The experiments on some tokamaks are simulated, where a fast response near the plasma centre was observed after cooling (by impurity ablation) or heating (by current rampup) at the edge. The canonical profiles transport model (CPTM) is modified for the simulations. The existing equations describing a slow relaxation of the real profile to the canonical profile are complemented by equations describing a fast evolution of the canonical profile. The problem of the canonical profile determination is linked to the transport set through the boundary conditions. Ohm's law and one of the Maxwell equations at the edge are used as boundary conditions for the canonical profile of μc (μ = 1/q, q is the safety factor). Therefore, a change of Te(r) profile near the edge leads to a redistribution of the μc(r) profile and to a jump in the electron and ion heat diffusivities over the whole plasma cross-section. The sign of the response (heating or cooling) is very sensitive to details of Te(r) and the evolution of its gradient at the edge. The model reasonably describes both the core heating in TFTR and TEXT, and the core cooling in JET.