I will just toss out a random idea for you to play with, since you are just looking to play a bit. Keeping your PNP transistor (not changing it to NPN) and also keeping the \$1\:\text{V}\$ source as an idea here (I kind of see why you added that diode and I think it's not necessary if you use just one voltage supply), I'd suggest running this:
\$C_1\$ is initially uncharged (due to the UIC mentioned in the .TRAN card) and \$R_1\$ then supplies it some current as \$Q_1\$ stays on. Over time, \$C_1\$ gradually charges up and pinches off \$Q_1\$.
A slight improvement would be to include a resistor from \$V_1\$ to \$Q_1\$'s base, like this:
That provides a weak galvanic bypass across the emitter-base junction. In idealized circuit simulations, it may not make much of a difference. But if I were to build this, for real, I may likely include some large-valued \$R_3\$ there.
