EI-Kuramoto model with distributed Jee and plasticity.

(A) Schematic representation of the pEI-Kuramoto model. Left: Oscillator units are excitatory (warm-colored) or inhibitory (cool-colored), each with its own natural frequency (shown by color gradients). Middle: Jee functions attractively between excUnits (red arrows). Kei is the attractive interaction strength from excUnit to inhUnit (magenta arrow). Kii represents the repulsive strength between inhUnits (blue arrows). Kie is the repulsive strength from inhUnit to excUnit (cyan arrow). Kei, Kii, and Kie are constants that are applied uniformly across all units. Right: Each Jee unit pair has its own coupling weight and undergoes plastic modification. (B) Distribution of Jee coupling weights with different σJee. μJee = 1, Nexc = 80. (C) Rexc time traces of synchronized, bistable, and desynchronized states with each σJee and no plasticity. (D) Exponential rules of Hebbian potentiation plasticity. The Jee weight of each pair of units is potentiated based on the phase difference between the units. (E) Evaluation of the Jee distribution at the last frame (t = 50 s) with various plasticity parameters. Skewness (top panels), kurtosis (middle), and dip values (bottom). (F) Jee distribution of exponential Hebbian plasticity rules at the last frame (t = 50 s) with various parameter settings. The parameter settings of the top, middle, and bottom panels correspond to those marked by magenta, green, and yellow circles in (E). (G) Evaluation of state synchrony with various plasticity parameters. E[Rexc]t and S[Rexc]t denote the time-averaged mean and standard deviation, respectively, of the order parameter for excUnits, Rexc. The parameters used for evaluation were Nexc = 80, Ninh = 20, μJee = 1, σJee = 0.1, Kei = 3, Kie = 3, Kii = 3, corresponding to a desynchronized state of the system without plasticity (E–G).

Changes in Jee fluctuation with Ki modification.

(A–D) (A) E{S[Jee]t}n, (B) S{S[Jee]t}n, (C) E[Rexc]t, and (D) S[Rexc]t as a function of Ki. Red crosses indicate each value in the Ki conditions shown in (E) and (F). (E) Time course of Jee weights under different Ki conditions. The order of unit pairs is sorted by E[Jee]t for each Ki value. (F) Upper panels show Rexc traces. The lower panels show the Jee weight matrix at selected time points indicated by the red and blue vertical lines in the top panels (red and blue are local maximum and local minimum, respectively, for Ki = 0.8 and 1.0; t = 24, 29, 34, 39, 44, and 50 for Ki = 0.1 and 1.5). The order of units is rearranged by the result of hierarchical clustering for the leftmost panel (around t: 24-25). E and F share the color bar. μJee = 1, σJee = 0.1, Kei = 3, α = 0.01, β = 0.5, γ = 2.

Relationship between the magnitude and fluctuation of Jee.

(A) E[Jee]t histogram. (B) S[Jee]t histogram. (C) E[Jee]t vs. S[Jee]t scatter plot for different Ki values. (D) E[ΔJee]t,n as a function of Jee for different Ki values. The right panel shows an enlarged view of the region enclosed by the dotted rectangle in the left panel. Black line indicates E[ΔJee]t,n = 0. Error bars indicate the standard error of the mean (SEM). (E) S[ΔJee]t,n as a function of Jee for different Ki values. At each time point, all pairs of ΔJee and the immediately preceding Jee values were collected, binned by the Jee values, and used to compute the statistics within each bin (D, E). μJee = 1, σJee = 0.1, Kei = 3, α = 0.01, β = 0.5, γ = 2.

Factors affecting the Jee fluctuation.

(A–C) E[Jee]t vs. S[Jee]t scatter plot with color coding of (A) the natural frequency difference Δω, (B) the initial phase difference Δθ, and (C) the initial Jee weight over different Ki. The red dotted line for each panel indicates E[Jee]t = 0.018. Left column, Ki = 0.8; middle column, Ki = 1.0; right column, Ki = 1.5. (D–F) Violin plots provide distribution densities based on whether the threshold E[Jee]t ≥ 0.18 or not for (D) natural frequency differences Δω, (E) initial phase differences Δθ, and (F) initial Jee weights across different Ki. The red and blue horizontal lines indicate the mean and median values, respectively. The red vertical lines indicate the standard errors. The numbers in the upper right corner of each panel are the p-values from the Mann–Whitney U-test. The numbers of unit pairs were as follows: for Ki = 0.8, E[Jee]t < 0.018: n = 5,968, E[Jee]t ≥ 0.018: n = 352; for Ki = 1.0, E[Jee]t < 0.018: n = 5,920, E[Jee]t ≥ 0.018: n = 400; for Ki = 1.5, E[Jee]t < 0.018: n = 5,852, E[Jee]t ≥ 0.018: n = 468. The statistics were as follows: (D, natural frequency difference), Ki = 0.8, z = 24.7, p < 0.001; Ki = 1.0, z = 26.8, p < 0.001; Ki = 1.5, z = 27.8, p < 0.001. (E, initial phase difference) Ki = 0.8, z = 9.20, p < 0.001; Ki = 1.0, z = 16.4, p < 0.001; Ki = 1.5, z = 16.7, p < 0.001. (F, initial Jee weight) Ki = 0.8, z = –0.046, p = 0.96; Ki = 1.0, z = 0.046, p = 0.96; Ki = 1.5, z = 0.17, p = 0.87. (G and H) Scatter plots of natural frequency differences Δω and initial phase differences Δθ with color coding of (G) E[Jee]t and (H) S[Jee]t (Ki = 1.5). μJee = 1, σJee = 0.1, Kei = 3, α = 0.01, β = 0.5, γ = 2.

Jee fluctuation changes with Ki cyclic modulation.

(A) Time trace of each Jee weight sorted by E[Jee]t. (B) Time traces of (top) Ki modulation, (middle) Rexc, and (low) E{S[Jee]t}n. μJee = 1, σJee = 0.1, Kei = 3, α = 0.01, β = 0.5, γ = 2.