A tridomain model for potassium clearance in optic nerve of Necturus

Abstract

Complex fluids flow in complex ways in complex structures. Transport of water and various organic and inorganic molecules in the central nervous system are important in a wide range of biological and medical processes. However, the exact driving mechanisms are often not known. In this work, we investigate flows induced by action potentials in an optic nerve as a prototype of the central nervous system. Different from traditional fluid dynamics problems, flows in biological tissues such as the central nervous system are coupled with ion transport. They are driven by osmosis created by concentration gradient of ionic solutions, which in turn influence the transport of ions. Our mathematical model is based on the known structural and biophysical properties of the experimental system used by the Harvard group Orkand et al. Asymptotic analysis and numerical computation show the significant role of water in convective ion transport. The full model (including water) and the electrodiffusion model (excluding water) are compared in detail to reveal an interesting interplay between water and ion transport. In the full model, convection due to water flow dominates inside the glial domain. This water flow in the glia contributes significantly to the spatial buffering of potassium in the extracellular space. Convection in the extracellular domain does not contribute significantly to spatial buffering. Electrodiffusion is the dominant mechanism for flows confined to the extracellular domain.

Figure 1

Optic nerve structure. (a) Key features of the optic nerve region and subarachnoid space (SAS); (b) longitudinal section of the optic nerve; (c) cross section of the optic nerve. To see this figure in color, go online.

Figure 2

Domain of the axial symmetry model. The optic nerve Ω_OP consist of axon compartment Ω_ax, glial compartment Ω_gl, and extracellular space ${\text{Ω}}_{ex}^{OP}$. The subarachnoid space only has extracellular space ${\text{Ω}}_{ex}^{SAS}$. R_a = 48 μm is the radius of optic nerve, and R_b = 60 μm is the radius from optical nerve center to the dura mater. To see this figure in color, go online.

Figure 3

Flowchart for simulation process.

Figure 4

Recording axon membrane potential, glial membrane potential, and extracellular K⁺ at center axis point (where r = 0 and z = L/2) when using the extracellular solution with 3 mM K⁺. To see this figure in color, go online.

Figure 5

Comparison between the experiment in (8) and simulation on the effect of nerve impulses on the membrane potential of glial cells. The solid symbols are resting potentials, and the open symbols are depolarization potentials with different K⁺ concentrations in ${\text{Ω}}_{ex}^{OP}$. To see this figure in color, go online.

Figure 6

(a–d) Potassium flux through M_S, E_T, M_NS, and G_T during a train of axon firing. (e–h) Cumulative potassium flux during axon firing period [0, T_sti]. To see this figure in color, go online.

Figure 7

Spatial distribution of potassium changes from the resting state. To see this figure in color, go online.

Figure 8

Spatial distribution of potassium changes from the resting state during and after a train of stimuli. To see this figure in color, go online.

Figure 9

(a–d) Average water velocity through M_S, E_T, M_NS, and G_T during a train of axon firing period [0, T_sti]. (e and f) The extracellular volume fraction variation in the stimulated region and nonstimulated region. To see this figure in color, go online.

Figure 10

(a) Schematic graph of the potassium flux when axon is stimulated. In the stimulated region, the potassium moves through the extracellular pathway and through the glial compartment by way of the glial membrane. In the nonstimulated region, the potassium leaks out to the extracellular space through the glial membrane. (b) Schematic graph of the water circulation when the inner part of the axon is stimulated. In the stimulated region, the glial transmembrane water flow goes from extracellular space into the glial compartment as the effect of osmosis difference. In the extracellular space, water goes from the nonstimulated region to the stimulated region in the radial direction. In the glial compartment, it goes in the opposite direction. Note that these graphs summarize outputs of large numbers of calculations solving partial differential equations in longitudinal and radial spatial directions and time. They do not represent a compartmental model. They are the output of a model distributed in space. To see this figure in color, go online.

Figure 11

(a–d) Potassium flux through M_S, E_T, M_NS, and G_T after a train of action potentials. (e–h) Cumulative potassium flux after axon firing. To see this figure in color, go online.

Figure 12

(a–d) Average water velocity through M_S, E_T, M_NS, and G_T after a train of axon firing. To see this figure in color, go online.

Figure 13

(a) Schematic graph of the potassium flux after the axon was stimulated. The potassium flux leaks into the glial compartment from the extracellular space through the glial membrane in both the stimulated and unstimulated regions. The potassium flux in the extracellular space and glial compartment is negligible. (b) Schematic graph of the water flux after the axon was stimulated. Note these graphs are graphs that summarize outputs of large numbers of calculations solving partial differential equations in longitudinal and radial spatial directions and time. They do not represent a compartmental model. They are the output of a model distributed in space. To see this figure in color, go online.

Figure 14

(a and b) Potassium concentration variation in the extracellular stimulated region and nonstimulated region. To see this figure in color, go online.

Figure 15

The stimulated radial segments in each case. The intervals with value 1 are stimulated segments, and the intervals with value 0 are unstimulated segments. To see this figure in color, go online.

Figure 16

(a–d) The comparison between the spatially random stimulated case with the spatially uniform radial (inner) case during a train of axon firing. (e and f) The cumulative flux comparison in M_S, E_T, M_NS, and G_T during a train of axon action potentials. To see this figure in color, go online.

Figure 17

(a–d) The comparison between the spatially random case with the uniform inner radial case after a train of action potentials. Vector directions are as defined previously. (e and f) The cumulative flux comparison in M_S, E_T, M_NS, and G_T after a train of action potentials. To see this figure in color, go online.

Figure 18

(a and b) Variation of potassium concentration in the extracellular simulated region and unstimulated regions. (c and d) Average glial compartment radial absolute velocity and extracellular space radial absolute velocity. To see this figure in color, go online.

Figure 19

(a–d) The cumulative flux comparison in M_S, E_T, M_NS, and G_T during a train of action potentials. (e–h) The cumulative flux comparison in M_S, E_T, M_NS, and G_T after a train of action potentials. To see this figure in color, go online.

Figure 20

(a and b) Extracellular potassium concentration variation comparison between the model with NKCC and baseline model (without NKCC). (c) Average potassium variation in the axon stimulated region. To see this figure in color, go online.

Figure 21

(a and b) Extracellular potassium concentration variation comparison between the model with nonselective pathway on pia boundary and baseline model (without nonselective pathway). (c) Total cumulative potassium flux through glial membrane. (d) Total cumulative potassium flux through pia boundary. To see this figure in color, go online.