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The temperature dependence of the nonlinear current-voltage ($I$-$V$)characteristics in highly underdoped La$_{2-x}$Sr$_{x}$CuO$_{4}$ ($x=0.07$ and0.08) thick films has been studied in both zero and perpendicular magneticfields $H$. Power-law behavior of $V(I)$ is found for both $H=0$ and $H \neq0$. The critical current $I_{c}$ was extracted, and its temperature andmagnetic field dependences were studied in detail. TheBerezinskii-Kosterlitz-Thouless physics dominates the nonlinear $I$-$V$ nearthe superconducting transition at $H=0$, and it continues to contribute up to acharacteristic temperature $T_x(H)$. Nonlinear $I$-$V$ persists up to an evenhigher temperature $T_{h}(H)$ due to the depinning of vortices.

Current-voltage characteristics and vortex dynamics in highly underdoped La$_{2-x}$Sr$_{x}$CuO$_{4}$