Fig 1.
Integrated Information decomposition framework (Φ-ID).
The Φ-ID framework describes how information is carried from present to future states of a system (information dynamics). By considering the simplest partition of a system—a bi-partition—and defining redundancy and double redundancy functions (see main text or [58]), we can compute all information atoms from the bottom to the top of the redundancy lattice (see at the bottom center of the panel). Each type of information atom captures a different mode in which information can be carried. The colors in the lattice indicate the type of atom: synergistic (red), unique (orange and yellow), and redundant (blue). Color combinations in certain atoms indicate time evolution of these atoms from one type in the present to another type in future.
Fig 2.
Measures to capture information dynamics.
Information measures computed from information atoms presented in the redundancy lattice. Each node has two PID atoms from the present to the future (α → β) indicated by a color code: synergistic (red), unique 1 and 2 (orange and yellow) and redundant (blue). The colored area indicates the atoms that make up each measure. Revised integrated information [37] (green), unique information (yellow) and non-synergetic redundancies (blue).
Fig 3.
Model for generation in silico of EEG brain rhythms: Local dynamics and network topology.
(A) The model implements a minimal neuronal circuit, where excitatory and inhibitory pulses are different and act in different timescales. (B) Schemes representing the network topology used in the present study as in [22–24]. Inhibitory neurons (blue dots) have 32 pre-synaptic excitatory neighbours (red dots inside the dashed circle) while each of them are pre-synaptic neighbours of 12 excitatory neurons (inside the solid line circle).
Table 1.
Meaning, symbols and values of the EEG model parameters.
Fig 4.
Measurement schemes used for the study of the information dynamics of the network.
(A) LPF-scheme: Five groups of 32 E and 9 I neurons (inside green circles) across the network were selected. The averaged membrane potential for each group was calculated, generating a five-channel EEG-like time series. (B) Scheme for Φ-ID analysis: 12 E neurons (E neurons inside the yellow circle) and 9 I (I neurons inside the green square) were selected. The states (active/inactive) of each of them were saved to obtain time series of discrete Boolean variables.
Fig 5.
Neuronal activity ρ features across the (μ, τrec) parameter space.
The white solid line on the left shows a second-order phase transition from a silent to an active state (AT). The white dashed line shows the emergence of an explosive first-order transition. The region between dashed lines is a meta-stable phase, as described in [24]. (A) Temporal average of excitatory and inhibitory neuronal activity 〈ρ〉: The region between the second-order and first-order transitions shows what appears to be a low activity intermediate phase (II.a), which gives place to a full active phase of high activity (II.b) for noise values μ > 5. (B) Temporal variance of excitatory and inhibitory neuronal activity: The inhibitory population shows complex activity patterns along the phase space with clear regions of high temporal variability of its activity (in phase II.b), whereas the excitatory population has, in general, low temporal variability in its activity along the phase space. (C) Temporal variance of Boolean states (X) of neurons: The maximum variance (gray region) appears to mark the transition line between the low activity intermediate phase II.a and the high activity phase II.b. Note that this finding occurs for both the excitatory and inhibitory neuronal populations.
Fig 6.
Phase diagram of emerging rhythms in the model.
Rhythms observed in the (A) excitatory and (B) inhibitory neuronal population across the phase diagram. Each color shows a specific band of brain rhythms. The regions were defined as areas in the parameter space where the maximum PSD in a given band exceeds 1011, except for γLow where the threshold is 1010. Additionally, for the inhibitory case, the threshold for δ waves is 2 × 1010. In the white regions, maximum PSD values in each band are below the corresponding thresholds, which means there is not a clear dominant rhythm. The colored stars indicate the point of highest maximum power for each rhythm. The detailed PSD for each band is presented in Fig A in S1 Appendix. The solid line indicates a second-order phase transition (absorbing-active phase transition); dashed line indicates a first-order phase transition. Dotted and dash-dotted lines indicate the onset of transition to a LAI phase, for the excitatory and inhibitory population respectively. Vertical and horizontal black arrows indicate the values of μ and τrec for which full spectrum is shown in Figs C and D in S1 Appendix respectively.
Fig 7.
Information dynamics across phase diagram for time delay τ = 1 bin (4ms).
The white solid line represents the previously explained second-order phase transition, while the white dashed line indicates the first-order phase transition present in the system. The white dotted (dash-dotted) line indicates the maximum variance in the states of the I(E) neurons (see Fig 5). The color code indicates the values of information measures. (A) Integrated information ΦR, (B) Redundant information and (C) Differentiated information
in excitatory and inhibitory groups respectively. The neuron groups consists of 12 E neurons and 9 I neurons as indicated in Fig 3B of Material and methods section.
Fig 8.
Relation between β rhythms and information transfer in inhibitory neuron population.
The figure only illustrate regions where β rhythms dominate. (A, B) Color maps show clear regions of maximum of PSD for β rhythms in Phase II.a, for both excitatory (A) and inhibitory (B) neuron populations. White regions correspond to other phases (without dominant β waves) that were ignored. Panel (C) shows the maximum transfer of information (see colorbar code on the right of panel D) between partitions of inhibitory neurons within the Phase II.a. Panel (D) illustrates the transfer of information of the bipartition that maximize information transfer in the inhibitory group. Maximum information transfer increases with increasing power of β waves in both neuron populations (region indicated by black arrow), showing that in our system, β waves are related with information transfer between inter-neurons. Panels (A, C, D) depict, however, that there are points where we see a high PSD in β, mainly in the excitatory population (red arrow), that does not relate with an increase in transfer information. Dotted black horizontal line indicates the value of τrec and range of μ used in Fig 9.
Fig 9.
Comparison of δ − β rhythms coexistence and emergence of dominant β rhythm shows different information dynamics profile.
Panels (A, B) depict maximum PSD in δ, θ, α and β bands and three information measures (Transfer, Redundancy and Differentiation) within Phase II.a for fixed τrec ≈ 180. (A) Excitatory and (B) Inhibitory neuron populations. As observed in Fig 8, information transfer (blue × symbol data), but also Redundancy (green × symbol data) in inhibitory group are maximum for the same μ that maximizes PSD for β rhythms in both populations (vertical dotted line in μ ≈ 4.5). The inhibitory differentiation (magenta × symbol data curve) increases considerable close to the discontinous transition between phase II.b and III (see 7C bottom). Here, we observe the beginning of this behaviour, which are not related to emerging waves in phase II.a. We also observe a first peak of PSD of β waves coexisting with slow waves—cf. panel (E) red PSD—that does not relate with a peak in information transfer (vertical dotted line for μ ≈ 1.8 in panel (B)). In this “waves coexistence” regime we observe a relatively small peak in Redundancy in excitatory neurons but no information transfer (see green × symbol curve and blue × symbol curve on panel (B)). Panels (C, D, E, F) illustrate the averaged membrane potential fluctuations and the corresponding PSDs for a group of excitatory and inhibitory neuron populations for μ ≈ 1.8 (panels C and E) and μ ≈ 4.5 (panels D and F). The membrane fluctuations features are clearly different and, while in (C) we have a rhythm with multiple frequency waves including slow and fast components (i.e. δ − β waves coexistence), in (D) we observe a clear dominant β rhythm regime.
Fig 10.
Relation between δ rhythm emergence in the meta-stable region and information transfer in excitatory neuron population.
Color maps in panels (A,B) show maximum PSD of δ rhythm in meta-stable region between II.a and III for excitatory (A) and inhibitory (B) neuronal populations. Panel (C) shows the maximum transfer of information between groups of excitatory neurons. White regions in panels (A-C) are part of the parameter space that we are not taking into account in the present analysis, as they belong to other phases. Panel (D) depicts the information transfer of a bipartition that maximizes information transfer in excitatory populations. Maximum information transfer increases with increasing power of the emerging δ waves in the excitatory population, while shows no clear correlation with δ waves in inhibitory population.
Fig 11.
Relation between information storage in excitatory and inhibitory neuron populations and rhythms emergence.
Information storage on phase II.a in (A) excitatory and (B) inhibitory neuronal groups. We observe higher values of information storage in excitatory population in the meta-stable region where δ rhythms have their higher PSD values (see Fig 10A), and also close to the LAI phase transition (see Fig 8A) where we observe the higher maximum PSD in the β band (as shown previously in Fig 8). Additionally, we can see that information storage in excitatory neurons seems to be an ubiquitous information dynamics for waves emergency in the region where we have coexistence between α, θ and β rhythms (as shown when 1 ⪅ μ ⪅ 4 in Fig 6A), where also there is also an increase in redundancy in excitatory neurons (see Fig 7B top). This indicates that, in phase II.a, high level of information storage in excitatory neuronal population is related with both, a clear rhythm emergence with strong power (as occurs for δ and β in the regions indicated) and coexistence of low frequency waves. Implications of this result will be discussed later in the text.
Fig 12.
Time-delayed mutual information and the emergence of fast γ rhythms.
(Top) PSD peak frequency of oscillations in excitatory (A) and inhibitory (B) neuron groups, where color opacity is directly proportional to the power spectrum density. (Bottom) Time-delayed mutual information in (A) excitatory and (B) inhibitory neuron groups. The solid black line indicates the region where high frequency excitatory oscillations γ emerge (see Fig 6A). The dashed black line indicates the region where inhibitory fast γ oscillations with PSD >1 × 109 coexist with excitatory oscillations with PSD >1 × 1011 (see Fig 6B). Panel (B) bottom depicts that there is a clear relationship between lower mutual information and inhibitory fast γ oscillations. Lower mutual information indicates that the system is less predictable, which means knowing pass states gives less information about the future of the system and vice versa. Dotted and dashed-dotted black lines indicate the line of maximum variance of the inhibitory and excitatory neuron states X, respectively.
Fig 13.
Two different regimes of γfast rhythms are discriminated by the information dynamics.
Regime 1 (left panels): representative (μ,τrec) points in phase II.b, where γfast dominates only in excitatory (E) neuron population. Regime 2 (right panels): representative (μ,τrec) points in phase II.b where γfast dominates in both E and inhibitory (I) populations. Panels (A,B) depict the dispersion of the points (shown also in the insets) along the Differentiated information in E neurons (x-axis) and Redundant information
in I neurons (y-axis) plane, for Regime 1 (A) and 2 (B). The inset of Panel (A) shows the maximum PSD of γfast band of E neuronal population in phase II.b, with exception of points corresponding to Regime 2 enclosed by a dashed black line and also appearing in the inset of panel (B). Panels (C,D) shows mean and standard deviation of the information measures values
in E and
in I obtained by grouping them into PSD intervals. The intervals are indicated in the discrete colorbar code of panel (D). Regime 1 (A and C) shows that the points with high intensity of excitatory γfast oscillations also have the highest
and
observed in the data. Regime 2 (B and D) shows less redundant information in the I population. Panel (C) clearly shows that, in Regime 1, higher
and
are associated with dominant high frequency γ oscillations (PSD>1010). However, panel (D) shows that, in Regime 2, there is almost no increase in redundant information as γfast band maximum PSD increases. Spider graphs on top of panels (A,B) show, for each γfast regime, information dynamic diagrams for two points with high values of maximum PSD in γfast band in the E neuronal population. The representative points were indicated with stars in panels (A,B) and their insets. There, the star’s color indicates the maximum PSD of γfast in excitatory population following the colormap code presented in the (A,B) panels.
Fig 14.
Emergence of γfast rhythms in I groups and information dynamics.
Colors indicate the level of maximum power density in the γfast band. The data points in (A) correspond to points of high activity ρI of the inhibitory population (region II.b) as shown in the inset (C) of the figure. The scatter plot of differentiated () and redundant (
) information in inhibitory groups, respectively, shows that the points of highest intensity of inhibitory fast γ oscillations have the lowest
and
observed in the data. These points are those where excitatory and inhibitory oscillations coexist (see Fig 13F). (B) By grouping data in power spectrum density (PSD) intervals and computing the mean and standard deviation of time-delayed mutual information (TDMI), we observe a clear inverse relation between inhibitory oscillations amplitude and mutual information. The black dashed lines in (B) are the averaged TDMI of all data.
Fig 15.
Emergence of γfast rhythms and the integrated information.
The data points correspond to the high activity ρI of the inhibitory population (region II.b) as in Figs 13 and 14. The scatter plot of these points shows the values of the integrated information (ΦR) in the excitatory and inhibitory groups at each point. The colors indicate the maximum power density level in the (A) excitatory and (B) inhibitory γfast band. Points with high power for γfast appear to concentrate in regions of lower integrated information in inhibitory groups.