By addressing the defects in classical nucleation theory(CNT),we develop an approach for extracting the free energy of small waterclusters from nucleation rate experiments without any assumptionsabout the form of the cluster free energy. For temperatures higherthan ∼250 K, the extracted free energies from experimentaldata points indicate that their ratio to the free energies predictedby CNT exhibits nonmonotonic behavior as the cluster size changes.We show that this ratio increases from almost zero for monomers andpasses through (at least) one maximum before approaching one for largeclusters. For temperatures lower than ∼250 K, the behaviorof the ratio between extracted energies and CNT’s predictionchanges; it increases with cluster size, but it remains below onefor almost all of the experimental data points. We also applied astate-of-the-art quantum mechanics model to calculate free energiesof water clusters (2–14 molecules); the results support theobserved change in behavior based on temperature, albeit for temperaturesabove and below ∼298 K. We compared two different model chemistries,DLPNO-CCSD(T)/CBS//ωB97xD/6-31++G** and G3, against each otherand the experimental value for formation of the water dimer. In this Letter, we propose an approach forcalculating the free energy of water clusters, from the experimentson water nucleation rate, without any assumptions about the form ofthe cluster free energy.Water is essential to life andthe most abundant substance on the earth’s surface, and thusunderstandably the most extensively studied substance in the historyof science. However, the more we sharpen our theoretical and experimentaltools, the more elusive the behavior of water seems, which leavesit as an enduring mystery.3 and Zeldovich.4 Thepopularity of CNT stems from its simplicity, achieved by controversialassumptions about the free energy of cluster formation, which rendersits prediction questionable and in many cases in obvious disagreementwith experiment. Therefore, we begin developing our approach by addressingCNT’s problems and some common but inaccurate views about CNT.It should be noted that although here the focus is on water, thisapproach can be used for other substances, as well. Using this approach,we probe the water nucleation experiments and show that at highertemperatures (∼250 K < T) the ratio ofthe extracted free energy to CNT’s prediction exhibits nonmonotonicbehavior with a change in cluster size: toward the smallest clusters,this ratio continuously decreases to almost zero for monomers, whileas the cluster size increases, this ratio surpasses one and has atleast one maximum before returning toward one for sufficiently largeclusters. For lower temperatures (T < ∼250K), this ratio stays below one for almost all experimental data, althoughit increases with cluster size.To do so, we use classical nucleationtheory (CNT), which stillprovides the most popular framework for predicting nucleation, despitedecades of research after its development by Becker and Döring5) isK isthe product of the Zeldovich factor4 andthe frequency of attachment of the monomer to the critical cluster, i* is the number of molecules in the critical cluster, S is the supersaturation [the ratio of vapor pressure tosaturation pressure (S = Pv/Ps)], and ni*s is the number density of critical clusters in the saturated vapor.The term Si* formulatesthe thermodynamic driving force for the phase transition from a supersaturatedvapor to a liquid cluster of size i*. For an arbitrarycluster size, the thermodynamic driving force is readily calculatedas S is experimentally measurable. In contrast to Si*, CNT resorts to severalassumptions to calculate nis. Following Boltzmann’slaw, nis is related to the cluster free energyat saturation ΔGiskB is Boltzmann’s constant and T isthe temperature. CNT presupposes that ΔGis is solely equal to surface work ΔGiθ and omits theoretical possibilities such as contributions of configurationaleffects and various degrees of freedom to the cluster free energy.CNT considers an i-mer as a spherical droplet witha sharp interface and properties identical to those of the bulk liquid.Thus, the surface work is calculated as ΔGiθ = θ∞A1i2/3, the product of planar surface tension θ∞ and cluster surface area A1i2/3, where A1 is the monomer surface area. One naturally expects that for monomers n1s = n1s. This requires that ΔG1θ = 0, whichis not true in CNT; hence, CNT appears to be self-inconsistent. Girshickand Chiu6 suggested a simple way to recoverself-consistency [termed the internally consistent classical theory(ICCT)] by correcting the surface work with the equation ΔGiθ = θA1(i2/3 – 1). The requirement that ΔG1θ =0 has been questioned stating that CNT treats monomers as single-moleculedroplets rather than vapor monomers.8 However, Saltz9 argued that (except for the lack of the 1/S factor) the theory is consistent because, on the basisof the ideal vapor assumption in CNT, n1s is equal to totalcluster number density ns. Although Saltz’sargument is correct in a strict sense, it is intriguing to conceivethis problem conversely. We can drop the ideal vapor assumption butthink of ns just as an approximation for n1s. Therefore, setting n1s ≈ ns = Ps/kBT, ΔG1θ = 0, which is also an approximation.We discuss below how this approximation improves the prediction offree energy for monomers and dimers.The kinetic aspect of CNT combinedwith its thermodynamic aspecttreats nucleation in the form of a transition from the metastablevapor to the liquid phase as clusters surmount a free energy barrier.The rate of this transition is called the nucleation rate and (applyingCourtney’s correctionΔGi = ΔGis – (i – 1)kBT ln(S), i* isfound through Gibbs free energy minimization, resulting in theso-called Gibbs–Thomson equationUpon determination of thefree energy barrier to nucleation as 10 which provides a model-independent way to calculate i* from the nucleation rate asJexp, the result is calledthe experimental critical size iexp*. Using 11 showedthe Gibbs–Thomson equation disagrees with the experiments forwater and several other substances. The comparison between iGT* and iexp* for nine nucleation experiments with wateris shown in iGT* and iexp* to a size-dependenterror in calculating ΔGiθ. Merikantoet al.12 also showed, through Monte Carlosimulations, that below a certain size threshold the error in ΔGiθ is size-dependent. Both works statedthat the stepwise free energy change upon addition of the monomer,ΔΔGiθ = ΔGiθ – ΔGi–1θ, is expected to approach the macroscopic prediction as the sizeof the droplet increases, not the total surface work. As i increases, the error in ΔΔGiθ decreases to zero, and the total error in ΔGiθ becomes independent of size and dependent on only temperature;i.e., after a size threshold, the actual cluster free energy can becalculated as θ∞A1i2/3 – D(T), where D(T) is a correctionterm. Consequently, if i* becomes larger than thesize threshold, the first nucleation theorem dictates that iGT* = iexp*. Therefore, a correction that depends on only temperature,as proposed by McGraw and Laaksonen,13 inevitablyrelies on the validity of the Gibbs–Thomson equation or assumesthat iexp* is larger than the threshold. However, thereported experimental critical sizes are in the subnanometer range,and it is unlikely that they surpassed the relevant threshold. Indeed,the deviation of iGT* from iexp* in iexp* in most of the experiments.The accuracy of theGibbs–Thomson equation can be checkedagainst the first nucleation theorem,iGT* and iexp* on nucleation rate is illustrated by meansof SiGT*–iexp* in Si* in JCNT/Jexp versus temperature. The deviations in JCNT/Jexp are too small to be indicativeof the huge errors associated with SiGT*–iexp*. The error caused by overpredictionof i* is compensated by the exponential decreasein ni*s due to the increase in ΔGiθ. This compensation mechanism in CNTalong with the oversimplified comparisons between CNT and experimentmay be the reason that the significance of the inaccuracy of the Gibbs–Thomsonequation has not been previously identified. Moreover, these comparisonshave been partly responsible for echoing the view that CNT exhibitsa strong temperature dependence without considering the other variables, where A and B are constants.This correction was later claimed25 tobe the experimental substantiation of the function D(T) envisaged by McGraw and Laaksonen.13 However, a similar correction can be easilyassumed in the form of exp(A′ + B′T)/S, which not only isequally successful in improving CNT’s prediction but also includesthe 1/S factor, which as explained in ref . This accidental success requiresrestraint before attaching a physical significance or insight to suchcorrections. We do not dismiss the role of a wrong temperature dependencein CNT’s failure but stress that each variable needs to bestudied in isolation. Here, we aim to include the cluster size effectin our examination for two reasons. First, the deviation of iGT* from iexp* indicates that the error in nucleation barriersin CNT is dependent on size. Second, following the recommendationof Gibbs27 and Tolman,28 we consider the curvature/size dependence of surface tension.It is noted that, in the case of microscopic cluster sizes, the surfacetension is reduced from a physical property to a mathematical quantity.The potential effect of the deviation between see refs , 20),  is unknown, we do not resort to thethermodynamics of nucleation to calculate Z and obtainthis factor kinetically without any assumption about the form of ΔG(i). In a saturated vapor, the principleof detailed balance dictatesEi is the frequency of detachment of the monomerfrom an i-mer. Using nis is defined in a recursive fashionQuestioning the validity of Gibbs–Thomson equation opensa new pathway for evaluating CNT in terms of the cluster size. Thispath requires an approach, immune from CNT’s assumptions incalculating l as a continuous variable and using Considering i at i = i* yields S = E(i*)/Cs(i*),where S defines the supersaturationunder a fictitious equilibrium condition, where i* would amount to the critical size. We emphasize that S = E(i*)/Cs(i*) derived here in a saturated vapor isequivalent to E(i*) = C(i*) for a supersaturated vapor. The latter as Kashchiev30 stated is the kinetic definition of the criticalcluster and was determined previously (see refs (E(i*) = Cs(i*)S and using Taking derivativesfrom both sides of see refs  and 32)i at i = Gs(i* – 1) is expandedas a Taylor series about i*Moreover, ΔTruncating the aboveseries after the second-order term and using 2ΔG(i)/∂i2]i=i* ≈ −2kBT ln(Ai*/Ai*–1), which when inserted into the relation for Z providesan approximation for this factor asAi* = A1i*σ (σ = 2/3). Finally,according to Gs(i) from the experiment asRewriting both sidesof Z by i* is given by the first nucleation theorem, which is model independent,and C(i) ∝ Ai ∝ iσ. Although the first proportionalityis reasonable, the second one with σ = 2/3 invokes the spherical assumption about the cluster shape. This assumptionwas shown to be inaccurate in the molecular dynamics simulations ofsimple fluids, resulting in nonspherical shapes and consequently largersurface areas for clusters as large as a 100-mer.34 Also, ref  given by 3/4 increases ΔGs(i) by only 0.06 kBT. The important point is Gs(i) and thusprovides an independent check on ΔGθ(i) in CNT.The calculation of lso, ref  proposedGs(i) obtainedfrom experiments, let us conduct a thought experimentto reveal the behavior of ΔGs(i)/ΔGθ(i) versus i. We expect thatby increasing i and surpassing the size thresholdone can calculate the actual cluster free energy as ΔGs(i) = ΔGθ(i) – D(T), which leads to the asymptotic behavior G1s must be close to zero because at temperaturesbelow the critical point the vapor is mainly monomers. The secondcondition relies on the average free energy of dimers calculated fromthe second virial coefficient in the water vapor virial equation ofstate. Below the critical point where the cluster–cluster interactionsare insignificant, direct relations can be established between thenumber density of the smallest clusters in the vapor and the coefficientsin the virial series. To calculate n2s, we follow therelations developed by Saltz9 that canalso be observed in Monte Carlo calculations36 . For temperatures from 200 to 375 K, the experimental38 and correlational42 values of the second virial coefficientsuggest that ΔG2s/ΔG2θ is muchless than one (∼0.5) (see Gs(i)/ΔGθ(i) increases from ∼0 at i = 1 to ∼0.5 at i = 2. Therefore, one canenvisage two general scenarios under which ΔGs(i)/ΔGθ(i) starts by an increasebetween i = 1 and i = 2 and thenapproaches one. In the first scenario, ΔGs(i)/ΔGθ(i) crosses unity at leastonce and manifests at least one maximum point, while in the secondscenario, ΔGs(i) always remains below ΔGθ(i) and ΔGs(i)/ΔGθ(i) may have zero to an infinite number ofextrema. In the simplest case fulfilling the first scenario, ΔGs(i)/ΔGθ(i) exhibitsa single maximum after crossing unity and before surpassing the sizethreshold. The simplest case fulfilling the second scenario is a curvemonotonically increasing from ∼0 and approaching unity frombelow, similar to ICCT.Before discussingΔ(see ref  fora si0.5) see 2. The fiG2s (which arediscussed below), show that CNT, in contrast to ICCT, greatly overpredictsthe free energy of dimers. It is noted that near the triple pointthe vapor compressibility factor is quite close to one and the contributionsof the higher-order virial coefficients to vapor pressure become relativelyunnoticeable. Thus, the data on the second virial coefficient forthe temperature range of interest in nucleation experiments (210–320K) either are not available or cover only the higher end of the temperaturerange. As a remedy to this problem, the relations for the second virialcoefficient are extrapolated down to 200 K in G2s/ΔG2θ is evidently smaller than one.With regard to the aforementioned conditions, incomparison toCNT, ICCT predicts the free energies of monomers and dimers closerto those obtained on the basis of monomer concentration and the secondvirial coefficient. Gs(i) from Gθ(i)by CNT versus i and T in panelsa and b, respectively, of Gs(i), because it is derived by combining modeling and fitting to thedifferent data sets, we attempt to provide reasonable estimate envelopesto consider the uncertainties. The error bars in panels a and b of iexp* and the estimate envelopes for ΔGs(i). To calculate the envelopebounds, the uncertainties in iexp* and C(i) are considered conservatively to achieve the largestbounds. For iexp*, from the reported uncertainties in refs , the main source of uncertainty is the sticking probability coefficient.Historically, a broad range of values (0.01–1.0) for αwere observed in experiments.44 However,recent reviews47 and experiments50 reported larger valuesfor α (0.2–1.0). In particular, ref (C(i) is underestimated by 53 Therefore, we assume C(i) maydeviate from The Δ in refs  and (20− in refs  the larg in refs ). Due tolar, ref  showed tiexp*. The pointsbelow one all belong to the smaller clusters observed at temperatureslower than ∼250 K. Leaving aside temperature, for these pointsas iexp* increases ΔGs(i)/ΔGθ(i) also increases. This pattern is also observed for thepoints on similar isotherms of 210, 220, and ∼230 K whose iexp* and ΔGs(i) errorbars do not overlap. Comparing point 1 with point 4, point 2 withpoint 5, point 3 with points 6–8, and point 6 with point 9shows the increase in ΔGs(i)/ΔGθ(i) on the aforementioned isotherms as thecluster size increases. The fact that ΔGs(i)/ΔGθ(i) always remains below onesuggests that the second scenario may be true in the case of the lowertemperatures. However, due to the lack of data for larger clusters,the existence of a crossover cannot be completely dismissed at thesetemperatures. For instance, we may conjecture that a crossover occursbetween points 6 and 9, which both approximately are at 230 K. Interestingly,the opposite pattern is observed for the data points above one. Comparingpoint 10 with point 12 (310 K) and point 11 with point 13 (320 K)reveals that as the size increases, ΔGs(i)/ΔGθ(i) decreases; recalling thefirst scenario, this suggests that these points are located aftera maximum.Even upon consideration of these uncertainties,many data pointslie above one in 54 and Du et al.55 , we also applystate-of-the-art quantum mechanics models to investigate the freeenergy of water clusters. After extensive conformational searchingusing Ogolem,58 geometry optimizationsand frequency calculations were computed with the ωb97xd densityfunctional59 and electronic energies werecorrected with the DLPNO-CCSD(T) method using the cc-pVNZ basis sets.61 Single-point calculations with N = D, T, and Q were extrapolatedto the complete basis set (CBS) limit and used to determine Gibbsfree energies.64 We determined the DLPNO-CCSD(T)/CBS//ωB97xD/6-31++G**average free energy change in cluster formation from monomers to clusters, iH2O > (H2O)i, using the lowest-Gibbs free energy (H2O)i clusters for eight different temperatures rangingfrom 216.65 to 310 K. In addition to determining the overall ΔG° values for cluster formation, we calculated thestepwise monomer addition (ΔΔG°)for each successive addition of a water monomer to the proceedingcluster for i = 2–10 [H2O + (H2O)i−1 > (H2O)i]. We note that the minimum energystructureschange as a function of temperature for i = 4, 6,8, 9, and 10 and that using the lowest-energy structure instead ofBotzmann averaging the ensemble of low-energy structures is less importantthan obtaining the most accurate possible CBS electronic energy (seethe Supporting Information and refs (Gis and ΔGis (see the Supporting Information). We then used the G3 method to determine the G3 free energies forevery structure within 1 kcal mol–1 from the DLPNO-CCSD(T)/CBS//ωB97xD/6-31++G**results. The Gibbs free energy changes in the stepwise monomer additionΔΔG°, using five different quantummechanical simulation methods at 298.15 and a standard state of 1atm, are listed in iH2O > (H2O)i] using the lowest-Gibbs free energy (H2O)i clusters. On the basis ofthe data in 2O)i=11–14 clusters. We then computed the G3 energies for the lowest-lyingDLPNO-CCSD(T)/CBS//ωB97xD/6-31++G** clusters for (H2O)i=11–14. For method d, simulationswere performed for eight different temperatures ranging from 216.65to 310 K. For method e, to reveal the effect of temperature, we coveredan extensive range including 16 temperatures from 200 to 373 K. Allof the details about methods e and d along with values of ΔΔG° and overall free energy change ΔG°, and their corresponding values at saturation pressure , are given in the Supporting Information. ΔG2s derived from the second virial coefficient and also the experimentalmeasurement of the water free energy change in dimerization. In addition,a similar comparison between the free energies derived from the thirdvirial coefficients and the results from methods d and e is presentedin Figure S2. This comparison also showsa better agreement 57)). Howand refs −64 for dGis/ΔΔGiθ as shown in T), forthe current cluster size range ΔGis/ΔGiθ exhibits a crossover forall temperatures, which is the evidence for the first scenario. Fortemperatures of <298.15 K, although ΔGis/ΔGiθ still generally increaseswith cluster size up to 13-mer, the rate of this increase decreasesas the temperature decreases and the cluster size increases. Therefore,for the lower temperatures, it seems that the second scenario maybe true. If by approaching the macroscopic sizes, ΔGis/ΔGiθ remains entirely below oneat lower temperatures , this suggests a shift from thesecond scenario to the first that is governed by only temperature.On the basis of the results from method d shownin panels a andb of 67 and molecularsimulations70 that surface tension shows a nonmonotonic dependenceon curvature.In summary, we developed an approach for extractingthe clusterfree energy from nucleation experiments independent of the form ofthe cluster free energy. We observed that for water at temperaturesabove ∼250 K the extracted cluster free energy behaves nonmonotonicallyrelative to CNT’s prediction: toward the lower end of the sizerange, the extracted cluster free energy is much smaller than CNT’sprediction, while an increase in size it increases it above that ofCNT and the ratio between them passes through at least one maximumbefore approaching the macroscopic prediction. For temperatures lowerthan ∼250 K, the ratio of extracted free energies versus CNT’sprediction behaves differently. For these lower temperatures, almostall of the extracted free energies from experiments lie below CNT’sprediction, although the gap between them is closing as the clustersize increases. We also calculated free energies of small water clustersusing state-of-the-art G3 model chemistry. We showed that G3 resultsare more in line with other quantum mechanical simulations and alsoin excellent agreement with dimers’ energy derived from thesecond virial coefficients of water for a wide range of temperatureand also with an experimental dimerization energy measurement. G3quantum mechanical simulations confirmed our observation of the behaviorchange of CNT’s prediction based on temperature, however, arounda different temperature (∼298 K). It is noteworthy that, ifthe surface work assumes the entire free energy, our observation forhigher temperatures qualitatively supports the emerging consensusbetween density functional theory calculations


Nanoporous membranes based on two dimensional materials are predicted to provide highly selective gas transport in combination with extreme permeance. Here we investigate membranes made from multilayer graphdiyne, a graphene-like crystal with a larger unit cell. Despite being nearly a hundred of nanometers thick, the membranes allow fast, Knudsen-type permeation of light gases such as helium and hydrogen whereas heavy noble gases like xenon exhibit strongly suppressed flows. Using isotope and cryogenic temperature measurements, the seemingly conflicting characteristics are explained by a high density of straight-through holes (direct porosity of ∼0.1%), in which heavy atoms are adsorbed on the walls, partially blocking Knudsen flows. Our work offers important insights into intricate transport mechanisms playing a role at nanoscale. 2D nanoporous membranes are predicted to provide highly selective gas transport in combination with extreme permeance. Here authors demonstrate gas separation performance and transport mechanisms through membranes of graphdiyne, a quasi 2D material with a graphene-like structure. This interest is due to the fact that atomic-scale thickness implies very fast molecular permeation, as compared to conventional 3D membranes that exhibit flow rates scaling inversely proportional to the membrane thickness. To prove this ultimately fast permeance, 2D membranes with relatively large pores having the effective size d0 larger than the kinetic diameter dk of sieved molecules have intensively been explored . This regime is well described by the classical Knudsen theory and allows a moderate selectivity that arises from differences in thermal velocities of gases with different molecular weights m2. On the other hand, gas selectivity can greatly be improved using membranes with angstrom-scale pores of d0 ≤ dk. In this case, molecules encounter substantial activation barriers for translocation through membranes, which leads to exponentially enhanced selectivity between gases having even marginally different dk5. Unfortunately, the presence of activation barriers also implies an exponential suppression of flow rates5. This tradeoff between permeability and selectivity is well known6 and motivates the search for novel nanoporous materials with optimal tradeoff characteristics.Porous membranes made on the basis of two-dimensional (2D) materials attract intense interest for their potential use in separation technologies5. An alternative approach, perhaps more realistic in terms of applications, is bottom-up synthesis of thin nanoporous membranes such as, e.g., laminates made of 2D materials8 and multilayer films of intrinsically porous crystals11. Such quasi-2D membranes with thicknesses comparable to the mean free path λ of gas molecules may still allow flow rates similar to those achievable using nanoporous strictly-2D crystals13. However, mechanisms governing gas permeation and separation by quasi-2D membranes remain poorly understood  as, for example, they may differ from simple Knudsen and activated transport models. One of the candidate crystals for envisaged high-performance molecular-sieving membranes is graphdiyne, a carbon allotrope that has intrinsic triangular pores of a few angstroms in size15. Its potential use in gas separation technologies has extensively been discussed through theory and simulations19 but the experimental assessment of its gas separation properties is still lacking.To create nanopores in 2D crystals, top-down fabrication is often utilized to introduce nanoscale defects in initially impermeable 2D materialsIn this work, we have investigated gas transport through graphdiyne-based nanoporous films. Fast and selective gas permeation are found in such quasi-2D membranes. Isotope experiments, cryogenic temperature measurements, and binary mixture experiments reveal that adsorption of heavy gases at the pores’ interior plays a critical role for gas transport at nanoscale.20. Details of the synthesis and characterization of the resulting material, using Raman and X-ray photoelectron spectroscopy, are provided in 5 Pa (see below). Our extensive examination of the films using both scanning and transmission electron microscopy (TEM) revealed no defects larger than >50 nm in size . However, at the bottom of each microwell we observed small regions with vanishing thickness. These regions are separated by typical distances of ∼100 nm as shown in Fig. 21.Our graphdiyne-based nanoporous films were synthesized via coupling reactions of hexaethynylbenzene molecules22, with device fabrication procedures detailed in Supplementary Fig. To investigate gas transport through the graphdiyne-based films, we suspended them over micrometer-sized apertures etched in silicon-nitride/silicon wafers  and hydrogen isotopes D2 and HD. Note H2 was avoided because of a large fluctuating background usually arising in mass spectrometry for this particular isotope, which did not allow sufficient accuracy required for purposes of this report. The chosen gases provided a wide range of m and dk values which allowed detailed characterization of molecular transport through our membranes. Examples of the measured gas flow rates Γ as a function of the applied pressure P are shown in Fig. Γ scaled linearly with P, gas transport through the graphdiyne-based films can be characterized by their permeance Γ* = Γ/P. If the same apertures were covered by multilayer graphene (impermeable to gases23) or if a ∼200 nm metal film was evaporated on top of graphdiyne post examination, no helium gas flow could be discerned within our detection accuracy of ∼10−14 mol s−1. This corroborates that graphdiyne membranes were the only pathway for gas transport in our experiments. Furthermore, to calibrate our mass spectrometers with respect to different gases, we used “bare-hole” devices made in the same manner but without placing a graphdiyne film over the apertures. The reference devices exhibited Γ* approximately 1000 times higher than that in the presence of graphdiyne-based membranes. This yields a porosity of ∼0.1% for graphdiyne films . For comparison, nanometer-thick membranes made from graphene oxide8, metal-organic frameworks, covalent-organic frameworks and zeolite11 exhibited one to two orders of magnitude lower porosity.First, we studied permeation of various gases at room temperature Γ* are summarized in Fig. 3He, 4He, D2, HD, and Ne, the observed permeance can accurately be described by the Knudsen behavior Γ* ∝ m−1/2 that is expected for free molecular flows (λ ≫ d0 ≫ dk). In contrast, heavy gases show pronounced deviations from the Knudsen dependence . Accordingly, if steric sieving was important, both hydrogen isotopes would show a notable suppression in Γ*, in contrast to the experiment . Within the accessible T range from 300 K down to 10 K for helium and down to 30 K for hydrogen (the latter condenses at lower T), Γ* was found to vary as ∝ T−1/2  to be satisfied over the whole range of T and P under investigation26. Using our lowest T of 10 K and highest applied P of 1 × 105 Pa, we find that d0 should be <5 nm. The validity of this analysis was crosschecked by measuring T dependence for helium flowing through apertures of 30–50 nm in diameter. The relatively large apertures exhibited clear nonlinearities in permeance caused by transition from free molecular flow into the viscous regime, if either P was increased or T decreased 1/2 and aperture dimensions27. For 4He at 10 K, λB reaches ∼4 Å. This value exceeds the kinetic diameters of the studied gases, which are given by the size of electron clouds around nuclei25. If holes in our graphdiyne films were smaller than 1 nm, a contribution of quantum-mechanical diffraction should have been noticeable in the measured T dependences or as an isotope effect for light gases29. The purely Knudsen flow observed experimentally implies that d0 should be considerably larger than dk + λB ≈ 7 Å. If we now take a theoretical perspective, our DFT calculations show that noble gases with dk close to d0 should experience very large (∼1 eV) energy barriers . On the other hand, no sign of activated transport is noticeable in Fig. kBT at cryogenic T. This again suggests pores in our graphdiyne-based films to be larger than at least 1 nm. The above estimate 1 nm < d0 < 5 nm is also consistent with the observed porosity and the described morphology of the graphdiyne films. Indeed, hole-like regions seen in our TEM images . The mixtures’ Γ exhibited profoundly nonlinear dependences as a function of the partial pressure of helium . In other words, if mixed, noble gases no longer flow independently through graphdiyne pores even at room T. To the best of our knowledge, such interaction between flows of noble gases has never been observed before. We attribute this phenomenon to adsorption of heavy atoms on inner walls of straight-through holes in the graphdiyne films. As the holes are only ∼10 times larger in size than typical dk and are expected to have a finite length , incident noble gas atoms would see the nanopores as partially filled with a heavy gas. This effectively reduces d0 and leads to a suppressed flow of helium. For the case of pure heavy gases like Kr and Xe, this partial filling should also play a role and can be interpreted as either partial blockade of pores by same-gas atoms or their longer translocation times. This explanation agrees well with the progressively weaker interaction effect observed for lighter noble gases . Supplementary Fig. 33, which enhances permeation by many orders of magnitude. In contrast, adsorption of heavy atoms on internal surfaces of the graphdiyne-based membranes reduces permeation. Moreover, the inner-pore adsorption gives rise to a counterintuitive effect of interacting flows of supposedly non-interacting, inert gases. No noticeable gas transport through the intrinsic mesh within graphdiyne’s crystal structure has been evidenced, due to their small effective diameter of <4 Å which yields high energy barriers (Supplementary for DFT simulations) and/or the non-aligned intrinsic meshes from adjacent atomic layers in ABC stacked multi-layer graphdiyne that blocks gas flows. To this end, carbon allotropes with larger unit cells could be better candidates for gas separation, if their mechanical stability can be achieved.To conclude, in comparison with the previously reported atomically-thin membranes with nanopores obtained by top-down fabrication, our quasi-2D membranes exhibit similar selectivities combined with high flow rates thanks to a relatively high pore density  for 64 h and after that, graphdiyne-based films were grown on the surface of copper foils. Note that the entire process should avoid solution/reactant in contact with oxygen (in air) and light. Consequently, the graphdiyne-based membranes were rinsed using heated acetone and N, N-dimethylformamide to remove any residues , and were dried under argon.Graphdiyne-based films were synthesized via cross-coupling reactions of the monomer 1,2,3,4,5,6-hexaethynylbenzene (HEB). In brief, first, HEB was synthesized by adding tetrabutylammonium fluoride to tetrahydrofuran solutions of hexakis[(trimethylsilyl)ethynyl]benzene at 0 °C and used after reaction time of 10 min. Next, HEB was dissolved in pyridine and was added slowly (in 8 h) to a mixture of treated copper foils immersed in pyridine at 110 °C under argon atmosphere, with the ratio of HEB quantity to area of copper foils ∼1 μg/cm2 trench was removed using focused Ga ion-beam  images were obtained using Zeiss Ultra SEM. By tilting the SEM stage at an angle, the interconnected nanometer-thick vertical walls and the microwell structures can be clearly seen  chambers. One chamber is filled with gases under investigation (or their binary mixtures for the measurements shown in Fig. m = 3, 4.To measure gas permeation through graphdiyne membranes, we use experimental set-up as shown in Supplementary Fig. −13 mol s−1. This 1–2 orders of magnitude higher background than that measured under ambient conditions was a consequence of He leakage through the fittings of the pipe-work into the atmosphere which was picked up by the sensitive leak detector. This background was however still at least one order of magnitude smaller than our smallest signals and thus has negligible influence to the accuracy of measurements.For measuring gas permeation at cryogenic temperatures, we use a home-made constant flow cooling system as shown in Supplementary Fig. Supplementary information


The obtained carbon particles also showed high surface area and hierarchical pore structure. Kraft lignin carbonized up to 1400 °C gives the highest BET surface area of 646 m2 g−1, which makes it a good candidate for electrode materials in energy storage applications. The energy storage application has been validated in a three-electrode set up device, and a specific capacitance of 97.2 F g−1 was obtained at a current density of 0.1 A g−1 while an energy density of 1.1 Wh kg−1 was observed at a power density of 50 W kg−1. These unique characteristics demonstrated the potential of kraft lignin-based carbon particles for electrochemical energy storage applications.Technical lignins, kraft, soda, lignoboost, and hydrolysis lignins were used for the production of carbon particles at different carbonization temperatures, 1000 °C and 1400 °C. The results showed that the lignin source and carbonization temperature significantly influenced the carbon quality and microstructure of the carbon particles. Soda lignin carbonized up to 1400 °C showed higher degree of graphitization and exhibited the highest electrical conductivity of 335 S·m Over the past decade, the development of biomass-based carbon materials and their conversion into different kind of carbon nanomaterials has gained immense attention in the research community and different biomass feedstocks have been converted into value added end products by researchers around the globe . Out of In the current study, four different commonly available technical lignins, namely kraft lignin (KL), soda lignin (SL), lignoboost lignin (LB), and hydrolysis lignin (HL), were carbonized at two different carbonization temperatures 1000 °C and 1400 °C. A simple, green, and low-cost strategy such as a direct carbonization of the lignin, was adopted. Neither activation steps nor any special templating agents, techniques, additional additives, or extra processing steps were used to enhance the porosity or microstructure of the carbon particles. Resulting carbon particles were systematically analyzed for their suitability for high value applications, such as supercapacitor electrodes and conductive graphitic carbon additives. The carbon particles with the highest specific surface area were used for making supercapacitor electrodes and their electrochemical performances were analyzed. Carbon particles were also evaluated for their electrical conductivities to determine technical lignin and temperature of carbonization for achieving highest degree of graphitization for their effective use as a conductive filler. Remarkable electrochemical and electrical properties were achieved, and the resulting graphitic carbon obtained in this study is another important step forward towards achieving functional graphitic carbon from biomass. Hence, this study presents the importance of types of lignin and their carbonization process, and how they impact on the final properties of carbon materials. This work gives new insights towards the effective utilization and value addition of an abundant and unique natural resource, reducing the carbon footprint, economic growth potential, and improving ecosystem sustainability.TM lignin (LB), based on softwood was kindly supplied by Domtar Plymouth pulp mill  and (4) hydrolysis lignin (HL), which was kindly supplied by a Finnish energy company, St1 biofuels . Sulphuric acid  and ethanol was purchased from VWR International AB . Polytetrafluoroethylene (PTFE) (60 wt.% aqueous suspension) was purchased from Sigma-Aldrich .Four different lignins were used; (1) a low sulfonate content kraft lignin (KL) supplied from Sigma-Aldrich ; (2) soda lignin (SL) Protobind 2000, from Greenvalue Enterprises LLC, ; (3) Lignoboost lignin, Biochoice2) atmosphere. The setup used was as follows: first one hour at 100 °C, then one hour at 400 °C, and then two hours at the final temperatures 1000 °C or 1400 °C. All the heating steps during the carbonization process were performed at a rate of 5 °C/min. After carbonization, the samples were cooled down to room temperature and taken out carefully for further characterizations. Before the carbonization, all the lignin powders were powdered and sieved using an 80 μm sieve size to the similar particle size (<80 μm). A horizontal tube furnace Nabertherm RHTC-230/15, Nabertherm GmbH  was used for the carbonization in nitrogen , and carbon black in ethanol, in the weight ratio 75:15:10, where PTFE acted as polymeric binder and carbon black as a conductive filler. This slurry was then left for 6 h in oven at 100 °C to evaporate ethanol (which was used to disperse the carbon particles and PTFE in the slurry) and water (from the aqueous suspension of PTFE). After drying, the working electrode was weighed (Approximately 20 mg) and used for the electrochemical measurements. 2) atmosphere, from room temperature to 1000 °C, maintaining a heating rate of 10 °C/min. TGA-Q500  was used for the thermo gravimetric analysis (TGA) in nitrogen (NEquation (1) was used for the calculation of carbon yield,The microstructure of the carbon particles was studied using scanning electron microscopy (SEM) JSM-IT300,  with an acceleration voltage of 20 kV under a high vacuum. The samples preparation was performed by diluting aqueous dispersions of the carbon particles to approximately 0.01 wt.%, the dispersion was drop casted onto a freshly cleaved mica sheet attached to the sample holder, and the dispersion was dried before sputter coating with platinum to a thickness of 15 nm. 2, . The elemental analysis of the carbon materials was carried out using energy dispersive X-ray spectroscopy (EDX) on the same SEM instrument equipped with a silicon drift detector Oxford X-MaxN 50 mmThe particle sizes were determined from SEM images taken at random locations of the samples. For each sample, the sizes of at least 200 particles were measured by the software ImageJ . λ = 532 nm) with a power of 5 mW, was used to study carbon structure of the prepared materials. The measurements were performed at a magnification of 20×, and the spectra were recorded from 50 to 3500 cm−1 with an acquisition time of 2 s. A Bruker Senterra Raman microscope, , using green laser , the XRD crystallite size was calculated. Thus, the radial expansion of the carbon crystal and the stacking height of graphene layers were determined using the Scherrer equation, as shown in Equation (2) [iL (nm) can take two indices: aL which is the radial expansion of the carbon crystal contributed by the (001) peak and cL which is the stacking height of the graphene layers can be determined from (002) peak. iK is a structural constant and values used were aK = 1.84 and cK = 0.89 taken from the literature [λ is the wavelength of αCu K radiation, λ = 0.1542 nm [iβ (rad) is the FWHM for each peak and iθ (rad) is the diffraction angle for each peak. X-Ray diffraction (XRD) was performed with a PANalytical EMPYREAN diffractometer  using Cu K alpha radiation. Intensity data were recorded in intervals of 0.03° between the scattering angles 2tion (2) .(2)Li=Kterature . λ is thd) between two graphene layers was calculated using Bragg’s law for n = 1,The distance (cL) with interplanar distance (d). The number of graphene layers per stack unit was calculated by dividing the stacking height (2 adsorbed at a relative pressure P/P0 = 0.99. Specific surface area (SSA), average pore size, and pore volume of the carbon particles were measured using Micromeritics Gemini VII 2390a Brunauer–Emmett–Teller (BET) analyzer at 77 K . The samples were degassed at 300 °C for 3 h prior the BET measurement in a Micromeritics FlowPrep060 to remove entrapped moisture and increase the accessibility to the pores. Micropore area was calculated using the standard t-plot method and total pore volume was determined from the amount of N2SO4) solution. To analyze the rate capabilities of the electrodes, cyclic voltammetry measurements (CV) were carried out at different scan rates between 2 and 100 mV s−1 in the potential range from 0 to 1 V. Specific capacitances obtained from CV tests can be calculated using Equation (4),C (F g−1) is the specific capacitance; m (mg) is the mass of active materials loaded in the working electrode; v (V s−1) is the scan rate; I (A) is the discharge current; V2 and V1 (V) are high and low potential limits of the CV tests. Further galvanostatic charge discharge method (GCD) was conducted using at different current densities in the range from 0.1 to 1 Ag−1 to investigate the electrochemical performance. Specific capacitance from GCD measurements can be calculated using Equation (5),C (F g−1) is the specific capacitance; I (A) is the discharge current; ∆t (s) is the discharge time; ∆V (V) is the potential window; and m (mg) is the total mass of electrode material. Electrochemical properties of KL1400 electrode were measured using a Princeton Applied Research VerstaSTAT 3 Potentiostat/Galvanostat  connected with a three-electrode cell kit . The working electrode was prepared as described in the previous section. Platinum was used as the counter electrode, Ag/AgCl as reference electrode and the electrolyte used was 1M sulphuric acid (HE) and specific power density (P), a practical supercapacitor was assembled using the prepared electrodes and two electrode method was used for the measurements and the details of which is given in supporting information. From the galvanostatic tests, E and P can be calculated using Equations (6) and (7),E (Wh kg−1) is the average energy density; C (F g−1) is the specific capacitance; ∆V (∆V) is the potential window; P (W kg−1) is the average power density and ∆t (s) is the discharge time. Electrochemical impedance spectroscopy (EIS) measurements were carried out in the frequency range of 10−2 to 105 Hz to determine the resistances offered by the electrode. To calculate the specific energy density . The conductivity values were recorded for frequencies ranging from 1 kHz to 5 MHz.2 atmosphere to 1000 and 1400 °C to obtain the different carbon particles as listed in max) in the temperature range between to 300 °C to 400 °C, as shown in 2 as major decomposition products during TGA and KL showed relatively stronger peaks between 600 °C and 700 °C, indicating the release of more CO and CO2 [ and CO2  which co and CO2 . Experimental carbon yield obtained after carbonization of lignin particles are represented in The morphology of carbon particles was analyzed using scanning electron microscopy (SEM) and the results are shown in 2, were released. Minor amounts of sodium (Na) and sulfur (s) were present in KL and SL based carbon particles, which were derived from the lignin isolation process. Minor amounts of potassium (K) and silicon (Si) were present in SL based carbon particles. These were due to the impurities from the wheat straw arising from the starting materials for isolating soda lignin. The presence of ash also can contribute to the presence of silicon in the carbonized lignins. A High purity of lignoboost lignin [Elemental analysis of carbon particles was performed using energy dispersive X-ray spectroscopy (EDX) and the results obtained are listed in t lignin  could bet lignin . EDX prot lignin . −1 (G-band) and at 1330 cm−1 (D-band). Vibrations of sp2 bonded carbon atoms in the 2D hexagonal lattice is the reason for G-band and in-plane terminated disordered tangling bonds in graphite, as represented by the D-band [D/IG ratio) [Raman spectra of carbon particles are represented in e D-band ,20. RamaG ratio) ,22. D/IG ratio and full width at half maximum (FWHM) of D-band of the carbon particles carbonized at 1000 °C and 1400 °C calculated from the Raman spectra. In D/IG ratio increased significantly when the temperature was increased from 1000 °C to 1400 °C. This clearly indicates the growth of aromatic clusters and graphitic layers in the carbon structure [tructure ,21,23. Itructure  was usedtructure . For alltructure . These rtructure . The positions of D and G bands are given in θ = 22–23° and the second peak was located between 2θ = 43–44°. The first diffraction peak (2θ = 22–23°) corresponds to the (002) plane which represents the reflections from stacked graphene layers. The second maximum (2θ = 43–44°) corresponds to the (100) plane, which originates from the aromatic ring structures present within the graphene layers [XRD patterns of carbon particles were recorded and represented in e layers . With thn = 1) and Scherrer-Equation were used and mean distance between graphene layers d, stacking height cL and radial expansion aL were calculated as shown in aL increased with increase in the carbonization temperature and the highest expansion (aL = 5.65 nm and cL = 1.25 nm) was observed for SL1400, which indicated the higher level of graphitization achieved, as discussed earlier. The distance between the stacked layers were calculated and the value is ranging between 0.377 nm–0.395 nm  and (100) peaks, Bragg’s Law , and pore volume (Vp) of the carbon particles were measured using Brunauer–Emmett–Teller (BET) analysis. Nitrogen (N2) adsorption isotherms for all the samples carbonized at 1000 °C and 1400 °C are represented in 2 g−1 among the samples produced at 1000 °C while KL1400 showed 646 m2 g−1 among the samples carbonized at 1400 °C. The specific surface area (SSA), average pore diameter  and SL (81 m2 g−1 at 1000 °C and 216 m2 g−1 at 1400 °C) showed quite similar adsorption behavior during the BET analysis, but SL showed the lowest adsorption and hence lowest SSA among all the samples for both carbonization temperatures. This could be due to the higher graphitization and the growth of crystallites [2 g−1) compared to HL1000 (290 m2 g−1) while the contribution from meso and macropores remained almost the same for both HL1000 and HL1400. To gain a better understanding of the SSA, pore size distributions of all carbon particles are shown in LB  solution as the electrolyte. CV measurements were carried out at different scan rates and the obtained results are shown in The electrochemical properties of the KL1400 based electrode were studied using cyclic voltammetry (CV) and galvanostatic charge discharge (GCD) measurements using a three-electrode system with 1M sulphuric acid  has been carried out in the frequency range between 10−2 and 105 Hz and the Nyquist plot for KL1400 is shown in −1 while KL1000 showed the lowest electrical conductivity of the order of 10−6 S m−1. The direct comparison of electrical conductivity values between different studies is difficult due to different factors, e.g., the difference in lignin resource, carbonization procedures, and conductivity measurement conditions, such as the equipment and conditions of the measurement. However, the conductivity values obtained for SL1400 is much higher than the conductivity values reported by Snowden et al. [−1) and that of lignin-based carbon fillers (142 S m−1) reported by Gindl-Altmutter et al. [r et al. . Table 8In the current study, carbon particles were prepared from four different types of technical lignins. Two different final carbonization temperatures were chosen to study the impact of carbonization temperature on the microstructural evolution. It has been concluded that, except for hydrolysis lignin, the specific surface area was increased with the increase in the carbonization temperature. This is attributed to the evolution of more volatiles at higher temperatures. Kraft lignin carbonized at 1400 °C showed the highest surface area and was used as electrode material together with PTFE as binder to show promising electrochemical properties and hence suitability as electrodes in supercapacitor. It was also observed that the extent of graphitization was influenced by the type of lignin and carbonization temperature. Soda lignin at 1400 °C showed the highest electrical conductivity and could be utilized as functional carbon additives in composites, e.g., in EMI shielding applications. Hence, the current study presents new insights regarding the importance of the proper selection of technical lignins and their carbonization conditions for achieving the requirements of specific applications.


Human brain organoids replicate much of the cellular diversity and developmental anatomy of the human brain. However, the physiology of neuronal circuits within organoids remains under-explored. With high-density CMOS microelectrode arrays and shank electrodes, we captured spontaneous extracellular activity from brain organoids derived from human induced pluripotent stem cells. We inferred functional connectivity from spike timing, revealing a large number of weak connections within a skeleton of significantly fewer strong connections. A benzodiazepine increased the uniformity of firing patterns and decreased the relative fraction of weakly connected edges. Our analysis of the local field potential demonstrate that brain organoids contain neuronal assemblies of sufficient size and functional connectivity to co-activate and generate field potentials from their collective transmembrane currents that phase-lock to spiking activity. These results point to the potential of brain organoids for the study of neuropsychiatric diseases, drug action, and the effects of external stimuli upon neuronal networks. Brain organoids replicate cellular organization found in the developing human brain. Here, the authors utilize microelectronics to map activity in brain organoids and assemble functional circuits that mirror complexity found in brain networks in vivo. Neurons within these three-dimensional assemblies generate action potentials upon depolarization6, display excitatory and inhibitory post synaptic currents7 and exhibit spontaneous network activity as measured by calcium imaging10 and by extracellular field potential recordings from small numbers of electrodes14. Progress in the development of flexible electronics enable three-dimensional readouts of electrical activity distributed across the surface15 and throughout the volume of an organoid16, and recent work extended the activity repertoire of organoids to include rhythmic activity over a range of oscillatory frequencies17.Under a variety of conditions18. These technical innovations have revealed an unprecedented vantage of spatiotemporal dynamics governing propagating neuronal activity occurring in cortical-hippocampal circuits19 and retinal network oscillations20. Continued advancements in CMOS-based MEA devices have increased the number of recording sites while retaining high signal-to-noise ratios21, which have enabled large-scale ex vivo mapping of cortical synaptic projections22, meanwhile the innovation of high-density three-dimensional electronic probes has opened the possibility of mapping neural activity in three-dimensions23. Here, we used state-of-the-art CMOS MEA technology to generate detailed electrical activity maps from 500 µm thick human brain organoid slices. These arrays have 26,400 routable electrodes, of which 1024 can be recorded from simultaneously. The surface area of these arrays (≈8 mm2) is comparable to the cross-sectional area of our brain organoids24. We constructed a network-level description of neuronal signaling in organoid slices by first constructing a spatial map of extracellular spikes generated by single neurons using densely tiled recording electrodes spaced at approximately the size of a neuronal soma (see Methods), thereby enabling robust and accurate assignment of single-unit activity by utilizing electrode redundancy and the characteristic waveform shapes determined by the neuron’s location relative to the recording electrode26. These technological advances offer a significant improvement over the more limited ascertainment of multi-unit activity (MUA) obtained from considerably sparser arrays used in previous studies. In addition, we used a 960 electrode Neuropixels CMOS shank probe27 with an electrode pitch (≈20 µm) approximately the size of a neuron soma, to record from whole intact organoids. We quantified the probability distributions of single neuron spiking activity and found a subset of interspike intervals (ISIs) that are exponentially distributed having Poisson-like spike trains and another subset with more regular firing patterns. This emergent feature in human brain organoids resembles one facet of functional brain organization28. Benzodiazepine application shifted the ISI probability distributions toward lower frequency modes. These features were accompanied by less variation in burst to burst population-level dynamics. We constructed a spatial connectivity map derived from pairwise correlations between spiking units measured by the thousand microelectrodes selected from the total set of 26,400 electrodes with the highest activity. A subnetwork based on single-unit spikes demonstrated a marked increase in pairwise spike correlation strengths following benzodiazepine application. We detected theta frequency oscillations supported by their coherence across the surface of the organoid and stereotypical sets of neuronal ensembles phase-locked to the theta oscillation. The relationship between single-unit spikes and LFPs demonstrated here, as well as the stereotypical network behavior following benzodiazepine application suggest ways organoids could be used to study a range of neurological conditions.Technological limitations have restricted acquisition of extracellular voltages to a relatively small numbers of neurons. Consequently, technical shortcomings have hindered attempts to determine functional connectivity among large neuronal networks, how the sum of their transmembrane conductances collectively synchronize with rhythmic activity and how pharmacologic perturbations affect these physiologic parameters. Brain organoids provide many heuristic opportunities to reveal mesoscale features of neuronal signaling by linking single-unit activity to large network-level population dynamics across spatiotemporal scales impossible to resolve in vivo. The development of complementary metal-oxide-semiconductor (CMOS)-based microelectrode array (MEA) technology has enabled high-resolution readouts of extracellular field potentials generated by single neurons, at network scales, simultaneously over thousands of recording sites12 and promotes extensive axon outgrowth11. In this study we present electrical activity from six organoids positioned on 2D CMOS arrays and performed pharmacological manipulations on a subset (n = 4) of that cohort. Additionally, extracellular recordings were performed on three whole organoids using CMOS shank electrodes. Once positioned on the MEA, spontaneous spiking occurred within ~2 weeks across organoids, followed by increasing firing rates in the form of synchronized bursts at about 6 months  resulting in a 72% ± 29% (mean ± STD) reduction in spiking . The remaining action potentials in the absence of excitatory synaptic input could reflect interneuron firing30. A saturating concentration of the sodium-channel blocker tetrodotoxin (1 µM) was subsequently applied to the same organoid slices, resulting in a 98% ± 1% (mean ± STD) reduction in spiking activity compared to control  and seated with a sterile custom harp slice grid for recording up to 6 months after placement on arrays  with a stringent R2-threshold set to 0.9. We next utilized the coefficient of variation (CV), defined as the ratio of the standard deviation (σ) in the ISI to its mean (µ) to map the variability of our measured ISIs  decreased the inter-burst intervals, burst duration and the number of spikes participating in the burst (burst amplitude) throughout the organoid ). Furthermore, upon the addition of diazepam (50 µM) the of ISIs fraction with a CV > 0.7 drops to 8% ± 10% (n = 4 organoids) from 42% ± 4% of the units under control conditions. Interestingly, the total number of spikes per unit remained similar  increased the distribution of ISIs toward longer intervals Fig.  as obserp < 1e−4).Concomitant changes in single-unit spiking activity were observed at the population level Fig. . TemporaρC − ρD)/(ρC + ρD). Here, ρC and ρD are the average population rate differences for control and 50 μM diazepam conditions, respectively, for the given time interval relative to the burst peak set by tstart and tstop. Overall, we observed increased burst-to-burst variability when comparing control conditions relative to diazepam (across the full burst window); however, there was a slight increase in diazepam variability for organoid L1 over the narrow interval  ms relative to the burst peak  burst peak Fig. , given b46, and to be inherently nosier than high-fidelity action potentials48. Quantification of functional couplings between neurons according to these criteria allowed us to derive a map of pairwise spike correlations between single-unit spikes and their latency distributions   Fig.  inset, wges Fig. . Edge st ms Fig.  inset, c ms Fig. . The reln = 14 measurements across n = 6 organoids). Based on the directionality, the in-degree (Din) and out-degree (Dout) were computed for each unit, defined as the number of incoming and outgoing edges respectively. Nodes with a high fraction of outgoing edges (Dout − Din)/(Dout + Din) > 0.8 were labeled “sender“ nodes /(Dout + Din) > 0.8 were labeled “receiver” nodes . A negative mean spike time latency compared to the reference unit indicated a predominant directionality toward the reference unit and vice versa Fig. . Bi-diredes Fig. , top. Nodes Fig. , middle.des Fig. , bottom,des Fig. .p ≥ 0.1, Supplementary Fig. p ≥ 0.3, Supplementary Fig. 50 and a gamma distribution fit experimental data well, with the gamma distribution outperforming the truncated power law /(fd + fc), where fc and fd are the fraction of total edges for control and diazepam (50 µM) respectively  reveal a more correlated and interconnected network with less variable single-unit and population-level neuronal dynamics in the presence of diazepam  which increased the relative fraction of strong edges while decreasing the relative abundance of weaker ones Fig. . The cha53. We examined the relationship between single-unit spiking, MUA and LFPs at the theta frequency in organoid slices.The transmembrane currents associated with action potentials are summed in local networks and are detectable as local field potentials (LFPs). Additionally, extracellular voltage fluctuations associated with synaptic transmission are thought to represent the coordinated activation of neuronal ensembles and occur within frequency domains characteristic of many LFPsA receptors along with TTX-sensitive sodium channels, the LFP amplitudes were no longer detected  and found a prominent topological overlap with electrode sites identified by cross-correlation analysis  , we simultaneously surveyed theta oscillations and spikes across the organoid slice to construct a phase-locking map for regions of highly correlated theta activity identified previously by the cross-correlation analysis  exhibit phase-locking to theta frequency oscillations in our organoids within the Rayleigh criteria for non-uniformity (p < 0.05), other frequency bands, such as gamma, were not investigated and could account for other distinct populations reported elsewhere that exhibit phase-locking in these frequency regimes65. Theta oscillations provide a temporal window for computation within local circuits in situ66, suggesting that organoids may display sufficiently complex circuitry for intrinsic computation.In the absence of sensory input, brain activity can entrain action potentials to theta frequency oscillationsase Fig.  and dete. µ Fig. . Interes27. Recordings were made by inserting the shank into the organoid using a motorized, low-drift micromanipulator. We measured single-unit spiking across hundreds of microns of organoid tissue within the z-plane in whole organoids 53. Finally, we confirmed phase-locking of spikes to theta across multiple sites in whole brain organoids  is one and converges to a normal distribution for large k, thus providing a useful measure of ISI-regularity similar to the CV28. Depending on architectonically defined brain regions with specialized cellular compositions and intrinsic circuitry, neurons process information differently69. Indeed, neuronal firing varies considerably across cortical regions of monkeys71. Therefore, different organizational features across the brain organoid may exhibit different dynamics to account for the observed ISI distributions. The minority fraction of irregular ISI distributions may be a feature of higher levels of entropy and circuit complexity and contain increased capacity for computation and information transfer as found in prefrontal cortex compared to more regular firing patters found in motor regions28.Human brain organoids are an intrinsically self-organized neuronal ensemble grown from three-dimensional assemblies of human-iPSCs. As shown here, brain organoids offer a window into the complex neuronal activity that emerges from intrinsically-formed circuits capable of mirroring aspects of the developing human brain53. Thus, we cannot rule out the possibility that intermediate undetected neurons may account for the coupling between two correlated units. The graph does not imply downstream or upstream routes of information transfer beyond the individual binary couplings. Importantly, what the network does demonstrate is a non-random pattern of a relatively small number of statistically strong (reliable) couplings against a backdrop of weaker couplings. As demonstrated in the murine brain52, high anatomical connection strength edges shape a non-random framework against a background of weaker ones . Consistent with minimal volume conduction effects, we validated theta oscillations by demonstrating that the imaginary part of coherency54 projected onto the same spatial locations identified by cross-correlation analysis . The control iPCS line F12442.4 was cultured in mTeSR1 medium (Stem Cell Technologies) on tissue culture plates coated with hESC-qualified Matrigel (Corning). mTeSR1 was exchanged every other day and iPSCs were routinely passed using ReLeSR (Stem Cell Technologies). Brain organoids were generated after the method by Lancaster et al.8 with minor modifications. iPSCs were incubated in 0.5 mM EDTA in D-PBS for 3 min before dissociation in Accutase for 3 min at 37 °C. After adding mTeSR1 and triturating to achieve a single-cell suspension, cells were pelleted for 3 min at 1000 rpm. Cells were resuspended in low bFGF hES media  supplemented with ROCK inhibitor (50 µM) and plated in U-bottom ultra-low attachment plates at 4500 cells per well. On day 2, low bFGS hES media with ROCK inhibitor was replaced. Another media change was performed on day 4, with omission of bFGF and Rock inhibitor. On day 5, embryoid bodies were transferred to neural induction media , and media was replaced on days 7 and 9. On day 10, each neuroepithelial structure was embedded in 15 μl hESC-qualified Matrigel followed by incubation in neural induction media for another 2 days. On day 12, neural induction media was replaced with NeuroDMEM-A media . From day 19 on, organoids were cultured in NeuroDMEM + A media  with media changes twice per week. From day 21 on, organoids were kept on an orbital shaker at 75 rpm. Mycoplasma testing was routinely performed on organoids and iPSCs using the MycoAlert Mycoplasma DetectionKit (Lonza).Human iPSCs were donated by Karch78.Organoid samples were fixed in 4% paraformaldehyde in 0.1 M sodium cacodylate buffer (pH: 7.4) overnight at 4 °C. After rinsing, samples were embedded 10% agarose and sectioned (100 µm) using a vibratome . Primary antibodies used were anti-Parvalbumin , anti-GAD65 , anti-SMI312 , anti-GFAP , anti-connexin 43 , anti-MAP2 , anti-synaptobrevin . Samples were viewed through a ×20 oil immersion lens (N.A. 0.85) and imaged using an upright Olympus Fluoview 1000 laser scanning confocal microscope  equipped with a motorized stage utilizing Olympus Fluoview software version 4.2. High-resolution wide-field mosaics were produced as described elsewhere2 with occasional agitation. After 30 min of incubation, organoids were triturated ten times with a fire-polished glass Pasteur pipette, followed by further incubation and trituration every 15 min until dissociation was complete. The total incubation time in papain/DNaseI solution was 90 min. After adding Inhibitor Solution and DNaseI in EBSS, the dissociated cells were pelleted by centrifugation for 3 min at 300 g. The cells were resuspended in 500 µl ice-cold PBS/0.01% BSA using a wide-bore plastic pipette tip and passed through a 35 µm cell strainer. An aliquot of the single-cell suspension was mixed with Trypan blue and analyzed for viability and cell concentration on a Countess II FL Automated Cell Counter (Thermo Fisher). Viability was >90% for all organoid samples.Individual brain organoids were subjected to dissociation using the Worthington Papain Dissociation System (Worthington). Solutions were prepared according to the manufacturer’s instructions. Organoids in NeuroDMEM + A (defined in Brain organoid generation) media were cut into pieces with a sterile scalpel blade and washed twice in PBS prior to dissociation. Incubation in papain/DNaseI solution was performed at 37 °C and 5% CO79. Briefly cells and barcoded beads  were run on an aquapel-treated microfluidic drop-seq device (FlowJEM) for co-encapsulation in nanoliter-sized droplets. After droplet-breakage, reverse transcription and Exonuclease I treatment, beads were counted, and 2500 beads were apportioned per PCR tube for cDNA amplification. The amplified cDNA libraries were purified using SPRI beads (Beckman Coulter) and quantified on a Fragment Analyzer (Agilent). Tagmentations were performed using the Illumina Nextera XT kit (Illumina), and the resulting libraries were purified in two consecutive rounds of SPRI beads-based size selection (0.6x beads to sample ratio followed by 1x beads to sample ratio). The size and concentration of the final libraries were measured on a Fragment Analyzer and a Qubit Fluorometer (Thermo Fisher), respectively. Libraries were sequenced on an Illumina Nextseq500 instrument.Following dissociation, cells were diluted to 110 cells/µl in PBS/0.01% BSA. Drop-seq was performed as described in Macosko et al.79. Briefly, raw reads were converted to BAM files, cell barcodes and UMIs were extracted, and low-quality reads were removed. Adapter sequences and polyA tails were trimmed, and reads were converted to Fastq for STAR alignment (STAR version 2.6). Mapping to the human genome (hg19 build) was performed with default settings. Reads mapped to exons were kept and tagged with gene names, beads synthesis errors were corrected, and a digital gene expression matrix was extracted from the aligned library. We extracted data from twice as many cell barcodes as the number of cells targeted (NUM_CORE_BARCODES = 2x # targeted cells). Downstream analysis was performed using Seurat 3.081 in R version 3.6.3. An individual Seurat object was generated for each sample, and objects were merged using the Seurat merge function. Cells with <300 genes detected were filtered out, as were cells with >10% mitochondrial gene content. Counts data were log-normalized using the default NormalizeData function and the default scale of 1e4. The top 3000 variable genes were identified using the Seurat FindVariableFeatures function , followed by scaling and centering using the default ScaleData function. Principal Components Analysis was carried out on the scaled expression values of the 3000 top variable genes, and the cells were clustered using the first 30 principal components (PCs) as input in the FindNeighbors function, and a resolution of 0.4 in the FindClusters function. Non-linear dimensionality reduction was performed by running UMAP on the first 30 PCs. Following clustering and dimensionality reduction, putative cell doublets were identified using DoubletFinder 2.0.382, assuming a doublet formation rate of 5% and calculating homotypic proportions based on clustering at resolution 0.4 as described above. A pk value of 0.61 was selected for doublet identification based on the results of paramSweep_vs, summarizeSweep and find.pK functions of the DoubletFinder package. After filtering out the doublets as identified by DoubletFinder, counts data were extracted, and a new Seurat object was generated. Normalization, variable gene selection, scaling, clustering and UMAP dimensionality reduction were performed as described above with the exception that 32 PCs were used as input for clustering and UMAP. Clustering was performed at different resolutions . Clustering at resolution 0.4 resulted in 9 clusters and was in good agreement with the expression of known marker genes for cell types found in brain organoids 2 incubator at 37 °C prior to placement and positioning on the CMOS MEAs.Whole brain organoids were allowed to mature to 4–6 months (see Brain organoid generation section) and were then embedded in 10% (w/v) low melting point agarose at ≈40 °C and allowed to cool at 2 °C for ≈10 min. Organoids embedded in the agarose gel were then sectioned into 500 μm thick slices with a vibratome (Leica VT1000S) using an advance speed of one and a vibration speed of nine. The gel block containing the organoid was mounted onto the surface of the vibratome cutting block with super glue and placed into the cutting reservoir filled with ice-cold cutting solution composed of 50% (v/v) DMEM (Thermo Fisher), 50% (v/v) Neurobasal (Thermo Fisher), 1x penicillin streptomycin (Thermo Fisher). The space surrounding the vibratome liquid reservoir was filled with ice to keep the cutting solution temperature constant during sectioning. The vibratome blade was allowed to advance through the front-face of the gel block and organoid tissue, while leaving ≈3 mm gel uncut and attached at the back of the gel block. This method allowed previous sections to remain attached to the gel block, which minimized torque on the organoid during subsequent sections and prevented deforming or releasing of the remaining uncut tissue embedded in the gel below. Next, the contiguous layer stack of 500 μm thick organoid sections remaining in the gel block  ethanol solution in deionized water (18.2 MΩ-cm) for 30 min. The liquid reservoir of the CMOS MEA was then rinsed thoroughly with sterile ultra-pure distilled water in a sterile hood (≈5 ml per CMOS MEA well), and the lids were allowed to air dry in the sterile hood. The recording surfaces of the MEAs were then coated by adding 0.5 ml of poly-l-lysine (PLL) (Sigma-Aldrich) solution (0.1 mg/ml in ultra-pure water) into the CMOS MEA reservoirs and then placed in an incubator for ≈1 h at 37 °C with the lids on. The arrays were then transferred to a sterile hood, the PLL solution was aspirated off and washed an additional 3x with ultra-pure water (1 ml volume for each wash). The CMOS MEA reservoir was then filled with 0.5 ml of BrainPhys medium and the lid was placed on it. The CMOS MEA was transferred to a sterile dissection hood equipped with a stereoscopic microscope. Organoid sections were transferred to the CMOS MEA well with a cut P1000 pipette tip and gently positioned over the recording electrode surface of the CMOS MEA with sterile forceps while visualizing with the stereoscopic microscope , and media was exchanged twice a week.High-density CMOS MEAs and custom machined (UCSB physics machine shop) liquid reservoir lids made from Delrin (McMaster-Carr), fitted with CO2 at 37 °C) using CMOS microelectrode array technology  containing 26,400 recording electrodes with a diameter of 7.5 μm at a center-to-center distance of 17.5 µm within a sensing area of 3.85 mm × 2.1 mm. A subset of which 1024 electrodes could be selected for simultaneous recording24. The low-noise amplifiers have a high-pass filter (0.5 Hz) to minimize drift. Electrical measurements were performed once a week (24 h after a media change), and took around 2 weeks after sectioning to start spiking with a marked increase in activity in the form of synchronized bursts at ≈6 months in age , we concatenated the filtered traces in time before applying spike sorting. All of the processing was performed using the SpikeInterface framework26.The raw extracellular recordings were automatically spike sorted to extract single-unit activity. The same spike sorting procedure was applied to the planar HD-MEA and the Neuropixels recordings. The raw traces were band-pass filtered between 300–6000 Hz before applying the Kilosort2 algorithm86. Delta (0.5-4 Hz), theta (4–8 Hz), alpha (8–13 Hz), beta (13–30 Hz) and gamma (30–50 Hz) bands were extracted using eefilt.m function from the UCSD EEGlab MATLAB toolbox  were imported into MATLAB (Mathworks). The LFP component was extracted by first low pass filtering the raw data (frequency cutoff of 500 Hz using 4th order Butterworth filter), and downsampled to 1 kHz. The LFP was then filtered into the traditional oscillatory sub-bands using a bi-directional finite impulse response filter27 was inserted into the organoid using a low drift (<10 nm/h), motorized precision micromanipulator . The Neuropixels probe was mechanically attached to the micromanipulator using a custom machined bracket (UCSB physics machine shop). Electrophysiology data were acquired using SpikeGLX . Action potential channel was sampled at 30 kHz. The local field channel (sampled at 2500 Hz) was low pass filtered at 500 Hz. Downstream processing and analysis were performed with methods similar to 2D CMOS data.Six-month-old brain organoids were transferred to BrainPhys media and were cultured for 30 days without shaking before recording. Organoids were then positioned using a cut p1000 pipette tip into a custom well (filled with BrainPhys media) to immobilize the organoid (kept at 37 °C on a temperature-controlled stage). A high-density CMOS shank with 960 electrode sites that tile along a 70 × 20 μm cross-section shank (10 mm long) where 384 electrodes can be programmably routed and recorded from simultaneously28. First, single-units were selected for analysis that had a minimum of 30 spikes measured over a 3-min interval and ISI intervals with a range of 8 ms to 100 ms were only considered for further analysis. An exponential (f(x) = a ∙ eλ∙x−) function was fit using a non-linear regression model (fitnlm function in MATLAB) to normalized histograms (histcounts function in MATLAB) of the single-unit ISI distributions (bin widths were set to 1/15th of the median ISI for a given unit). Here, f(x) represents the normalized bin counts for a given ISI bin designated by x. The constants λ and a were free fitting parameters. Regression fits were selected that had R2 values >0.9 to classify ISI distributions as exponentially distributed. A two-sample KS test was performed to assess differences between ISI distributions under control and 50 µM diazepam conditions. All statistical tests performed in the manuscript were two-sided unless specified otherwise.The selection and filtering of single-unit activity was based on the work of Maimon and AssadρC − ρD)/(ρC + ρD), where ρC and ρC are the population rate differences for control and diazepam, respectively. This analysis was performed using a range of time windows with respect to the peak of the MUA activity. Window start times were selected between −200 ms until 0 ms with respect to the MUA peak with a step size of 10 ms. Window end times were selected between 0 ms until 500 ms with respect to the MUA peak with a step size of 10 ms. The average ratio over all the different time windows was computed to compare the control and diazepam treatments for the different organoids.For each recording, a population averaged firing rate vector was computed for the spike sorted single-units by summing the spiking activity over all units for each recording frame and averaging these summed values with a 5 ms moving average. For each burst, a burst population rate (5 ms moving average and 5 ms Gaussian) was selected by taking a window with respect to the peak of the MUA (defined below). For each pair of bursts, the similarity in the population rates were assessed by taking the average of the absolute difference between the two burst population rate vectors. This difference was then averaged over all burst pairs. Burst-to-burst similarity between control and diazepam was compared by computing the ratio in the average population rate vector difference between control and diazepam (87. Detected spike times (rounded to the nearest ms) were then stored as a binary spike time matrix (columns represent electrodes and rows represent time). MUA population-averaged spike rates were calculated by first summing all detected spikes  occurring within a one ms time window to create a “population spike” vector. The “population spike” vector was then averaged over a 20 ms sliding window and smoothed with a 100 ms Gaussian kernel, yielding a population-averaged spike rate as measured across the CMOS array. Next, population bursts peaks were defined when the population-averaged spike rate exceeded 2x its rms value (using the findpeaks function with a minimum peak distance of 1 s). Population burst durations were defined by time windows surrounding the burst peak where the burst amplitude was attenuated by 90% of its maximum value.MUA was extracted by band-pass filtering (cutoff frequencies of 0.3 and 7 kHz using a 4th order Butterworth filter) the raw voltage signal, and spike detection was set at 5x the rms-noise for a given electrode49. A publicly available MATLAB script11 was used to compute the STTC. A correlation window (Δt) of 20 ms was used to compute a score of functional connectivity strengths across all units. The remaining correlations between single unit pairs were further filtered based on the spike time latency distributions of the spike trains. All electrode pairs with a multimodal latency distribution (Hartigans’ dip test p < 0.1), a full width at half maximum larger than 15 ms were removed. To rule out the influence of outliers, these measures were computed over all spike time latencies within a  ms range. Using the STTC scores of the remaining unit pairs, functional connectivity networks were generated.Correlated spiking activity between single-unit spike trains was compared using the STTCn = 6 organoids), 93.2% ± 14.7% (mean ± STD) percent of the STTC connectivity edges strengths were below the 0.35 threshold across 13 separate measurements; organoid L6 had no pairwise spike correlations (edges) after spike train randomization and was excluded from the statistics above . The size of the largest and second largest component, defined as the number of nodes present in the largest and second largest structure of connected nodes. The size of the second largest component was <2% of the size of the largest component for L1, L2, L3 and L4 and <10% for L5 and L6 for 7-month-old organoids.Global and local connectivity metrics for the connectivity networks were analyzed using the MIT strategic engineering network analysis toolbox for MATLABDin) and out-degree (Dout) were computed per unit, defined as the number of incoming and outgoing edges respectively. “Receiver” nodes were defined as (Din − Dout)/(Din + Dout) > 0.8, while “sender” nodes were defined as (Dout − Din)/(Din + Dout) > 0.8. All other nodes were labeled “brokers“. The directionality per node was calculated for the largest component of interconnected nodes. To evaluate the significance of “sender” and “receiver” nodes, networks were randomized by performing double edge swaps (ten times as many swaps were performed as edges in the control network) and contained the same number of nodes and the same edge density. Randomized networks contained no “sender” or “receiver” nodes.Based on the mean of the spike time latency distributions, directionality was assigned to the edges in the functional connectivity map. A negative mean spike time latency compared to the reference unit indicated a directionality toward the reference unit and vice versa. Based on the directionality, the in-degree (n = 4) were then allowed to sit undisturbed for 15 min in the incubator (37 °C 5% CO2) before recording. Organoid dose responses were carried out over 3, 10, 30 and 50 µM diazepam concentrations  was solubilized in DMSO (Sigma) at 1000 times the target concentration and 1 µl was added to the 1 ml CMOS MEA reservoir containing culture media and gently mixed with a p100 pipette. The cultures (y = a · xb), an exponential fit (y = a · ex−c *  + d) a truncated power law fit (y = a · xb · ex-c· ) and a Gamma distribution fit (y = (a/(Γ(b) · cb)) · xb-1 · ex/c− were performed on the binned edge strength (pairwise spike train correlations determined by the spike time tile coefficient) distribution (bin size = 0.05). The goodness of the fit was assessed using the Akaike information criterion (AIC)89 /(fd + fc), where fc and fd are the fraction of total edge strength distribution fits to the gamma distribution for control and diazepam (50 µM), respectively. Pairwise spike correlation distributions were split into three unique sets of edges. The first set contained edges that were present in both the control and the diazepam recording. The second set contained edges that were silenced by diazepam, meaning that they were only present under control conditions. The third set contained edges that were induced by diazepam, meaning that they were only present under the diazepam condition. Pairwise spike train correlation distributions and connectivity maps were generated for the three different sets of edges.A gamma distribution was found to be the optimal fit  was calculated for each burst period. The results were averaged over all bursts within a recording centered about the population burst peak. Secondly, the peak MUA population rate was averaged over all bursts within a recording and compared between organoids at different ages. The average inter-burst interval, defined as the average time between burst peaks was compared. Functional connectivity maps based on the STTC were generated for the organoids at the different time points.Electrical recordings were made from organoids spaced 4 h apart. The number of spike sorted singe-units detected in each recording were compared for two separate organoids  for the theta band time series using the findpeaks function with a minimum peak height of 10 µV . Theta phase angle spread was determined by calculating the circular standard deviation of the theta phase relative to the population burst peaksTo visualize spatiotemporal changes in the theta amplitude during the onset of a population burst, a spatial window of ≈140 μm × 140 μm (corresponding to 5 × 5 electrode grid routing with a 35 μm center-to-center spacing) was selected to calculate spatial averages centered around each sampled recording electrode. The theta signals were then averaged over all the electrodes within this spatial window sampled over the duration of the recording. If no electrodes were sampled within the spatial window, the centered electrode was discarded. Next, spatially averaged theta filtered signals were averaged over all burst events relative to the peak in population activity (as described above). Spatial plots were made of the averaged theta signal at different time points with respect to the burst peak to visualize the spatial evolution of theta amplitudes during population bursts.Cross-correlation analysis of the theta band time series, as described above, was performed on the whole time series and as cross correlations on theta time series occurring within population bursts (detailed in “Population-averaged spike rate and population bursts” section of “Methods”). The number of correlation scores of at least 0.2 was compared between whole theta time series data and the theta time series occurring during population bursts. Next, for each electrode that had a correlation score of at least 0.15 with the reference site for the population burst time series, the envelope of the theta filtered LFP was generated and the rms of the envelope was computed. The fraction of the recording for which the envelope exceeded 1.5 times the rms was computed for the recording frames within burst periods and the recording frames outside burst periods. The number of electrodes that showed at least a 10% increase in the fraction of time during which the envelope was above the rms threshold during bursts periods compared to non-burst periods was counted. Similarly, the number of electrodes with a 10% increase during non-burst periods compared to burst periods was counted.54 was computed using a window function with the length of 0.5 s and 25% overlap. The connectivity strength at each electrode, regional imaginary coherence, was estimated by averaging across all Fisher’s Z-transformed imaginary coherence values91. After generating the imaginary coherence matrix for the full 1020 × 1020 electrode pairs, we thresholded this data-matrix at the 90th percentile of the imaginary coherence values. We selected the nodes with >200 connections in this limited matrix, which resulted in seven highly coherent electrodes . The clustering algorithm with its highest stability, identified two clusters, and separated a small cluster with 54 electrodes, which was locally distributed on the array  was solubilized in DMSO, the NMDA receptor blocker R-CPP (Abcam) and the GABA receptor blocker Gabazine (Abcam) were prepared in ultrapure distilled water (Life Tech) at 1000x the desired working concentration. The working concentrations were 10, 20 and 10 μM for NBQX, R-CPP and Gabazine respectively. The sodium channel blocker tetrodotoxin (TTX) citrate (Abcam) was solubilized in ultrapure water at 1000x the working concentration (1 μM). Recordings were made from organoids silenced with synaptic blockers and TTX. Spatial maps of the median firing rate per electrode were generated from organoids treated with synaptic blockers, synaptic blockers in combination with TTX and for the control recordings of untreated organoids. For each treatment, the number of actively spiking electrodes was counted. This was defined as having at least 30 spikes over a 120 s window.92. The rms of the theta amplitude envelope was calculated for recordings of the organoids silenced with synaptic blockers and TTX. Subsequently, the fraction of the total recording time that the theta envelope exceeded a threshold of 1, 1.5 or 2 times the rms was computed for each electrode, as well as for the burst periods and the non-burst periods of the same electrode during the control recording. This fraction of time above the threshold during either the burst period or the non-burst period of the control recording was compared to the blockers and TTX recording. The percent change with respect to the blockers and TTX recording was computed and averaged over all electrodes that spiked at least 30 times during the control recording. Similarly, signal envelopes were obtained for the delta, theta, alpha, beta and gamma frequency bands. For each frequency band in each electrode, the rms was computed from the organoids silenced with synaptic blockers and TTX. The fraction of time that the envelope exceeded 2x the rms was computed for the whole time range of the recording treated with blockers and TTX, as well as for the burst periods of the control recording. The percent change of the fraction of time that the envelope exceeded the threshold was computed for each frequency band in each electrode. In addition, the constant decrease in the LFP amplitude over all frequencies after treatment with blockers and TTX was further demonstrated by comparing the power spectral density plots of the same electrode during control recordings and during recordings after treatment with blockers and TTX. The LFP power spectrum scaled inversely with frequency (~1/f)93 and was above the electronic noise floor in control conditions  for blocker + TTX experiments , was determined by taking the standard Hilbert transformation, H(t), of the theta band oscillation and calculating the angle between the real and imaginary components of H(t) in MATLAB92. The theta phase was then counted at each spike event time for all spike-sorted units. The Rayleigh criterion was used to test the non-uniformity of spikes distributed across the circular phase angles 90. Spikes were considered to be phase-locked to theta if they passed the Rayleigh criteria test for non-uniformity (p < 0.05).The time series theta phase, Further information on research design is available in the Supplementary InformationPeer Review FileDescription of Additional Supplementary FilesSupplementary Data 1Reporting Summary