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Averaged equations for electrical potential and ion transport in brain tissue

University of British Columbia

The accurate modelling of bulk passive properties of neural tissue is essential to the modelling -of macroscopic phenomena in the brain such as spreading depression and epilepsy. Properties which characterise the passive and active flows of ions or electric current through tissue are referred to as transport properties. Such properties associated with passive flows include bulk conductivity, bulk diffusion coefficient, and those associated with electrically mediated ionic flux which is called 'spatial buffer flux'. While models for such transport properties of cortical tissue have been published, each of these models contained different assumptions about the structure of the tissue. Recent data on potassium transport through neural tissue are important for the construction of a unified model (i.e., based on a consistent set of assumptions) because they provide measurements of the amount of bulk electric current passing through cell membranes. In this thesis the Nernst-Planck equation is used as the governing equation for ion transport and electric potential, with specification of the jump conditions at the cell membrane. An asymptotic expansion and averaging procedure is -described which reduces the computation of bulk properties to a calculation for a single cell. The idea of transport numbers (a proportionality constant between ion transport and electric field vectors) in electrolytes is introduced and it is shown that this idea applies to bulk tissue. Estimates of the coefficients in the averaged equations are computed numerically for different geometries and a range of microscopic parameter values including cell size, membrane conductance, intracellular conductivity, extracellular space fractional volume. An important finding is that theoretical transcellular current, i.e., the bulk current flow through disconnected cells, is significant and relatively insensitive to several of these parameters, in particular cell size and membrane conductance. The role of electrotonic parameters (the parameters involving electrical constants) in the tissue model is discussed and a formal analogy between transcellular current and electrostatic polarization is introduced as an aid to physical understanding of the transport properties of arrays of disconnected (physically separated) cells. Asymptotic analyses of the electrotonic parameters are performed in order to supplement the numerical solutions with qualitative results, and it is shown how to incorporate asymptotic assumptions about these parameters into an asymptotic model. The properties of steady solutions to the averaged equations are discussed and it is shown that some coefficients of the equations cannot be estimated in a steady experiment. It is argued that the general model proposed here is simpler and more appropriate than cable theory for bulk tissue. For example, it is concluded that specialized transfer cells are unnecessary to -explain transcellular flux and spatial buffering, that disconnected cells cannot be neglected, and that cells of differing sizes may contribute significantly to transcellular flux. Since transcellular flux is significant and insensitive to geometry and intracellular conductivity in our model, our results imply that spatial buffering occurs very generally. This model is chosen to include most measurable quantities such as extracellular potential and extracellular K⁺ concentration, and to be mathematically simple. Since it is shown that the bulk parameters of the model are relative insensitive to many of the microscopic parameters of the tissue, the resulting governing equations should be applicable to many physiological situations.

Text

2010-10-16

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http://hdl.handle.net/2429/29213

Mathematics

University of British Columbia

Graduate