Fig 1.
Cortical activity across the sleep-wake cycle is characterized by intermittent irregular transitions between brain rhythms with dominant spectral power.
(a) (Top panel) Spectrogram obtained from cortical EEG signal of a control group rat over a 2 h segment of 12-hour lights-on period (when rats predominantly sleep). Spectral power is calculated in non-overlapping time windows w = 5 s, and is color coded over a range (0-20 Hz) of physiologically-relevant frequencies. Segments in red indicate bursts of prominent activity in the low frequency band (0-4 Hz, corresponding to δ waves) and intermediate frequency band (4-8 Hz, corresponding to θ waves). (Bottom panel) Ratio Rθδ = S(θ)/S(δ) of the spectral power in the θ and δ band in logarithmic scale obtained for each window w from the spectrogram shown in top panel. Values Rθδ above a threshold Th = 0 indicate predominance of θ rhythm (in red), while values below the threshold Th = 0 correspond to predominance of δ rhythm (in blue). (b) Smoothed ratio Rθδ of the spectral power in the θ and δ band during a 30 min segment of 12-hour dark (lights-off) period for a control rat (top panel) and a PZ-lesioned rat (bottom panel). Rθδ is calculated in non-overlapping windows w = 5 s; smoothing is performed using a 5 point moving average. θ- and δ-bursts are defined as sequences of consecutive windows w where either the power in θ or δ band is dominant.
Fig 2.
Durations of θ- and δ-bursts across the 24 h sleep-wake cycle follow distinct statistics that are robust, do not change with lesion of the PZ neurons, and are typical for non-equilibrium systems at criticality.
(a) Probability density distribution of θ-burst durations for control (open circles) and PZ-lesioned rats (full triangles) over the 24 h period (pooled data). The distribution exhibits a power-law behavior in both groups (red tick line), with exponent αctrl = 2.34 ± 0.06 for control rats and αPZ = 2.31 ± 0.07 for PZ. The power-law exponent value for control and PZ rats do not show significant difference (t-test, p > 0.99). Lesion of the PZ area does not cause significant variations in the power-law exponent α, indicating a robust scale-invariant temporal organization of θ-bursts. (b) Probability density of δ-burst durations for control and PZ-lesioned rats over 24 h period (pooled data). In contrast to the statistics of θ-bursts, δ-bursts durations follow a Weibull distribution (stretched exponential tail with a characteristic time scale, Eq 2). Parameters of the Weibull functional form are not significantly affected by lesion of the PZ (βctrl = 0.59, λctrl = 0.16; βPZ = 0.54, λPZ = 0.13). Black line is a Weibull fit of the distribution for control rats. All durations are calculated using window size w = 5 s for the spectrogram and threshold Th = 1 on the ratio Rθδ (Fig 1). Error bars are calculated for each value and where not shown are smaller than the symbol size. Error bars calculation and binning procedure are described in Materials and methods, Data analysis.
Fig 3.
Critical behavior represented by duality of power-law and Weibull distribution for θ- and δ-bursts characterizes cortical activity during both dark and light periods in control and PZ-lesioned rats.
(a) Probability distributions of θ-burst durations for control (circles) and PZ-lesioned (triangles) rats over the 12 h dark lights-off period (pooled data) follow a power-law with an exponent αctrl = 2.44 ± 0.06 and αPZ = 2.40 ± 0.06 (higher than for the 24 h sleep-wake cycle, Fig 2), where the line shows a power-law fit for the control group. (b) Probability distributions of δ-burst durations for control and PZ-lesioned rats over 12 h dark period (pooled data) follow a Weibull form, with no significant differences in the fitting parameters (βctrl = 0.59, λctrl = 0.16; βPZ = 0.54, λPZ = 0.14), where the line shows a Weibull fit for the control group. (c) Probability distributions of θ-burst durations for control and PZ-lesioned rats over the 12h lights-on period (pooled data) also follow a power-law but with smaller exponent αctrl = 2.28 ± 0.07 and αPZ = 2.24 ± 0.08 compared to the dark period, indicating higher probability for longer durations. (d) Probability distributions of δ-burst durations for control and PZ-lesioned rats over the 12 h light period (pooled data) follow Weibull behavior for both groups with no significant differences in the fitting parameters (βctrl = 0.54, λctrl = 0.12; βPZ = 0.56, λPZ = 0.14). Lines in (c) and (d) show fits for the distributions of the control group. All durations are calculated using window size w = 5 s for the spectrogram and threshold Th = 1 on the ratio Rθδ (Fig 1). Error bars are calculated for each value and where not shown are smaller than the symbol size. Error bars calculation and binning procedure are described in Materials and methods, Data analysis. The power-law exponent value for control and PZ rats do not show significant difference for both light and dark periods (t-test, p > 0.46).
Fig 4.
Critical characteristics in temporal dynamics of bursts in dominant rhythms are a fundamental feature of cortical activity across the sleep-wake cycle independent of thresholds utilized to define bursts.
The distribution functional forms of power-law for θ-bursts durations and Weibull for δ-bursts durations remain preserved for different threshold values Th imposed on the ratio Rθδ (Fig 1). (a) Distributions of θ-burst durations for control rats over a 24h period (pooled data) evaluated using different Th values consistently follow the same power-law behavior (red line), with an exponential cut-off that is controlled by Th. With increasing Th the distribution cut-off shifts towards shorter burst durations, a finite size effect typically observed in systems at criticality. Inset: data for different Th collapse onto a single universal function fθ when we plot P(d)dα versus Thϵd, with α = 2.35 and ϵ = 0.8. (b) Rescaled distributions of δ-burst durations for control rats over a 24 h period (pooled data) obtained for different Th values, consistently follow the same Weibull form. Distributions are rescaled by 〈dδ〉η, where 〈dδ〉 is the mean δ-burst duration and η = 1.2. After rescaling, distributions collapse onto a single function following a Weibull behavior, f(d; λ, β) (black line) with λ = 0.55 and β = 0.59. Inset: Distributions Pδ for different thresholds Th (not rescaled). (c) Distributions of θ-burst durations for PZ-lesioned rats over a 24 h period (pooled data) evaluated using different threshold values Th consistently follow the same power-law behavior (red line), with an exponential cut-off controlled by Th. Inset: Data collapse onto a universal function fθ by plotting P(d)dα versus Thϵd with α = 2.35 and ϵ = 0.8 (same as for control rats in a). (d) Rescaled distribution of δ-burst durations for PZ-lesioned rats over a 24 h period (pooled data) obtained for different Th values follow the Weibull form. Distributions are rescaled by 〈dδ〉η, with 〈dδ〉 mean δ-burst duration and η = 1.2. After rescaling, the distributions collapse onto a single Weibull distribution f(d; λ, β) (black line) with λ = 0.44 and β = 0.54. Inset: Distributions Pδ for different thresholds Th (not rescaled). Results in all panels are obtained for a fixed scale of analysis, keeping the window size w = 5 s (Fig 1). Results are consistent when considering separately light and dark periods (Fig B and Fig C in S1 File).
Fig 5.
Critical characteristics in temporal dynamics of bursts in dominant cortical rhythms are independent of the scale of analysis.
Distributions of θ- and δ-burst durations for different scales of observation defined by the window size w (Fig 1). (a) Rescaled distribution Pθ of θ-burst durations for control rats over a 24 h period (pooled data). Distributions obtained for different scales of observation are rescaled by the window size w and consistently show the same power-law behavior with α = 2.35, as proven by the data collapse (red line). Small deviations observed on the tail of Pθ for w > 6 are due to coarse-graining effect at large-window sizes. Inset: Distributions Pθ for different window sizes w (not rescaled). (b) Rescaled distribution of δ-burst durations for control rats over a 24 h period (pooled data). Distributions obtained for different window sizes w are rescaled by 〈dδ〉ξ, where 〈dδ〉 is the mean δ-burst duration and ξ = 1.2, and collapse onto a single function that is well described by a Weibull distribution f(d; λ, β) with λ = 0.55 and β = 0.59 (black line). Inset: Distributions Pδ for different window sizes (not rescaled). (c) Rescaled distribution of θ-burst durations for PZ-lesioned rats over a 24h period (pooled data). Distributions obtained for different window sizes w are rescaled by the corresponding window size, and consistently show the same power-law behavior with α = 2.35, as proven by the data collapse (red line). Small deviation observed on the tail of Pθ for w > 6 are due to large-window effects. Inset: Distributions Pθ for different window sizes (not rescaled). (d) Rescaled distribution of δ-burst durations for PZ-lesioned rats over a 24 h period (pooled data). Distributions are rescaled by 〈dδ〉ξ, with ξ = 1.2, and collapse onto a single function following a Weibull behavior f(d; λ, β) (black line) with λ = 0.44 and β = 0.54. Inset: Distributions Pδ for different window sizes (not rescaled). Results in all panels are obtained for fixed threshold Th = 1 on the ratio Rθδ (Fig 1). Results are consistent when considering separately light and dark periods (Fig D and Fig E in S1 File).
Fig 6.
Self-similar structure in quiet times between consecutive θ-bursts indicates coupling between time of occurrence and burst duration.
(a) Schematic diagram of quiet time Δt between consecutive θ-bursts. A quiet time Δti is the time elapsed from the end of burst θi to the beginning of the following burst θi+1. (b) Top: Time series of θ-burst durations for about 600 min recording of a control rat. Middle: A 60 min segment from the sequence shown in top panel. Bottom: Sequence comprised only of the θ-burst durations longer than D0 = 15 s that are present in the 600 min time series shown in the top panel. Selecting only bursts longer than D0 = 15 s, the temporal pattern at the scale of 600 min looks similar to pattern at smaller scale of 60 min, indicating self-similar structure in quiet times. (c) Distribution of quiet times for different thresholds D0 on θ-burst durations over a 24 h period in control rats (blue symbols). When rescaled by 〈Δt〉 (main panel), distributions obtained for different D0 collapse onto a unique function that is well described by a generalized Gamma distribution G(x; b, ν, p) (solid green line), with b = 0.15, ν = 0.31, and p = 0.91. Applying the same procedure to a sequence of randomly reshuffled θ-burst durations, thus eliminating information about the timing of θ-bursts, leads to distributions that collapse onto an exponential function (dashed lines). Inset: Distributions of quiet times for different thresholds D0 before rescaling. (d) Distributions of quiet times for different thresholds D0 on θ-burst durations over a 24 h period in PZ-lesioned rats. Distributions collapse onto a unique function when rescaled by 〈Δt〉 (main panel). Similar to control rats, this function is well described by a generalized Gamma function G(x; b, ν, p) (solid green line), with b = 0.17, ν = 0.24, and p = 0.83. Distribution of quiet times obtained from a sequence of randomly reshuffled θ-burst durations collapse onto an exponential distribution (dashed lines). Insets: Distributions of quiet times in PZ-lesioned rats for different thresholds D0 before rescaling. Results are consistent when considering separately light and dark periods (Fig F in S1 File).
Fig 7.
Long-range power-law correlations in sequences of consecutive θ- and δ-burst durations indicate a dynamical system at criticality.
Detrended fluctuation analysis (Materials and methods: Data analysis) for sequences of θ- and δ-burst durations from control and PZ-lesioned rats. Burst durations are calculated for window size w = 5 s and threshold Th = 1 on the ratio Rθδ (Fig 1), and are analyzed separately for 12 h dark and light periods. The root mean square (r.m.s.) fluctuation function F(n) is obtained averaging over all rats in the control (a) and PZ-lesioned group (b), respectively. Log-log plots of F(n) vs the time scale of analysis n, where n is the number of consecutive burst durations, show power-law correlations over a broad range of scales n. The scaling exponents are significantly larger than 0.5, both in light and dark periods, indicating presence of positive (persistent) long-range correlations in θ-bursts for both control and PZ-lesioned rats. Similar results are found in sequences of δ-bursts for (c) control and (d) PZ-lesioned rats. The observed difference in the correlation exponents between θ- and δ-bursts is significant for both control and PZ rats during light and dark periods (t-test, p < 0.05).
Fig 8.
Coupling between δ- and θ-burst durations indicates a common mechanism regulating the activity of these rhythms in relation to sleep micro-architecture.
Scatter plots and rank correlation analysis demonstrate coupling between consecutive δ- and θ-burst durations. (a) Scatter plot of δ-burst ranks vs following θ-burst ranks in the 24h period for control rats. Each dot represents a pair formed by a δ-burst and the following θ-burst, with burst durations separately ranked among the δ-bursts and the θ-bursts (longest duration corresponding to highest rank). (b) Scatter plot of δ-burst ranks vs following θ-burst ranks in the 24h period for PZ lesioned rats. For each rat group, ranks are calculated separately for each rat and then plotted together. (c) Average Spearman’s cross-correlation coefficient for control and PZ-lesioned rats in dark, light and 24h periods. Anti-correlations between consecutive θ- and δ-bursts are stronger during light than during dark periods in each of the two rat groups. Comparing dark vs light periods, the Student’s t-test gives p = 0.651 for control rats and p = 0.461 for PZ-lesioned rats. Importantly, PZ-lesioned rats generally exhibit stronger anti-correlations than the control group, in particular during dark periods, where the strength of anti-correlated coupling increases with ≈ 30% compared to control rats (control vs PZ t-test: 24h, p = 0.158; dark, p = 0.121; light, p = 0.064). All correlation coefficients calculated in both groups are significantly different from the corresponding values obtained in the surrogates (red bars) after randomly reshuffling the original order of θ- and δ-bursts (t-test: p < 0.001). All durations are calculated using a window w = 5 s and threshold Th = 1 on the ratio Rθδ (as in Fig 1). This finding of anti-correlated coupling between θ- and δ-bursts durations is further supported by an independent analysis based on conditional probability (Fig G in S1 File).
Fig 9.
Schematic diagram of a phenomenological model to generate sequences of θ- and δ-burst durations with varied degree of anti-correlated coupling.
(a) First, burst durations dθ and dδ are randomly drawn from the empirically obtained distributions (power law and Weibull, Fig 2) and separately ranked. Durations d = n * w are a multiple of the scale of analysis (window size w = 2 s). (b) Ranks of θ- and δ-burst durations are then paired to form an anti-correlated sequence: if the rank(dθ) of a θ-burst is large, than the rank(dδ) of the following dδ-burst is selected to be smaller, and vice versa. Repeating this process leads to a sequence of generated dθ and dδ durations with a certain degree of anti-correlation. (c) This newly generated anti-correlated time series is binarized, i.e. ‘+’/’-’ is assigned to each window w that belongs either to a dθ (red, ‘+’) or dδ (blue, ‘-’) duration, respectively. The binary time series is then coarse grained according to a majority rule applied over a window Δ = 5w. From the resulting coarse-grained (CG) binary series, consecutive θ durations, , and δ durations,
, are extracted. Details of the model are given in Materials and methods: Model of anti-correlated burst coupling.
Fig 10.
Anti-correlations between consecutive θ- and δ-bursts durations are essential for emerging critical behavior with duality of power-law and Weibull dynamics.
Probability distributions of θ- and δ-burst durations from 24 h control and PZ-lesioned rat data coarse-grained (CG) over a window Δ = 10 s, are compared with the distributions obtained from the model-generated coarse-grained binary time series of θ- and δ-bursts durations with anti-correlations and without correlations (random pairing of θ- and δ-bursts, Fig 9). (a) Distributions Pθ(d) of θ-burst durations for: (i) 24 h control rats data (red diamonds), (ii) model-generated time series of θ- and δ-bursts durations with anti-correlations (green circles), and (iii) model-generated time series with random pairing of θ- and δ-bursts durations (magenta dashed line). Inset shows results from same analysis on Pθ(d) for the group of PZ-lesioned rats. (b) Distribution Pδ(d) of δ-burst durations for: (i) 24h control rats data (blue diamonds), (ii) model-generated time series with anti-correlations (green circles), and (iii) model-generated time series with random pairing of θ- and δ-bursts durations (magenta dashed line). Inset shows results from same analysis on Pδ(d) for the group of PZ-lesioned rats. In both (a) and (b), durations are in units of Δ, which is the window size used to coarse grain the sequences of θ- and δ-bursts durations. The distributions obtained from the model using anti-correlated dθ and dδ pairing (green circles) closely match the duration distributions for the original data (diamonds)—power law for Pθ(d) and Weibull for Pδ(d)—for both control and PZ-lesioned rats. In contrast, a random pairing of dθ and dδ produces duration distributions following the Poisson functional form (magenta dashed lines) that significantly deviates from the original data.