I was playing with oscillators today and I’m not sure what functionality recurrent_tau has. Regardless of the value chosen, it doesn’t seem to affect anything at all, which isn’t that surprising when you consider the derivation of the oscillator. Am I missing something? Has analysis been done on this?
Additionally, is this only the case with the low-pass filter synapse model? Is it possible to choose better synapses so that the variables of the synapses actually affect the operation of the oscillator?
Principle 3 is designed to “convert” the synapse into an integrator, regardless of the time-constant. Therefore, yes; if you are implementing it ideally (i.e., the same $\tau$ on both the incoming and recurrent connection), then the $\tau$ does not “affect anything at all”. There are some caveats though, which is that the correctness of Principle 3 assumes: (1) the filtered spiking model mimics its filtered rate counterpart and, (2) perfect decoding. In practice when we have a small number of spiking neurons, a longer $\tau$ makes these assumptions more reasonable (the noise becomes smoothed over time).
For your question about using other synapses, the lowpass is already ideal in theory (ignoring noise). Other synapses would only help if you were able to say that one filters the noise better than another (e.g., an Alpha(0.1) will filter high frequencies better than Lowpass(0.1)). But usually it’s easier to just increase $\tau$ until you get the ideal behaviour.