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Applied Math Colloquium

Friday, Dec 7, 2012, 11:30 AM
Cullimore Lecture Hall, Lecture Hall II
New Jersey Institute of Technology

Is Our Sensing Compressed?

Gregor Kovacic

Department of Mathematics, Rensselaer Polytechnic Institute

Abstract

In the retina, the number of receptors (rods and cones) exceeds the number of the retinal ganglion cells immediately downstream from them by a factor of 100. How is it possible that not too much information is lost in this bottleneck? The answer may perhaps come from the theory of compressed sensing. The classic Shannon-Nyquist theorem states that a signal of a given bandwidth must be sampled at a rate at least twice this bandwidth in order to be accurately reproduced. However, if the signal is sparse in some appropriate Fourier domain, this sampling rate may be reduced considerably without degrading the reproduction, provided it is sampled sufficiently randomly. The talk will discuss how much information is lost/retained by neuronal networks of integrate-and-fire type, whose neurons sample stimuli in a random fashion. A modification of the Candes-Tao reconstruction algorithm, together with an appropriate linearization of the transformation of the stimulus into neuronal firing rates, will be used to determine how well the initial stimulus can be reconstructed from the given network's firing rates. The dependence of the reconstruction quality on the network parameters such as sparsity of connectivity will be described. A brief introduction to the relevant concepts from the theory of integrate-and-fire neuronal networks and their mean-field representations will also be given.