Local field potentials (LFPs) are associated with various aspects of animal cognition and behavior, including attention, volition, and learning1,2,3,4, and are also used as biomarkers of pathological states5,6. One of the major components of LFPs in the hippocampi of humans7,8,9 and rodents10,11,12 is theta (θ) oscillations (3–10 Hz). θ oscillations occur mainly during active exploration, exploratory sniffing, and rapid eye movement sleep and are essential for the normal functioning of the hippocampus12,13,14. Indeed, θ oscillations are correlated with memory performance15,16,17,18,19, and reduced θ oscillations are associated with memory deficits19,20, whereas enhanced θ oscillations facilitate cognitive function21,22.

As a component of LFPs, θ oscillations are thought to reflect extracellular currents that arise from the collective dynamics of subthreshold membrane potentials (Vm)23,24,25, including synaptic activity, dendritic integration, action potentials, afterpotentials, and other channel-dependent neuronal events, in a large number of neurons26,27. However, it remains unclear how θ oscillations are associated with the Vm dynamics of multiple neurons in vivo. Previous studies have examined the relationship between LFPs and the Vm of a single pyramidal cell and reported both similarities and discrepancies23,28,29. For example, one study with simultaneous LFP and single intracellular recordings from the hippocampi of anesthetized rabbits reported that intracellular θ oscillations (θVm) occurred together with LFP θ oscillations (θLFP) at similar oscillation frequencies23, suggesting that θVm serve as a source of θLFP; however, not all neurons exhibited θVm during θLFP states, and the amplitudes of θVm were also heterogeneous across cells. Thus, it remains unclear how similar θVm rhythms are exhibited in different cells during a θLFP period. A more recent study examined the relationships between θLFP and θVm in head-fixed running mice28, demonstrating that although θVm tended to occur together with θLFP, these two types of oscillations often had inconsistent phases and frequencies. This inconsistency is functionally important for θ phase precession because a θVm rhythm that is faster than the θLFP rhythm allows the neuron to fire at earlier phases of θLFP during spatial movement. Nonetheless, it remains unclear how the diverse θVms of multiple hippocampal neurons are reflected in unidimensional θLFP. These questions can be answered, at least in part, by simultaneously recording the Vms of multiple neurons.

In the present study, we simultaneously recorded the Vms of up to three CA1 pyramidal cells in mice to directly compare their Vm dynamics to LFPs. We found that θVm occurred intermittently, often together with θLFP. In some but not all LFP-cell pairs and cell‒cell pairs, θ oscillations showed similar changes in power and frequency over time. Specifically, the θLFP power increased when more CA1 pyramidal neurons simultaneously emitted θVms. When neurons had similar θVm frequencies, the θLFP power increased, and the θLFP frequency approached the average value of the neuron θVm frequencies. As a result, θLFP was predictable from the θVm dynamics of three CA1 pyramidal cells using deep learning. These findings provide fundamental insights into how LFPs are associated with multineuronal Vms in vivo, demonstrating their collective potential for predicting LFPs.


Coordinated θLFPs across the dorsal hippocampal CA1 area

To ensure the θLFPVm correlation analyses below, we first verified whether θLFPs were uniform within our targeted area. LFPs were simultaneously recorded from four sites in the dorsal CA1 area of the mouse hippocampus (Fig. 1a-c). The area enclosed by the four sites encompassed the area for LFP and patch-clamp recordings in the following experiments. LFPs spontaneously alternated between periods with and without θLFP at frequencies ranging from 3 to 10 Hz (Fig. 1d, e). The θLFP power was reduced after intraperitoneal administration of 50 mg/kg atropine, a muscarinic receptor antagonist (Supplementary Fig. 1). Therefore, the θLFP represented type 2 θ oscillations30.

Fig. 1: Synchronized θLFPs across the dorsal hippocampal CA1 area.
figure 1

a Schematic illustration of simultaneous in vivo LFP recordings from four sites in the dorsal hippocampal CA1 area. b Representative top view schematic of the hippocampus (yellow), hippocampal window (square), recording sites (black dots), and directions of the inserted glass pipettes used to acquire the data shown in d. c Fluorescence image of the track of an LFP electrode visualized by DiI (red). The histological section was counterstained with fluorescent Nissl stain (blue). d Representative traces of LFPs recorded simultaneously from four sites during a θLFP period (left) and a non-θLFP period (right). e Wavelet spectrograms of four simultaneously recorded LFP traces, parts of which are shown in d. Blue and gray bars indicate θLFP and non-θLFP periods, respectively. f Temporal evolution of the relationship between the θLFP powers of LFP1 and LFP2 shown in e. For a given 1-s segment, the θLFP powers of two LFPs are plotted as a single dot in the space of LFP1 and LFP2; temporally adjacent dots are connected by gray lines. The significant positive correlation indicates that the two θLFP powers changed similarly over time (R = 0.79, P < 10−323, t17,992 = 172.8, t-test for correlation coefficients, n = 17,994 segments). The black lines indicate the lines of best fit with ordinary least-squares regression. g Cumulative probability distribution of the correlation coefficients between pairs of θLFP powers, as calculated in f. The correlations were statistically significant for all 30 pairs of 20 LFP traces recorded from 5 mice. h Cross-correlograms of the 14 pairs of simultaneously recorded LFP traces bandpass filtered at 3–10 Hz (gray line) and their mean (black). Only the pairs of LFPs for which the relative positions of the electrodes along the mediolateral axis could be identified were included. In the inset, the time scale is expanded near 0 ms, and the plot indicates that the θLFPs recorded from relatively medial recording sites preceded their counterparts. i Same as h, but for the 15 pairs of LFPs for which the relative positions of the electrodes along the anteroposterior axis could be identified. Cross-correlations peaked at 0-ms time lags. j Time lags between pairs of θLFPs were calculated for the mediolateral (ML) and anteroposterior (AP) pairs shown in h and i, respectively. θLFP propagation was observed in the medial to lateral direction (P = 0.0023 vs. time 0, t14 = −3.7, Student’s t-test, n = 15 pairs from 5 mice) but not along the anteroposterior axis (P = 0.095 vs. time 0, t13 = 1.8, Student’s t-test, n = 14 pairs from 5 mice).

The LFP traces were divided into a time series of 1-s segments, and the mean θLFP power was calculated for each segment. The time-dependent changes in the mean θLFP power were plotted in the space of each pair of simultaneously recorded LFP traces (Fig. 1f). The more similar the changes in the θLFP power of the two recording sites were, the closer the data points were to the identity line in the plot. To quantify this similarity, we calculated the correlation coefficient for each plot. For all 30 LFP pairs from 5 mice, the correlation coefficients were significant and positive (Fig. 1g), indicating that θLFP emerged simultaneously at all four recording sites. Considering that the recording time affects the correlation, the entire recording period was divided into 1-, 2-, 3-, 5-, 10- or 15-min time windows, and the correlation was evaluated for each time window. While the correlation increased slightly with increasing time window lengths, a significant positive correlation was observed for >90% of the entire recording period, regardless of the length of the time window (Supplementary Fig. 2, all time window lengths, P = 1.1 × 10−4, χ2 = 25.6; time windows >2 min, P = 0.049, χ2 = 9.6; time windows >3 min, P = 0.11, χ2 = 6.11, chi-square test, n = 900, 450, 300, 180, 90, and 60 time windows for 1, 2, 3, 5, 10, and 15 min, respectively). Thus, the time changes in θLFP power were synchronized among the four recording sites and were minimally affected by the recording time. To examine the phase relationship between simultaneously recorded θLFPs, the LFP traces were bandpass filtered between 3 and 10 Hz, and the cross-correlations were computed. The time lags were calculated by referencing the θLFPs recorded from relatively medial or anterior locations for individual pairs so that the θLFP propagations along the longitudinal axis of the hippocampus11,31 could be extracted (Fig. 1h, i). Overall, the θLFPs were highly synchronous between LFP pairs, regardless of their relative positions (Fig. 1h, i). The θLFPs recorded from medial positions significantly preceded their counterparts (Fig. 1j ML, P = 0.0023 vs. time 0, t14 = −3.7, Student’s t-test, n = 15 pairs from 5 mice), while no such effects were observed between θLFP pairs along the anteroposterior axis (Fig. 1j AP, P = 0.095 vs. time 0, t13 = 1.8, Student’s t-test, n = 14 pairs from 5 mice), which is consistent with a previous report11. Therefore, θLFPs were synchronized across the entire hippocampal windows targeted in our study, with mediolateral propagation on a fine time scale.

Next, we divided the entire recording times into θLFP and non-θLFP periods using a threshold based on the standard deviation (SD) of the background noise against the power of the θ frequencies in the wavelet spectrogram (Supplementary Fig. 3). To determine an appropriate SD threshold, we categorized putative θLFP and non-θLFP periods using various multiples of the SD (1 SD, 2 SDs, 3 SDs, and 4 SDs) and calculated the Dice similarity coefficients to estimate the strength of synchronization of the detected θLFP periods between two recording sites (Supplementary Fig. 3a). The distributions of the Dice similarity coefficients were compared to their statistical chance levels obtained from 10,000 surrogate data samples in which the detected θLFP periods were randomly shuffled across time within each recording site. The difference in the original and surrogate data distributions was assessed according to the D value of a two-sample Kolmogorov‒Smirnov test, and the D value reached the maximum at 2 SDs (Supplementary Fig. 3b). Therefore, in the following analyses, we defined a θ period as a time period during which the θ power continued to exceed 2 SDs against the background noise.

Characteristics of Vm activity during θVm and non-θVm periods

To examine the θLFPVm relationship, we first characterized the θVm dynamics. We patch-clamped pyramidal cells in the hippocampal CA1 region of urethane-anesthetized mice (Fig. 2a). Recordings were excluded from subsequent analyses if post hoc biocytin-based visualizations, their recording sites, and/or their firing properties failed to identify the recorded neurons as CA1 pyramidal cells (Fig. 2b). As a result, we recorded Vms from 220 patch-clamped cells in a total of 112 mice. The recording periods ranged from 30 s to 2097 s (median = 180 s).

Fig. 2: Variable θVm frequencies in a CA1 pyramidal cell.
figure 2

a Schematic illustration of whole-cell current-clamp recordings from a CA1 pyramidal cell. b Representative confocal image of a recorded pyramidal neuron visualized with intracellular biocytin (red) and Nissl counterstain (blue). c A raw trace of Vm in a CA1 pyramidal cell (top trace) was bandpass filtered between 3 and 10 Hz (second row trace) and divided into θVm periods (blue) and non-θVm periods (gray) based on the 3–10 Hz oscillation power (third row trace). The bottom plot shows the wavelet spectrogram of the Vm trace. d Representative time course of the peak θVm frequencies during individual θVm periods, demonstrating that a single CA1 pyramidal cell exhibited θVm at various frequencies. e Cells fired spikes around the peaks of θVm cycles. P < 10−323, Z = 3.7 × 103, Rayleigh test, n = 4655 spikes from 220 cells. f The firing rates increased with increases in the θVm power. To pool data from different cells, the firing rates were Z-standardized on a logarithmic scale across the entire recording period of each cell. Each dot indicates the average value in a single θVm period, and the average values of the Z-standardized parameters are superimposed on a cell-by-cell basis. The black line indicates the line of best fit based on a generalized linear mixed model. β = 0.22, P = 8.0 × 10−9, t1032 = 5.8, t-test of the correlation coefficient, n = 1034 θVm periods from all 136 cells that fired at least one spike. g Same as f, but for the θVm frequencies. β = 0.10, P = 0.035, t1032 = 2.1, n = 1034 θVm periods from 136 cells.

Similar to LFPs, spontaneous Vm responses alternated between periods with and without θ oscillations (Fig. 2c); of the total recording time, ~34 ± 26% represented θ periods (mean ± SD of 220 cells, ranging from 0.84 to 95%). The frequency of θVm was not unique to a given cell but varied among θVm periods (Fig. 2d). Spikes from pyramidal cells were entrained to θVm cycles such that the firing rates peaked at the θVm phase of 0° (Fig. 2e; P < 10−323, Z = 3.7 × 103, Rayleigh test, n = 4655 spikes from 220 cells). Analysis using a generalized linear mixed model demonstrated that the mean firing rates during θVm periods increased as a function of θVm power (Fig. 2f, β = 0.22, P = 8.0 × 10−9, t1032 = 5.8, n = 1,034 θ periods from 220 cells). The mean firing rates also increased with increasing θVm frequency (Fig. 2g, β = 0.10, P = 0.035, t1,032 = 2.1, n = 1034 θ periods from 220 cells). The firing rates and power were Z-standardized in each cell and pooled across all recorded cells.

Weakly correlated θLFP and θVm

We next compared θ oscillations between Vm and LFPs. While recording Vm from a single CA1 pyramidal cell, we recorded LFPs from a single site in the CA1 region in the same hippocampal window (Fig. 3a–e). The θLFP power was not correlated with the firing rates of patch-clamped cells (Supplementary Fig. 4). To examine whether θVm and θLFP occurred simultaneously, we plotted the time-dependent changes in the θ powers of Vm and LFPs every 100 ms (Fig. 3c, e). The correlation coefficients varied among cells; the θVm power exhibited a significant positive correlation with the θLFP power in 66 (41%) cells out of 160 neurons, whereas the remaining 94 (59%) cells did not exhibit significant correlations (Fig. 3f). Therefore, the θVm power was correlated, at least in part, with the θLFP power; however, this correlation was not robust. The correlation coefficients were not associated with the distances between the LFP recording sites and the patch-clamped cells (Fig. 3g, R = 0.087, P = 0.43, t-test for correlation coefficients, n = all 86 cells whose loci were confirmed post hoc), indicating that even if the cells were located near the LFP recording site or generated strong θVm, their influence on θLFP was not necessarily large. In addition, there were no significant relationships between the correlation coefficients and the recording time length (Supplementary Fig. 5a, R = 0.11, P = 0.19, t-test for correlation coefficients, n = 160 cells), the mean θVm power (Supplementary Fig. 5b, R = −0.042, P = 0.60, n = 160 cells), the mean firing rate (Supplementary Fig. 5c, R = −0.081, P = 0.31, n = 160 cells), the mean Vm (Supplementary Fig. 5d, R = −0.14, P = 0.076, n = 160 cells), and the standard deviation (SD) of Vm (Supplementary Fig. 5e, R = 0.068, P = 0.39, n = 160 cells). Furthermore, neither the cell locations along the proximodistal (Supplementary Fig. 5f, P = 0.62, one-way analysis of variance (ANOVA), n = 27, 41, 45, 11, 11 for CA1a, a/b, b, b/c, c, respectively) nor the radial (Supplementary Fig. 5g, P = 0.27, t131 = 1.1, Student’s t-test, n = 80 and 53 cells for deep and superficial, respectively) axes were significantly related to the correlation coefficients. Therefore, the engagement of each cell with the ongoing θLFP might not be intrinsically predetermined but rather be flexibly modifiable.

Fig. 3: Weak correlations between θLFP and θVm.
figure 3

a Schematic illustration of simultaneous recordings of LFPs and Vm from a CA1 pyramidal cell. b Representative raw traces of simultaneously recorded CA1 LFPs and Vms of CA1 pyramidal cells. The middle of the action potentials was omitted to enlarge the changes in Vm, and a full action potential for each trace in b and c is shown on the right. c Temporal relationships in the power of θLFP and θVm. The θ power was plotted every 100 ms over the entire recording period in each dataset. No significant correlation was observed (R = 0.0011, P = 0.96, t-test for correlation coefficients, n = 1999 1-s segments). The black lines indicate the lines of best fit based on least-squares regression. d, e Same as b, c, but for a dataset in which a significant positive correlation was observed (R = 0.41, P < 10−323, n = 6999 1-s segments). f Cumulative probability distribution of the correlation coefficients between θLFP and θVm powers for all 160 recorded datasets. Red dots indicate cells exhibiting significant positive correlations. g The correlation coefficients between the θLFP and θVm powers (calculated in d) were plotted against the spatial distance between the tip of the LFP recording electrode and the patch-clamped cell. R = 0.087, P = 0.43, t-test for correlation coefficients, n = all 86 cells whose loci were confirmed post hoc. h Relationships between the frequencies of θLFP and θVm during co-θ periods when θ oscillations occurred simultaneously in LFPs and Vm. Each dot indicates a single co-θ period. R = 0.29, P < 10−323, t-test for correlation coefficients, n = 2659 co-θ periods from 160 cells. i The difference in the θ frequencies of θLFP and θVm in a co-θ period was negatively correlated with the geometric average of their powers. Each dot represents a co-θ period. The black line indicates the line of best fit based on least-squares regression. R = −0.094, P = 9.8 × 10−7, n = 2702 co-θ periods from 160 cells. j, k Circular distribution of the θ phase difference between LFPs and Vm when θLFP and θVm occurred simultaneously at similar frequencies (Δ frequency < 0.01 Hz). Because θLFP propagates along the mediolateral axis, the datasets were divided into two groups, in which the locations of the recorded cells were medial to the LFP recording sites (j) and vice versa (k). Red lines show the mean θ phase differences (−74° and 30° for left and right panels, respectively). The distribution was significantly nonuniform (j P = 8.1 × 10−8, Z = 16.0, Rayleigh test, n = 172 periods from 30 cells; k P = 0.038, Z = 3.25, Rayleigh test, n = 46 periods from 16 cells).

During a co-θ period when θ oscillations occurred simultaneously in Vm and LFPs, the θVm frequency was positively correlated with the θLFP frequency (Fig. 3h, R = 0.29, P < 10−323, t-test for correlation coefficients, n = 2659 co-θ periods from 160 datasets). However, there were also many outlier data points distant from the identity line in Fig. 3h, indicating that θVm and θLFP did not always share common frequencies. Consistent with this notion, the θVm frequencies were, on average, slightly higher than the θLFP frequencies (P = 8.8 × 10−10, t2,658 = −6.2, paired t-test, n = 2659 co-θ periods). Interestingly, the differences between θVm and θLFP frequencies during each co-θ period decreased as the mean θVm and θLFP powers increased (Fig. 3i, R = −0.094, P = 9.8 × 10−7, n = 2,703 co-θ periods from 160 cells). Moreover, θVm and θLFP were not in phase. We focused on the coherent periods, during which the difference between θLFP and θVm frequencies was <0.01 Hz, and calculated their phase differences. Considering the θLFP propagation along the mediolateral axis described in Fig. 1j, the phase differences were plotted separately for the datasets in which the recorded cells were located medial to LFP recording sites (Fig. 3j) or vice versa (Fig. 3k). The phase difference between θLFP and θVm was not uniformly distributed, and the θVm of the cells medial to the LFP recording sites preceded the θLFP by 74° on average (Fig. 3j, P = 8.1 × 10−8, Z = 16.0, Rayleigh test, n = 172 periods from 30 cells). Anatomically reversed datasets showed the opposite result, i.e., the θVm recorded at lateral positions to the LFP recording sites followed the θLFP by 30° on average (Fig. 3k, P = 0.038, Z = 3.25, Rayleigh test, n = 46 periods from 16 cells). The phase relationship between θLFP and θVm did not differ depending on the cell locations along the radial axis in the pyramidal cell layer (Supplementary Fig. 5h, i, P > 0.1, K = 1.8 × 103, Kuiper test, n = 151 and 74 periods from 26 and 20 deep and superficial cells, respectively).

We repeated the same series of experiments using awake, head-fixed mice (Supplementary Fig. 6a, b, n = 22 cells from 17 mice). Neither the durations nor the frequencies of the θLFP differed between urethane-anesthetized and awake mice (Supplementary Fig. 6c, d), indicating that type 2 θ oscillations dominated under our experimental conditions, in which mice were forced to be immobile. The data were analyzed, as shown in Fig. 3d, revealing that 10 (40%) of the 22 cells showed significant positive correlations in θ power changes (Supplementary Fig. 6e). This ratio did not differ from that of the anesthetized mice (P = 0.71, χ2 = 0.14, Pearson’s χ2 test). The differences between θVm and θLFP frequencies decreased as the mean θVm and θLFP powers increased (Supplementary Fig. 6f, β = −0.022, P = 0.0018, t2587 = −3.13, n = 2589 θ periods from 22 cells).

Therefore, we concluded that θVm and θLFP are partially correlated with each other and that the correlation strength depended on the oscillation states, such as θ power or θ frequency.

Weakly correlated θVms in two CA1 pyramidal cells

The weak θLFPVm coupling described above led to the inference that θVms of cell pairs are coherent. We obtained simultaneous patch-clamp recordings from two CA1 pyramidal cells (Supplementary Fig. 7a) and collected 125 dual patch-clamp datasets (Vm1 and Vm2) from 82 mice. The recording periods ranged from 36 s to 2097 s (median = 150 s). We applied the same analyses as in Fig. 3 to the dual patch-clamp recording datasets. We first plotted the time changes in the θVm powers of two cells (Supplementary Fig. 7b–e) and found that 80 (64%) of the 125 cell pairs showed significant positive correlations in their θVm power changes (Supplementary Fig. 7c, f), whereas the remaining 45 cell pairs did not (Supplementary Fig. 7e, f). Compared with the LFP-cell pair results in Fig. 3e, the proportion of significantly correlated cell pairs was high, which might reflect the different LFP states in separate datasets. However, the proportion of cell pairs with positive correlations did not change significantly from the proportion calculated for only the datasets used in Fig. 3 (Supplementary Fig. 8a, b, P = 0.49, χ2 = 0.47, chi-square test, n = 125 (all data) and 98 (data in Fig. 3 only) cell pairs, respectively). Cell pairs whose somata were physically closer had a stronger correlation (Supplementary Fig. 7g, R = −0.24, P = 0.033, n = 78 cell pairs). The θ frequencies during each co-θ period were significantly correlated between the two cells (Supplementary Fig. 7h, R = 0.25, P < 10−323, n = 2027 θ periods from 125 cell pairs), although they were not fully consistent. The differences between θVm frequencies decreased as the mean θVm powers of the two cells increased (Supplementary Fig. 7i, β = −0.011, P < 10−323, t34,628 = −47.3, n = 34,630 θ periods from 125 cell pairs). Therefore, θVms were correlated between adjacent cells, but again, these correlations were only partial.

Correlated θ power changes among LFPs and multiple cells

We expanded our experiments to triple patch-clamp recordings with CA1 LFP recordings (Fig. 4a, b). We collected 21 triple-patching and LFP datasets (Vm1, Vm2, Vm3, and LFP) from 15 mice. The recording periods ranged from 31 s to 900 s (median = 80 s). For the sake of simplicity, we first investigated the time-dependent changes in the θ power by focusing on the pairwise correlations between LFPs and the average value of three Vms (Fig. 4c–e). Of 21 datasets, 9 (43%) had significant positive correlations between the θLFP power and mean θVm power (Fig. 4c, e), whereas the other 12 did not (Fig. 4d, e).

Fig. 4: Stronger correlations of θLFP for more correlated θVms.
figure 4

a Schematic illustration of simultaneous recordings of LFPs and Vms of three CA1 pyramidal cells (Vm1, Vm2, and Vm3). b Representative raw traces of simultaneously recorded LFPs and Vms of three CA1 pyramidal cells. c, d The θLFP power during a 1-s segment plotted against the geometric mean of the θVm powers in three cells as a function of time. The triple recording in c exhibited a significant positive correlation (R = 0.51, P < 10−323, t-test for correlation coefficients, n = 740 segments), whereas the dataset in d did not (R = 0.065, P = 0.66, n = 470 segments). The black lines indicate the lines of best fit based on least-squares regression. e Cumulative probability distribution of the pairwise correlation coefficients between θLFP powers and the mean θVm powers of three cells for all 21 datasets. Each red dot indicates a dataset with a significant positive correlation. f The θLFP power increased as a function of the number of cells that simultaneously emitted θVms. Each gray line indicates a single dataset, and the black line represents the mean. P = 0.0060, Z = 2.51, Jonckheere trend test, n = 13 datasets. g The θLFP powers were positively correlated with the similarity of three θVm frequencies. The similarity was defined as the squared inverse of the coefficients of variance (1/CV2) of three θ frequencies for co-θ periods during which LFPs and three cells simultaneously exhibited θ oscillations. Each dot indicates a single co-θ period. R = 0.17, P = 0.014, t-test for correlation coefficients, n = 206 co-θ periods from 13 cell triplets. The black line indicates the line of best fit based on least-squares regression. h The difference between the θLFP frequency and the geometric mean of three θVm frequencies was negatively correlated with the similarity of three θVm frequencies. Each dot indicates a single dataset. R = −0.78, P < 10−323, n = 206 co-θ periods from 13 cell triplets. The black line indicates the line of best fit based on least-squares regression.

We next focused on the instantaneous θLFPVm correlations. The θLFP power became stronger when more cells simultaneously exhibited θVms (Fig. 4f, P = 0.0060, Z = 2.51, Jonckheere trend test, n = 13 datasets that included at least one period during which all recordings simultaneously exhibited θ oscillations). When all three cells simultaneously exhibited θVms, the θLFP power was positively correlated with the squared inverse of the coefficients of variance (1/CV2) of the three θVm frequencies (Fig. 4g, R = 0.17, P = 0.014, n = 206 co-θ periods from 13 datasets). This result indicates that the θLFP power increased when the three cells exhibited θVms with more similar frequencies. Moreover, the difference between the θLFP frequency and the mean frequency of the three θVms was negatively correlated with 1/CV2 of the three θVm frequencies, indicating that the θLFP frequency approached the mean θVm frequency when the three θVm frequencies were more similar (Fig. 4h, R = −0.78, P < 10−323, t204 = 17.7, n = 206 co-θ periods from 13 datasets).

Machine learning-based prediction of LFPs

These instantaneous θLFPVm correlations motivated us to hypothesize that the dynamics of θLFP at each moment can be estimated, at least in part, from the θVms of three cells. To test this hypothesis, we sought to predict the θLFP waveforms from three θVm waveforms using a machine learning model. Assuming that θVms are associated with θLFP in a nonlinear manner, we employed a deep neural network (DNN) with a convolutional layer (Fig. 5a, Supplementary Fig. 9). Among the 21 triple patch-clamp recording datasets, we selected 8 datasets with recording periods >3 min to ensure sufficient sample sizes for training the DNN. In each dataset, the Vm and LFP traces were bandpass filtered between 3 and 10 Hz, divided into 10 subsets in the recording time, and further divided into 1-s segments. We trained the DNN using 1-s segments in 9 subsets of the recorded data to predict the θLFP waveforms during 1-s segments in the remaining subset (1/10 subsets), which were not used to train the DNN (Fig. 5b, real). For each prediction, the prediction error between the original and predicted θLFP traces was quantified by the root mean square error (RMSE). We also trained the DNN using randomized pseudodata, in which 1-s segments from the training period were shuffled within each cell. We then predicted the θLFP waveform (Fig. 5b, shuffle) and computed the RMSEs for the original θLFP waveform and the θLFP waveform predicted by the shuffled data. We repeated this procedure so that all segments in the entire recording period were targeted for prediction. The RMSEs were pooled in a cumulative plot (Fig. 5c). The prediction performance was evaluated using the D value of a two-sample Kolmogorov‒Smirnov test; in all 8 datasets, the RMSEs of the real data were significantly lower than those of the shuffled data (Fig. 5d, 3 cells). These results indicate that the DNN predicted θLFP waveforms based on the θVm dynamics of as few as three cells with accuracy significantly higher than chance. However, the DNN significantly predicted zero or only two of the 24 datasets based on the θVm dynamics of only one or two cells, respectively (Fig. 5d, 1 cell, 2 cells). The D-values of the predictions based on the θVm dynamics of three cells were significantly higher than those based on the θVm dynamics of fewer cells, indicating that the θLFP dynamics reflect the collective features of multiple θVm dynamics, which needed to be collected from at least three cells. Supplementary Fig. 10 summarizes the D-values and their significance in all datasets.

Fig. 5: Prediction of θLFP from θVms.
figure 5

a Architecture of our neural network model. The numbers indicate the channel features (bin) in each layer. Conv: convolutional, FC: fully connected. The model input was three simultaneously recorded Vms (Vm1, Vm2, and Vm3) that were bandpass filtered between 3 and 10 Hz (left). The model was trained to output the corresponding bandpass-filtered LFPs (right). In d, one or two Vms were used as the inputs. b Representative traces of three Vms, the original LFPs (black), and the LFPs predicted from the real data (blue) or shuffled data (gray). The Vm and the original LFP traces were bandpass filtered between 3 and 10 Hz. The shuffled data were created by randomizing the temporal order of all 1-s segments within each cell and used to train the neural network. c Cumulative probability distribution of the root mean squared errors (RMSEs) between the original LFP waveforms and the LFP waveforms predicted from real or shuffled data. P = 1.5 × 10−86, D = 0.19, two-sample Kolmogorov‒Smirnov test, n = 2310 1-s segments in a single dataset. d Cumulative probability distribution of the D-values calculated as in c for all 8 datasets with LFPs and Vms of 3 cells, as well as 24 datasets with LFPs and Vms of 1 or 2 cells. Each dot indicates a single dataset, and red dots indicate significant D-values. The LFP waveforms were better predicted from Vms of 3 cells than those of 1 or 2 cells. D1 vs. 2 cells = 0.25, P1 vs. 2 cells = 0.39, D1 vs. 3 cells = 1.0, P1 vs. 3 cells = 2.3 × 10−6, D2 vs. 3 cells = 1.0, P2 vs. 3 cells = 2.3 × 10−6, two-sample Kolmogorov‒Smirnov test.

The same analysis was conducted using bandpass-filtered traces at 60–100 Hz (high gamma), 25–55 Hz (low gamma), and 0.5–1 Hz (slow oscillations). In contrast to the findings with the θ frequency band, significantly higher accuracies were obtained in only a small proportion of the datasets, regardless of the number of cells used for the prediction (Supplementary Fig. 11, high gamma: 5/24 datasets for 1 cell and 2/24 datasets for 2 cells; low gamma: 1/24 dataset for 1 cell and 2/24 datasets for 2 cells; slow oscillations, 4/24 datasets for 1 cell, 4/24 datasets for 2 cells and 2/8 datasets for 3 cells). For all three frequency bands, increases in the number of cells used for the predictions did not correspond to an increase in the prediction accuracy (Supplementary Fig. 11a, D1 vs. 2 cells = 0.17, P1 vs. 2 cells = 0.86, D1 vs. 3 cells = 0.50, P1 vs. 3 cells = 0.066, D2 vs. 3 cells = 0.46, P2 vs. 3 cells = 0.11; Supplementary Fig. 11b, D1 vs. 2 cells = 0.21, P1 vs. 2 cells = 0.62, D1 vs. 3 cells = 0.25, P1 vs. 3 cells = 0.79, D2 vs. 3 cells = 0.78, P2 vs. 3 cells = 0.29; Supplementary Fig. 11c, D1 vs. 2 cells = 0.46, P1 vs. 2 cells = 0.0082, D1 vs. 3 cells = 0.33, P1 vs. 3 cells = 0.43, D2 vs. 3 cells = 0.50, P2 vs. 3 cells = 0.066; two-sample Kolmogorov‒Smirnov test, n = 24, 24, and 8 datasets for 1, 2, and 3 cells, respectively). To examine the relationships among LFPs and Vms in frequency bands other than θ, we calculated the cross-correlograms between pairs of bandpass-filtered LFPs, the correlation coefficients between LFP and Vm powers of single cells, and the correlation coefficients between LFPs and the mean Vm power of three simultaneously recorded cells for all three frequency bands. Overall, the correlations among LFPs and Vms were comparable to those in the θ frequency band (Supplementary Fig. 12a–i). However, the average power spectra of the LFPs and Vms over the entire recording period showed that θ power dominated in all other frequency bands (Supplementary Fig. 12j, k). Therefore, the predictability of LFP traces from Vm traces was applicable only for physiologically dominant oscillations, suggesting that the DNN could extract biologically prominent signals.

If the θLFP dynamics reflected the simple summation of multiple θVms, the linear summations of the θVm traces were expected to become more similar to the θLFP traces as the number of cells increased. As shown in Supplementary Fig. 13a, we analyzed the correlation between the θLFP traces and the mean θVm traces of 1, 2, or 3 cells for each 1-s segment used in the θLFP prediction. However, contrary to expectations, the correlation coefficients decreased as the number of cells increased (D1 vs. 2 cells = 0.029, P1 vs. 2 cells = 6.5 × 10−42, D1 vs. 3 cells = 0.049, P1 vs. 3 cells = 4.1 × 10−59, D2 vs. 3 cells = 0.021, P2 vs. 3 cells = 1.3 × 10−12, two-sample Kolmogorov‒Smirnov test, n = 112,740, 112,740, and 37,580 1-s segments from 1, 2, and 3 cells, respectively, in 8 mice). This result indicates that the mean θVm traces of more cells were less similar to the θLFP trace. The θLFP and θVm powers were also calculated for each 1-s segment, and the correlation between the dynamics of the θLFP power and the dynamics of the mean θVm power of 1, 2, or 3 cells were analyzed in each dataset (Supplementary Fig. 13b). There were no significant differences among the three distributions, indicating that the similarity between the dynamics of the θLFP power and the mean θVm power of multiple cells did not change significantly depending on the number of cells (D1 vs. 2 cells = 0.17, P1 vs. 2 cells = 0.86, D1 vs. 3 cells = 0.25, P1 vs. 3 cells = 0.79, D2 vs. 3 cells = 0.17, P2 vs. 3 cells = 0.99, two-sample Kolmogorov‒Smirnov test, n = 24, 24, and 8 datasets from 1, 2, and 3 cells, respectively, in 8 mice). These results suggest that the ability to predict θLFP traces from the θVm traces of three cells is not explained solely by the linear sum of multiple θVms. Taken together, the DNN could predict θLFP waveforms from the θVm dynamics of three cells based on their nonlinear relationships.


We investigated the temporal correlations in extracellular and intracellular θ oscillations by directly comparing LFPs with the Vms of multiple CA1 pyramidal cells in vivo. We found that in terms of θ powers and frequencies, the θVms of hippocampal CA1 pyramidal cells were loosely correlated with each other and with θLFP. These correlations were not explained by the anatomical locations and likely arose due to functionally coherent network activity. Consistent with this idea, we demonstrated, using a machine learning technique, that the θLFP waveforms could be predicted from the θVm waveforms of only three pyramidal cells. Given that LFPs usually reflect the activity of considerably more than three neurons, our results suggest that many neurons simultaneously emit partially correlated θVms and that this loose simultaneity constitutes hippocampal θ states and is experimentally captured as θLFP.

We first demonstrated that the temporal evolution of θLFP power was similar across our target area in the dorsal CA1 region. Previous studies have reported that θLFP travel along the longitudinal axis of the hippocampus11,31,32 and that θLFP phase shifts monotonically as a function of the distance along the longitudinal axis, reaching ~180° between the septal and temporal poles31. Consistent with a previous report11, mediolateral propagation of the θLFP was observed in our study instead of anteroposterior propagation, despite the different animal conditions, i.e., awake and anesthetized conditions in the previous and present study, respectively. Therefore, in the present study, we considered the phase shift when analyzing the phase difference between θLFP and θVm (Fig. 3h), and the overall transition of the θLFP power was considered synchronous within our target area. The difference in the type of θLFP between awake and anesthetized animals should also be noted in considering the functional significance of our findings. The θLFP we recorded from urethane-anesthetized mice were atropine-sensitive type 2 θLFP, which usually have frequencies in the 4–7 Hz range; note that type 1 θLFP is resistant to atropine, have higher frequencies of ~8 Hz, and occur mainly during active locomotion and rapid eye movement sleep13,14. We also recorded θLFP in awake mice. Although we did not examine the effect of muscarinic receptor antagonists, the θLFP were also likely type 2 because the mice were immobile under head-fixed conditions, and their θLFP frequencies did not differ from those of anesthetized mice. Under anesthesia, the CA1 region may receive less input from the entorhinal cortex, but in both urethane-anesthetized and awake mice, cholinergic and GABAergic inputs from the medial septum are essential to generate θ oscillations30,33,34,35,36,37. Although type 1 and 2 θLFPs have different atropine sensitivities and θ frequency ranges, they share many common features; for example, both types depend on medial septal afferents and have similar θ phase distributions in the dorsal hippocampus33,37,38. Therefore, we believe that the mechanisms underlying the observed cell-to-LFP or cell-to-cell correlations could be partially shared by type 1 θLFP, at least within the local hippocampal circuit. Supporting this theory, pairwise θVm coherences during type 1 θLFP in awake, behaving mice have been reported to range from 0 to 0.8, regardless of the cell-to-cell distance39, which is consistent with our findings on type 2 θLFP. However, the anesthetized condition causes distinct activity in the medial septum40 and the entorhinal cortex41,42. As a result, instructive signals for learning-dependent neuronal activity are lacking, as observed during type 1 θLFP in the hippocampus43,44,45,46. Future experiments recording intracellular activity from multiple hippocampal neurons in awake behaving animals should clarify the subthreshold coordination of behavior-relevant cell assemblies in θLFP state.

LFPs are shaped by collecting a myriad of electrical currents arising from neural events, such as synaptic inputs and action potentials. Therefore, intuitively, the relationship between LFPs and Vms might be influenced by the physical distance from the cell body. However, we did not find that the cell-to-LFP correlations in the θ power or frequency changed with the cell-to-LFP distance, at least within a radius of ~1,000 µm. One possible explanation for this discrepancy is that synaptic inputs are received by dendrites distal from the cell bodies. The dendrites of many neurons intersect, generating spatially overlapping synaptic currents. The complexity of individual current sources may blur the dependency for cell-to-LFP distances. Another but more plausible possibility is that cell-to-cell coherence arises from intrinsic cell assembly dynamics. We recorded from only three cells out of numerous neurons in the dorsal hippocampus. Even though these cells were selected in a pseudorandom and blinded manner, their θVms were partially correlated, enough for our machine learning model to predict θLFP. Therefore, we believe that the θLFP-to-θVm correlations observed in the present study did not arise from a direct causal relationship produced by the neurons we recorded but rather from the activity of other neurons located closer to the LFP recording site that behaved in a similar manner to the recorded neurons.

The coordination of θLFP and θVms could be achieved by the interplay of inhibitory inputs from various types of interneurons47,48,49 and excitatory inputs mainly from CA3 pyramidal cells under our anesthetized condition46,50,51, each of which phase-locks to a specific θLFP phase. Among the various types of hippocampal CA1 interneurons, soma-targeting parvalbumin-positive basket cells (PV-BCs) and cholecystokinin-positive basket cells (CCK-BCs) effectively regulate θVm37,46,52. Considering that PV- and CCK-BCs differentially innervate CA1 pyramidal cells in the deep and superficial layers51, i.e., PV-BCs and CCK-BCs preferentially project onto deep and superficial cells, respectively, the phase relationships between θLFP and θVm may differ for deep and superficial cells53. However, Supplementary Fig. 5h indicates that deep and superficial cells showed similar θLFP phase preferences53. Because deep cells receive more inputs from the medial entorhinal cortex than superficial cells, this discrepancy might indicate that excitatory inputs from the medial entorhinal cortex, which are weakened under anesthesia, contribute to the shifted phase preference of deep cells. In addition, the variable holding Vm may have affected the phase lag between the peaks of θLFP and θVm46, while the overall phase lags were consistent with those in previous intracellular studies37,54. Although we did not note any anatomical bias in the relationship between θLFP and θVm, presumably due to the anesthesia and the large distance between the LFP and Vm recording sites (φ < 1,000 µm), the temporally precise and partially shared excitatory and inhibitory inputs among CA1 pyramidal cells may organize multicellular coordination in the θLFP state.

Using a DNN, we predicted the θLFP traces from the θVm traces of three CA1 pyramidal neurons. Previous in silico studies have simulated LFPs based on modeled neuronal activity in the neocortex55,56,57. However, no studies to date have attempted to predict real LFPs based on the membrane potentials of multiple neurons in vivo. This lack of data is partly due to the lack of simultaneous recordings of LFPs and Vms from multiple neurons, which is technically difficult to accomplish, especially in the hippocampus. Therefore, the present study provides the first evidence that θLFP dynamics are predictable from the raw θVm traces of three neurons in the hippocampus in vivo. It remains unclear why the predictability increased in a nonlinear manner when the number of cells used for θLFP prediction was increased from two to three. One possible explanation is composite factors among the θVms of multiple neurons, as implied in our triple-patching datasets (Fig. 4); possible combinations among these factors could increase in a nonlinear manner as the number of cells increases. Further investigations are needed to elucidate latent factors shared across θVms that collectively predict θLFP dynamics.

One limitation of this study is that our prediction approach did not reflect the diversity of pyramidal cells embedded in anatomically and physiologically nonuniform hippocampal circuits51,58,59. While our study did not observe any anatomical or physiological biases within our recorded cells (Supplementary Fig. 5), the θVms of individual pyramidal cells should show distinct relationships with θLFP according to different input patterns and intrinsic properties in awake behaving animals53,60. The DNN used to predict θLFP from θVms may be improved by considering these biased innervation and intrinsic properties. Furthermore, the prediction could benefit by including not only the θVms of pyramidal cells but also the activity of PV- and CCK-BCs37,46,52 as the inputs to the DNN. On the other hand, the fact that the DNN significantly predicted θLFP from θVms of as few as three pseudorandomly selected pyramidal cells might suggest that the DNN could accommodate the spatiotemporally biased circuit structure, confirming the potential ability of the DNN to process higher-order information. Future studies should combine cell-type-specific recordings of excitatory and inhibitory neurons with computational approaches such as deep learning-based analyses to better understand the circuit mechanisms involved in coordinating network activity at the single-cell level.


Animal ethics

Animal experiments were performed with the approval of the animal experiment ethics committee at the University of Tokyo (approval numbers: P29-9) and in accordance with the University of Tokyo guidelines for the care and use of laboratory animals. These experimental protocols were conducted in accordance with the Fundamental Guidelines for the Proper Conduct of Animal Experiments and Related Activities in Academic Research Institutions (Ministry of Education, Culture, Sports, Science and Technology, Notice No. 71 of 2006), the Standards for Breeding and Housing of and Pain Alleviation for Experimental Animals (Ministry of the Environment, Notice No. 88 of 2006) and the Guidelines on the Method of Animal Disposal (Prime Minister’s Office, Notice No. 40 of 1995).

Surgery and animal preparation

Whole-cell recordings through the hippocampal window were obtained from male ICR mice (Japan SLC, Shizuoka, Japan) that were 28–45 days old61. Mice were anesthetized with urethane (2.25 g/kg, intraperitoneal [i.p.]). Anesthesia was confirmed by the absence of paw withdrawal, whisker movement, and eyeblink reflexes. The skin was subsequently removed from the head, and the animal was implanted with a metal head-fixation plate. A craniotomy (2.5 × 2.0 mm2) was performed, centered at 2.0 mm posterior to the bregma and 2.5 mm ventrolateral to the sagittal suture, and neocortical tissues above the hippocampus were aspirated62,63,64,65. The exposed hippocampal window was covered with 1.7% agar at a thickness of 1.5 mm. To obtain recordings from unanesthetized mice, mice were implanted with metal head-holding plates under short-term anesthesia with 2–3% isoflurane. After full recovery, the mice received head-fixation training on a custom-made stereotaxic fixture for 1–2 h per day. The training continued for up to 5 days until the mice learned to remain calm.

In vivo electrophysiology

Patch-clamp recordings were obtained from neurons in the dorsal CA1 stratum pyramidale (AP: -2.0 mm; ML: 2.0 mm; DV: 1.1–1.3 mm) using borosilicate glass electrodes (4–7 MΩ). Pyramidal cells were identified by their regular spiking properties and post hoc intracellular visualization. For current-clamp recordings, the intrapipette solution consisted of the following reagents (in mM): 120 K-gluconate, 10 KCl, 10 HEPES, 10 creatine phosphate, 4 MgATP, 0.3 Na2GTP, 0.2 EGTA (pH 7.3), and 0.2% biocytin. Liquid junctions were corrected offline. Cells were discarded when the mean resting potential exceeded −50 mV or the action potentials did not exceed -20 mV. LFPs were obtained from the CA1 stratum pyramidale using tungsten electrodes (UEWMGCSEKNNM, FHC, USA) coated with 1,1′-dioctadecyl-3,3,3′,3′-tetramethylindocarbocyanine (DiI). The tungsten electrode location was detected by post hoc observation of fluorescent DiI tracks. Simultaneous quadruple LFP recordings were obtained by using four glass electrodes (0.5-2.5 MΩ) filled with artificial cerebrospinal fluid (aCSF), including DiI (4% w/v), Evans Blue (2%), or Trypan Blue (2%), for post hoc identification of the electrode locations (Fig. 1). The electrode locations estimated during the stereotaxic experiments were approximately the same as the post hoc visualized positions. Only the LFPs recorded in the pyramidal cell layer were included in the analyses. The signals from the four glass electrodes were amplified using a MultiClamp 700B amplifier, whereas the signals from the tungsten electrode were amplified using a DAM80 AC differential amplifier (World Precision Instruments). All signals were digitized at a sampling rate of 20 kHz using a Digidata 1440 A digitizer (Molecular Devices) that was controlled by pCLAMP 10.3 software (Molecular Devices). To confirm the type of θLFP (Supplementary Fig. 1), atropine was intraperitoneally injected at a dose of 50 mg/kg at least 120 s after the beginning of the LFP recordings. The effect of atropine was examined 960–1,080 s after the injection (a 120-s period).


Following each experiment, the electrode was carefully withdrawn from the hippocampus. The mice were transcardially perfused with 4% paraformaldehyde followed by overnight postfixation. The brains were sagittally sectioned at a thickness of 100 μm using a vibratome. The sections were incubated with 2 μg/ml streptavidin-Alexa Fluor 594 conjugate and 0.2% Triton X-100 for 4 h, followed by incubation with 0.4% NeuroTrace 435/455 blue fluorescent Nissl stain (Thermo Fisher Scientific; N21479) for 4 h. The tracks of the LFP electrodes were also detectable via DiI fluorescence. Fluorescence images were acquired using an FV1200 confocal microscope (Olympus, Tokyo, Japan) and subsequently merged. The depth of the soma and the tips of the tungsten electrodes were estimated in the Z-scan series of an FV1200 confocal microscope. More specifically, the CA1 area was roughly divided into three subareas along the proximodistal axis, and the locations of the soma and the tips of tungsten electrodes were visually classified as one of the following five positions: CA1a, the border between CA1a and b, CA1b, the border between CA1b and c, or CA1c. Then, the XYZ coordinates of the soma and the tips of tungsten electrodes in the brain were determined based on the origin located at the bregma, with the X- and Y-axes representing the mediolateral and anteroposterior axes, respectively.

Statistics and reproducibility

Data analyses were performed using MATLAB (R2017b, Natick, Massachusetts, USA), and the summarized data are reported as the mean ± SD unless otherwise specified. P < 0.05 was considered statistically significant. Sample size (the number of triple-patching datasets and single whole-cell recordings in awake condition) was determined by referencing previous publications (Jouhanneau et al., 2018, 2019). No statistical methods were used to predetermine the sample size. All conclusions of this study are based on recordings of populations of cells.

Data analysis

To detect periods with θLFP, wavelet transformations were first conducted for each LFP recording, the sampling rate of which was reduced to 500 Hz. Any period was defined as a θLFP period if the mean absolute value of its wavelet coefficients between 3‒10 Hz exceeded the mean ± 2 SDs of the values between 1–100 Hz. When the duration of θLFP or non-θLFP periods was <1 s, we regarded the duration as a non-θLFP or θLFP period, respectively. The 2 SD threshold for detecting the θLFP period was determined after ascertaining that the similarity of θLFP periods in the four simultaneously recorded LFPs at this threshold was most distinguishable from the surrogate data (Supplementary Fig. 2). θVm periods were also detected using the same threshold. The similarity of θLFP periods in simultaneously recorded LFP pairs was quantified using the Dice similarity coefficient, the double union of two independent sets divided by the sum of the two sets. The surrogate data were created by randomly shuffling the timings of the individual θLFP periods without changing the duration of each period. The peak frequency was calculated based on the wavelet power as the absolute value of the wavelet coefficient. The wavelet power at each frequency was averaged across the target period, and the frequency at which the mean wavelet power reached its maximum value was calculated. The Z score powers of θLFP and θVm were also calculated based on the wavelet power. We averaged the wavelet power for each frequency across 1-s bins. The maximum power for each bin and the Z scores of all bins were subsequently calculated.

Prediction of LFPs from Vms

A custom DNN model was constructed to predict LFPs from up to three simultaneously recorded Vms using Python. Our DNN had an encoder-decoder structure. The encoder compressed the input (i.e., 1, 2, or 3 Vms) to a lower dimensional representation and extracted features from the input, whereas the decoder reconstructed the final output (i.e., LFP) from the compressed vector. In our study, the Vms of 1, 2, or 3 cells over 1-s windows (size 500, 2-ms bins) were first passed through a convolutional layer. In this operation, the input was transformed into a feature matrix of reduced size through convolution by kernel matrices. This kernel processing enabled the extraction of meaningful features from the input into a smaller number of parameters66. Then, the layer output was flattened to a one-dimensional vector (size 32,000). The one-dimensional vector was passed through four fully connected layers and finally compressed to a size of 100. This lower dimensional representation was reconstructed to a size of 500 via three fully connected layers. Our DNN was implemented using the Python deep learning library Keras and the TensorFlow backend. The network was optimized by adaptive moment estimation (Adam) with a learning rate of 0.001. The parameters for the optimizer Adam were as follows: β1 (an exponential decay rate for the first moment estimates) = 0.9, β2 (an exponential decay rate for the second moment estimates) = 0.999, ε = 10−7, and decay = 0.0001; the default values were used for the other parameters. Because our focus was θ oscillations, raw LFP and Vm traces (20,000 Hz) were downsampled to 500 Hz and bandpass filtered between 3 and 10 Hz in Fig. 5. In Supplementary Fig. 7, the downsampled traces were bandpass filtered at 60–100 Hz, 25–55 Hz, and 0.5–1 Hz for high gamma, low gamma, and slow oscillations, respectively. Datasets with recording durations >3 min were used for prediction (n = 8). Our DNN model was trained to produce 1-s θ-filtered LFPs (500 bins) from up to three Vms of the corresponding time bins. To assess the model performance on the entire dataset, 10-fold cross-validations were used. Each dataset was equally divided into 10 subsets, and during each training session, one subset was used as test data, while the remaining 9 subsets were used as training data. To obtain sufficient numbers of training data, 1-s segments (500 bins) were extracted by shifting a 1-s time window at a step of 2 ms (1 bin) across the training data. For test data, 1-s segments were extracted by shifting a 1-s time window at a step of 100 ms. For each cross-validation, the training lasted 50 epochs with a batch size of 256, and the RMSEs were calculated to assess how well the model predicted new data that had not been used for training. As a randomized control, surrogate data were produced by shuffling the combinations of three Vms; that is, the time labels for segments were exchanged within each cell. The DNN was also trained using the shuffled data, and RMSEs were calculated. To evaluate the significance for prediction, the RMSEs of all 1-s segments in all 10 subsets were pooled for either original or shuffled data in each dataset, and a two-sample Kolmogorov‒Smirnov test was used to calculate the D-values and P-values for each dataset.

Model and parameter tuning

The model architecture and training parameters were optimized on a different dataset before analyzing the main dataset. First, we compared four model structures with distinct characteristics (Supplementary Fig. 9a). In Model 1, which became our final model, the input is first passed to a convolutional layer and then to a series of fully connected layers. The structure of Model 2 is similar to that of Model 1, but without the first convolutional layer. Model 3 has only one fully connected layer. Finally, Model 4 has a convolutional layer and fully connected layers but does not have an encoder-decoder structure.

All models were trained on data that were specially prepared for model tuning (tuning datasets). Model 1, which was chosen as the final architecture in this work, showed the lowest RMSE value among all models (Supplementary Fig. 9b, c). In our model, the convolutional layer learns filters in the temporal dimension. Since convolutional layers are used to extract meaningful local structures, it is possible that convolutional layers successfully extracted local oscillations in the input Vms, which were important for predicting LFPs. The convolutional layer was followed by a series of deep layers. It is important to note that these fully connected layers have encoding and decoding architecture. By implementing this feature, the model is forced to extract and learn only the important features for predicting LFPs.

After the model architecture was selected, the model parameters were optimized based on the performance of the tuning data. As representative data, the results from four sets of parameters are shown in Supplementary Fig. 9d, e. Dropout rate of the dropout layers and learning rate were also optimized to 0.5 and 0.01 so that the RMSE takes the minimum value; the RMSE was 0.0203, 0.0229, 0.0205 when dropout rate was 0.5, 0.7, 0.9, and it was 0.0194, 0.112, 0.0201 when learning rate was 0.01, 0.001, 0.1. The optimal number of epochs was determined similarly based on the learning curve. The average of all traces shows that the RMSE value of the validation data hit the lowest at 5 epochs. After that, overfitting was observed as the RMSE of the validation data began to increase while the value of the training data continued to decrease.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.