Brain systems span a wide range of spatial and temporal scales, from individual neurons firing every few milliseconds to whole-brain connectivity changing over many years. Network models offer a mathematical framework for describing, predicting, and controlling a diverse array of brain-related phenomena. Regardless of the particular system of study, it is possible to characterize the range of observed variability in system behavior over time by calculating topological and statistical metrics from inferred networks. In this thesis, tools from network science, computational neuroscience, and statistics are applied to data sets spanning several systems and spatiotemporal scales to characterize patterns of variability in brain activity and behavior within and between individuals over time.

First, electrophysiological activity of dissociated rat hippocampal neurons in vitro is analyzed to reveal the emergence and persistence of stereotyped activity patterns over time in the absence of external stimuli. These discoveries motivate further use of in vitro hippocampal cultures as model systems for studying developmental phenomena related to learning and memory.

Next, fruit fly behavior is modeled using tools from computational ethology and statistics to define and compare the syntax of grooming, a common behavior in Drosophila. We identify duration dependence in syntax for the first time, suggesting that central pattern generators contribute to organizing behavior in a partially sensory-independent manner.

We also identify inter- and intra-species variation in grooming syntax, but do not observe a significant correspondence between genetic heterogeneity and grooming variability. We also examine optogenetically-stimulated flies and find variability in grooming responses despite careful control of sensory presentation. Taken together, these results demonstrate that genetic differences can produce variation in grooming behavior, but that sensory experiences and stochastic effects contribute more strongly than genetics to the natural range of expressed grooming phenotypes.

Finally, we characterize the stability and changes in human brain network connectivity over the course of a complete menstrual cycle in a ``dense sampling" experimental paradigm. We identify stable communities that persist for the majority of the menstrual cycle. However, we also discover a correspondence between sex hormones and transient large-scale brain network reorganization, implicating hormones in shaping network topology.

Despite the differences in systems and phenomena of interest in this thesis, the work contained within is unified by a concern for variability in brain activity and behavior. Across contexts, we demonstrate the utility of network models in describing changes in complex systems over time.

Over the last two decades an association between microbiome composition and some human diseases has been unambiguously established, and this discovery has provoked clinical interest into microbiome-based medical therapies. The advent of high-throughput sequencing has produced vast amounts of microbial abundance data, but mechanistic models that describe and predict this complex microbial ecosystem are not yet established.

In my doctoral research I employ novel theoretical approaches and tractable experimental systems to study simplified instances of the complex microbial dynamics of the microbiome. This research is motivated by a desire to inform the mechanism of action and development of microbiome-based bacteriotherapies. We use generalized Lotka-Volterra (gLV) models, a class of model that exhibits prototypical ecological behaviors, as a theoretical proxy for true microbial dynamics. We develop a numerical framework to predict how introduced foreign microbes (direct bacteriotherapy) and a modified microbiome environment (indirect bacteriotherapy) affect the composition of a microbiome. Additionally, we derive the dimensionality-reduction technique "steady-state reduction" (SSR), which compresses bistable dynamics in a high-dimensional gLV system into a reduced two-dimensional system. The insights gained from this reduced system inform the microbial dynamics of high-dimensional gLV systems, and therefore contribute to our knowledge of how bacteriotherapies function.

Additionally, in experimental work I construct simple models from first principles that describe how the measured physiological traits depend on microbiome composition in the fruit fly Drosophila melanogaster. The fruit fly microbiome naturally hosts only a few species of commensal bacteria. In our research we cultivate a core group of five commensal bacteria, and associate each possible combination of the five bacteria with a set of germ-free flies. Since the flies were identically reared, the flies associated with each of these 32 bacterial combinations carry a distinct microbiome-affiliated phenotype (e.g. lifespan or fecundity) that is a function of the complex interactions in the microbiome. We found that the complexity of these fly phenotypes was often reducible: we could approximate the phenotype of flies associated with more than one bacterial species by averaging the phenotypes of the flies with the corresponding single-species associations. Thus, we found that simple models can describe complex behaviors in the fly microbiome.

The human brain is a complex system in which interactions of billions of neurons give rise to behavior. fMRI allows researchers to measure the functional activity of the working brain, allowing both the localization of specific functions within the brain and the investigation of multivariate patterns of functional activation. These patterns have been found to correspond both to short-term brain states such as focused attention or daydreaming, and to characteristics such as age or disease. Functional patterns also show substantial variation across individuals. Understanding the correspondence of distributed functional activity to these various factors is an ongoing research area.

Network science is a valuable tool for representing complex brain function, providing a framework for quantifying multivariate activity as a network of interactions. Here, we build upon recent advances in dynamic network science, using time-evolving networks to investigate how the organization of brain dynamics is related to demographics and brain states.

We use \textit{hypergraphs} to analyze brain network dynamics during different cognitive tasks and the transitions between them. We identify the presence of \textit{hyperedges}, groups of functional interactions that fluctuate coherently in strength over time both within and across brain states. We develop metrics to quantify the variation of hyperedge structure between tasks and across individuals. We find that the spatial location of hyperedges is relatively consistent across individuals, serving as a signature of a cognitive task, while hyperedge size exhibits variation across individuals but remains consistent between tasks.

We also investigate the variation of brain dynamics across the human lifespan, using both hypergraphs and dynamic clusters, or \textit{communities}, of brain regions with similar activity. We find significant relationships between age and dynamic organization: younger subjects tend to have larger hyperedges, as well as less fragmented and more coherent communities, and their brain regions tend to switch between communities less often. Further, the dynamics of different cognitive brain systems respond differently to aging.

Finally, we propose and evaluate a method of targeted node removal during the data-driven detection of communities, using synthetic and fMRI-derived networks to show that the method can improve identification of multi-scale community structure, and help to resolve key features of community dynamics.