We investigate transport properties of superconducting TiN films in the vicinity of the superconductor-insulator transition (SIT). We show that the current-voltage (I-V) characteristics are mirror-symmetric with respect to the SIT and can be switched to each other by the applied magnetic field. In both superconducting and insulating states, the low-temperature I-V characteristics have pronounced diode-like threshold character, demonstrating voltage/current jumps over several orders of magnitude at the corresponding critical current or threshold voltage. We have found that for both states, the theory developed for Josephson junction arrays offers a quantitative description of the experimental results. (C) 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4789510]