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Be- cause of the generally complicated mathematical treatment required by complex 130 G order doxepin overnight anxiety network. Berger equations doxepin 10 mg on-line anxiety symptoms psychology, the ﬁnal step will be the numerical resolution of these equations on a computer generic 10 mg doxepin with visa anxiety nervousness. We may observe that this resolution, based on the rigorous methods of numerical analysis, occurs only in the terminal phase of the modeling process. In the best case, these neurons correspond to a preliminary discretization of space. If each neuron is an elementary circuit, then discretization is done at each point in space where a given neuron exists. In contrast, with mathematical modeling, the resolution of equations is carried out in a continuous space, and discretization does not depend on the position of neurons, only on the mathematical constraints of resolution. However, as we will see, with more complicated models, only the mathematical approach is appropriate. Indeed, mathematical modeling does more than establish relationships between observables. In a correctly adapted representation, not only simpliﬁcations, but also a certain type of organization, a functional order, may appear. Determining the space, that is, the eigenvectors, in which a matrix is trans- formed into a diagonal matrix (in which only the diagonal numbers, the eigenvalues, are not null) puts the response of the system in a direct relationship to the input. By using this mathematical transformation, the state variables are kept distinct. There is a decoupling in the subsystems, each of them being represented by a single state vari- able. Similarly, when the matrix is reduced to diagonal blocks, several state variables describe the subsystem. As we see in this simple case, the new representation has led to new properties for the couplings between subsystems. More generally, with this type of representation we may obtain a new interpretation and discover new properties speciﬁc to the phe- nomenon observed. We have chosen hierarchical structural and functional representations, which pro- vide new laws for the functional organization of biological systems. Because the formulation calls for complex mathematical techniques, the equations have been grouped in appendices. Here, the interested reader will ﬁnd part of the mathematical reasoning behind the theory. Two kinds of neural networks, artiﬁcial and real, will be presented ﬁrst, followed by the theoretical framework. In the concluding section, we discuss the technique appropriate for neuromimetic circuits. Mathematical Modeling of Neuromimetic Circuits 131 What Is an Artiﬁcial Neural Network? The ﬁeld of artiﬁcial neural networks has been extensively developed in the past few years. Each artiﬁcial neuron is a mathematical entity possessing two properties: (1) the output Y is the sum of the inputs Xi, weighted by the synaptic e‰cacies mi; and (2) the variation of the synaptic e‰cacy is proportional to the input signal Xi and the output signal Y. In the case of a network of n neurons connected to a given neuron, these properties are mathematically represented by a nonlinear dynamic system: " Xn Y ¼ F miXi i¼1 ð7:1Þ dmi ¼ aiX Yi ; i ¼ 1;... The second equation of this system is known as the learning rule of the neural network. With a given connectivity between neurons, the problem is to determine the math- ematical properties of the network related to the learning and memorization of pat- terns. All these net- works possess speciﬁc mathematical properties that unfortunately do not correspond to biological reality. Another di‰culty arises from the nonlinearity of the mathematical systems and the impossibility of ﬁnding an analytical solution for a dynamic system involving synap- tic weighting. The true complexity of the problem will be readily appreciated when we consider that the artiﬁcial neuron and its corresponding network are extremely simple compared with the real neuron surrounded by nervous tissue. From the biological point of view, the complexity of the phenomena involved is es- sentially the same whether we consider a real, isolated neuron or a network of artiﬁ- cial neurons. This idea stimulated the search for a representation incorporating the properties of a real neural network (G. Over the past few years, much headway has been made in the mathematical description of a real biological 132 G. The hierarchical organization of biological structural units from the cellular to the organismal levels (cell organelles, nuclei, neurons, synapses, neural groups, nervous tissue, and cerebral organs), naturally suggested a hierarchical representa- tion of a system. However, the hierarchical aspect of the corresponding functional organization is far from evident. The novel three-dimensional representation of a biological system that one of us has proposed (G. Chauvet, 1996a), with axes for space scales, time scales, and structural units, allows visualization of the coupling between the structural and functional organizations. This representation is based es- sentially on the determination of the time scales of the dynamic systems describing physiological functions. This functional hierarchy is useful for determining the phys- iological functions associated with nervous structures. In the case of real neural net- works, there are at least two physiological functions: the propagation of membrane potential on a time scale on the order of milliseconds, and the modiﬁcation of synap- tic e‰cacy on a time scale on the order of seconds or even hours. Thus, the func- tional order has its origin in a functional hierarchy that is evidently a manifestation of molecular mechanisms. Typically, the artiﬁcial neural networks generally studied have several neuron layers. The structure-function rela- tionship is more evident in this representation than in any other one. The hierarchical network is fundamentally di¤erent and, in particular, possesses speciﬁc emergent properties, that is, properties that appear at a higher level in a new structure. An im- Inputs Outputs (behaviors) Sensory Hierarchical Motor neurons circuit neurons Figure 7. Properties emerge from a lower level and appear at a higher level inside a new structure. This new structure is called a functional unit if, and only if, it has a speciﬁc function. Mathematical Modeling of Neuromimetic Circuits 133 portant advantage of the hierarchical representation is that it o¤ers a rigorous ap- proach to the notion of a functional unit that may now be deﬁned as a structural unit with a speciﬁc function at a higher level of organization (G. The functional unit, possessing its own time scale, incorporates a new function that can be derived mathematically from the lower levels of organization in a biological system.    