Table 1.
List of abbreviations.
Fig 1.
Proposed neural mass model of the cortical column.
A) Layer distribution of the four neuronal types. The excitatory populations are the intrinsically bursting (IB), and the regulatory spiking (RS). The inhibitory populations are low-threshold spiking (LTS) and fast spiking (FS). B) Connectivity matrix values used for coupling the 14 populations modeled. Negative values correspond to inhibitory connections.
Fig 2.
A) The model comprises three neuronal populations labelled as ‘1’, ‘2’, and ‘3’, coloured in blue, red and green, respectively. This color legend is used across all panels in the figure. B) Connectivity matrix. C) Temporal dynamics of the three neuronal populations. D) Spectral density. The low frequency (4.40 Hz) is modulating the higher frequencies (50 and 57.8 HZ) which is demonstrated by the appearance of secondary peaks at frequencies 50Hz ± 4.40 Hz and 57.8Hz ± 4.40 Hz on both sides of the main peaks. The secondary peaks are indicated with arrows. E) Spectral density when substituting the sigmoid function with the linear function S(x) = x.
Fig 3.
A) Midx, ESC, cMI and TE. B) Midx, ESC, cMI and cTE when the noise is increased (σ1 = σ2 = σ3 = 10).
Fig 4.
Simulated temporal evolution of the postsynaptic potentials for all populations for one realization of the model.
Excitatory populations are depicted in red and inhibitory ones in green.
Fig 5.
Normalized spectral density (nSD) of the postsynaptic potentials shown in Fig 3 obtained by subtracting the mean of the spectral density vector and dividing by the standard deviation.
Excitatory populations are depicted in red and inhibitory ones in green. nSD coloured in black show the results when the connections between populations are set to zero.
Fig 6.
Laminar distribution of average LFP.
A) Temporal dynamics in layers 2/3 (L2/3), 4(L4), 5(L5) and 6(L6). B) Spectral density (SD).
Fig 7.
Phase-amplitude coupling (PAC) corresponding to the simulation presented in Fig 4.
Non-significant values were set to zero and are depicted in white. Black dots indicate existing anatomical connections (see Fig 1).
Fig 8.
Exploring the parameter space for three different PAC combinations.
A) Average cTE values for delta-gamma (orange), theta-gamma (green), and alpha-gamma (blue) PAC when considered 100 different values for nine different parameters: 1) a multiplying factor η = 0.03:0.03:3 controlling the global strength of the connectivity matrix (Γnm = ηΓnm), 2) the reciprocal of the time constant of RS populations (kRS = 5:5:500s−1), 3) the reciprocal of the time constant of IB populations (kIB = 5:5:500s−1), 4) the reciprocal of the time constant of LTS populations (kLTS = 5:5:500s−1), 5) the reciprocal of the time constant of FS populations (kFS = 5:5:500s−1), 6) the external input to the L4RS population (), 7) the external input to the L4FS population (
), 8) the gains of the six excitatory populations (GE ≡ G1 = G2 = G5 = G8 = G9 = G12 = 0.2:0.2:20mV), and 9) the gains of the eight inhibitory populations (GI ≡ G3 = G4 = G6 = G7 = G10 = G11 = G13 = G14 = 0.5:0.5:50mV). B) Plot of cTE versus η. C) Average cTE values for direct and indirect PAC connections. Labels ‘d’, and ‘i’ correspond to direct and indirect PAC connections.
Fig 9.
The link between local topological measures and PAC.
A) local clustering coefficient (Cm), B) local efficiency (Em), C) local betweenness centrality (Bm). In all panels, labels ‘d’, and ‘i’ correspond to direct and indirect PAC connections, respectively. Populations can send and/or receive PAC interactions, or they can be not involved in the generation of PAC.
Fig 10.
A) number of delta-gamma PAC connections, B) number of theta-gamma PAC connections, C) number of alpha-gamma PAC connections, D) topological measures: local clustering coefficient (Cm), local efficiency (Em), and local betweenness centrality (Bm). In all panels, connections can be direct or indirect, and populations can send and/or receive PAC interactions, or they can be not involved in the generation of PAC.
Fig 11.
Amplitude-phase coupling (APC) corresponding to the simulation presented in Figs 4 and 5.
Non-significant values were set to zero and are depicted in white. Black dots indicate existing anatomical connections (see Fig 1).
Fig 12.
Mechanisms mediating indirect PAC connections (Case I).
A) Three population toy model comprising three neuronal populations labelled as ‘1’, ‘2’, and ‘3’, oscillating in the theta (θ), delta (δ), and gamma (γ) bands. B) PAC involving the phase of theta in population 1 and the amplitude of gamma in population 3 () obtained when varying the connectivity parameters between populations 1 and 2 (Γ12 = 30:4:1000) and between populations 2 and 3 (Γ23 = 30:4:1000). Panels C to R, displays the 16 predictors used in the ten models explored (Table 2). S) Coefficient of determination (R2) for the ten models explored (Table 2). T) Correlation coefficient between the 16 predictors.
Fig 13.
Mechanisms mediating indirect PAC connections (Case II).
A) Three population toy model comprising three neuronal populations labelled as ‘1’, ‘2’, and ‘3’, oscillating in the theta (θ), beta (β), and gamma (γ) bands. B) PAC involving the phase of theta in population 1 and the amplitude of gamma in population 3 () obtained when varying the connectivity parameters between populations 1 and 2 (Γ12 = 30:4:1000) and between populations 2 and 3 (Γ23 = 30:4:1000). Panels C to R, display the 16 predictors used in the ten models explored (Table 2). S) Coefficient of determination (R2) for the ten models explored (Table 2). T) Correlation coefficient between the 16 predictors.
Table 2.
Indirect PAC modeled as a cascade of direct CFC and SFC in a three population network.
Two cases were considered: population 2 oscillates in the delta (δ) band (Case I), and population 2 oscillates in the beta (β) band (Case II). Populations 1 and 3 always oscillate in the theta (θ) and gamma (γ) bands, respectively.