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

McLaughlin

9971813

The investigator and his colleagues study complex nonlinear

dynamics for two extended systems: (i) Nonlinear Dispersive Waves

and (ii) Primary Visual Cortex. Project (i) concerns the random

behavior of nonlinear waves -- specifically chaos for

near-integrable soliton partial differential equations. For these

equations temporal chaos is now rather well understood; however,

far less is known about the more severe behavior of

spatial-temporal chaos. Both its onset, as well as the

description of its macroscopic temporal evolution, are studied

through a combination of asymptotic averaging and numerical

experiments. Project (ii) is to develop a theoretical and

computational model of the primary visual cortex (V1). A point

neuron model of V1 is developed, with a realistic coupling

architecture between cortical neurons. The model is constrained,

whenever possible, by physiological and anatomical measurements.

It has a multi-layered architecture and focuses upon dynamical

behavior of the cortical network.

Nonlinear systems can have solutions that behave randomly,

even though the system itself is strictly deterministic. When

these systems describe waves that are extended spatially, this

chaotic behavior is not well understood today. Such extended

waves are prevalent throughout nature -- including the turbulent

behavior of air masses and ocean currents that influence both

weather and climate. Clearly, the prediction of dynamical

behavior for such chaotic waves has fundamental importance. The

investigator carries out mathematical studies of fundamental

chaotic behavior for idealized nonlinear wave equations. The

brain is an extremely complex extended system whose dynamical

response to stimulation is a central problem for modern neural

science. He and his colleagues, an interdisciplinary group of

neural scientists and mathematicians at New York University, are

developing a computational model of area V1 of the Primary Visual

Cortex of the Macaque monkey, the first cortical region along the

visual pathway at which cortical processing of visual information

is thought to occur. Their numerical experiments are designed to

simulate specific laboratory observations of V1 -- with the

results of both numerical simulation and laboratory experiments

under continual comparison. It is easier to identify possible

mechanisms for cortical processing through numerical experiments

than in the laboratory experiments. Such interdisciplinary

collaborations between experimental biologists and computational

applied mathematicians unveil how the cortex processes visual

information, and will lead to a deeper understanding of the

functioning of the brain. The project is supported by the Applied

Mathematics program in the Division of Mathematical Sciences and

by the Computational Neuroscience program in the Division of

Integrative Biology and Neuroscience.