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State space representation and global descriptors of brain electrical activity in Visual Studio .NET Print PDF 417 in Visual Studio .NET State space representation and global descriptors of brain electrical activity




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9 State space representation and global descriptors of brain electrical activity use .net vs 2010 qr bidimensional barcode generation toget qr-code in .net Basice Knowlege of iReport Structure of the EEG state space Vector space structure The elements of a stat e space defined on the basis of measurements are generally specified by m-tuples of real numbers. However, this numeric representation does not necessarily imply that the familiar arithmetic operations are meaningfully defined. For the EEG state space we may assume the algebraic structure of a linear vector space, for the following reasons: EEG measurements are voltages, i.

e. differences within the electric potential distribution generated by the brain (see 1). Voltages generated by the combination of several independent field sources add up, while a differently strong activation of the same field source leads to multiplication of voltages by a common factor.

These physical processes are captured by the vector space operations of (element-wise) addition of data vectors and multiplication by a real number. The elements u of an m-dimensional vector space can be represented with respect to a set of linearly independent basis vectors e 1 , e 2 , . .

. , e m by m-tuples (u1 , u2 , . .

. , um ) of real numbers such that u = u1 e 1 + u2 e 2 + + um e m . For EEG, the original data vectors already have this form, which can be interpreted as their representation in the standard basis e 1 = (1, 0, .

. . , 0), e 2 = (0, 1, .

. . , 0), etc.

. Metric structure The structure of the E EG state space can be further enriched by introducing a binary operation between vectors, called inner product, resulting in a real number; we choose it as the dot product of data vectors. u v :=. ui vi Based on this, a measu Visual Studio .NET QR re of the magnitude of a vector, usually called its norm, is defined as u := u u (2) The norm of a difference between two vectors, or equivalently, the distance between state space points, u and v, is then. u v = (ui vi )2. The distance in the st ate space thus naturally provides a single-valued measure of the global difference between momentary electrical brain states represented by data vectors u, v. A given vector u defines a direction in the state space; the set of all multiples au (where a > 0) constitutes a ray in the direction of u. Since the magnitude of a vector is irrelevant for the direction, it is meaningful to represent directions by the normalized vector u := u , u (for u = 0) (4).

whose magnitude is uni QR Code 2d barcode for .NET ty. Then, for two non-zero vectors u, v, the inner product u v = u v u v (5).

Electrical Neuroimaging attains a value fr .NET qr codes om 1 to +1. A number [0, ] such that cos = u v is the angle between the vectors u and v (or, generally, between their respective directions).

For two vectors pointing into directions perpendicular to each other, the inner product becomes u v = 0; those vectors are called orthogonal. In contrast to the vector space structure of EEG data, which is immediately motivated by the physical properties of EEG measurements, the metric structure introduced via the inner product has to be seen as a heuristic choice which is justified by its utility for the further analysis and characterization of data. In particular, normalization (Eqn 4) allows a separation of the shape and the strength of an electric field distribution, while the definition of angle and distance facilitates a comparison of the shape of different distributions.

An important aspect of this choice is the fact that the definition of norm and angle via the dot product of EEG data vectors implies a dependence on the reference underlying the voltage measurements. While it is true that the landscape of each single field distribution, as defined in terms of relative differences between measurement loci, is not altered by a change of the reference by an additive term, the geometry of the entire data-set with respect to the inner product is changed by the transformation. In the following we assume that the EEG data have been transformed to the average reference ( 2).

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