How do neurons limit their input current?
Current is Voltage (driving potential) times conductance.  As the membrane potential approaches the Nernst potential of the conductance, the current approaches 0.  Conductances can be turned on or off through receptor binding, but there is no such thing as voltage-independent current injection (at least under physiologically relevant conditions).  

Indeed, it is the case that if you artificially inject large amplitudes of current into neurons, you observe voltage traces similar to your simulations.  This will very quickly lead to cell death.