Modeling Rhythm Generation in Swim Central Pattern Generator of Melibe Leonina

Central pattern generators (CPGs) are neural networks to produce a rich multiplicity of rhythmic activity types like walking, breathing and swim locomotion. Basis principles of the underlying mechanisms of rhythm generation in CPGs remain yet insufficiently understood. Interactive pairing experimental and modeling studies have proven to be vital to unlocking insights into operational and dynamical principles of CPGs and support the consensus that the most of essential structural and functional elements in vertebrate and invertebrate nervous systems are shared. We have developed a family of highly-detailed, biologically plausible CPG models using the extensive data intracellularly recorded from constituent interneurons of the swim CPG of the sea slug {\it Melibe leonina}. We also have deduced fundamental properties needed for the devised Hodgkin-Huxley type neuronal models with specific slow-fast dynamics to become qualitatively and quantitatively similar to biological CPG interneurons and their responses to parameter and external perturbations. We have studied the onset and robustness of rhythmogenesis of network bursting the CPG circuits comprised of tonic spiking interneurons coupled with mixed inhibitory/excitatory, slow chemical synapses. We have shown that the mathematical CPG model can be reduced functionally from an 8-cell circuit to a 4-cell one using the calibration of timing and weights of synaptic coupling between CPG core interneurons. We demonstrate that the developed mathematical network meets all the experimental fact-checks obtained for the biological Melibe swim CPG from a variety of state-of-the-art experimental studies including dynamic-clamp recordings, external pulses perturbations as well as from its forced behaviors under applications of neuro-blockers such as curare and TTX. Our model and developed mathematical approaches and computational methodology allow for laying down theoretical foundations necessary for devising new detailed and phenomenological models of neural circuits and for making testable predictions of dynamics of rhythmic neural networks from diverse species

Biological Neuron Voltage Recordings, Driving and Fitting Mathematical Neuronal Models

The manual process of comparing biological recordings from electrophysiological experiments to their mathematical models is time-consuming and subjective. To address this problem, we have created a blended system that allows for objective, high-throughput, and computationally inexpensive comparisons of biological and mathematical models by developing a quantitative measure of likeness (error function). Voltage recordings from biological neurons, mathematically simulated voltage times series, and their transformations are inputted into the error function. These transformations and measurements are the action potential (AP) frequency, voltage moving average, voltage envelopes, and the probability of post-synaptic channels being open. The previously recorded biological voltage times series are first, translated into mathematical data to input into mathematical neurons, creating what we call a blended system. Using the sea slug Melibe Leonina\u27s swimming central pattern generator (CPG) as our circuit to compare and the source of our biological recordings, we performed a grid search of the conductance of the inhibitory and excitatory synapse found that a weighted sum of simple functions is required for a comprehensive view of a neuron\u27s rhythmic behavior. The blended system was also shown to be able to act as rhythm directors like pacemakers and drivers of Dendronotus Iris swimming interneuron (Si) cells and was able to replicate the perturbations of biological recordings. After verification steps using different configurations, calculated mean and variance of rhythmic characteristics, as well as recordings created from data augmentation. The form of data augmentation introduced can be generalized to other biological recordings or any time series. With all these tools developed and expanding the parameter dimensions a hypothesis was posited that there is a contralateral electric synapse not previously included in the Melibe CPG model

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

Rhythmogenesis and Bifurcation Analysis of 3-Node Neural Network Kernels

Central pattern generators (CPGs) are small neural circuits of coupled cells stably producing a range of multiphasic coordinated rhythmic activities like locomotion, heartbeat, and respiration. Rhythm generation resulting from synergistic interaction of CPG circuitry and intrinsic cellular properties remains deficiently understood and characterized. Pairing of experimental and computational studies has proven key in unlocking practical insights into operational and dynamical principles of CPGs, underlining growing consensus that the same fundamental circuitry may be shared by invertebrates and vertebrates. We explore the robustness of synchronized oscillatory patterns in small local networks, revealing universal principles of rhythmogenesis and multi-functionality in systems capable of facilitating stability in rhythm formation. Understanding principles leading to functional neural network behavior benefits future study of abnormal neurological diseases that result from perturbations of mechanisms governing normal rhythmic states. Qualitative and quantitative stability analysis of a family of reciprocally coupled neural circuits, constituted of generalized Fitzhugh–Nagumo neurons, explores symmetric and asymmetric connectivity within three-cell motifs, often forming constituent kernels within larger networks. Intrinsic mechanisms of synaptic release, escape, and post-inhibitory rebound lead to differing polyrhythmicity, where a single parameter or perturbation may trigger rhythm switching in otherwise robust networks. Bifurcation analysis and phase reduction methods elucidate qualitative changes in rhythm stability, permitting rapid identification and exploration of pivotal parameters describing biologically plausible network connectivity. Additional rhythm outcomes are elucidated, including phase-varying lags and broader cyclical behaviors, helping to characterize system capability and robustness reproducing experimentally observed outcomes. This work further develops a suite of visualization approaches and computational tools, describing robustness of network rhythmogenesis and disclosing principles for neuroscience applicable to other systems beyond motor-control. A framework for modular organization is introduced, using inhibitory and electrical synapses to couple well-characterized 3-node motifs described in this research as building blocks within larger networks to describe underlying cooperative mechanisms

Locomotor Network Dynamics Governed By Feedback Control In Crayfish Posture And Walking

Sensorimotor circuits integrate biomechanical feedback with ongoing motor activity to produce behaviors that adapt to unpredictable environments. Reflexes are critical in modulating motor output by facilitating rapid responses. During posture, resistance reflexes generate negative feedback that opposes perturbations to stabilize a body. During walking, assistance reflexes produce positive feedback that facilitates fast transitions between swing and stance of each step cycle. Until recently, sensorimotor networks have been studied using biomechanical feedback based on external perturbations in the presence or absence of intrinsic motor activity. Experiments in which biomechanical feedback driven by intrinsic motor activity is studied in the absence of perturbation have been limited. Thus, it is unclear whether feedback plays a role in facilitating transitions between behavioral states or mediating different features of network activity independent of perturbation. These properties are important to understand because they can elucidate how a circuit coordinates with other neural networks or contributes to adaptable motor output. Computational simulations and mathematical models have been used extensively to characterize interactions of negative and positive feedback with nonlinear oscillators. For example, neuronal action potentials are generated by positive and negative feedback of ionic currents via a membrane potential. While simulations enable manipulation of system parameters that are inaccessible through biological experiments, mathematical models ascertain mechanisms that help to generate biological hypotheses and can be translated across different systems. Here, a three-tiered approach was employed to determine the role of sensory feedback in a crayfish locomotor circuit involved in posture and walking. In vitro experiments using a brain-machine interface illustrated that unperturbed motor output of the circuit was changed by closing the sensory feedback loop. Then, neuromechanical simulations of the in vitro experiments reproduced a similar range of network activity and showed that the balance of sensory feedback determined how the network behaved. Finally, a reduced mathematical model was designed to generate waveforms that emulated simulation results and demonstrated how sensory feedback can control the output of a sensorimotor circuit. Together, these results showed how the strengths of different approaches can complement each other to facilitate an understanding of the mechanisms that mediate sensorimotor integration

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described