Self-Organizing Circuit Assembly through Spatiotemporally Coordinated Neuronal Migration within Geometric Constraints

Background Neurons are dynamically coupled with each other through neurite-mediated adhesion during development. Understanding the collective behavior of neurons in circuits is important for understanding neural development. While a number of genetic and activity-dependent factors regulating neuronal migration have been discovered on single cell level, systematic study of collective neuronal migration has been lacking. Various biological systems are shown to be self-organized, and it is not known if neural circuit assembly is self-organized. Besides, many of the molecular factors take effect through spatial patterns, and coupled biological systems exhibit emergent property in response to geometric constraints. How geometric constraints of the patterns regulate neuronal migration and circuit assembly of neurons within the patterns remains unexplored. Methodology/Principal Findings We established a two-dimensional model for studying collective neuronal migration of a circuit, with hippocampal neurons from embryonic rats on Matrigel-coated self-assembled monolayers (SAMs). When the neural circuit is subject to geometric constraints of a critical scale, we found that the collective behavior of neuronal migration is spatiotemporally coordinated. Neuronal somata that are evenly distributed upon adhesion tend to aggregate at the geometric center of the circuit, forming mono-clusters. Clustering formation is geometry-dependent, within a critical scale from 200 µm to approximately 500 µm. Finally, somata clustering is neuron-type specific, and glutamatergic and GABAergic neurons tend to aggregate homo-philically. Conclusions/Significance We demonstrate self-organization of neural circuits in response to geometric constraints through spatiotemporally coordinated neuronal migration, possibly via mechanical coupling. We found that such collective neuronal migration leads to somata clustering, and mono-cluster appears when the geometric constraints fall within a critical scale. The discovery of geometry-dependent collective neuronal migration and the formation of somata clustering in vitro shed light on neural development in vivo.


Introduction
The brain is composed of hundreds of nuclei densely populated with neuronal somata, while the rest is packed with interconnecting neurites. The assembly of the neural circuits in vivo is achieved through a cascade of processes involving neuronal migration to define the somata locations of neurons [1,2,3,4].
A range of molecular [5] and activity-dependent [6] factors have been elucidated in regulating neuronal migration, mostly at the single cell level [7]. As neurons are dynamically connected with each other through neurite adhesion during development, the migratory behaviors of adjacent neurons within a circuit are coupled, making it a dynamic system. In this regard, systematic analysis of neuronal migration and circuit assembly has been lacking. As a variety of coupled biological systems give rise to emergent self-organization [8,9,10], we wondered if self-organization also exists in collective neuronal migration and circuit assembly.
Understanding the collective behavior of neuronal migration and its regulation through geometric constraints may be an important step in understanding circuit assembly in the complex settings of the brain. Geometric constraints can regulate collective behavior of some coupled biological systems [11], we also wondered if the same case takes place in neural circuits.
Neurons and networks in culture have been widely employed for studying formation and function of the nervous systems recently [12,13,14,15,16]. We set out to study collective neuronal migration and circuit assembly using an in vitro model system with dissociated neuronal culture.

Results
Neuronal migration at specific developmental stage on coated surfaces in vitro Primary hippocampal neurons from Sprague Dawely (SD) rats (See Materials and Methods for details on primary neuron culture) migrate actively on Matrigel (MG) coated gold substrates (gold substrate was used for assembling SAMs throughout the paper, see Materials and Methods for details on SAMs).
We found that the speed of migration depends on the surface treatment and donor animal age ( Figure 1, see Materials and Methods for details on measurement of migration). Neurons migrate very slowly on polymeric surfaces such as poly-D-lysine (PDL), poly-L-lysine (PLL), polyethyleneimine (PEI), as well as the extracellular matrix protein fibronectin (FN) ( Figure 1A, Table 1). While they do migrate on laminin (LN); the speed of migration is significantly lower than those on MG-treated surfaces ( Figure S1). For example, the speed of migration on MG-coated surfaces (1.0660.08 mm/min, n = 24) is significantly higher than that on PDL (0.0560.02 mm/min, n = 8, P,0.0001, One-way ANOVA followed by Tukey's post hoc test, Figure 1A). As the age of the donor animal or cell culture increases, the speed of migration decreases ( Figure 1B). The speed of migration of neurons from E16 rat pups on Matrigel is the highest and relatively stable (with the lowest coefficient of variation, Table 1), and was used throughout the paper.
We plated neurons onto patterned SAMs [17,18,19,20,21] throughout the paper (refer to the Materials and Methods for details) to apply geometric constraints on neuronal culture, in order to control the geometry of the neuronal circuit [22,23,24,25,26]. Briefly, patterns promoting neuron adhesion were formed through micro-contact printing (mCP) with alkanethiols terminated with methyl groups (-CH 3 ) on gold substrates evaporated on glass cover slips. The gold substrates were immersed in poly-ethylene glycol (-EG 6 ) terminated alkanethiols to make the rest of the surfaces anti-fouling. The constraint could assume arbitrary geometry by design to limit the number of neurons present on an island. We treated the surfaces with MG. Neuronal migration is spatially constrained within the restrictions. In combination with time-lapse imaging, this method facilitates spatiotemporal analysis of the collective migratory behavior of all neuronal cells that constitute an independent neuronal circuit.

Correlated neurite fasciculation and soma migration
Migration of individual neurons is gradually coupled into collective migration as neurons adhere to each other through neurite fasciculation. At first, soon after adhesion (0 DIV), neurons are unconnected and neuronal migration is disordered. Accompanying the development of the neurites, adjacent neurites come into contact with each other, through migration and neurite development, and depart again. After a period of dynamic interactions, some of the neurites adhere to each other through cell adhesion molecules [27], leading to neurite fasciculation.  Figure S1 for details). The speed of migration is determined by time-lapse imaging at a rate of 10 seconds per frame and calculating the speed by dividing the displacement by the time interval. Note that the long term displacement is zero for any surface coating other than MG and LN (refer to the Materials and Methods section for details). B, The speed of migration of neurons at different developmental stages. The speed of migration decreases against the age of the donor animals and the culture. Neurite fasciculation at this stage is mechanically vulnerable and dynamic, and its impact on neuronal migration is still not obvious. Fasciculated neurites gradually become strengthened and affect neuronal migration [28]. We used a cross-shaped geometric constraint ( Figure 2A) to illustrate the relationship between neurite fasciculation and somata migration at 3 DIV. The somata are confined to migrate within the two branches of the cross, while neurites become fasciculated and not restricted by the geometric constraint (note that neurites do not adhere on areas outside of the geometry). Gradually, the formation of fasciculated neurites ( Fig. 2A, red arrows) accompanies the formation of cell assemblies (yellow dots) on the two branches of the cross. This correlation is more clearly seen in the superimposed silhouettes in Figure 2B. We further quantified the number of neurons within each cluster (161 to 1162 during the 15 hrs time window, n = 5, 20 data points for each time point, Figure 2C) and the width of fasciculated neurites (0.9104760.1003 mm to 4.3017360.84857 mm, n = 5, 20 data points for each time point, Figure 2D), both showing a trend of increase with respect to time. Increasing number of neurons in each cluster corresponds to continuous clustering while increasing width of neurites indicates continuous fasciculation. We found that these two processes are temporally correlated ( Figure 2E). Collective migration on two-dimensional boundless surfaces is self-organized with somata clustering and neurite fasciculation. Similar results have been reported [29,30]. In this paper, we focus on how geometric boundary conditions affect the dynamics of collective neuronal migration, and therefore the self-organizing circuit assembly.

Spatiotemporally coordinated collective neuronal migration on neuronal circuits with defined geometric boundaries
We studied the effect of the boundary condition on the dynamics of collective neuronal migration through SAMs-based geometric constraints. Figure 3A shows a series of observations on a square-shaped geometric constraint over 12 days. We found that small clusters ( Figure 3A, red arrows) finally merge into each other, forming large clusters. We found that DIV 2-3 is a critical time window in neurite fasciculation and cluster formation. Before this time point, collective neuronal migration is largely uncorrelated ( Figure S2), although neurons show a trend of slowing down in migration. Since the critical time point, collective neuronal migration undergoes drastic change and become correlated. We studied neuronal migration at this critical stage within a circle ( Figure 3B and 3C). Figure 3B shows the migration of clusters over 24 hrs at 2 DIV. Each blue shaded area covers a cluster while red arrowheads show the migration direction of the clusters. Several small clusters gradually migrate into each other and finally merge into fewer numbers of clusters. Under a specific geometric scale, the clusters merge into a single one, as will be fully discussed in the next section. We analyzed a network on which the cells were plated at a very low density so that each cell was clearly discernable ( Figure 3C). We traced the migration of all neurons in the circuit. We use an arrow to indicate the general direction and distance of the migratory pathway of each neuron, where each arrow begins from the original location of a neuron and pointing to the final location. While the cells were evenly distributed throughout the geometric constraints at the beginning, they finally become clustered into a few spots highlighted by pink shades. Clustering takes place consistently, regardless of their initial density. To quantify the trend of migration of all neurons in a network, we calculated the cumulative distance of all neurons of the network. The cumulative distance is the summary of all distances between each pair of neurons (refer to Materials and Methods for description). We calculated the cumulative distance on geometries of various shapes. As the normalized cumulative distance curves show in Figure 3D, the cumulative distance generally decreases over 24 hrs regardless of the shapes of the geometric constraints.
We found that the migration of a specific neuron depends on its spatial location within the geometric constraint. Figure 3E shows the three-dimensional (3D) plot of collective migratory trajectory. This image is generated by serially stacking the position of all neurons in a network at different time points into a 3D plot. The horizontal axis corresponds to the position of a specific neuron, the vertical axis indicates specific time points. The starting time of image capture is set to 0 and the value on the vertical axis increases over time. The migration of all neurons tends to be oriented towards the center (Fig. 3E), indicating spatial coordination. Temporally, the speed of neuronal migration is synchronized in that neurons tend to speed up and slow down during migration at the same pace, indicating that the temporal dynamics of collective neuronal migration is also coordinated ( Figure 3F, P,0.001, Pearson's x 2 test, compared with the temporally shuffled random data, see Materials and Methods for description of the statistical analysis). Taken together, we demonstrate that collective neuronal migration within geometric constraint is spatiotemporally coordinated.
We also studied the heterogeneity of collective migration, by performing covariance analysis of collective neuronal migration ( Figure S3).We classified the neurons into two populations according to their covariance coefficients (k-means clustering, Figure S3B), and found two types of distinct dynamics of neuronal migration. Figure S3B is the planar projection of Figure 3E. One population of neurons (blue) generally locate in the peripheral zone, spanning across significantly longer ranges ( Figure S3C, P,0.001, Kolmogorov-Smirnov test) and assuming higher migratory speed; while the other population of the neurons (red) tend to be located in the center, showing a centripetal pattern of migration.
Emergence and centrality of mono-cluster is selforganized and sensitive to geometry When this clustering process takes place on geometric constraints, the final location of the cluster and the trajectory of migration depend on the shape of the geometric constraints. The clusters generally locate in the geometric center on isotropic restrictions. On anisotropic restrictions, the locations of the clusters tend to be in the central region of the geometric constraint, albeit with a relatively large range of distribution ( Figure 4D). This is entirely different from the situation where neurons grow in unrestrained culture. On un-constrained gold substrates (25625 mm 2 ) that could be considered an infinitely large two-dimensional plane with respect to the scale of a single neuron (approximately 6 orders of magnitude larger), the networks are dominated by randomly located clusters ( Figure 4A -C). We plotted the locations of the mono-clusters on circular, square and rectangular restrictions ( Figure 4E) and analyzed their mean displacements with respect to their geometric centers respectively (864 mm (circle, n = 5, 60 data points); 966 mm (square, n = 5, 60 data points) and 3269 mm (rectangle, n = 5, 80 data points). The locations of the clusters fall around the geometric centers of the circuit. They form a band along the longer axis of the rectangle.
In order to verify the robustness of the mono-cluster and centrality property, we extracted the location of the clusters on an array of circular and rectangular restrictions of the same area (see Figure S4 for details on the experimental setup). In all geometries we analyzed, centralized mono-clusters were consistently observed. Figure 4F and 4G are box plots of standard deviation of X and Y values respectively (The mean displacements for different shapes are: large rectangles, X = 53612 mm, Y = 80634 mm; small rectangle, X = 3168 mm, Y = 58629 mm; large circle, X = 65623 mm, Y = 69625 mm; small circle, X = 41611 mm, Y = 3869 mm). As shown here, the fluctuations along Y on rectangles are always larger than that on circles (1, n = 8, 0.01,P,0.1. 11,n = 8, P,0.001. One-way ANOVA followed by Tukey-Kramer's post-hoc multiple comparisons test). Larger geometric constraints also accommodate wider range of fluctuations (SD X : *, n=8, P,0.001. **, n = 8, P,0.001. SD Y : *, n = 8, 0.01,P,0.1. **, n = 8, P,0.001. One-way ANOVA followed by Tukey-Kramer's post-hoc multiple comparisons test). By contrast, there is much less fluctuation along the X axis than along the Y axis on rectangular shape (large rectangles: P,0.001, small rectangles: P,0.001, Kolmogorov-Smirnov test).
The exact location of the cluster depends on the geometry of the restriction through mechanical interaction between constituent subunits including cells and sub-clusters. By assuming the pair-wise mechanical interactions between sub-clusters to be independent of their distances, and the distribution of the sub-clusters within the geometry to be statistically homogeneous, we theoretically estimated the direction of the resultant force of all elements in the circuit acting on arbitrary sub-cluster to be pointing towards the centroid of the geometry of the circuit (refer to Materials and Methods for details). Therefore, the coordinated migration is centripetal. Consequently, the mono-cluster, where it emerges, is located on the geometric center of the constraint. This is in agreement with our experimental observations ( Figure 4E, Figure 5A and 5B). When the geometric centroid lies outside of the constraint ( Figure 5C), the cluster tend to locate as close to the centroid as possible within the constraint.
In addition to shape, the emergence of mono-cluster its location depends on the scale of the network. Using circularshaped constraints with gradually increasing diameters, we found that mono-cluster has the largest probability to form under 600 mm ( Figure 6A, Figure S6A). We performed a linear regression to the cluster data and found a good fit between the diameters of the geometric constraint and the number of clusters (R = 0.9992). In addition, the occurrence of mono-cluster also depends on the diameters of the geometric constraints. On networks smaller than 200 mm, clusters do not always appear, while there might be more than one cluster on larger networks ( Figure 6B). Finally, we performed relative neighborhood density [31] analysis on networks of different size ( Figure S6B). Quantitative analysis indicates that cells are highly clustered on small networks where neurons are clustered at the center. On larger networks, the curve has a heavier tail, indicating a higher proportion of neurons that are not fully clustered. On free cultures such as those in Figure 4A (25000 mm), the curve has a very mild slope.
In an effort to test the generality of the critical scale, we designed a series of complex networks of identical overall area but composed of decreasing number of units of increasing areas (Figure S5A -C). As shown in the immunofluorescence photomicrographs ( Figure S5D and S5G), decreasing the size of the units of the geometric constraints increases the orderliness of the cluster number and location. The number of clusters after long term culture remains a single one on smaller constraints but begins to increase above a threshold range. The threshold appears somewhere between 400 mm and 600 mm, similar to the simple scenario as discussed above ( Figure 6). On networks with subunits above the critical scale, the basic network structure is similar to network motifs on unconfined surfaces ( Figure 4A -C).

Cell-type specific pattern of neuronal distribution within the network
We found that the cluster is composed of a high density of neuronal somata ( Figure 7A, 7B). As functional neuronal networks are composed of a balanced combination of excitatory and inhibitory neurons, we set out to analyze the cellular types on networks. Using immunocytochemistry against excitatory and inhibitory neuron markers, a-Ca 2+ /Calmodulin dependent protein kinase II (CaMK II), and c-aminobutyric acid (GABA), respectively ( Figure 7C), we found that the ratio of excitatory versus inhibitory neurons is maintained ( Figure 7G) regardless of the number of neurons of the circuit as the area of the constraint grows, with 70% of excitatory neurons and 30% of inhibitory neurons, in accordance with previous reports [32,33]. There are some fluctuations of the exact value, but as the restriction area increases the value converges towards the value on unrestricted culture ( Figure 7H). The clusters (in the center of the geometric constraint) are mostly composed of excitatory neurons ( Figure 7D) with inhibitory neurons in the periphery. We frequently found inhibitory neurons far away from the cluster, while excitatory neurons are seldom seen outside the cluster. On a fully mono-clustered network ( Figure 7E), the cluster is composed of mostly excitatory neurons while a few inhibitory neurons could be observed in the periphery of the geometric constraint far away from the cluster ( Figure 7F). We analyzed the distance between arbitrary pair of cells, and sorted the distances into two groups, one group is the distance between a pair of excitatory neurons and the other the distance between an inhibitory neuron and an arbitrary neuron. As the histogram shows ( Figure 7I), the excitatory neurons tend to cluster together while inhibitory neurons stay far away from each other. This result is also evident in Figure 7C. Interestingly, the peak of the histogram overlaps with the range of conventional soma diameters (data acquired from neurons in the same culture experiment), indicating that most of the excitatory neurons are clustered.  Figure 2C) and the width of neurites ( Figure 2D). The horizontal axis represents the temporal shift between the two variables, and the highest value is achieved with no time shift, meaning these two events are temporally correlated. Scale bar: 200 mm. doi:10.1371/journal.pone.0028156.g002

Discussion
We demonstrate that neural circuits within critically-scaled geometric constraints exhibit spatiotemporally coordinated pattern of migration ( Figure 3) and mono-clustered network structure ( Figure 4 and Figure 5) on MG-coated SAM surfaces, in contrast to disordered neuronal migration and circuit structure ( Figure 5A -5C) in the absence of geometric constraints. Besides, neural circuits adapt to various geometric inputs in a circuit-autonomous manner. (Figures 4, 5, 6, Figure S5). Taken together, these results indicate that developing neural circuits are self-organized within geometric constraints. The assembly of neural circuits is regulated by both genetics [34,35] and neural activity [6,36]. However, molecular cues and neural activity take effects in the form of spatiotemporal patterns onto neural circuits. Geometric constraints have been found to regulate non-neuronal cell migration in vitro [18,37]. How geometric distribution of molecular cues and neural activity regulates circuit assembly remains an open question [38]. Our in vitro study shows geometric control of circuit assembly, which could be a developmental strategy readily implementable in vivo.
We found that neuronal migration and neurite fasciculation are correlated ( Figure 2E). Fasciculated neurites linking clustered somata tend to be straight (Figure 2, Figure 4A and 4B), indicating that the neurites are in tension. The tension could be caused by traction force reciprocally exerted by two clusters onto each other, and the traction force mediated through these fasciculated neurites would in turn affect neuronal migration. When most of the neurons are connected through these fasciculated neurites in tension, the circuit would be a mechanically coupled entity ( Figure 2, Figure 4A, 4B, 4C), and neuronal migration within the circuit become globally coordinated.
We built a working model based on the experimental observations ( Figure S7). On an infinitely large network populated with neurons connected through neurites under tension, the network would be stable if the neurons are stationary and neurites under balanced tension force. However, the cells are migrating and undergoing development, leading to transient fluctuations in tension force, thereby breaking the symmetry of mechanical equilibrium in the neurites, and the cell exerting stronger force would drag adjacent neurons closer, facilitating adhesion and clustering. This pair-wise process scales up on a network with a large amount of neurons to recruit and multiple fasciculated neurites for each neuron, causing the formation of hierarchically larger clusters ( Figure S7B). Accompanying the formation of hierarchically larger clusters, the migration of neurons in a network might be coordinated into the migration of hierarchically larger clusters, via mechanical interactions between the clusters linked by fasciculated neurites. Based on this working model, we built a simple theoretical model (see Materials and Methods) and predict that mono-clusters tend to locate on the centroid of the geometric constraint ( Figure  S7D), which is in agreement with experimental results (Figure 4, 5). However, our theoretical model assumes that the force between sub-clusters is independent of the distance, which would be violated over longer distances ( Figure S7C), and above a critical scale, where reliable propagation and coordination fails, leaving local clusters un-centralized ( Figure 4A and 4B). The application of geometric constraint defines the range of mechanical coupling, and therefore different patterns of collective migration and circuit assembly (Figures 4, 5, 6, Figure  S5). Similar results of the dependence of emergent property on a critical scale has been demonstrated in other coupled systems such as the microbial oscillators [11] in which microfluidic channel played a similar role as SAMs-based geometric constraints here. While mechanical coupling has long been hypothesized to influence circuit assembly [39], and several groups have addressed this issue through experimental and theoretical approaches [40,41,42,43,44,45,46,47], our work proposes a potential role played by mechanical tension in coordinating collective neuronal migration within geometric constraint, thereby regulating circuit assembly.
Clustering permeates neural networks in vivo and in vitro at synapses [48] and neurites [49]. Soma clustering exists in a variety of central nervous systems, in which somata are grouped together into grey matter while neurites white matter. This segregation may be evolutionarily advantageous due to economy [50]. Our results indicate that circuit-autonomous self-organization in response to geometric control is sufficient for the realization of similar structures through neuronal migration in vitro ( Figure 7A and 7B). Our discovery of self-organized formation of clusters under specific geometric constraints possibly indicates that clustered somata distribution is the most economical pattern of wiring [51]. Self-organization has been found in various physical [52] and biological subjects [10]. It has been implicated in the morphogenesis of tissues ranging from hair [9], pancreatic islets [53], optic cup [8] to the brain [49,54,55]. As geometric constraints have been shown to affect circuit function [56], the interaction between geometry and wiring, as a case study of formation and function, could be a general strategy for wiring optimization during evolution [57]. Besides, neuronal morphology with elongated neurites might be a structural basis for wiring optimization [39]. . Collective dynamics of neuronal migration within geometric constraints is spatiotemporally coordinated. A, Time-lapse imaging of a circuit maintained over 12 DIV. The red arrows indicate clusters. As shown here, the clusters gradually come together and finally merge into a mono-cluster. B, Collective neuronal migration over 24 hrs on a circular SAMs geometric constraint (depicted by green dashes) at 2 DIV. Transparent blue shades highlight the clustered somata that are coordinated in migration, red arrows designate the orientation of the migrating clusters. As time elapses, smaller clusters merge into larger ones and gradually approach the geometric center of the geometric constraint. C, The behavior of migration of all 36 neurons (at a lower density) on a circle (black dashes) over 24 hrs at 2 DIV at an interval of 5 min. The blue arrow begins from the original position of the cells at 0 hr, and ends at the final location at 24 hr. Neurons that were evenly distributed throughout the geometric constraint at the beginning finally migrate into a few clusters (transparent pink shades) in the end. D, Cumulative distance curves of all neurons on geometric constraints of different shapes (circular, triangular, square and hexagonal) over 24 hrs at 2 DIV at an interval of 5 min (see Materials and Methods for details on the definition of the curve). E, Collective pattern of neuronal migration on a square-shaped constraint over 12 hrs at 4 DIV at an interval of 5 min. The X-Y axes indicate the spatial locations of all neurons at a specific time, which is represented by the vertical axis. A trace of a particular color corresponds to one neuron. F, The migratory speed of neurons on a square-shaped constraint over 12 hrs at 4 DIV at an interval of 5 min as in panel E is temporally coordinated (P,0.001, Pearson's x 2 test, compared with the temporally shuffled random data, see Materials and Methods for details on data analysis). Each black block represents the speed of the specific neuron (indexed by the vertical axis of the block) at a specific time when the speed of migration (indexed by the horizontal axis of the block) is high (above 30%). The vertical axis is the serial number of neurons. The yellow shades highlight time points when most of neurons are migrating at a higher speed. The red curve above is the   Figure S4 for the original image and data extraction procedure). The mean displacements for different We found cell-type specific clustering of excitatory and inhibitory neurons ( Figure 7C -7I). This is possibly correlated with different patterns of neuronal migration ( Figure S3). In fact, excitatory neurons migrate along a trajectory different from inhibitory neurons in the brain. As excitatory-inhibitory interaction and balance is critical in neural physiology and pathophysiology [58,59], cell-type specific clustering might be the structural basis of neural computation within local microcircuits [60,61].
Precise engineering of the circuit map is a prerequisite for constructing functional neural circuits for neural prosthetics, regenerative medicine and fundamental research [62,63]. Controlling the relative position of the neuronal somata within the networks is a key step in neural circuit engineering. The elucidation of the presence of critical scale is important for engineering two-dimensional neural circuits, and shed light on the engineering of three-dimensional neuronal networks through scaffold with critically-scaled geometric features [64]. The model system established here could be used for establishing disease models of neurological disorders [65] and for quantitative analysis of complex systems [66,67].

Microfabrication and surface chemistry
We employed standard protocols of soft lithography for fabricating the stamps for micro-contact printing. Briefly, masks were designed with AutoCAD (Autodesk) and generated with high-resolution printing, SU-8 (SU-8 2025 and 2100, MicroChem) mold were made through standard photolithography with a mask aligner (MJB4, Suss MicroTec), soft lithography [20,68] of polydimethylsiloxane (PDMS, Dow Corning) was performed to obtain the elastic stamps for mCP.

Primary culture of rat hippocampal neurons
Methods used for culturing hippocampal neurons follows traditional approaches with adaptions [72,73]. Neonatal (P0, P1) and embryonic (E16, E18, E19) Sprague-Dawley rats are anesthetized, and hippocampi were isolated and chopped into pieces, treated in 0.25% Trypsin (GIBCO) supplemented with DNase I (Sigma) for 30 min at 37uC, and triturated. Dissociated neurons were seeded onto the surfaces in DEME/F12 medium supplemented by 10% horse serum. After cell adhesion, the surfaces were rinsed gently with DEME/F12 medium twice, and replaced with Neurobasal medium supplemented with 2% B27 and 1% GlutaMax-1 (all from GIBCO) without antibiotics. The medium was replaced by half every four days. All animal experiments were approved by Institutional Animal Care and Use Committee of National Center for Nanoscience and Technology (2009-0214-01).
Each parallel experiment was carried out with new microfabrication and surface chemistry as well as new primary culture. This corresponds to a single count in terms of the number of experiments (designated by n in statistics).

Imaging
Time-lapse imaging was performed on an AF6000 live cell imaging workstation (Leica Microsystems) based on the inverted fluorescence microscope DMI 6000B (Leica Microsystems). For migratory speed analysis as in Figure 1, we used an acquisition interval of 10 seconds between adjacent frames. This particular time interval was determined by analyzing the smallest discernable displacement through the wide-field optical microscope. We found 20 seconds to be the lower bound for migratory speed variation, and 10 seconds was used according to Shannon's sampling theorem.
Fluorescence photomicrographs were taken on a LSM 710 laser scanning confocal microscope (Carl Zeiss).
Enhancement and processing of photomicrographs was done with Photoshop (Adobe).

Data analysis and statistics
ImageJ (NIH) and Image Pro Plus (Media Cybernetics) were used for semi-automatic quantitative measurements on image data.
Data analysis was performed in Matlab (The MathWorks). Statistical analysis was performed by R (The R Foundation for Statistical Computing). For groups of two conditions, statistical tests were performed using two-tailed Student's t test, Mann-Whitney U test or Kolmogorov-Smirnov test. For groups of three or more conditions, one-way ANOVA or Kruskal-Wallis' ANOVA were used, followed by Tukey-Kramer's or Bonferroni's post hoc multiple comparisons tests. Polynomial regression was used for curve fitting. Descriptive statistics were presented as mean6S.E.M. unless otherwise noted. All error bars designate S.E.M. Monotonicity was tested via Spearman's rho test or Mann-Kendall's tau test. Kmeans clustering was employed for clustering analysis.
Enhancement and processing of vector graphs was done with Illustrator (Adobe).  Figure S6 for the original data points. B, The percentage (probability) of the appearance of mono-clusters (red) varies with the diameter of the geometric constraint. The highest probability occurs when the diameter of the circular geometric constraint is either 250 or 300 mm. Note that for small geometric constraints there is sometimes no clusters (green) at all, while for large geometric constraints there are multiple clusters (blue). The same set of data as Figure 6A

Measurement of migration
The measurement of migration is based on position analysis on image series acquired during time-lapse imaging. Specifically, the geometric center of the cell is manually determined (the size of the neuronal soma is relatively small and therefore manual analysis will not introduce much error), and the displacement between adjacent frames was calculated. The speed of migration is defined by the displacement divided by the time interval for image acquisition. The exact time interval depends on the analysis. For the comparison analysis on the speed of migration as in Figure 1, high sampling rate of 10 seconds per frame was employed. For long term monitoring in the rest of the paper, a sampling rate of 5 min per frame was used to reduce the size of the data. Similar methods were used for analyzing acceleration and angle in Figure  S2.

Calculation of normalized cumulative distance
Suppose a network composed of N neurons is studied within time duration [0, T]. The time variant cumulative distance D c (t) is calculated as the sum of all distances between each pair of neurons: Theoretical prediction of cluster location In a circuit with geometric shape R populated with N subclusters, for arbitrary sub-cluster i within R, if we designate the tension force mediated through neurite fasciculation between subcluster i and j as f I ij , where we assume f I ij to be independent of the distance between i and j, such that: in which, f is the magnitude of f I ij and r I ij is the unitary vector with the direction from i pointing to j.
Then for the resultant force of all sub-clusters within R acting on sub-cluster i: We assume that the N sub-clusters are evenly distributed throughout R, which is likely to be true statistically. Then the term P N j~1 r I ij is a vector pointing from i to the centroid of the geometric constraint.
Therefore, the direction of the resultant force of all elements in the circuit acting on sub-cluster i points toward the centroid of the geometry of the circuit.

Relative neighborhood density
Relative neighborhood density quantifies the cluster distribution.
For a surface with N clusters, the distribution function of cluster n(r) is a variable that depends on the location of the plane, for arbitrary cluster i, we define its specific neighborhood density as: in which A(r) is the area of an annuli with a distance of r from cluster i, with width d, we then have the relative neighborhood density by averaging over all N clusters on the plane: In this paper, the width of the annuli d, which defines the resolution of the analysis, is set to 1.
For an infinitely large two-dimensional plane, suppose the distribution of the neurons is homogenous. Consequently, the distribution function of the neurons n(r) is a constant independent of r. Suppose the density of clusters is d, then we have: The specific neighbor density is: the relative neighborhood density is: which is a constant independent of r. Temporal synchronization analysis For Figure 3F, the speed of each neuron is normalized across time. A binarization algorithm is then applied with a threshold of 30% of the highest speed of the neuron. The peak of the temporal bands above threshold is extracted as the time point for synchronization analysis (black block). The time intervals between the time points are then shuffled across cells as the random level. The P values of synchrony level (defined as a probability that the same or higher level of coactivation could occur by chance in 1000 interval-shuffled surrogates [74]) are then compared between the original data and the shuffled random series, and subject to statistical test (Pearson's x 2 test). The raster plot is collapsed into the yellow dots, which is then smoothened with a Gaussian kernel to generate the histogram (red curve). Figure S1 Multiple Comparisons test on the speed of neuronal migration on gold substrates with various surface coatings. Red error bars denote significant difference (P,0.05), while blue error bars denote non-significant difference. The speed of neuronal migration on MG coating is significantly higher than any other coatings (One-way ANOVA followed by Tukey's post hoc test).  Figure 3E. The numbers are the serial numbers of the cells. The color of each dot in the matrix represents the covariance coefficient between the two cells designated by the horizontal and vertical index of the matrix respectively. The color bar relates different colors to the covariance coefficients (between 0 and 1). B, The projection of the three-dimensional figure in Figure 3E onto the horizontal plane. We categorized the cells according to their covariance coefficients (panel A) through k-means clustering, cells with larger covariance coefficients are labeled blue while the rest are labeled red. The two populations of neurons exhibit two modes of migration. Neurons with red-colored trajectories are coordinated by the centripetal movement, while neurons with blue-colored trajectories show edge-philic movement. C, Box chart showing the cumulative distance that the red and blue group of neurons travel (* P,0.001, Kolmogorov-Smirnov test, Central: n = 56, Peripheral: n = 13). Scale bar: 200 mm.  Figure S7 Working model for geometric control over coordinated collective migration via mechanical coupling. A, Fasciculation and tension generation between two adjacent neurons. Contact of adjacent neurites (light blue) facilitates adhesion (deep blue velcro) mediated neurite fasciculation, while the somata (yellow balls) are still subject to active migration. The neurites are then subject to tension force (red arrow), possibly caused by the development of neurite fasciculation. Finally, the fasciculated neurites under tension drag the somata closer. B, Formation of larger clusters on a random network composed of small clusters (dark blue circles) fully connected through fasciculated neurites (light blue lines). The break of symmetry on a neurite will result in directed forces along a neurite, leading to somata migration. C, Size dependent difference in network structure. On very small circuits (dashed circle with smallest diameter), there is no cluster. On circuit above critical scale (dashed circle with largest diameter) there would be multi-clusters. Spatiotemporally coordinated migration and monocluster appears when the geometry of the circuit falls in between (dashed circle with medium-sized diameter). The blue balls are stabilized clusters under equilibrium in the absence of geometric constraints. The size of the geometric constraint (green dashes) with respect to the critical scale (red circle) determines the potential number of clusters the circuit would host. D, Geometry regulates cluster location. Filled blue circles represent sub-clusters. Blue lines represent fasciculated neurites between the clusters. Red arrows represent traction forces exerted onto the clusters. Filled yellow circles (and yellow ring in the case of concentric ring) represent final locations of clusters. Green dashed lines are the boundaries of the geometric constraints. The centers of the clusters are located on the geometric centers. When the centripetal movement is blocked by a concentric ring, the cluster forms around the ring. (TIF)