How the optic nerve allocates space, energy capacity, and information

Abstract

Fiber tracts should use space and energy efficiently, because both resources constrain neural computation. We found for a myelinated tract (optic nerve) that astrocytes use nearly 30% of the space and >70% of the mitochondria, establishing the significance of astrocytes for the brain's space and energy budgets. Axons are mostly thin with a skewed distribution peaking at 0.7 microm, near the lower limit set by channel noise. This distribution is matched closely by the distribution of mean firing rates measured under naturalistic conditions, suggesting that firing rate increases proportionally with axon diameter. In axons thicker than 0.7 microm, mitochondria occupy a constant fraction of axonal volume--thus, mitochondrial volumes rise as the diameter squared. These results imply a law of diminishing returns: twice the information rate requires more than twice the space and energy capacity. We conclude that the optic nerve conserves space and energy by sending most information at low rates over fine axons with small terminal arbors and sending some information at higher rates over thicker axons with larger terminal arbors but only where more bits per second are needed for a specific purpose. Thicker axons seem to be needed, not for their greater conduction velocity (nor other intrinsic electrophysiological purpose), but instead to support larger terminal arbors and more active zones that transfer information synaptically at higher rates.

Figure 1.

Retinal ganglion cell axons are mostly thin. A, Myelinated axons in the optic nerve vary in diameter by ∼10-fold and are separated from each other by astrocyte processes (electron micrograph). Boxed region is enlarged in B. B, Higher magnification shows mitochondria (mit) in axons and astrocyte processes (a). C, Allocation of space in the optic nerve: 28% to astrocytes. D, Optic axons reconstructed from inner diameters. Over this length scale (12 μm), each axon varies markedly in caliber (max/min = 2.0 ± 0.6; n = 1200). Arrows mark constrictions. None of these constrictions were nodes of Ranvier. E, Distribution of diameters is skewed with thin axons predominating. Shaded area includes 95% of the total and corresponds to the range (0.5–1.5 μm) where probability values were >10% of the peak. Solid line is a lognormal fit. Inset, Distribution of diameters along the reconstructed segments (D) for a subset of axons with mean diameter 0.55 μm (n = 1100) and 1.55 μm (n = 500). Solid lines are Gaussian fits.

Figure 2.

Energy capacity distributes evenly along the axon. A, Mitochondria distribute evenly along the axon [except at constrictions (B) and nodes of Ranvier (Fig. 3A)]. Mean distance from ATP source to axonal membrane (site of Na/K-ATPase) is <1 μm. B, Mitochondria avoid the narrowest constrictions (<0.3 μm) but maintain a constant concentration in axons whose narrowest regions ≥0.6 μm. Dashed line indicates mean mitochondrial concentration across all axons. C, EM section through a mitochondrion. Inner membrane forms numerous tubules that expand its surface area. Inner membrane cut perpendicularly forms a sharp image in this section (∼90 nm thick), and these regions were traced to obtain a measure of inner membrane area. D, Mitochondrial volume correlates strongly with inner membrane area [which sets the number of respiratory units and thus energy capacity (Weibel, 2000)].

Figure 3.

Mitochondria avoid nodes of Ranvier. A, Longitudinal section through a node of Ranvier. B, Mitochondria are present in astrocytes, but they do not cluster at the node (nr). This point is quantified in D. C, Mitochondria are rare in nodal axoplasm but reach the mean level (−) within ∼4 μm. D, A “node-centered average” of astrocytic mitochondria was obtained by aligning the three-dimensional mitochondrial distribution, around 51 nodes, and collapsing them to one dimension. Thus, astrocytic mitochondria avoid the nodes of Ranvier. E, Sodium diffuses longitudinally from its entry point at the node, falling by 90% within a millisecond (inset).

Figure 4.

Energy capacity versus axon caliber. A, Mitochondria in myelinated axons thicker than ∼0.7 μm occupy ∼1.5% of the cytoplasm—independently of axon diameter. Profiles thinner than 0.7 μm had lower mitochondrial concentrations. Mitochondria in unmyelinated segments (within the retina) that are thicker than ∼0.6 μm occupy ∼4% of cytoplasm. Horizontal error bars indicate SD for axon diameter (adaptive binning; see Materials and Methods); vertical bars indicate SEM. B, Mitochondrial volume per unit axonal length rises linearly with axon diameter for small profiles (d <0.7 μm) and quadratically for larger axons. Solid curves indicate quadratic polynomial fits.

Figure 5.

Unmyelinated axons and retinal glia. A, Ganglion cell axons within the retina are unmyelinated and run in dense bundles toward the optic disk. The bundles are wrapped by processes of Muller glia. gc, Ganglion cell. B, Higher magnification of boxed region in A. Retinal axons contain mitochondria (mit), but the Muller processes do not. Instead, they are dark from accumulated glycogen. C, Distribution of diameters for unmyelinated segments is skewed like the distribution for myelinated segments (Fig. 1E). Shaded area (95%) corresponds to the range (0.3–1.1 μm) where probability values were >10% of the peak. Solid line, lognormal fit. D, Subcellular distribution of mitochondria seen as red fluorescence after staining with TMRE. Mitochondria are numerous in ganglion cell somas (gc) distinguished by their eccentrically located nuclei (n) (Kao and Sterling, 2006) and in ganglion cell dendrites and axon bundles (ab). Muller cell processes are dark, as expected from electron microscopy (A). dpa, Displaced amacrine cell.

Figure 6.

Retinal ganglion cells mostly fire at low rates. A, Multiple ganglion cells responding to naturalistic movies were recorded simultaneously on a multielectrode array (shown here are 2 examples). The smallest cells (local-edge) had the lowest mean and peak rates, whereas largest cells (brisk-transient) had the highest rates (Koch et al., 2006). B, Distribution of firing rates shows mostly low rates and skew toward high rates. Solid line indicates lognormal fit. C, Distribution of firing rates compared with distribution of axon diameters from Figure 1 by assuming a linear relation between rate and diameter. The match seems close, especially considering that the sample sizes differ by two orders of magnitude.

Figure 7.

Metabolic cost of spiking. A, Sodium entry for one action potential based on capacitance calculations and a compartmental model (see Materials and Methods; see supplemental material, available at www.jneurosci.org). Sodium entry per spike for myelinated axons is constant but for unmyelinated axons rises with axon diameter. B, Energy capacity per spike, calculated by dividing the mitochondrial volume for each axon diameter (Fig. 4B) by the corresponding estimated firing rate [Fig. 6C or r = 10 (d − 0.46) for myelinated axons, and r = 13.4 (d − 0.32) for unmyelinated axons]. C, Information rises more slowly than energy capacity, giving a law of diminishing returns. Information rate was calculated from firing rates associated with different axon calibers (Koch et al., 2006).

Figure 8.

Larger axons are not required for sodium buffering. A, Compartmental model correctly reproduces the action potential shape and conduction delays between successive nodes of Ranvier. B, Model reproduces the known relationships between axon diameter and conduction velocity for myelinated and unmyelinated axons. C, Model shows that a train of spikes in a fine, myelinated axon (d = 0.2 μm) raises the local concentration of sodium (gray line). However, when the model allows diffusion to the internodes (τ ∼ 1 ms), sodium does not accumulate (black line).

Figure 9.

The distribution of axon diameters in conserved across species. A, The distribution of optic nerve axon diameters is conserved, with mostly small diameters, peaking near 0.7 μm and skewed. This is despite a 10-fold difference in conduction distance. In fact, human, with the longest conduction distance, has the finest fibers. Most axons in the human optic nerve derive from midget ganglion cells with input from a single cone. Thus, these cells should have low-firing and low-information rates consistent with their fine caliber. B, Optimizing information with respect to energy cost can select the most common axon diameter. The total power associated to a unit length of axon was assumed to be E = V_m + C, where V_m is the mitochondrial volume per unit length (Fig. 4B) and C is an additional constant cost, expressed in the same units, that is independent of firing rate. Bits per energy was computed by dividing the information rate associated to an axon of a given diameter by E (Materials and Methods). The constant C was adjusted to give an extremum at 0.7 μm.