Extended Data Fig. 6: Simulations supporting the PAC neuron selection approach.

We argue that a neuron that fires randomly with respect to theta phase and gamma power could still be selected as a PAC neuron if only the GLM interaction term between theta phase and gamma amplitude is considered. In addition to selecting neurons whose FRs are better explained by a model including an interaction term as compared to a model with no interaction term, we therefore introduced a second criterion by comparing the full model against a model that lacks the gamma amplitude term. The simulations presented here are meant to visualize our reasoning. In (a), we simulate theta (6 Hz) and gamma (80 Hz) signals, where gamma amplitudes perfectly couple to theta phase. This highly artificial LFP signal only serves to simplify visualization. We also include illustrations in (b) to (d) using an originally recorded LFP channel from our dataset (filtered between 3–7 Hz and 70–140 Hz) that shows strong levels of PAC. This is to show that the same arguments also hold for real data. For the purpose of these illustrations, we used an LFP signal of roughly 160 s length and simulated 300 spike timestamps (black ticks), of which 9 s are plotted. (a) In this simulation, we modelled random spike timestamps with respect to theta phase and gamma amplitude (upper panel). According to our GLM selection approach, we grouped spikes in 10 theta phase bins and 2 gamma amplitude bins and determined spike counts in each bin (lower panels). As can be seen from the histograms in the lower panels, the theta phase distribution of spike counts differs between low and high gamma amplitudes, resulting in a highly significant interaction term between theta phase and gamma amplitude. The reason for this is that gamma amplitude itself is already perfectly coupled to theta phase. Separating spikes into low and high gamma will therefore also result in different theta distributions among the two spike count groups. Thus, when testing a model that contains theta phase and gamma amplitudes as well as their interaction against a model without the interaction, spike counts will be highly significantly better explained by the full model, as was the case in this example (p < 0.001; see likelihood-ratio test results on the right). However, since the time stamps are random, we should not observe a difference in overall spike counts between low and high gamma amplitudes, which was also the case in this example (p = 0.98). Introducing such a gamma term comparison as a second selection criterion thus ensures that this simulated random neuron would not have been selected. In 1000 repetitions of this simulation, our approach would have selected only 1.8% of such randomly spiking neurons (see text on right side). (b) Similar to (a) but using a real LFP recording from our dataset that shows strong levels of PAC. 300 spike timestamps were again modelled randomly with respect to theta phase and gamma amplitude. Similar albeit weaker statistics were observed in these simulations. (c) Using the same LFP as in (b) but now simulating 300 spike timestamps that prefer high gamma amplitudes and a theta phase of 0 (i.e., PAC spiking plus 10% noise). Here, as desired, the full model explains spike counts significantly better than both the other models and this neuron would be selected as a PAC neuron. (d) In this example, we simulate a “gamma neuron”, i.e., a neuron whose FR follows gamma amplitude, but not theta phase. In most cases (79.3% of 1000 repetitions), these gamma neurons were successfully rejected. Since we did not control for theta phase in these simulations using a strong LFP channel, however, around 20% of the simulations modelled PAC rather than pure gamma spiking.