Opera 3d Vector Fields

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The essential issue of electroshock therapy (ECT) is the activity of physical stimulus, i.e., the electric current, on the disturbed structures of the brain. ECT sessions--when chronically applied for evoking antidepressive effects--are responsible for the appearance of excessive incitement in the neuronal net in the brain tissue in a form of self-sustaining after-discharge (SSAD) (convulsive attack characteristic for ECT). The study presents the computer research on basic biophysical phenomena of electroshock therapy (flow of electric current in the structures of the head just before convulsive attack). Five-layer 3-D model of the head was created in OPERA-3D (Vector Fields Ltd., Oxford), general 3 dimensional issues solver. Geometrical dimensions and electrophysical properties of each layer correspond with natural properties. The model was subjected to the action of electric stimulation (parameters identical to those applied in clinical conditions). Analysis of the flow in particular layers revealed the crawling/spreading effect present not only in the scalp layer but also in the layer of cerebrospinal fluid. The effect is conditioned by "deeper situated" lesser conduction of electricity-respectively skull bones, brain tissue. Crawling effect is the reason why only 5-15% of the electricity applied on the surface of the head reaches the surface of the brain. Electro-stimulation examinations also showed that the values of the so called density of the current in layers of brain tissue balanced between 1-10 mA/mm2. The current parameters of ECT were effective in evoking subsequent convulsive attack and safe for the brain tissue. The model was subjected to the action of magnetic stimulation according to the parameters of neurologic technique of transcranial magnetic stimulation (TMS). ELECTRA module was used to solve wire-current issues. The examination showed more regular distribution of current vectors in all layers of the head. The density of cerebral cortex was 0.1-1 mA/mm2, confirming markedly lesser current charge than that observed during ECT. The problem of magnetic stimulation efficacy in irritating deep structures of the brain demands further studies.

The Laplacian occurs in many differential equations describing physical phenomena. Poisson's equation describes electric and gravitational potentials; the diffusion equation describes heat and fluid flow, the wave equation describes wave propagation, and the Schrödinger equation in quantum mechanics. In image processing and computer vision, the Laplacian operator has been used for various tasks, such as blob and edge detection. The Laplacian is the simplest elliptic operator and is at the core of Hodge theory as well as the results of de Rham cohomology.

The Laplace operator itself has a physical interpretation for non-equilibrium diffusion as the extent to which a point represents a source or sink of chemical concentration, in a sense made precise by the diffusion equation. This interpretation of the Laplacian is also explained by the following fact about averages.

In fact, the algebra of all scalar linear differential operators, with constant coefficients, that commute with all Euclidean transformations, is the polynomial algebra generated by the Laplace operator.

Another generalization of the Laplace operator that is available on pseudo-Riemannian manifolds uses the exterior derivative, in terms of which the "geometer's Laplacian" is expressed as Δ f = δ d f . {\displaystyle \Delta f=\delta df.}

Here δ is the codifferential, which can also be expressed in terms of the Hodge star and the exterior derivative. This operator differs in sign from the "analyst's Laplacian" defined above. More generally, the "Hodge" Laplacian is defined on differential forms α by Δ α = δ d α + d δ α . {\displaystyle \Delta \alpha =\delta d\alpha +d\delta \alpha .} 2b1af7f3a8