Multi-scale organization in communicating active matter

The emergence of collective motion among interacting, self-propelled agents is a central paradigm in non-equilibrium physics. Examples of such active matter range from swimming bacteria and cytoskeletal motility assays to synthetic self-propelled colloids and swarming microrobots. Remarkably, the aggregation capabilities of many of these systems rely on a theme as fundamental as it is ubiquitous in nature: communication. Despite its eminent importance, the role of communication in the collective organization of active systems is not yet fully understood. Here we report on the multi-scale self-organization of interacting self-propelled agents that locally process information transmitted by chemical signals. We show that this communication capacity dramatically expands their ability to form complex structures, allowing them to self-organize through a series of collective dynamical states at multiple hierarchical levels. Our findings provide insights into the role of self-sustained signal processing for self-organization in biological systems and open routes to applications using chemically driven colloids or microrobots.


Supplementary Notes 1 Derivation of the hydrodynamic equations through a Boltzmannlike kinetic approach
In this section we show how the set of hydrodynamic equations, can be derived from a Boltzmann-like approach for the probability density P (r, ϕ, t) of finding a particle at position r with orientation ϕ at time t; the particle's orientation is signified by the unit vector n = (cos ϕ, sin ϕ) T . The equation accounts for center-of-mass diffusion, particle self-propulsion, rotational diffusion, alignment with the signaling field, and interactions between particles: The advection term together with the rotational diffusion describe the self-propelled motion of the particles combined with the angular noise as in the agent-based model. The fourth term corresponds to a probability flux directed towards orientations that are aligned with the local gradients of the signaling field c with sensitivity parameter ω(c) and ϕ c ≡ tan −1 (∂ y c/∂ x c) = angle (∇c). The interaction contributions will be discussed further below. We follow the standard approach for deriving hydrodynamic equations from a Boltzmanntype of equation by expanding the probability density function in Fourier modes for the spatial orientation of the director n in two-dimensional space 1,2 , whereby, for the sake of brevity, we suppress the time dependency here and in the following. The corresponding Fourier coefficients follow from the forward transform We define the particle density ρ and the density-weighted polar order p by relating them to the harmonics via the Fourier expansion, Eq. (3): To describe the intrinsic states of the communicating active matter, we introduce a probability density P s (s) of particles in a given signaling state s and assume that the total probability density P tot (r, ϕ, s) = P s (s) P (r, ϕ) factorizes in a part for the signaling state and the distribution for the agent's positions and orientations. Thus, the density-weighted signaling state of the agents is given bȳ In the following, the different contributions to the Boltzmann equation, Eq.
(2), are analyzed separately. First, in order to derive expressions for the diffusive contributions in the hydrodynamic equations we use the projection onto the m-th harmonic, which gives the m-th Fourier coefficient to the expansion above, Eq. (3). Applying the projection operator, Eq. (8), onto the corresponding term in Eq. (2) one obtains for the dynamics of the density. One would obtain the same dynamics for the center-of-mass diffusion in the polar order field, but contributions from interaction kernels, representing elasticity of the polarity field, can lead to similar terms, which is why we assume a different coefficient D p for the polar field. Continuing with the advective term, (i.e. ∼ v 0 ), the projection onto the modes yields With the definitions, Eqs. (5) and (6), we obtain for the field variables Since a Boltzmann-approach is by design a low-density approximation, these results must be interpreted as such and require for an extension to assure well-behavedness at higher densities. Notably, this applies to the coupling of the polarity field to density gradients, At low densities, this term accounts for an effective pressure, increasing with increasing particle densities. At higher densities, other cooperative effects emerging from anisotropic interactions can dominate the coupling of the polarity field to density gradients, counteracting the repulsion dominating at low densities. In addition, at a critical maximum density, which we set to ρ = 2, the effective pressure increases significantly due to the finite volumes of the agents. Therefore, steric interactions dominate the cooperative interactions for ρ → 2. We account for these effects by extending the terms ∼ −∂ i ρ by a density-dependent prefactor Q(ρ) which is proportional to v 0 and has the following form: The function Q(ρ) captures the repulsion at low densities which decays for intermediate densities due to cooperative effects. Moreover, it limits the maximum density to values ρ ≈ 2 taking into account the steric repulsion at dense packing of the agents. The presented results do not qualitatively depend on the particular choice of the function Q(ρ). The scalar field corresponding to the agent's signaling activity, Eq. (7) is directly associated with the agents. Hence, in the same way as the particles it is advected with the polar flow and exhibits center-of-mass diffusion. From the definition, Eq. (7), we obtain and with the definition of the polarity field, Eq. (6), Thus, the complete diffusive and advective contributions to the dynamics of the density weighted signaling states = ρs are given by Correspondingly to the agent-based model, we re-express the state fields in terms of the 'state concentration', i.e., the local state normalized by the particle density, s by replacing s =s/ρ in Eq. (15); one obtains where we neglected cross-gradient contributions in the density ρ and the field s. Next, we turn to the contribution of the angular noise to the dynamics of the polar field. Fourier-expanding the corresponding term ∼ D R in Eq. (2) and projecting it onto the j th harmonic according to Eq. (8), yields the equation and, thus, with the definition of the polar field, Eq. (6), Finally, regarding the alignment of the agents' orientation vectors with gradients of the signaling field c, we want to briefly highlight the origin of the corresponding terms, ∼ ω, in the Boltzmann equation (2) starting from the proposed underlying Langevin dynamics with the particle position vector r and the angle of the chemical gradient ϕ c = angle(∇c).
The chemotaxis contributes to the Boltzmann equation, Eq.
(2), directly as the angular drift term Expanding the probability density in the Fourier harmonics as in Eq. (3), one obtains and after integration by parts Using the definitions, Eqs. (5), (6), and neglecting contributions of the second harmonics, the response of the dynamics of p to the signaling stimulus is given by where we chose a linear dependence of the alignment strength on the signaling amplitude c, namely ω(c) = 4π ωc. The contributions arising from particles' interactions can be motivated as done in Refs. [3][4][5] . As such, we include for completeness an elasticity like contribution and a self-propulsion in the model. Both terms may arise from anisotropic interactions, e.g., for elongated particles. They are not included in the agent-based model and we set the corresponding parameters D p and χ to small values as the effects are not crucial for the reported behavior of signaling active matter. Altogether we obtain the set of hydrodynamic equations

Reduced model without decision making
To highlight the role of the individual decision making for the multi-scale aggregation process, for comparison we also investigate the behavior of a system lacking such a mechanism. In particular, we modify the source dynamics given in main text Eq. (3), such that it becomes independent of the agents' internal state, The polar agents with dynamics given by main text Eqs. (1), (2), and supplementary information Eq. (27), are assumed to contribute as persistent sources of the signaling field. Similar to what has been reported in reference 6 , we observe aster-like stationary cluster formation with interface controlled ripening, see Supplementary Fig. 4a. Moreover, the interplay between self-propulsion and attraction towards a local aggregation center can give rise to short-lived ring-like structures and vortices which eventually tend to dissolve into a few aster-like aggregates as depicted in Supplementary Fig. 4b. Since in the modified model there is only local interactions mediated by the comparably slow diffusion of the signaling field, it does not exhibit a collective long-range organization of aggregation centers. In contrast to a system with active decision making, here the established smaller aggregates collide and merge upon random encounters.

Model parameters
The supplementary tables 1-3 provide an overview of the system parameters used in the numerical simulations shown in the main text as well as in the supplementary figures and movies. We measure densities in units of the critical density for the isotropic-polar transition.