Ivancevic_Applied-Diff-Geom

582 Applied Differential Geometry: A Modern Introduction(2) Excitatory and inhibitory unsupervised learning (ω)–dynamics (4.87–4.89) generated by random differential Hebbian learning process (4.90–4.92), defined respectively by contravariant synaptic tensor–field ω ij =ω ij (t) : M → T T M N im and covariant synaptic tensor–field ω ij = ω ij (t) :M → T ∗ T ∗ M, representing cross–sections of contravariant and covarianttensor bundles, respectively.The system equations are defined asẋ i = A i (q) + ω ij f j (y) − x i , (4.84)ẏ i = B i (p) + ω ij f j (x) − y i , (4.85)A i (q) = K q (q i − q i R), (4.86)B i (p) = K p (p R i − p i ), (4.87)˙ω ij = −ω ij + I ij (x, y), (4.88)˙ω ij = −ω ij + I ij (x, y), (4.89)I ij = f i (x) f j (y) + ˙ f i (x) ˙ f j (y) + σ ij , (4.90)I ij = f i (x) f j (y) + ˙ f i (x) ˙ f j (y) + σ ij , (4.91)u i = 1 2 (δ ij x i + y i ), (i, j = 1, . . . , N). (4.92)Here ω is a symmetric 2nd order synaptic tensor–field; I ij = I ij (x, y, σ)and I ij = I ij (x, y, σ) respectively denote contravariant–excitatory andcovariant–inhibitory random differential Hebbian innovation–functions withtensorial Gaussian noise σ (in both variances); fs and fs ˙ denote sigmoidactivation functions (f = tanh(.)) and corresponding signal velocities( f ˙ = 1 − f 2 ), respectively in both variances;A i (q) and B i (p) are contravariant–excitatory and covariant–inhibitoryneural inputs to granule and Purkinje cells, respectively; u i are the correctionsto the feedback–control 1−forms on the cerebellar FC–level.Nonlinear activation (x, y)−-dynamics (4.84–4.87), describes a two–phase biological neural oscillator field, in which excitatory neural fieldexcites inhibitory neural field, which itself reciprocally inhibits the excitatoryone. (x, y)−-dynamics represents a nonlinear extension of a linear,Lyapunov–stable, conservative, gradient system, defined in local neural coordinatesx i , y i ∈ V y on T ∗ M asẋ i = − ∂Φ∂y i= ω ij y j − x i ,ẏ i = − ∂Φ∂x i = ω ijx j − y i . (4.93)The gradient system (4.93) is derived from scalar, neuro-synaptic action