Stochastic cycle selection in active flow networks

Abstract

Active biological flow networks pervade nature and span a wide range of scales, from arterial blood vessels and bronchial mucus transport in humans to bacterial flow through porous media or plasmodial shuttle streaming in slime molds. Despite their ubiquity, little is known about the self-organization principles that govern flow statistics in such nonequilibrium networks. Here we connect concepts from lattice field theory, graph theory, and transition rate theory to understand how topology controls dynamics in a generic model for actively driven flow on a network. Our combined theoretical and numerical analysis identifies symmetry-based rules that make it possible to classify and predict the selection statistics of complex flow cycles from the network topology. The conceptual framework developed here is applicable to a broad class of biological and nonbiological far-from-equilibrium networks, including actively controlled information flows, and establishes a correspondence between active flow networks and generalized ice-type models.networks | active transport | stochastic dynamics | topology B iological flow networks, such as capillaries (1), leaf veins (2) and slime molds (3), use an evolved topology or active remodeling to achieve near-optimal transport when diffusion is ineffectual or inappropriate (2, 4-7). Even in the absence of explicit matter flux, living systems often involve flow of information currents along physical or virtual links between interacting nodes, as in neural networks (8), biochemical interactions (9), epidemics (10), and traffic flow (11). The ability to vary the flow topology gives networkbased dynamics a rich phenomenology distinct from that of equivalent continuum models (12). Identical local rules can invoke dramatically different global dynamical behaviors when node connectivities change from nearest-neighbor interactions to the broad distributions seen in many networks (13-16). Certain classes of interacting networks are now sufficiently well understood to be able to exploit their topology for the control of input-output relations (17, 18), as exemplified by microfluidic logic gates (19,20). However, when matter or information flow through a noisy network is not merely passive but actively driven by nonequilibrium constituents (3), as in maze-solving slime molds (6), there are no overarching dynamical self-organization principles known. In such an active network, noise and flow may conspire to produce behavior radically different from that of a classical forced network. This raises the general question of how path selection and flow statistics in an active flow network depend on its interaction topology.Flow networks can be viewed as approximations of a complex physical environment, using nodes and links to model intricate geometric constraints (21-23). These constraints can profoundly affect matter transport (24-27), particularly for active systems (28-30) where geometric confinement can enforce highly ordered collective dynamics (20,(31)(32)(33)(34)(35)(36)(37)(38)(39)(40). In s...

Flowing Active Liquids in a Pipe: Hysteretic Response of Polar Flocks to External Fields

“…Finally, we have effectively exploited the bistability of active flows to engineer active-fluid oscillators with frequency and amplitude set by the geometry of the container. Together with the virtually unlimited geometries accessible to microfabrication, the intrinsic nonlinearity of active flows offer an effective framework for the design of emergent microfluidic functions [22][23][24]. Our experiments are based on colloidal rollers, see [4].…”

Information transmission and signal permutation in active flow networks

“…Indeed, the established Toner-Tu model of self-organised flow [34,37,38] reduces to that of overdamped diffusion in a double-welled potential when averaged along a narrow channel [15], yielding Boltzmann statistics as per a Landau theory; even if real-world AFNs do not obey exact equilibrium statistics in coarse-grained variables, as is likely, the intrinsic propensity of active matter toward flowing states at characteristic velocities at the heart of the Toner-Tu model suggests that we should expect statistics at least similar to the pseudo-energy fluctuations encoded in the Boltzmann distribution. The result is a form of vertex model on general graphs in the same family as ice-type or loop models [39][40][41][42], endowed with input-output capability, which qualitatively replicates the full continuous lattice field model of [28] (SM Text).…”

“…A non-zero flow 1 e f = | | indicates self-organised unidirectional flow along e at the typical velocity of the active matter system under consideration, normalised to unity, while f e =0 corresponds to a quiescent, overturning or turbulent state within the channel with zero net flux. This discretisation of flow states is a simplification of velocities fluctuating within a double-welled potential [28,34], modelling the tendency of active suspensions to adopt either a unidirectional flow state at a preferred velocity or, failing that, a qualitatively different state [19].…”

Topology-dependent density optima for efficient simultaneous network exploration

“…Finally, we generalised to a biased, non-equilibrium process and uncovered a qualitative change in the topological dependence of optimal densities. This evidences how the transport process itself must be taken into account when optimising random search, and leads us to ask whether there is an efficient statistical characterisation of the topology-transport interplay for general active networked processes 33,[48][49][50] . More broadly, our work paves the way towards new strategies for topology optimisation in process allocation and flow transport problems across physical and biological networked systems.…”