Spikes as projections: Representation and processing of sensory stimuli in the time domain

This thesis presents a general theoretical framework for the representation and processing of sensory stimuli in the time domain, with a population of spiking neural circuits. For a large number of neural circuits it is shown that neural spiking, as a response to a sensory stimulus, amounts to taking a series of generalized measurements on the stimulus, in the form of inner products, rendering representation in the spike domain at largely equivalent with traditional amplitude domain representation.;The spiking mechanisms considered are of integrate-and-fire or threshold-and-fire type, with the existence of spike triggered feedback. Pulse coupled neural pairs of the aforementioned types are also considered. It is formally shown that for bandlimited stimuli a perfect stimulus recovery from the train of spikes is possible, provided that the population spike density is above the stimulus Nyquist rate. Explicit recovery algorithms are provided. These results are then extended for vector-valued and space-time (video) bandlimited input stimuli. The mathematical tools for these results include the theory of frames and sampling in Hilbert spaces.;In the case where the bandwidth assumption is dropped, the recovery problem is formulated as one of optimization based on appropriate cost functionals. Its solution comes from the theory of splines and is presented both in the noiseless as well as in the case where noise is present, in the form of random thresholds. It is demonstrated that the quality of the reconstruction improves as more neurons are employed to sense the stimulus. Increasing the number of neurons to achieve a finer representation of the sensory world is consistent with basic neurobiological thought.;All of the theory is accompanied by detailed examples based on neural circuits that operate akin to circuits found in peripheral sensory systems (auditory, visual). The algorithms are tested with input signals as complex as natural scenes, and implications to neurobiological modeling as well as neuromorphic engineering are discussed.;Finally, a neural architecture is presented that allows the implementation of operations, such as translations, rotations and zooming on visual stimuli, in the spike domain. These operations can be realized with the same basic stimulus decoding algorithm by dynamically modulating the connectivity between the encoding and decoding blocks. Examples are given and the emergence of invariant representations in the time domain is investigated.