parkinson's disease. While numerous mathematical models have been proposed for these systems, there has been very little mathematical analysis of these models. This is because the models are highly nonlinear, they exhibit an extremely complex structure of spatio-temporal dynamic behaviors and they depend in often unintuitive ways on the numerous parameters in the model. This research develops mathematical and computational tools for analyzing a general class of neuronal models. In particular, the research: (i) helps explain mechanisms underlying complex spatiotemporal patterns seen in recent experiments on early sensory processing in the insect antennal lobe; (ii) develops a novel model for working memory based on calcium dynamics and excitatory-inhibitory interactions within the prefrontal cortex; and (iii) develops new mathematical and computational tools for studying calcium and membrane potential wave propagation in spatial domains with complex geometries. The issues considered arise in many neuronal systems throughout the brain and the new mathematical tools will be very useful in the study of these other systems as well.Everything that the brain does depends on the firing properties of neurons. This includes the control of movements, learning, memory, emotion and sensory processing. Changes in neuronal firing patterns are associated with memory and aging, and pathological patterns have been implicated in neurological diseases such as schizophrenia and Parkinson's disease. With the development of new sophisticated experimental techniques, neuroscientists are now beginning to better understand the functional roles of these firing patterns in normal brain processing, the biophysical mechanisms underlying these patterns and what is responsible for changes in these patterns during learning, aging and disease. However, it is becoming increasingly clear that mathematical modeling and computational methods are critically important in accounting for the massive amounts of new data, testing new hypotheses and understanding how complicated processes interact to generate complex brain rhythms. Novel mathematical tools are developed for analyzing detailed models that arise in numerous brain systems. In particular, models for working memory are constructed and analyzed; this corresponds to our ability to store and manipulate information for a short time in order to carry out complex tasks. Problems with working memory is associated with several neurological diseases including schizophrenia and the project explores how changes in neurotransmitters such as dopamine lead to pathological rhythms associated with these neurological diseases.