Dynamical aspects of mean field plane rotators and the Kuramoto model

The Kuramoto model has been introduced in order to describe synchronization phenomena observed in groups of cells, individuals, circuits, etc... We look at the Kuramoto model with white noise forces: in mathematical terms it is a set of N oscillators, each driven by an independent Brownian motion with a constant drift, that is each oscillator has its own frequency, which, in general, changes from one oscillator to another (these frequencies are usually taken to be random and they may be viewed as a quenched disorder). The interactions between oscillators are of long range type (mean field). We review some results on the Kuramoto model from a statistical mechanics standpoint: we give in particular necessary and sufficient conditions for reversibility and we point out a formal analogy, in the N to infinity limit, with local mean field models with conservative dynamics (an analogy that is exploited to identify in particular a Lyapunov functional in the reversible set-up). We then focus on the reversible Kuramoto model with sinusoidal interactions in the N to infinity limit and analyze the stability of the non-trivial stationary profiles arising when the interaction parameter K is larger than its critical value K_c. We provide an analysis of the linear operator describing the time evolution in a neighborhood of the synchronized profile: we exhibit a Hilbert space in which this operator has a self-adjoint extension and we establish, as our main result, a spectral gap inequality for every K>K_c.Comment: 18 pages, 1 figur

Uncertainty Principle for Control of Ensembles of Oscillators Driven by Common Noise

We discuss control techniques for noisy self-sustained oscillators with a focus on reliability, stability of the response to noisy driving, and oscillation coherence understood in the sense of constancy of oscillation frequency. For any kind of linear feedback control--single and multiple delay feedback, linear frequency filter, etc.--the phase diffusion constant, quantifying coherence, and the Lyapunov exponent, quantifying reliability, can be efficiently controlled but their ratio remains constant. Thus, an "uncertainty principle" can be formulated: the loss of reliability occurs when coherence is enhanced and, vice versa, coherence is weakened when reliability is enhanced. Treatment of this principle for ensembles of oscillators synchronized by common noise or global coupling reveals a substantial difference between the cases of slightly non-identical oscillators and identical ones with intrinsic noise.Comment: 10 pages, 5 figure

The what and where of adding channel noise to the Hodgkin-Huxley equations

One of the most celebrated successes in computational biology is the Hodgkin-Huxley framework for modeling electrically active cells. This framework, expressed through a set of differential equations, synthesizes the impact of ionic currents on a cell's voltage -- and the highly nonlinear impact of that voltage back on the currents themselves -- into the rapid push and pull of the action potential. Latter studies confirmed that these cellular dynamics are orchestrated by individual ion channels, whose conformational changes regulate the conductance of each ionic current. Thus, kinetic equations familiar from physical chemistry are the natural setting for describing conductances; for small-to-moderate numbers of channels, these will predict fluctuations in conductances and stochasticity in the resulting action potentials. At first glance, the kinetic equations provide a far more complex (and higher-dimensional) description than the original Hodgkin-Huxley equations. This has prompted more than a decade of efforts to capture channel fluctuations with noise terms added to the Hodgkin-Huxley equations. Many of these approaches, while intuitively appealing, produce quantitative errors when compared to kinetic equations; others, as only very recently demonstrated, are both accurate and relatively simple. We review what works, what doesn't, and why, seeking to build a bridge to well-established results for the deterministic Hodgkin-Huxley equations. As such, we hope that this review will speed emerging studies of how channel noise modulates electrophysiological dynamics and function. We supply user-friendly Matlab simulation code of these stochastic versions of the Hodgkin-Huxley equations on the ModelDB website (accession number 138950) and http://www.amath.washington.edu/~etsb/tutorials.html.Comment: 14 pages, 3 figures, review articl

Simple, Fast and Accurate Implementation of the Diffusion Approximation Algorithm for Stochastic Ion Channels with Multiple States

The phenomena that emerge from the interaction of the stochastic opening and closing of ion channels (channel noise) with the non-linear neural dynamics are essential to our understanding of the operation of the nervous system. The effects that channel noise can have on neural dynamics are generally studied using numerical simulations of stochastic models. Algorithms based on discrete Markov Chains (MC) seem to be the most reliable and trustworthy, but even optimized algorithms come with a non-negligible computational cost. Diffusion Approximation (DA) methods use Stochastic Differential Equations (SDE) to approximate the behavior of a number of MCs, considerably speeding up simulation times. However, model comparisons have suggested that DA methods did not lead to the same results as in MC modeling in terms of channel noise statistics and effects on excitability. Recently, it was shown that the difference arose because MCs were modeled with coupled activation subunits, while the DA was modeled using uncoupled activation subunits. Implementations of DA with coupled subunits, in the context of a specific kinetic scheme, yielded similar results to MC. However, it remained unclear how to generalize these implementations to different kinetic schemes, or whether they were faster than MC algorithms. Additionally, a steady state approximation was used for the stochastic terms, which, as we show here, can introduce significant inaccuracies. We derived the SDE explicitly for any given ion channel kinetic scheme. The resulting generic equations were surprisingly simple and interpretable - allowing an easy and efficient DA implementation. The algorithm was tested in a voltage clamp simulation and in two different current clamp simulations, yielding the same results as MC modeling. Also, the simulation efficiency of this DA method demonstrated considerable superiority over MC methods.Comment: 32 text pages, 10 figures, 1 supplementary text + figur

Blood-Labyrinth Barrier Permeability in Menière Disease and Idiopathic Sudden Sensorineural Hearing Loss: Findings on Delayed Postcontrast 3D-FLAIR MRI

Background and purposeMenière disease and idiopathic sudden sensorineural hearing loss can have overlapping clinical presentation and may have similar pathophysiology. Prior studies using postcontrast 3D-FLAIR MR imaging suggest abnormal blood-labyrinth barrier permeability in both conditions, but the 2 diseases have not been directly compared by using the same imaging techniques. We hypothesized that delayed postcontrast 3D-FLAIR MR imaging would show differences in blood-labyrinth barrier permeability between Menière disease and idiopathic sudden sensorineural hearing loss.Materials and methodsPatients with unilateral Menière disease (n = 32) and unilateral idiopathic sudden sensorineural hearing loss (n = 11) imaged with delayed postcontrast 3D-FLAIR MR imaging were retrospectively studied. Signal intensities of the medulla and perilymph of the cochlear basal turns of both ears in each patient were measured in a blinded fashion. Cochlea/medulla ratios were calculated for each ear as a surrogate for blood-labyrinth barrier permeability. The ears were segregated by clinical diagnosis.ResultsCochlea/medulla ratio was higher in symptomatic ears of patients with Menière disease (12.6 ± 7.4) than in patients with idiopathic sudden sensorineural hearing loss (5.7 ± 2.0) and asymptomatic ears of patients with Menière disease (8.0 ± 3.1), indicating increased blood-labyrinth barrier permeability in Menière disease ears. The differences in cochlea/medulla ratio between symptomatic and asymptomatic ears were significantly higher in Menière disease than in idiopathic sudden sensorineural hearing loss. Asymptomatic ears in patients with Menière disease showed higher cochlea/medulla ratio than symptomatic and asymptomatic ears in patients with idiopathic sudden sensorineural hearing loss.ConclusionsIncreased cochlea/medulla ratio indicates increased blood-labyrinth barrier permeability in Menière disease compared with idiopathic sudden sensorineural hearing loss. Increased cochlea/medulla ratio in asymptomatic ears of patients with Menière disease also suggests an underlying systemic cause of Menière disease and may provide a pathophysiologic biomarker

Common perceptions of periodontal health and illness among adults : A qualitative study

Peer reviewe

High Mobility Group box-1 (HMGB1) Protein As a Biomarker for Acute Cholecystitis

Background: Acute cholecystitis is defined as gallbladder inflammation caused by obstruction of the cystic duct. The pro-inflammatory cytokine, high mobility group box-1 (HMGB1), has been found to hold critical roles in the pathogenesis of several different inflammatory diseases. This study aimed to determine the relationship between HMGB1 and acute cholecystitis, and examine the potential for this cytokine as a biomarker for clinical diagnosis. Methods: The serum of 23 patients with severe acute cholecystitis, 45 patients with mild acute cholecystitis and 35 healthy subjects was collected and isolated from peripheral blood. The serum levels of HMGB1, CRP, amylase, lipase and the number of white blood cells were measured prior to the patient's cholecystectomy and 48 hours following the procedure. Results: A significant increase in the levels of HMGB1 were observed in both patient groups with mild or severe acute cholecystitis compared with normal group. ROC analysis determined a cut-off point of 2.34 for HMGB1 serum levels to discriminate between the normal group and acute cholecystitis patients with a sensitivity of 79.41 and a specificity of 54.3. The area under the ROC curve was 0.71. Furthermore, a positive correlation was observed between CRP and HMGB1 levels and no significant difference in the levels of amylase and lipase was observed between groups. Conclusions: These findings suggest a potential role for HMGB1 as an effective biomarker in improving the diagnostic accuracy of acute cholecystitis when used in conjunction with the standard diagnostic tests

Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators

We study the dynamics of the large N limit of the Kuramoto model of coupled phase oscillators, subject to white noise. We introduce the notion of shadow inertial manifold and we prove their existence for this model, supporting the fact that the long term dynamics of this model is finite dimensional. Following this, we prove that the global attractor of this model takes one of two forms. When coupling strength is below a critical value, the global attractor is a single equilibrium point corresponding to an incoherent state. Conversely, when coupling strength is beyond this critical value, the global attractor is a two-dimensional disk composed of radial trajectories connecting a saddle equilibrium (the incoherent state) to an invariant closed curve of locally stable equilibria (partially synchronized state). Our analysis hinges, on the one hand, upon sharp existence and uniqueness results and their consequence for the existence of a global attractor, and, on the other hand, on the study of the dynamics in the vicinity of the incoherent and synchronized equilibria. We prove in particular non-linear stability of each synchronized equilibrium, and normal hyperbolicity of the set of such equilibria. We explore mathematically and numerically several properties of the global attractor, in particular we discuss the limit of this attractor as noise intensity decreases to zero.Comment: revised version, 28 pages, 4 figure