Fig. 5: Bifurcation theory characterizes wavefront propagation and appearance of new clusters over K-dimensional geometrical channels.
From: Simplicial cascades are orchestrated by the multidimensional geometry of neuronal complexes
We consider 2D simplicial threshold model (STM) cascades over a noisy ring complex (recall Fig. 2b) for various T and either (left) Δ = 0.1, (center) Δ = 0.5, or (right) Δ = 0.9. a Bifurcation diagrams depict the critical thresholds \({T}_{0}^{WFP}\) and \({T}_{0}^{ANC}\) given by Eqs. (3) and (4), respectively, for different T, d(NG) and d(G). We find four regimes that are characterized by the absence/presence of wavefront propagation (WFP) and appearance of new clusters (ANC). Observe that increasing Δ suppresses ANC, and the regime that exhibits ANC with no WFP disappears under higher-order interactions with Δ > 0.1. Vertical gray lines and horizontal colored marks identify the values d(NG)/d(G) = 0.25 and T ∈ {0.05, 0.1, 0.275, 0.35, 0.5}, and in panels b and c, we show for these values that the spatio-temporal patterns of STM cascades are as predicted. b Colored curves indicate the sizes q(t) of STM cascades versus time t, averaged across all possible initial conditions with cluster seeding. c Colored curves indicate the average number C(t) of cascade clusters, and one can observe a peak only when ANC occurs. Three scenarios give rise to WFP and ANC: (Δ, T) ∈ {(0.1, 0.05), (0.1, 0.1), (0.5, 0.05)}. Four scenarios give rise to no spreading: (Δ, T) ∈ {(0.1, 0.5), (0.5, 0.5), (0.9, 0.35), (0.9, 0.5)}. The other selected values of Δ and T yield WFP and no ANC, in which case q(t) grows linearly, dq/dt = 2(j + 1) for \(T\in [{T}_{j+1}^{WFP},{T}_{j}^{WFP})\). d Black symbols and gray curves indicate observed and predicted values, respectively, of cascade growth rates, dq/dt, for STM cascades exhibiting WFP and no ANC for a noisy ring complex with d(NG) = 0 and d(G) ∈ {6, 12, 24, 48, 96} (i.e., channel dimensions K ∈ {3, 6, 12, 24, 48}). Combining high-dimensional channels with higher-order interactions allows cascade growth rates to have a nonlinear sensitivity to changes for the threshold T.
