The structure of the extracellular space (ECS) of the brain can be approximated by two parameters, the volume fraction (α) and the tortuosity (λ) (Nicholson & Phillips, 1981). Volume fraction is the ratio of the volume of ECS to total tissue volume in a representative elementary volume of brain tissue. Tortuosity is defined as λ=√(*D*/*D**) where *D* (cm^{2} s^{−1}) is the free (aqueous) diffusion coefficient of the molecule and *D** (cm^{2} s^{−1}) is the effective (apparent) diffusion coefficient in brain tissue. Tortuosity is related to the hindrance imposed on a diffusing particle by the cellular elements, and possibly the extracellular matrix (Nicholson & Syková, 1998; Nicholson, 2001). A quantitative understanding of the structure of the ECS is important for many issues, including volume transmission (Fuxe & Agnati, 1991; Agnati *et al.* 2000), the interpretation of diffusion-weighted magnetic resonance imaging (MRI) (Nicolay *et al.* 2001; Norris, 2001), the treatment of brain ischaemia and other traumas (Syková, 1997) and the delivery of drugs and therapeutic agents (Saltzman, 2001). Here we used osmotic stress to study the relationship between volume fraction and tortuosity.

Both α and λ can be measured by analysing the diffusion of molecules that remain predominantly in the ECS after release from a point source (Nicholson & Phillips, 1981; Nicholson & Syková, 1998; Nicholson, 2001). For a specific brain structure, in the sense that the geometry is preserved, one would anticipate an inverse relationship between α and λ, because a reduction in the volume of the ECS might be expected to increase hindrance to diffusion. This study used osmotic stress, induced by varying the NaCl content of the artificial cerebrospinal fluid (ACSF) to change the size of the ECS. It is known (KrizTortuosity and volume fraction were measured with the real-time ionophoretic (RTI) method using tetramethylammonium (TMA^{+}, 74 *M*_{r}), the small extracellular cationic probe (Nicholson & Phillips, 1981; Nicholson, 1993). Measurements of extracellular field potentials and water content provided supporting data. Rat neocortical slices were used because they provided a homogenous, isotropic tissue and good control of the extracellular ionic environment. We emphasise that our goal was not to study the effect of osmotic stress on tissue but to use the challenge to explore a fundamental question about the relationship between structural parameters of brain tissue.

Over the range of osmolalities employed, we found that there was no simple relationship between α and λ. Tao (1999) measured λ under similar osmotic challenge but using 3000 *M*_{r} dextran and the integrative optical imaging (IOI) method. Comparing the results from that paper with the data reported here leads to new insights into the structural constraints imposed by the ECS and has led to a novel theoretical model (Chen & Nicholson, 2000). Some results have been published in abstract form (Nicholson *et al.* 1998).

The study by Krizaj et al. (1996) used the whole turtle cerebellum whereas the present study employs slices of a mammalian neocortex and therefore is more relevant to other studies of slice physiology. The turtle study also did not explore hypotonic challenges extensively. The paper by Tao (1999) looked at a similar range of osmotic stresses in the same slice preparation as used here but used a significantly larger molecule and IOI, a technique that permits measurement of the effective diffusion coefficient, and hence tortuosity, but not volume fraction. The present study has used the RTI method with TMA^{+}, as frequently used in other studies; importantly, this enables both tortuosity and volume fraction to be measured simultaneously. The theoretical paper by Chen & Nicholson (2000) is based on the data reported in the present paper and therefore complements this study. Because the theory is quite complex and likely to be applicable widely, it was published separately.