Asymptotic Analysis of Buffered Calcium Diffusion near a Point Source

The domain calcium (Ca2+) concentration near an open Ca2+ channel can be mod- eled as buffered diffusion from a point source. The concentration profiles can be well approximated by hemispherically symmetric steady-state solutions to a system of reaction-diffusion equations. After nondimensionalizing these equations and scaling space so that both reaction terms and the source amplitude are 0(1), we identify two dimensionless parameters, Cc and Eb, that correspond to the diffusion coefficients of dimensionless Ca2+ and buffer, respectively. Using perturbation methods, we derive approximations for the Ca2+ and buffer profiles in three asymptotic limits: (1) an excess buffer approximation (EBA), where the mobility of buffer exceeds that of Ca2+ (Eb \u3e Ec) and the fast diffusion of buffer toward the Ca2+ channel prevents buffer saturation (cf. Neher [Calcium Electrogenesis and Neuronal Functioning, Exp. Brain Res. 14, Springer-Verlag, Berlin, 1986, pp. 80-96]); (2) a rapid buffer approximation (RBA), where the diffusive time-scale for Ca2+ and buffer are comparable, but slow compared to reaction (ec \u3c\u3c 1, Eb reaction (ec \u3c\u3c 1, Eb reaction (ec \u3c\u3c 1, E

Lattice Boltzmann equation models for migration of ions in brain and their applications

The brain is divided into two parts by cellular membranes: intracellular space (ICS) and extracellular space (ECS). The brain-cell microenvironment is usually identified with the ECS. The structure of the brain resembles a porous medium. The objective of this research has been to develop quantitative methods for the study of the migration of ions in the brain, including movement between the ICS and the ECS. In the brain-cell microenvironment, the movement of ions such as tetramethylammo-nium (TMA) and tetraethylammonium (TEA) is by diffusion when there is neither any electrical activity in the cells nor an externally applied electric field. The diffusion process is constrained by the geometrical factors of the medium, especially tortuosity and volume fraction. The tortuosity and the volume fraction are lumped parameters that incorporate geometrical properties such as connectivity and pore size. It is difficult to study the effects of the geometrical properties on the tortuosity and the volume fraction by using conventional methods. Therefore, we build a lattice cellular automata (LCA) model for ion diffusion within the brain-cell microenvironment and perform numerical simulations on this model by using the corresponding lattice Boltzmann equation (LBE). In the model, particle injection is introduced to match the experimental situation of ion injection through a microelectrode. As in porous media theory, the LBE model can accurately describe ion diffusion in the ECS of brain tissue. As an application of the model, we combine the results from the simulations with porous media theory to compute tortuosities and volume fractions for various regular and irregular porous media, and a possible relationship between the volume fraction and the tortuosity also is investigated. The correlation of the results for the relationships between the tortuosity and the volume fraction for various porous media with experimental results on brain tissues suggests that the small change of the tortuosity during ischemia, hypoxia, and postnatal development is due to the small change of the basic geometrical properties of the brain, whereas the large change of the tortuosity after x-irradiation is due to the change of the geometrical properties as a result of cell death and cell damage caused by the x-irradiation. In the brain, potassium dynamics is constrained not only by extracellular diffusion, but also by intracellular diffusion and by active and passive transport of ions across the cell membrane. The movement of electrically charged potassium ions also is subject to electrical gradients and the spatial buffering mechanism. In addition, the geometrical factors of the brain-cell microenvironment can impose constraints on the diffusion process. It is difficult to study such a complex system using conventional methods. Therefore, we build an LBE model for this system. The evolution of the system via this model consists of three successive operations: particle injection, collision, and propagation. Those mechanisms affecting the movement of potassium are incorporated into the model by suitable choices of the injection and the collision operations, while the geometrical factors such as tortuosity and volume fraction are incorporated into the model by a suitable choice of the brain tissue as a porous medium based on our previous results for tortuosity and volume fraction. Numerical simulations on this model are performed, and the numerical results on the artificial brain as a porous medium reproduce qualitatively the behavior of potassium ions obtained from experiments with brain tissue. As applications of the model, we study the effects of each specific mechanism on clearance of potassium within the ECS. We found that both active and passive transport of ions across the membrane affect the dispersal of injected potassium ions. However, active transport plays a more important role than the passive transport. With a very brief injection, the difference between their effects is not as large as that with a prolonged continuous injection. The geometrical factors of the media also affect the movement of potassium. Irregularly shaped media slow down the movement of potassium since higher tortuosity makes it more difficult for the ions to move. Larger volume fraction makes the accumulated [K+]₀ disperse faster. This result suggests that age-related potassium clearance is perhaps due to age-related brain geometry changes. The clearance of the accumulated extracellular potassium might depend on the specific animal and region we are studying, and some pathological conditions such as hypoxia and ischemia might affect the clearance of the accumulated ECS potassium and affect the time needed for restoring the accumulated ECS potassium to its resting level. The results also imply that young animals disperse the accumulated [K+]₀ faster and consequently might prevent hypoxia, seizure, and spreading depression more efficiently than adults. Further, the above LBE model is extended to model the migration of elevated potassium through brain tissue when there is an electric current flow. The flux is caused mainly by the iontophoretic potassium injection; the current flow is due partially to a voltage gradient through the tissue.Science, Faculty ofMathematics, Department ofGraduat

Analysis of a micro piezoelectric vibration energy harvester by nonlocal elasticity theory

A theoretical model of a micro piezoelectric energy harvester is proposed based on the nonlocal elasticity theory, which is operated in the flexural mode for scavenging ambient vibration energy. A nonlocal scale is defined as the product of internal characteristic length and a constant related to the material. The dependences of performance of the harvester upon the nonlocal scale and the scale ratio of the nonlocal scale to the external characteristic parameter are investigated in detail. Numerical results show that output power of the harvester decreases, and resonance frequency reduces gradually at first then increases rapidly when nonlocal scale increases. The results of nonlocal elasticity theory are compared with that of classic beam theory. All the results are helpful for material and structure design of the micro piezoelectric energy harvester

Multiple low-frequency broad band gaps generated by a phononic crystal of periodic circular cavity sandwich plates

International audienceWe propose a new type of phononic crystal (PnC) composed of a periodic alternation of circular cavity sandwich plates. In the low-frequency regime, the crystal can modulate the propagation of flexural waves. Governing equations are deduced basing on the classical theory of coupled extensional and flexural vibrations of plates. The dispersion relation of the infinite PnC is calculated by combining the transfer matrix method with Bloch theory. The dynamic response of the PnC with finite unit cells is further studied with finite element analysis. An experiment is carried out to demonstrate the performance of the PnC in vibration isolation. Numerical results and experimental results both illustrate that the proposed PnC can generate several wide low-frequency Bragg band gaps providing strong attenuation. The dependence of band gaps upon geometric and material parameters is also analyzed in detail in view of vibration isolation applications

An Enhanced Plane Wave Expansion Method to Solve Piezoelectric Phononic Crystal with Resonant Shunting Circuits

An enhanced plane wave expansion (PWE) method is proposed to solve piezoelectric phononic crystal (PPC) connected with resonant shunting circuits (PPC-C), which is named as PWE-PPC-C. The resonant shunting circuits can not only bring about the locally resonant (LR) band gap for the PPC-C but also conveniently tune frequency and bandwidth of band gaps through adjusting circuit parameters. However, thus far, more than one-dimensional PPC-C has been studied just by Finite Element method. Compared with other methods, the PWE has great advantages in solving more than one-dimensional PC as well as various lattice types. Nevertheless, the conventional PWE cannot accurately solve coupling between the structure and resonant shunting circuits of the PPC-C since only taking one-way coupling from displacements to electrical parameters into consideration. A two-dimensional PPC-C model of orthorhombic lattice is established to demonstrate the whole solving process of PWE-PPC-C. The PWE-PPC-C method is validated by Transfer Matrix method as well as Finite Element method. The dependence of band gaps on circuit parameters has been investigated in detail by PWE-PPC-C. Its advantage in solving various lattice types is further illustrated by calculating the PPC-C of triangular and hexagonal lattices, respectively

Two methods to broaden bandwidth of a nonlinear piezoelectric bimorph power harvester

International audienceWe propose two methods to broaden the operation bandwidth of a nonlinear pinned-pinned piezoelectric bimorph power harvester. The energy-scavenging structure consists of a properly poled and electroded flexible bimorph with a metallic layer in the middle, and is subjected to flexural vibration. Nonlinear effects at large deformations near resonance are considered by taking the in-plane extension of the bimorph into account. The resulting output powers are multi-valued and exhibit jump phenomena. Two methods to broaden the operation bandwidth are proposed: The first method is to extend the operation frequency to the left single-valued region through optimal design. The second method is to excite optimal initial conditions with a voltage source. Larger output powers in the multi-valued region of the nonlinear harvester are obtained. Hence the operation bandwidth is broadened from the left single-valued region to the whole multi-valued region

Genomic evolution of 11 type strains within family Planctomycetaceae.

The species in family Planctomycetaceae are ideal groups for investigating the origin of eukaryotes. Their cells are divided by a lipidic intracytoplasmic membrane and they share a number of eukaryote-like molecular characteristics. However, their genomic structures, potential abilities, and evolutionary status are still unknown. In this study, we searched for common protein families and a core genome/pan genome based on 11 sequenced species in family Planctomycetaceae. Then, we constructed phylogenetic tree based on their 832 common protein families. We also annotated the 11 genomes using the Clusters of Orthologous Groups database. Moreover, we predicted and reconstructed their core/pan metabolic pathways using the KEGG (Kyoto Encyclopedia of Genes and Genomes) orthology system. Subsequently, we identified genomic islands (GIs) and structural variations (SVs) among the five complete genomes and we specifically investigated the integration of two Planctomycetaceae plasmids in all 11 genomes. The results indicate that Planctomycetaceae species share diverse genomic variations and unique genomic characteristics, as well as have huge potential for human applications