Asterios Charisiadis

Dr Emmanouil Giannoudis

Zoi Pournara

Aimilia Kosma

Vasilis Nikolaou

Georgios Charalambidis

Dr Vincent Artero

Murielle Chavarot-Kerlidou
email: murielle.chavarot-kerlidou@cea.fr

Dr Athanassios G Coutsolelos

Dr Z A Charisiadis

A Pournara

Dr V Kosma

Dr G Nikolaou

Prof Charalambidis

Synthesis and Characterization of a Covalent Porphyrin-Cobalt Diimine-Dioxime Dyad for Photoelectrochemical H-2 Evolution

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Inspired by photosynthesis, many researchers have developed artificial photosynthetic systems achieving efficient lightharvesting and solar energy conversion into an electrochemical potential. [1] One step further, the construction of dye-sensitized photoelectrochemical cells (DSPECs) has recently emerged as a highly attractive approach to convert and store solar energy into solar fuels. [2] Water splitting tandem DSPECs are in particular the subject of intense research efforts to produce hydrogen from sunlight and water. Their main limitation lies however in the low efficiency of the photocathodes reported so far, compared to the photoanodes counterpart. Thus, the design of novel H2-evolving photocathode architectures with high catalytic performance is crucial and a large number of research groups has been motivated and focused their studies towards this target. [2c, 2d, 3] The first essential component in a dye-sensitized photocathode is the photosensitizer, which should be cost-effective, photo-stable and absorb irradiation to relatively wide part of the sunlight spectrum, possessing mobility and high carrier lifetime as well. Porphyrin derivatives are ideal candidates for that purpose since apart from being analogues to the natural occurring chlorophyll compounds, they also possess all the above features. [4] Additionally, porphyrins can be easily structurally modified through straightforward synthetic procedures, [START_REF] Hiroto | [END_REF] rendering them attractive components for the preparation of covalently linked photosensitizer-catalysts dyads [6] and for the construction of dyesensitized photocathodes for H2 production. [7] Recent literature reports have described the first functional H2evolving photocathodes based on a noble metal free covalent dye-catalyst assembly. [8] Herein, we report the synthesis and full characterization of the first example of a covalent porphyrin dyecatalyst assembly based on the same cobalt diimine-dioxime catalyst (ZnP-Co) and its integration in an H2-evolving photocathode (Figure 1). The NiO valence band edge potential at this pH value was estimated from the 0.37 V vs NHE value determined at pH 7. [9] Based on our previous work, the photosensitizer and catalyst moieties were connected via a Cu-catalyzed click reaction. [8a] This reaction offers facile experimental conditions and in addition, the triazole bridge is reported to provide enhanced electronic communication in porphyrin-based molecular assemblies. [10] For the preparation of the ZnP-Co dyad, ZnP porphyrin (Figure 2) was used as the main building block. This derivative was also used as reference compound and is functionalized on one side with an alkyne group for the coupling reaction with the catalyst and on the other side with a carboxylic acid group for the anchoring of the dyad onto the NiO surface. We also prepared the methyl-ester analogues of the porphyrin precursor and of the final dyad (Figure 2), in order to achieve enhanced solubility in organic solvents, namely THF, thus enabling their complete characterization in solution.

Results and Discussion

The synthetic route that was followed for the preparation of ZnP-Co is outlined in Scheme 1 and consisting of six steps. In the first step, the acid-catalyzed condensation between dipyrromethane 3 [11] and the aldehydes 1 and 2 took place yielding the free-base porphyrin derivative 4. Consequently, the macrocycle was metallated with zinc resulting in the formation of COOMe-ZnP and then the methylester group was hydrolyzed under basic conditions to provide compound ZnP. In the following step, ZnP was coupled with the copper diimine-dioxime complex 5 [12] via copper(I)-catalyzed azide-alkyne cycloaddition (CuAAC) according to our previously reported protocole. [8a, 12] Then, metal exchange was performed through reaction with an excess of CoCl2 in an acetone/THF (1:1) mixture under continuous air bubbling to obtain a stable Co(III) complex. This intermediate was purified through silica-gel flash chromatography and subsequently precipitated in NaBr-saturated aqueous solution to yield the desired dyad (ZnP-Co). The procedures that were performed for the preparation of the methyl-ester dyad (COOMe-ZnP-Co) are depicted in Scheme S1. A click reaction was also carried out in order to connect COOMe-ZnP with complex 5, followed by a metal exchange reaction. The purity of all intermediates as well as both final dyads (ZnP-Co and COOMe-ZnP-Co) was confirmed by MALDI-TOF mass spectrometry and NMR spectroscopy ( 1 H and 13 C) (Figures S1-S21). Both dyads (ZnP-Co and COOMe-ZnP-Co) presented in their 1 H NMR spectra the characteristic signal for the proton of the oxime bridge in the pseudo-macrocyclic cobalt diimine-dioxime catalyst at 20.07 and 19.28 ppm, respectively; the peak of the newly formed triazole ring is observed at 8.34 and 9.28 ppm, respectively. Moreover, the peak corresponding to the carboxylic acid anchoring group was observed at 13.37 ppm for ZnP-Co. The photophysical characterization of the final dyad as well as the reference porphyrin was performed in their ester analogues via absorption and emission spectroscopies. The absorption spectra of COOMe-ZnP and COOMe-ZnP-Co in THF solutions are presented in Figure S22. The reference derivative illustrates typical zinc porphyrin-based features, i.e. one Soret (425 nm) and two Q bands (557 and 598 nm). These characteristics are also observed in the dyad accompanied by some weak contributions at lower wavelengths, which could be attributed to the cobalt diimine-dioxime catalyst. [8a] Since no significant shift was observed for any of the absorption peaks, but only a slightly decreased absorption coefficient for the Soret band in the dyad, we can safely assume that the two constituent entities interact weakly in the ground state of the dyad.

In an effort to gain some insight into the photoinduced processes that can take place in our dyad, fluorescence experiments were conducted and the emission spectra of COOMe-ZnP and COOMe-ZnP-Co in THF were compared (Figure 3).

Both the reference compound (COOMe-ZnP) and the dyad (COOMe-ZnP-Co) exhibited characteristic emission peaks for a zinc porphyrin, located at around 608 and 656 nm. In the case of the dyad, the intensity of both bands is partly quenched upon selective excitation of the porphyrin unit at 555 nm. Two processes might account for this observation: either a thermodynamically allowed electron transfer from the porphyrin excited state to the Co(III) center (ΔG o = -1.13 eV, see below), or an energy transfer from the first singlet excited state of porphyrin towards the catalyst unit. The electrochemical properties of COOMe-ZnP-Co were determined by cyclic voltammetry (Figure 4) in freshly distilled and deoxygenated tetrahydrofuran with tetrabutylammonium hexafluorophosphate (TBAPF6) as supporting electrolyte. All the electrochemical data are summarized in Table 1. Three reduction processes and two oxidative waves can be observed in the case of the dyad (Figure 4, bottom). The first two reductive processes at -0.63 V vs Fc and -1.00 V vs Fc were attributed to the Co III/II and Co II/I couples of the catalytic moiety, respectively, by comparison with the cyclic voltammogram of the Co-N3 precursor (Figure 4, middle) and with the previously reported dye-catalyst assembly, which displayed similar processes at -0.55 and -1.02 V vs Fc. [8a] In Co-N3, the second process is less reversible and slightly shifted to a more cathodic potential, a behaviour we attribute to some conformational constraints created by the azido substituent. As it has already been reported, this shift of approximately 100 mV of the irreversible reductive wave (Co II/I ) is assigned to the formation of the triazole ring in the dyad. [13] Table 1. Summary of the electrochemical redox data of COOMe-ZnP, Co-N3 and COOMe-ZnP-Co in THF. All potentials are reported vs. Fc/Fc + which was used as the internal standard. E00 stands for the energy of the singlet excited state of the ZnP estimated from the intercept of the absorption and emission spectra as 2.1 eV.

ZnP-Co was then anchored onto homemade F108-templated NiO films. A dyad loading of 6.4 ± 0.9 nmol.cm -2 (Table S1) was determined by desorption of the dyad from freshly-sensitized films in a THF solution of 1 M phenylphosphonic acid. [14] Linear sweep voltammograms were recorded in a 2-(Nmorpholino)ethanesulfonic acid (MES) buffer electrolyte at pH 5.5 [8a] under dark, chopped-light and continuous light irradiation conditions (Figure 5), using the ZnP-Co-sensitized photocathode as working electrode in a three-electrode setup. The experimental conditions that were followed herein are based on our previous light-driven H2 evolution studies with covalent dye-catalyst assemblies. [8a] Cathodic photocurrents are clearly generated under visible light irradiation (Figure 5) from +0.94 V vs RHE, reaching up to 11 µA.cm -2 (dark current subtracted). To assess whether these photocurrents are correlated with photoelectrocatalytic hydrogen production, two hours chronoamperometric measurements coupled to hydrogen detection were carried out (Figure S23) at -0.4 V vs Ag/AgCl for direct comparison with our previously-reported photocathodes based on the same cobalt diimine dioxime catalyst. [8]

E (V) vs RHE

At the end of the experiments, the amount of hydrogen produced was measured in the headspace and in the electrolyte solution, [8b] reaching 8 ± 1% Faradaic Efficiency (F.E.) and 7 ± 1 nmol.cm -2 . Similarly low F.E., ranging from 6 to 13 %, were also obtained for the previously-reported dye-sensitized H2-evolving photocathodes based on the same cobalt diimine-dioxime catalyst; [8,15] the reason behind this is still not clearly understood although it might be related to the reduction of some O2 trapped within the mesoporous NiO structure or to some degradation pathways. By comparison, control experiments performed with ZnP-sensitized NiO films under identical conditions produced only 3 nmol.cm -2 (Table S1).

Post-operando characterization of the dyad was performed after desorption from the film in a 1 M phenylphosphonic acid solution in THF. The UV-vis spectrum of the solution displays an overall lower intensity of the porphyrin characteristic signals by comparison with the spectrum recorded after desorption of the freshly-sensitized film (Figure 6); this indicates that some desorption of ZnP-Co from the NiO surface (estimated to ≈ 40 %) occurred during the course of the two-hours photoelectrochemical experiment. In addition, a clear increase in the number of Q bands is observed. This can be attributed either to the demetallation of the porphyrin core, [16] or to the formation of chlorin species due to the reduction/hydrogenation of the porphyrin ring. [17] As demetallation of porphyrins is reported to take place under acidic conditions, [16] the photoelectrocatalytic activity of the films was also assessed at pH 7. Although the ZnP-Co-sensitized film displays improved stability at this less acidic pH (desorption ≈ 25 %, see Figure S24), the activity was not enhanced and the same amount of hydrogen was produced (Table S1). The overall performances of this novel dyad are slightly lower than the ones previously reported for a NiO-based photocathode based on the same cobalt diimine-dioxime complex covalently coupled with a push-pull organic dye. [8a] In order to ascertain whether electrons are transferred to the catalytic center within the dyad, a UV-vis-monitored photolysis experiment was performed in a THF solution of COOMe-ZnP-Co, using triethanolamine (TEOA) as sacrificial electron donor (Figure S25). Unfortunately, in all conducted experiments we could neither observe the accumulation of the Co(II) state nor the one of the Co(I) state of the catalyst, which is in sharp contrast with our previous dyads. [8a], [18] The newly-formed peak at approximately 620 nm is ascribed to the reduced form of the porphyrin macrocycle, generated upon reductive quenching of the excited state of the latter by TEOA. [19] Accumulation of the reduced porphyrin during the course of the photolysis experiment thus clearly indicates that electron transfer to the catalyst is not efficient although this process is thermodynamically favored. Compared to our dye-catalyst assemblies based on a push-pull organic dye, a lack of directionality for the electron transfer from the porphyrin core to the catalytic center can be invoked to support these observations. Indeed, the electron-withdrawing ability of the carboxylate anchoring group most probably stabilizes the electron on the opposite side of the cobalt complex, preventing the formation of the catalytically active species and accounting for the low photoelectrocatalytic activity of the ZnP-Co sensitized photocathode.

Conclusion

To sum up, a new noble metal-free dyad was synthesized by employing click chemistry reaction in order to covalently connect a zinc-porphyrin sensitizer with a H2-evolving cobalt diimine dioxime catalyst. We have also prepared the methyl-ester analogue of the dyad to determine the electrochemical features of our novel system. In the cyclic voltammogram of COOMe-ZnP-Co we observed five redox potentials, which were assigned to either the chromophore or catalytic unit by comparison with the corresponding voltammograms of both precursors (COOMe-ZnP and Co-N3, respectively). Additionally, we performed photolysis experiments in the COOMe-ZnP-Co dyad in order to further evaluate the electronic communication between the zinc porphyrin and the catalyst. However, we did not observe the formation of the desired Co(I) species, hence suggesting that an electron shift from the porphyrin to the cobalt diimine-dioxime moiety does not take place easily. Finally, dye-sensitized photocathodes were constructed incorporating both the ZnP-Co dyad and the ZnP dye and their photoelectrochemical activity was assessed under pertinent conditions. This is the first attempt to prepare a noble metal free covalently connected porphyrin-cobalt dyad targeting its utilization in PEC for H2 production. All the results in this study highlight the importance of developing novel photosensitizer-catalyst entities for photo-electrocatalytic reduction reactions of interest in the context of a solar-driven chemistry. Several modifications should be explored in future research works examining crucial factors that impact the catalytic efficiency as well as the stability of the dyad onto the photocathode. Namely, the insertion of a nonconjugated methylene spacer between the anchoring group and the porphyrin ring will decrease the electronic coupling with NiO, and in turn slow down unwanted recombination processes. [20] In addition, introducing electron donating moieties on the periphery of the porphyrin macrocycle could possibly assist the charge transfer process from the sensitizer to the catalyst. Finally, a thorough investigation regarding different anchoring groups of the photosensitizer, such as phosphonic acid, hydroxamic acid or alkoxysilane, is required to increase the stability of the dyad onto the photocathode and enhance the catalytic activity. Alternatively, atomic layer deposition (ALD) deposition of a thin layer of alumina would prevent the dyad release in solution. Optimizing all the above mentioned parameters will allow to develop better performing photocathodes for H2 evolution based on noble metal free porphyrin light harvesting units.

Experimental Section

Materials and techniques. Reagents and solvents were purchased as reagent grade from usual commercial sources and were used without further purification, unless otherwise stated. NMR spectra were recorded on a Bruker AVANCE III-500 MHz spectrometer using solutions in deuterated solvents and the solvent peak was chosen as the internal standard. A Bruker UltrafleXtreme matrix assisted laser desorption ionization time-offlight (MALDI-TOF) spectrometer was used to record the mass spectra and trans-2-[3-(4-tert-butylphenyl)-2-methyl-2propenylidene] malononitrile (DCTB) was selected as matrix.

Photophysical Measurements. Absorption spectra were obtained using a Shimadzu UV-1700 spectrophotometer and steady-state emission spectra were recorded on a JASCO FP-6500 fluorescence spectrophotometer.

Electrochemistry. Cyclic voltammetry experiments were carried out at room temperature using an AutoLab PGSTAT20 potentiostat. All measurements were carried out in freshly distilled and deoxygenated THF in the presence of 0.1 M of tetrabutylammonium hexafluorophosphate (TBAPF6) as the supporting electrolyte, at a scan rate of 100 mV s -1 . A threeelectrode cell setup was used with a glassy carbon working electrode, an Ag/AgCl reference electrode, and a platinum wire as a counter electrode. In all measurements the potential of the reference electrode was calibrated vs the electrolyte ferroceniumferrocene (Fc +/0 ) couple, by adding in the supporting electrolyte of each experiment, ferrocene as an internal reference.

Photoelectrochemical measurements.

For a typical photoelectrocatalytic experiment, irradiation was carried out with a 300W ozone-free Xe lamp operated at 280 W equipped with a water-filled filter for elimination of IR irradiation (Spectra-Physics 6123NS) and a UV cut-off filter (Spectra-Physics 59472, λ > 400 nm). The power density was calibrated to 65 mW•cm -2 (≈1 sun) using a 60% optical density filter using a Newport PM1918-R power-meter. The experiments were performed in a threeelectrode cell, using the NiO-sensitized film as working electrode (3.3 to 3.3 cm 2 , accurately measured for each experiment), Ag/AgCl as the reference electrode and a Pt wire as the auxiliary electrode. The supporting electrolyte was 0.1 M 2-(Nmorpholino)ethanesulfonic acid (MES) / 0.1 M NaCl buffer at pH 5.5 or Britton-Robinson buffer at pH 7. The volume of supporting electrolyte (around 5 mL) and the headspace volume (around 2 mL) were accurately measured for each single experiment. The different compartments of the cell and the electrolyte were degassed for 30 minutes before the PEC experiments. The amounts of evolved hydrogen were determined after two hours of chronoamperometry run by sampling 50 μL of the headspace in Perkin Elmer Clarus 580 gas chromatograph equipped with a molecular sieve 5 Å column (30m -0.53 mm) and by micro-clark electrode in the electrolyte solution. [8b] Linear sweep voltammograms (LSV). Prior to the long-term chronoamperometric experiments, linear sweep voltammograms were recorded for each sensitized NiO film under dark conditions (lamp off), then continuous irradiation (lamp on), then chopped light irradiation (lamp on-off), screening from + 0.5 V to -0.5 V vs Ag/AgCl with a scan rate of 10 mV.s -1 .

Film sensitization. F108-templated NiO films were prepared according to our previously reported procedure. [8a] Sensitization was achieved by soaking them for 24h in 0.1 mM ZnP-Co or ZnP THF solutions. Finally, the films were soaked in pure THF to remove physiosorbed compounds from the surface and dried under air flow.

Determination of the surface concentration in ZnP-Co (or ZnP). Every film was cut in half after sensitization. One half was used to determine the surface concentration and the other half was used for the assessment of the PEC activity. The freshly grafted half of the film was dipped for 4 hours in 5 mL of 1 M phenylphosphonic acid THF solution in order to desorb the dyad from the surface, according to a previously reported procedure. [14] The UV-Visible absorption spectrum of the desorption solution was then recorded to determine the amount of grafted complex from the molar extinction coefficient at the Soret band. Knowing the surface concentrations in ZnP-Co allowed us to calculate turnover numbers (TON) based on the amount of catalyst present at the surface of the film. A similar desorption procedure was applied to the other half of the film after the PEC measurements for postoperando analysis. Synthesis of 4. In a two-neck round bottom flask, dipyrromethane 3 (500 mg, 1.9 mmol), 4-ethynylbenzaldehyde 1 (124 mg, 0.95 mmol) and methyl-4-formylbenzale (2) (156 mg, 0.95 mmol) were dissolved in 250 mL of CHCl3. The reaction mixture was subsequently bubbled under N2 for 15 min followed by addition of BF3OEt2 (75 μL) and stirring for 24 h. Then, 300 mg of DDQ were added and the mixture was additionally left stirring for 20 h. The resulting product was filtered through a silica pad and afterwards purified via column chromatography (silica gel, Hexane/CH2Cl2 in 7:3 ratio) providing 150 mg of porphyrin 4 (yield = 20%). 1 H NMR (500 MHz, CDCl3): δ = 8.79 (d, J = 4.7 Hz, 2H), 8.75 (d, J = 4.7 Hz, 2H), 8.72 (d, J = 4.7 Hz, 4H), 8.43 (d, J = 8.4 Hz, 2H), 8.32 (d, J = 8.4 Hz, 2H), 8.20 (d, J = 8.3 Hz, 2H), 7.89 (d, J = 8.3 Hz, 2H), 7.29 (s, 4H), 4.11 (s, 3H), 3.32 (s, 1H), 2.63 (s, 6H), 1.85 (s, 12H), -2.64 (s, 2H) ppm. 13 C NMR (75 MHz, CDCl3): δ = 167. 5, 147.0, 142.7, 140.5, 139.5, 138.9, 138.3, 138.2, 138.0, 134.7, 134.5, 130.7, 129.3, 128.6, 128.0, 127.9, 121.7, 118.8, 118.7, 118.3, 118.2, 83.8, 78.4, 52.6, 21.8, 21.6 ppm. MS (MALDI-TOF): calculated for C54H44N4O2 [M] + 780.3464; found 780.3452. Synthesis of COOMe-ZnP. To a 40 mL CH2Cl2 solution of 4 (78 mg, 0.10 mmol), a MeOH (6 mL) solution containing Zn(CH3COO)2 . 2H2O (215 mg, 1 mmol) was added and the reaction mixture was stirred at room temperature (RT) overnight. The solvents were evaporated under reduced pressure and the resulting residue was purified by column chromatography (silica gel, using Hexane/CH2Cl2 as eluent in a 7:3 ratio). The formed metallated porphyrin derivative COOMe-ZnP was obtained in quantitative yield, namely 98% (80 mg). 1 H NMR (500 MHz, CDCl3): δ = 8.87 (d, J = 4.6 Hz, 2H), 8.83 (d, J = 4.6 Hz, 2H), 8.80 (m, 4H), 8.40 (d, J = 8.3 Hz, 2H), 8.32 (d, J = 8.3 Hz, 2H), 8.21 (d, J = 8.2 Hz, 2H), 7.88 (d, J = 8.2 Hz, 2H), 7.29 (s, 4H), 4.08 (s, 3H), 3.31 (s, 1H), 2.64 (s, 6H), 1.84 (s, 12H) ppm. 13 C NMR (75 MHz, CDCl3): δ = 167. 6, 150.2, 150.1, 149.8, 149.6, 147.9, 143.6, 139.3, 139.0, 138.0, 137.7, 134.6, 134.5, 132.3, 132.1, 131.8, 131.6, 131.2, 131.1, 130.5, 129.9, 129.3, 127.8, 121.4, 119.7, 119.5, 119.2, 119.0, 83.9, 78.3, 52.5, 21.8, 21.6 ppm. MS (MALDI-TOF): calculated for C54H43N4O2Zn [M+H] + 843.2677; found 843.2689. Synthesis of ZnP. To a THF (17 mL) and MeOH (8 mL) mixed solution of COOMe-ZnP (65 mg, 0.08 mmol), an aqueous solution (9 mL) of KOH (430 mg, 7.70 mmol) was added. The reaction mixture was stirred at room temperature for two days and then the solvents were evaporated under vacuum and distilled H2O (10 mL) was added to the resulting residue. The mixture was acidified using HCl(aq) 1 M resulting in the precipitation of derivative as a purple solid which after being filtered, washed with distilled water and dried, was collected (62 mg, 97%). 1 H NMR (500 MHz, CDCl3): δ = 8.80 (m, 8H), 8.51 (db, J = 5.9 Hz, 2H), 8.38 (d, J = 7.4 Hz, 2H), 8.21 (d, J = 7.5 Hz, 2H), 7.89 (d, J = 6.9 Hz, 2H), 7.29 (s, 4H), 3.31 (s, 1H), 2.64 (s, 6H), 1.84 (s, 12H) ppm. 13 C NMR (125 MHz, CDCl3): δ = 171.2, 150. 3, 150.2, 149.9, 149.6, 148.8, 143.3, 139.3, 139.0, 137.7, 134.7, 134.5, 132.4, 132.1, 131.3, 130.5, 130.3, 128.5, 127.9, 121.4, 119.8, 119.7, 118.9, 83.9, 78.3, 52.5, 21.8, 21.6 ppm. MS (MALDI-TOF): calculated for C53H40N4O2Zn [M] + 828.2443; found 828.2451.

Synthesis of 6: ZnP (62 mg, 0.07 mmol) and copper diaminedioxime complex 5 (26 mg, 0.07 mmol) were solubilized in CH2Cl2 (5 mL) under Ar. A solution of CuSO4 . 5H2O (12 mg, 0.05 mmol) and sodium ascorbate (28 mg, 0.14 mmol) in Ar-purged water (2 mL) was added. Then, Ar-degassed CH3OH (5 mL) was added to the biphasic mixture until a single phase was obtained. The reaction mixture was stirred at RT under Ar. After 24 h the solution was concentrated under vacuum yielding a precipitate, which was filtered and washed with water. The solid was purified by flash chromatography on silica gel using as eluent a mixture CH3CN/aqueous KNO3 solution at 10% of the saturating concentration. The pure product was obtained after evaporation of CH3CN in the combined fractions and precipitation by addition of a KPF6-saturated aqueous solution. The precipitate was filtered, washed with water and dried under vacuum to give dyad 6 as redbrown brick solid (65 mg, 74% yield), which was directly used for the next step.

Synthesis of ZnP-Co. Dyad 6 (65 mg, 0.05 mmol) and CoCl2•6H2O (132 mg, 0.55 mmol) were solubilized in an acetone/THF (1:1) mixture (30 mL) and stirred overnight under air bubbling. After the reaction reached completion, the solution was evaporated under vacuum and the solid was purified by flash chromatography on silica gel using as eluent a mixture of CH3CN and aqueous KNO3 solution at 10% of the saturating concentration, then a solution of acetone saturated in NaI to completely desorb the product from the silica gel. The desired fractions were combined and the organic solvents were evaporated under vacuum, yielding a precipitate that was filtered and washed with water. The solid was re-solubilized in an acetone/THF (1:1) mixture. To this solution a NaBr-saturated aqueous solution (a third of the final volume) is subsequently added. The mixture was stirred at RT for five days to allow the organic solvents to slowly evaporate, giving a precipitate. This precipitate was filtered, washed with water and dried under vacuum to obtain the expected ZnP-Co dyad as a purple solid (18 mg, 33% yield). 1 H NMR (500 MHz, DMF-d7): δ = 20.07 (s, 1H), 13.37 (sb, 1H), 8.94 (m, 2H), 8.84 (d, J = 4.5 Hz, 2H), 8.69 (m, 4H), 8.47 (d, J = 7.5 Hz, 2H), 8.39 (d, J = 7.7 Hz, 2H), 8.34 (sb, 1H), 8.16 (d, J = 8.0 Hz, 2H), 8.02 (m, 2H), 7.38 (s, 4H), 6.14 (m, 1H), 4.61 (db, J = 14.3 Hz, 2H), 4.48 (m, 2H), 2.63 (s, 6H), 2.49 (s, 6H), 2.28 (s, 6H), 1.86 (m, 12H) ppm. 13 C NMR (125 MHz, DMF-d7): δ = 174. 2, 168.0, 156.4, 150.2, 149.9, 149.7, 149.6, 148.1, 139.8, 139.0, 137.6, 134.9, 134.3, 132.5, 132.0, 130.5, 130.2, 127.9, 126.1, 125.5, 123.7, 118.7, 58.9, 53.5, 21.3, 20.9, 17.2, 12.6 ppm. MS (MALDI-TOF): calculated for C64H56Br2CoN11O4Zn [M] + 1323.1507; found 1323.1522.

Entry for the Table of Contents

Herein, we report the synthesis of the first covalently linked porphyrin-cobalt diimine-dioxime dyad (ZnP-Co). Using various spectroscopic techniques for the characterization of the dyad, we were observed that a photoinduced electron transfer from ZnP* to the Co(III) center of the catalyst takes place in our system. Based on these findings, the photosensitizer-catalyst dyad was later tested in photoelectrochemical cells towards H2 evolution.

Figure 1 .

 1 Figure 1. Energy diagram at pH 5.5 and working principle of the NiO photocathode based on ZnP-Co. Potentials were converted from Fc to NHE considering E°(Fc +/0 ) = +0.53 V vs NHE.The NiO valence band edge potential at this pH value was estimated from the 0.37 V vs NHE value determined at pH 7.[9]

Figure 2 .

 2 Figure 2. Structures of the two reference compounds (COOMe-ZnP and ZnP) along with the two dyads (COOMe-ZnP-Co and ZnP-Co) reported herein.

Figure 3 .

 3 Figure 3. Emission spectra of isoabsorbing solutions of compound COOMe-ZnP (black) and COOMe-ZnP-Co (red) in THF upon excitation at 555 nm.

Figure 4 .

 4 Figure 4. Cyclic voltammograms of COOMe-ZnP (top) compared with those of Co-N3 (middle) and of the dye precursor COOMe-ZnP-Co (bottom).

  wave along with both oxidation peaks were assigned to the porphyrin core by comparing the voltammograms of our dyad and the reference dye(Figure 4, top). Both oxidation peaks are slightly cathodically shifted in the dyad compared to the reference dye (approximately 0.1 V), due to the different peripheral substituents of the porphyrin ring, namely triazole ring and ethynyl group, respectively.Based on the electrochemical data, we estimated a driving force ΔG° of -1.13 eV for the photoinduced electron transfer from ZnP* to the Co(III) center, according to the simplified Rehm-Weller equation: ΔG° = Eox (ZnP/ZnP + ) -E00 (ZnP*) -Ered(Co III/II ).

Figure 5 .

 5 Figure 5. Linear sweep voltammograms of ZnP-Co-sensitized NiO electrodes, recorded in a pH 5.5 MES 0.1 M/NaCl 0.1 M buffer under dark conditions (black line), continuous visible light irradiation (red line) or chopped light irradiation (blue line).

Figure 6 .

 6 Figure 6. UV-vis absorption spectra of desorption solutions (1 M phenylphosphonic acid in THF) of a fresh ZnP-Co-sensitized NiO film (black line) and the same sample (second half of the same sensitized film; red line) after a two-hours chronoamperometric measurement under continuous light irradiation at an applied potential of -0.4 V vs Ag/AgCl in MES buffer pH 5.5.

Acknowledgements

This research was funded by the General Secretariat for Research and Technology (GSRT) and Hellenic Foundation for Research and Innovation (HFRI; project code: 508). This research has also been co-financed by the European Union and Greek national funds through the Operational Program Competitiveness, Entrepreneurship, and Innovation, under the call RESEARCH-CREATE-INNOVATE (project code: T1EDK-01504). In addition, this research has been co-financed by the European Union and Greek national funds through the Regional Operational Program "Crete 2014-2020," project code OPS:5029187. Moreover, the European Commission's Seventh Framework Program (FP7/2007-2013) under grant agreement no. 229927 (FP7-REGPOT-2008-1, Project BIO-SOLENUTI), the Special Research Account of the University of Crete and the French National Research Agency in the framework of the "Investissements d'avenir" program (ANR-15-IDEX-02, Labex ARCANE and CBH-EURGS, ANR-17-EURE-0003) are gratefully acknowledged for the financial support of this research. Keywords: Porphyrin-cobalt diamine dioxime dyad • photosensitizer-catalyst dyad • "click" chemistry • H2 evolution • photoelectrochemical cells (PEC)


Parisa Rahmani

Fernando Peruani

Pawel Romanczuk
email: pawel.romanczuk@hu-berlin.de

Flocking in complex environments -attention trade-offs in collective information processing

The ability of biological and artificial collectives to outperform solitary individuals in a wide variety of tasks depends crucially on the efficient processing of social and environmental information at the level of the collective. Here, we model collective behavior in complex environments with many potentially distracting cues. Counter-intuitively, large-scale coordination in such environments can be maximized by strongly limiting the cognitive capacity of individuals, where due to self-organized dynamics the collective self-isolates from disrupting information. We observe a fundamental trade-off between coordination and collective responsiveness to environmental cues. Our results offer important insights into possible evolutionary trade-offs in collective behavior in biology and suggests novel principles for design of artificial swarms exploiting attentional bottlenecks.

INTRODUCTION

Consensus formation, coordination and collective response to environmental cues are important aspects in collective behavior of many interacting agents in biology, physics, robotics and computational social science. The understanding of these processes is fundamental for a better comprehension of collective intelligence that confers groups the ability to solve problems collectively using strategies which are beyond the reach of single individuals [1]. Crucial for understanding benefits of collective behavior, is the understanding of the mechanisms underlying collective information processing: How new information is acquired, how information is shared and combined within the collective, and how the collective deals with conflicting information are among the most compelling and elusive questions on the selforganization of collectives. Generic flocking models (see e.g. [2][START_REF] Vicsek | Novel type of phase transition in a system of self-driven particles[END_REF][START_REF] Grünbaum | Schooling as a strategy for taxis in a noisy environment[END_REF][START_REF] Iain D Couzin | Collective memory and spatial sorting in animal groups[END_REF]) allow to study the interplay between emergent collective behaviors and collective information processing in a dynamical system setting. For example, using flocking models it has been demonstrated, that only a small fraction of informed individuals is sufficient to accurately guide large collectives [START_REF] Iain D Couzin | Effective leadership and decision-making in animal groups on the move[END_REF][START_REF] Cucker | Flocking with informed agents[END_REF][START_REF] Guttal | Social interactions, information use, and the evolution of collective migration[END_REF]. It was also shown that groups are able to collectively track dynamic environmental gradients not detectable by individuals [START_REF] Grünbaum | Schooling as a strategy for taxis in a noisy environment[END_REF][START_REF] Berdahl | Emergent sensing of complex environments by mobile animal groups[END_REF], or that groups can make efficient consensus decisions in conflict situations without any implicit knowledge about the majority-minority relationships [START_REF] Iain D Couzin | Effective leadership and decision-making in animal groups on the move[END_REF][START_REF] Iain D Couzin | Uninformed individuals promote democratic consensus in animal groups[END_REF]. Only recently, predictions of such models on fundamental decision bifurcations in spatial movement decisions have been confirmed in the collective migration of baboon groups [START_REF] Strandburg-Peshkin | Shared decision-making drives collective movement in wild baboons[END_REF].

A fundamental aspect of the self-organization of collectives -from collective decisionmaking to consensus formation, including coordinated movements -is that individuals are limited in terms of perception and cognition. Without direct access to the state of the whole group they must rely on local information [START_REF] Moussaïd | Experimental study of the behavioural mechanisms underlying self-organization in human crowds[END_REF][START_REF] Moussaïd | How simple rules determine pedestrian behavior and crowd disasters[END_REF][START_REF] Strandburg-Peshkin | Visual sensory networks and effective information transfer in animal groups[END_REF][START_REF] Daniel Jg Pearce | Role of projection in the control of bird flocks[END_REF]. The emergent collective patterns in agentbased models have been shown to depend strongly on the field of view of individuals [START_REF] Strömbom | Collective motion from local attraction[END_REF][START_REF] Barberis | Large-scale patterns in a minimal cognitive flocking model: incidental leaders, nematic patterns, and aggregates[END_REF], and more generally on what local information individuals pay attention to [START_REF] Romanczuk | Swarming and pattern formation due to selective attraction and repulsion[END_REF][START_REF] Romensky | Tricritical points in a vicsek model of self-propelled particles with bounded confidence[END_REF]. Furthermore, even for a strongly limited field of view, the sensory input of individual agents may contain a large number of social and non-social cues. However, social interaction in animal groups appear to be restricted to a rather low number of neighbors [START_REF] Ballerini | Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study[END_REF][START_REF] Jiang | Identifying influential neighbors in animal flocking[END_REF]. On the one hand, this suggests additional cognitive constraints on the processing of available sensory information, which is also in-line with a wide range of experimental results on limited capacity for visual tracking of multiple objects in animals and humans [START_REF] Intriligator | The spatial resolution of visual attention[END_REF][START_REF] Todd | Capacity limit of visual short-term memory in human posterior parietal cortex[END_REF][START_REF] Cavanagh | Tracking multiple targets with multifocal attention[END_REF][START_REF] Green | Enumeration versus multiple object tracking: The case of action video game players[END_REF][START_REF] George | How many objects can you track?: Evidence for a resource-limited attentive tracking mechanism[END_REF]. On the other hand, it has been shown in generic flocking models that the emergent, large scale collective behavior depends strongly on how many neighbors a given individual can pay attention to [START_REF] Ginelli | Relevance of metric-free interactions in flocking phenomena[END_REF][START_REF] Chou | Kinetic theory for systems of self-propelled particles with metric-free interactions[END_REF][START_REF] Shang | Influence of the number of topologically interacting neighbors on swarm dynamics[END_REF]. However, to our knowledge the explicit role of cognitive constraints on collective information processing has not been systematically explored.

Up-to-date most theoretical and empirical research focused on information sharing and collective decision making, in idealized, laboratory-like environments, including the works mentioned above, discussing cognitive and sensorial limitations (see e.g. [START_REF] Iain D Couzin | Effective leadership and decision-making in animal groups on the move[END_REF][START_REF] Iain D Couzin | Uninformed individuals promote democratic consensus in animal groups[END_REF][START_REF] Bh Lemasson | Collective motion in animal groups from a neurobiological perspective: the adaptive benefits of dynamic sensory loads and selective attention[END_REF][START_REF] Ashley | Fast and accurate decisions through collective vigilance in fish shoals[END_REF]). However, collectives in realistic scenarios need to cope with complex environments with a large number of potentially informative or distracting environmental cues (see Fig. 1a and [START_REF] Strandburg-Peshkin | Habitat and social factors shape individual decisions and emergent group structure during baboon collective movement[END_REF]). Whereas recently it was shown using minimal flocking models that self-organized, collective behaviors are strongly affected by complex environments [START_REF] Chepizhko | Optimal noise maximizes collective motion in heterogeneous media[END_REF][START_REF] Chepizhko | Active particles in heterogeneous media display new physics[END_REF], it remains open how complex environments impact collective information processing, which is the main question we focus on here. More specifically, we investigate the emergence of collective behaviors in a generic model of socially interacting agents in a complex environment containing many potentially dangerous sites. Individuals try to avoid these sites, while at the same time trying to coordinate with their neighbors. In addition, some informed individuals have also private information on a global preferred direction of migration, which may be in conflict with the local environmental cues. Our main aim is to explore how the cognitive, and/or sensory constraints of the individuals affect group-level coordination, information exchange and collective response to environmental cues.

Our results show that in heterogeneous environments, strongly limited attention capability of individual agents results in higher accuracy with respect to large-scale coordination, which is in stark contrast to previous results obtained in simple environments [START_REF] Chou | Kinetic theory for systems of self-propelled particles with metric-free interactions[END_REF][START_REF] Shang | Influence of the number of topologically interacting neighbors on swarm dynamics[END_REF]. This is caused by a dynamical, spatial "echo chamber"-like effect, where individual attention becomes saturated by social information and non-social cues are largely ignored. However, if these non-social cues provide important information about potential environmental dangers, the emergent dynamical "echo chambers" become strongly detrimental to the ability of the collective to safely navigate the environment. Note that information exchange through social interactions is typically believed to be beneficial for the collective [START_REF] Berdahl | Emergent sensing of complex environments by mobile animal groups[END_REF][START_REF] Sirot | Social information, antipredatory vigilance and flight in bird flocks[END_REF]. Here, our analysis shows that below a critical threshold in attention capacity, groups perform worse at acquiring new information about the environment than non-interacting agents. This is due to the emergent self-isolation from environmental cues, which is exactly what facilitates group coordination in complex environments. Our findings not only suggest a fundamental trade-off in collective behavior in natural systems, but also provide important insights for the design of communication in artificial, distributed systems, such as robotic swarms.

RESULTS

Model.

We consider a flocking model consisting of N agents moving in a two-dimensional environment of size L × L with periodic boundary conditions. In addition, we assume the presence of N inf informed individuals with private information about a preferred direction of motion ûp . The informed fraction of the collective is

R inf = N inf /N .
The environment contains non-social cues, which represent features of the environment that solitary agents in general try to avoid, as they, for example, signal potential dangers. We will refer to these disrupting cues as "danger sites" (or distraction sites) in short as DS. Each DS is surrounded by an effective repulsion zone of radius r = 1. The environment contains N DS randomly distributed DSs at fixed positions. In particular at high DS densities ρ DS = N DS /L 2 , the corresponding repulsion zones may overlap (see Fig. 2a,b). We note that the agents and the danger sites (DS) are assumed to be point-like, as in many agentbased models for collective movement, e.g. Vicsek-type models [START_REF] Vicsek | Novel type of phase transition in a system of self-driven particles[END_REF]. This corresponds to the scenario where the sensory ranges are large compared to the physical size of moving agents and DSs.

Based on experimental observations, it was suggested that animals interact with a limited number of conspecifics [START_REF] Strandburg-Peshkin | Visual sensory networks and effective information transfer in animal groups[END_REF][START_REF] Ballerini | Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study[END_REF][START_REF] Jiang | Identifying influential neighbors in animal flocking[END_REF]. Motivated by these findings, and with intention of explicitly studying the impact of limited attention in a generic flocking model, we assume that each agent can pay attention only to k nearest objects (kN O) in its vicinity, irrespective whether it is another agent (social cue) or a DS (non-social cue). Thus, k can be interpreted as the number of available attention slots for each agent. The parameter k quantifies the individual attention capacity.

Each agent moves with a constant speed v 0 and reacts to neighbors and DSs through corresponding changes in its direction of motion ûi = (cos ϕ i , sin ϕ i ) T defined by a polar orientation angle ϕ i . All agents -informed and uninformed -turn away from DSs when two conditions are met simultaneously: 1) the agent perceives the DS -that means that the DS in question, say l, is within its k nearest objects -and 2) the distance between the agent and DSs (d il = | x i -x l |) is smaller than the radius of its repulsion zone. In all other cases, individuals ignore the DSs and coordinate their motion with other individuals within their kNO by aligning their direction of motion with the average direction of their neighbors. Informed individuals exhibit an additional bias to orient towards the preferred direction of motion that we denote ûp . Throughout this work, the preferred direction of motion of informed individuals will be along the x-axis: ûp = (1, 0) T . We emphasize that the response to DSs, once detected, dominates all other behavioral responses of individuals, whether informed or uninformed. The specific formulation of the mathematical model in terms of stochastic differential equations, together with the parameters used, are given in Methods.

The finite attention capacity to k nearest objects leads to a natural competition between social and non-social cues: If the k nearest objects of the focal agent are other agents, it will be not capable to "sense" a DS l even if d il < 1 (see Fig. 1b). Note that in the case of vanishing density of DSs ρ DS → 0 (homogeneous environment), the model reduces to a simple flocking model with so-called metric-free alignment interaction with k-nearest neighbors (see e.g. [START_REF] Ballerini | Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study[END_REF][START_REF] Chou | Kinetic theory for systems of self-propelled particles with metric-free interactions[END_REF][START_REF] Peshkov | Continuous theory of active matter systems with metric-free interactions[END_REF]) with the additional feature of informed individuals. If instead of topological, metric interaction are used, then the model reduces in the limit of ρ DS → 0 and R inf = 0 to a Vicsek-type alignment model [START_REF] Vicsek | Novel type of phase transition in a system of self-driven particles[END_REF], which has been extensively discussed in [START_REF] Vicsek | Collective motion[END_REF][START_REF] Marchetti | Hydrodynamics of soft active matter[END_REF]. For R inf > 0 it is closely related to the model explored in [START_REF] Iain D Couzin | Effective leadership and decision-making in animal groups on the move[END_REF], while for R inf = 0 and ρ DS > 0 it reduces to the model studied in [START_REF] Chepizhko | Optimal noise maximizes collective motion in heterogeneous media[END_REF][START_REF] Chepizhko | Active particles in heterogeneous media display new physics[END_REF].

Interaction Networks & Collective Accuracy. In order to quantify the emergent collective dynamics and study the effect of varying attention capability, we have performed systematically numerical simulations of the above model for varying attention limit k, DS density ρ DS , and the ratio of informed individuals R inf (see Methods for details). By neglecting the directed nature of inter-individual links, the entire agent system can be viewed as a time-dependent, undirected interaction network. For all DS densities, we obtain the same qualitative picture in the stationary state: For low k, we observe strongly fragmented dynamical networks, which at given time t are characterized by a large number of small, disconnected sub-groups (see Fig. 2a and Supp. Fig. S1). Each such cluster corresponds to an isolated connected component. These components are not static: We observe continuous fission-fusion of clusters over time due to randomness in individual motion and interactions with DSs (see Movie S1). By increasing the attention limit k, we observe a fast decrease in the number of disconnected clusters that results in an increase in average cluster size (see Fig. 2b and Supp. Fig. S2a). Eventually, by increasing k above a critical value, we can obtain fully connected networks with a single connected component. Correspondingly, the average life-time of a connection between specific agents grows strongly with increasing attention limit k, whereas an increase in DS density ρ DS reduces the life-time of individual edges in the network (see Supp. Fig. S2b). In general, as one would intuitively expect, the network of social interactions becomes more tightly connected with increasing attention capacity k.

We measure the accuracy of the collective migration through the average agreement between the heading ûj of individuals and the preferred direction of motion ûp :

C = 1 N j ûj • ûp (1)
with • indicating the temporal average in the stationary state. If all agents move perfectly along the preferred direction, which is available only to informed individuals, then we obtain C = 1, whereas for disordered movement we observe C ≈ 0. For collective behavior in homogeneous environments it has been shown that increasing the connectivity of the interaction networks is beneficial for coordination [START_REF] Shang | Influence of the number of topologically interacting neighbors on swarm dynamics[END_REF][START_REF] Bh Lemasson | Collective motion in animal groups from a neurobiological perspective: the adaptive benefits of dynamic sensory loads and selective attention[END_REF], which is also in line with general results on synchronization in (dynamic) networks of oscillators [START_REF] Belykh | Blinking model and synchronization in small-world networks with a time-varying coupling[END_REF][START_REF] Dörfler | Synchronization in complex oscillator networks and smart grids[END_REF]. In particular for few informed individuals, one intuitively expects that a strongly connected information network ensures that information about the preferred direction of motion diffuses more efficiently across large parts of the collective. Therefore, the natural prediction would be that collective accuracy increases with increasing attention capacity k. This is indeed the case in the limit of vanishing density of DSs ρ DS = 0 (homogeneous environment, see Fig. 3a), where we observe a monotonous increase in accuracy C with the attention capacity k, for a fixed ratio of informed individuals: Whereas for k = 1, in order to achieve an accuracy of C > 0.9, we require the majority of the collective to be informed (R inf ≥ 0.6), for k = 6 it is already sufficient to have a small fraction R inf = 0.2 of informed individuals to achieve the same level of collective accuracy.

The situation completely reverses for high DS densities (see Fig. 3b). For ρ DS = 0.2 we observe a monotonous decrease in the collective accuracy with increasing k for all values R inf > 0. In order to achieve a certain level of collective accuracy (e.g. C = 0.5) for larger k we need a larger fraction of the system to be informed. In other words, stronger connected flocks become more difficult to guide. Even more dramatically, for the largest attention capacity investigated (k = 24) the maximal attainable average accuracy for a fully informed system (R inf = 1) is C ≈ 0.62. This is lower than the average collective accuracy of C ≈ 0.66 for collectives at minimal attention capacity (k = 1) with only a tiny fraction of informed individuals R inf ≈ 0.013 (1.3% of the entire collective). In fact, it appears that for k = 1, the collective accuracy C versus R inf depends only very weakly on the DS density, whereas the accuracy for high k is massively decreased for all R inf > 0. Thus, in complex environments, strongly limited attention resulting in a very sparse, fragmented -yet dynamic -interaction networks, turns out to be highly beneficial for global consensus formation (see Fig. 3c). This counter-intuitive effect can be understood through an analysis of information flows and how environmental information is processed by the collective, for example through analysis of the (stationary) probability distribution of agents having a particular combination of in and out-degree. The out-degree D out quantifies how many individuals a focal individual pays attention to, whereas the in-degree D in is the number of others paying attention to the focal individual. Agents directly responding to a DS have a social out-degree D out = 0 -they ignore their neighbors. However, their neighbors can still pay attention to them so their social in-degree D in can be larger than zero (see Supp. Information and Fig. 2c,d), in this case they "broadcast" information on the DS through their evasion behavior to others. Therefore, environmental cues affect collective behavior directly through the individuals directly responding to a DS, as well as, indirectly through information transmitted to other agents via social interactions. High consensus in complex environments requires effective self-isolation of the collective from distracting environmental cues. It can emerge due to two mechanisms: 1) The number of direct responders remains small; 2) Their influence on others is weak due to a low D in , in particular compared to the in-degree of non-responders. In our case both effects play a role: For low k, the fraction of agents directly responding to DSs remains low even at very high ρ DS . At low k, aligned individuals move together in dense sub-groups, and even if one or more agents enter a repulsion zone, there is a high probability that there are k neighbors closer than the DS, which prevents the detection of the latter. In addition, the in-degree of agents responding to DSs is on average significantly lower than that of agents not responding to the DSs (Fig. 2c,d). Agents evading a DS, have a high probability to move away from their neighbors, which in turn decreases the probability that they will be within the kN O of others. Thus, in particular for low k, the indirect response to DSs, mediated through social interactions, is strongly inhibited. These emergent small, dense agent clusters at low k permanently merge and split up over time, which leads to exchange of directional information across the collective on long time scales, eventually leading to global consensus and high coordination levels. This situation changes with increasing attention capacity k. The chance of an agent to detect a DS increases strongly, as the distance to the k nearest objects is an increasing function of k. As soon as this distance becomes larger than the distance to the next DS (d il 1), agents are capable to detect the environmental cues reliably, and react to them if they are within their repulsion zone. In addition, larger k also increases the number of other agents paying attention to a direct-responder. The motion of the emergent sub-groups is still well coordinated at a local scale (see Supplementary Movie S2). However, these sub-groups interact now predominantly with the environment by changing their direction of motion and by complex fission-fusion dynamics directly triggered by the DSs. This results in quick loss of directional information across time and space, and in a vanishing impact of the directional information provided by the informed individuals, which yields a strong decrease in collective accuracy C.

The contribution of both effects to the emergent self-isolation for low k is shown in Fig 4a : The fraction of direct responders r d is much lower for low k and shows only a slow increase with increasing DS density ρ DS . The same holds for the fraction of first-order indirect responders r i , defined by agents paying attention to at least one direct responder. For k = 2, r i < 0.1 for all DS densities studied, whereas for k = 24 at high DS densities, it saturates at more than half of the entire collective (r i ≈ 0.55).

Collective Avoidance of Repulsion Zones. While limited attention is beneficial for consensus formation in complex environments, it may be very detrimental to the collective if the environmental cues (DSs) provide reliable informations about environmental dangers (e.g. predators).

In order to quantify the performance of the collective to respond to environmental cues, we introduce in the following a DS avoidance measure. First, we compute the average fraction of agents in "safe" areas, i.e. outside the DS repulsion zone. As a reference, let us estimate the same quantity in a control numerical experiment with the same number of agents but who are only interacting exclusively with the environment and not with other agents. Our DS avoidance measure A is then defined as the ratio between these two quantities (see Methods for details). This measure A allows us not only to compare the relative collective performance of the flock at different attention capacities k, but also to compare it with the performance of solitary agents without any social interactions. For A > 1, social interactions provide a benefit with respect to solitary individuals. For A < 1, a collective performs on average worse than solitary agents in avoiding the potentially dangerous areas. Fig. 4b shows clearly that for small values of k, the collective performs much worse than solitary individuals in avoiding the repulsion zones. Here, increasing k results in a monotonous increase in A. However, we observe A > 1 only for sufficiently large k 12. For lower k it turns out that social interactions are detrimental with respect to DS avoidance. The observed behavior is directly linked to the emergent local echo-chamber effect at small k. Socially interacting individuals can only outperform solitary agents in responding to environmental cues, if sufficient amount of information is able to enter the interaction network and spread effectively through it.

We can quantify the emergent trade-off between migration accuracy (quantified by C) and DS avoidance (measured by A) through a global fitness function F (C, A) (see Methods and Supplementary Information for details). While F depends implicitly on the attention limit k through A and C, we introduce the parameter δ to quantify the relative benefits of avoidance versus accuracy. Whereas in safe environments, benefit of avoidance may be negligible (δ ≈ 0), in environments where local cues provide important information about potential dangers one can assume δ 1. Fig. 4c shows F as a function of k and δ for collectives For low attention capacity (k = 1, e), individuals ignore the structure of the environment and align with the preferred direction of migration. At high attention capacity (k = 16, f), the collective behavior is dominated by the environmental structure and collective migration breaks down. f: Normalized DS avoidance A in structured environment depicted in d, e versus attention limit k. A = 1 is the DS avoidance of solitary agents in the same environment. R inf = 0.1 in all the panels in this figure .   in random DS fields. We note that F = 0 corresponds to the average fitness expected for solitary individuals. For low δ, we observe a single maximum of F at low attention capacities k ≈ 2. For increasing benefits of DS avoidance δ, a second maximum emerges at large k, while at low k, the socially interacting collectives are on average outperformed by solitary individuals.

This trade-off between accuracy and avoidance, or "responsiveness", becomes particularly prominent if we consider structured, inhomogeneous environments containing free paths and voids in a landscape otherwise filled with high density of DSs, instead of homogeneous, random environments. At low k the agents completely ignore the environmental structure and their dynamics is dominated by social interactions with high directional accuracy. At high k, the situation reverses and the collective dynamics is dominated by the environment, where agents track the environmental structure, staying preferentially in areas with low DS density, while ignoring the preferred direction of migration (see Fig. 4d-f, see also Supp. Fig. S3 and Supp. Movie S3, S4).

Model Variations & Generality of Results. So far, we have considered social interactions which do not pay special attention to neighbors responding to DSs. This is motivated by the idea of social interactions being based on observations of behavior of others but absence of direct communication about the cause of their particular behavior. In order to test the robustness of our results, we explored an extension of the basic model, by introducing active signaling about potential danger by agents interacting with a DS. In this case, all neighboring agents connected to the signaler put their full attention on the signaling agent and respond only to it, while ignoring other non-signaling agents. As expected, the additional signaling improves the collective DS avoidance due to increased saliency of the corresponding cues within the network. However, the general results -in particular the coordination and responsiveness trade-off -remain unchanged (see Supp. Fig. S4).

Furthermore, the general trade-off between consensus and responsiveness to DSs does not depend on the presence of informed individuals. For R inf = 0 and ρ DS = 0, our system reduces to a Vicsek model with topological interactions in homogeneous environments [START_REF] Ginelli | Relevance of metric-free interactions in flocking phenomena[END_REF][START_REF] Chou | Kinetic theory for systems of self-propelled particles with metric-free interactions[END_REF][START_REF] Peshkov | Continuous theory of active matter systems with metric-free interactions[END_REF]. With R inf = 0 and ρ DS > 0, i.e. without the bias provided by informed individuals, the model becomes a topological Vicsek-like model in heterogeneous environments. All the phenomena discussed above hold also in this case, if we replace the collective accuracy C by the (normalized) average velocity C = ( i ûi )/N , which quantifies the overall degree of (orientational) order in the system (see Supp. Information and Supp. Fig. S5). It is worth stressing that there exist fundamental differences with metric Vicsek-like models in heterogeneous environments (cf. [START_REF] Chepizhko | Optimal noise maximizes collective motion in heterogeneous media[END_REF][START_REF] Chepizhko | Active particles in heterogeneous media display new physics[END_REF]), where agents are not subject to any cognitive limitation and display exclusively a limited perception capacity. Finally, we note that variations of the topological interaction mechanism do not affect the reported results. Specifically, we study the behavior of a model where the k-nearest objects are selected from the Voronoi neighborhoods (Supp. Fig. S6). This version of the model resembles the spatially balanced topological flocking algorithm proposed in [START_REF] Camperi | Spatially balanced topological interaction grants optimal cohesion in flocking models[END_REF] and represents a better approximation of visual networks [START_REF] Strandburg-Peshkin | Visual sensory networks and effective information transfer in animal groups[END_REF]. All this suggests that the fundamental coordination-responsiveness trade-off discussed here is independent on the specific choice of the social interaction model.

DISCUSSION

Using a generic flocking model we have demonstrated the importance of finite attention capacity of individuals for collective information processing in complex environments. In our model, agents dynamically allocate their limited attention to process social and environmental stimuli, whereby the saliency of different stimuli is governed by their spatial vicinity. We demonstrated, that contrary to the general intuition, large-scale coordination and, as a result, the accuracy of collective migration in complex environments is maximized for strongly limited individual attention capacity. High levels of accuracy for agents which can pay attention only to few stimuli at a time, are a direct consequence of a self-isolation from distracting environmental cues through social interactions. On the other hand, the increased ability of agents to respond collectively to environmental cues for high attentional capacity leads to a breakdown of collective accuracy for a minority of informed individuals as the strong information inflow through local distractions overrides the information on the global preferred direction of motion available only to a minority of informed individuals. This demonstrates a fundamental trade-off between large-scale coordination and collective accuracy on the one hand, and the dynamical response to local environmental cues in complex environments, on the other hand. We would like to emphasize that our general finding of weaker connected networks achieving higher global accuracy in complex environments is diametrically opposed to widely accepted and intuitive knowledge in network science that more connections lead more effective information exchange and thus higher levels of synchronization (see e.g. [START_REF] Shang | Influence of the number of topologically interacting neighbors on swarm dynamics[END_REF][START_REF] Bh Lemasson | Collective motion in animal groups from a neurobiological perspective: the adaptive benefits of dynamic sensory loads and selective attention[END_REF][START_REF] Belykh | Blinking model and synchronization in small-world networks with a time-varying coupling[END_REF][START_REF] Dörfler | Synchronization in complex oscillator networks and smart grids[END_REF]). We recover this intuitive result for flocking in empty environments, which demonstrates how taking into account environmental disturbances may dramatically change the collective behavior of self-organizing systems.

Our results suggest a specific link between cognitive and sensory capabilities of flocking animals and the ecological context. For example, for migrating animals, with high fitness benefits associated with coordination and information sharing on a preferred migration direction, with no (or very low) fitness costs of ignoring local environmental cues, strongly limited attention appears to be beneficial. However, if collective response to environmental cues is highly relevant for individual fitness but global coordination is not, as for example in foraging reef fish [START_REF] Andrew M Hein | Conserved behavioral circuits govern high-speed decision-making in wild fish shoals[END_REF], then being able to pay attention to many stimuli simultaneously becomes important. This yields testable hypotheses, on how the attention capability of different species exhibiting grouping behavior should co-vary with ecological niche, or how individuals within the same species should modulate their attention capabilities across contexts. Here, we note that a recent analysis of collective behavior in Hemigrammus rhodostomus, a strongly schooling fish species, suggests that an individual fish appears to pay attention only to one or two neighbors at a time [START_REF] Jiang | Identifying influential neighbors in animal flocking[END_REF].

Interestingly, being social offers an advantage over solitary behavior with respect to response to DSs only above a critical attention capacity. This poses some fundamental questions regarding the co-evolution of social behavior and individual attention capacity, especially taking into account potential developmental costs of higher attention capacity. Overall, our results point towards a complex interrelation between pre-existing attention capability, evolution of grouping behavior and the ecological niche.

We note that in our minimal model we varied only a single dimension of cognitive capabilities, namely the total number of objects an individual can pay attention to. There are other more complex cognitive processes, which affect the individual processing of a large number of social and non-social stimuli. For example object recognition and classification, may enable individuals to dynamically vary the relative saliency of social and non-social stimuli, which is not considered in this work but could allow individuals to adapt to different behavioral contexts. In general, the strength of the observed trade-off will depend on model choice and model details. For example, making the agents more likely to detect danger sites, will make the collectives more responsive to the environmental cues. However, if the overall attention capacity has an upper bound -as assumed here -it must come at the cost of decreasing social interactions. Hence, the existence of the general effects discussed here, in particular the surprising increase in collective accuracy with decreasing attention capabilities, is not restricted to particular model choice. The qualitative results should hold as long as the following three conditions are met: 1) the attention capacity is limited, 2) the salience of social cues has some distance-based component, and 3) the structure of the interaction emerges naturally from spatial self-organization. Overall, this work demonstrates the fundamental importance of potential constraints in sensory and cognitive abilities of individuals on emergence and function of collective behavior.

We considered explicitly the case of spatial flocking behavior, however the coordinationresponsiveness trade-off as a general principle should be observable in different collective behaviors. Only recently it was shown that Guinea baboon exhibit stronger response to known social cues than to novel ones. It was hypothesised that this unexpected behavior can be linked to the complex social environments in which Guinea baboon groups live, and the corresponding necessity to filter out irrelevant or distracting information ("social noise") [START_REF] Faraut | How life in a tolerant society affects the attention to social information in baboons[END_REF].

Our findings have also implications for the design of interaction networks in artificial distributed systems, such as robotic swarms that operate in complex environments. Instead of continuously increasing the sensory and computational abilities of individual agents in order to cope with the consensus problems in complex environments, it may be promising to think about constraining the "cognition" of swarming robots. By generating specifically tailored attentional bottlenecks, resulting in emergent self-isolation as observed here, one can facilitate coordination and exchange of relevant information in complex environments. Attentional bottlenecks based on static features, e.g. colors, which can be easily distinguished from the background, are widely used in swarm robotics [START_REF] Brambilla | Swarm robotics: a review from the swarm engineering perspective[END_REF]. Here, we demonstrated that dynamical features (as opposed to static ones), like relative distance, or relative speed [START_REF] Romanczuk | Swarming and pattern formation due to selective attraction and repulsion[END_REF], could provide effective means of coordination in complex environments, where "filtering" based on static features is difficult or not feasible.

Last but not least, our work yields potentially interesting implication for social sciences, where "echo-chambers" have received considerable attention recently. In human social networks, this effect is typically linked to homophily and confirmation bias [START_REF] Dandekar | Biased assimilation, homophily, and the dynamics of polarization[END_REF][START_REF] Del Vicario | The spreading of misinformation online[END_REF]. Our results show collective self-isolation from conflicting external information as agents moving in the same direction form tightly interacting social groups which ignore environmental cues. This can be viewed as an "echo-chamber"-like effect, which emerges naturally even in the absence of such explicit biases and self-sorting mechanisms. On the one hand, this suggests that these self-isolation tendencies may be much more prevalent and easier to obtain for agents with limited attention capacity. On the other hand, our results provide support for the evolution of proximate, socio-psychological mechanism facilitating the formation of echo chambers, such as homophily, by demonstrating how an emergent "echo chambers"-like effect strongly increase intra-group synchronization for a collective in a complex environment.

METHODS

Agent-based Model. We consider a system of N self-propelled agents and N DS danger sites DSs in a two dimensional domain of size L × L with periodic boundary conditions (torus). The agent and DS densities are thus ρ = N/L 2 and ρ DS = N DS /L 2 . The agents move with a fixed speed v 0 = 0.5 and respond to other agents and DSs by changing their direction of motion ûi = (cos ϕ i (t), sin ϕ i (t)) T . The behavior of each agent is mathematically described by following stochastic equations of motion, whereby dϕ i /dt corresponds to the turning rate of agent i:

d x i dt = v i (t) = v 0 ûi (t) = v 0 cos ϕ i (t) sin ϕ i (t) (2)
dϕ i dt = (1 -g i (t)) γ s n s j∈kNO sin(ϕ j -ϕ i ) -γ p sin(ϕ i ) +g i (t) γ l n l l∈kNO sin(α i,l -ϕ i ) + ηξ i (t) (3)
The first term in the turning response equation 3, is the alignment interaction. A focal individual aligns with the strength γ s = 1 with neighbors j, which are part of its kNO-set. In addition, informed individuals have a (weak) tendency γ p = 0.1 to move in a preferred direction, here +x = (1, 0) T (γ p = 0 for non-informed individuals). The turning away (repulsion) from the DS l, which are in the kNO-set is given by the third term (γ l = 1). Here, α i,l is the spatial position angle of the focal agent relative to the DS l. Both interactions are normalized by the respective number of agents and DSs, respectively (n y = y∈kNO 1). The function g i (t) determines whether the agent responds to DSs, or whether it aligns with its neighbors, and, in the case of informed individuals, biases its motion towards the preferred direction of motion. It is defined as

g i (t) = 1 for l ∈ kNO and d il < r 0 else . (4)
Note that an agent responding to a non-social cue (g(t) = 1), will not interact socially with other agents until its interaction with the DS is terminated, either because it leaves the repulsion zone or the site falls out of its k-nearest object set. The last term in equation 3 accounts for the stochasticity in the motion of individuals, with η being the angular noise strength and ξ i (t) a normally distributed Gaussian white noise with standard deviation 1. In all the simulations discussed, η = 0.25 and the average density of particles is fixed at 1 in a box of linear size L = 25 (N = 625). We have confirmed, that a detailed choice of v 0 , η, and L does not change the qualitative observations discussed here. Avoidance Parameter. We quantify avoidance of repulsion zones through à = 1 -N rz (t)/N . Here N rz (t) is the number of agents, which are within at least one repulsion zone at time t, and • represents temporal average in the stationary state. à will always decrease with DS density, as more and more space is occupied by repulsion zones. In order to control for this trivial effect, we rescale à by the corresponding value for non-interacting agents A = Ã/ Ãni . Thus, A = 1 indicates same average performance of the flock as solitary agents without any social interactions. Here, solitary individuals are always responding to a DS, once they are within the corresponding repulsion zone.

Fitness Function. We can quantify the emergent trade-off between collective accuracy and responsiveness, through the following global fitness function depending on the collective accuracy C and DS avoidance:

F (C, A) = C + δ(A -1) , (5)
with δ being the relative benefits of DS avoidance with respect to migration accuracy. The above function was defined in a way so that a value F = 0 corresponds to the behavior of solitary individuals, where the average accuracy vanishes (C = 0) and the DS avoidance is A = 1, according to the definition above. Thus, only for F > 0 the collectives perform better than single individuals (see Supp. Information for more details).

Research Foundation) under Germanys Excellence Strategy -EXC 2002/1 "Science of Intelligence -project number 390523135. P. Rahmani acknowledges support from the Ministry of Science, Research and Technology of Iran. F. Peruani was supported by the Agence Nationale de la Recherche via project BactPhys, Grant No. ANR-15-CE30-0002-01. We thank Ana Sofía Peruani for the artwork in Fig. 1a.

where δ = b a /b c represents the relative benefits of DS avoidance versus coordination. Please note that F is not fitness in a strict evolutionary sense (selection at the level of individuals), as it is a function collective variables A and C. Here, "fitness" refers rather to a collective utility function, where (local) maxima correspond to (local) optimal collective strategies for homogeneous collectives. For a given set of parameters, A and C are calculated in a stationary regime of a stochastic system. Thus both variables reflect the average ability of individuals to avoid DSs and to coordinate with their neighbors in homogeneous groups.

For δ = 1 both behaviors yield equal benefits, whereas for δ > 1 (δ < 1) DS avoidance is more (less) beneficial than coordination. For a wide range of DS densities and angular noise, we observe two distinct maxima of F in the δ, k-plane. For δ 1 coordination is much more beneficial than DS avoidance and the maximum of F is located at small k where coordination is highest. With increasing δ this maximum decreases and we eventually see a rise of a second maximum at large k, leading to two local maxima at intermediate δ.

Eventually, for large δ, where DS avoidance becomes far more important than coordination, we observe a single global maximum at large k.

Numerical Implementation and Experiments

The mathematical model was implemented in C/C++ with the k nearest object interaction implemented using the kd-tree implementation in the CGAL library [1]. The kNN interaction implementation in periodic boundary conditions was combining the standard kd-tree based algorithm with generation of mirror images of the environment to account for the interactions on a torus.

The stochastic equations of motion were numerically integrated using a standard Euler-Maruyama scheme [2]. The numerical time step was set dt = 0.1, whereby also smaller time steps were tested to show that the general results do not depend on the time step.

All the data were obtained by averaging over 20 realizations, each including 5 × 10 4 relaxation time steps and 10 5 stationary time steps. For low k(= 3) the network is sparse, composed of many small connected components, whereas for large k = 12, the network is highly connected with less components.

FIG. 1 .

 1 FIG. 1. a: Schematic visualisation of attention trade-off in collective behavior in complex environments. The focal individual can only pay attention to k = 3 nearest objects -other agents or non-social environmental features -simultaneously. b: Visualization of different situations that may occur in the model. The arrows indicate the velocity vectors v of the different agents. The small black circles indicate the location of danger sites l and l with their repulsion zones shown in blue. Agent i (red) reacts to the danger site (DS) l as two conditions are met simultaneously:DS l is in i's kNO, and agent i is also within the corresponding repulsion zone. Agent j (magenta) does not react to DS l since it's attention slot is already filled with three other agents (one blue and two gray). Agent k (blue) perceives DS l but does not react to it, because it is outside of the repulsion zone. It only reacts to two other neighbors (gray and magenta) and aligns with them.

12 FIG. 2 .

 122 FIG. 2. a,b: Examples of social interaction networks for k = 3 (a) and k = 12 (b) at ρ DS = 0.25. The black symbols indicate socially interacting agents, whereas the red symbols indicate agents responding to a DS. The lines indicate the (non-directed) interaction network. Filled circles represent uninformed agents, empty circles indicate agents informed about the preferred direction of migration. The DS positions shown by blue dots, are surrounded by a disc-like repulsion zones (light blue). For clarity, only a portion of the respective simulation box is represented here, see Supplementary Figure S1 for the full snapshots. c,d: In-out degree distributions for the emergent social interaction networks for low attention limit k = 3 (c) and high attention limit k = 12 (d) at low and high DS densities (ρ DS = 0.05, and ρ DS = 0.25). The vertical dashed lines are for visual guidance to distinguish the subpopulations with D out = 0 corresponding to agents responding to DSs (left of the vertical line). At high density of DSs this distribution is clearly bimodal with two peaks at D out = 0 and D out = k. By increasing DS density number of agents with D out = 0 increases. These agents have a lower in-degree compared to non-responders, which contributes to the self-isolation of the collective from environmental cues at low k values. R inf = 0.1 in all the panels in this figure.

2 FIG. 3 .

 23 FIG. 3. a, b: Collective accuracy C of migration along the preferred direction versus the ratio of informed individuals for different attention limits k, for environments with no danger sites ρ DS = 0 (a), and environments with high DS density ρ DS = 0.2 (b). The red arrows show the direction of increasing k. c: Collective accuracy C versus attention limit k for different DS densities ρ DS at R inf = 0.1.

FIG. 4 .

 4 FIG. 4. a:The fraction of agents responding to DS directly r d (direct responders, solid lines), or indirectly via social interaction with direct responders r i (indirect responders, dashed lines), for k = 2 (blue) and k = 24 (red) versus DS density ρ DS . b: Normalized DS avoidance A versus attention limit k for different DS densities ρ DS . A = 1 corresponds to the DS avoidance of solitary (non-interacting) agents. c: Global fitness versus attention limit k and relative benefits of DS avoidance δ at ρ DS = 0.25. Red (blue) regions correspond to better (worse) performance of a collective than isolated individuals according to the fitness function used. d and e: Example snapshots of emergent collective behavior in structured environments with a circular, DS-free path. For low attention capacity (k = 1, e), individuals ignore the structure of the environment and align with the preferred direction of migration. At high attention capacity (k = 16, f), the collective behavior is dominated by the environmental structure and collective migration breaks down. f: Normalized DS avoidance A in structured environment depicted in d, e versus attention limit k. A = 1 is the DS avoidance of solitary agents in the same environment. R inf = 0.1 in all the panels in this figure.

FIG. S1 .

 S1 FIG. S1. Snapshots of the (undirected) social interaction network in random environments with ρ DS = 0.25 for k = 3 upper panel, and k = 12 lower panel, at R inf = 0.1. Black agents are socially interacting, and red agents react to DSs. Informed and uninformed individuals are represented by empty and filled circles, respectively. Light blue circles are repulsion zones of DSs specified with blue dots. For the sake of clarity, the links between agents interacting with their periodic neighbors are removed. The black squares depict the close-ups shown in panels a, b of Fig.2(main text). For low k(= 3) the network is sparse, composed of many small connected components, whereas for large k = 12, the network is highly connected with less components.

  FIG.S2. Temporal interaction networks: Average number of connected components of the interaction network versus attention limit k (a). For all DS densities ρ DS , we observe a fast decay of the number of connected components, which due to constant number of agents N is equivalent to the growth of the average connected component size, indicating a more tightly connected temporal network. The average life time of an edge in the interaction network decreases with increasing density of DSs (b). However with increasing k for a fixed DS density we observe longer life times due to increased connectivity in the interaction network.

  FIG. S4. Attention trade-off in a group of agents with active signalers. Each agent connected to another individual signalling direct interaction with a DS (direct responder), pays only attention to the signaller(s) and ignores other social cues. Accuracy C (a) and normalized DS avoidance A (b) versus attention limit k for different DS densities at R inf = 0.1.

  FIG.S5. Emergence of global order in the system with no informed individuals, R inf = 0. a: Coordination C (directional order) versus attention limit k for different DS densities. b: Normalized DS avoidance versus attention limit k. A = 1 corresponds to non-interacting agents. The qualitative behavior with a coordination-responsiveness trade-off is similar to the model with informed individuals, but here instead of a specific direction, the emergent consensus direction is a random (spontaneous symmetry breaking).

  FIG. S6. Collective motion of agents with Voronoi-based kNN interaction network. Here, k nearest agents are selected from first shell of Voronoi neighbors. If the number of neighbors in first layer is smaller than k, then depending on k, the second Voronoi shell is considered. It is defined by the Voronoi neighborhood of the (direct) Voronoi neighbors of the focal agent. Accuracy C (a) and normalized DS avoidance A (b) versus attention limit k at R inf = 0.1.

ACKNOWLEDGMENTS P. Romanczuk & P. Rahmani acknowledge support by the German Science Foundation (DFG) through RO 4766/2-1 and the German Academic Exchange Service (DAAD). P. Romanczuk acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German

SUPPLEMENTARY INFORMATION

Stationary distribution of in and out-degrees

From our numerical simulations, we can extract the stationary probability distribution f (D out , D in ) for an agent having a particular combination of out-degree D out and in-degree D in .

In homogeneous environments, ρ DS → 0, the out-degree of each agent is directly set by the attention capacity k. Thus, the distribution of individual in and out-degrees f (D out , D in ) is sharply peaked at D out = k. The in-degree is not fixed: Whereas a focal individual i pays attention only to k nearest neighbors, the number of other agents m paying attention to it may be lower or higher. Therefore, we observe a spreading out of f (D out , D in ) along the indegree axis. This pattern holds also for finite number of DSs, however, now in addition, there is also a finite probability that an agent interacting with a DS ignores its neighbors (see Fig. 2c,d). In this case, its out-degree with respect to social interactions is zero: D out = 0, while the in-degree can assume many possible values depending on how many other agents pay attention to it at a given time. In general, for a finite DS density, the combined distribution f (D out , D in ) shows a bimodal distribution with two maxima at D out = 0 and D out = k.

For low k, the fraction of agents responding to DSs remains low even at very high ρ DS . The average out-degree as well as the average in-degree increases as expected with increasong k. However, the expected in-degree of direct responders with D out = 0 is always lower then the expected in-degree of other agents with D out > 0. As a results, for low k < 3, DS responders become often completely isolated from other agents by having D in = D out = 0.

Emergence of global order in the absence of informed individuals

In the absence of informed individuals R inf = 0, there is no preferred direction of motion. Here, instead of collective accuracy, coordination can be quantified using average polarization of the flock

which is equivalent to the ferromagnetic order parameter in physics. By replacing C by C, we observe the same fundamental coordination-responsiveness trade-off as discussed in the main text (Fig. S5).

Quantifying the Coordination-Responsiveness Trade-Off -Global Fitness Function

We can quantify the emergent trade-off between coordination and responsiveness, by introducing the following global fitness function depending on the collective accuracy C and DS avoidance A:

with b c being the benefits per unit of directional consensus, and b a the benefits per unit of DS avoidance. Without loss of generality, the equation can be rescaled, by b c to

SUPPLEMENTARY MOVIES

Each supplementary movie shows the spatial dynamics in the main panel (left), the corresponding accuracy C versus time (right top) and the corresponding DS avoidance A versus time (bottom right). The black line at A = 1 corresponds to the DS avoidance of non-interacting agents (see Materials and Methods). The first part shows the initial development of the system (t = 0 -200), the second part shows the stationary state at large times (t = 5200 -5400).

Supplementary Movie 1: Collective behavior at high density of DSs for low attention capacity k = 1 characterized by high accuracy of collective migration. Supplementary Movie 2: Collective behavior at high DS densities for high attention capacity k = 24 characterized by efficient response to environmental cues. Supplementary Movie 3: Collective behavior in structured environment with a circular DS free region for low attention capacity k = 1.

Supplementary Movie 4: Collective behavior in structured environment with a circular DS free region for high attention capacity k = 24.

SUPPLEMENTARY FIGURES