Nonlinear Dynamics of Two Extended Systems: (i) Nonlinear Dispersive Waves, (ii) Primary Visual Cortex

McLaughlin9971813研究员和他的同事研究了两个扩展系统的复杂非线性动力学：(I)非线性色散波和(Ii)初级视觉皮质。项目(I)涉及非线性波的随机行为--特别是近可积孤子偏微分方程解的混沌。对于这些方程，时间混沌现在已经被很好地理解了；然而，对于时空混沌更严重的行为，人们所知的要少得多。通过渐近平均和数值实验相结合的方法，研究了它的发生以及它的宏观时间演变的描述。项目(II)是开发初级视皮层(V1)的理论和计算模型。建立了V1的穴位神经元模型，皮质神经元之间具有真实的耦合结构。该模型尽可能地受到生理和解剖学测量的限制。它具有多层结构，并关注大脑皮层网络的动态行为。非线性系统可以有随机行为的解，即使系统本身是严格确定的。当这些系统描述在空间上延伸的波时，这种混沌行为在今天还没有被很好地理解。这种扩展波普遍存在于整个自然界--包括影响天气和气候的气团和洋流的湍流行为。显然，对这种混沌波的动力学行为的预测具有根本的重要性。研究者对理想化的非线性波动方程的基本混沌行为进行了数学研究。大脑是一个极其复杂的扩展系统，其对刺激的动态反应是现代神经科学的中心问题。他和他的同事们，一个由纽约大学的神经科学家和数学家组成的跨学科小组，正在开发猕猴初级视觉皮质V1区域的计算模型，这是视觉通路上第一个被认为发生视觉信息皮质处理的皮质区域。他们的数值实验是为了模拟V1的特定实验室观测而设计的--数值模拟和实验室实验的结果都是在不断比较下得出的。通过数值实验比在实验室实验中更容易识别可能的皮层处理机制。实验生物学家和计算应用数学家之间的这种跨学科合作揭示了大脑皮层是如何处理视觉信息的，并将导致对大脑功能的更深入了解。该项目得到了数学科学部的应用数学计划和综合生物学和神经科学部的计算神经科学计划的支持。

McLaughlin9971813 The investigator and his colleagues study complex nonlineardynamics for two extended systems: (i) Nonlinear Dispersive Wavesand (ii) Primary Visual Cortex. Project (i) concerns the randombehavior of nonlinear waves -- specifically chaos fornear-integrable soliton partial differential equations. For theseequations temporal chaos is now rather well understood; however,far less is known about the more severe behavior ofspatial-temporal chaos. Both its onset, as well as thedescription of its macroscopic temporal evolution, are studiedthrough a combination of asymptotic averaging and numericalexperiments. Project (ii) is to develop a theoretical andcomputational model of the primary visual cortex (V1). A pointneuron model of V1 is developed, with a realistic couplingarchitecture between cortical neurons. The model is constrained,whenever possible, by physiological and anatomical measurements.It has a multi-layered architecture and focuses upon dynamicalbehavior of the cortical network. Nonlinear systems can have solutions that behave randomly,even though the system itself is strictly deterministic. Whenthese systems describe waves that are extended spatially, thischaotic behavior is not well understood today. Such extendedwaves are prevalent throughout nature -- including the turbulentbehavior of air masses and ocean currents that influence bothweather and climate. Clearly, the prediction of dynamicalbehavior for such chaotic waves has fundamental importance. Theinvestigator carries out mathematical studies of fundamentalchaotic behavior for idealized nonlinear wave equations. Thebrain is an extremely complex extended system whose dynamicalresponse to stimulation is a central problem for modern neuralscience. He and his colleagues, an interdisciplinary group ofneural scientists and mathematicians at New York University, aredeveloping a computational model of area V1 of the Primary VisualCortex of the Macaque monkey, the first cortical region along thevisual pathway at which cortical processing of visual informationis thought to occur. Their numerical experiments are designed tosimulate specific laboratory observations of V1 -- with theresults of both numerical simulation and laboratory experimentsunder continual comparison. It is easier to identify possiblemechanisms for cortical processing through numerical experimentsthan in the laboratory experiments. Such interdisciplinarycollaborations between experimental biologists and computationalapplied mathematicians unveil how the cortex processes visualinformation, and will lead to a deeper understanding of thefunctioning of the brain. The project is supported by the AppliedMathematics program in the Division of Mathematical Sciences andby the Computational Neuroscience program in the Division ofIntegrative Biology and Neuroscience.