The dynamic behavior of spiral waves in stochastic Hodgkin–Huxley neuronal networks with ion channel blocks

Chemical blocking is known to affect neural network activity. Here, we quantitatively investigate the dynamic behavior of spiral waves in stochastic Hodgkin–Huxley neuronal networks during sodium- or potassium-ion channel blockages. When the sodium-ion channels are blocked, the spiral waves first become sparse and then break. The critical factor for the transition of spiral waves (x _Na) is sensitive to the channel noise. However, with the potassium-ion channel block, the spiral waves first become intensive and then form other dynamic patterns. The critical factor for the transition of spiral waves (x _K) is insensitive to the channel noise. With the sodium-ion channel block, the spike frequency of a single neuron in the network is reduced, and the collective excitability of the neuronal network weakens. By blocking the potassium ion channels, the spike frequency of a single neuron in the network increases, and the collective excitability of the neuronal network is enhanced. Lastly, we found that the behavior of spiral waves is directly related to the system synchronization. This research will enhance our understanding of the evolution of spiral waves through toxins or drugs and will be helpful to find potential applications for controlling spiral waves in real neural systems.     

Boundary element modeling for defect characterization potential in a wave guide

Wave scattering analysis implemented by boundary element methods (BEM) and the normal mode expansion technique is used to study the sizing potential of two-dimensional shaped defects in a wave guide. Surface breaking half-elliptical shaped defects of three opening lengths (0.3, 6.35 and 12.7 mm) and through-wall depths of 10–90% on a 10 mm thick steel plate were considered. The reflection and transmission coefficients of both Lamb and shear horizontal (SH) waves over a frequency range 0.05–2 MHz were studied. A powerfully practical result was obtained whereby the numerical results for the S₀ mode Lamb wave and n₀ mode SH wave at low frequencies showed a monotonic increase in signal amplitude with an increase in the defect through-wall depth. At high frequency (usually above the cut-off frequency of the A₁ mode for Lamb waves and the n₁ mode for SH waves, respectively), the monotonic trend does not hold in general due to the energy redistribution to the higher order wave modes. Guided waves impinging onto an internal stringer-like an inclusion were also studied. Both the Lamb and SH waves were shown to be insensitive to the stringer internal inclusions at low frequency. Experiments with piezoelectric Lamb wave transducers and non-contact SH wave electro-magnetic acoustic transducers (EMAT) verified some of the theoretical results.     

Effects of impedance mismatch on the strength of waves in layered solids

This research program was conducted to study the effects of acoustic-impedance mismatch between materials in a layered elastic solid on the amplitudes of the head waves generated at the interface as a stress wave develops and propagates in one of the layers. Dynamic photoelasticity methods were employed. The isochromatic-fringe patterns used for analysis were recorded with a Cranz-Schardin multiple-spark camera operating at a framing rate of approximately 188,000 exposures per second. Acoustic-impedance ratios from a low of 1.7∶1 to a high of 17.4∶1 were studied. Small charges of lead azide were used to generate the original dilatational (P ₁) wave. Results of the study confirm the existence of all waves predicted by theory except for theP ₁ P ₁ waves reflected from the free surface and from the interface near the source in the low-impedance layer. In the region near the explosive detonation, the head waves are important since they have significant magnitudes for certain impedance ratios and they appear to attenuate at a rate much lower than the rate associated with the incidentP ₁ wave or the reflectedP ₁ S ₁ waves.     

On the Existence and Conditional Energetic Stability of Solitary Gravity-Capillary Surface Waves on Deep Water

This paper presents an existence and stability theory for gravity-capillary solitary waves on the surface of a body of water of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy E{{\mathcal E}} subject to the constraint I=?2m{{\mathcal I}=\sqrt{2}\mu}, where I{{\mathcal I}} is the wave momentum and 0 < m << 1{0 < \mu \ll 1} . Since E{{\mathcal E}} and I{{\mathcal I}} are both conserved quantities a standard argument asserts the stability of the set D _μ of minimisers: solutions starting near D _μ remain close to D _μ in a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are modelled as solutions of the nonlinear Schr?dinger equation with cubic focussing nonlinearity. We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of this model equation as mˉ 0{\mu \downarrow 0} .     

Dissipative interface waves and the transient response of a three-dimensional sliding interface with Coulomb friction

We investigate the linearized response of two elastic half-spaces sliding past one another with constant Coulomb friction to small three-dimensional perturbations. Starting with the assumption that friction always opposes slip velocity, we derive a set of linearized boundary conditions relating perturbations of shear traction to slip velocity. Friction introduces an effective viscosity transverse to the direction of the original sliding, but offers no additional resistance to slip aligned with the original sliding direction. The amplitude of transverse slip depends on a nondimensional parameter η=c_sτ₀/μv₀, where τ₀ is the initial shear stress, 2v₀ is the initial slip velocity, μ is the shear modulus, and c_s is the shear wave speed. As η→0, the transverse shear traction becomes negligible, and we find an azimuthally symmetric Rayleigh wave trapped along the interface. As η→∞, the inplane and antiplane wavesystems frictionally couple into an interface wave with a velocity that is directionally dependent, increasing from the Rayleigh speed in the direction of initial sliding up to the shear wave speed in the transverse direction. Except in these frictional limits and the specialization to two-dimensional inplane geometry, the interface waves are dissipative. In addition to forward and backward propagating interface waves, we find that for η>1, a third solution to the dispersion relation appears, corresponding to a damped standing wave mode. For large-amplitude perturbations, the interface becomes isotropically dissipative. The behavior resembles the frictionless response in the extremely strong perturbation limit, except that the waves are damped. We extend the linearized analysis by presenting analytical solutions for the transient response of the medium to both line and point sources on the interface. The resulting self-similar slip pulses consist of the interface waves and head waves, and help explain the transmission of forces across fracture surfaces. Furthermore, we suggest that the η→∞ limit describes the sliding interface behind the crack edge for shear fracture problems in which the absolute level of sliding friction is much larger than any interfacial stress changes.     

Non-linear modulation of SH waves in a two-layered plate and formation of surface SH waves

Non-linear modulation of shear horizontal (SH) waves in a two-layered elastic plate of uniform thickness is considered. Both layers are assumed to be homogeneous, isotropic and incompressible elastic and having different mechanical properties. The problem is investigated by a perturbation method and in the analysis it is assumed that between the linear shear velocities of the top layer, c₁, and the bottom layer, c₂, the inequality c₁<c₂ is valid. In the layered structure then an SH wave exists if the wave velocity c of the wave satisfies either the condition c₁<c?c₂ or the one c₁<c₂?c. Here the problem is examined under the former condition and it is shown that the non-linear modulation of SH waves is governed by a non-linear Schrödinger equation. In this case the formation of surface SH (Love) waves is also revealed if the top layer is thinner when compared with the bottom layer. Then the stability condition is discussed and the existence of bright (envelope) and dark solitons are manifested.     

Cylindrical elastic-plastic waves due to discontinuous loading at a circular cavity

The plastic waves in rate-independent, isotropically work-hardening media obeying the von Mises yield condition generated by radial stress uniformly applied at a circular cavity of radius r = r₀, are studied. Both plane stress and plane strain motions are considered. The radial stress and its time derivative at the cavity may be discontinuous at time t = t₀. If the applied radial stress is continuous while its time derivative is not, the discontinuity at (r₀, t₀) propagates into r >r₀ along the characteristics and/or the elastic-plastic boundaries. If the applied radial stress itself is discontinuous, the discontinuity in stress may propagate into r >r₀ in the form of a shock wave, or a pseudo centered simple wave, or a combination of both. This is a systematic study on the nature of solutions in the neighborhood of (r₀, t₀) for all possible combinations of discontinuous loadings applied at (r₀, t₀). The special cases of linear work-hardening and perfectly-plastic media are also discussed. Finally, the corresponding problem for materials obeying the Tresca yield condition is studied briefly.     

Dynamics of spiral waves driven by a dichotomous periodic signal

We study the effects of a dichotomous periodic force on meandering and rigidly rotating spiral waves. For a meandering state, the periodic forcing induces more modulating frequencies according to the rules of frequency-locked relations and linear combinations. It can also generate some unique closed tip orbits. On the modulating period T-axis, there exist all kinds of resonant entrainment bands. Arnold tongues exist in the period-amplitude space. The width of entrainment bands is affected by the symmetry of positive and negative parts in each signal unit. In addition, appropriate choices of T-value can be used to eliminate spiral waves. For a rigidly rotating state, the periodic forcing can induce a transition toward meandering spiral waves via generating a transitive bidirectional spiral wave. It is very interesting that, after the transition, the meandering spiral wave has the same primary rotating period as the free meandering states.     

Study on stress wave propagation in fractured rocks with fractal joint surfaces

This paper presents the experimental and theoretical investigation of property of stress wave propagation in jointed rocks by means of SHPB technique and fractal geometry method. Our aim focuses on the influence of the rough joint surface configuration on stress wave propagation. The comparison of behavior of reflection and transmission waves, deformation and energy dissipation of a rough joint surface characterized by its fractal feature with that of a smooth plane joint has been carried out. It has shown that the rough joint surface distinctly affects the stress wave propagation and energy dissipation in the jointed rocks. The rougher the joint surface was, the more permanent deformation occurred and the more attenuation stress wave took place as well. A nonlinear relationship between the normalized energy dissipation ratio W_J/W_I of the jointed rock and the joint roughness in terms of the fractal dimension has been formulated. It seems that the ratio W_J/W_I, presenting how much energy has been dissipated in the joint, nonlinearly increased with the increment of the fractal dimension D of the jointed surface. The ratio W_J/W_I of a roughly jointed rock, however, tends to be the same as that of a smoothly jointed rock if the fractal dimension is less than a critical value D_c = 2.20. The energy dissipation ratio at the critical point D_c seem to be a constant, not dependent of rock type but fractal joint configuration.     

On the energy conservation and critical velocities for the propagation of a ＂steady-shock＂ wave in a bar made of cellular material

The propagation of shock waves in a cellular bar is systematically studied in the framework of continuum solids by adopting two idealized material models, viz. the dynamic rigid, perfectly plastic, locking （D-R-PP-L） model and the dynamic rigid, linear hardening plastic, locking （D-R-LHP-L） model, both considering the effects of strain-rate on the material properties. The shock wave speed relevant to these two models is derived. Consider the case of a bar made of one of such material with initial length L 0 and initial velocity v i impinging onto a rigid target. The variations of the stress, strain, particle velocity, specific internal energy across the shock wave and the cease distance of shock wave are all determined analytically. In particular the ＂energy conservation condition＂ and the ＂kinematic existence condition＂ as proposed by Tan et al. （2005） is re-examined, showing that the ＂energy conservation condition＂ and the consequent ＂critical velocity＂, i.e. the shock can only be generated and sustained in R-PP-L bars when the impact velocity is above this critical velocity, is incorrect. Instead, with elastic deformation, strain-hardening and strain-rate sensitivity of the cellular materials being considered, it is appropriate to redefine a first and a second critical impact velocity for the existence and propagation of shock waves in cellular solids. Starting from the basic relations for shock wave propagating in D-R-LHP-L cellular materials, a new method for inversely determining the dynamic stress-strain curve for cellular materials is proposed. By using e.g. a combination of Taylor bar and Hopkinson pressure bar impact experimental technique, the dynamic stress-strain curve of aluminum foam could bedetermined. Finally, it is demonstrated that this new formulation of shock theory in this one-dimensional stress state can be generalized to shocks in a one-dimensional strain state, i.e. for the case of plate impact on cellular materials, by simply making proper replacements of the elastic and plastic constants.     

Generation of internal waves by a turbulent jet in a stratified fluid

The process of generation of internal waves by an initially cylindrical, turbulent jet with a Gaussian profile of the average horizontal velocity component in a fluid with stable linear density stratification is investigated by direct numerical simulation. It is shown that on time intervals Nt < 30, where N is the buoyancy frequency, the vertical velocity pulsations collapse, which is accompanied by the generation of internal waves whose spatial period is close to the wavelength of the spiral mode of jet instability in a homogeneous fluid. The wave dynamics and kinematics can be satisfactorily described by the linear theory for a pulsed source and their parameters are in good agreement with the parameters of the “coherent” internal waves generated by a stratified wake in a laboratory experiment. At large times the wave generation ceases and the variations of the fluid density are localized in the neighborhood of the centers of large-scale vortices formed in the horizontal plane in the neighborhood of the jet.     

Transient radiation by a submerged fluid-filled cylindrical shell

The radiation by a submerged fluid-filled cylindrical shell in response to a transient external pressure pulse is considered, and a semi-analytical model based on the Reissner–Mindlin shell theory is employed to simulate the interaction numerically. Two types of radiated waves that have been previously seen in experimental images for a submerged evacuated cylindrical shell are observed in both the external and internal fluids, the symmetric Lamb waves S₀ and the antisymmetric Lamb (or pseudo-Rayleigh) waves A₀. The third type of radiated waves is also observed that has not been explicitly imaged either experimentally or numerically for a submerged evacuated cylindrical shell, and it is demonstrated that these waves are the Scholte–Stoneley waves A. The effect that the complex structure of the radiated field has on the wave phenomena in the internal fluid is analyzed for shells of several different thicknesses, and the results of this analysis are summarized in the form of diagrams suitable for the use at the pre-design stage.     

Surface waves in an exponentially graded, general anisotropic elastic material under the influence of gravity

In a recent paper Destrade [1] studied surface waves in an exponentially graded orthotropic elastic material. He showed that the quartic equation for the Stroh eigenvalue p is, after properly modified, a quadratic equation in p² with real coefficients. He also showed that the displacement and the stress decay at different rates with the depth x₂ of the half-space. Vinh and Seriani [2] considered the same problem and added the influence of gravity on surface waves. In this paper we generalize the problem to exponentially graded general anisotropic elastic materials. We prove that the coefficients of the sextic equation for p remain real and that the different decay rates for the displacement and the stress hold also for general anisotropic materials. A surface wave exists in the graded material under the influence of gravity if a surface wave can propagate in the homogeneous material without the influence of gravity in which the material parameters are taken at the surface of the graded half-space. As the wave number k → ∞, the surface wave speed approaches the surface wave speed for the homogeneous material. A new matrix differential equation for surface waves in an arbitrarily graded anisotropic elastic material under the influence of gravity is presented. Finally we discuss the existence of one-component surface waves in the exponentially graded anisotropic elastic material with or without the influence of gravity.     

Analogy between soap film and gas dynamics. II. Experiments on one-dimensional motion of shock waves in soap films

This paper presents an experimental investigation of one-dimensional moving shock waves in vertical soap films. The shock waves were generated by bursting the films with a perforating spark. Images of propagating shock waves and small disturbances were recorded using a fast line scan CCD camera. An aureole and a shock wave preceding the rim of the expanding hole were clearly observed. These images are similar to the x-t diagrams in gas dynamics and give the velocities of shock and sound waves. The moving shock waves cause jumps in thickness. The variations of the induced Mach number, M₂ and the ratio of film thickness across the shock wave, δ ₂/δ ₁, are plotted versus the shock Mach number, M _s. Both results suggest that soap films are analogous to compressible gases with a specific heat ratio of γ≅1.0. Published online: 15 October 2002     

Wave jumps in media described by a modified Schrödinger equation

The nonlinear Schrödinger equationA _t ±iA _xx+i‖A‖² A=0 describes an envelope of periodic waves with slowly varying parameters on water, in plasmas, and in nonlinear optics [1]. This equation can also be applied to steady periodic waves (the wave amplitude and wave number do not depend on time, the variablest andx are replaced by the variables of a horizontal coordinate system on the surface of the fluid [2]). In the present paper the properties of a modified Schrödinger equation involving the third and higher derivatives are studied. Solutions describing transition regions between uniform wave states are obtained numerically. If the structure of the transition region whose extent increases with time is not considered, these solutions may be interpreted as jumps.     

Spectral analysis of the Pekeris operator in the theory of acoustic wave propagation in shallow water

ThePekeris differential operator is defined by $$Au = - c^2 (x_n )\rho (x_n )\nabla \cdot \left( {\frac{1}{{\rho (x_n )}}\nabla u} \right),$$ wherex=(x ₁,x ₂,...x _n )∈R _n ,?=(?/?x ₁, ?/?x ₂,...?/?x _n ), and the functionsc(x _n),σ(x _n) satisfy $$c(x_n ) = \left\{ \begin{gathered} c_1 , 0 \leqq x_n< h, \hfill \\ c_2 , x_n \geqq h, \hfill \\ \end{gathered} \right.$$ and $$\rho (x_n ) = \left\{ \begin{gathered} \rho _1 , 0 \leqq x_n< h, \hfill \\ \rho _2 , x_n \geqq h, \hfill \\ \end{gathered} \right.$$ wherec ₁,c ₂,? ₁,? ₂, andh are positive constants. The operator arises in the study of acoustic wave propagation in a layer of water having sound speedc ₁ and density? ₁ which overlays a bottom having sound speedc ₂ and density? ₂. In this paper it is shown that the operatorA, acting on a class of functions u (x) which are defined for x_n≧0 and vanish for x_n=0, defines a selfadjoint operator on the Hilbert space whereR ₊ ⁿ ={x∈R ⁿ :x _n >0} anddx =dx ₁ dx ₂...dx _n denotes Lebesgue measure in R ₊ ⁿ . The spectral family ofA is constructed and the spectrum is shown to be continuous. Moreover an eigenfunction expansion for A is given in terms of a family of improper eigenfunctions. Whenc ₁≧c ₂ each eigenfunction can be interpreted as a plane wave plus a reflected wave. When c₁< c₂, additional eigen-functions arise which can be interpreted as plane waves that are trapped in the layer 0n h by total reflection at the interface x_n=h.     

Non-linear wave modulation in a prestressed viscoelastic thin tube filled with an inviscid fluid

In the present work, the propagation of weakly non-linear waves in a prestressed thin viscoelastic tube filled with an incompressible inviscid fluid is studied. Considering that the arteries are initially subjected to a large static transmural pressure P₀ and an axial stretch λ_z and, in the course of blood flow, a finite time-dependent displacement is added to this initial field, the governing non-linear equation of motion in the radial direction is obtained. Using the reductive perturbation technique, the propagation of weakly non-linear, dispersive and dissipative waves is examined and the evolution equations are obtained. Utilizing the same set of governing equations the amplitude modulation of weakly non-linear and dissipative but strongly dispersive waves is examined. The localized travelling wave solution to these field equations are also given.     

A gaseous shock wave theory of the galactic spiral structure

In this paper, the galactic spiral structure is studied by the galactic shock wave of interstellar gas with self-gravitation. The perturbed gravitation of stars is not a necessary condition for the existence of such shock. It is proved first of all that there exists solution of local shock wave even if the perturbed gravitation is absent. The condition |₀|> is required for such solution. The spiral structure can only be explained by the shock solution when the difference of density between the regions of arm and interarm is larger. The grand design of shock wave with self-gravitation is obtained by the iterative method. The features of shock wave can be analyzed qualitatively in the velocity plane for a special perturbed gravitation which is used to simulate the self-gravitation of interstellar gas. As the mass distribution in proto- galactic disk is irregular initially, the grand design of the galactic shock wave was developed by the processes of winding, growth of instability and overlapping of waves. Hence, it gives a complete figure about the origin, evolution and persistance. A lot of observed phenomena and classificational features of the galactic spiral structure can be explained by adopting these ideas.     

Chaotic diffusion in steady wavy vortex flow—Dependence on wave state and correlation with Eulerian symmetry measures

The dimensionless effective axial diffusion coefficient, D_z, calculated from particle trajectories in steady wavy vortex flow in a narrow gap Taylor-Couette system, has been determined as a function of Reynolds number (R=Re/Re_c), axial wavelength (λ_z), and the number of azimuthal waves (m). Two regimes of Reynolds number were found: (i) when R<3.5, D_z has a complex and sometimes multi-modal dependence on Reynolds number; (ii) when R>3.5, D_z decreases monotonically.Eulerian quantities measuring the departure from rotational symmetry, ?_θ, and flexion-free flow, ?_ν, were calculated. The space-averaged quantities and were found to have, unlike D_z, a simple unimodal dependence on R. In the low R regime the correlation between D_z and ?_θ?_ν was complicated and was attributed to variations in the spatial distribution of the wavy disturbance occurring in this range of R. In the large R regime, however, the correlation simplified to for all wave states, and this was attributed to the growth of an integrable vortex core and the concentration of the wavy disturbance into narrow regions near the outflow and inflow jets.A reservoir model of a wavy vortex was used to determine the rate of escape across the outflow and inflow boundaries, the size of the ‘escape basins’ (associated with escape across the outflow and inflow boundaries), and the size of the trapping region in the vortex core. In the low R regime after the breakup of all KAM tori, the outflow basin (γ_O) is larger than the inflow basin (γ_I), and both γ_O and γ_I are (approximately) independent of R. In the large R regime, with increasing Reynolds number the trapping region grows, the outflow basin decreases, and the inflow basin shows a slight increase. This implies that the growth of the integrable core occurs at the expense of the outflow escape basin. Finally, it is shown that the variation of the weighted escape rates (γ_Or_O,γ_Ir_I) with Reynolds number was in excellent qualitative agreement with the variation of .     

A proof of the Benjamin-Feir instability

The existence and linear stability problem for the Stokes periodic wavetrain on fluids of finite depth is formulated in terms of the spatial and temporal Hamiltonian structure of the water-wave problem. A proof, within the Hamiltonian framework, of instability of the Stokes periodic wavetrain is presented. A Hamiltonian center-manifold analysis reduces the linear stability problem to an ordinary differential eigenvalue problem on ℝ⁴. A projection of the reduced stability problem onto the tangent space of the 2-manifold of periodic Stokes waves is used to prove the existence of a dispersion relation Λ(λ,σ, I ₁, I ₂)=0 where λ ε ℂ is the stability exponent for the Stokes wave with amplitude I ₁ and mass flux I ₂ and σ is the “sideband’ or spatial exponent. A rigorous analysis of the dispersion relation proves the result, first discovered in the 1960's, that the Stokes gravity wavetrain of sufficiently small amplitude is unstable for F ε (0,F₀) where F ₀ ≈ 0.8 and F is the Froude number.