On The Dynamics Of Systems With Vastly Different Frequencies

The method of direct partition of motion (DPM) has been widely used to study the dynamics of non-autonomous oscillatory systems subject to external high frequency excitation. In this work, we explore the non-trivial dynamics that arise in autonomous systems with vastly different frequencies, in which the fast excitation is intrinsic to the system and possibly influenced by the slow dynamics. Using three model problems, we illustrate how DPM could be useful for the analysis of such systems and when combined with WKB can serve to capture the strong modulation of a fast oscillator due to coupling to a much slower one -a phenomenon that is not readily handled using standard perturbation methods. First, we study a system of three coupled limit cycle oscillators, which when uncoupled, have the frequencies [omega]1 = O (1), [omega]2 = O(1/[epsilon]) and [omega]3 = O(1/[epsilon]2), respectively, where [epsilon] << 1. Approximate expressions for the limit cycles of oscillators 1 and 2 are found in terms of Jacobi elliptic functions. For coupling strengths exceeding critical bifurcation values, the limit cycle of oscillator 1 or 2 is found to disappear. In the second problem, we consider a simple pendulum coupled to a horizontal mass-spring system of a much higher frequency. In contrast to the first problem, the coupling here allows the slow oscillator to affect the leading order dynamics of the fast oscillator. This calls for a rescaling of fast time, inspired by the WKB method, to be employed in conjunction with DPM. We obtain a critical energy value at which a pitchfork bifurcation of periodic orbits is found to occur, giving rise to non-local periodic and quasi-periodic orbits in which the pendulum oscillates about an angle between zero and [pi]/2 from the downwards position. Finally, the developed method is utilized to explain the non-trivial dynamics arising in a model of a thin elastica presented by Cusumano and Moon in 1995. We observe that for the corresponding experimental system, the ratio of the two natural frequencies of the system was [ALMOST EQUAL TO] 44 which can be considered to be of O (1/[epsilon]) where [epsilon] << 1. Hence, the system is best viewed as one with vastly different frequencies. The method leads to an approximate expression for the non-local modes of the type observed in the experiments as well as the bifurcation energy value at which these modes are born. The formal approximate solutions obtained for these problems are validated by comparison with numerical integration

On the dynamics of a spinning top under high-frequency excitation: part I—pivot point under vertical harmonic vibration

International audienc

On the dynamics of a spinning top under high-frequency excitation. Part II: pivot point under horizontal harmonic vibration

International audienc

A mathematical model of ephaptic interactions in neuronal fiber pathways: Could there be more than transmission along the tracts?

International audienceWhile numerous studies of ephaptic interactions have focused on either axons of peripheral nerves or on cortical structures, no attention has been given to the possibility of ephaptic interactions in white matter tracts. Inspired by the highly organized, tightly packed geometry of axons in fiber pathways, we aim to investigate the potential effects of ephaptic interactions along these structures that are resilient to experimental probing. We use axonal cable theory to derive a minimal model of a sheet of N ephaptically coupled axons. Numerical solutions of the proposed model are explored as ephaptic coupling is varied. We demonstrate that ephaptic interactions can lead to local phase locking between adjacent traveling impulses and that, as coupling is increased, traveling impulses trigger new impulses along adjacent axons, resulting in finite size traveling fronts. For strong enough coupling, impulses propagate laterally and backwards, resulting in complex spatiotemporal patterns. While common large-scale brain network models often model fiber pathways as simple relays of signals between different brain regions, our work calls for a closer reexamination of the validity of such a view. The results suggest that in the presence of significant ephaptic interactions, the brain fiber tracts can act as a dynamic active medium

On the dynamics of a spinning top under the influence of rotation: Resonant relative equilibrium states

International audienc

Entropy, free energy, symmetry and dynamics in the brain

International audienceNeuroscience is home to concepts and theories with roots in a variety of domains including information theory, dynamical systems theory, and cognitive psychology. Not all of those can be coherently linked, some concepts are incommensurable, and domain-specific language poses an obstacle to integration. Still, conceptual integration is a form of understanding that provides intuition and consolidation, without which progress remains unguided. This paper is concerned with the integration of deterministic and stochastic processes within an information theoretic framework, linking information entropy and free energy to mechanisms of emergent dynamics and self-organization in brain networks. We identify basic properties of neuronal populations leading to an equivariant matrix in a network, in which complex behaviors can naturally be represented through structured flows on manifolds establishing the internal model relevant to theories of brain function. We propose a neural mechanism for the generation of internal models from symmetry breaking in the connectivity of brain networks. The emergent perspective illustrates how free energy can be linked to internal models and how they arise from the neural substrat

In pursuit of the epileptogenic zone in focal epilepsy:a dynamical network biomarker approach

International audienc

Nontuberculous Mycobacterial Ocular Infections: A Systematic Review of the Literature

Nontuberculous or atypical mycobacterial ocular infections have been increasing in prevalence over the past few decades. They are known to cause periocular, adnexal, ocular surface and intraocular infections and are often recalcitrant to medical therapy. These infections can potentially cause detrimental outcomes, in part due to a delay in diagnosis. We review 174 case reports and series on nontuberculous mycobacterial (NTM) ocular infections and discuss etiology, microbiology, risk factors, diagnosis, clinical presentation, and treatment of these infections. History of interventions, trauma, foreign bodies, implants, contact lenses, and steroids are linked to NTM ocular infections. Steroid use may prolong the duration of the infection and cause poorer visual outcomes. Early diagnosis and initiation of treatment with multiple antibiotics are necessary to achieve the best visual outcome

Symmetry breaking organizes the brain's resting state manifold

International audienceSpontaneously fluctuating brain activity patterns that emerge at rest have been linked to brain's health and cognition. Despite detailed descriptions of the spatio-temporal brain patterns, our understanding of their generative mechanism is still incomplete. Using a combination of computational modeling and dynamical systems analysis we provide a mechanistic description of the formation of a resting state manifold via the network connectivity. We demonstrate that the symmetry breaking by the connectivity creates a characteristic flow on the manifold, which produces the major data features across scales and imaging modalities. These include spontaneous high amplitude co-activations, neuronal cascades, spectral cortical gradients, multistability and characteristic functional connectivity dynamics. When aggregated across cortical hierarchies, these match the profiles from empirical data. The understanding of the brain's resting state manifold is fundamental for the construction of task-specific flows and manifolds used in theories of brain function such as predictive coding. In addition, it shifts the focus from the single recordings towards brain's capacity to generate certain dynamics characteristic of health and pathology

Nontuberculous mycobacterial ocular infections: a systematic review of the literature

Nontuberculous or atypical mycobacterial ocular infections have been increasing in prevalence over the past few decades. They are known to cause periocular, adnexal, ocular surface and intraocular infections and are often recalcitrant to medical therapy. These infections can potentially cause detrimental outcomes, in part due to a delay in diagnosis. We review 174 case reports and series on nontuberculous mycobacterial (NTM) ocular infections and discuss etiology, microbiology, risk factors, diagnosis, clinical presentation, and treatment of these infections. History of interventions, trauma, foreign bodies, implants, contact lenses, and steroids are linked to NTM ocular infections. Steroid use may prolong the duration of the infection and cause poorer visual outcomes. Early diagnosis and initiation of treatment with multiple antibiotics are necessary to achieve the best visual outcome