Pattern formation in Hamiltonian systems with continuous spectra; a normal-form single-wave model

Pattern formation in biological, chemical and physical problems has received considerable attention, with much attention paid to dissipative systems. For example, the Ginzburg--Landau equation is a normal form that describes pattern formation due to the appearance of a single mode of instability in a wide variety of dissipative problems. In a similar vein, a certain "single-wave model" arises in many physical contexts that share common pattern forming behavior. These systems have Hamiltonian structure, and the single-wave model is a kind of Hamiltonian mean-field theory describing the patterns that form in phase space. The single-wave model was originally derived in the context of nonlinear plasma theory, where it describes the behavior near threshold and subsequent nonlinear evolution of unstable plasma waves. However, the single-wave model also arises in fluid mechanics, specifically shear-flow and vortex dynamics, galactic dynamics, the XY and Potts models of condensed matter physics, and other Hamiltonian theories characterized by mean field interaction. We demonstrate, by a suitable asymptotic analysis, how the single-wave model emerges from a large class of nonlinear advection-transport theories. An essential ingredient for the reduction is that the Hamiltonian system has a continuous spectrum in the linear stability problem, arising not from an infinite spatial domain but from singular resonances along curves in phase space whereat wavespeeds match material speeds (wave-particle resonances in the plasma problem, or critical levels in fluid problems). The dynamics of the continuous spectrum is manifest as the phenomenon of Landau damping when the system is ... Such dynamical phenomena have been rediscovered in different contexts, which is unsurprising in view of the normal-form character of the single-wave model

Mechanical Control of Sensory Hair-Bundle Function

Hair bundles detect sound in the auditory system, head position and rotation in the vestibular system, and fluid flow in the lateral-­‐‑line system. To do so, bundles respond to periodic, static, and hydrodynamic forces contingent upon the receptor organs in which they are situated. As the mechanosensory function of a hair bundle varies, so too do the mechanical properties of the bundle and its microenvironment. Hair bundles range in height from 1 μμm to 100 μμm and in stiffness from 100 μμN·∙m-­‐‑1 to 10,000 μμN·∙m-­‐‑1. They are composed of actin-­‐‑filled, hypertrophic microvilli—stereocilia—that number from fewer than 20 through more than 300 per bundle. In addition, bundles may or may not possess one true cilium, the kinocilium. Hair bundles differ in shape across organs and organisms: they may be isodiametric, fan-­‐‑shaped, or V-­‐‑shaped. Depending on the organ in which they occur, bundles may be free-­‐‑standing or they may be coupled to a tectorial membrane, otolithic membrane, cupula, or sallet. Because all hair bundles are comprised of similar molecular components, their distinct mechanosensory functions may instead be regulated by their mechanical loads. Dynamical-­‐‑systems analysis provides mathematical predictions of hair-­‐‑bundle behavior. One such model captures the effects of mechanical loading on bundle function in a state diagram. A mechanical-­‐‑load clamp permits exploration of this state diagram by robustly controlling the loads—constant force, load stiffness, virtual drag, and virtual mass—imposed on a hair bundle. Upon changes in these mechanical parameters, the bundle’s response characteristics alter. Subjected to particular control parameters, a bundle may oscillate spontaneously or remain quiescent. It may respond nonlinearly to periodic stimuli with high sensitivity, sharp frequency tuning, and easy entrainment; or it may respond linearly with low sensitivity, broad tuning, and reluctant entrainment. The bundle’s response to a force pulse may resemble that of an edge-­‐‑detection system or a low-­‐‑pass filter. Finally, a bundle from an amphibian vestibular organ can operate in a manner qualitatively similar to that from a mammalian auditory organ, implying an essential similarity between hair bundles. The bifurcation near which a bundle’s operating point resides controls its function: the state diagram provides a functional map of mechanosensory modalities. Auditory function is best tuned near a supercritical Hopf bifurcation, whereas vestibular function is captured by a subcritical Hopf bifurcation and a cusp bifurcation. Within the proposed region vestibular responsiveness, a hair bundle exhibits mechanical excitability analogous to the electrical excitability of neurons. This behavior implies for the first time a direct relationship between the mechanical behaviors of sensory organelles and the electrical behaviors of afferent neurons. Man-­‐‑made detectors function in limited capacities, each designed for a unique purpose. A single hair bundle, on the other hand, evolved to serve multiple purposes with the requirement of only two functional traits: adaptation and nonlinear channel gating. The remarkable conservation of these capabilities thus provides unique insight into the evolution of sensory systems

Bifurcation contrasts between plane Poiseuille flow and plane Magnetohydrodynamic flow

The stability characteristics of an incompressible viscous pressure-driven flow of an electrically conducting fluid between two parallel boundaries in the presence of a transverse magnetic field are compared and contrasted with those of Plane Poiseuille flow (PPF). Assuming that the outer regions adjacent to the fluid layer are perfectly electrically insulating, the appropriate boundary conditions are applied. The eigenvalue problems are then solved numerically to obtain the critical Reynolds number Rec and the critical wave number ac in the limit of small Hartmann number (M) range to produce the curves of marginal stability. The non-linear two-dimensional travelling waves that bifurcate by way of a Hopf bifurcation from the neutral curves are approximated by a truncated Fourier series in the streamwise direction. Two and three dimensional secondary disturbances are applied to both the constant pressure and constant flux equilibrium solutions using Floquet theory as this is believed to be the generic mechanism of instability in shear flows. The change in shape of the undisturbed velocity profile caused by the magnetic field is found to be the dominant factor. Consequently the critical Reynolds number is found to increase rapidly with increasing M so the transverse magnetic field has a powerful stabilising effect on this type of flow

Intermediate Stable Phase Locked States In Oscillator Networks

The study of nonlinear oscillations is important in a variety of physical and biological contexts (especially in neuroscience). Synchronization of oscillators has been a problem of interest in recent years. In networks of nearest neighbor coupled oscillators it is possible to obtain synchrony between oscillators, but also a variety of constant phase shifts between 0 and pi. We coin these phase shifts intermediate stable phase-locked states. In neuroscience, both individual neurons and populations of neurons can behave as complex nonlinear oscillators. Intermediate stable phase-locked states are shown to be obtainable between individual oscillators and populations of identical oscillators.These intermediate stable phase-locked states may be useful in the construction of central pattern generators: autonomous neural cicuits responsible for motor behavior. In large chains and two-dimenional arrays of oscillators, intermediate stable phase-locked states provide a mechanism to produce waves and patterns that cannot be obtained in traditional network models. A particular pattern of interest is known as an anti-wave. This pattern corresponds to the collision of two waves from opposite ends of an oscillator chain. This wave may be relevant in the spinal central pattern generators of various fish. Anti-wave solutions in both conductance based neuron models and phase oscillator models are analyzed. It is shown that such solutions arise in phase oscillator models in which the nonlinearity (interaction function) contains both higher order odd and even Fourier modes. These modes are prominent in pairs of synchronous oscillators which lose stability in a supercritical pitchfork bifurcation

Dynamics and precursor signs for phase transitions in neural systems

This thesis investigates neural state transitions associated with sleep, seizure and anaesthesia. The aim is to address the question: How does a brain traverse the critical threshold between distinct cortical states, both healthy and pathological? Specifically we are interested in sub-threshold neural behaviour immediately prior to state transition. We use theoretical neural modelling (single spiking neurons, a network of these, and a mean-field continuum limit) and in vitro experiments to address this question. Dynamically realistic equations of motion for thalamic relay neuron, reticular nuclei, cortical pyramidal and cortical interneuron in different vigilance states are developed, based on the Izhikevich spiking neuron model. A network of cortical neurons is assembled to examine the behaviour of the gamma-producing cortical network and its transition to lower frequencies due to effect of anaesthesia. Then a three-neuron model for the thalamocortical loop for sleep spindles is presented. Numerical simulations of these networks confirms spiking consistent with reported in vivo measurement results, and provides supporting evidence for precursor indicators of imminent phase transition due to occurrence of individual spindles. To complement the spiking neuron networks, we study the Wilson–Cowan neural mass equations describing homogeneous cortical columns and a 1D spatial cluster of such columns. The abstract representation of cortical tissue by a pair of coupled integro-differential equations permits thorough linear stability, phase plane and bifurcation analyses. This model shows a rich set of spatial and temporal bifurcations marking the boundary to state transitions: saddle-node, Hopf, Turing, and mixed Hopf–Turing. Close to state transition, white-noise-induced subthreshold fluctuations show clear signs of critical slowing down with prolongation and strengthening of autocorrelations, both in time and space, irrespective of bifurcation type. Attempts at in vitro capture of these predicted leading indicators form the last part of the thesis. We recorded local field potentials (LFPs) from cortical and hippocampal slices of mouse brain. State transition is marked by the emergence and cessation of spontaneous seizure-like events (SLEs) induced by bathing the slices in an artificial cerebral spinal fluid containing no magnesium ions. Phase-plane analysis of the LFP time-series suggests that distinct bifurcation classes can be responsible for state change to seizure. Increased variance and growth of spectral power at low frequencies (f < 15 Hz) was observed in LFP recordings prior to initiation of some SLEs. In addition we demonstrated prolongation of electrically evoked potentials in cortical tissue, while forwarding the slice to a seizing regime. The results offer the possibility of capturing leading temporal indicators prior to seizure generation, with potential consequences for understanding epileptogenesis. Guided by dynamical systems theory this thesis captures evidence for precursor signs of phase transitions in neural systems using mathematical and computer-based modelling as well as in vitro experiments

A novel technique for high-resolution frequency discriminators and their application to pitch and onset detection in empirical musicology

This thesis presents and evaluates software for simultaneous, high-resolution time-frequency discrimination. Whilst this is a problem that arises in many areas of engineering, the software here is developed to assist musicological investigations. In order to analyse musical performances, we must first know what is happening and when; that is, at what time each note begins to sound (the note onset) and what frequencies are present (the pitch). The work presented here focusses on onset detection, although the representation of data used for this task could also be used to track the pitch. A potential method of determining pitch on a sample-to-sample basis is given in the final chapter. Extant software for onset detection uses standard signal processing techniques to search for changes in features like the spectrum or phase. These methods struggle somewhat, as they are constrained by the uncertainty principle, which states that, as time resolution is increased, frequency resolution must decrease and vice versa. However, we can hear changes in frequency to a far greater time resolution than the uncertainty principle would suggest is possible. There is an active process in the inner ear which adds energy and enables this perceptual acuity. The mathematical expression which describes this system is known as the Hopf bifurcation. By building a bank of tuned resonators in software, each of which operates at a Hopf bifurcation, and driving it with audio, changes in frequency can be detected in times that defy the uncertainty relation, as we are not seeking to directly measure the time-frequency features of a system, rather it is used to drive a system. Time and frequency information is then available from the internal state variables of the system. The characteristics of this bank of resonators - called a 'DetectorBank' - are investigated thoroughly. The bandwidth of each resonator ('detector') can be as narrow as 0.922Hz and the system bandwidth is extended to the Nyquist frequency. A nonlinear system may be expected to respond poorly when presented with multiple simultaneous input frequencies; however, the DetectorBank performs well under these circumstances. The data generated by the DetectorBank is then analysed by an OnsetDetector. Both the development and testing of this OnsetDetector are detailed. It is tested using a repository of recordings of individual notes played on a variety of instruments, with promising results. These results are discussed, problems with the current implementation are identified and potential solutions presented. This OnsetDetector can then be combined with a PitchTracker to create a NoteDetector, capable of detecting not only a single note onset time and pitch, but information about changes that occur within a note. Musical notes are not static entities: they contain much variation. Both the performer's intonation and the characteristics of the instrument itself have an effect on the frequency present, as well as features like vibrato. Knowledge of these frequency components, and how they appear or disappear over the course of the note, is valuable information and the software presented here enables the collection of this data

Nonlinear amplification by active sensory hair bundles

The human sense of hearing is characterized by its exquisite sensitivity, sharp frequency selectivity, and wide dynamic range. These features depend on an active process that in the inner ear boosts vibrations evoked by auditory stimuli. Spontaneous otoacoustic emissions constitute a demonstrative manifestation of this physiologically vulnerable mechanism. In the cochlea, sensory hair bundles transduce sound-induced vibrations into neural signals. Hair bundles can power mechanical movements of their tip, oscillate spontaneously, and operate as tuned nonlinear amplifiers of weak periodic stimuli. Active hair-bundle motility constitutes a promising candidate with respect to the biophysical implementation of the active process underlying human hearing. The responsiveness of isolated hair bundles, however, is seriously hampered by intrinsic fluctuations. In this thesis, we present theoretical and experimental results concerning the noise-imposed limitations of nonlinear amplification by active sensory hair bundles. We analyze the effect of noise within the framework of a stochastic description of hair-bundle dynamics and relate our findings to generic aspects of the stochastic dynamics of oscillatory systems. Hair bundles in vivo are often elastically coupled by overlying gelatinous membranes. In addition to theoretical results concerning the dynamics of elastically coupled hair bundles, we report on an experimental study. We have interfaced dynamic force clamp performed on a hair bundle from the sacculus of the bullfrog with real-time stochastic simulations of hair-bundle dynamics. By means of this setup, we could couple a hair bundle to two virtual neighbors, called cyber clones. Our theoretical and experimental work shows that elastic coupling leads to an effective noise reduction. Coupled hair bundles exhibit an increased coherence of spontaneous oscillations and an enhanced amplification gain. We therefore argue that elastic coupling by overlying membranes constitutes a morphological specialization for reducing the detrimental effect of intrinsic fluctuations