Ivancevic_Applied-Diff-Geom

158 Applied Differential Geometry: A Modern Introductioning Kosko BAM and RBAM systems (see [Kosko (1992)]). Mathematically,the GBAM is a tensor–field system (q, p, W ) defined on a manifold M calledthe GBAM manifold. The system (q, p, W ) includes two nonlinearly coupled(yet non–chaotic and stable) subsystems (see Figure 3.4): (i) activation(q, p)−-dynamics, where q and p represent neuronal 1D tensor–fields, and(ii) self–organized learning W −-dynamics, where W is a symmetric synaptic2D tensor–field.Fig. 3.4Architecture of the GBAM neurodynamical classifier.GBAM Activation DynamicsThe GBAM–manifold M can be viewed as a Banach space with aC ∞ −smooth structure on it, so that in each local chart U open in M,an nD smooth coordinate system U α exists.GBAM–activation (q, p)−-dynamics, is defined as a system of two coupled,first–order oscillator tensor–fields, dual to each other, in a local Banachchart U α , α = 1, ..., n on M:1. An excitatory neural vector–field q α = q α (t) : M → T M, being across–section of the tangent bundle T M; and2. An inhibitory neural 1–form p α = p α (t) : M → T ∗ M, being across–section of the cotangent bundle T ∗ M.To start with conservative linear (q, p)–system, we postulate the GBAM