Our results indicate that neocortical white matter axons can be divided into three populations: unmyelinated, myelinated, and giant myelinated fibers wider than approximately 2 μm. As brain diameter increases, myelinated and giant myelinated fibers become more prevalent. Unmyelinated, high-capacitance fibers decrease in abundance in larger brains, consistent with the lower metabolic rates seen in a variety of tissues as animals become larger (West et al., 1997
). Among myelinated fibers, the largest axons scale up in size with brain diameter, indicating that these very fast axons maintain short cross-brain conduction times. The variation seen among myelinated axons suggests that white matter is composed of populations separable not only by size but also by functional role.
Because they have very slow conduction velocities, unmyelinated axons are poorly suited to convey information in the form of precise spike timing. Transmission speeds can increase or decrease by up to 20% if an axon has fired recently (Swadlow, 2000
), presumably because of changes in the activation/inactivation state of voltage-gated channels or accumulation of intracellular sodium. For axons with long conduction times, changes in transmission speed introduce potentially large amounts of timing jitter. For instance, an action potential traveling at less than 0.3 m/s across a 15 cm human brain would take over half a second to arrive at its destination, with variation in arrival time of up to 100 ms. In contrast, arrival times in myelinated axons would vary at most by a few milliseconds. Thus unmyelinated long-distance neocortical axons may encode signals in the form of firing rates and brief bursts rather than precisely timed spikes, and represent a case in which a premium is placed on volume minimization with the accompanying penalties of higher metabolic rate and slower speed.
At the other size extreme, large myelinated axons are likely to play functional roles requiring speed or precise timing. These axons constitute only a small fraction of the total number present in white matter, suggesting that they constitute highly specific highways for neural activity. Their low capacitance and intrinsically precise timing allows them to transmit information with much less energy expenditure than unmyelinated axons. Larger axons may also be able to support more extensive terminal arborizations, thus making more synapses and enhancing their influence in long-distance communication. In the case of callosal axons, which originate from neurons in both supragranular and infragranular layers, communication over rapidly conducting axons may be necessary to synchronize activity between distant cortical areas (Swadlow, 2000
). Computational models (Bush and Sejnowski, 1996
) suggest that during oscillatory activity, the generation of synchronous firing requires the participation of axons that have conduction delays of less than 5 ms. However, the great majority of midline-crossing axons have much longer conduction times (Ringo et al., 1994
). Thus, although large axons are costly to build, they may be necessary for synchrony at the level of the whole neocortex (Varela et al., 2001
Another possible role for fast transmission is the preservation of precise spike timing. In the optic nerve, submillisecond conduction times preserve the timing of action potentials relative to one another as they propagate down wide axons. Similarly, the extrastriate areas MT and MST, which are thought to help process moving visual stimuli, are heavily myelinated. MT receives a population of very large (2–3 μm) axons originating from primary visual cortex (Rockland, 2002
) with exceptionally short conduction times of 1–2 ms (Movshon and Newsome, 1996
), which would minimize the timing jitter in action potentials. Fast conduction all the way from the retina through MT would allow spike timing to convey differences in stimulus timing in a Reichardt-style motion detector (DeFelipe et al., 2002
Application of our measurements to neocortical white matter scaling shows a possible means of filling gaps in several theoretical analyses based on wiring architecture (Zhang and Sejnowski, 2000
; Changizi, 2001
; Harrison et al., 2002
). Zhang and Sejnowski (2000)
used the principle of wire length minimization to account for the disproportionate increase in white matter volume in large brains. However, their extrapolation from wiring length to wiring volume required the assumption that the mean axon cross-sectional area is constant, since unlike idealized wires, axons have nonzero thickness and therefore occupy an amount of space that cannot be specified by length-minimization arguments alone. Since the mean axon cross-sectional area varies by at least eight-fold, this particular scenario is invalid. Replacement of another assumption by Zhang and Sejnowski, that the neuron density per unit area of neocortical sheet is constant (Rockel et al., 1980
), a claim that is superseded by stereologically more rigorous measurements (Changizi, 2001
; Tower, 1954
) suggests that per-surface-area neuron density in fact declines with brain size by the same factor as the increase in axon cross-section. Our measurements are consistent with this revised assumption (Changizi, 2001
; Harrison et al., 2002
Although the distribution of firing rates among neocortical neurons is not known (Shoham et al. 2006
), the ratio of the predicted per-action-potential and observed per-gram energy consumption corresponds to a supportable mean firing rate of ~14 Hz if all metabolic energy is converted to action potential firing. However, a substantial fraction of the brain’s metabolism continues in the absence of electrical activity; the rate of glucose consumption drops by no more than half under conditions such as coma, barbiturate anesthesia, and ouabain blockade of sodium pump activity (reviewed in Sokoloff, 1996
). A more conservative assumption, that half the metabolic energy in white matter is available for generating action potentials, indicates that the maximum supportable firing rate, averaged across all neurons, is 7 ± 2 Hz for brain sizes ranging from shrew to macaque. This estimated maximum rate carries many uncertainties due to the assumptions necessary in making the calculation, and should be directly testable by future cellular-level physiological investigations in awake, behaving animals (Dombeck et al., 2007
The availability of enough metabolic energy to support an estimated firing rate of 7 Hz is broadly consistent with the mean rate of 4 Hz assumed by Attwell and Laughlin (2001)
in their estimates of the energetic costs of signaling processes in rat neocortical gray matter. Another calculation based on metabolic rate has suggested the possibility that in human neocortex the mean firing rate might be less than 1 Hz (Lennie, 2003). However, that calculation used mouse anatomical parameters for axonal and dendritic structure. Both our current findings and previous observations (reviewed in Wittenberg and Wang, 2007) illustrate that interspecies variations in neuron structure are considerable and must be taken into account in energetic calculations. Therefore the calculation of energetic costs in human neocortical tissue is a topic worth revisiting. Energetic cost breakdowns in white matter are of particular interest since, compared with gray matter, calculation is simplified by the white matter’s predominantly axonal composition. Such a calculation will require high-quality ultrastructural measurements. Since human brain metabolism is somewhat higher than expected from simple extrapolation of the scaling law seen in smaller brains (see Supplementary Information
), it is likely that human neocortical white matter is capable of supporting firing rates equal to or higher than 7 Hz.
Unmyelinated axons should consume 2.5 to 10-fold more energy per spike than myelinated axons. However, this concept is contradicted by a previous claim. Ritchie (1967)
suggested that the per-fiber cost of firing was similar for unmyelinated and myelinated axons. He arrived at this conclusion by comparing per-axon oxygen consumption for C fibers, which are unmyelinated, with an estimated value for myelinated axons obtained by dividing the oxygen consumption of whole sciatic nerve (Brink et al., 1952
) by the total number of axons (Dunn, 1909
; Gasser and Erlanger, 1927
). However, anatomical work since that time has established that two-thirds of the axons in adult sciatic nerve are unmyelinated (for instance see Jenq et al. 1986). Thus, in both C fibers and sciatic nerve, oxygen consumption is likely to be dominated by contributions from unmyelinated axons, leaving the claim unsupported.
The parameters of speed, timing, and energetic cost can be combined to estimate the cost of transmitting information. Although an exact calculation of information would require experimental knowledge that is currently unavailable, such as the distribution of firing rates and patterns among neocortical axons, an upper bound can be estimated. The maximum possible amount of information carried by spike timing can be expressed in terms of the bit content of a single action potential (Brenner et al. 2000
). For a mean firing rate of r a single action potential arriving at a random time during an interval Δt can carry up to log2
(1/rΔt) bits of information. For example, if the interval Δt represents variability in transit time and is ±10% of the total conduction time, in a 1 cm-long axon with a mean firing rate of 7 Hz the maximum information content is between 4 and 10 bits per spike depending on the axon diameter, with the most information per spike per axon carried by the widest myelinated axons.
If transmission time is variable, as has been observed (Swadlow et al., 1980
; Swadlow, 2000
), timing precision should decrease with increasing transmission distance, thus leading to the loss of information. For a 10-fold longer conduction distance the maximum information content decreases by log2
(10)=3.3 bits per spike. Thus the amount of spike timing-based information potentially lost over long-distance transmission is a substantial fraction of the total information. In unmyelinated axons the fractional loss of timing-based information is particularly great, favoring the transmission of information via changes in firing rate (e.g. C fibers; Basbaum and Jessell, 2000
). Finally, in all cases energetic costs scale up linearly with conduction distance. Taking these factors into account, sending information costs considerably more energy in a large brain than in a small brain.
We suggest that neocortical white matter architecture must follow a limited energy budget within which to optimize both volume (Hsu et al., 1998
; Chklovskii, 2004
) and conduction time (Ringo et al., 1989). Our work suggests that the mean firing rate of long distance-projecting neocortical pyramidal neurons may be relatively unvarying across species. Moreover, within a species, white matter metabolic rates at other neocortical locations appear to be similar: among neocortical regions that include both callosal and noncallosal tissue, the range of observed values is approximately 1.2-fold in mouse (Nowaczyk and Des Rosiers, 1981
), in rat (Sokoloff et al., 1977
; Collins et al., 1987
), and in macaque (Kennedy et al., 1978
; Shapiro et al., 1978
). Metabolic rate may be limited by the energy-supplying capacity of the circulatory network (West et al., 1997
). If this is so, then the mean firing rate and the distribution of axon sizes are strongly linked to one another by a general constraint, the metabolic rate. In this hypothesis, higher firing rates are achievable in white matter, but only if a structure contains fewer unmyelinated axons than what we report here for the corpus callosum. Conversely, predominantly unmyelinated structures might only be able to support low-frequency and/or sparse firing. In this context, our measurements provide a view of neocortical architecture that is centered on metabolic constraints on axon composition. Further characterization of the sizes, origins, and targets of neocortical white matter axons will help in understanding system-wide dynamics of neural activity.