We identified and characterized FR during non-REM sleep (Figure S1) in a cohort of 46 patients implanted with stereo-EEG (SEEG), which was required for pre-surgical evaluation. The focality of the patients’ seizures varied, as did the surgical interventions, and the patients’ post-operative seizure outcomes (Table S1). We examined the temporal dynamics and interactions of FR generating sites to understand the mechanisms that underlie FR networks and to relate them to interictal spikes and sharp waves.

## Characterizing fast ripple propagation

For each patient we compared FR onset times between each pair of electrode contacts (*i.e.*, nodes) to determine which pairs exhibited unidirectional statistically significant FR propagation (*i.e.*, edges; sign test, p < 0.005, FDR corrected, see methods). We excluded FR that coincided with epileptiform spikes, which occur during inter-ictal15 and ictal epochs11. Overall, per patient, FR propagation was rare, and correlated with the mean number of FR per electrode contact (R2 = 0.435, 0.105, p < 0.05, Figure S2A). In the SOZ (*i.e.*, SOZ:SOZ), 0.86% ± 0.33% of edges (n = 29) with FR mutual information (MI) greater than zero had statistically significant propagation. In the non-SOZ (*i.e.*, NSOZ:NSOZ), 0.24% ± 0.13% (n = 25) of edges had significant propagation (Kruskal Wallis d.f.=1, Χ2 = 0.01, p > 0.05). In the SOZ:NSOZ (or NSOZ:SOZ), 0.064% ± .18% of the edges (n = 11) had propagation. Visual inspection confirmed FR propagation for significant SOZ:SOZ edges and demonstrated that at the out-node (*i.e.*, FR origin), and sometimes also the in-node (*i.e.*, FR destination), FR precede epileptiform spikes and sharp waves (Fig. 1A,S3A,S7A). For NSOZ:NSOZ, NSOZ:SOZ, and SOZ:NSOZ edges with significant FR propagation, visual inspection was confirmatory but the associated events in the broadband intracranial EEG were diverse (Fig. 1B, S4, S5, S6 S7B). For SOZ:SOZ edges we found a mean FR propagation velocity of 1.54 mm/sec13, but velocities across edges varied more than previously reported13 (Fig. 1C). FR Propagation in the NSOZ:NSOZ, SOZ:NSOZ, and NSOZ:SOZ edges was at a slower velocity than the SOZ:SOZ edges (Fig. 1C). The neuroanatomical location of the edges showing propagating FRs were most often in limbic and frontal regions (Fig. 1D,S2C).

To assess whether FR propagation results from a common synaptic driver inducing responses at the in- and out-node near-simultaneously, we compared the spectral frequency, duration, and power of each FR event. We assumed that if this driver was responsible for the measured pseudo-propagation the properties of the propagating FR events at the in- and out-node should be similar. We defined propagating FR as FR in either the in- or out-node that coincide with a FR in its paired node by < 250 ms in edges with significant propagation. For predicting FR frequency, the four-way interaction of whether the node was out- or in- relative to the propagation direction (*i.e.*, node status), if the FR was propagating (*i.e.*, propagation status), the mean delay of propagation, and whether the contact detecting the fast ripple was in the seizure onset zone (*i.e.*, SOZ status) was significant (generalized linear mixed effects model, GLMM, p < 1e-4, Table S2). Only in the SOZ:SOZ edges did edges with a relatively shorter propagation delay and propagation distance exhibit lower FR frequencies when they were non-propagating FR events in the out-node (Fig. 2A). This suggests that the FR propagation was not due to volume conduction. Power was predicted by the three-way interaction of the node status, propagation status, and SOZ status (GLMM, p < 1e-4, Table S3) and propagated events at the out-node were highest in power (Fig. 2B). Furthermore, the propagating FR power at the out-node was largest at longer propagation delays and propagation distances (GLMM, p < 1e-3, Table S3, Fig. 2B3). Therefore, because of the distinct frequency and power of the propagating FR in the out-node, it is more likely that FR propagate through slow and likely polysynaptic mechanisms to the in-node as opposed to pseudo-propagation from a common driver. Notably, pseudo-propagation could have resulted in differences in FR properties in the out and in node, and even differences in “propagating” events due to variation in the neuronal elements influenced by the driver, but pseudo-propagation cannot easily account for the interaction of node status and propagation status, and the correlation of propagation delay with FR frequency and power. In the case of NSOZ:NSOZ, NSOZ:SOZ, and SOZ:NSOZ edges, the differences in FR frequency (Fig. 2A2,3) and FR duration (GLMM Table S4, Fig S8) for propagating and non-propagating FR events in the in- and out-node were distinct from the SOZ:SOZ edges, suggesting different mechanisms.

## Delta oscillation and fast ripple coupling and propagation

Although the FR properties analysis support FR propagation, a synaptic driver to FR generating nodes has been found in studies of the kainic acid model of mesial temporal lobe epilepsy16,17. Delta oscillations (3–5 Hz) have been shown to entrain multi-unit activity and FR during the peak of the wave16,17. We found in patients, nodes with FR coupled to delta at or around the peak were common in the SOZ (Rayleigh Z, p < 0.001; Fig. 3A,B1). In the NSOZ, delta coupling occurred at more diverse preferred phase angles (Fig. 3B2). To relate the FR-Delta coupling to propagation we predicted the sign test Z value (*i.e.* the propagation significance measure) with the interaction of out- and in-node FR-delta coupling strength (*i.e.*, Rayleigh Z value the phase locking strength measure), and the relative location of the nodes in the SOZ or NSOZ (GLMM, p < 1e-9, Table S5). In this model, the predictive power of delta coupling in the out-node was greater than the in-node (Table S5). The SOZ:SOZ edges with FR-delta coupling in both the out- and in-node were the most likely to exhibit propagation (Fig. 3C).

To disentangle the effect of FR-Delta coupling from FR propagation we examined the preferred phase angle distributions of propagating and non-propagating FR with respect to delta. Overall, in the SOZ and NSOZ out-node and in-node, propagating FR, across pooled edges, showed a preferred phase angle near the peak of delta (Fig. 3D, Table S6). In the out-node, the preferred phase angle of the FR-Delta coupling was during the ascending portion of the wave slightly before the peak, as compared with the in-node where the preferred phase angle was nearer to the peak. For FR recorded from the SOZ in limbic structures, propagated FR occurred earlier before the peak and precessed more of the delta wave at the in-node as compared with the non-propagating distributions (Bayesian mixed-effect regression model for circular data, bpnreg18, Bayes Factor (BF) 2 of 2 for interaction of propagation status and node status compared to node status for predicting FR-Delta phase angle, Fig. 3D1-2, Table S6). In the frontal lobe and the parietal lobe SOZ, node status predicted the FR-Delta preferred phase angle more than propagation status, or the nodal status and propagation interaction (bpnreg, BF 1.98/2 frontal, BF 2/2 parietal node status compared to propagation status, Fig. 3C1-2, Table S6). Pooling of individual edges may have contributed to regions where the phase angle distribution showed weak or no FR-Delta phase locking. In our bpnreg models, edge number was used as the random effect to control for such variability.

In the NSOZ, limbic FR-Delta angle distributions were not phase locked in the non-propagating FR (Figure S9, Table S7). However, in other regions, phase locking was observed for non-propagating FR in the in-node (Figure S9, Table S7). In the case of propagating FR in the NSOZ, FR-Delta phase locking was observed at both the in-node and out-node (Fig. 3C4, Table S7). Despite these observations at the pooled group level, our model indicated in the NSOZ, the interaction between node status and propagation status predicted FR-Delta phase angle distributions as well as node status and propagation status alone (bpnreg, Table S7).

Since recording the iEEG in either a referential or bipolar montage can influence both the properties of FR and FR-Delta coupling, we examined the effects of montage in our GLMMs (Table S2-S4) and our bpnreg models (Table S6, S7). Montage accounted for a small portion of the outcome for the GLMMs (Table S2-S4). In the bpnreg models, montage accounted for almost none of the phase angle prediction in the limbic SOZ (BF 0.04/2, Table S6), but in the temporal SOZ was a significant confound (BF 2/2, Table S6).

Most of the FR in the analysis were recorded from limbic SOZ regions (Fig. 3C). Limited sample size and sampling bias could have influenced our observations in the other neuroanatomical locations and the NSOZ. In the limbic SOZ, FR-delta coupling does appear to drive FR generation, and be an important factor promoting propagation, independent of whether the FR is propagated. However, propagation appears to be a distinct mechanism because it is associated with a larger delta wave phase precession to the peak.

## Spikes and sharp waves follow propagated fast ripples at the out node

Having established that, in patients with epilepsy, delta oscillations in the limbic SOZ putatively drive the generation of FR, and that FR propagation is an event associated with, but distinct from, this coupling, we next sought to quantify our visual observations (Fig. 1A,S3A,S7A) and determine if FR propagation is associated with the generation of epileptiform spikes and sharp-waves in the intracranial EEG. To quantify changes in the iEEG and epileptiform activity following propagating FR, we accounted for the fact that power spectral densities (PSD) and time-frequency spectrograms of epileptiform spikes demonstrate a broad band increase in power associated with the sharp transients of an epileptiform spike’s positive and negative going components19. The broad band power increases have also been associated with increased neuronal activation20–22 and measured using the aperiodic offset of the fit power spectral density23.

To assess if FR propagation is associated with the initiation of epileptiform spikes and sharp waves, we separated the propagated from non-propagated FR and created one second iEEG trials with the FR onset time aligned at 250 ms. Visual inspection confirmed our automated detection system methodology assuring that in each trial, the FR onset occurred before the initial positive or negative phase of the spike15 (Fig. 1A, S3A,S7A). From these trials we measured the FR-related modulation of the post-FR iEEG signal. As part of this analysis, the occurrence of the FR in the out-node was also used as a trigger for generating one second iEEG trials in the in-node to measure cross-modulation and confirm propagation. Additionally, for each set of propagated, or non-propagated, FR-triggered iEEG trials from the in-node or out-node we calculated the average fit power spectral density23. We found that 8 of the 29 SOZ:SOZ edges exhibited propagating FR triggered epileptiform spikes in the out- and sometimes also in-node, but in these cases, we did not observe spikes associated with non-propagating FR (Fig. 4A,B). In another 12 of the 29 SOZ:SOZ edges, a FR triggered spike or sharp-wave was seen following both propagated and non-propagated FR at the out- and in-node, but the modulation of the spike was larger following the propagated fast ripples (Fig. 4C,D,S10,S11). In one patient, who had longer propagation delays (Fig. 1C1) and longer duration FR (Figure S3B), FR propagation was associated with the onset of a sharply contoured delta wave (Figure S12). To quantify these differences, we examined the aperiodic and periodic parameters of the PSD fits and compared propagating FRs with non-propagating FRs in the out- and in-node. We found that in the out-node, but not the in-node, the FR-iEEG trials associated with propagating FRs had a larger aperiodic offset and peak frequency (paired t-tests, d.f.=28, Benjamani-Hochberg adj-p = 0.03, 0.02, Fig. 4E1, E3, Table S8). As a control, we also performed an identical comparison of the FR-iEEG trials one second before the aligned FR events and found no significant differences (adj-p > 0.05, Fig. 4E, Table S8). These quantitative results support that propagating FR events in the SOZ precede epileptiform spikes and sharp-waves.

Since the propagating and non-propagating FR exhibited distinct properties (Fig. 2, Table S2-S4), it was required to determine whether the FR propagation status or the FR properties were more predictive of an after-going epileptiform spike. We used a GLMM to predict the aperiodic offset of individual iEEG trials in the out-node, for events in SOZ:SOZ edges, using the accompanying FR’s propagation status, power, frequency, and duration. We found that the propagation status of the FR predicted the aperiodic offset magnitude (GLMM, p < 1e-26, Table S9), but not FR power, frequency, duration, or the interaction of FR power with propagation status (GLMM, p > 0.05, Table S9).

When this cross-modulation and PSD fitting approach was applied to NSOZ:SOZ, SOZ:NSOZ, and NSOZ:NSOZ edges heterogenous results were found. In one NSOZ:SOZ edge propagating FR resulted in a more strongly modulated spike in the in-node but not the out-node (Figure S13), but this was not the case for other edges bordering the two territories. For the NSOZ:NSOZ edges, most commonly no epileptiform spike or sharp wave followed FR propagation (Fig. 5A,B), although in one edge a sharp wave was seen in the out-node (Figure S14). Another patient with NSOZ:NSOZ edges exhibited extremely long FR (Figure S4) but no strong cross-modulation (Figure S15). Across all the NSOZ:SOZ, SOZ:NSOZ, and NSOZ:NSOZ edges, in the out- or in-node, the aperiodic and periodic parameter differences of the PSD fits between propagating and non-propagating events did not reach significance (adj-p > 0.05, Fig. 5C, Table S10) indicating that FR propagation outside of the SOZ appears to be by distinct mechanisms that, in contrast to the SOZ:SOZ edges, do not usually precede epileptiform spikes.