Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 595(2) Action, defined as a 1−cell:ACTION✲Initial(3) Locomotion, defined as a 2−cell:Sustain ❄✲◆✻✍MonitorLOCOMOTIONNow, each causal arrow in (4.110), say f : A → B, stands for a generic‘neuro–morphism’, representing a self–organized, oscillatory neurodynamicsystem. We define a generic neuro–morphism f to be a nonlinear tensor–field (x, y, ω)−-system (4.111–4.116), acting as a bidirectional associativememory machine on a ND Riemannian manifold M N of the human cortex.It is formed out of two distinct, yet nonlinearly–coupled neural subsystems:(1) Activation (x, y)−-dynamics (4.111–4.112), defined as an interplay ofan excitatory vector–field x i = x i (t) : M N → T M, representing across–section of the tangent bundle T M, and and an inhibitory 1−formy i = y i (t) : M N → T ∗ M, representing a cross–section of the cotangentbundle T ∗ M.(2) Excitatory and inhibitory unsupervised learning (ω)–dynamics (4.113–4.116) generated by random differential Hebbian learning process(4.115–4.116), defined respectively by contravariant synaptic tensor–field ω ij = ω ij (t) : M N → T T Mim N and covariant synaptic tensor–field ω ij = ω ij (t) : M N → T ∗ T ∗ M, representing cross–sections of contravariantand covariant tensor bundles, respectively.(x, y, ω)−-system is analytically defined as a set of N coupled neurodynamicequations:ẋ i = A i + ω ij f j (y) − x i , (4.111)ẏ i = B i + ω ij f j (x) − y i , (4.112)˙ω ij = −ω ij + I ij (x, y), (4.113)˙ω ij = −ω ij + I ij (x, y), (4.114)I ij = f i (x) f j (y) + ˙ f i (x) ˙ f j (y) + σ ij , (4.115)I ij = f i (x) f j (y) + ˙ f i (x) ˙ f j (y) + σ ij , (4.116)(i, j = 1, . . . , N).Here ω is a symmetric, second–order synaptic tensor–field; I ij = I ij (x, y, σ)and I ij = I ij (x, y, σ) respectively denote contravariant–excitatory and