Stochastic Accumulation by Cortical Columns May Explain the Scalar Property of Multistable Perception

The timing of certain mental events is thought to reflect random walks performed by underlying neural dynamics. One class of such events—stochastic reversals of multistable perceptions—exhibits a unique scalar property: even though timing densities vary widely, higher moments stay in particular proportions to the mean. We show that stochastic accumulation of activity in a finite number of idealized cortical columns—realizing a generalized Ehrenfest urn model—may explain these observations. Modeling stochastic reversals as the first-passage time of a threshold number of active columns, we obtain higher moments of the first-passage time density. We derive analytical expressions for noninteracting columns and generalize the results to interacting columns in simulations. The scalar property of multistable perception is reproduced by a dynamic regime with a fixed, low threshold, in which the activation of a few additional columns suffices for a reversal.

Figure 1

Match between theoretical and simulated higher-order statistics of FPT: (a) coefficient of variation $c_v$ and (b) skewness ${\gamma }_1$ of FPT distributions, as a function of the initial condition $X_0$ and distribution means $⟨T⟩$. Each symbol (circles) represents 160 000 simulated FPTs, with error bars indicating SEM. The curves represent theoretical values from Eqs. (7)–(9). Transition rates ${\nu }_{+}$ and ${\nu }_{-}$ were varied to span $X_{\infty }-\theta /N\in [-0.2,+0.2]$. Threshold $\theta =25$, number of nodes or columns $N=50$, relaxation time $\tau =({\nu }_{+}+{\nu }_{-}{)}^{-1}=1$.

Figure 2

Two dynamical regimes with scalar property. (a) Coefficient of variation $c_v$ of FPT distribution as a function of the steady state $X_{\infty }$ (a measure of sensory input) and threshold $\theta$. Curves with identical $c_v$ are marked in gray. Population size $N=100$ and initial condition $X_0=0.2$. (b) Ranges with $c_v\in [0.5,0.6]$ for different initial conditions $X_0=\{0,0.2,0.4\}$ (from dark to light shades, respectively). Dotted lines mark values of $X_0$, dashed line marks steady-state activity $N{X}_{\infty }$. (c) Time evolution of density $P_{NX}(t)$ for $X_0=0.2$ and $X_{\infty }=0.9$ [20 000 simulated realizations of $X(t)$]. Shadings indicate 10 percentile steps. The inset magnifies the initial dynamic. Either a suprathreshold (dashed red) or a subthreshold (dashed blue) yields a $c_v=0.55$. (d) The FPT distribution is significantly less skewed in the suprathreshold (red histogram) than in the subthreshold regime (blue histogram, see text). (e) Threshold values $\theta /N$ required to obtain $c_v\in [0.5,0.6]$ with different mean FPTs $⟨T⟩$ in the supra- and subthreshold regimes (red and blue, respectively). Shading represents different values of $X_0=\{0,0.2,0.4\}$. (f) Ratio between skewness and $c_v$ of FPT distribution for ranges with $c_v\in [0.5,0.6]$ and different $X_0$, as in (e).

Figure 3

Higher-order statistics of FPT for populations of interacting columns. (a) Top: comparison of coefficients of variation for interacting columns, $c_{v,\text{int}}$, and independent columns, $c_{v,\text{free}}$, for various coupling values ${\nu }_{+}^{(1)}$. Bottom: comparison of mean FPT, $⟨T_{\text{int}}⟩$ and $⟨T_{\text{free}}⟩$. Colors distinguish supra- and subthreshold regimes and hues indicate different initial conditions $X_0$, as in Figs. 2 and 2. (b) Difference in the shape of FPT distribution shape between sub- and suprathreshold regimes, as expressed by $\mathrm{\Delta }{\gamma }_1/c_v$. Shading distinguishes initial conditions. The baseline rates ${\nu }_{\pm }^{(0)}$ were chosen to obtain $X_{\infty }=0.8$ and $\tau =1$ with zero coupling [${\nu }_{+}^{(1)}=0$]. In the subthreshold regime, ${\nu }_{+}^{(1)}>0$ were compensated by increasing the baseline inactivation ${\nu }_{-}^{(0)}$, such as to maintain $X_{\infty }=0.8$. For each dynamical regime, $\theta$ was set to maintain $c_{v,\text{free}}\approx 0.6$. Symbols represents mean and SEM from 5000 simulated realizations.