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

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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.