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Section DYNAMICS IN LIFE SCIENCES, NEUROSCIENCE APPLICATIONS WORKSHOP
Towards Bifurcation Theory for Rhythmogenesis in Neural Networks
A. Shilnikov1 *, D. Alacam1, J. Collens1, A. Kelley1 and J. Schwabedal2
1 Neuroscience Institute,Department of Mathematics & Statistics, Georgia State University, Atlanta USA;
2 Max Planck Institute for the Physics of Complex Systems, Dresden, Germany. * Presenting e-mail: ashilnikov@gsu.edu
Rhythmic motor behaviors 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 microdrcuit of cells whose interactions can autonomously generate an array of polyrhythmic patterns of activity that determine motor behaviors in animals and humans. Modeling studies 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 remains unclear how the CPGs achieve the level of robustness and stability observed in nature. Whereas a dedicated CPG generates a single pattern robustly, a multifunctional CPG can flexibly produce distinct rhythms, such as temporally distinct swimming and versus crawling locomotion, and alternation of direction of blood circulation in leeches. Switching between various attractors of a CPG network causes switching between locomotion behaviors. Each attractor is associated with a definite rhythm running on a specific time scale with well-defined and robust phase lags among the constituting neurons. The emergence of synchronous rhythms in neural networks is closely related to temporal characteristics of coupled neurons due to intrinsic properties and types of synaptic coupling, which can be inhibitory, excitatory and electrical, fast and slow.
We are interested in exploring repetitive dynamics generated by constituent building blocks, or "motifs" that make up more complex CPG circuits, and the dynamic principles underlying more general multi-stable rhythmic patterns. We have considered the range of basic motifs comprising three and four biophysical cells and their synapses, chemical inhibitory, excitatory and electrical, and how those relate, and can be understood and generalized onto from the known principles of minimal motifs.
We have developed a novel dynamical and bifurcation framework combining analytical approaches and computational tools to in-detail study oscillatory networks constituted by endogenously bursting, tonic spiking neurons and network bursters. The approaches let us reduce the problem of the stability and existence of bursting and other oscillatory rhythms generated by networks to bifurcation analysis of fixed points and invariant circles of Poincare return maps measuring the phase lags between the burst initiations in the neuro. The structure of the phase space of the map reflects all significant characteristics of the state space of the given network. Equipped with the powerful apparatus of such return maps we are able to predict and identify the set of robust bursting outcomes of CPGs, differentiated by phase-locked or periodically varying lags that correspond to stable fixed point and/or invariant circle attractors of the map. Comprehensive simulations of the transient phasic relationships in the network are based on the delayed release of cells from a suppressed, hyperpolarized state, and allow for thorough exploration of network dynamics with spiking and bursting cell.
Transient and Periodic Spatiotemporal Structures in a Reaction-Diffusion-Mechanics System
V.A. Kostin1'2, G.V. Osipov1'2 *
1 Lobachevsky State University of Nizhny Novgorod, 23 Gagarin ave, 603950 Nizhny Novgorod, Russia;
2 Institute of Applied Physics, Russian Academy of Sciences, Ylyanova str., 46, Nizhny Novgorod 603950, Russia. * Presenting e-mail: osipov@vmk.unn.ru
The reaction-diffusion-mechanics models are the models used to describe self-consistent electromechanical activity in a cardiac muscle. Such models couples two mechanisms of signal spreading in the tissue: the slow (reaction-diffusion) spreading of electrical excitation and the fast (almost instantaneous) spreading of mechanical deformations. This coupling may significantly modify the electrical excitation spreading and corresponding contractile activity with emergence of new spatiotemporal structures and patterns, which modification is not yet completely understood even in the one-dimensional case of a single muscle fiber. We propose clear convenient model which allows one to study the electromechanical activity of such a fiber in relation to the mechanical parameters of fiber fixation (such as stiffness of tissue fixation and the applied mechanical load, which can be easily controlled in experiments). Using this model, we determine and analyze the physical
OM&P
48 Opera Med Physiol 2016 Vol. 2 (S1)
