Alternating ON and OFF with logic gates in a simulator
One way could be to build your own CMOS using a PMOS and an NMOS. Look up CMOS logic or something like that to get the gist. 

If you wanna do it your way, i'd hook the gates up to the same current source - one with the inverter, one without. Please note that I'm not at all familiar with the simulation environment you're using.