Fig 1.
Schematic of the cortical motif.
Coupled populations of excitatory (red) and inhibitory (blue) neurons. (a) Mean-field neural mass model with synaptic feedforward and feedback connections. Each node represents a population. (b) Schematic of the corresponding spiking AdEx neuron network with connections between and within both populations. Both populations receive independent input currents with a mean and a standard deviation
across all neurons of population α ∈ {E, I}.
Fig 2.
Bifurcation diagrams and time series.
Bifurcation diagrams depict the state space of the E-I system in terms of the mean external input currents to both subpopulations α ∈ {E, I}. (a) Bifurcation diagram of mean-field model without adaptation with up and down-states, a bistable region bi (green dashed contour) and an oscillatory region LCEI (white solid contour). (b) Diagram of the corresponding AdEx network with N = 50 × 103 neurons. (c) Mean-field model with somatic adaptation. The bistable region is replaced by a slow oscillatory region LCaE. (d) Diagram of the corresponding AdEx network. The color in panels a—d indicate the maximum population rate of the excitatory population (clipped at 80 Hz). (e) Example time series of the population rates of excitatory (red) and inhibitory (blue) populations at point A2 (top row) which is located in the fast excitatory-inhibitory limit cycle LCEI, and at point B3 (bottom row) which is located in the slow limit cycle LCaE. (f) Time series at corresponding points for the AdEx network. All parameters are listed in Table 1. The mean input currents to the points of interest A1-A3 and B3-B4 are provided in Table 2.
Table 1.
Summary of the model parameters.
Parameters apply for the Mean-Field model and the spiking AdEx network.
Table 2.
Values of the mean external inputs to the excitatory () and the inhibitory population (
) in units of nA for points of interest in the bifurcation diagrams Fig 2.
Fig 3.
Transition from multistability to slow oscillation is caused by somatic adaptation.
Bifurcation diagrams depending on the external input currents to both populations α ∈ {E, I} for varying somatic adaptation parameters a and b. The color indicates the maximum rate of the excitatory population. Oscillatory regions have a white contour, bistable regions have a green dashed contour. (a) Bifurcation diagrams of the mean-field model. On the diagonal (bright-colored diagrams), adaptation parameters coincide with (b). (b) Bifurcation diagrams of the corresponding AdEx network, N = 20 × 103. All parameters are listed in Table 1.
Fig 4.
State-dependent population response to time-varying input currents.
Population rates of the excitatory population (black) with an additional external electrical stimulus (red) applied to the excitatory population. (a, b) A DC step input with an amplitude of 60 pA (equivalent E-field amplitude: 12 V/m) pushes the system from the low-activity fixed point into the fast limit cycle LCEI. (c, d) A step input with amplitude 40 pA (8 V/m) pushes the system from LCEI into the up-state. (e, f) In the multistable region bi, a step input with amplitude 100 pA (20 V/m) pushes the system from the down-state into the up-state and back. (g, h) Inside the slow oscillatory region LCaE, an oscillating input current with amplitude 40 pA and a (matched) frequency of 3 phase-locks the ongoing oscillation. (i, j) A slow 4 Hz oscillatory input with amplitude 40 pA drives oscillations if the system is close to the oscillatory region LCaE. For the AdEx network, N = 100 × 103. All parameters are given in Table 1. The parameters of the points of interest are given in Table 2.
Fig 5.
Frequency entrainment of the population activity in response to oscillatory input.
The color represents the log-normalized power of the excitatory population’s rate frequency spectrum with high power in bright yellow and low power in dark purple. (a) Spectrum of the mean-field model parameterized at point A2 with an ongoing oscillation frequency of f0 = 22 Hz (horizontal green dashed line) in response to a stimulus with increasing frequency and an amplitude of 20 pA. An external electric field with a resonant stimulation frequency of f0 has an equivalent strength of 1.5 V/m. The stimulus entrains the oscillation from 18 Hz to 26 Hz, represented by a dashed green diagonal line. At 27 Hz, the oscillation falls back to its original frequency f0. At a stimulation frequency of 2f0, the ongoing oscillation at f0 locks again to the stimulus in a smaller range from 43 Hz to 47 Hz. (b) AdEx network with f0 = 30 Hz. Entrainment with an input current of 40 pA is effective from 27 Hz to 33 Hz. Electric field amplitude with frequency f0 corresponds to 2.5 V/m. (c) Mean-field model with a stimulus amplitude of 100 pA (7.5 V/m at 22 Hz). Green dashed lines mark the driving frequency fext and its first and second harmonics f1H and f2H and subharmonics f1SH and f2SH. Entrainment is effective from the lowest stimulation frequency up to 36 Hz at which the oscillation falls back to a frequency of 20 Hz. New diagonal lines appear due to interactions of the endogenous oscillation with the entrained harmonics and subharmonics. (d) AdEx network with stimulation amplitude of 140 pA (8.75 V/m at 30 Hz). For the AdEx network, N = 20 × 103. All parameters are given in Tables 1 and 2.
Fig 6.
Phase locking of ongoing oscillations via weak oscillatory inputs.
The left panels show heatmaps of the level of phase locking for (a) the mean-field model and (b) the AdEx network for different stimulus frequencies and amplitudes. Dark areas represent effective phase locking and bright yellow areas represent no phase locking. Phase locking is measured by the standard deviation of the Kuramoto order parameter R(t) which is a measure for phase synchrony. White dashed lines correspond to electric fields with equivalent strengths in V/m. (c) Time series of four points indicated in (a) with the excitatory population’s rate in black and the external input in red (upper panels). In the lower panels, the Kuramoto order parameter R(t) is shown, measuring the phase synchrony between the population rate and the external input. Constant R(t) represents effective phase locking (phase difference between rate and input is constant), fluctuating R(t) indicates dephasing of both signals, hence no phase locking. (d) Corresponding time series of points in (b). Both models are parameterized to be in point A2 inside the fast oscillatory region LCEI. Insets show zoomed-in traces from 15 to 16 seconds. For the AdEx network, N = 20 × 103. All parameters are given in Table 1.
Fig 7.
Precomputed quantities of the linear-nonlinear cascade model.
(a) Nonlinear transfer function Φ for the mean population rate (Eq 15) (b) Transfer function for the mean membrane voltage (Eq 10) (c) Time constant τα of the linear filter that approximates the linear rate response function of AdEx neurons (Eq 7). The color scale represents the level of the input current variance σα across the population. All neuronal parameters are given in Table 1.
Fig 8.
Conversion between electric field amplitudes and equivalent input currents.
Each curve shows the frequency-dependent amplitude of an equivalent input current in pA to an exponential integrate-and-fire neuron with parameters as defined in Table 1 when the electric field amplitude acting on an equivalent ball-and-stick neuron is held constant. Electric field amplitudes in V/m are annotated for each curve.