Modern neurotechnologies enable the recording of neural activity at the scale of entire brains and with single-cell resolution. However, the lack of principled approaches to extract structure from these massive data streams prevent us from fully exploiting the potential of these technologies. This thesis, divided in three parts, introduces new statistical machine learning methods to enable the large-scale analysis of some of these complex neural datasets. In the first part, I present a method that leverages Gaussian quadrature to accelerate inference of neural encoding models from a certain type of observed neural point processes --- spike trains --- resulting in substantial improvements over existing methods. The second part focuses on the simultaneous electrical stimulation and recording of neurons using large electrode arrays. There, identification of neural activity is hindered by stimulation artifacts that are much larger than spikes, and overlap temporally with spikes. To surmount this challenge, I develop an algorithm to infer and cancel this artifact, enabling inference of the neural signal of interest. This algorithm is based on a a bayesian generative model for recordings, where a structured gaussian process is used to represent prior knowledge of the artifact. The algorithm achieves near perfect accuracy and enables the analysis of data hundreds of time faster than previous approaches. The third part is motivated by the problem of inference of neural dynamics in the worm C.elegans: when taking a data-driven approach to this question, e.g., when using whole-brain calcium imaging data, one is faced with the need to match neural recordings to canonical neural identities, in practice resolved by tedious human labor. Alternatively, on a bayesian setup this problem may be cast as posterior inference of a latent permutation. I introduce methods that enable gradient-based approximate posterior inference of permutations, overcoming the difficulties imposed by the combinatorial and discrete nature of this object. Results suggest the feasibility of automating neural identification, and demonstrate variational inference in permutations is a sensible alternative to MCMC.

The approaches available for unlocking a neural circuit – deciphering its algorithm’s means and ends – are restricted by the biological characteristics of both the circuit in question and the organism in which it is studied. The cerebellum has long appealed to circuits neuroscientists in this regard because of its simple yet evocative structure and physiology. Decades of efforts to validate theories inspired by its distinctive characteristics have yielded intriguing but highly equivocal results. In particular, the general spirit of David Marr and James Albus’s models of cerebellar involvement in associative learning, now almost 50 years old, continues to shape much research, and yet the resulting data indicates that the Marr-Albus theories cannot, in their original incarnations, be the whole story. In efforts to resolve these mysteries of the cerebellum, researchers have pushed the advantages of its simple circuit even further by studying it in model organisms with complimentary methodological advantages. Much early work for example was conducted in monkeys and humans taking advantage of the mechanically simple and precise oculomotor behaviors at which these foveates excel. Then, as genetic tools entered the scene and became increasingly powerful, neuroscientists began porting what had been learned into mouse, a model system in which these tools can be deployed with great sophistication. This was effective in part because cerebellum is highly conserved across vertebrates so complimentary insights can be made across different model systems. Today genetic prowess has been further augmented by rapid advances in optical methods for visualizing and manipulating genetically targeted components. The promise of these new capabilities provides grounds for exploring additional model organisms with characteristics particularly suited to harnessing the power of modern methodology. In the following chapters I explore the promise and challenges of adding a new organism to the current pantheon of most commonly studied cerebellar model organisms. In chapter 1, I introduce the cerebellar circuit and a sampling of the historically equivocal outcomes met by efforts to test Marr-Albus theories in the context of a classical cerebellar learning paradigm: vestibulo-ocular reflex adaptation. In chapter 2, I detail my efforts to establish a method for population calcium imaging in cerebellar granule cells (GCs) of the weakly electric mormyrid fish, gnathonemus petersii. The unusual anatomical placement of GCs in this organism, directly on the surface of the brain, is ideal for optical methods, which require the ability to illuminate structures of interest. Furthermore, in the mormyrid, GCs play analogous role in two circuits -- the cerebellum and a purely sensory structure, the electrosensory lobe, which has a cerebellum-like structure. This latter circuit is unusually well-characterized and appears to employ a Marr-Albus style associative learning algorithm. This could provide a helpful context for interpreting the purpose of GC processing, shared by this circuit and the cerebellum proper. However, taking advantage of these qualities will require overcoming methodological hurdles presented by imaging in this as-yet not genetically tractable organism. While I was able to load and image evoked transients in these cells, and twice observed spontaneous transient, I did not find a loading method that allowed routine observation of spontaneous levels of activity. In chapter 3, I introduce the larval zebrafish, danio rerio, an organism in which optical and genetic methods are already quite established. The zebrafish is genetically tractable and orders of magnitudes smaller than other vertebrate model systems, making it extremely accessible to optical monitoring and manipulation of neural activity. However, in contrast to the mormyrid, very little is known about the physiology of the cerebellar circuit components in this organism or the behaviors to which they contribute. In chapter 4 I detail my efforts to contribute to this modest foundational knowledge by characterizing the electrophysiological activity of Purkinje cells of larval zebrafish during the optomotor response (OMR)—a behavior with similarities to cerebellar-dependent visual stabilization behaviors that have been studied extensively in mammals. I observe a diversity of structured motor and visual activity that suggests that Purkinje cells could contribute to adjusting swim speed during the OMR and other behaviors. In chapter 5, I outline some of the upfront work that remains before cerebellar researchers are likely to fully harness the power of optical and genetic methods in the zebrafish as well as the types of experiments that may become possible if we do.