The project will develop the dynamical principles of multistability of bursting patterns of polyrhythmic activity and its control for multifunctional Central Pattern Generators. Multistability enhances the flexibility of nervous systems and has far reaching implications for motor control, dynamic memory, information processing, and decision making. The Investigator and his students will identify and study generic nonlocal bifurcations of bursting rhythms in realistic models of single and networked interneurons, as well as create a dynamical systems classification for the bursting genesis in CPGs. The research team will create a suite of new methods and computational tools based on the theory of dynamical systems and global bifurcations to examine complex transformations of bursting patterns in high-order Hodgkin-Huxley type models and networks. The Investigator and his students will enhance the existing mathematical technique by creating transparent computational tools for the detection and prediction of transformations of complex oscillatory solutions in neuronal models with multiple time scales. This includes the novel approaches of reducing neuronal dynamics to a complete, equation-free family of onto Poincaré mappings for membrane potentials, and the phase-difference mappings for bursting CPG circuits. The reduction will yield a clear understanding of the dynamics of a high-order, multiple-time scale neuron model, as well as provide with a control of the multistability by revealing the hidden centers that govern globally the dynamics of a mutlifunctional CPG network. Having the extensive knowledge of dynamical properties of networked busting interneurons will allow the team to derive precise phase models to replicate the dynamics of their high-dimensional models. These reduced models will be used to examine larger and more complex realistic models of the specific excitatory-inhibitory CPG circuits.
The ability of distinct anatomical circuits, like Central Pattern Generators, to generate multiple patterns of neural activity to control several locomotion types, like cardiac beating, waking, swimming etc, is widespread among vertebrate and invertebrate species. Understanding generic mechanisms of the evolution of neuronal connectivity and transitions between different patterns of neural activity and modeling these processes are the fundamental challenges for applied mathematics and computational neuroscience. This project is a genuinely cross-disciplinary research, bridging state-of the art mathematics, more specifically the theory of applied dynamical systems and nonlocal bifurcations, with life sciences. It shall extend and generalize our understanding of dynamical principles of neural systems; specifically mechanisms regulating polyrhythms of multifunctional Central Pattern Generators. Multistability enhances the flexibility of nervous systems and has far reaching implications for motor control, dynamic memory, information processing, and decision making of humans and animals. The Investigator and his students will identify and study generic bifurcations of bursting rhythms in realistic models of single and networked interneurons, as well as create a dynamical systems classification for the bursting genesis in multifunctional neural circuits.
Rhythmic motor behaviors, such such as heartbeat, respiration, chewing, and locomotion on land and in water are produced by networks of cells called Central Pattern Generators (CPGs) . A CPG is a neural microcircuit of such cells whose synergetic interactions can autonomously generate an array of multicomponent/polyrhythmic bursting patterns of activity that determine vital motor behaviors in animals. Modeling studies, phenomenologically mathematical and exhaustively computational, have proven to be useful to gain insights into operational principles of CPGs. Although various models, reduced and feasible, of specific CPGs have been developed, it still remains unclear how the CPGs achieve the level of robustness and stability observed in nature. Many abnormal neurological phenomena are perturbations of normal functions of the underlying mechanisms governing the animal behaviors, specifically movements. Repetitive behaviors are often associated hypothetically with the phenomenon of rhythmogenesis in small networks that are able autonomously to generate or continue, after induction a variety of activity patterns without further external input, abrupt or not. The goal of this modeling study was to identify decisive components of a biologically plausible models of 3-cell central pattern generators Due to the recurrent nature of such bursting patterns of self-sustained activity, we employ Poincare return maps defined on phases and phase-lags between burst initiations in the interneurons to study quantitative and qualitative properties of CPG rhythms and corresponding attractors. The proposed approach was specifically tailored for various studies of neural networks in neuroscience, computational, and experimental. Development of such tools and our understanding of such CPGs can be applied to gain insight into governing principles of neurological phenomena in higher order animals and can aid in treating anomalies associated with neurological disorders associated with CPG arrhythmia. There is a growing consensus in the community of neurophysiologists and computational researchers that some basic structural and functional elements are likely shared by CPGs of both invertebrate and vertebrate animals. Before we can study the mechanisms of disorders at the level of individual neurons and CPG circuits in mammals, we therefore first seek to develop better tools and techniques in the context of much simpler animals. However, the ultimate aim of developing our tools and approach to understanding CPGs in lower animals is to make them applicable to studying the governing principles of neurological phenomena in higher animals, and so could potentially assist in treating neurological disorders associated with CPG arrhythmia. This project was intended as a a pilot yet comprehensive study that demonstrated the effectiveness of our analytical approach that connects exploratory mathematical models to experimental data in the context of known behavioral patterns A strong working assumption in this study was that a CPG is composed of (nearly) identical elements - interneurons or bursting HCOs, which are interconnected through chemical synapses and gap junctions of equal conductivity. Due to alterations in the reciprocal wiring, such a homogeneous CPG, can be adapted to become dedicated to a single rhythm or multifunctional. We might presume that, through iterative processes of learning and evolution, real CPG might develop a heterogeneous structure as specific connections become stronger or weaker, so that it can become better adapted to performing specific functions in specific animals. Certainly, we are all aware of examples where, through learning and exercise, mammalian motor systems become "multifunctional" and are able to quickly transition between several dynamic functions on demand: for instance, the diverse swimming styles that have been cultivated by humans, including the in-phase breaststroke and butterfly, and the anti-phase crawl and backstroke. For now, we showed that there is a multifunctional, and presumably heterogeneous, CPG network underlying these specific swimming rhythms that determines the phase relationships between rhythmic muscle control signals. In general, our insights allow us to predict both quantitative and qualitative transformations of the observed patterns whenever the network configurations are altered. The nature of these transformations provides a set of novel hypotheses for biophysical mechanisms about the control and modulation of rhythmic activity. A powerful aspect to our analytical technique is that it does not require knowledge of the equations that model the system. Thus, we believe that developed a universal approach to studying both detailed and phenomenological models of oscillatory networks is also applicable to a variety of rhythmic biological phenomena beyond motor control. This project provided interdisciplinary training and educational opportunities for two graduate students and summerresearch experience for five undergraduate students. PI’s results was included in recent textbooks on mathematical neuroscience. His students received the best poster award at the SIAM Applied Dynamical Systems meeting in 2013 and Dynamics Days in 2014. Multimedia supplements for this project are available on YouTune, see http://youtu.be/QNwmp9r279M and http://youtu.be/gqP1qn4JJB0 and http://youtu.be/-myVW67YVMk for open public access and in-class demonstrations.
