Look at the circuit again - the NPN transistor's collector is always at high voltage whether its on or off - it can't saturate.
The proposed circuit is based on the following assumptions:
The collector of Q2 is feeding the anode of a Nixie tube. One of the Nixie's cathodes goes to approximately zero volts to light up a particular digit on that tube.
For purposes of analysis, I assume that the Nixie tube is a Zener diode with a voltage of 135 volts. (I know, I know, that's not what a real Nixie tube looks like, but the tube I am recalling for this example had a voltage of about 135 volts when it was illuminated, and at 135 volts the current was about
2 mA and it "looked good.") A real Nixie has a "soft knee" in its Voltage/Current characteristics at the point it starts to illuminate and it has varying dynamic resistance (and capacitance and maybe some other stuff) that I didn't try to model since I am only looking at DC operating points for the two logic levels out of the Arduino.
Here are the back-of-the napkin calculations that led me to the values that I showed:
With a logic 1 from the Arduino, the base of Q1 gets something close to half a milliamp of current. With the values shown for R2 (and R3), for Hfe values greater than 10 or so, Q1 will be saturated, and the collector of Q1 goes to Vcesat, a few tenths of a volt. Note that the voltage across R3 will not be more than a few tenths of a volt with this circuit, since it is across the forward-biased base-emitter junction of Q2.
With Q1 saturated, the base of Q2 is driven by the voltage divider consisting of R2 and R3 going between 200 Volts and (approximately) ground. With the values shown, this gives a base current in Q2 of something between 4 and 5 mA, so Q2 will be saturated, and almost all of the 200 volts is applied to the Nixie. The Nixie fires somewhere around 135 volts, and R4 limits the current to something a little less than 2 mA, which was the target. These calculations may be performed more precisely for any given Nixie, but if you are going to multiplex them you will more than likely have to determine the value of R4 experimentally to obtain the desired visual results anyhow. (Unless I had extremely detailed device characteristics and had done extremely precise calculations, I would usually start with circuit values that give something less than nominal current to the device and work my way up.)
With logic 0 from the arduino, Q1 does not conduct. There is no current (other than the very small leakage current---less than 0.1 microamps) through Q1, so there's essentially no current through R2. That means that the base of Q2 is pulled up to 200 Volts by R3 (0.1 uA through R3 is not enough to bias Q2 into any kind of meaningful conduction). Since the base of Q2 is at essentially the same potential as its emitter, Q2 does not conduct, and there is no current through the Nixie.
I hate to repeat myself, but I haven't actually used this to do Nixie multiplexing. (I have used circuits like this to switch higher voltages in other applications.) My hope was (and is) that the stuff that I posted may be useful as a starting point using cheap and readily available semiconductors in what I think is a practical circuit. There are other ways...[/begin Important edit]
Resistor R2 in the diagram will dissipate more than a Watt when the Arduino applies logic 1. I won't go back and change it more appropriate values, but it can definitely be improved. And it really should be improved --- see my quick take on a (possibly) better starting point in my post a couple of replies down from this.
I regret the "little lapse" in my analysis.
Thanks to MarkT for pointing this out![/end Important edit]
The transistors dissipate very little power when they are cut off or when they are saturated. Now, a Nixie tube fires and extinguishes rather slowly (compared to transistor switching time), and the transistors may dissipate a little during transitions. For multiplexing a reasonable number of things that people are going to look at (each device visually "refreshed" a few tens of times a second), the multiplexing frequency will be low enough that I wouldn't expect any anode's Q2 would spend enough time in the transition region to cause concern. However (do I have to say it again?):