B thoracic and 1st lumbar vertebrae, inferiorly by an B1-B9 Cell Groups (Serotonergic Cell imaginary transverse line through the 4th sacral Groups) segment, and laterally by a line tangential to the lateral border of the erector spinae and a line between the posterior superior iliac spine and the inferolateral corner Definition of the sacrum. B1-B9 is the original designation of nine seratonin- Etiology containing cell groups visualized in the brainstem by Causes the use of fluorescent histochemical methods. Some of ▶ these serotonergic groups project caudally to the spinal Low Back/Spine Pain cord and others rostrally to different parts of the forebrain. Backache Babinski Reflex (Babinski Response, Babinski Sign) ▶Low Back/Spine Pain Definition Reflex response elicited preferably by a blunt rodlike instrument stroked along the lateral footsole from heel Back-Propagation in Neurons to toes. In neurologically normal adult persons, the stimulus elicits a flexor of foot and toes. In infants and adult patients with lesions of the ▶pyramidal tract, the Definition toes are spread apart, the big toe slowly extends and the ▶Action Potential Propagation other toes flex. Lesions limited to the pyramidal tract produce a Babinski sign and paresis (i.e., negative symptoms such as temporary weakness and loss of dexterity), but neither spastic ▶dystonia nor permanent weakness. Back-Propagation Learning in ▶Pyramidal Tract Neural Networks Definition The backpropagation algorithm is one of the most Back Pain popular procedures for training multi-layer artificial neural networks. Neurons in all layers change their synaptic weights based on the gradient descent of the Definition sum of squared differences between the network output Pin perceived anywhere in a region bounded superiorly and the target vector, where this sum is propagated by an imaginary transverse line between the 12th backwards through the net to adjust weights deep within 332 Bacterial Artificial Chromosome (BAC) the network. This algorithm is capable of solving many a certain domain, of an extensive physical quantity. In practical problems that are nonlinearly separable. continuum mechanics, in particular, physical content is attributed only to a material body or a part thereof ▶Connectionism (as opposed to a field theory such as electromagnetism, ▶Neural Networks where a vacuum can be a carrier of fields). The time rate of change of the content of a physical quantity Ψ in the body is postulated to be entirely due to only two causes, a production Π within the body and a flux Φ through its boundary: DÉ Bacterial Artificial Chromosome (BAC) ¼ Å þ È: ð1Þ Dt To convert this general statement of balance into a more Definition specific form can be done in either the referential Bacterial artificial chromosome (BAC) is a DNA vector (“Lagrangian”) or spatial (“Eulerian”) setting. The total whose basis is a fertility plasmid (F-plasmid), which content can be represented in terms of the referential contains sites necessary for DNA to be handled and volume element dΩ or its spatial counterpart dω as replicated as a bacterial chromosome. BACs are used follows: for cloning in E. coli, and because their insert sizes can Z Z – be large (100 300 kb), have proven useful in the É ¼ É0 d ¼ É d!; ð2Þ identification of gene regulatory sequences that can ! involve huge pieces of DNA upstream of a gene. Ψ Ψ Human and yeast artificial chromosomes are also where 0 and represent the content density per unit available. referential and spatial volume, respectively. These two densities are related by: É0 ¼ JÉ; ð3Þ where J is the determinant of the deformation gradient F, as defined in the kinematics of deformation (q.v.). Bait Shyness Similarly, the production is assumed to be expressible as: Z Z Å ¼ ¼ !; ð Þ ▶Aversive Taste Memory 0 d d 4 ! with 0 ¼ J: ð5Þ Afluxisassumedtobedefinednotonlythroughthe Balance Laws external boundary of the body, but also through any internal boundary between sub-bodies. Moreover, the flux is assumed to be given by an integral (over the MARCELO EPSTEIN area of interest) of some density. But, unlike the case Schulich School of Engineering, University of Calgary, of the volume integrals, now it is not allowed to Calgary, AB, Canada simply claim that this density is a mere function of position on the surface. A simple example will clarify Definition this situation. If sun-tanning on the beach, it makes a The statement of the causes of change of content of an difference whether one is lying horizontally or extensive physical quantity (such as mass, energy, standing up. The heat energy flows at a maximum momentum, entropy and so on) over a certain domain. rate when the exposed surface is perpendicular to the The quantitative aspects of balance laws are expressed rays. This is as true for the outer boundary as for an in terms of balance equations. internal point. When considering an interior point of the endodermis, the flux at that point will vary Description of the Theory depending on the orientation of the imagined surface A balance equation in continuum physics is a rigorous element considered at the point. In other words, accounting of the causes for the change in content, over the flux density must depend not only on position but Balance Laws 333 also on some other characteristic of the boundary. The integral statement of the generic law of balance in The simplest assumption is that the flux depends Lagrangian form is finally: on the boundary only through its (local) normal, Z Z Z D namely through its orientation alone (as opposed É d ¼ d þ H Á N dA; ð9Þ B Dt 0 0 to its curvature, for example). Having made these @ assumptions: and in Eulerian form: Z Z Z Z Z È ¼ f ðX; N; Þ ¼ fðx; n; Þ ; ð Þ D 0 t dA t da 6 É d ¼ d þ h Á n da: ð10Þ Dt @ @! ! ! @! where ∂ stands for “boundary of ” and where the Although these statements may suffice for computational notation of kinematics of deformation (q.v.) is used. To applications (such as the “finite-volume” method), under obtain the relation between the referential and spatial suitable assumptions of smoothness their local (differen- f f flux densities ( 0 and , respectively), (4) of kinematics tial) forms can be obtained by using the divergence of deformation (q.v.) might be invoked. However, in theorem to convert the flux integral into a volume integral this equation the dependence on the normal is and invoking the fact that the resulting three integrals explicitly linear, whereas the assumed dependence must be identically equal regardless of the domain of of the flux densities on the normal is so far arbitrary. It integration within the body. The results are, therefore, would appear that the only way out of the impasse @É would be to assume such linearity (a logical thing to 0 ¼ 0 þ Div H; ð11Þ do, particularly considering the example of the sun @t rays, whose effect on the skin is governed by a simple and projection on the normal). It is a rather remarkable DÉ fact, however, that such an assumption is superfluous, þ É div v ¼ þ div h; ð12Þ since it can be derived from the general statement of Dt the balance law. This result is called Cauchy’s where the last equation necessitated the application of theorem, since it was Cauchy who, in a series of Reynolds’ transport theorem. The operators Div and div papers published in the 1820’s, first established its stand for the referential and spatial divergence, respectively. validity in the context of his theory of stress. The Applying the general prescription just obtained to the five proof is based in the so-called tetrahedron argument. fundamental quantities to be balanced in a traditional Assuming that the general balance law, (1), is valid, continuum mechanics treatment gives the following. with the same densities for any sub-body (thereby essentially ruling out scale effects such as surface Conservation of Mass tension) and applying it to a tetrahedron with three A balance law is said to be a conservation law if the faces aligned with the coordinate system and the production and the flux vanish identically. This is the fourth face with its vertices lying on the surface of case for the mass of a classical continuum. (In modern interest, the mean value theorem is applied to each theories of biological growth, however, or in theories of term of (1) and the far vertex of the tetrahedron is chemically reacting mixtures (when looking at each allowed to approach the surface. Observing that the component of the mixture), conservation of mass does two volume integrals approach zero with the cube of a not hold, and specific mass production and/or flux terms typical line dimension of the tetrahedron, while the are to be included). Denoting by ρ0 and ρ the referential surface integral does so with the square, thereby being and spatial mass densities, respectively, the Lagrangian the only “survivor” in the limit, the theorem follows. and Eulerian differential balances are obtained as: Therefore: @ Z Z 0 ¼ ; ð Þ @ 0 13 È ¼ HðX; tÞN dA ¼ hðx; tÞn da; ð7Þ t @ @! and H h D where and are the Lagrangian and Eulerian flux þ div v ¼ 0: ð14Þ vectors, respectively. If the quantity to be balanced is Dt vectorial in nature, these become flux tensors. The latter (Eulerian) version is known in hydrodynamics Invoking now (4) of kinematics of deformation (q.v.), as the continuity equation. the following relation between the Lagrangian and Eulerian flux densities is obtained: Balance of Linear Momentum This balance is a statement of Newton’s second law as À1 H ¼ J F h: ð8Þ applied to a deformable continuum.