5,429 research outputs found
Chaos at the border of criticality
The present paper points out to a novel scenario for formation of chaotic
attractors in a class of models of excitable cell membranes near an
Andronov-Hopf bifurcation (AHB). The mechanism underlying chaotic dynamics
admits a simple and visual description in terms of the families of
one-dimensional first-return maps, which are constructed using the combination
of asymptotic and numerical techniques. The bifurcation structure of the
continuous system (specifically, the proximity to a degenerate AHB) endows the
Poincare map with distinct qualitative features such as unimodality and the
presence of the boundary layer, where the map is strongly expanding. This
structure of the map in turn explains the bifurcation scenarios in the
continuous system including chaotic mixed-mode oscillations near the border
between the regions of sub- and supercritical AHB. The proposed mechanism
yields the statistical properties of the mixed-mode oscillations in this
regime. The statistics predicted by the analysis of the Poincare map and those
observed in the numerical experiments of the continuous system show a very good
agreement.Comment: Chaos: An Interdisciplinary Journal of Nonlinear Science
(tentatively, Sept 2008
Mechanism, dynamics, and biological existence of multistability in a large class of bursting neurons
Multistability, the coexistence of multiple attractors in a dynamical system,
is explored in bursting nerve cells. A modeling study is performed to show that
a large class of bursting systems, as defined by a shared topology when
represented as dynamical systems, is inherently suited to support
multistability. We derive the bifurcation structure and parametric trends
leading to multistability in these systems. Evidence for the existence of
multirhythmic behavior in neurons of the aquatic mollusc Aplysia californica
that is consistent with our proposed mechanism is presented. Although these
experimental results are preliminary, they indicate that single neurons may be
capable of dynamically storing information for longer time scales than
typically attributed to nonsynaptic mechanisms.Comment: 24 pages, 8 figure
Chaotic exploration and learning of locomotion behaviours
We present a general and fully dynamic neural system, which exploits intrinsic chaotic dynamics, for the real-time goal-directed exploration and learning of the possible locomotion patterns of an articulated robot of an arbitrary morphology in an unknown environment. The controller is modeled as a network of neural oscillators that are initially coupled only through physical embodiment, and goal-directed exploration of coordinated motor patterns is achieved by chaotic search using adaptive bifurcation. The phase space of the indirectly coupled neural-body-environment system contains multiple transient or permanent self-organized dynamics, each of which is a candidate for a locomotion behavior. The adaptive bifurcation enables the system orbit to wander through various phase-coordinated states, using its intrinsic chaotic dynamics as a driving force, and stabilizes on to one of the states matching the given goal criteria. In order to improve the sustainability of useful transient patterns, sensory homeostasis has been introduced, which results in an increased diversity of motor outputs, thus achieving multiscale exploration. A rhythmic pattern discovered by this process is memorized and sustained by changing the wiring between initially disconnected oscillators using an adaptive synchronization method. Our results show that the novel neurorobotic system is able to create and learn multiple locomotion behaviors for a wide range of body configurations and physical environments and can readapt in realtime after sustaining damage
Multimodal oscillations in systems with strong contraction
One- and two-parameter families of flows in near an Andronov-Hopf
bifurcation (AHB) are investigated in this work. We identify conditions on the
global vector field, which yield a rich family of multimodal orbits passing
close to a weakly unstable saddle-focus and perform a detailed asymptotic
analysis of the trajectories in the vicinity of the saddle-focus. Our analysis
covers both cases of sub- and supercritical AHB. For the supercritical case, we
find that the periodic orbits born from the AHB are bimodal when viewed in the
frame of coordinates generated by the linearization about the bifurcating
equilibrium. If the AHB is subcritical, it is accompanied by the appearance of
multimodal orbits, which consist of long series of nearly harmonic oscillations
separated by large amplitude spikes. We analyze the dependence of the
interspike intervals (which can be extremely long) on the control parameters.
In particular, we show that the interspike intervals grow logarithmically as
the boundary between regions of sub- and supercritical AHB is approached in the
parameter space. We also identify a window of complex and possibly chaotic
oscillations near the boundary between the regions of sub- and supercritical
AHB and explain the mechanism generating these oscillations. This work is
motivated by the numerical results for a finite-dimensional approximation of a
free boundary problem modeling solid fuel combustion
Nearly inviscid Faraday waves in containers with broken symmetry
In the weakly inviscid regime parametrically driven surface gravity-capillary waves generate oscillatory viscous boundary layers along the container walls and the free surface. Through nonlinear rectification these generate Reynolds stresses which drive a streaming flow in the nominally inviscid bulk; this flow in turn advects the waves responsible for the boundary layers. The resulting system is described by amplitude equations coupled to a Navier-Stokes-like equation for the bulk streaming flow, with boundary conditions obtained by matching to the boundary layers, and represents a novel type of pattern-forming system. The coupling to the streaming flow is responsible for various types of drift instabilities of standing waves, and in appropriate regimes can lead to the presence of relaxations oscillations. These are present because in the nearly inviscid regime the streaming flow decays much more slowly than the waves. Two model systems, obtained by projection of the Navier-Stokes-like equation onto the slowest mode of the domain, are examined to clarify the origin of this behavior. In the first the domain is an elliptically distorted cylinder while in the second it is an almost square rectangle. In both cases the forced symmetry breaking results in a nonlinear competition between two nearly degenerate oscillatory modes. This interaction destabilizes standing waves at small amplitudes and amplifies the role played by the streaming flow. In both systems the coupling to the streaming flow triggered by these instabilities leads to slow drifts along slow manifolds of fixed points or periodic orbits of the fast system, and generates behavior that resembles bursting in excitable systems. The results are compared to experiments
New dynamics in cerebellar Purkinje cells: torus canards
We describe a transition from bursting to rapid spiking in a reduced
mathematical model of a cerebellar Purkinje cell. We perform a slow-fast
analysis of the system and find that -- after a saddle node bifurcation of
limit cycles -- the full model dynamics follow temporarily a repelling branch
of limit cycles. We propose that the system exhibits a dynamical phenomenon new
to realistic, biophysical applications: torus canards.Comment: 4 pages; 4 figures (low resolution); updated following peer-review:
language and definitions updated, Figures 1 and 4 updated, typos corrected,
references added and remove
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