
Neural Coding My longterm research on neural coding is an interdisciplinary study in which I have been developing mathematical and statistical models to understand how the brain codes (or represents) information and how we can decode neural signals to rebuild internal or external behaviorals. My work in this field has focused on classical methods in dynamic systems such as Kalman filters, hidden Markov models, and point processes. Recently, I have started to investigate neural coding in the function space of neural spike trains, and this research has been a close collaboration with Anuj Srivastava. We are proposing a datadriven framework by treating each spike train as one point in an infinite dimensional "spike manifold". In this framework we aim to construct new tools for: 1) quantifying differences in spike trains using a "Euclidean distance", 2) based on this "Euclidean distance", computing summary statistics such as the mean and covariance of spike trains, and 3) performing statistical inferences such as confidence intervals, hypothesis tests, regressions, and PCA in the spike train space. This new set of statistical tools is expected to provide an alternative pathway to the commonlyused modelbased methods (i.e. rate models and temporal models) in the neural coding community. Current and Past Collaborators:Anuj Srivastava, FSU, Statistics Robert J. Contreras, FSU, Psychology Aaron Wilber, FSU, Psychology David Mumford, Brown, Applied Mathematics Michael J. Black, Brown, Computer Science John P. Donoghue, Brown, Neuroscience Elie Bienenstock, Brown, Applied Mathematics/Neuroscience Nicholas Hatsopoulos, U Chicago, Neuroscience Zhiyi Chi, U Connecticut, Statistics Liam Paninski, Columbia, Statistics My work in functional data anlaysis has been a close collaboration with Anuj Srivastava who motivated my interest in this field. In this research we propose to develop a comprehensive framework for a joint registration and analysis of functional data. The term functional data is used here very generally as it encompasses a variety of situations including registration of realvalued functions, Euclidean curves, parametrized surfaces, 2D and 3D images, timeindexed paths on nonlinear manifolds and so on. In case of shape analysis of curve and surfaces, the registration maps are actually called the reparameterization functions, while in cases of images the registrations are more commonly termed deformations. We also perform theoretical investigations on consitency and template estimation. Current Collaborators:Anuj Srivastava, FSU, Statistics Jinfeng Zhang, FSU, Statistics Eric Klassen, FSU, Mathematics I have been collaborating with Frank Johnson, Richard Bertram, and Rick Hyson on birdsong analysis. We investigate the functional integration of three distinct brain pathways (auditory, premotor, and striatal) during juvenile learning and adult recitation of songbird vocal patterns. My work focuses on statistical methods to analyze experimental data and developing models to characterize song production. Current Collaborators:Frank Johnson, FSU, Psychology Richard Bertram, FSU, Mathematics Rick Hyson, FSU, Psychology 