Those ringing oscillations start at the same time your pulse starts a positive excursion.
Then the ringing fades. Characteristic of an LC tank circuit.
Do you have any coils in your power supplies? In the waveform generator?
How are you producing 0.6V at right? Is it by means of a high impedance?
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There's another possibility. Your positive pulse begins when the op amp sends positive bias to the 2N2222. However notice that no bias current flows. The bias current meets very high resistance at the gate of the mosfet. And the 2N3906 is turned off at that moment.
However it works somewhat (according to your plot) in the sense that the 2N2222 manages to turn on the mosfet, even though the 2N2222 is being operated without current flowing through it.
Then I also notice you put a load in the source leg of the mosfet. With current going through the load, it generates a voltage across it.
Whatever that voltage is, it has the effect of reducing net voltage at the mosfet's gate. (Because an N-mosfet has its gate V referenced to the source terminal.) This has a feedback effect back to the 2N2222, which in turn has a feedback effect to the op amp and its internal transistor.
I think the oscillations result from these feedbacks causing a bouncing effect (ringing) in the biasing current (or should it be called bias voltage?).
I think you can stop the ringing if you install a 1k to 100k ohm resistor from the mosfet gate to zero ground. That should give you more stable operation as the 2N2222 starts to respond to the input pulse.