Filaments are key for star formation models. As part of the study carried out by the Herschel GCC Programme, here we study the filament properties presented in GCC.VII in context with theoretical models of filament formation and evolution. A conservative sample of filaments at a distance D<500pc was extracted with the Getfilaments algorithm. Their physical structure was quantified according to two main components: the central (Gaussian) region (core component), and the power-law like region dominating the filament column density profile at larger radii (wing component). The properties and behaviour of these components relative to the total linear mass density of the filament and its environmental column density were compared with theoretical models describing the evolution of filaments under gravity-dominated conditions. The feasibility of a transition to supercritical state by accretion is dependent on the combined effect of filament intrinsic properties and environmental conditions. Reasonably self-gravitating (high Mline-core) filaments in dense environments (avsim3mag) can become supercritical in timescales of tsim1Myr by accreting mass at constant or decreasing width. The trend of increasing Mline-tot (Mline-core and Mline-wing), and ridge Av with background also indicates that the precursors of star-forming filaments evolve coevally with their environment. The simultaneous increase of environment and filament Av explains the association between dense environments and high Mline-core values, and argues against filaments remaining in constant single-pressure equilibrium states. The simultaneous growth of filament and background in locations with efficient mass assembly, predicted in numerical models of collapsing clouds, presents a suitable scenario for the fulfillment of the combined filament mass-environment criterium that is in quantitative agreement with Herschel observations.
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