All the trajectories of an extended averaged Hebbian learning equation on the quantum state space are the e-geodesics

In this paper, two families of trajectories on the quantum state space (QSS) originating from a synaptic-neuron model and from quantum information geometry meet together. The extended averaged Hebbian learning equation (EAHLE) on the QSS developed by the author and Yuya (Far East Journal of Applied Mathematics, vol.47, pp.149-167, 2010) from a Hebbian synaptic-neuron model is studied from a quantum-information-geometric point of view. It is shown that all the trajectories of the EAHLE are the e-geodesics, the autoparallel curves with respect to the exponential-type parallel transport, on the QSS. As a secondary outcome, an explicit representation of solution of the averaged Hebbian learning equation, the origin of the EAHLE, is derived from that of the e-geodesics on the QSS.