Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1039Note that p ij can be a function of variables such as i or j. For example, inthe 1D chain model, p ij is 1 if |i − j| = 1, else it is 0. The average over allnetworks can be expressed as〈 ⎡⎤exp ⎣ ∑ 〉∑i∆t¯x ik a ij g(x i,k−1 , x j,k−1 ) ⎦ =ikj[} ]∏p ij exp i∆t¯x ik g(x i,k−1 , x j,k−1 ) + 1 − p ij ,so we getij{ ∑k〈e −S 〉 = exp(−S 0 ) ∏ ij[ { ∑p ij expki∆t¯x ik g(x i,k−1 , x j,k−1 )}+ 1 − p ij],whereS 0 = ∑ ikσ 2 ∆t¯x 2 ik + i¯x ik {x ik − x i,k−1 − ∆tf i (x i,k−1 )}.2This expression can be applied to the dynamics of any complex networkmodel. [Ichinomiya (2005)] applied this model to analysis of the Kuramototransition in random sparse networks.6.3.8 Application: Dissipative Quantum Brain ModelThe conservative brain model was originally formulated within the frameworkof the quantum field theory (QFT) by [Ricciardi and Umezawa (1967)]and subsequently developed in [Stuart et al. (1978); Stuart et al. (1979);Jibu and Yasue (1995); Jibu et al (1996)]. The conservative brain model hasbeen recently extended to the dissipative quantum dynamics in the work ofG. Vitiello and collaborators [Vitiello (1995); Alfinito and Vitiello (2000);Pessa and Vitiello (1999); Vitiello (2001); Pessa and Vitiello (2003);Pessa and Vitiello (2004)].The canonical quantization procedure of a dissipative system requiresto include in the formalism also the system representing the environment(usually the heat bath) in which the system is embedded. One possible wayto do that is to depict the environment as the time–reversal image of thesystem [Celeghini et al. (1992)]: the environment is thus described as thedouble of the system in the time–reversed dynamics (the system image inthe mirror of time).Within the framework of dissipative QFT, the brain system is describedin terms of an infinite collection of damped harmonic oscillators A κ (the