My thesis is a collection of theoretical and practical techniques for mapping functional or effective connectivity in cellular neuronal networks, at the cell scale. This is a challenging scale to work with, primarily because of the difficulty in labeling and measuring the activities of networks of cells. It is also important as it underlies behavior, function, and complex diseases. I present methods to measure and quantify the dynamic activities of cells using the optical flow technique, which can identify activity and directions of information processing using calcium fluorescence measurements. I present a unified framework for simulation and estimation of neuronal activity, tailored towards interpretation of experimental data, and implemented in a fully parallel fashion on graphics processor unit (GPU) cards. This framework permits experimenters to estimate hidden quantities in collected data, using any neuronal or astrocyte model. I introduce a technique for mapping functional connectivity in neuronal networks, using experimental data and an arbitrary state space model. The technique makes some simplifications that reduces the dimensionality of the estimation problem, and shows excellent performance for networks of up to 30 possible independent incoming connections. While the framework and mapping algorithms use a state space, parametric representation of individual cell dynamics, I've also developed a time-embedded, nonparametric technique for estimating input-output relationships, and applied it to estimating current from voltage measurements and spikes from fluorescent calcium. Without any knowledge of the underlying neuronal dynamics, this technique can reconstruct a current signal from measured voltage in mouse pyramidal neurons with an R-value of 0.9. Finally, I present my findings and theoretical perspectives acquired while developing the framework and methods. Optimization as a means of estimating functional weights is especially challenging due to the topology of the parameter space, with small perturbations in weights resulting in drastically different simulated dynamics. High-dimensional spaces are prone to the curse of dimensionality, and network states represented in such spaces are not likely to be stable or typical. Finally, the effects of the concentration of measure, as I believe I've observed when mapping large networks, makes it unlikely that real-world networks have more than about 7 independent functional inputs at any given time

An optical flow gradient algorithm was applied to spontaneously forming networks of neurons and glia in culture imaged by fluorescence optical microscopy in order to map functional calcium signaling with single pixel resolution. Optical flow estimates the direction and speed of motion of objects in an image between subsequent frames in a recorded digital sequence of images (i.e., a movie). Computed vector field outputs by the algorithm were able to track the spatiotemporal dynamics of calcium signaling patterns. We begin by briefly reviewing the mathematics of the optical flow algorithm, and then describe how to solve for the displacement vectors and how to measure their reliability. We then compare computed flow vectors with manually estimated vectors for the progression of a calcium signal recorded from representative astrocyte cultures. Finally, we applied the algorithm to preparations of primary astrocytes and hippocampal neurons and to the rMC-1 Muller glial cell line in order to illustrate the capability of the algorithm for capturing different types of spatiotemporal calcium activity. We discuss the imaging requirements, parameter selection and threshold selection for reliable measurements, and offer perspectives on uses of the vector data.

Retinal Müller glial cells, in addition to providing homeostatic support to retinal neurons, have been shown to engage in modulation of neuronal activity and regulate vasomotor responses in the retina, among other functions. Calcium-mediated signaling in Müller cells has been implicated to play a significant role in the intracellular and intercellular interactions necessary to carry out these functions. Although the basic molecular mechanisms of calcium signaling in Müller cells have been described, the dynamics of calcium responses in Müller cells have not been fully explored. Here, we provide a quantitative characterization of calcium signaling in an in vitro model of Müller cell signaling using the rMC-1 cell line, a well-established line developed from rat Müller cells. rMC-1 cells displayed robust intracellular calcium transients and the capacity to support calcium transient-mediated intercellular calcium waves with signaling dynamics similar to that reported for Müller cells in in situ retinal preparations. Furthermore, pharmacological perturbations of intracellular calcium transients with thapsigargin and intercellular calcium waves with purinergic receptor antagonists and gap junction blockers (PPADS and FFA, respectively) suggest that the molecular mechanisms that underlie calcium signaling in rMC-1 cells has been conserved with those of Müller cells. This model provides a robust in vitro system for investigating specific mechanistic hypotheses of intra- and intercellular calcium signaling in Müller cells.