Figure 4. Separation of timescales and metastable regime.

(a) Examples of bistable activity. (i, iv,i) - time courses; (ii, v) - histograms of unit’s value across time; (iii, vi) - histograms of dwell times. (a–i, ii, iii) An example of a probe unit ${\displaystyle x_2}$ with ${\displaystyle s_2=5}$, embedded in a neural network with ${\displaystyle N=1000}$ units, ${\displaystyle N_1=N-1}$ units with ${\displaystyle s_1=1}$ and ${\displaystyle g=1.5}$. (a–iv, v, vi) An example of a probe unit driven by white noise. Note the differences in the x-axis scalings of the timecourses (a–i vs. a–iv and a–iii vs. a–vi).(b) The unified colored noise approximation stationary probability distribution (dark blue curve, Equation 19, its support excludes the shaded gray area) from the effective potential ${\displaystyle U_{eff}}$ (dashed blue curve) captures well the activity histogram (light blue area; same as (a–ii)); whereas the white noise distribution (dark green curve, obtained from the naive potential ${\displaystyle U}$, dashed green curve) captures the probe unit’s activity (light green area; same as (a–v)) when driven by white noise, and deviates significantly from the activity distribution when the probe is embedded in our network. (c) Average dwell times,${\displaystyle ⟨T⟩}$, in the bistable states. Simulation results, mean, and 95%CI (blue curve and light blue background, respectively; An example of the full distribution of the dwell times is given in (a-iii)). Mean-field predictions (purple curve) were generated by calculating the average dwell times from a trace of ${\displaystyle x_2}$, which was produced by solving the mean-field equations; Equation 2 simultaneously and consistently with Equation 3 with ${\displaystyle n_1=1}$ and ${\displaystyle n_2=0}$. The mean first passage time from the unified colored noise approximation (Equation 22, black curve), and for a simplified estimate thereof (Equation 4, gray dashed line) capture well ${\displaystyle ⟨T⟩}$. When driven by white noise (green curve and light green curve are simulation results and simplified estimate using Equation 4, respectively), the probe’s average dwell times are orders of magnitude shorter than with colored noise, exhibiting substantial support of the probe distribution in the region where the crossing between wells happens (allowing frequent crossing,(a-iv) green line at ${\displaystyle x=0}$) and, equivalently, the low value of the potential around its maxima ((b) green dashed line at ${\displaystyle x=0}$). Comparison of white and colored noise demonstrates the central role of the self-consistent colored noise to achieve long dwell times.

Figure 4—figure supplement 1. Validation of universal colored noise approximation (UCNA) approach to estimate escape times.

’Escape points’ from one metastable state to the next were estimated from simulations as follows: for a transition from the ${\displaystyle x^{-}=-s_2}$ well towards the ${\displaystyle x^{+}=x_2}$ well (Left panel; analogous calculation for transitions ${\displaystyle {\displaystyle x^{+}\to x^{-}}}$, Center panel), the escape point ${\displaystyle x_{esc}}$ was defined as the point where the trajectory starts accelerating towards the target well (positive second derivative). The distribution of escape points (Right panel) predominantly lied outside of the forbidden region (93.8% of escape points had ${\displaystyle {\displaystyle |x_{esc}|>x_c}}$). Parameters: same as in Figure 4a–i, ${\displaystyle s_2=5}$. See Methods ('Universal colored-noise approximation to the Fokker-Planck theory') for further details.