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Brain extracellular space is the narrow microenvironment that surrounds every cell of the central nervous system. It contains a solution that closely resembles cerebrospinal fluid with the addition of extracellular matrix molecules. The space provides a reservoir for ions essential to the electrical activity of neurons and forms an intercellular chemical communication channel. Attempts to reveal the size and structure of the extracellular space using electron microscopy have had limited success; however, a biophysical approach based on diffusion of selected probe molecules has proved useful. A point-source paradigm, realized in the real-time iontophoresis method using tetramethylammonium, as well as earlier radiotracer methods, have shown that the extracellular space occupies ∼20% of brain tissue and small molecules have an effective diffusion coefficient that is two-fifths that in a free solution. Monte Carlo modeling indicates that geometrical constraints, including dead-space microdomains, contribute to the hindrance to diffusion. Imaging the spread of macromolecules shows them increasingly hindered as a function of size and suggests that the gaps between cells are predominantly ∼40 nm with wider local expansions that may represent dead-spaces. Diffusion measurements also characterize interactions of ions and proteins with the chondroitin and heparan sulfate components of the extracellular matrix; however, the many roles of the matrix are only starting to become apparent. The existence and magnitude of bulk flow and the so-called glymphatic system are topics of current interest and controversy. The extracellular space is an exciting area for research that will be propelled by emerging technologies.

The narrow spaces between the cells of the brain are a highway for diffusing molecular traffic. In this issue, Xiao et al. (1) use experiments and Monte Carlo modeling to show that molecular transport in the granular layer of the rat cerebellum must be described by anomalous diffusion mediated by anatomical structures called glomeruli. The extracellular space (ECS) of the brain comprises all the territory that resides outside the neurons and glia cells, most importantly the interstitial space between neighboring cells.

It has long been assumed that one important mechanism for the dissipation of local potassium gradients in the brain extracellular space is the so-called spatial buffer, generally associated with glial cells. To date, however, there has been no analytical description of the characteristic patterns of K+ clearance mediated by such a mechanism. This study reanalyzed a mathematical model of Gardner-Medwin (1983, J. Physiol. (Lond.). 335:393–426) that had previously been solved numerically. Under suitable approximations, the transient solutions for the potassium concentrations and the corresponding membrane potentials of glial cells in a finite, parallel domain were derived. The analytic results were substantiated by numerical simulations of a detailed two-compartment model. This simulation explored the dependence of spatial buffer current and extracellular K+ on the distribution of inward rectifier K+ channels in the glial endfoot and nonendfoot membranes, the glial geometric length, and the effect of passive KCl uptake. Regarding the glial cells as an equivalent leaky cable, the analyses indicated that a maximum endfoot current occurs when the glial geometric length is equal to the corresponding electrotonic space constant. Consequently, a long glial process is unsuitable for spatial buffering, unless the axial space constant can match the length of the process. Finally, this study discussed whether the spatial buffer mechanism is able to efficiently transport K+ over distances of more than several glial space constants.

This paper describes the theory of an integrative optical imaging system and its application to the analysis of the diffusion of 3-, 10-, 40-, and 70-kDa fluorescent dextran molecules in agarose gel and brain extracellular microenvironment. The method uses a precisely defined source of fluorescent molecules pressure ejected from a micropipette, and a detailed theory of the intensity contributions from out-of-focus molecules in a three-dimensional medium to a two-dimensional image. Dextrans tagged with either tetramethylrhodamine or Texas Red were ejected into 0.3% agarose gel or rat cortical slices maintained in a perfused chamber at 34 degrees C and imaged using a compound epifluorescent microscope with a 10 x water-immersion objective. About 20 images were taken at 2-10-s intervals, recorded with a cooled CCD camera, then transferred to a 486 PC for quantitative analysis. The diffusion coefficient in agarose gel, D, and the apparent diffusion coefficient, D*, in brain tissue were determined by fitting an integral expression relating the measured two-dimensional image intensity to the theoretical three-dimensional dextran concentration. The measurements in dilute agarose gel provided a reference value of D and validated the method. Values of the tortuosity, lambda = (D/D*)1/2, for the 3- and 10-kDa dextrans were 1.70 and 1.63, respectively, which were consistent with previous values derived from tetramethylammonium measurements in cortex. Tortuosities for the 40- and 70-kDa dextrans had significantly larger values of 2.16 and 2.25, respectively. This suggests that the extracellular space may have local constrictions that hinder the diffusion of molecules above a critical size that lies in the range of many neurotrophic compounds.

Ca2+ diffusion in brain extracellular space (ECS) regulates local Ca2+ concentration and influences synaptic transmission and neuronal excitability. Ca2+ diffusion is controlled by ECS geometry and extracellular matrix, a major component of which is chondroitin sulfate (CS). Previous experiments (Hrabetova et al., J. Physiol. 2009, 587:4029) show that Ca2+ movement is reduced when interacting with CS, however the process still obeys the diffusion equation. This implies that Ca2+-CS interaction can be modeled as a fast equilibrium bimolecular reaction (FEBR; Nicholson et al. Comput. Visual Sci., 2012, 10.1007/s00791-012-0185-9). The FEBR would provide an alternative to solving the Poisson-Boltzmann equation for electrostatic interactions between negatively charged CS and mobile cations.Our objectives were 1) to test whether the FEBR model describes previous experimental data, 2) if so, to determine the dissociation constant Kd, (ratio between backward and forward rate constants), 3) to explore the effect of background Ca2+ and/or Na+ on Ca2+ diffusion and 4) use a Monte Carlo simulator (MCell; www.mcell.org) to verify results.We developed analytical expressions for the Ca2+ effective diffusion coefficient in the presence of CS and compared results to experimental data with different background Ca2+ and CS concentrations (Magdelenat et al., Biopolymers, 1974, 13:1535; Maroudas et al., Biophys. Chem., 1988, 32:257). This validated the FEBR approach and provided estimates of Kd (0.01 to 0.1 mM) in agreement with literature data. Kd values were resolved into forward and backward rate constants using the Damkohler formula and the results further tested with MCell. Finally, we extended the work to include additional background Na+.We conclude that the FEBR model captures the main features of Ca2+ diffusion in CS matrix and can be extended to interactions involving multiple cations. Supported by NIH/NINDS grant R01-NS-28642.

A major source of diffusion anisotropy in the central nervous system is the presence of oriented fibers (e. g. corpus callosum; cerbellar molecular layer). The diffusion properties may be explored experimentally using a point-source paradigm based on tetramethylammonium (TMA; Sykova & Nicholson, Physiol. Rev. 2008, 88, 1277). To better understand such results the MCell Monte Carlo program (www.mcell.psc.edu) was used to model diffusion from a point source in bundles of rectangular fibers with extracellular volume fractions (α) ranging from 0.02 - 0.8.The simulation employed 64 rectangular rods each of 0.6 x 0.6 μm in cross-section with surfaces that reflected the particles. Rods were spaced appropriately to obtain the required α and the length of each rod was chosen to make the rods into a cubic ensemble. A population of 250,000 point particles was released into the center of the ensemble with free diffusion coefficient (D) equal to that for TMA and the simulations run with a time-step of 10 ns for 1 ms. These parameters ensured that the particles remained within the ensemble.It was found that the component of the effective diffusion tensor, D∗, aligned with the direction of the bundles was equal to D so the tortuosity (λ = (D/D∗)1/2) was unity in this axis. For the two symmetrical components of D∗ in the orthogonal axes, λ was a function of α that could be accurately describe by an expression derived by Bell & Crank (J. Chem. Soc. Faraday Trans., 2, 1974, 70, 1259). These results are compared to those already obtained for an assembly of cubes (Tao & Nicholson, J. Theoret. Biol. 2004, 229, 59). Supported by NIH/NINDS Grant R01-NS-28642.

There are a limited number of methods available to quantify the extracellular diffusion of macromolecules in an anisotropic brain region, e.g., an area containing numerous aligned fibers where diffusion is faster along the fibers than across. We applied the integrative optical imaging method to measure diffusion of the fluorophore Alexa Fluor 488 (molecular weight (MW) 547) and fluorophore-labeled flexible random-coil dextran polymers (dex3, MW 3000; dex75, MW 75,000; dex282, MW 282,000; dex525, MW 525,000) in the extracellular space (ECS) of the anisotropic molecular layer of the isolated turtle cerebellum. For all molecules, two-dimensional images acquired an elliptical shape with major and minor axes oriented along and across, respectively, the unmyelinated parallel fibers. The effective diffusion coefficients, D*(major) and D*(minor), decreased with molecular size. The diffusion anisotropy ratio (DAR = D*(major)/D*(minor)) increased for Alexa Fluor 488 through dex75 but then unexpectedly reached a plateau. We argue that dex282 and dex525 approach the ECS width and deform to diffuse. In support of this concept, scaling theory shows the diffusion behavior of dex282 and dex525 to be consistent with transition to a reptation regime, and estimates the average ECS width at approximately 31 nm. These findings have implications for the interstitial transport of molecules and drugs, and for modeling neurotransmitter diffusion during ectopic release and spillover.

We investigated the conditions under which inhomogeneity in electrical conductivity may significantly modify the magnetic evoked field (MEF) due to primary currents (i.e., neuronal currents) in the brain. In the case of an isolated turtle cerebellum immersed in a large bath of physiological saline, our theoretical analysis showed the cerebellar surface to significantly enhance the MEF due to a primary current, by a factor of as much as two, for experimentally determined values of the conductivities of the cerebellar tissue and saline. A further parametric investigation of the conductivity effect revealed that conductivity boundaries may significantly modify the MEF due to neuronal currents located within 1 mm of a conductivity boundary, as would be the case for active neurons near an edema, an anoxic fringe such as might occur during stroke, or a ventricle in the human head. For a stationary neural source, conductivity boundaries may modify the magnitude of its MEF without affecting its temporal waveform. However, this boundary effect was found to be small for a model geometry locally approximating cortical sources in a sulcus or a fissure, where the boundary effects from adjacent sulcal walls tend to cancel each other.