Tissue source and electron microscopy conditions

The EM image stack used here is centered on a large apical dendrite of a CA1 neuron (Fig. 2A) and has been previously described (V3) by Mishchenko et al. (2010). All procedures were approved by the University of Texas at Austin Institutional Animal Care and Use Committee and the Institutional Biosafety Committee and followed the National Institutes of Health guidelines for the humane care and use of laboratory animals. No new animals were prepared for this study. The image stack came from hippocampal area CA1 of a perfusion-fixed brain that had previously been obtained from a male rat of the Long–Evans strain weighing 310 g (postnatal day 77; Harris and Stevens, 1989). It was located in the middle of s. radiatum about 150–200 μm from the hippocampal CA1 pyramidal cell bodies. A series spanning 160 sections was cut according to our published methods (Harris et al., 2006), 100 of which were used in this study. Briefly, a diamond-trimming tool (EMS, Electron Microscopy Sciences, Fort Washington, PA) was used to make small trapezoidal areas that were about 200 μm wide by 30–50 μm high. Serial thin sections were cut at ~45–50 nm on an ultramicrotome and then mounted on Pioloform on a Synaptek slot grid. The sections were counterstained with saturated ethanolic uranyl acetate, followed by Reynolds lead citrate, each for 5 minutes. Individual grids were placed in grid cassettes and stored in numbered gelatin capsules. The cassettes were mounted in a rotating stage to obtain uniform orientation of the sections on adjacent grids, and the series were photographed at 5,000× on a JEOL 1200EX electron microscope (JEOL, Peabody, MA).

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Figure 2

Sheet-and-tunnel model of extracellular space allows exploration of ECS lacunarity. A: Normalized histogram of extracellular width for sheets and tunnels, and arbitrary EM image from stack partially segmented with contours. Inset numbers represent median values of the distributions. The median extracellular width for tunnels was 17 nm (4,965 measurements) for sheets, and 18 nm (151 measurements) for tunnels, somewhat wider than the sheets, as observed by Van Harreveld et al. (1965). B: Virtual slice through the raw reconstruction with ICS rendered black and ECS as white. The median sheet extracellular width distribution was 8 nm (N = 1,756,847), smaller than observed in EM images, suggesting a bias in contour placement toward the extracellular side. All virtual slices in B–H are at the same location in the stack as A. C–H: Virtual slices through six reconstructions, built from the raw reconstruction, covering a range of ECS lacunarity values, but with nearly identical extracellular volume fraction of 0.2 (0.202 ± 0.003). C,E,G: Half of the reconstructions were scaled by 15% along three orthogonal axes to account for shrinkage associated with the tissue preparation protocol used here. The reconstructions in B–D had similar median values for sheet and tunnel extracellular width (ECW) distributions. These provide a lower bound on ECS lacunarity. Expanding the tunnel ECS volume (E–H) at the expense of the sheet ECS volume, to maintain constant extracellular volume fraction, increased the lacunarity of the ECS and broadened the tunnel extracellular width distributions. The integral of each sheet (red) and each tunnel (black) relative frequency histogram was normalized to one. Scale bar = 1 μm in A–H.

EM image stack registration

The 100 serial-section EM images of adult rat hippocampus (4,096 × 4,096 pixels, 2.3-nm resolution, gray-scale) were co-registered by using the software program RECONSTRUCT (Fiala, 2005), which is freely available from http://synapses.clm.utexas.edu. Images were optimized for brightness and contrast. Manual alignment was obtained by placing five or more fiducial points (e.g., cross-sectioned mitochondria or microtubules) that were in the same location on adjacent pairs of serial sections. The minimal algorithm (usually linear) was applied in RECONSTRUCT to perform section-to-section alignment while blending the adjacent images. Pixel size was calibrated relative to a diffraction grating replica (Ernest F. Fullam, Latham, NY) photographed with each series, and section thickness was computed by dividing the diameters of longitudinally sectioned mitochondria by the number of sections they spanned (Fiala and Harris, 2001; Harris et al., 2006). Every structure was then traced through serial sections and identified as a dendrite, spine, axon, glial process, synapse, etc. until the entire volume was complete (total = 180 μm³).

3D reconstruction of an EM image stack

The process for building 3D reconstructions of neuropil from electron microscopy images is described in detail by Kinney (2009). Briefly, the outer surface of the plasma membrane of every dendritic, axonal, and glial process in each section was manually traced from the micrographs to create contours at a resolution of 2.3 nm per pixel by using RECONSTRUCT software. The sets of contours for each process were stitched together by using the ContourTiler software, which generates 3D surfaces by constructing triangles between contours on pairs of adjacent sections (Bajaj et al., 1996, 1999). Because the tiling algorithm only operates on pairs of sections at a time, the program need only load a small amount of data into memory at any one moment, making the program easily extendable to the reconstruction of forests of neurons in the software program VolumeRover, a suite of tools for reconstruction, visualization, and analysis of high-quality meshes, (Edwards and Bajaj, 2011; http://cvcweb.ices.utexas.edu/cvcwp/?page_id=100).

The 3D computer model of neural tissue consists of water-tight, triangulated surfaces representing the outer surface of the lipid bilayer of each process in the volume. In Mishchenko et al. (2010) this EM image stack was studied with a focus on the location of synapses, and the ECS was 1 pixel wide by default following semiautomated reconstruction. Here we use the same EM image stack to investigate the morphology of the ECS.

Measurements of extracellular width in EM images

We chose to measure the extracellular width in EM images at places where plasma membranes lie orthogonal to the plane of section (Fig. 3). Such places were identified by using the aforementioned 3D computer model of the neural tissue. Specifically, regions of the EM images were chosen as candidates for extracellular width measurement if both opposing membranes of the ECS had normal vectors lying within 4.5 degrees of the plane of section. Of the more than 2.3 million vertices in the reconstruction, 10% were tagged as orthogonal. Furthermore, the candidate regions must have (upon inspection of the image) clearly delineated dark staining of the membranes with abrupt transitions to minimal staining of the cytoplasm.

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Figure 3

Contour location error accounts for one-quarter of ECS loss observed in raw reconstruction. In the raw reconstruction 8% of the total volume is ECS, compared with 20% in vivo. The discrepancy is partly explained by contour placement error during EM image tracing. Contour location error (yellow) was estimated by measuring the extracellular width (red) in regions where the cell membranes lies orthogonal to the plane of section. The raw reconstruction has a median sheet extracellular width of 7.92 nm, well below the EM image stack sheet median extracellular width of 17.32 nm (Table 1).

In the sheet regions of the ECS, the width was measured as the shortest perpendicular distance between the opposing membranes. Straight lines (red) were drawn from the intracellular edge of one membrane, across the membrane, across the ECS, to the intracellular edge of the opposing membrane. This approach was adopted to handle accurately cases in which the extracellular width was near zero. The extracellular width was calculated as the length of the measurement line minus twice the membrane width. The membrane width was measured to be 7.5 nm, consistent with the membrane staining width in EM images (Rosenbluth, 1962; Lehninger, 1968). In the tunnel regions of the ECS, the extracellular width was measured directly, because the tunnel width was always greater than zero. Extracellular width measurements were made on odd numbered EM images (50 in total) to distribute the measurements across the stack of 100 images.

Calculating the volume of ECS on the nanometer scale

Here we describe a method to parcel the ECS into small, discrete subregions, associated with surface patches on neighboring cells. In this way, the heterogeneity of the ECS on the nanometer scale is measured. To begin, the ECS around each reconstructed object is imagined as a shell encapsulating the object (Fig. 4A). The total volume V_ECS of the ECS is the sum of all ECS shells (one shell per object for N objects): V_ECS = Σ_NV_shell,i. The volume V_shell,i of the shell of object i is calculated by summing the volume ΔV_j of small shell elements over the entire surface S of object i: V_shell,i = Σ_SΔV_j. Beginning from a small area A_j on the surface of the object, a shell element extends out radially a distance equal to one-half of the extracellular width ECW_j. The surface mesh of each reconstructed object is not entirely flat like a plane, but also includes some concave regions and other convex regions (Fig. 4B). On a small scale the curvature of the surface can be considered constant and modeled as the surface of a sphere with a radius R_j fit to the small patch of surface under the shell element. The volume ΔV_j of a shell element is calculated as an integral of differential elements shaped like spherical shells: ΔV_j = ∫dA_jdr. The area of a sphere of radius r is equal to 4πr². The area dA_j of the differential element is a fraction f_j of the total area of the sphere where 0 < f_j < 1: dA_j = f_j4πr². Because the shell element extends radially from the surface, f_j is constant for all differential elements in ΔV_j and is calculated from the surface area A_j of the shell element and the radius of curvature $R_j:f_j=A_j/4\pi R_j^2$.

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Figure 4

Accurate calculations of extracellular space volume using radial shell elements. A: The ECS volume is approximated as the sum of many small shell elements (one per vertex). The curvature of the element is estimated by fitting a sphere of radius R to the constellation of local vertices that describe the surface patch of area A under the element with volume ΔV. B: The extracellular width (ECW) is measured as the distance between the vertex V and closest point P. The shell element extends halfway across the extracellular width. C: A sample patch of a surface mesh illustrates how the surface was partitioned by using a median mesh into distinct domains around each vertex. Edges (solid black) connect vertices (white circles). Median lines (dashed) were drawn between vertices and the midpoint of the opposite edge. This particular construction divides the triangles into six smaller triangles with equal area. One-third of the area of each large triangle is assigned to its vertices. This method has three advantages. First, the surface domain (red, gray, blue, yellow) of each vertex surrounds the vertex. Second, the surface area of the domain is simply one-third the cumulative area of all neighbor triangles of the vertex. Third, the domains perfectly tile the surface. The shape of the shell element surface domain does not influence the calculation; only the scalar area is measured and used.

Finally, the volume ΔV_j of a shell element is calculated as follows:

This expression for the volume ΔV_j of a shell element behaves well for all values of radius of curvature R_j. For shell elements over flat regions of the object surface (R_j → ±∞), the element volume reduces to the expression for a simple prism(A_jECW_j/2). It can also be shown that for convex surface regions where 0 < R_j < ∞ (e.g. R_j = ECW_j) the shell element volume is greater than the volume for a flat region as expected. Likewise, for concave surface regions where −∞ < R_j < 0, e.g. R_j = −ECW_j, the shell element volume is less than the volume for a flat region as expected. The local radius of curvature at each vertex was measured by generating the least-squares fitted sphere to the patch of surface around each vertex (Coleman et al., 2005). Specifically, each sphere was fit to the collection of the primary vertex and immediately adjacent vertices. The radius of the sphere was taken as the curvature measure.

To assess the accuracy of these equations, we compared the volume of the entire ECS in each reconstruction, computed by subtracting the ICS of each cell from the volume of bounding box of the reconstruction, to the volume calculated using these equations. An algorithm for calculating the volume of the ECS in reconstruction is to assign an ECS shell element to each vertex in the model:

For each vertex j in each object i, calculate 1) the local radius of curvature R_j, 2) the surface area A_j of the mesh region around the vertex (Fig. 4C), and 3) the extracellular width ECW_j between the vertex and its closest point. Use these three values to compute the volume contribution ΔV_j of the shell element over the vertex j to the entire ECS volume. Error associated with this algorithm should converge to zero as surface patch size shrinks, i.e., number of vertices increases in mesh for a given surface. This model had an ECS volume error fraction of 0.009 ± 0.015 (mean ± SD, max = 0.027) across all reconstructions. A planar model of ECS (assuming zero curvature) had an ECS volume error fraction of 0.108 ± 0.033 (mean ± SD, max = 0.162) and was not used.

Tortuosity

On the micron scale, the morphology of the ECS is dictated by the twisting and turning of the cell membranes. Looking much like a plate of spaghetti, the convoluted structure of neuropil hinders the rate of neurotransmitter diffusion by virtue of its constricted pore space and the farther distance between two locations in the ECS compared to a free environment, attributes that we refer to as micron-scale, or geometric, tortuosity (λ_g). The micron-scale tortuosity of each reconstruction was estimated by measuring the diminished rate of diffusion in MCell simulations (Stiles and Bartol, 2001) of molecules in the ECS. The effective rate of diffusion of 10,000 molecules through the ECS was recorded by using concentric sampling boxes centered on the release site (Tao and Nicholson, 2004). The largest sampling box (8.5 μm on a side) was located fully within the reconstruction. To minimize boundary effects, the MCell simulation ended when 4% of the molecules had diffused out of the largest sampling box (Fig. 5).

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Figure 5

Measuring micron-scale tortuosity in a reconstruction of a 5 × 6 × 6 μm volume of neural tissue. The diffusion of glutamate in the ECS was simulated with MCell to measure the diffusional impedance due to micron-scale geometry of each reconstruction. Seventeen concentric sampling boxes (cyan) recorded the time course of glutamate (white) concentration during diffusion through the reconstructed tissue (not shown). The largest sampling box is 8.5 μm on a side. The glutamate count in each sampling box over time was used to estimate a diminished rate of diffusion (Tao and Nicholson, 2004). The resulting estimate of micron-scale tortuosity (Fig. 8C,D) had a fairly tight confidence interval, suggesting that a tissue volume of 1,000 μm³ would yield similarly well-constrained estimates of tortuosity without mirroring (see Materials and Methods). Reconstructions smaller than this are not representative of the neuropil because the volume is too small to average out the heterogeneity seen in the cellular structure.

The particles must sample a large subset of the reconstruction geometry by diffusing several microns to yield accurate and precise estimates of the micron-scale tortuosity. However, the particular tissue volume used here contains a large apical dendrite centered in the volume, which precluded central glutamate release. Our only recourse was to release peripherally and use three mirror planes to artificially expand the size of the reconstruction eightfold around the site of glutamate release to improve the estimate of micron-scale tortuosity. Thus we effectively oversampled the data, possibly introducing biases into our estimates of micron-scale tortuosity. Anisotropy of micron-scale tortuosity was not explicitly measured in our reconstruction (because the tissue volume was so small); however, the presence of the large apical dendrite would likely bias the rate of diffusion to be faster in the direction of the dendrite’s medial axis. The site of molecule release was located 250 picometers away from the point of intersection of the three mirror planes so that the cloud of molecules underwent simulated diffusion as if from a single point source. We confirmed that our choice of time step (1.0 × 10⁻⁶ seconds) was sufficiently small to avoid diffusion artifacts by reducing the time step by 50 (to 0.5 × 10⁻⁹ seconds) after which only a small change in rate of diffusion was measured (data not shown).

The rate of molecule diffusion through the ECS is also diminished relative to a free environment due to macromolecules obstructing the diffusion path on the nanometer scale and drag effects from the membrane walls. The diffusion constant of a molecule in the ECS can be calculated from its free diffusion constant in water, e.g., 7.5 cm² per second at 25°C for glutamate (Longsworth, 1953), and the ratio of geometric to total tortuosity in the ECS, e.g., 1.45 for glutamate in rat hippocampal CA1 (Sykova and Nicholson, 2008):

Quantitative topological analysis of the ECS

We performed a quantitative topological analysis of the ECS and decomposed the ECS into sheets and tunnels. There are a number of ways to do a comprehensive topological analysis, e.g., Morse graphs or stable manifolds (Shinagawa et al., 1996; Chazal and Lieutier; Goswami et al., 2006). Most of them essentially reduce to constructing and analyzing the medial axis set of the ECS. This is equivalent to skeletonizing a bounded volume with surface boundaries. For the ECS, the surface boundaries are surface patches of the axial/dendritic/glial membranes that come close together in the neuropil. The medial axis set is usually sheet-like when two surface patches of the ECS come close together, and when three or more such membrane boundaries come together in close proximity, they often induce a tubular or tunnel-like region, whose medial axis is curve-like and forms the axis path of the tunnel.

Using this decomposition, we classified a surface patch around each vertex in the surface meshes as belonging to either sheets or tunnels, according to the number of closest neighbor surface points each vertex had, 2 and 3+ respectively. The algorithm has several steps: For each vertex, an axis is defined through the vertex by averaging the normal vectors of the neighboring polygons. Around this axis, a cone of angle θ (e.g., 60 degrees) and radius r (e.g., 250 nm) is searched by ray tracing along the axis itself and eight evenly spaced rays along the perimeter of the cone. Intersection of the rays and any surface reveals the presence of a neighboring surface patch (Fig. 6A). The number and arrangement of the closest surface patch points determines the topological identity of the vertex and its neighborhood.

Manipulating the ECS on the nanometer scale

After the ECS was decomposed into sheets and tunnels, we varied the distribution of ECS between the two regions, allowing for the creation of a family of reconstructions, all with identical extracellular volume fraction but with different distributions of tunnel and sheet volumes (Fig. 2). Varying the lacunarity of the ECS according to topology, as such, is one way of introducing variance into the ECS distribution. However, this latter morphing process was underconstrained on the micron scale, so we made the simplifying assumption of uniform ECS width within the sheet region and within the tunnel region, and built this assumption into the morphing algorithm. However, in fact, the ECS width was never perfectly uniform in any of the reconstructions (Fig. 3); the spring model implicitly captured the lipid membrane’s resistance to bending, so the ECS width histograms of sheets and tunnels have some spread.

We wrote custom software to control the ECS by manipulating the location of the reconstructed surfaces on the nanometer scale. The software implements a simple model of forces on the membrane surface involving springs between neighboring regions on a surface and between surfaces on different neighboring processes. The collection of all springs describes a potential energy system, which is relaxed to determine the final location of the surfaces. The spring forces will tend to generate an ECS with uniform width controllable by a user-specified target value.

From the two corrected reconstructions (Fig 2C,D), four additional versions of the reconstruction were created by varying the allocation of the ECS volume between sheets and tunnels (Fig. 2E–H). All six of the new reconstructions had the same extracellular volume fraction (0.202 ± 0.003), but the uniformity of the extracellular width in the reconstructions varied dramatically. The distributions of extracellular width in the sheets and tunnels were nearly identical in the more uniform reconstructions (Fig. 2C,D; Table 1) but diverged as the sheet volume shrank and the tunnel regions expanded (Fig. 2E–H). The variances of the extracellular widths for sheets and tunnels increased along with their medians.

The sheets and tunnels were non-uniformly distributed

We examined the distribution of sheets and tunnels around axons, dendrites, and astrocytic processes and found that axons were surrounded by more tunnel-like spaces and astrocytes tended to be near sheet-like volumes (Fig. 7B). We attribute these findings to differences in cellular geometry: Axons have a smaller cross-sectional area than dendrites (Mishchenko et al., 2010), which promotes the formation of tunnels in the ECS. In contrast, the vellate structure of astrocytes is more conducive to having sheet-like ECS (Online Movie at http://cnl.salk.edu/~jkinney/waltz/waltz.mp4).

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Figure 7

Astrocytes and synapses were enriched with sheet-like extracellular space (ECS) and the perisynaptic region with tunnel-like ECS. The subcellular distribution of ECS volume was measured for each ECS-corrected reconstruction, including the fraction of ECS in synaptic clefts, perisynaptic regions, and surrounding axons, dendrites, and astrocytes. The relative amount of tunnel- and sheet-like volume in these functionally defined regions of ECS was also calculated. A: Perisynaptic regions have the highest fraction of ECS volume in tunnels. In contrast, synaptic clefts have an ECS more sheet-like than other regions of membrane. The perisynaptic region was defined to be twice as wide as synaptic clefts to adequately sample ECS around synapses. This implies the perisynaptic region will be eightfold larger than cleft area (assuming concentric, circular geometry). Perisynaptic regions and synaptic clefts accounted for 12.3% and 3.5% of the ECS volume, respectively. B: The ECS was also decomposed into three fractions corresponding to axonal, dendritic, and astrocytic regions accounting for 60%, 30%, and 10% of the ECS volume, respectively. Axonal regions of ECS are consistently more tunnel-like than dendrites and astrocytes. Conversely, the ECS around astrocytes is enriched with sheets.

By specifying exact patches of the reconstruction surface, both axonal and dendritic, as belonging to glutamatergic synaptic clefts and perisynaptic regions, the ECS volume belonging to these two groups was identified and measured (Fig. 7A). As expected, synaptic clefts were enriched with sheet-like ECS; indeed, the classical definition of a synapse as being an apposition of presynaptic and postsynaptic membrane matches our topological definition of sheet. The fraction of ECS volume in synaptic clefts in our reconstructions was 3.5% of the ECS, consistent with previous findings (Rusakov et al., 1998). Non-glutamatergic synapses accounted for less than 2% of all synapses and were excluded from the analysis.

The ECS in perisynaptic regions was more tunnel-like than all other membrane types, consistent with the fact that glutamatergic synapses in hippocampus occur on dendritic spines. The perisynaptic ECS volume fraction was 12.3% on average, fourfold larger than the synaptic cleft volume. It is generally accepted that the width of the synaptic cleft is tightly controlled in the 15–25-nm range (Savtchenko and Rusakov, 2007) owing to the high concentration of synaptic adhesion molecules in the cleft (Latefi and Colman, 2007). In our reconstructions the synaptic cleft width was not explicitly controlled, but instead was allowed to vary according to local topology of the neuropil. The mean synaptic cleft extracellular widths in the ECS-corrected reconstructions were 39.2, 33.8, 45.2, 34.4, 47.9, and 38.1 nm for reconstructions C–H, respectively.

To probe further the relationship between synapses and tunnels, we computed the mean straight-line distance from each vertex to the centroid of the nearest synapse (Table 2). In all models the tunnels were closer to the synapses than sheets: On average the tunnels were closer by 26.7 ± 4.7 nm. Coupled with the result that diffusion is faster in the tunnels than the sheets, this finding suggests that the ECS may be functionally organized into regions.

Grant sponsor: National Science Foundation to the Center for Theoretical Biological Physics; Grant number: PHY-0822283 (to T.M.B. and T.J.S.; National Institutes of Health to the Center for Theoretical Biological Physics; Grant numbers: MH079076, GM068630, and P01-NS044306 (to J.P.K., T.M.B., and T.J.S.); Grant sponsor: the Howard Hughes Medical Institute (to T.J.S.); Grant sponsor: National Institutes of Health; Grant numbers: EB002170, NS074644, and NS21184 (to K.M.H.) and R01-GM074258 and R01-EB004873 (to C.B.); Grant sponsor: the UT-Portugal CoLab project (to C.B.).

The authors thank Daniel Keller, Rex Kerr, Vladimir Mitsner, Ben Regner, Jed Wing, Mary Kennedy, and Dennis McKearin for their contribution of ideas and constructive criticism. The authors also acknowledge Bryan Nielsen, Jorge Aldana, and Chris Hiestand for their generous computer support of this project.

ROLE OF AUTHORS

All authors had full access to all the data in the study and take responsibility for the integrity of the data and the accuracy of the data analysis. Acquisition of data: J.S. and K.M.H. Analysis and interpretation of data: J.P.K., J.S., T.M.H., C.L.B., K.M.H., and T.J.S. Drafting of the manuscript: J.P.K., T.J.S., and K.M.H. Critical revision of the manuscript for important intellectual content: J.P.K., J.S., T.M.H., C.L.B., K.M.H., and T.J.S.