Analysis Of Rhythm Generation In The Caenorhabditis Elegans Motor Circuit

Understanding the neuronal control of movement has been a central goal of neuroscience for decades. In many organisms, chains of neural oscillators underlie the generation of coordinated rhythmic movements. However, the sheer complexity of spinal locomotor circuits has made understanding the mechanisms underlying rhythmic locomotion in vertebrates challenging. The roundworm C. elegans generates rhythmic undulatory movements that resemble those of swimming vertebrates, but using only a few hundred neurons. The relative simplicity of this organism has allowed a complete synaptic map of the nervous system to be developed. Moreover, C. elegans has a three-day life cycle and is amenable to a powerful battery of genetic techniques that allow the molecular basis of circuit functions to be probed much more rapidly than is possible in more complex organisms. Because of these advantages, C. elegans offers the possibility of understanding the network, cellular, and molecular principles of rhythmic locomotion in deeper detail than has been possible in any other model organism. However, it is currently unclear where in the C. elegans motor circuit rhythms are generated, and whether there exists more than one rhythm generator. I used optogenetic and lesioning experiments to probe the nature of rhythm generation in the locomotor circuit. I found that rhythmic activity in different parts of the body can be decoupled by both methods, implying that multiple sections of forward locomotor circuitry are capable of independently generating rhythms. By perturbing different components of the motor circuit, I localized at least two rhythmic sources to a network of cholinergic motor neurons that are distributed along the body. Moreover, I used rhythmic optogenetic manipulations to show that imposed rhythmic signals in any portion of the motor circuit can entrain oscillatory activity in the rest of the body, suggesting bidirectional coupling within the motor circuit. This organization, in which distributed oscillating circuits exist along the body but are closely linked by bidirectional coupling, is found in wide range of vertebrate and invertebrate animals. My results show that the functional architecture of the C. elegans motor circuit is highly analogous to that of much more complex organisms

Parameter identification in networks of dynamical systems

Mathematical models of real systems allow to simulate their behavior in conditions that are not easily or affordably reproducible in real life. Defining accurate models, however, is far from trivial and there is no one-size-fits-all solution. This thesis focuses on parameter identification in models of networks of dynamical systems, considering three case studies that fall under this umbrella: two of them are related to neural networks and one to power grids. The first case study is concerned with central pattern generators, i.e. small neural networks involved in animal locomotion. In this case, a design strategy for optimal tuning of biologically-plausible model parameters is developed, resulting in network models able to reproduce key characteristics of animal locomotion. The second case study is in the context of brain networks. In this case, a method to derive the weights of the connections between brain areas is proposed, utilizing both imaging data and nonlinear dynamics principles. The third and last case study deals with a method for the estimation of the inertia constant, a key parameter in determining the frequency stability in power grids. In this case, the method is customized to different challenging scenarios involving renewable energy sources, resulting in accurate estimations of this parameter

<p>Pattern Formation in Coupled Networks with Inhibition and Gap Junctions</p>

In this dissertation we analyze networks of coupled phase oscillators. We consider systems where long range chemical coupling and short range electrical coupling have opposite effects on the synchronization process. We look at the existence and stability of three patterns of activity: synchrony, clustered state and asynchrony. In Chapter 1, we develop a minimal phase model using experimental results for the olfactory system of Limax. We study the synchronous solution as the strength of synaptic coupling increases. We explain the emergence of traveling waves in the system without a frequency gradient. We construct the normal form for the pitchfork bifurcation and compare our analytical results with numerical simulations. In Chapter 2, we study a mean-field coupled network of phase oscillators for which a stable two-cluster solution exists. The addition of nearest neighbor gap junction coupling destroys the stability of the cluster solution. When the gap junction coupling is strong there is a series of traveling wave solutions depending on the size of the network. We see bistability in the system between clustered state, periodic solutions and traveling waves. The bistability properties also change with the network size. We analyze the system numerically and analytically. In Chapter 3, we turn our attention to a very popular model about network synchronization. We represent the Kuramoto model in its original form and calculate the main results using a different technique. We also look at a modified version and study how this effects synchronization. We consider a collection of oscillators organized in m groups. The addition of gap junctions creates a wave like behavior

Encoding of Coordinating Information in a Network of Coupled Oscillators

Animal locomotion is driven by cyclic movements of the body or body appendages. These movements are under the control of neural networks that are driven by central pattern generators (CPG). In order to produce meaningful behavior, CPGs need to be coordinated. The crayfish swimmeret system is a model to investigate the coordination of distributed CPGs. Swimmerets are four pairs of limbs on the animal’s abdomen, which move in cycles of alternating power-strokes and return-strokes. The swimmeret pairs are coordinated in a metachronal wave from posterior to anterior with a phase lag of approximately 25% between segments. Each swimmeret is controlled by its own neural microcircuit, located in the body segment’s hemiganglion. Three neurons per hemiganglion are necessary and sufficient for the 25% phase lag. ASCE DSC encode information about their home ganglion’s activity state and send it to their anterior or posterior target ganglia, respectively. ComInt 1, which is electrically coupled to the CPG, receives the coordinating information. The isolated abdominal ganglia chain reliably produces fictive swimming. Motor burst strength is encoded by the number of spikes per ASCE and DSC burst. If motor burst strength varies spontaneously, the coordinating neurons track these changes linearly. The neurons are hypothesized to adapt their spiking range to the occurring motor burst strengths. One aim of this study was to investigate the putative adaptive encoding of the coordinating neurons in electrophysiological experiments. This revealed that the system’s excitation level influenced both the whole system and the individual coordinating neurons. These mechanisms allowed the coordinating neurons to adapt to the range of burst strengths at any given excitation level by encoding relative burst strengths. The second aim was to identify the transmitters of the coordinating neurons at the synapse to ComInt 1. Immunohistochemical experiments demonstrated that coordinating neurons were not co-localized with serotonin-immunoreactive positive neurons. MALDI-TOF mass spectrometry suggested acetylcholine as presumable transmitter

Entrainment and chaos in a pulse-driven Hodgkin-Huxley oscillator

The Hodgkin-Huxley model describes action potential generation in certain types of neurons and is a standard model for conductance-based, excitable cells. Following the early work of Winfree and Best, this paper explores the response of a spontaneously spiking Hodgkin-Huxley neuron model to a periodic pulsatile drive. The response as a function of drive period and amplitude is systematically characterized. A wide range of qualitatively distinct responses are found, including entrainment to the input pulse train and persistent chaos. These observations are consistent with a theory of kicked oscillators developed by Qiudong Wang and Lai-Sang Young. In addition to general features predicted by Wang-Young theory, it is found that most combinations of drive period and amplitude lead to entrainment instead of chaos. This preference for entrainment over chaos is explained by the structure of the Hodgkin-Huxley phase resetting curve.Comment: Minor revisions; modified Fig. 3; added reference

Higher order approximation of isochrons

Phase reduction is a commonly used techinque for analyzing stable oscillators, particularly in studies concerning synchronization and phase lock of a network of oscillators. In a widely used numerical approach for obtaining phase reduction of a single oscillator, one needs to obtain the gradient of the phase function, which essentially provides a linear approximation of isochrons. In this paper, we extend the method for obtaining partial derivatives of the phase function to arbitrary order, providing higher order approximations of isochrons. In particular, our method in order 2 can be applied to the study of dynamics of a stable oscillator subjected to stochastic perturbations, a topic that will be discussed in a future paper. We use the Stuart-Landau oscillator to illustrate the method in order 2

Computational Study of the Mechanisms Underlying Oscillation in Neuronal Locomotor Circuits

In this thesis we model two very different movement-related neuronal circuits, both of which produce oscillatory patterns of activity. In one case we study oscillatory activity in the basal ganglia under both normal and Parkinsonian conditions. First, we used a detailed Hodgkin-Huxley type spiking model to investigate the activity patterns that arise when oscillatory cortical input is transmitted to the globus pallidus via the subthalamic nucleus. Our model reproduced a result from rodent studies which shows that two anti-phase oscillatory groups of pallidal neurons appear under Parkinsonian conditions. Secondly, we used a population model of the basal ganglia to study whether oscillations could be locally generated. The basal ganglia are thought to be organised into multiple parallel channels. In our model, isolated channels could not generate oscillations, but if the lateral inhibition between channels is sufficiently strong then the network can act as a rhythm-generating ``pacemaker'' circuit. This was particularly true when we used a set of connection strength parameters that represent the basal ganglia under Parkinsonian conditions. Since many things are not known about the anatomy and electrophysiology of the basal ganglia, we also studied oscillatory activity in another, much simpler, movement-related neuronal system: the spinal cord of the Xenopus tadpole. We built a computational model of the spinal cord containing approximately 1,500 biologically realistic Hodgkin-Huxley neurons, with synaptic connectivity derived from a computational model of axon growth. The model produced physiological swimming behaviour and was used to investigate which aspects of axon growth and neuron dynamics are behaviourally important. We found that the oscillatory attractor associated with swimming was remarkably stable, which suggests that, surprisingly, many features of axonal growth and synapse formation are not necessary for swimming to emerge. We also studied how the same spinal cord network can generate a different oscillatory pattern in which neurons on both sides of the body fire synchronously. Our results here suggest that under normal conditions the synchronous state is unstable or weakly stable, but that even small increases in spike transmission delays act to stabilise it. Finally, we found that although the basal ganglia and the tadpole spinal cord are very different systems, the underlying mechanism by which they can produce oscillations may be remarkably similar. Insights from the tadpole model allow us to predict how the basal ganglia model may be capable of producing multiple patterns of oscillatory activity