INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY CONCLUSIONS APPENDIX A REFERENCES

The phenomenon of spatial potassium concentration gradients in extracellular space (ECS) and their buffering by glial cells was first described by Kuffler et al. (1966) and Orkand et al. (1966), who demonstrated that depolarization of glial cells in the leech and the mud puppy was synchronized with neuronal activity. It was hypothesized that the depolarization was mediated by potassium released from spiking neurons. Since then the existence of spatial buffering (SB) has been demonstrated, or at least implied, in numerous experiments (to name a few, Coles and Orkand, 1983; Gardner-Medwin et al., 1981; Gardner-Medwin and Nicholson, 1983; Immel and Steinberg, 1986; Karwoski et al., 1989; Oakley et al., 1992) in various brain regions of vertebrates and invertebrates. Glial cells are interposed between virtually all neurons and axons, with a K⁺-dominated resting membrane potential ~20 mV more negative than neurons. The functional roles played by glial cells are far from being understood, but it is generally agreed that their unique membrane properties are involved in the regulation of the extracellular potassium concentration, [K⁺]_o. The stability of [K⁺]_o is essential during prolonged neuronal activity; otherwise there would be uncontrolled variations in neuronal excitability (Barres, 1991; Syková, 1983; Walz, 1989).

In addition to the ubiquitous process of diffusion, at least three different mechanisms are also involved in the clearance of excessive K⁺ in the ECS (Amédée et al., 1997; Ballanyi et al., 1987; Coles and Orkand, 1983; Dietzel et al., 1989; Walz, 1989): 1) current-mediated K⁺ entry via K⁺ channels, especially inward rectifiers; 2) enhanced K⁺ transport by Na⁺/K⁺-ATPase after an increase in intracellular Na⁺; and 3) passive KCl uptake through inward rectifier K⁺ channels and voltage-gated Cl channels. The first mechanism, spatial buffering by K⁺ channels, is the focus of this work. When, as a consequence of enhanced neuronal activity, excessive K⁺ ions are released into the interstitial clefts, the local [K⁺]_o level rises. This causes local depolarization of glial membrane potential that can spread electrotonically through cytoplasm, and possibly gap junctions, to more distal regions. The asymmetrical spatial distribution of potential difference across the glial membrane elicits a local circuit current that, because of the high K⁺ permeability, mediates an influx of K⁺ into the cell in the region where [K⁺]_o is raised, and an efflux of K⁺ into the ECS from distal glial processes whose surrounding [K⁺]_o is still low. This mechanism of dissipating [K⁺]_o spatial gradients in the brain ECS via glial intracellular pathways, termed "spatial buffering" (Orkand et al., 1966), is passive, energy-independent, and in most cases more efficient than diffusion through the interstitium (Gardner-Medwin, 1983a, 1986; Gardner-Medwin and Nicholson, 1983).

Inward rectifier K⁺ channels

Many lines of evidence (Kettenmann et al., 1983; Kuffler et al., 1966; Lothman and Somjen, 1975; Newman, 1985, 1993) have confirmed that the glial membrane is selectively permeable to K⁺ at rest and passively obeys the Nernst equation over a wide range of [K⁺]_o after taking into account the intracellular K⁺ activity. Although many voltage-gated channels, previously identified in neurons, have now been found to exist in glia (Barres et al., 1990; Sontheimer, 1994), these channels only contribute a small fraction of the whole membrane conductance. At least four different voltage-dependent K⁺ channels (inward rectifier, Kir; delayed rectifier, Kd; transient A-type, K_A; Ca²⁺-activated, K_Ca) have been identified in glial cells (Sontheimer, 1994). Among them, the dominant K⁺ channel is Kir, which retains a high open probability at the resting potential and is easily activated at more negative levels. Kir is known to exhibit an asymmetrical response to hyperpolarization (high conductance) compared to depolarization (low conductance). Therefore K⁺ enters the cell more readily than it leaves. The inward rectifier improves the efficiency of SB (Newman, 1993) by enhancing the K⁺ influx in regions of elevated [K⁺]_o and by spreading the membrane depolarization to more remote parts of the glial cell (Amédée et al., 1997). The hypothesis that the outward rectifier, Kd, facilitates K⁺ efflux at depolarized regions during SB, though attractive, is not feasible, except in pathological conditions, because activation requires significant depolarization (50 mV; Sontheimer, 1994).

Nonuniform Kir distribution and glial endfeet

It has repeatedly been demonstrated (Brew et al., 1986; Newman, 1984, 1985, 1993; Skatchkov et al., 1995, 1999) that glial (Müller) cells in the retina of amphibians have a markedly nonuniform distribution of Kir channels with a predominance at the endfeet regions facing the vitreous humor. The nonuniform distribution of membrane conductance appears to be most effective in directing a large K⁺ efflux through the high-conductance region for situations involving distances of only a few space constants (Brew and Attwell, 1985; Eberhardt and Reichenbach, 1987). A refined mechanism, termed "potassium siphoning," was proposed and numerically simulated (Newman, 1993; Odette and Newman, 1988) to show how a specialized endfoot in the Müller cell can direct most K⁺ efflux to the vitreous humor. Based on histological evidence, it is highly possible that astrocytes also have regionally specialized K⁺ conductance in their multiple processes. Newman (1986) provided further support for this idea, and this led to the conjecture that excessive K⁺ in the ECS could be siphoned via astrocytic endfeet processes to regulate blood vessel dilation and cerebral blood flow (Paulson and Newman, 1987). In Schwann cells, Kir is highly localized in the microvilli (Mi et al., 1996), which implies its possible involvement in K⁺ regulation in the peripheral nervous system.

Glial syncytium

Many of the in vivo or in vitro experiments involve only a single cell or a few coupled in a syncytium. Thus the geometric length scale in these experiments was no more than 200 µm (the vertical distance of the rather elongated Müller cell in rabbit retina; Reichenbach and Robinson, 1995). To test the effectiveness of the glial syncytium in SB, another type of experiment is required. Such an experiment was devised by Gardner-Medwin (1983a,b) and Gardner-Medwin and Nicholson (1983), who ran the passive SB in reverse by applying an external electrical current across the cerebellum of the rat to evoke a local change in [K⁺]_o. In other relevant work Dietzel et al. (1982, 1989) measured the field potential changes, caused by the SB K⁺ current and cotransport of other ions, across the extent of the cerebral cortex in cats. Because these experiments usually involved hundreds of glial cells, presumably interconnected by gap junctions, the length and time scales in these experiments are, respectively, millimeters and minutes, and any nonuniformity in the distribution of membrane conductance in an individual cell may have been unimportant. This is especially true when the boundary of the tissue is not terminated by glial endfeet. Instead, introducing a statistically averaged membrane conductance, analogous to the tortuosity factor characteristic of the macroscopic structure of the ECS (Nicholson and Syková, 1998), seems more appropriate for understanding the overall SB of a glial syncytial network.

In this paper, we have focused on a macroscopic scale, in the same sense as defined by Gardner-Medwin (1983a,b), by viewing the brain cell microenvironment as a homogeneous continuum and the glial syncytium as an equivalent cable, as shown in Fig. 1. The homogeneous assumption is a necessary simplification when the fate of [K⁺]_o within a glial syncytium is simulated. To obtain a practical macroscopic model of K⁺ diffusion, a tortuosity factor must be introduced that embodies the macroscopic structure and connectivity of the ECS and neglects differences in the individual properties of the glial cells. Nevertheless, the governing equations and their solutions should also be able to describe a single glial cell, preferably in a cylindrical shape, after a suitable modification of the boundary conditions.

View larger version (47K): [in this window] [in a new window]   FIGURE 1   (a) Schematic drawing showing the simultaneous K+ transport by spatial buffer (SB) via the glial cells (represented by the shaded rectangles, possibly interconnected by gap junctions) and interstitial diffusion. Arrows indicate K+ flows due to either diffusion or SB. (b) When the macroscopic transport is effectively one-dimensional, the complex intracellular and extracellular geometry in the brain tissue is simplified as a unit rectangular slab, in which an equivalent cylindrical cable represents one single glial cell or a glial syncytium. The rest of the space is composed of neurons and other tissue cells and extracellular space (ECS). For simplicity, neuronal cells were not drawn here. The ECS is regarded as homogeneous with an effective diffusion coefficient DK/o2 for potassium. The diameter of the equivalent cable and the size of the slab were arranged to achieve the approximate volume ratio between glial cells and the ECS (i:o = 0.4:0.2; Dietzel et al., 1989). At both boundaries highly permeable endfeet may be imposed on the cross-sectional area of the glial cylinder. Arrows indicate SB intracellular current and transmembrane K+ current.

	THEORY

TOP ABSTRACT INTRODUCTION THEORY CONCLUSIONS APPENDIX A REFERENCES

In this section we introduce a two-compartment model, consisting of the ECS and the intracellular glial compartment, together with the coupled partial differential equations that govern the temporal and spatial distributions of intra- and extracellular K⁺. To estimate how the glial intracellular [K⁺]_i varies with [K⁺]_o, the dynamics of [Cl]_i and [Cl]_o will be modeled under the assumption of passive KCl uptake, which is considered the most important uptake mechanism for glial cells (Amédée et al., 1997). The Na⁺/K⁺/2Cl cotransporter, the active Na⁺/K⁺ pump, and the change in ECS volume due to osmotic shifts induced by ionic movement between compartments were not considered here. To do so would require a full accounting of the movement of other ions (such as Na⁺) in both compartments and necessitate several other sets of nonlinear differential equations. However, it should be pointed out that the ECS is known to shrink when [K⁺]_o is raised well above its resting level (Dietzel et al., 1989; Ransom et al., 1985).

The glial membrane potential _m (mV) is defined as the difference between the intracellular and extracellular potentials, _i  o. Regarding the equivalent cylinder in Fig. 1 as an idealized glial cell, the governing equation for the membrane potential _m of the cylindrical cable at location (m) can be described by conventional cable theory (Gardner-Medwin, 1983b) as

(1)

where î_m (A m²) is the net outward ionic current density across the glial membrane (outward means positive charges move across cell membrane from cytoplasm to ECS) per unit area of glial membrane surface. The averaged glial membrane surface area per unit volume of tissue, available for K⁺ exchange, is designated by a (m¹). In Eq. 1, r_i and r_o ( m) are the apparent intra- and extracellular resistivity, respectively. (Here r_i and r_o are based on the cross-sectional area of the whole tissue. For the conventional definitions of intracellular and extracellular resistivity, r'_i and r'_o, we assumed that r_i = r'_ii²/_i and r_o = r'_oo²/_o, where _o, _i are the volume fractions of the ECS and glial cells in the brain tissue, respectively, and _o, _i are the tortuosity factors in the corresponding compartments.) Because the typical characteristic time constant of the membrane capacitance is ~1 ms (Newman, 1985), which is too short to be significant in comparison to the typical time scale for diffusion (seconds), the transient term in Eq. 1 has been omitted. We assumed that the transmembrane current î_m is strictly carried by K⁺ because of the exclusive K⁺ permeability of the glial membrane. Furthermore, we expect a negligible contribution to changes in î_m to result from the passive entry of Cl ions, through independent Cl channels.

The K⁺ concentration in the extracellular compartment, _K,o (moles per unit volume of ECS, mol m³), is governed by the mass conservation equation in the ECS modified for interactions with the intracellular compartment,

(2)

where _o and _i (compartmental space per unit tissue volume) are the volume fractions of the ECS and the intracellular space of the glial cells, respectively. The K⁺ flux per unit area of the ECS is _K,o (mol m² s¹), and F is the Faraday constant. The term P (mol m² s¹) represents the net rate of passive influx of KCl into glial cells, following the model of Boyle and Conway (1941). According to this model, a fluctuation in the concentration of K⁺ or Cl in either compartment will result in a passive KCl redistribution across the membrane, with the final concentrations in both compartments obeying a Donnan equilibrium. The rate of the KCl uptake is assumed to be proportional to the concentration difference,

(3)

in which (s) is the first-order uptake constant; _K,i and _Cl,i represent [K⁺]_i and [Cl]_i, respectively; and _K,i is the [K⁺]_i concentration at Donnan equilibrium, estimated from local K⁺ and Cl concentrations in both compartments.

Assuming approximately equal mobility for K⁺ and other ions (mostly Na⁺ and Cl) in the ECS, the effective K⁺ flux in the ECS is described by the generalized Nernst-Planck equation,

(4)

This equation shows that D_K (m² s¹), the K⁺ diffusion coefficient in the bulk saline, is reduced by the square of the tortuosity factor, _o, associated with the ECS (Nicholson and Phillips, 1981; Nicholson and Syková, 1998). The symbol  (mV) designates a standard potential in the units of RT/F (R is the gas coefficient and T is the absolute temperature). The potential driving force in the ECS, d_o/d, can be further expressed in terms of d_m/d and the external electric current, Î (A m²), per unit area of the tissue (Gardner-Medwin, 1983b). Additional source/sink terms (such as artificial K⁺ injections or extra K⁺ released by stimulated neurons) within the domain were not considered in the analysis. These situations will be studied later by numerical simulations.

Transmembrane K⁺ flux

The constitutive relation between the transmembrane K⁺ current, î_m, and the membrane potential is usually empirical; thus in Gardner-Medwin (1983b) the nonlinear Goldman-Hodgkin-Katz model was employed, whereas here we adopt the simpler Ohm's law,

(5)

where ê_K (mV) is the glial equilibrium potential described by the Nernst equation, ê_K =  ln(_K,o/_K,i). The resting potential ê_K^o (85.2 mV) is obtained when _K,o = _K,o^o and _K,i = _K,i^o. The g_K (S m²) is the specific glial membrane K⁺ conductivity and may vary with many factors. Although various K⁺ channels in glial cells are found to be voltage-gated with rectification, ATP-sensitive, or activated by neurotransmitters (see reviews by Barres, 1991; or Walz, 1989) or other cations (mostly intracellular Ca²⁺ or Na⁺), we only considered the Kir channel, because of its dominance. Newman (1993) empirically determined the Kir conductivity in retinal Müller cells as

(6)

and we adopt his result here. In the above equation, g_K^o is the membrane conductivity at rest (_K,o = _K,o^o, _m = ê_K^o, and  = 0). The first term in the parentheses describes how the rectification changes with , and the second term signifies the open channel probability that increases with the membrane potential. For simplicity, the concentration dependence and the two probabilistic functions are lumped together as a rectification factor f_Kir. At rest f_Kir = 1. When the membrane hyperpolarizes ( < 0) or [K⁺]_o is elevated above the resting value, f_Kir > 1. In contrast, when depolarization occurs or [K⁺]_o is below the resting value, f_Kir < 1.

Nondimensionalization

To generalize the theoretical as well as the numerical results, it is convenient to express the system of equations in terms of the following dimensionless variables:






in which  (m) is the glial electrotonic space constant, defined as  = [g_K^oa(r_i + r_o)]^1/2, (m) is the length of the finite domain, _i is the intracellular tortuosity factor within the equivalent glial cable, and g_K,L^o and g_K,R^o are the basal K⁺ conductivity in the left (L) and right (R) endfeet, such that g_K,L^of_Kir and g_K,R^of_Kir represent the total Kir conductivity of each endfoot, respectively. The endfeet, when present, are represented by the cross-sectional area at the ends of the cylindrical cable in Fig. 1 b.

The dimensionless Nernst equation becomes

(7)

and Eqs. 1 and 2 become, respectively,

(8)

and

(9a)

Similarly, the governing equation for intracellular [K⁺]_i can be written in dimensionless form as

(9b)

The terms on the right-hand side of Eqs. 9a and 9b respectively indicate contributions from i) SB transmembrane K⁺, ii) diffusion, iii) ionic movement driven by SB loop current, iv) externally applied electrical current, and v) KCl uptake. We also modified the uptake pathway, using the same inward rectification property, f_Kir; this will cause KCl entry into glial cells to be further enhanced. Because we assumed that the transmembrane current is composed of K⁺ and that the only pathway by which Cl can enter glial cells is via passive KCl uptake, the governing equations for [Cl]_o and [Cl]_i can be similarly written as

(10a)

and

(10b)

with the dimensionless c_Cl,o and c_Cl,i defined as _Cl,o/_Cl,o^o and _Cl,i/_Cl,i^o, respectively, and where D_Cl is the diffusion coefficient of Cl in saline.

The initial conditions are c_K,o(x, 0) = c_K,i(x, 0) = c_Cl,o(x, 0) = c_Cl,i(x, 0) = 1 and v(x, 0) = e_K(x, 0) = 0. The extracellular [K⁺]_o and [Cl]_o are assumed to be fixed at c_K,o(0, t) = c_K,o^L, c_K,o(l, t) = c_K,o^R, and c_Cl,o(0, t) = c_Cl,o(l, t) = 1. For the membrane potential v and the intracellular ions, the boundary conditions to be imposed depend upon the nature of the end constraints. From cable theory, the boundary conditions for the membrane potential v can be set up as

(11)

where dv/dx, v, and e_K are evaluated at the boundaries, _L and _R are the dimensionless endfoot K⁺ conductivities at x = 0 and x = l, respectively, and e_K(0) and e_K(l) denote the values of e_K at x = 0 and l, respectively. If _L or _R is nonzero, the corresponding end is also referred to as an open-end boundary. Otherwise it is called a sealed-end boundary. For c_K,i, the intracellular electrochemical K⁺ flux at the boundaries must be equal to the transmembrane K⁺ flux through endfeet, i.e.,

(12)

Considering [Cl]_i, we note that the glial endfeet do not possess particularly large Cl conductance; therefore,

(13)

Interpretation of dimensionless parameters

Obviously, the two most important dimensionless parameters are  and . The space constant  estimates how far the glial membrane depolarization can spread axially. Therefore,  is defined in terms of the membrane conductivity in the nonendfoot area. There are two alternative interpretations of the increase in the dimensionless length l: it can mean either an actual increase in the glial geometric length or a reduction in the space constant . The parameter  represents the relative strength of the SB-mediated K⁺ transport compared to the ECS diffusion, measured over a distance of several space constants. The total SB current will be i_m multiplied by . However, changing the glial surface area density a or the specific membrane conductivity g_K^o does not affect  because this change will be canceled out by the corresponding modification in ². Changing  without affecting the dimensionless length l is possible only through the effective ECS diffusion coefficient, the ECS volume fraction, or the baseline [K⁺]_o level. Therefore, a small ECS volume fraction or a low resting [K⁺]_o can boost the SB influence relative to interstitial diffusion. Similarly, a high resting intracellular K⁺ concentration minimizes the relative fluctuations of [K⁺]_i and hence augments the effectiveness of SB. The low [K⁺]_o in the ECS and the high [K⁺]_i in glial cells and the selective K⁺ permeability make glial cells perfectly qualified for the task of SB (Amédée et al., 1997). The resistance ratios _o and _i indicate the ease with which ionic movements can be induced in each compartment. In addition to diffusion gradients, ionic flux is driven in part by electrical potentials, represented by the terms 3) and 4) in the above dimensionless governing equations. These current-induced ionic movements can also be expressed in the form of ionic transport numbers (Gardner-Medwin, 1983b).

The parameters _L, _R indicate the relative weighting of the membrane conductance at the endfeet to those on the cylindrical cable surface of a unit space constant. For a cylindrical glial cell shape (e.g., the Müller cells in retina), assuming the cell diameter to be d and _i = 1, the specific area density a can be written as (d/d²/4)_i. Then _L (or _R) can be shown to be equal to [(d²/4)g_K,L^o]/[(d)g_K^o] = (/)(G_Kef^o/G_K^o), where G_Kef^o and G_K^o are the total membrane K⁺ conductances in the endfoot and nonendfoot areas, respectively. Assuming of Müller cells to be approximately the same length as  and adopting the known ratio of G_Kef^o/G_K^o  20 (Newman, 1985), _L (or _R) can be as large as 20.

It should be noted that the dimensionless time t and the spatial axis x are defined relative to the space constant , a parameter that encapsulates the membrane properties of glial cells. This way of defining t and x may make it difficult to compare the distributions of ionic concentrations and membrane potentials in absolute units, but it does make the comparisons in dimensionless forms more meaningful. Very often one is interested in comparing how the [K⁺]_o distribution changes subject to changes in a specific membrane property, say, the glial membrane conductivity g_K^o or the intracellular resistivity r_i. Changes in these parameters automatically modify  and therefore affect the dimensionless t and x. Assume now two cases with the same but different space constants: one is , regarded as a control, and the other is 2. If we extend the domain of the case with 2 to twice its original length, 2, the two cases will have the same relative decay property for the membrane potential (i.e., we keep the dimensionless x fixed). In evaluating the response to the increased domain length, we must compare the [K⁺]_o distribution curve at the time 4 (i.e., we keep the dimensionless time t fixed) to the corresponding curve of the control case with  and at time . Without SB, the two cases should produce exactly the same curve when the results are displayed on the relative scales. If SB comes into play, however, the comparisons and evaluation of the buffering effect in this dimensionless situation are more objective (because we have excluded the possible influences from diffusion and the potential decay in the equivalent cable). Further comments on the dimensionless scaling, when results are compared in relative scales, can be found in Gardner-Medwin (1983b).

Membrane potential distribution

If we restrict the problem to a constant membrane conductance, it is possible to obtain an explicit expression for the membrane potential distribution v(x) in terms of an arbitrary Nernst potential distribution e_K(x). Although the effect of Kir will be investigated numerically later, to obtain analytical results, we first consider the case of constant conductivity with f_Kir = 1. A constant f_Kir has been used previously by other investigators (Immel and Steinberg, 1986; Newman and Odette, 1984; Odette and Newman, 1988; Eberhardt and Reichenbach, 1987) studying the clearance of K⁺. Because we aim to exploit analytical properties and features of SB, employing a constant g_K is sufficient. Letting f_Kir = 1, the general solution of Eq. 8, subject to the prescribed boundary conditions in Eq. 11, is given by

(14)

where the Green's function G(x, ), in which x is the location of a pulse perturbation and  is an independent spatial variable, is defined to be
In Eq. 14, the second term only exists when I is nonzero, and the third term only exists when the open-end boundary condition is used. Equation 14 indicates that the potential v(x) at x is not solely a function of local properties but is also influenced by the extracellular K⁺ disturbance far away, with the influence decaying exponentially in terms of some space constant. This solution agrees with the statement made by Gardner-Medwin (1983b) that the membrane potential v in glial cells will be determined not only by the local extracellular K⁺ concentrations, but also by those in neighboring regions. The shape of G(x, ) may be better understood graphically in Fig. 2, where the Green's function distributions along  at different locations of x are depicted. The distribution shows exponential decays on both sides of the pulse location x. When we consider the endfeet, the influence of a nonzero _L or _R penetrates from the boundaries to the inner domain, also exponentially. If l is large enough, the shape of G in the inner domain remains unaffected. This exponential decay in G and the fact that the influence of any perturbations cannot extend beyond a few space constants is characteristic of the passive cable.

View larger version (23K): [in this window] [in a new window]   FIGURE 2   Distributions of the Green's function G(x, ) along  at various pulse locations of x. G(x, ) is viewed as a weighting function centered on the K+ pulse at x for determining the membrane potential distribution. Generally, the distribution G decays exponentially along both sides of the pulse. When the endfoot exists, with elevated conductance, the boundary condition for the membrane potential will be modified, which will also affect the G distribution for the pulse close to the endfeet.

Simplified linear analyses

We shall only consider the situation of a domain bounded by two parallel surfaces; practically this can be thought of as a typical brain slice. Assume that a slice of brain tissue with a thickness l is initially perfused in a physiological saline containing K⁺ at normal physiological concentration (~3 mM for mammalian brains) at both sides under resting conditions. At time zero, the K⁺ concentrations of the perfusates at the two sides are set at two different fixed levels. We are interested in the time courses of various ionic concentrations and membrane potential distributions brought about by the interplay of diffusion and SB. We have already simplified the problem by assuming zero rectification (f_Kir = 1) and _o = _i = 0 to eliminate their associated nonlinear terms. For an analytic solution to be feasible, we limit ourselves to the situation of small K⁺ disturbances, i.e., c_K,o  1, in the ECS such that we can ignore the uptake and linearly expand the logarithmic Nernst potential. It is true that active and passive K⁺ uptake into glial cells has been shown to occur during neuronal activity (Ballanyi et al., 1987; Coles and Orkand, 1983; Coles and Tsacopoulos, 1979). However, assuming the [K⁺]_o fluctuation is only small and recognizing that [K⁺]_i is much higher than [K⁺]_o, it should be a good enough approximation to ignore the uptake by assuming that c_K,i = 1 and c_Cl,o = 1. In reality, the extracellular K⁺ level can be as high as c_K,o  4 (because the ceiling level [K⁺]_o  12 mM; Syková, 1983) during intense neuronal activity, although this is not commonly seen. This ratio can go even higher under pathological conditions (Syková, 1983). Thus the assumption of a constant c_K,i and a linear expansion of the logarithmic term is not always valid. However, the analytical solutions, realizable only after the linear form of ln c_K,o  c_K,o  1 is adopted, while failing to be quantitatively correct, can still provide useful qualitative information about the behavior of [K⁺]_o and SB currents in space and time, and so offer valuable insights into the mechanism of SB. Most importantly, it allows us to arrive at linear governing equations that can be solved analytically. To summarize, we give analytical solutions of the following simplified linear equations:

(15)

(16)

in which the nondimensional logarithmic Nernst potential, ln c_K,o, is replaced by c_K,o  1, and only the SB term and the diffusion term (terms i and ii in Eq. 9a) are considered. Comparisons of results that take into account varying c_K,i, c_Cl,i, Kir, and the external current I will be studied numerically, using the fully coupled nonlinear equations, later in the paper.

Profiles at steady states

It is possible to achieve a steady state that differs from a uniform distribution as long as the perfusates on both sides act as a stable K⁺ source and sink, respectively. Upon letting the left-hand side of Eq. 16 be zero, the steady-state profiles, v* and c^*_K,o, of Eqs. 15 and 16 are found to be

(17a)

and

(17b)

in which c_K,o  c_K,o^L  c_K,o^R, and u denotes . ₁, ₂, and ₃ denote, respectively,


Both solutions consist of two components: the linear component representing the diffusion, and the exponential component representing the effect of SB from the glial core conductor. Because the only source and sink in the domain are assumed to be located at the boundaries, the exponential terms decay toward the inner region. However, if any source/sink appears within the domain, one should expect a corresponding exponential term appearing at the location of each source/sink in the solutions, as justified from the distribution of Green's function shown previously. If   0, ₂ and ₃ go to zero but ₁ remains finite, reducing c^*_K,o to its linear form with the slope c_K,o/l, characteristic of diffusion. If  is large, the linear slopes of both c^*_K,o and v* gradually decrease, resulting in the formation of thin boundary layers for c^*_K,o at both ends.

The steady-state [K⁺]_o distributions, in terms of the dimensionless Nernst potential, together with the membrane potential profiles at various values of , for the sealed-end case (_L = _R = 0), are shown in Fig. 3. The profiles of the dimensionless Nernst potential e^*_K (= ln c^*_K,o = c^*_K,o  1, for the simplified case) and the glial membrane potential v* differ substantially only in the regions near the boundaries, meaning that the transmembrane current i_m is nearly zero in the interior and SB action mainly takes place in the proximity of the source and the sink. This trend is generally valid regardless of the inclusion of the Kir properties or the varying [K⁺]_i. In the interior region, the buffering term is negligible because of the small potential driving force, and thus diffusion dominates, resulting in a linear distribution. At the outer regions, enhanced diffusion and SB activity balance each other. As  increases, the SB driving force also increases, as indicated by the more significant potential difference in ln c^*_K,o and v*.

View larger version (20K): [in this window] [in a new window]   FIGURE 3   Analytical solutions from the simplified linear model (Eqs. 15 and 16): steady-state distributions for the dimensionless Nernst equilibrium potential e*K (= ln c*K,o) (solid lines) and the glial membrane potential (dashed lines) in a finite domain (l = 10). The K+ source is located at x = 0 and the sink at x = l. Because in the simplified linear model the e*K is approximated by cK,o  1, the solid lines can also represent the ECS K+ distributions. As the SB strength indicator  increases, more K+ in the ECS is driven into glial cells at the left end (provided by the source at x = 0), and an equivalent amount of intracellular K+ is driven into the ECS at the right end, which then diffuses toward the sink. The transmembrane SB currents at both boundary regions are perpendicular to the axial direction of glial cells. At the center, the diffusion in the ECS and the electrotonic K+ movement in the intracellular space are parallel to the x axis. The consequence is that the K+ distribution in the ECS still remains linear in the center (but with a reduced slope) and forms two sharp boundary regions with steep concentration gradients close to both ends.

The effect of the relative domain length l on the distributions of ln c^*_K,o and v* is shown in Fig. 4. Because the dimensionless l is made relative to , a change in the dimensionless l can mean either a change in the geometric dimension representing the glial cells (or the syncytium) or the glial electrotonic space length , due to changes in the membrane K⁺ conductance. Thus it is expected that the characteristic thickness of the boundary layers will be independent of the dimensionless l as long as l is far greater than 1. Fig. 4 demonstrates this point by showing that increasing l mainly elongates the interior region, where v* approaches the Nernst equilibrium potential as l increases.

View larger version (23K): [in this window] [in a new window]   FIGURE 4   Analytical solutions from the simplified linear model (Eqs. 15 and 16): steady-state distributions for the dimensionless Nernst equilibrium potential e*K (= ln c*K,o) (solid lines) and the glial membrane potential (dashed lines) at different geometric lengths. Even though  is large, when a low-resistance endfoot (characterized by a high L) is added at the left end, most K+ is redirected to enter glial cells through this endfoot at x = 0, thus reducing the transmembrane current in the nearby nonendfoot area. The diffusion gradient at x = 0 therefore becomes smaller. This results in a higher K+ content in the ECS and a higher membrane depolarization. Both diffusion and endfoot current in the left region are parallel to the x axis. Intracellular K+ is forced to enter the ECS at the right end, because of the lack of endfeet in that region, and then diffuses toward the sink at x = l.

Fig. 4 also compares the influence of the endfoot conductance on the membrane potential distributions. When the open-end boundary condition with a finite endfoot conductance is used, the endfoot provides an additional pathway for SB current. This makes the corresponding v* and ln c^*_K,o in the vicinity of the endfoot conform more completely with each other, thus reducing the transmembrane i_m in the nearby nonendfoot area. It is shown in Fig. 4 that adding an endfoot at the left-end source allows the membrane potential v* to depolarize more toward its Nernst potential e^*_K,o by directing most of the transmembrane K⁺ current through the left endfoot. Comparing the curves of the same l with and without an endfoot at x = 0, we also see that adding an endfoot facing the source (x = 0) brings in more K⁺ from the source, thus increasing the [K⁺]_o content in the tissue. If the endfoot faces the sink (at x = l) instead, more K⁺ will be driven out through this endfoot, thus increasing the depletion of the [K⁺]_o content. This dependence of SB performance on the endfoot conductance will be further illustrated later.

The K⁺ diffusion flux entering/leaving the boundary surface via the ECS pathway at either end can be evaluated by dc^*_K,o/dx as _K,oo²/(_K,o^o D_K) = (c_K,o/l)Y, with Y given by

(18)

If   0, we have Y = 1. Then the overall K⁺ flux through the ECS reduces to the expected expression c_K,o/l for pure diffusion, implying that the ratio of the diffusion fluxes with and without SB is simply Y.

The relationship of Y to l, , and endfeet is shown in Fig. 5. Without endfeet (_L = _R = 0; Fig. 5 a), the diffusion flux ratio Y at the boundary increases with ; this increase becomes more significant at a larger l. If an endfoot exists at x = 0 (_L = 5, but _R = 0), the diffusion ratio Y at the same side as the endfoot shows a decreasing trend with  at l  1 (Fig. 5 b), which forms a minimum ECS diffusion ratio along l. This is because at l  1 the maximum SB current through the endfoot is achieved, which reduces the corresponding ECS diffusion rate to a minimum. Fig. 5 c ( = 5, _R = 0) shows that the ECS diffusion ratio Y measured at x = 0 decreases when the endfoot conductivity at the same side (