Title: Massively Parallel Spiking Neural Circuits
Description: This thesis presents a class of massively parallel spiking neural circuit architectures in which neurons are modeled by dendritic stimulus processors cascaded with spike generators. We investigate how visual stimuli can be represented by the spike times generated by the massively parallel neural circuits, how the spike times can be used to reconstruct and process visual stimuli, and the conditions when visual stimuli can be faithfully represented/reconstructed. Functional identification of the massively parallel neural circuits from spike times and its evaluation are also investigated. Together, this thesis offers a comprehensive analytic framework of massively parallel spiking neural circuit architectures arising in the study of early visual systems. In encoding, modeling of visual stimuli in reproducing kernel Hilbert spaces is presented, recognizing the importance of studying visual encoding in a rigorous mathematical framework. For massively parallel neural circuits with biophysical spike generators, I/O characterization of the biophysical spike generators becomes possible by introducing phase response curve manifolds for the biophysical spike generators.

I/O characterization of the entire neural circuit can then be interpreted as generalized sampling in the Hilbert space. Multi-component dendritic stimulus processors are introduced to model visual encoding in stereoscopic color vision. It is also shown that encoding of visual stimuli by an ensemble of complex cells has the complexity of Volterra dendritic stimulus processors. Based on the I/O characterization, reconstruction algorithms are derived to decode, from spike times, visual stimuli encoded by these massively parallel neural circuits. Decoding problems are first formulated as spline interpolation problems. Conditions on faithful reconstruction are presented, allowing the probe of information content carried by the spikes. Algorithms are developed to qualify the decoding in massively parallel settings. For stereoscopic color visual stimuli, demixing of individual channels from an unlabeled set of spike trains is demonstrated. For encoding with complex cells, decoding problems are formulated as rank minimization problems.

It is shown that the decoding algorithm does not suffer from the curse of dimensionality and thereby allows for a visual representation using biologically realistic neural resources. The study of visual stimuli encoding and decoding enables the functional identification of massively parallel neural circuits. The duality between decoding and functional identification suggests that algorithms for functional identification of the projection of dendritic stimulus processors onto the space of input stimuli can be formulated similarly to the decoding algorithms. Functional identification of dendritic stimulus processors of neurons carrying stereoscopic color information as well as that of energy processing in complex cells is demonstrated. Furthermore, this duality also inspires a novel method to evaluate the quality of functional identification of massively parallel spiking neural circuits. By reconstructing novel stimuli using identified circuit parameters, the evaluation of the entire identified circuit is reduced to intuitive comparisons in stimulus space.

The use of biophysical spike generators advances a methodology in the study of intrinsic noise sources in neurons and their effects on stimulus representation and on precision of functional identification. These effects are investigated using a class of nonlinear neural circuits consisting of both feedforward and feedback Volterra dendritic stimulus processors and biophysical spike generators. It is shown that encoding with neural circuits with intrinsic noise sources can be interpreted as generalized sampling with noisy measurements. Effects of noise on decoding and functional identification are derived theoretically and were systematically investigated by extensive simulations. Finally, the massively parallel neural circuit architectures are shown to enable the implementation of identity preserving transformations in the spike domain using a switching matrix regulating the connection between encoding and decoding. Two realizations of the architectures are developed, and extensive examples using continuous visual streams are provided.

Implications of this result on the problem of invariant object recognition in the spike domain are discussed.