We develop a biophysical model of neurosensory integration in the model organism Caenorhabditis elegans. Building on experimental findings on the neuron conductances and their resolved connectome, we posit the first full dynamic model of the neural voltage excitations that allows for a characterization of network structures which link input stimuli to neural proxies of behavioral responses. Full connectome simulations of neural responses to prescribed inputs show that robust, low-dimensional bifurcation structures drive neural voltage activity modes. Comparison of these modes with experimental studies allows us to link these network structures to behavioral responses. Thus the underlying bifurcation structures discovered, i.e., induced Hopf bifurcations, are critical in explaining behavioral responses such as swimming and crawling.

Resting state networks (RSNs) extracted from functional magnetic resonance imaging (fMRI) scans are believed to reflect the intrinsic organization and network structure of brain regions. Most traditional methods for computing RSNs typically assume these functional networks are static throughout the duration of a scan lasting 5–15 minutes. However, they are known to vary on timescales ranging from seconds to years; in addition, the dynamic properties of RSNs are affected in a wide variety of neurological disorders. Recently, there has been a proliferation of methods for characterizing RSN dynamics, yet it remains a challenge to extract reproducible time-resolved networks. In this paper, we develop a novel method based on dynamic mode decomposition (DMD) to extract networks from short windows of noisy, high-dimensional fMRI data, allowing RSNs from single scans to be resolved robustly at a temporal resolution of seconds. After validating the method on a synthetic dataset, we analyze data from 120 individuals from the Human Connectome Project and show that unsupervised clustering of DMD modes discovers RSNs at both the group (gDMD) and the single subject (sDMD) levels. The gDMD modes closely resemble canonical RSNs. Compared to established methods, sDMD modes capture individualized RSN structure that both better resembles the population RSN and better captures subject-level variation. We further leverage this time-resolved sDMD analysis to infer occupancy and transitions among RSNs with high reproducibility. This automated DMD- based method is a powerful tool to characterize spatial and temporal structures of RSNs in individual subjects.

Using a computational model of the Caenorhabditis elegans connectome dynamics, we show that proprioceptive feedback is necessary for sustained dynamic responses to external input. This is consistent with the lack of biophysical evidence for a central pattern generator, and recent experimental evidence that proprioception drives locomotion. The low-dimensional functional response of the Caenorhabditis elegans network of neurons to proprioception-like feedback is optimized by input of specific spatial wavelengths which correspond to the spatial scale of real body shape dynamics. Furthermore, we find that the motor subcircuit of the network is responsible for regulating this response, in agreement with experimental expectations. To explore how the connectomic dynamics produces the observed two-mode, oscillatory limit cycle behavior from a static fixed point, we probe the fixed point’s low-dimensional structure using Dynamic Mode Decomposition. This reveals that the nonlinear network dynamics encode six clusters of dynamic modes, with timescales spanning three orders of magnitude. Two of these six dynamic mode clusters correspond to previously-discovered behavioral modes related to locomotion. These dynamic modes and their timescales are encoded by the network’s degree distribution and specific connectivity. This suggests that behavioral dynamics are partially encoded within the connectome itself, the connectivity of which facilitates proprioceptive control. The nematode C. elegans lives a complex and rich life despite having only 302 neurons. The full connectivity between these neurons (its “connectome”) has been measured, making it an ideal model system for understanding how neural processing generates behavior. However, unlike most animals, it doesn’t appear to have neural circuits dedicated to generating rhythmic motion. Even the simple rhythmic behavior of forward locomotion is believed to be ultimately driven by stretch-receptive proprioception as the worm changes body shape. Computational models of connectome dynamics can capture behavioral responses, making them valuable for studying neural dynamics. We consider a model of the full-connectome dynamics which, without external input, is in a static neural state. We show that proprioception-like input can bring the system into a rhythmic state similar to locomotion. We then show that arbitrary external inputs across all neurons can only excite some combination of six neural patterns, and that the shape and dynamics of these patterns are determined by the connectome. Furthermore, two of these patterns correspond to forward locomotion, such that the connectome facilitates locomotion-like responses. We thus find that the connectome itself is not random, but is ideally tuned for generating behavior.

The neural dynamics of the nematode Caenorhabditis elegans are experimentally low-dimensional and may be understood as long-timescale transitions between multiple low-dimensional attractors. Previous modeling work has found that dynamic models of the worm's full neuronal network are capable of generating reasonable dynamic responses to certain inputs, even when all neurons are treated as identical save for their connectivity. This study investigates such a model of C. elegans neuronal dynamics, finding that a wide variety of multistable responses are generated in response to varied inputs. Specifically, we generate bifurcation diagrams for all possible single-neuron inputs, showing the existence of fixed points and limit cycles for different input regimes. The nature of the dynamical response is seen to vary according to the type of neuron receiving input; for example, input into sensory neurons is more likely to drive a bifurcation in the system than input into motor neurons. As a specific example we consider compound input into the neuron pairs PLM and ASK, discovering bistability of a limit cycle and a fixed point. The transient timescales in approaching each of these states are much longer than any intrinsic timescales of the system. This suggests consistency of our model with the characterization of dynamics in neural systems as long-timescale transitions between discrete, low-dimensional attractors corresponding to behavioral states.

Inferring and comparing complex, multivariable probability density functions is fundamental to problems in several fields, including probabilistic learning, network theory, and data analysis. Classification and prediction are the two faces of this class of problem. This study takes an approach that simplifies many aspects of these problems by presenting a structured, series expansion of the Kullback-Leibler divergence—a function central to information theory—and devise a distance metric based on this divergence. Using the Möbius inversion duality between multivariable entropies and multivariable interaction information, we express the divergence as an additive series in the number of interacting variables, which provides a restricted and simplified set of distributions to use as approximation and with which to model data. Truncations of this series yield approximations based on the number of interacting variables. The first few terms of the expansion-truncation are illustrated and shown to lead naturally to familiar approximations, including the well-known Kirkwood superposition approximation. Truncation can also induce a simple relation between the multi-information and the interaction information. A measure of distance between distributions, based on Kullback-Leibler divergence, is then described and shown to be a true metric if properly restricted. The expansion is shown to generate a hierarchy of metrics and connects this work to information geometry formalisms. An example of the application of these metrics to a graph comparison problem is given that shows that the formalism can be applied to a wide range of network problems and provides a general approach for systematic approximations in numbers of interactions or connections, as well as a related quantitative metric.

Using a model for the dynamics of the full somatic nervous system of the nematode C. elegans, we address how biological network architectures and their functionality are degraded in the presence of focal axonal swellings (FAS) arising from neurodegenerative disease and/or traumatic brain injury. Using biophysically measured FAS distributions and swelling sizes, we are able to simulate the effects of injuries on the neural dynamics of C. elegans, showing how damaging the network degrades its low-dimensional dynamical responses. We visualize these injured neural dynamics by mapping them onto the worm’s low-dimensional postures, i.e. eigenworm modes. We show that a diversity of functional deficits arise from the same level of injury on a connectomic network. Functional deficits are quantified using a statistical shape analysis, a procrustes analysis, for deformations of the limit cycles that characterize key behaviors such as forward crawling. This procrustes metric carries information on the functional outcome of injuries in the model. Furthermore, we apply classification trees to relate injury structure to the behavioral outcome. This makes testable predictions for the structure of an injury given a defined functional deficit. More critically, this study demonstrates the potential role of computational simulation studies in understanding how neuronal networks process biological signals, and how this processing is impacted by network injury. Neurodegenerative diseases such as Alzheimer’s disease, Creutzfeldt-Jakob’s disease, HIV dementia, Multiple Sclerosis and Parkinson’s disease are leading causes of cognitive impairment and death worldwide. Similarly, traumatic brain injury, the signature injury of the Iraq and Afghanistan wars, affects an estimated 57 million people. All of these conditions are characterized by the presence of focal axonal swellings (FAS) throughout the brain. On a network level, however, the effects of FAS remain unexplored. With the emergence of models which simulate an organism’s full neuronal network, we are poised to address how neuronal network performance is degraded by FAS-related damage. Using a model for the full-brain dynamics of the nematode Caenorhabditis elegans, we are able to explore the loss of network functionality as a function of increased neuronal swelling. The relatively small neuronal network generates a limited and tractable set of functional behaviors, and we develop metrics which characterize how these behaviors are impaired by network injuries. These metrics quantify the severity of TBI and/or neurodegenerative disease, and could potentially be used to construct diagnostic tools capable of identifying various cognitive deficits. Additionally, we apply classification trees to our results to make predictions about the structure of an injury from specific cognitive deficits.