Network States Classification based on Local Field Potential Recordings in the Awake Mouse Neocortex

Simultaneous intracellular and extracellular dynamics in the somatosensory cortex of awake mice: variability and network states of wakefulness

We performed simultaneous closely-located (∼200–250 μm) recordings of the LFP and the membrane potential (V_m) of pyramidal cells (see traces from two example recordings in Fig. 1a) in the superficial layers of the barrel cortex (S1). Recordings were performed in awake head-fixed mice habituated to sit quietly on the recording rig (Materials and Methods; see also Poulet and Petersen, 2008). Recordings had a duration of 5.1 ± 2.0 min (n = 14 from 4 mice). Before proceeding to use these data to define indices of network states from the LFP, we document some of its basic properties.

First, we found a notable heterogeneity across recordings (compare recording #2 with #11 in Fig. 1a). In particular, the relationship between extracellular population (LFP) and intracellular (V_m) signals was highly variable (Fig 1c, recordings were sorted by their level of absolute correlation, note the large range of observed correlation values) and this variability could not be explained by cellular properties of the recorded cell (such as the membrane resistance and action potential threshold; see Extended Data Fig. 1-1). Second, intracellular and extracellular dynamics had a rich repertoire of activity patterns (illustrated in Fig. 1b). As previously reported under similar conditions (Poulet and Petersen, 2008; McGinley et al., 2015a; Vinck et al., 2015; Chen et al., 2017; Einstein et al., 2017), both LFP and V_m traces displayed epochs of rhythmic activity in the δ-band (defined as time samples with high [2,4] Hz envelope, see the V_m and LFP spectrums in Fig. 1d,h, respectively, peaks were observed at 3.0 ± 0.3 Hz for the V_m and 3.3 ± 0.5 for the LFP, n = 14 recordings). Those rhythmic epochs presented a high synchronization between LFP and V_m (correlation between V_m and LFP δ envelopes across time samples: 0.55 ± 0.11, one-sample t test for positive correlation, p = 2e-10, n = 14). Examples epochs of rhythmic activity are shown in traces number 1 and 2 of recording #11 (Fig. 1b). Next, confirming previous observations (McGinley et al., 2015a; Neske et al., 2019; Zerlaut et al. 2019; Nestvogel and McCormick, 2022), we found nonrhythmic epochs (here defined as time samples with a V_m δ envelope lower than 6 mV; see Fig. 1e) at different depolarization levels (see the population histogram on Fig. 1f ranging from ∼−80 mV hyperpolarization levels to ∼−45 mV depolarization levels). Example epochs of nonrhythmic activity patterns at different depolarization levels (increasing from epoch 3 to 7) are shown in Figure 1b. Taken together, the epochs 1–7 of Figure 1b recapitulate the different states described by the “U-model” of cortical states (McGinley et al., 2015b), that is rhythmic states of δ-band activity and nonrhythmic states at various membrane potential depolarization levels (from hyperpolarized to depolarized). Finally, recordings also differed not only in terms of the average strength of δ rhythmicity, but also in terms of the distribution of V_m levels over time (Fig. 1f). Recordings with lower average δ envelope tended to have lower variability of V_m levels over time in nonrhythmic states (Fig. 1g, Pearson correlation between mean V_m δ envelope per recording and V_m SD of nonrhythmic time samples, c = 0.83, p = 2e-4). The diversity of recordings thus filled a continuum between two qualitatively different cases of either recordings displaying mostly nonrhythmic states at an almost constant depolarization level (e.g., recording #2 in Fig. 1a, see V_m histogram in Fig. 1f showing a low variability of mean V_m over time) and recordings exhibiting overall stronger time-averaged δ V_m envelope and a much wider variation of the V_m depolarization levels (e.g., recording #11 in Fig. 1a, see V_m histogram on Fig. 1f). These latter cases exhibited a complex dynamics with the alternation of oscillatory δ-band activity together with nonrhythmic activity at very different V_m depolarization levels (see example epochs of recording #11 in Fig. 1b). In the next sections, we investigate how to quantitatively classify and differentiate these network states based on the extracellular LFP.

Limitations of the existing spectral analysis of LFP for the characterization of network states with different degrees of membrane potential rhythmicity and depolarization

Because of the high impact of different strength of V_m rhythmicity and V_m depolarization levels on behavior and sensory function, we next considered how to determine a quantitative index of networks states with such V_m features from the more easily accessible LFP signal. Ideal properties of this index would include: (1) the ability to predict the rhythmicity and depolarization of V_m from the LFP; (2) a U-shaped dependence of the index on depolarization levels of membrane potential and of firing of local neural populations to directly map onto the U-model of network states.

We first evaluated whether existing methods based on simple spectral properties, could be used to characterize in this way, using only the LFP, the diversity of network states observed in the awake neocortex. Previous LFP-based characterization of different network states relied on the ratio between δ and γ power (Cheng-yu et al., 2009; Saleem et al., 2010). We therefore computed the time-varying δ [2,4] Hz and γ [30,80] Hz envelope of the LFP, and the γ-to-δ envelope ratio over time samples (see example single recording and population histograms in Fig. 1i). We investigated the ability of the γ-to-δ ratio to differentiate between epochs of activity that have dynamical features resembling the network states previously documented with V_m and described by the U-model of cortical states (McGinley et al., 2015b). To gain intuition, we first considered the example epochs 1–7, which were sorted according to the γ-to-δ ratio of their LFP (Fig. 1i, see the corresponding time-varying δ and γ envelopes for those epochs in Fig. 1b). While this ratio could distinguish the strongly rhythmic epochs (1,2) from the high-γ and highly depolarized nonrhythmic epoch 7 (Fig. 1i), it could not distinguish well different V_m depolarization levels within the nonrhythmic sets of epochs. Epoch 6 had a mean V_m depolarization value >15 mV higher than that of epochs 3, 5, but all these three epochs had similar γ-to-δ ratios (Fig. 1i). Moreover, a nonrhythmic epoch (4) had similar γ-to-δ ratio to the two rhythmic epochs (1, 2). Overall, when quantifying the dependence of mean depolarization level on the γ-to-δ ratio across all epochs for either the example recording #11 (Fig. 1j) or across all sessions (Fig. 1k), it was apparent that, using the γ-to-δ LFP ratio, it would be very difficult to distinguish between rhythmic states and nonrhythmic hyperpolarized states, and to distinguish between hyperpolarized and depolarized states within the nonrhythmic range. Next, because states with nonrhythmic and hyperpolarized V_m are accompanied by a reduced level of spiking activity (McGinley et al., 2015a; Neske et al., 2019; Zerlaut et al., 2019; Nestvogel and McCormick, 2022), we also analyzed the level of population firing by computing the MUA from the extracellular recordings (see Materials and Methods). Similar to what we observed with the average V_m depolarization (Fig. 1k), we found that the γ-to-δ LFP ratio had a very weak predictive power with regard to the population spiking activity (Fig. 1l). Thus, the γ-to-δ ratio could not be used to identify states of reduced spiking network activity during nonrhythmic activity. Reduced depolarization levels and spiking activity are important as they identify specific network states which are characterized by different properties of sensory information processing during wakefulness (Reimer et al., 2014; McGinley et al., 2015a; Vinck et al., 2015; Neske et al., 2019).

We concluded that the γ-to-δ LFP ratio poorly differentiated rhythmic and nonrhythmic states and, more importantly, did not enable to evidence different levels of depolarization within the nonrhythmic states observed in cortical dynamics under awake condition. In the next section, we introduce a processing step of the LFP that allows such characterization.

Using the time-varying high-γ envelope of the LFP for a richer network state characterization

Because the membrane potential V_m is the reference signal for cortical state classification (Poulet and Petersen, 2008; Polack et al., 2013; Reimer et al., 2014; McGinley et al., 2015a; Einstein et al., 2017; Arroyo et al., 2018), and because we wanted to capture from the LFP the finer features of V_m dynamics (including the variations in rhythmicity and depolarization levels posited by the U model) that cannot be captured by the simple δ-to-γ ratio, we next investigated whether simple mathematical transformations of the LFP displayed temporal fluctuations more qualitatively similar to those of the membrane potential.

The inverted LFP (–LFP) displays high correlation values (cc ∼0.5) with the membrane potential in awake animals (Poulet and Petersen, 2008; Arroyo et al., 2018) and could thus potentially provide a basis signal for the characterization of network states. However, the amplitude of the LFP is strongly dependent on the depth of the recording (Sakata and Harris, 2009; Kajikawa and Schroeder, 2011; Lindén et al., 2011; Smith et al., 2012; Herreras et al., 2015) and is subjected to drifts over short (<10 s, Fig. 2a; epochs i and ii) and long (>1 min) time scales. These factors limit the similarity between V_m and –LFP, and they were shown in previous work to prevent robust classification of network state during slow wave (<1 Hz) activity (Mukovski et al., 2007).

Guided by the previous findings in anesthetized animals (Mukovski et al., 2007), we hypothesized that the high-frequency content (f > 40 Hz, including the γ band activity) of the LFP would provide a good predictor of the depolarization level V_m. We therefore applied a wavelet transform to the extracellular LFP and identified the frequency band maximizing the cross-correlation with the simultaneously recorded membrane potential in our dataset (Fig. 2b). This was performed by independently varying a root frequency f and a width factor w, yielding the frequency band [f/w, f·w]. For each frequency band, we divided the band into 20 evenly spaced wavelet frequencies, we computed the mean over frequencies of the wavelet envelope of the LFP (resulting in the time-varying envelope shown in Fig. 2a), and we analyzed the correlation between this transformed LFP trace and the V_m trace after averaging over recordings (see the individual values per recording in Fig. 2c sorted by recording index in the inset). We found that the band maximizing this correlation was achieved for f_opt = 72.8 Hz and w_opt = 1.83, i.e., the [39.7,133.6] Hz band (see Fig. 2b). We also found that a temporal smoothing of the LFP envelope (with an optimal value T_opt = 42.2 ms; see Fig. 2d) enhanced its correlation with the V_m signal. We refer in the following to such smoothed high-γ envelope as the pLFP (in analogy with the terminology by Mukovski et al., 2007). Following previous literature, we interpreted this quantity as an approximation to the time-varying recruitment of synaptic activity from a local region (diameter: ∼100–200 μm) surrounding the extracellular electrode (Ray et al., 2008; Katzner et al., 2009; Lindén et al., 2011; Mazzoni et al., 2011; Ray and Maunsell, 2011; Buzsáki et al., 2012; Gaucher et al., 2012; Einevoll et al., 2013).

After this transformation of the LFP, we observed a qualitative match between the previously reported V_m signatures of network states (McGinley et al., 2015b) and specific features of the pLFP signal. We illustrate this finding on the recording shown in Figure 3a. We observed rhythmic activity at different envelope levels (example epochs #1, #2) as well as nonrhythmic fluctuations at various mean levels of pLFP signal (example epochs #3–7). This similarity encouraged us to develop a quantitative NSI based on the pLFP.

Designing a NSI from the pLFP

To better discriminate specific network states of wakefulness from the LFP, we thus developed a quantitative index from the pLFP: the pLFP-based NSI (NSI_pLFP). The rationale and procedure to compute the NSI from the pLFP are described in the following text and is sketched graphically with example data in Figure 3a.

To capture the slow fluctuations of network activity over time, we computed the sliding mean Y(t) of the pLFP over a slow time scale (T_mean = 500 ms). Because the pLFP signal had a nonzero value at all points, we quantified the baseline of the raw pLFP signal p₀ (set as the lowest 100th percentile of the pLFP distribution, and a measure of the level of baseline noise in the extracellular signal). We then analyzed pLFP fluctuations relative to this baseline level. We classified the pLFP fluctuations at such slow time scale as either rhythmic or nonrhythmic. Rhythmicity was quantified by the time-varying low-frequency envelope of the pLFP fluctuations δ_env(t) using a wavelet transform. We observed that establishing the rhythmic condition by only thresholding δ_env(t) would be misleading. Indeed, the δ-band envelope was strongly co-modulated by the mean activity level Y(t) even for nonrhythmic epochs (correlation coefficients between Y(t) and δ_env(t) in the nonrhythmic epochs defined as NSI_pLFP > 0: c = 0.39 ± 0.16 across the n = 14 recordings, significance of a positive correlation: p = 3.3e-10, one-sample t test). This indicated that the δ envelope could reach high values, and cross an arbitrary threshold in absence of strong rhythmicity. This phenomenon is visible on Figure 3a: the envelope in epoch 7 is equivalent to the envelope in epoch 2 without exhibiting the clear rhythmicity in the pLFP or in the V_m signal that characterized epoch 2 (middle and top plots, respectively). We therefore introduced a simple model-based criterion for evaluating rhythmicity based on the following reasoning. In a noiseless, purely rhythmic setting defined by pLFP(t) = p₀ + δ_env · (1 + sin(2·π·f_δ·t)), where δ_env is the envelope of the oscillation, we have Y(t) = p₀ +δ _env. If Y(t) has an additional nonrhythmic component, we get Y(t) > p₀ + δ_env(t) (i.e., the oscillation alone does not account for the mean level of the signal). We adapted this last relation to build our rhythmicity criterion: a higher signal mean Y(t) than the mean expected from the δ component implies nonrhythmicity. Given the nonideal nature of the signal and to compensate for the misestimation of rhythmicity in fluctuating regimes, we rescaled the slow oscillation with a parameter α (see the next section for the determination of α), i.e., we introduced the time-varying quantity X(t) = p₀ + α·δ_env(t). We compared the estimate of the rhythmic contribution, X(t), with the mean activity, Y(t), to quantify rhythmicity: if X(t) ≥ Y(t) activity was set as “rhythmic” (because the slow oscillation pattern is able to account for the mean activity level). Activity was defined as “nonrhythmic” otherwise. Finally, we quantified the amount of the pLFP activity in the two regimes. In the rhythmic regime, the amount of network activity was captured by the amplitude of the oscillation, 2·δ_env(t). In the nonrhythmic regime, the pLFP deviations from baseline Y(t)-p₀ estimated the level of ongoing activity.

The pLFP-derived NSI was defined as the amount of pLFP activity projected on the negative and positive axis for the rhythmic and nonrhythmic regimes respectively. Such a definition resulted in a continuous index where states with stronger δ components had negative values while nonrhythmic states with stronger high-γ components had higher positive values (Fig. 3a). As highlighted by the colored area (Fig. 3a, bottom plot, purple and kaki colors for rhythmic and nonrhythmic epochs, respectively), the classification into the two states relied on the sign of the difference between the X(t) and Y(t) signals (Fig. 3a, middle plot) followed by a projection on either the negative part of the axis weighted by the oscillation amplitude 2·δ_env(t) for rhythmic epochs, or the positive part of the axis weighted by the increase from baseline Y(t)-p₀ nonrhythmic epochs (Fig 3a, bottom plot). The described procedure for the computation of the NSI is formalized in Equation 4 in Materials and Methods.

As the time-varying signal NSI(t) might exhibit fluctuations due to noise in both the Y(t) and X(t) quantities (X and Y are derived from the noisy LFP signal and their difference might amplify noise, see for example the signal jumps in epochs 2, 3 in Fig. 3a), we added a consistency criterion to NSI(t) to obtain a robust state classification of individual epochs. We first defined network state “episodes” with a window of T_state = 400 ms and an update every T_state/2 = 200 ms. The motivation behind the choice of this time scale T_state was that it offered a good compromise between two constraints: T_state was long enough to get well defined states (e.g., more than half a cycle for oscillations in the [2,4] Hz range) and it was short enough to catch the fast and frequent switches of network states during wakefulness (McGinley et al., 2015b). The consistency criterion for episode classification required that, within a given time window of duration T_state, the fluctuations of the NSI signal remained within a fluctuation threshold, equal to the pLFP noise level p0 (because this noise level provided an estimate of the amount of signal below which a variation is not a robust signal variation). If this stability condition was met, a network state in this time window was labelled as “validated.” The “validated” states are highlighted with brown dots over some sample epochs in Figure 3a. If this stability condition was not met, a network state at a well-defined level could not be assessed and the network state was labelled as “unclassified.” The consistency criterion prevented state validations in the presence of strong fluctuations in the time-varying NSI signal (epochs 3 and 4 in Fig. 3a, bottom).

Calibration of the rhythmicity threshold in the NSI definition

The parameter α in the pLFP-based NSI weights the propensity to classify the network activity as rhythmic. Its effect is illustrated in Figure 3b. Increasing α increased the proportion of rhythmic states from ∼0% at α < 1 to ∼100% at α > 6. We used simultaneous V_m recordings to optimize α to ensure that the classification of states as rhythmic using the LFP actually finds states that would be defined as rhythmic based on the membrane potential. In Figure 3c, we show, as a function of α, the average across all episodes classified as rhythmic of the [2,4] Hz δ envelope of the membrane potential V_m. The δ envelope of V_m of the states classified as rhythmic decreased exponentially when α increased. We thus set α to a value α^opt = 2.87 that was equal to the decay constant of the V_m envelope as a function of α. This choice ensures we detect a large enough number of rhythmic states with a genuine amount of V_m rhythmicity. When classifying rhythmic states using such a value in the pLFP-based NSI algorithm, we indeed obtained that the states classified as rhythmic had V_m δ envelope systematically larger than the states classified as nonrhythmic (paired t test, p = 1.2e-7, n = 14 recordings). Importantly, such a α value was found to be very close to the value maximizing the fraction of unclassified states (α = 2.95, Fig. 3b, thin grey dashed line), thus corresponding to the most conservative setting to classify network states.

Electrophysiological signatures of NSI_pLFP-defined network states

We then analyzed additional electrophysiological features of network regimes defined by the NSI_pLFP measure. We show the relationship between the NSI_pLFP level and the mean V_m depolarization value for a single recording in Figure 3d and across all recordings in Figure 3e. In Figure 3f, we show the relationship between the NSI_pLFP level and the MUA across recordings.

States of robust rhythmicity (high δ envelope, e.g., epoch 1 in Fig. 3a) showed strongly negative NSI_pLFP values and were associated to intermediate depolarization and MUA levels (see population data in Fig. 3e,f). When the rhythmicity was not present (low values of pLFP δ envelope, as, e.g., in epochs 3 and 4 in Fig. 3a), both the depolarization and the MUA levels had values close to their minimum (see population data in Fig. 3e,f). States for which the δ component did not significantly contribute to the network activity (quantified by the pLFP) were classified as nonrhythmic (i.e., NSI_pLFP > 0) and both their depolarization and MUA levels strongly increased with the mean level of network activity as captured by the NSI_pLFP (epochs 4–7 in Fig. 3a and population data in Fig. 3e,f). Importantly, the values of mean depolarization and of mean spiking activity in Figure 3e,f had a much lower SE for any given value of the NSI_pLFP than the one that was found when considering the dependence of mean depolarization and of mean spiking activity on the γ-to-δ ratio (Fig. 1k,l), suggesting that the NSI_pLFP is a much tighter predictor of membrane potential and of spiking dynamics than the γ-to-δ ratio of the LFP.

We conclude that the NSI_pLFP, has several strengths, especially when compared to previous indices. The NSI_pLFP captured key features of V_m-defined network states during wakefulness (McGinley et al., 2015b): it enabled extraction of membrane potential activity regimes ranging from δ-band activity, to asynchronous regimes at low activity levels and asynchronous regimes at high activity levels (Fig. 3e,f). The NSI_pLFP therefore provided a quantitative measure of network states that allowed extracting the U-shape nature of cortical states from LFP recordings previously documented with intracellular recordings (Fig. 3e,f).

Evaluation of the pLFP-based NSI accuracy in estimating membrane potential-based features

The above considerations suggest that it should be possible to use the NSI_pLFP, a measure only based in LFPs, to identify reasonably well states that have either rhythmic or nonrhythmic membrane potential properties, and to identify among the states with nonrhythmic membrane potentials, those that have either depolarization or hyperpolarization of membrane potential. In this section, we quantified the accuracy of such state characterization using the pLFP-based NSI.

To this aim, we first computed the NSI on the V_m signal. The lower bound of the signal p0 was translated into the V_m0 value by taking the first percentile of the V_m distribution (see Figs. 2a, 4a). We computed the sliding mean and the time-varying low-frequency envelope with the same parameters as for the pLFP signal. We derived the X(t) and Y(t) and the “V_m-defined NSI” (NSI_Vm) according to Equation 4 (see Materials and Methods). This V_m-defined NSI has (by construction) negative values for the states with V_m rhythmicity, low positive values for states with hyperpolarized nonrhythmic V_m, and high positive values for states with depolarized nonrhythmic V_m. Thus, comparing the value of the V_m-defined and pLFP-defined NSI during the validated network states enables a simple quantification of how good is the pLFP-defined NSI at identifying states of rhythmic, nonrhythmic, depolarized and hyperpolarized membrane potential.

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Figure 4.

The pLFP-based NSI across multiple recordings: variability and estimated accuracy. We show seven recordings (from i to vii) covering the whole range of observed values of correlations between the intracellular (V_m) and extracellular (pLFP) signals. a, A 60-s sample of the simultaneous LFP (gray) and V_m (black) signals. At the bottom, we show the validated network states with their NSI value both for the “pLFP-defined NSI” (NSI_pLFP, brown dots) and “V_m-defined NSI” (NSI_Vm, black dots). b, Histogram of the NSI_pLFP over the whole recording length for each recording. The color code per recording (from red to blue, see also Fig. 2c) represents the value of the correlation coefficient between the V_m and pLFP signals. c, Scatter plots of the “V_m-defined NSI” (NSI_Vm) and the “pLFP-defined NSI” (NSI_pLFP) values for all validated episodes in a given recording (i.e., extending before and after the recording sample shown in a). We highlight the correct area with a green color and the incorrect area with a red color. The large red circles give examples of the different sort of rejections that may happen during cross-validation (see main text). See Extended Data Figure 4-1 for the accuracy estimate with different dataset segmentation.

Figure 4a shows the time course over different recording sessions of the NSI computed either on the pLFP or on V_m. From these plots, it is apparent that the two NSI indices are remarkably well matched over the time epochs of the recordings, with the occasional presence of episodes in which the two indices were mismatched (e.g., those marked by red circles in Fig .4a). To assess when the pLFP-based NSI did and did not correctly predict the NSI measured on the membrane potential, we implemented the criterion displayed in Figure 4c. We first determined the scaling factor F between the V_m-based NSI and the pLFP-based NSI, by performing the linear fit of the data predicting either both rhythmicity or both nonrhythmicity (i.e., on the lower left or the upper right of the plots in Fig. 4c). This linear relationship yielded a prediction for the NSI_Vm value from a NSI_pLFP value. Then, at a given time-point t_i, the prediction was considered as correct if the difference between the predicted and observed value of NSI_Vm lied within a tolerance interval defined by two free parameters, ptol and Vmtol . We set the value of ptol as the average noise level of the pLFP signal across recordings, i.e., pnoise = 2.85 μV. For the tolerance value Vmtol , we took Vmtol = 2 mV to be well above the noise level of V_m recordings (∼0.1 mV) and still have a high resolution in the [0–35] mV range of observed depolarization levels (see Fig. 3e). Correct detection thus occurred when the following conditions were met: F·(NSIVm(ti)+Vmtol)<NSIpLFP(ti)+ptol and F·(NSIVm(ti)−Vmtol)>NSIpLFP(ti)−ptol . The strictness of this criterion is illustrated in Figure 4c. States were taken as incorrect when there was a mismatch between the rhythmic versus nonrhythmic classification (as shown for cases ii, iv–vii in Fig. 4, see the nonmatching events highlighted with a red circle). Importantly, state classification was also taken as incorrect when the graded level of the rhythmicity or the nonrhythmicity was not predicted well enough (see Fig. 4, i, iii). For example, in recording #11 (Fig. 4, i), the NSI_Vm value did not display a high enough value to be linearly related to NSI_pLFP level according to the relationship F.

Using the tolerance criteria defined in the above paragraph, the accuracy of detection of a V_m-based NSI value from the pLFP-based NSI value was 79.7 ± 10.2% (mean ± SEM over the n = 14 recordings and all validated episodes). Decreasing the tolerance parameter values to ptol = 1 μV and Vmtol = 1 mV (corresponding to extremely strict matching criteria) led to an accuracy of 57.2 ± 15.1%, meaning that, in more than half of the cases, the network state could be identified even with such remarkably high precision. We checked that accuracy was not artificially inflated by overfitting by splitting the dataset into different training and test sets. In this control analysis, we found very similar parameters and we did not find any significant differences in the measured accuracies (see Extended Data Fig. 4-1).

We noted that the misclassifications were not homogeneously distributed across different states (see Table 2). A specific set of network state misclassifications represented 65.3% of the misclassifications over the merged episodes across all recordings (i.e., 13.1% of all episodes). In this set, rhythmic activity predicted from the V_m (NSI_Vm ≤ 0) was misclassified as nonrhythmic activity from the pLFP (NSI_pLFP > 0). We analyze in the next section the reasons behind this finding.

Graded aspect of network states in terms of pLFP and V_m fluctuations: rhythmic versus nonrhythmic states

The above analysis revealed a stronger tendency to classify network states as rhythmic from the V_m than from the pLFP fluctuations. This difference in classification suggested that the NSI measures revealed previously unexplored asymmetries between the characteristics of fluctuations of pLFP and V_m signal across different network states. We therefore analyzed more closely the correspondence between pLFP and V_m fluctuations for different levels of pLFP-based NSI and how this impacted our classification results.

For nonrhythmic activity (NSI_pLFP > 0), the level of the NSI_pLFP was given by the mean pLFP deflection (Y-p₀) in the time window T_state = 400 ms. We thus compared the positive NSI values to the mean membrane potential depolarization level in the same window T_state. For rhythmic activity (NSI_pLFP ≤ 0), the level of the NSI was proportional to the δ envelope of the pLFP. We thus compared the negative NSI values to the δ envelope in the V_m signal. We show the relationship between those pLFP-defined and V_m-defined levels for two recordings in Figure 5a,b (for recording #1 and recording #11, respectively) and for the population data on Figure 5c.

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Figure 5.

Relationship between pLFP-derived population activity (NSI_pLFP) and single-cell depolarization (V_m) in rhythmic and nonrhythmic regimes. a, Relationship, at the single recording level (shown for recording #1), between the NSI_pLFP levels and the properties of the V_m fluctuations. For rhythmic states (NSI_pLFP ≤ 0, purple color), we show the relationship between the NSI level and the V_m amplitude of the δ-band envelope. For nonrhythmic states (NSI_pLFP > 0, kaki color), we show the link between the NSI level and the mean V_m depolarization level over a T_state = 400 ms window. We highlight with gray dots the values of the single episodes visible on Figure 3a (reproduced on the top inset, V_m in black and pLFP in brown). We show the linear regressions for the rhythmic and nonrhythmic data (dashed red line). Note that the plain curve does not reach episode 3 because the minimum number of episodes for averaging is not reached at that level (see Materials and Methods). b, Same than a for recording #11. c, Reproducing the analysis of a, b over all n = 14 recordings (see main text). We show the mean relations over recordings (thin gray lines) and the mean (wide curve) and SD (shaded area) across recordings. Evaluated only for the NSI_pLFP levels displayed by multiple recordings (i.e., n = 6 recordings for rhythmic activity and n = 13 recordings for nonrhythmic activity). For each recording, we perform a linear regression with respect to the NSI_pLFP levels, we compute the mean across recordings 〈s〉 and the probability of a deviation from the 0-slope hypothesis p (paired t test). d, Relationship between the α parameter (that sets the proportion of nonrhythmic episodes; see Fig. 3c) and the mean slope over cells (top, 〈s〉_cells) together with the p-value testing the significance of a non-zero slope (bottom), i.e., reproducing the analysis of c for different α values.

Overall, we found (Fig. 5) that the membrane depolarization exhibited a strong dependency on the pLFP-based NSI level for nonrhythmic activity (NSI_pLFP > 0). We showed single episodes of increasing NSI_pLFP levels (numbered 3, 4, 5, 6 in Fig. 5a,b) and their respective episode averages at all NSI_pLFP > 0 levels for those two recordings (Fig. 5a,b, kaki curves). The ∼15 μV variability in terms of pLFP based NSI levels corresponded to a ∼30 mV variability of depolarization level with a clear monotonic relationship for those two sample recordings. This behavior was confirmed at the population level (Fig. 5c). We standardized the analysis across all recordings by binning the NSI_pLFP levels from 0 to 30 μV (a range covering all observed values) in bins of 1 μV. We found that all recordings exhibited depolarizations with a steep dependency on the NSI_pLFP level 1.5 ± 0.7 mV/μV, significantly deviating from the null hypothesis of a zero slope (p = 1.2e-5, n = 13 recordings, paired t test). On the other hand, we found that the value of pLFP-based NSI for rhythmic activity (NSI_pLFP ≤ 0) had a much lower impact on the membrane depolarization level (see Fig. 5). The ∼10 μV variability in terms of pLFP based NSI levels translated into a weakly modulated V_m oscillation with a 5 to 10 mV amplitude (Fig. 5a,b, purple curves). We highlight this weak dependency on the selected samples shown in Figure 5a,b. We extended the analysis to all recordings, after standardizing the data by binning the pLFP-based NSI levels from –30 to 0 μV. At the population level, we confirmed that the mean membrane depolarization had a weak dependency on the pLFP-based NSI level (–0.2 ± 0.3 mV μV), which was not significantly deviating from the null hypothesis of a zero slope (p = 0.14, over the n = 6 recordings displaying rhythmic activity within multiple NSI_pLFP bins, paired t test). It should be noted that the lack of graded V_m variations for rhythmic activity was not related to our “rhythmicity threshold” limiting the set of rhythmic samples to a potentially-biased subset. When varying the rhythmicity-factor α up to α = 5 (where the occurrence of rhythmic NSI_pLFP ≤ 0 states reaches ∼80%; see Fig. 3b), the depolarization level still showed a very weak dependency to the NSI level compared to that for nonrhythmic activity (Fig. 5d, top) with similar statistical significance values (see Fig. 5d, bottom).

We concluded that the graded levels of neural activity captured by the NSI_pLFP had a strong correlate in terms of membrane potential depolarization for nonrhythmic activity (NSI_pLFP > 0), while the various pLFP-based NSI levels of rhythmic activity (NSI_pLFP ≤ 0) rather corresponded to a stereotypical V_m oscillation with ∼5 to 10 mV amplitude.

Those observations explained the results of our cross-validation analysis (Table 2). The high precision of the classification in the jointly nonrhythmic case (“NSI_pLFP > 0 and NSI_Vm > 0” in Table 2, only 5.4% of the misclassifications) resulted from the strong relationship between the pLFP and V_m signals during nonrhythmic states (Fig. 5a–c, kaki curves). On the other hand, the existence of a few episodes showing a high envelope δ oscillation in the V_m with a low δ envelope in the pLFP (cases such as episode 2 in Fig. 5a,b) created an ambiguous situation for the classifier because rhythmicity was hard to establish from the pLFP signal in those episodes. The cases with mixed rhythmic/nonrhythmic predictions indeed represented 90.3% of the misclassifications (“NSI_pLFP > 0 and NSI_Vm ≤ 0” and “NSI_pLFP ≤ 0 and NSI_Vm > 0” in Table 2). In particular, predicting rhythmicity from the V_m and nonrhythmicity from the pLFP was the prevalent misclassification case (65.3% for the case “NSI_pLFP ≤ 0 and NSI_Vm > 0”), consistent with the stronger representation of the δ pattern in the V_m than in the pLFP (NSI_pLFP ≤ 0 range in Fig. 5a–c). We confirmed that such misclassification cases originated from episodes of low δ envelope in the pLFP signal: the mean δ_env in misclassified cases was significantly lower than in accurately classified cases (1.5 ± 0.3 vs 2.7 ± 0.5 μV, p = 3.3e-10 paired t test, in the n = 13 recordings displaying both conditions).

Using the NSI_pLFP to quantify network state distributions in the somato-sensory cortex of awake head-fixed mice

We analyzed the NSI_pLFP state distribution in the whole dataset (n = 14 simultaneous V_m and LFP recordings in the somato-sensory cortex of awake head-fixed awake mice). Figure 4 shows eight example recordings representative of the dataset, spanning the full range of correlations between the pLFP and V_m fluctuations (see Fig. 2c). We show a 60-s sample of the intracellular and extracellular recordings together with the time-varying NSI_pLFP (Fig. 4a) and the histogram of the NSI_pLFP levels across the whole recording (Fig. 4b).

Overall, the dataset was dominated by nonrhythmic activity with a fraction of nonrhythmic states: FNSI>0 = 81.1 ± 15.9%. The fraction of rhythmic activity exhibited a large variability with n = 3 recordings below FNSI≤0 = 2% and n = 3 recordings above FNSI≤0 = 35% (peaking at FNSI≤0 = 46.8% for recording #1, see Fig. 4b).

The mean absolute NSI_pLFP value over rhythmic periods (i.e., NSI_pLFP ≤ 0) was 5.65 ± 1.02 μV (mean ± SEM over n = 14 recordings) and 6.67 ± 1.34 μV for nonrhythmic periods (i.e., NSI_pLFP > 0), yielding a significant increase from rhythmic to nonrhythmic states (p = 3.0e-5, paired t test). The increased amplitude in terms of pLFP signal (i.e., high-frequency content of the LFP) suggested that, as an average, synaptic activity was stronger in the nonrhythmic periods that in the rhythmic ones (see Discussion). This increased range of pLFP level was also true for the maximum level displayed by single recordings with 9.52 ± 2.25 versus 12.06 ± 3.4 μV (p = 9.8e-3, paired t test) for rhythmic and nonrhythmic activity, respectively. During nonrhythmic periods, we found a mild but significant increase of the variability (SD, from 1.29 ± 0.32 to 1.56 ± 0.71 μV, p = 8.9e-2, paired t test) and skewness of the pLFP distributions (from 0.35 ± 0.33 to 0.7 ± 0.41, p = 8.2e-2, paired t test).

Network state variability within individual recordings predicts the average correlation between population and single-cell signals

We next analyzed whether the NSI_pLFP state distributions could explain the variability of the correlation cc(V_m, pLFP) between the time-varying population signal pLFP(t) and the single-cell signal V_m(t) in our recordings (see Fig. 4a,b). We reduced the NSI_pLFP distribution per recording to a few components (detailed below) and we used univariate and multivariate linear regressions to analyse how the variability of those components across recordings explained the variability of the observed correlation values.

From top to bottom in Fig. 4b (i.e., from high to low correlation recordings), the distribution of NSI_pLFP levels across recordings was quantitatively different. At the top [high correlations, cc(V_m, pLFP) > 0.4, panels i–v], distributions were bimodal with two high peaks both in the rhythmic (NSI_pLFP ≤ 0) and nonrhythmic (NSI_pLFP > 0) areas. At the bottom [low correlations, cc(V_m, pLFP) < 0.4, panels vi, vii], distributions were dominated by nonrhythmic episodes with rather narrow range of NSI_pLFP values within the (NSI_pLFP > 0) domain. To quantify these features within the single recording distributions, we decomposed each distribution into the following quantities: (1) the mean NSI_pLFP > 0 over the whole recording μ_NSI; (2) the variability (SD) of the full NSI distribution σ_NSI; (3) the fraction of rhythmic episodes FNSI≤0 ; (4) the mean in NSI_pLFP values restricted to nonrhythmic episodes μNSI>0 ; (5) the mean in NSI_pLFP values restricted to rhythmic episodes μNSI≤0 ; (6) the variability in NSI_pLFP values restricted to nonrhythmic episodes σNSI>0; (7) the variability in NSI_pLFP values restricted to rhythmic episodes σNSI≤0 .

We show the results of a linear regression analysis for each factor in Figure 6a (ordered by explained variance) and their relationship with the correlation value in Figure 6b. The factor explaining the highest percentage of the variability (55.27% of the full variability) was the SD of the NSI distribution σNSI . Strikingly, the fraction of (synchronous) rhythmic states was only the third most important factor with an explained variance of 33.89%. A more important factor was found to be the variability of network states within nonrhythmic states σNSI>0 with an explained variance of 41.5%. Recording #14 in Figure 4ii provided an example of these observations. It exhibited strong variability in terms of nonrhythmic states and relatively low occurrence of rhythmic activity (e.g., lower than recording #1). However, it still displayed high correlation coefficient [with cc(Vm , pLFP) = 0.66]. Other individual factors had weak statistical significance (p > 0.04; see Fig.6a). However, using a multiple linear regression including all factors, we found that different components of the NSI distributions had complementary contributions in shaping the average correlation per recording. The full linear model indeed yielded an explained variance of 69.17% (after correction by the number of linear factors, f test p = 3.1e-2). Reducing the dimensionality of the linear model (up to three components), we found that combining the significant subcomponents of the NSI variability (σNSI>0 and FNSI≤0 ) with the variability within rhythmic states σNSI≤0 produced a statistically-significant model (f test, p = 6.6e-3) predicting 59.84% of the variability (corrected by the number of factors).

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Figure 6.

Features in the distribution of the pLFP-based NSI explain the diversity over recordings of the correlation value between the population (pLFP) and single-cell (V_m) signals. a, We perform a linear-regression between the correlation coefficient cc(V_m,pLFP) and the following quantities characterizing the NSI_pLFP distribution: (1) the variability σ_NSI (SD) of the NSI_pLFP values across the whole recording, (2) the variability in NSI_pLFP values restricted to nonrhythmic episodes σ_NSI>0 , (3) the fraction of rhythmic episodes F_NSI≤0 , (4) the mean NSI_pLFP over the whole recording μ_NSI, (5) the variability in NSI_pLFP values restricted to rhythmic episodes σ_NSI≤0 , (6) the mean in NSI_pLFP values restricted to nonrhythmic episodes μ_NSI>0 , (7) the mean in NSI values restricted to rhythmic episodes μ_NSI≤0. We show the explained variance for all quantities on the x-axis and the statistical significance of those linear models (p-values). We also performed multiple linear regressions with all those factors (dark gray bar) and we show the best three component model (light gray bar: σ_NSI>0, F_NSI≤0,σ_NSI≤0). We report the variance corrected by the number of linear factors (adjusted R²). b, Scatter plot between the value of a given factor and the V_m-pLFP correlation coefficient across individual recordings. Shown for the first five factors of the NSI_pLFP distribution with the highest percentage of explained variance (see variance explained and p-values in a). The color code of individual recordings matches that of Figure 2c.

We concluded that, in the present dataset, the average correlation between a single-cell signal (V_m) and the population signal (pLFP) could be largely explained (up to ∼70%) by features of the NSI_pLFP distributions. State variability among all NSI_pLFP-defined states (σNSI ) was a critical factor in determining such a variability (Fig. 6a, blue). We decomposed this state variability and found that the two most prominent factors are the variability within nonrhythmic states (σNSI>0 ; Fig 6a, orange) followed by the occurrence of synchronous rhythmic activity (FNSI≤0 ; Fig. 6a, green).

Computation of NSI_pLFP indices in mouse visual cortex

Finally, we demonstrate the general applicability of the method on a publicly available dataset from the Allen Institute for Brain Science (Siegle et al., 2021). We analyzed neural activity recorded with Neuropixels probes (Jun et al., 2017) in the V1 of awake mice on a steering wheel during spontaneous activity (grey screen presentation). We extracted a V1-located LFP signal from the probes (Materials and Methods) and we computed the time-varying population firing rate summing spikes from all well-isolated single units (Materials and Methods). We first re-performed all calibration steps of the method on the new dataset. We found that, with respect to our V_m and LFP S1 dataset analyzed in previous figures, the δ oscillation in V1 was clearly shifted up in frequency (Fig. 7a, peak at 5.7 ± 0.6 Hz, n = 11 sessions). We therefore shifted the δ-band to [4,8] Hz, and we decreased the temporal smoothing (T_smoothing = 30 ms) to avoid smoothing out the faster temporal dynamics of the δ oscillation. We next determined the optimal rhythmicity parameter α by analyzing the amount of δ envelope in rhythmic states when varying α (Fig. 7b, analysis as in Fig. 3c). We found an optimal value of α=3.07, close to the α= 2.85 value of the S1 dataset with V_m recordings. Optimal parameters for the V1 dataset are summarized in Table 3.

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Figure 7.

Application of the NSI analysis to the “Visual coding – Neuropixels” dataset of the Allen Institute (Siegle et al., 2021). a, pLFP envelope in the δ range for the Allen dataset in V1 and in our S1 dataset for comparison. b, Mean δ envelope in the spiking rate (Materials and Methods) across all pLFP-defined rhythmic episodes as a function of the α parameter. c, Network state variability in the Allen dataset captured by the pLFP-defined NSI. Thin lines represent single session data. d, Histogram of pLFP-defined NSI values depending on running conditions. The “running” condition corresponds to episodes when the absolute speed is >5cm/s and “still” when the absolute speed is <5 cm/s. e, Relationship between pLFP-defined NSI levels and average pupil area at those levels. f, Example episodes (from left to right) in session 774875821. From top to bottom, The single unit spikes over time, the time-varying rate computed from those spikes (thick transparent line: sliding mean), the raw LFP in the selected channel (see Materials and Methods), the pLFP signal (thick transparent line: sliding mean), and the pLFP-defined NSI with the validated episodes (validated episodes as dots). g, Same as in f for session 768515987. h, Relationship, at the single session level, between the NSI_pLFP levels and the rate fluctuations (shown as mean ± SEM over episodes). For rhythmic states (NSI_pLFP ≤ 0, purple color), we show the relationship between the NSI level and the rate amplitude of the δ-band. For nonrhythmic states (NSI_pLFP > 0, kaki color), we show the link between the NSI level and the mean rate level over a T_state = 400 ms window. We highlight with black dots the values of the single episodes shown in g. We show the linear regressions for the rhythmic and nonrhythmic data (dashed red line). i, Same as in h for session 774875821. j, Reproducing the analysis of h, i over all n = 11 sessions. Evaluated only for the NSI_pLFP levels displayed by multiple recordings (i.e., n = 9 recordings for rhythmic activity and n = 11 recordings for nonrhythmic activity; Materials and Methods). For each recording, we performed a linear regression with respect to the NSI_pLFP levels, we computed the mean across recordings 〈s〉 and the probability of a deviation from the 0-slope hypothesis p (paired t test). In a,b,c,d,e,j, data are shown as mean ± SEM (thick line with shaded area) over the sessions of the dataset.

Similarly to our S1 dataset, we observed that network activity in V1 during wakefulness was characterized by a strong variability of network activity patterns, including δ oscillation and nonrhythmic episodes at various spiking activity levels (see raw data in Fig. 7f,g and histograms in Fig. 7c). The activity was overall dominated by nonrhythmic episodes with a percentage of 86.7 ± 3.3% of all validated episodes (see Fig. 7c), i.e., activity was ∼5% less rhythmic than in our dataset. State distributions were less variable across recordings than in our S1 dataset (e.g., rhythmic activity fraction only varies from a minimum of 10.2% to a maximum of 22.5%), likely due to the increased sampling durations (20 min here vs ∼5 min in our dataset). We used this dataset to analyze the relationship between NSI_pLFP values, behavioral state and population firing rates. If the NSI_pLFP index well captures the U-shaped relationship between network states and behavioral states, it should satisfy three properties: (1) high values during running; (2) increasing values with pupil size; (3) a U-shape relationship with firing rates. We examined next how well the NSI_pLFP index matches these expectations.

We first computed the network state distribution in “running” (mean absolute running speed during the episode larger than 5cm/s) and “still” conditions (Fig. 7d). As predicted based on previous studies (Niell and Stryker, 2010; Ayaz et al., 2013; Polack et al., 2013; Vinck et al., 2015), we found a significant shift of the mean network state toward the higher values of NSI_pLFP (mean NSI_pLFP in running 31.3 ± 9.6 vs 18.9 ± 4.7 μV in still conditions, p = 2e-4, paired t test, n = 11 sessions). Next, we found a strong and monotonic relationship between NSI and pupil area (Fig. 7e, Pearson correlation, c = 0.85, p = 8e-6) as reported in previous studies (Reimer et al., 2014; McGinley et al., 2015a), suggesting that the NSI_pLFP is a good index of the states expected by the U-model (McGinley et al., 2015b). Finally, we investigated the dependence of population firing rates on the NSI_pLFP index. We found that the relationship between pLFP and spiking rates significantly deviated from the null hypothesis of 0-slope both in the rhythmic and nonrhythmic regimes (one sample t test with slope values, p < 0.05 in both cases, see values in Fig. 7j), with a positive slope for positive NSI_pLFP values and a negative slope for negative NSI_pLFP values. Thus, the NSI_pLFP had a U shape relationship with the population firing rate.