Fig 1.
Data, theories, and models of the EEG alpha rhythm.
(A) Alpha oscillations are most strongly observable in the occipital lobe of the cerebral cortex (A1), where they are characterized by a peak in the power spectrum between 8 and 12 Hz (A2). Panel A3 summarizes the role alpha plays in cognitive processes, as well as abnormal alpha rhythm features observed in various diseases (see refs in main text). (B) Summary of the different theories that have been proposed to explain the alpha rhythm (thalamic image reference (B1) [18] and eigenmodes reference (B3) [19]). We focus on theories emphasizing the importance of interactions between neural populations (B2). (C) Alpha rhythm theories are clarified and concretized by mathematical formulations, allowing numerical and analytical investigation of their predictive and explanatory scope. The principal class of models used to date are neural population (neural mass and neural field) models (C1 and C2), which are the focus of the present work (reference for neurons in C4 [20]).
Fig 2.
Schematic depiction of two candidate theories of alpha rhythmogenesis.
(A) Cortico-cortical columnar microcircuit model, representing the generation of alpha rhythm through interconnected macrocolumns. (B) Cortico-thalamic model, involving thalamic neural populations in the process of alpha genesis.
Fig 3.
Foundational components of convolution-based NMMs.
Neural populations are composed of (A) A rate-to-potential operator describing the postsynaptic potential generated by the firing rates of the presynaptic neurons; and (B) a potential-to-rate operator, typically expressed as a nonlinear function, to relate the membrane potential of the neurons to their spiking activity.
Fig 4.
JR model topography, schematic, numerical mathematical expression, and alpha simulation results.
(A) General structure of the model, along with a detailed schematic that includes the operators and representations of the connectivities. (B) Numerical mathematical expression for each neural population (neuron hand-drawn based on Fig 1 in [45]). (C) Simulation outputs of the model with standard parameters (time series, power spectrum estimated from the time series).
Fig 5.
MDF model topography, schematic, numerical mathematical expression and alpha simulation results.
(A) Composed of three neural populations with similar wiring structure to JR with the addition of an inhibitory self-connection. (B) Numerical mathematical expression for each neural population (neuron hand-drawn based on Fig 1 in [45]). (C) Simulation outputs of the model with modified parameters to generate alpha oscillations (time series, power spectrum estimated from the time series).
Fig 6.
LW model topography, schematic, numerical mathematical expression, and alpha simulation results.
(A) The general structure of the model is two neural populations each with a self-connection. In the detailed schematic, compared to the other models, a third block is introduced to transform PSP into soma membrane potential. (B) Numerical mathematical expression for each neural population (neuron hand-drawn based on Fig 1 in [45]). (C) Simulation outputs of the model with standard parameters (time series, power spectrum estimated from the time series).
Fig 7.
RRW model topography, schematic, numerical and analytical mathematical expression, and alpha simulation results.
(A) Three main populations are broadly described: the cortex (composed of excitatory and inhibitory neurons) and two thalamic populations (reticular nucleus and relay nuclei). Delays are included to take into account long range connections from the cortex to the thalamus. (B) Numerical mathematical expression for each neural population; Numerical mathematical expression for each neural population. (C) Simulation outputs of the model with standard parameters (time series, power spectrum estimated from the time series.
Fig 8.
Simulation results with standard parameter settings to generate characteristic resting state alpha oscillations features.
(A) Power spectra with characteristic occipital alpha rhythm from empirical EEG time series (left), from numerical simulation results (middle), and from analytical simulations (right). The red zone in the simulated results corresponds to the alpha range. All models generate an alpha oscillation with variations in specific features (peak frequency, presence of harmonics, 1/f shape). (B) Simulation results for EC and EO in JR, LW and RRW. The difference from EC to EO is an attenuation in the amplitude of the alpha rhythm.
Table 1.
Evaluating model performance against empirical EEG features
Fig 9.
Effect of rate constants on dominant frequency of oscillation for the JR, MDF, and LW models.
(A) Example time series and power spectra of a set of specific rate constant values to show the slowing in frequency as the values of the excitatory and inhibitory rate constant increase. (B) Heatmap presenting the dominant frequency of oscillation as a function of the rate constants of the JR model. (C) Three heatmaps for the JR, MDF and LW with the dominant frequency of oscillation as a function of the rate constants. For JR and MDF and
are varied from 2 ms to 60 ms. For LW,
changes from 1.72 ms to 5 ms, and
from 10 to 50ms to generate oscillatory behaviour.
Fig 10.
Frequency of oscillation parameter spaces as a function of E-I connectivities.
(A) Schematic of the models with their principal E-I loop highlighted. These are the parameters that are going to be varied. (B1 and B2) Time series and corresponding power spectra for specific combinations of E-I, showing different dynamics. (C) Heatmaps presenting the dominant frequency of oscillation as a function of E-I connectivity. The dark region presents non-oscillatory or non-physiological time series. JR and LW have a clearly defined regime of lower frequency of oscillations being generated (purple and red region), whereas RRW quickly tends to produce signals of lower amplitude, or higher frequency of oscillations. In RRW, the dark blue regime indicates that the system is still oscillating but at a higher amplitude and higher frequency as the system is starting to explode. In the light blue regime, the dominant frequency of oscillation is in the beta regime. In the three models, white or orange areas correspond to alpha or higher oscillations.
Fig 11.
Fixed points and corresponding phase planes of JR and LW at specific connectivity values with high and low noise.
By performing stability analysis, the stability of the fixed points of JR and LW is determined for connectivity values intersecting across the parameter space (yellow arrow). For JR, (A1) are the fixed points and (A2) is the phase plane for specific values of connectivity. Similarly to JR, in (B1) the fixed points of LW are presented with the corresponding phase plane in (B2). Unstable fixed points are red, whereas stable fixed points are blue. The light orange area corresponds to the optimal connectivity parameter setting to generate alpha oscillations in each model.
Table 2.
Common parameters across models based on their biological interpretation.
Fig 12.
Sigmoid curve of each model with firing rate against voltage with different firing threshold.
The sigmoids differ in terms of the maximum value and the voltage at which the inflection point occurs which is modulated by the firing threshold.
Fig 13.
Impulse response of excitatory and inhibitory population with varying rate constant.
Top: EPSP; Bottom: IPSP; except for RRW which uses the same dendritic response curve for EPSP and IPSP. The general shape of EPSP and IPSP between the models is consistent and mainly differ in terms of amplitude. Rate constant is varied for the first three models and for RRW, the different curves correspond to varying decay times.
Table 3.
Global evaluation of the models