Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1091by the index j). From (6.141), we can further calculate all thermodynamic–like and statistical properties of MD & CD (see e.g., [Feynman (1972)]), asfor example, transition entropy S = k B ln Z, etc.From cognitive perspective, our adaptive path integral (6.139) calculatesall (alternative) pathways of information flow during the transitionIntention −→ Action.In the language of transition–propagators, the integral over histories(6.139) can be decomposed into the product of propagators (i.e., Fredholmkernels or Green functions) corresponding to the cascade of the four motivationalphases (*)∫〈Action|Intention〉 paths = Σ dx F dx I dx M dx T K(F, I)K(I, M)K(M, T ),(6.142)satisfying the Schrödinger–like equation (see e.g., [Dirac (1982)])i ∂ t 〈Action|Intention〉 paths = H Action 〈Action|Intention〉 paths , (6.143)where H Action represents the Hamiltonian (total energy) function availableat the state of Action. Here our ‘golden rule’ is: the higher the H Action ,the lower the microscopic fatigue.In the connectionist language, our propagator expressions (6.142–6.143)represent activation dynamics, to which our Monitor process gives a kind ofbackpropagation feedback, a version of the basic supervised learning (6.136).Mechanisms of Decision–Making under Uncertainty. The basicquestion about our local decision making process, occurring underuncertainty at the intention formation faze F, is: Which alternative tochoose? (see [Roe et al. (2001); Grossberg (1982); Grossberg (1999);Grossberg (1988); Ashcraft (1994)]). In our path–integral language thisreads: Which path (alternative) should be given the highest probabilityweight w? Naturally, this problem is iteratively solved by the learning process(6.135–6.136), controlled by the MONIT OR feedback, which we termalgorithmic approach.In addition, here we analyze qualitative mechanics of the local decisionmaking process under uncertainty, as a heuristic approach. This qualitativeanalysis is based on the micro–level interpretation of the Newtonian–likeaction S[x], given by (6.137) and figuring both processing speed ẋ andLTM (i.e., the force–field ϕ(x), see next subsection). Here we considerthree different cases: