Exact solution of linear transport equations in fractal media—I. Renormalization analysis and general theory

Local energy on demand: Are 'spontaneous' astrocytic Ca2+-microdomains the regulatory unit for astrocyte-neuron metabolic cooperation?

Astrocytes are a neural cell type critically involved in maintaining brain energy homeostasis as well as signaling. Like neurons, astrocytes are a heterogeneous cell population. Cortical astrocytes show a complex morphology with a highly branched aborization and numerous fine processes ensheathing the synapses of neighboring neurons, and typically extend one process connecting to blood vessels. Recent studies employing genetically encoded fluorescent calcium (Ca²⁺) indicators have described 'spontaneous' localized Ca²⁺-transients in the astrocyte periphery that occur asynchronously, independently of signals in other parts of the cells, and that do not involve somatic Ca²⁺ transients; however, neither it is known whether these Ca²⁺-microdomains occur at or near neuronal synapses nor have their molecular basis nor downstream effector(s) been identified. In addition to Ca²⁺ microdomains, sodium (Na⁺) transients occur in astrocyte subdomains, too, most likely as a consequence of Na⁺ co-transport with the neurotransmitter glutamate, which also regulates mitochondrial movements locally - as do cytoplasmic Ca²⁺ levels. In this review, we cover various aspects of these local signaling events and discuss how structural and biophysical properties of astrocytes might foster such compartmentation. Astrocytes metabolically interact with neurons by providing energy substrates to active neurons. As a single astrocyte branch covers hundreds to thousands of synapses, it is tempting to speculate that these metabolic interactions could occur localized to specific subdomains of astrocytes, perhaps even at the level of small groups of synapses. We discuss how astrocytic metabolism might be regulated at this scale and which signals might contribute to its regulation. We speculate that the astrocytic structures that light up transiently as Ca²⁺-microdomains might be the functional units of astrocytes linking signaling and metabolic processes to adapt astrocytic function to local energy demands. The understanding of these local regulatory and metabolic interactions will be fundamental to fully appreciate the complexity of brain energy homeostasis as well as its failure in disease and may shed new light on the controversy about neuron-glia bi-directional signaling at the tripartite synapse.

A Mathematical Model of a Novel 3D Fractal-Inspired Piezoelectric Ultrasonic Transducer

Piezoelectric ultrasonic transducers have the potential to operate as both a sensor and as an actuator of ultrasonic waves. Currently, manufactured transducers operate effectively over narrow bandwidths as a result of their regular structures which incorporate a single length scale. To increase the operational bandwidth of these devices, consideration has been given in the literature to the implementation of designs which contain a range of length scales. In this paper, a mathematical model of a novel Sierpinski tetrix fractal-inspired transducer for sensor applications is presented. To accompany the growing body of research based on fractal-inspired transducers, this paper offers the first sensor design based on a three-dimensional fractal. The three-dimensional model reduces to an effective one-dimensional model by allowing for a number of assumptions of the propagating wave in the fractal lattice. The reception sensitivity of the sensor is investigated. Comparisons of reception force response (RFR) are performed between this novel design along with a previously investigated Sierpinski gasket-inspired device and standard Euclidean design. The results indicate that the proposed device surpasses traditional design sensors.

Description of the Convective Air-Drying of a Food Model by Means of the Fractal Theory

A slab-shaped model food prepared using glucose solutions and agar as jellifying agent was subjected to drying in an experimental drier. Drying kinetics and surface temperature (ST) distribution along drying were evaluated. When fractal analysis was applied to ST distributions it was possible to observe three stages: the first one, at the beginning of the process, was very short and could not be associated with a fractal dimension. The second one, by far the longest, had a constant value of the fractal dimension of the ST distribution and towards the end of the process, as temperature of the surface of the material tended to homogenise, a final linear stage was found which corresponded to equilibrium conditions. Images of the slab along drying were recorded and showed an increasing heterogeneous appearance as drying proceeds. Grey level intensity plots corresponding to these images also showed an increasing irregularity (higher values of fractal dimension) with drying time. Fractal analysis probed to be a useful tool for describing drying kinetics and for characterising images of samples subjected to dehydration.

Hydrodynamic dispersion in a hierarchical network with a power-law distribution of conductances

Dispersion is studied on a two-dimensional hierarchical pore network with a power-law distribution of conductances, i.e., P(g) approximately g(mu-1), with gepsilon(0,1), and mu is the disorderliness parameter (mu > 0). A procedure for computing tracer dispersion transport on hierarchical networks was developed. The results show that the effective diffusion coefficient of the network scales similarly as conduction on the same lattice. This means that the disorder length scales for conduction and diffusion processes are the same, and can be predicted from percolation theory. The dispersivity, xi identical with D(||)/U, was found to diverge rapidly as mu-->0. The result is in agreement with the model developed by Bouchaud and Georges (C.R. Acad. Sci. (Paris) 307 1431, 1988). A limiting value of mu approximately 0.45 was found, below which the convection-dispersion equation is no longer valid.

Evolution of fractal particles in systems with conserved order parameter

Computer simulations of the evolution of fractal aggregates in systems with conserved order parameter are described in this work. The aggregates are generated by diffusion-limited aggregation. This model describes such important processes as annealing of dendrite inclusions in solids, healing of cracks in ceramics, temperature-induced transformations in composites, relaxation of rough surfaces, aging of colloid particles, etc. It is shown that the evolution in fractal media differs significantly from that occurring in initially homogeneous systems and leads to different values of the scaling exponent. A relationship between the fractal dimension, mechanism of relaxation, and scaling exponent was also derived.