A general structure of stimulus-response formula is presented to specify the interacted network dynamics in a day-to-day time scale under the assumption of a learning and adaptive behavioral process. By taking the time derivative of system variable as a reaction in stimulus-response equation the evolution of whole system is formulated as a dynamical system. Issue of stability for the proposed differential equations are briefly discussed. Approximation of time dependent route-choice model is derived from the addressed path flow dynamics. The threshold effect on path flow dynamics is developed in term of defining a discontinuous stimulus. Quasi user equilibrium is researched to be a steady state when all users feel indifferent between experienced travel time and predicted travel time provided by Advanced Traveler Information Systems. The proposed quasi user equilibrium is reduced to Wardrop's user equilibrium if threshold effect is vanishing.