Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 161where the three terms from the left to the right denote respectively thenew–update value, the old value and the innovation of the synaptic tensorW . In this case the nonlinear (usually sigmoid) innovation functions Φ αβand Φ αβ are defined by one of following four Hebbian models:Signal Hebbian learning, with innovation in both variance–forms:Φ αβ = S α (q α ) S β (p β ),Φ αβ = S α (q α ) S β (p β ); (3.25)Differential Hebbian learning, with innovation in both variance–forms:Φ αβ = S α (q α )S β (p β ) + Ṡα(q α )Ṡβ(p β ),Φ αβ = S α (q α )S β (p β ) + Ṡα (q α )Ṡβ (p β ), (3.26)where Ṡ−terms denote the so–called ‘signal velocities’ (for details see[Kosko (1992)]).Random signal Hebbian learning, with innovation in both variance–forms:Φ αβ = S α (q α ) S β (p β ) + n αβ ,Φ αβ = S α (q α ) S β (p β ) + n αβ , (3.27)where n αβ = {n αβ (t)}, n αβ = {n αβ (t)} respectively denote covariant andcontravariant additive, zero–mean, Gaussian white–noise processes independentof the main innovation signal; andRandom differential signal Hebbian learning, with innovation in bothvariance–forms:Φ αβ = S α (q α )S β (p β ) + Ṡα(q α )Ṡβ(p β ) + n αβ ,Φ αβ = S α (q α )S β (p β ) + Ṡα (q α )Ṡβ (p β ) + n αβ . (3.28)Total GBAM (q, p, W )−neurodynamics and biological interpretationTotal GBAM tensorial neurodynamics is defined as a union of the neuraloscillatory activation (q, p)−-dynamics (3.22) and the synaptic learning