Table 1.
Default parameters of the Wilson-Cowan model.
Fig 1.
Phase portraits of the Wilson-Cowan model and detection of fixed points.
(A) Phase portraits of the Wilson-Cowan model under different levels of external input Iext. Red and blue lines represent the rE and rI nullclines, respectively, black lines represent the trajectory of the systems with initial condition (0, 0), and arrows represent the flux dictated by and
. (B) Detection of fixed points of the Wilson-Cowan model under different levels of external input Iext.
Fig 2.
Different implementations of excitatory-inhibitory homeostasis in the Wilson-Cowan model.
For each mode of homeostasis, we present a diagram of the Wilson-Cowan model, on which we highlight the model components that are modulated by E-I homeostasis. For plasticity of intrinsic excitability, we explore two methods which either modulate the threshold (μE) of the input-output function, or adapt its threshold and slope (σE) in a coordinated manner.
Fig 3.
Mean firing rate as a function of fixed point rE for the Wilson-Cowan model under homeostasis of GE and Iext = 2.
While the solid blue line represents the mean firing rate as a function of , the dashed black line corresponds to
. Red dots show the iterations of the procedure. On the right, we include a zoom-in on the plot around the chosen value of
corresponding to ρ.
Fig 4.
Behavior of the Wilson-Cowan model under single modes of homeostasis as a function of Iext and target firing rate ρ.
(A) Homeostatic value of GE for different combinations of ρ and Iext (B) Homeostatic value of cEI for different combinations of ρ and Iext (C) Homeostatic value of μE for different combinations of ρ and Iext under μE homeostasis (D) Homeostatic value of μE for different combinations of ρ and Iext under coupled μE and σE homeostasis. For all models, σE = KμE. In all plots, dashed lines represent the transition from stable fixed point to stable spiral, while solid lines show the Andronov-Hopf bifurcation between a stable spiral and a limit cycle. The parameters were estimated using a timestep of 0.1 ms. For each mode of homeostasis, we include a diagram of the Wilson-Cowan model, indicating which model components are modulated by homeostatic plasticity. Blank areas relate to combinations of parameters for which the system has no solution (i.e. not possible to find a stable fixed point corresponding to ρ).
Fig 5.
Behavior of the Wilson-Cowan model under multiple modes of homeostasis as a function of Iext and target firing rate ρ.
(A) Homeostatic values of GE (Left) and cEI (Right) (B) Homeostatic values of GE (Top left), cEI (Top right) and μE (Bottom) under homeostasis of GE, cEI and μE (C) Homeostatic values of GE (Top left), cEI (Top right) and μE (Bottom) under homeostasis of GE, cEI, μE and σE, with σE = KμE. In all plots, dashed lines represent the transition from stable fixed point to stable spiral, while solid lines show the Andronov-Hopf bifurcation between a stable spiral and a limit cycle. The parameters were estimated using a timestep of 0.1 ms. For each combination of modes, we include a diagram of the Wilson-Cowan model, indicating which model components are modulated by homeostatic plasticity. Blank areas relate to combinations of parameters for which the system has no solution (i.e. not possible to find a stable fixed point corresponding to ρ).
Fig 6.
Effect of E-I homeostasis on oscillation frequency.
(A) Peak frequency of oscillation of the Wilson-Cowan Model under Different Modes of Homeostasis as a Function of Iext and target firing rate ρ. We present the peak frequency of oscillation across the parameter space for all modes of homeostasis. The colormap is centered around 40 Hz, the frequency of oscillation of the default model. In addition, we also display the bifurcations between the stable fixed point and stable spiral regimes (dashed line) and between the stable spiral and limit cycle (solid line). (B) Oscillation frequency as a function of cEI for different modes of homeostasis. We present the relationship between the intrinsic frequency of oscillation and the strength of inhibition cEI in models with cEI (Top Left), GE + cEI (Top Right), GE + cEI + μE (Bottom Left) and GE + cEI + μE + σE (Bottom Right) homeostasis. In addition, we plot the relationship in models with different target firing rates (ρ), ranging from 0.1 to 0.14. In all plots, each dot represents a model with a different level of incoming input Iext.
Fig 7.
Hopf-Bifurcation as a function of the integration time step (dt).
For each mode of homeostasis, dashed black lines represent the analytical bifurcation between the stable spiral and limit cycle regimes as a function of ρ and Iext. In addition, we present the bifurcation lines computed numerically using the Euler method with different integration time steps, ranging from 10−5 to 10−3 seconds.
Fig 8.
Hopf-Bifurcation as a function of the time constant of the inhibitory population (τI).
For each mode of homeostasis, we present the analytical Hopf-bifurcation as a function of ρ and Iext, in models with τI ranging from 5 to 100 ms. In all simulations, the excitatory time constant (τE) was set to 2.5 ms.