Mode Selection in Compressible Active Flow Networks

Coherent, large-scale dynamics in many nonequilibrium physical, biological, or information transport networks are driven by small-scale local energy input. Here, we introduce and explore an analytically tractable nonlinear model for compressible active flow networks. In contrast to thermally driven systems, we find that active friction selects discrete states with a limited number of oscillation modes activated at distinct fixed amplitudes. Using perturbation theory, we systematically predict the stationary states of noisy networks and find good agreement with a Bayesian state estimation based on a hidden Markov model applied to simulated time series data. Our results suggest that the macroscopic response of active network structures, from actomyosin force networks to cytoplasmic flows, can be dominated by a significantly reduced number of modes, in contrast to energy equipartition in thermal equilibrium. The model is also well suited to study topological sound modes and spectral band gaps in active matter.

Figure 1

Activity can select a single dominant oscillation mode on hierarchically weighted networks. (a) The edges in the graph simulated in (b) and (c) are given weights decreasing exponentially with their distance from the central red path. (b) Oscillations in pressure and flux develop primarily along the central high-weight path (Movie 1). (c) Edge fluxes ${\varphi }_e$ settle into steady synchronized oscillations as exemplified for two edges indicated in (b), one on (${\varphi }_{17}$) and one off (${\varphi }_{59}$) the path. (d) Plotting the time-dependent amplitude of each analytically determined flow eigenmode confirms the selection of a single oscillatory mode. The ten modes with the highest average amplitude in this simulation run are pictured; the marked top two rows are oscillatory modes, while the remaining rows are cyclic modes. See Fig. S6 for all modes. Simulation parameters were $ϵ=0.1$, $\mu =1$, and $D=10^{-4}$.

Figure 2

First-order perturbation theory accurately predicts the stable states on small trees. (a) A five-vertex tree possessing four nontrivial modes, as illustrated. (b) On the tree in (a), mode amplitudes settle into one of two stable stationary states, as seen in simulations for three different initial conditions. Modes are ordered by frequency from high (top) to low (bottom). (c) Simulated mode trajectories (rainbow) in (b) match our analytic predictions (blue streamlines) in the subspaces of activated modes. There are three possible arrangements of nonzero critical points in each 2D subspace: a saddle point on one axis and a stable node on the other axis (left), a stable node on each axis and a saddle point in the middle (center), or a saddle point on each axis and a stable node in the middle (right; Movie 2). Higher-order effects cause both the convergence to a point with $A_2>0$ in the left and middle plots and the oscillations in the trajectories. Parameters used are $ϵ=0.5$, $\mu =1$, and $D=0$.

Figure 3

States on larger trees possess surprisingly few active modes, which can be inferred from time series with nonzero noise. (a) The mean number of stationary states of Eq. (2) grows exponentially with edges $E$ as $1.77^E\approx (2^E{)}^{4/5}$ (solid orange line), close to the upper bound of $2^E$ states (dashed black line), while the mean number of stable states grows as $1.21^E\approx (2^E{)}^{1/4}$ (solid blue line). We counted states on all nonisomorphic trees with $E\le 14$ edges (solid circles) and on a random sample of $\sim 175$ trees per point for $15\le E\le 24$ (open circles). Averages are over trees with a fixed number of edges. (b) As $E$ increases, both the mean and the variance of the distribution of trees with each number of stable states increase rapidly. (c) Distribution of the average number of modes active in a stable state. The mean over trees scales like $0.26E\approx E/4$ (solid line), significantly below $E/2$ expected if modes were selected randomly. (d) Two example trees indicated in (a)–(c) by the corresponding colored symbols. Stable states on paths ($\times$) always activate only one mode; complex trees ($+$) have more modes active. (e) Noisy networks ($D>0$) transition stochastically between stable states, exemplified by an amplitude-time trace for the tree shown. Modes are ordered by frequency from high (top) to low (bottom). Simulation parameters are $ϵ=0.5$, $\mu =1$, and $D=5\times 10^{-3}$. (f) States found by vbFRET from simulations on the tree in (e) (Supplemental Material [25]). The second, first, and fifth columns are states seen in (e), indicated by the colored bars above. (g) States predicted by Eq. (2) for the tree in (e). The first five states in (f) match those in (g); the sixth column in (f) is likely a transient combination of analytically stable states.