I don't know how Toyota's systems work, because Toyota doesn't sell diesel engines in North America. I've read some of the VW technical papers on how their system works, so that's the one I'm explaining. But, from what I can tell, all of these systems are more or less similar in concept of operation. The main differences are the type of de-NOx catalyst (VW Jetta uses a lean de-NOx catalyst, others use SCR with AdBlue fluid.)
The temperature inside the DPF catalyst itself is well below the NOx formation threshold, even during regeneration, if *that* is what you are asking.
Raising the exhaust temperature to initiate DPF regeneration is normally done by a combination of retarding injection timing (tends towards less NOx) and injecting fuel late in the power stroke which ends up being oxidized in the oxidizing catalyst, raising its temperature - but again, the oxidizing catalyst operates well below the NOx formation threshold, even during regeneration. The predominant effect is increased fuel consumption during this mode, not necessarily higher NOx emissions.
The VW system has several different regeneration modes, because each of the DPF, the de-NOx catalyst, and the de-H2S catalyst all require regeneration by different methods and different frequencies. The de-NOx catalyst is regenerated every few minutes by running the engine at stoichiometric (unusually rich for a diesel engine) which feeds the de-NOx catalyst a small amount of hydrocarbons to reduce the stored NOx. (This is done by temporarily increasing the EGR rate through the low pressure EGR system.) The DPF is regenerated every few hundred kilometers by raising the exhaust temperature in lean-burn mode. I don't recall how the de-H2S catalyst is regenerated, but it is at an even lower frequency.