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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: [email protected]
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: [email protected]
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
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Section DYNAMICS IN LIFE SCIENCES, NEUROSCIENCE APPLICATIONS WORKSHOP
origin of the primary dynamical effects which can be caused by electromechanical coupling and mechanoelec-trical feedback in a cardiac tissue.
On the basis of the reaction-diffusion-mechanics model with the self-consistent electromechanical coupling, we have numerically analyzed the emergence of structures and wave propagation in the excitable contractile fiber for various contraction types (isotonic, isometric, and auxotonic) and electromechanical coupling strengths. We have identified two main regimes of excitation spreading along the fiber: (i) the common quasi-steady-state propagation and (ii) the simultaneous ignition of the major fiber part and have obtained the analytical estimate for the boundary between the regimes in the parameter space. The uncommon oscillatory regimes have been found for the FiteHugh-Nagumo-like system: (i) the propagation of the soliton-like waves with the boundary reflections and (ii) the clusterized self-oscillations. The single space-time localized stimulus has been shown to be able to induce long-lasting transient activity as a result of the after-excitation effect when the just excited fiber parts are reexcited due to the electromechanical global coupling. The results obtained demonstrate the wide variety of possible dynamical regimes in the electromechanical activity of the cardiac tissue and the significant role of the mechanical fixation properties (particularly, the contraction type), which role should be taken into consideration in similar studies. In experiments with isolated cardiac fibers and cells, these parameters can be relatively easily controlled, which opens a way to assess electrical and mechanical parameters of the fibers and cells through analysis of dynamical regimes as dependent on fixation stiffness and external force. In real heart, high blood pressure and hindered blood flow play similar role to the applied external force and increased fixation stiffness. Our results provide a hint of how such global (i.e., associated with the large areas of the heart tissue) parameters can affect the heart electrical and contraction activity.
Chaos & Biological Information Processing: Coarse-Graining, Rough Set Approximations and Quantum Cognition in Decision Making
V. Basios*
Interdisciplinary Centre for Nonlinear Phenomena and Complex systems, & Dept. de Physique des Systèmes Complexes et Mécanique Statistique, University of Brussels, Brussels, Belgium. * Presenting e-mail: [email protected]
Aims
The role of chaos in biological information processing has been established as an important breakthrough of nonlinear dynamics, after the early pioneering work of J.S. Nicolis [1] (and notably in neuroscience by the work of Walter J. Freeman and co-workers spanning more than three decades, see Chapter 13 by Walter J. Freeman in [1]). Yet the models describing apprehension, judgment and decision making in various populations of biological systems, be it a large collective of neural networks, a colony of ants, a hive of bees or other communities of "agents', do not readily accommodate such an insight. With this work we aim at bridging this gap by considering recent advances in apprehension and judgment (see Chapter 15, by T. Arrechi in [1]). We propose a scheme [2] that underlies the mechanism of classification in judgememt and decision making, under uncertainty and conflict, by utilizing coarse-graining techniques from chaotic dynamics [5] based in "rough-sef theory with self-referential, non-linear, feed-back loops.
Methods
Our methods derive from an interdisciplinary framework combining tools from statistical mechanics, dynamical system theory and in particular coarse-graining (via 'rough-sef approximation) computational techniques. We use data coming from experiments targeted on recording populations of neurons under controlled decision-making processes. We have identified the basis of this scheme as compatible with the principles of quantum cognition [4] and investigated the properties of it's logical structure as an orthomodular lattice known from Quantum Logic. Bayesian inference based on the upper/lower approximation selects the modification and/or replacement of the algorithms for decision-making by a composition of two different equivalence relations.
Results
At this stage we have identified a minimal model for apprehension and judgment and interpreted data from human subjects [2]. The proposed 'non-algorithmic jumps' reveal the associated quantum-like effects reported in the literature. The composition of the two kinds of equivalence relations, leads to a logic structure expressed as an orthomodular lattice. Conversely it
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