In many neurophysiological studies and brain-inspired computation paradigms, there is still a need for new spike train analysis and learning algorithms because current methods tend to be limited in terms of the tools they provide and are not easily extended. This chapter presents a general framework to develop spike train machine learning methods by defining inner product operators for spike trains. They build on the mathematical theory of reproducing kernel Hilbert spaces (RKHS) and kernel methods, allowing a multitude of analysis and learning algorithms to be easily developed. The inner products utilize functional representations of spike trains, which we motivate from two perspectives: as a biological-modeling problem, and as a statistical description. The biological-modeling approach highlights the potential biological mechanisms taking place at the neuron level and that are quantified by the inner product. On the other hand, by interpreting the representation from a statistical perspective, one relates to other work in the literature. Moreover, the statistical description characterizes which information can be detected by the spike train inner product. The applications of the given inner products for development of machine learning methods are demonstrated in two

One of the fundamental difficulties in neural assembly studies is the lack of an effective, high resolution measure of the spatio-temporal structure in spike trains obtained from a single realization. In this chapter a systematic approach to estimate the cross-correlation (CC) of spike trains, over time and in only one realization, is proposed. The solution lies in an alternate definition of cross-correlation which suggests that, rather than time averaging as is current practice, we should use ensemble averaging. This observation suggests a natural instantaneous CC estimator as required for high temporal resolution and real-time ensemble analysis and decoding. Results are shown in simulated datasets and neural activity of rat motor cortical neurons during a behavioral task. 1

It would be only appropriate to thank my advisor Dr. Jose ́ Carlos Santos Carvalho Pŕıncipe first for his guidance and lessons not only for research but for life in general. A lot of people helped me get through my journey of graduate school, and perhaps my attempt to properly thank them all will fail miserably, but I have to try. Dr. Thomas B. DeMarse helped me enormously especially by letting me perform experiments, and he has been emotionally supporting my research as well. I owe my deepest gratitude to Dr. Murali Rao for bringing mathematical rigor to my clumsy ideas. My committee members Dr. Arunava Banerjee, Dr. Bruce Wheeler and Dr. Justin Sanchez supported me and kept me motivated. Dr. John Harris’s kind support allowed me to make friends and connections around the world. Dr. Purvis Bedenbaugh brought me a special Christmas gift of auditory spiking data in 2009. I am indebted to many of my colleagues; without their support this dissertation would not have been possible. António Rafael da Costa Paiva has been a great friend and colleague for developing spike train based signal processing algorithms. Jianwu Xu and Weifeng Liu gave me great intuitions for reproducing kernel Hilbert spaces. Dongming Xu enlightened me on dynamical systems. Brain storming with Karl Dockendorf was always a pleasure. I learned so much from the discussions with Steven Van Vaerenbergh and Luis Sanchez. Among all the most fruitful collaboration was with Sohan Seth. He has been a great friend, and brought joy to my work. I greatly appreciate all the support my friends gave me in a number of ways. I only mention a few of them here: Pingping Zhu the operator operator, Jason Winters