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

研究者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）初级视觉皮层。项目(i)关注非线性波的随机行为——特别是近可积孤子偏微分方程的混沌。对于这些方程，时间混沌现在已经相当清楚了；然而，对于更严重的时空混沌行为，我们所知甚少。通过渐近平均和数值实验的结合，研究了它的开始，以及它的宏观时间演化的描述。项目（ii）是建立初级视觉皮层（V1）的理论和计算模型。建立了V1的点神经元模型，该模型具有真实的皮质神经元之间的耦合结构。只要有可能，该模型就会受到生理和解剖学测量的约束。它具有多层结构，重点研究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扩展的波时，这种混沌行为在今天还没有得到很好的理解。这种延伸波在自然界中普遍存在，包括影响天气和气候的气团和洋流的湍流行为。显然，预测这种混沌波的动力行为具有根本的重要性。研究者对理想非线性波动方程的基本混沌行为进行数学研究。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核心问题。他和他的同事，一个由纽约大学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层V1区域的计算模型，这是视觉通路上的第一个皮层区域，视觉信息的皮质处理被认为发生在这里。他们的数值实验旨在模拟V1的特定实验室观测结果，并将数值模拟和实验室实验的结果进行持续比较。通过数值实验比在实验室实验中更容易确定皮层处理的可能机制。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皮层如何处理视觉信息，并将导致对大脑功能的更深层次的理解。该项目由数学科学部的应用数学项目和综合生物学与神经科学部门的计算神经科学项目支持。

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.