What signaling pathways mediate responses of cortical axons to Wnt5a?Wnts exert their effects through both Ryk and Frizzled (Fz) receptors (Salinas and Zou, 2008) which are expressed on developing hamster cortical neurons (Li et al., 2009) at P0–P3 when axons are projecting into corticospinal and callosal pathways (Reh and Kalil, 1981; Norris and Kalil, 1992). Early postnatal mouse cerebral cortex also expresses seven different Fz receptors (Shimogori et al., 2004). In Fz3 knockout mice cortical axon tracts including the corpus callosum and corticospinal pathway failed to develop (Wang et al., 2002) providing the first evidence that Fz signaling plays an important role in cortical axon development. We blocked or knocked down Ryk receptors which prevented increased cortical axon outgrowth by Wnt5a (Li et al., 2009). In contrast, blocking Wnt/Fz interactions (Rodriguez et al., 2005), did not prevent increased axon outgrowth (**Figure 5**). These results showed that Wnt5a increases in axon outgrowth are mediated by Ryk but not Fz receptors whereas knocking down Ryk and blocking Fz receptors prevents Wnt5a repulsive turning (**Figure 5**). Thus both Ryk and Fz receptors mediate repulsive cortical axon guidance by Wnt5a. Fz receptors are involved in Wnt/calcium signaling during morphogenic events and in cell motility (Kohn and Moon, 2005) but the involvement of Wnt/calcium signaling in axon growth and guidance are unknown. Further, the downstream components of Ryk signaling have not been identified. Wnt5a induced global or local Ca**2**+ transients involving both Ryk and Fz receptor types (Li et al., 2009). The source of intracellular calcium is important for its function (Henley and Poo, 2004; Gomez and Zheng, 2006) and Ca**2**+ release from intracellular stores regulates axon growth and guidance (Ooashi et al., 2005; Jacques-Fricke et al., 2006). To determine the role of this Ca**2**+ signaling mechanism in effects of Wnt5a we blocked IP3 receptors on the endoplasmic reticulum which blocked calcium release from intracellular stores. This partially attenuated Wnt5a induced Ca**2**+ activity. Calcium influx through TRP (transient receptor potential) channels also plays an important role during axon growth and guidance (Li et al., 2005; Shim et al., 2005; Wang and Poo, 2005). We therefore blocked TRP channels, which also partially inhibited Wnt5a induced calcium activity. Simultaneously blocking calcium release from stores and calcium influx through TRP channels virtually silenced Wnt5a induced calcium activity. Blocking Ca**2**+ release from stores prevented Wnt5a induced increase in axon outgrowth but the same axons showed repulsive turning away from the source of Wnt5a (Li et al., 2009). In contrast, blocking TRP channels prevented both increased axon outgrowth and repulsion by Wnt5a. As in netrin-1 calcium signaling (Tang and Kalil, 2005) CaMKII is an essential downstream component of Wnt5a/calcium signaling. These results suggest a model (**Figure 6**) in which increased axon outgrowth induced by Wnt5a involves activation of Ryk receptors, calcium release from intracellular stores and activation of CaMKII signaling. Growth cone repulsion by a gradient of Wnt5a is mediated by both Ryk and Fz receptors and requires calcium entry through TRP channels but not calcium release from intracellular stores. In support of our results that different calcium

signaling mechanisms (perhaps in different regions of the growth cone) mediateWnt5a axon growth vs. guidance behaviors, a recent study (Nicol et al., 2011) showed that in *Xenopus* spinal neurons different calcium and cyclic AMP signaling localized to central vs. filopodial growth cone regions mediate netrin-1 evoked axon outgrowth vs. steering. Thus Wnt5a can simultaneously activate different receptors and different calcium signaling pathways at the growth cone to elicit growth and guidance behaviors required for wiring of cortical circuits during development.

#### **AXON GROWTH AND GUIDANCE IN THE CORPUS CALLOSUM IS REGULATED BY Wnt/CALCIUM SIGNALING**

Thus far we have discussed mechanisms of axon growth and guidance in dissociated cortical neurons. However, it is important to test the relevance of these mechanisms in a more complex cellular environment that mimics development *in vivo.* For example, in dissociated cultures knocking down CaMKI decreased hippocampal and cortical axon elongation (Ageta-Ishihara et al., 2009; Davare et al., 2009; Neal et al., 2010) whereas knockdown of CaMKI *in vivo* during activity-dependent cortical wiring decreased callosal axon branching into cortex without affecting rates of callosal axon elongation (Ageta-Ishihara et al., 2009). To test the *in vivo* function of Wnt/calcium signaling mechanisms we used living slices of developing hamster sensorimotor cortex that contained the entire callosal pathway (Hutchins et al., 2011). This permitted live cell imaging of intact callosal axons extending along their entire trajectory (Halloran and Kalil, 1994). We electroporated plasmids, encoding the cytoplasmic fluorescent marker DsRed2, into one hemisphere of cortical slices from P0 brains and after 48 h used confocal microscopy to image fluorescently labeled callosal axons (**Figure 7**), which exhibited continual growth cone motility and forward advance during the 90-min imaging sessions. Importantly, axons before the midline (precrossing) advanced more slowly than those that have crossed the midline (postcrossing). To determine whether callosal axons expressed Ca**2**+ transients correlated with rates of

axon outgrowth we electroporated a genetically encoded calcium indicator (GCaMP2) into one hemisphere and measured frequencies of Ca**2**+ transients in pre- and post-crossing axons for up to 30 min. (**Figure 8**). Calcium transients were expressed in axons and growth cones (**Figure 8**) in frequencies closely related to rates of callosal axon outgrowth. Thus higher frequencies of Ca**2**+

**(B)** Tracings of trajectories of P0 axons exposed to Wnt5a gradients following Ryk knockdown, which prevents increased axon outgrowth as well as repulsion by Wnt5a. **(C)** Bar graphs quantifying turning angles (top) and outgrowth rates (bottom) of axons in **(A)** and **(B)**. Scale bar, 5μm. In all histograms \*p < 0.05, \*\*p < 0.01; n.s., not significant. (Reprinted from Li et al., 2009; with permission from the Society for Neuroscience).

transients are well correlated with higher rates of callosal axon outgrowth, similar to our results for dissociated cortical neurons (Hutchins and Kalil, 2008).

The corpus callosum is surrounded by a Wnt5a gradient secreted by midline glial structures (Keeble et al., 2006). Since callosal axons express Ca**2**+ transients correlating with rates of axon outgrowth we reasoned that in the corpus callosum the Wnt5a gradient could mediate axon growth and guidance by calcium signaling mechanisms. To determine whether Ca**2**+ release from

**FIGURE 6 | Schematic summary showingWnt receptors and calcium signaling components involved in cortical growth cone behaviors in response toWnt5a.** At left, increased cortical axon outgrowth is induced by Wnt5a, which involves activation of Ryk receptors, calcium entry through TRP channels, calcium release from intracellular stores via IP3 receptors on the endoplasmic reticulum, and activation of PLC and CaMKII signaling. At right, growth cone repulsion by a Wnt5a gradient is mediated by both Ryk and Fz receptors and requires calcium entry through TRP channels. Symbols in the growth cone are shown in the box. (Reprinted from Li et al., 2009; with permission from the Society for Neuroscience).

**FIGURE 7 | Imaging of individual callosal axons and their growth cones extending through the corpus callosum. (A)** A low power confocal image of a cortical slice at 3DIV, after electroporation of cortical neurons with DsRed2 in the slice from P0 sensorimotor hamster cortex. Individual efferent axons are clearly visible. Arrow indicates location of

the cortical growth cone imaged at higher power in the time lapse sequence in **(B)** which shows filopodia and lamellipodia of growth cones during outgrowth and turning. Times are in minutes and seconds. Scale bar, 10μm. Symbols show reference points. (Reprinted from Hutchins et al., 2011).

the growth cone measured in **(B,C)** at time points indicated by the arrowheads in **(C)**. **(E)** Within 20 min of the onset of calcium activity shown in **(B)** the axon begins to advance rapidly through the contralateral callosum. **(F)** Examples of single calcium transients measured by ratiometric imaging of

SKF. Scale bars 10μm except **(I)** which is 5μm. Pseudocolor calibration bars indicate fluorescence intensity **(D)** or ratio of GCaMP2 to DsRed2 fluorescence intensities **(F)** in arbitrary units. (Reprinted from Hutchins et al., 2011).

stores plays a role in regulating callosal axon outgrowth we blocked IP3 receptors (on the endoplasmic reticulum) pharmacologically to inhibit calcium release from stores. Using live cell imaging, we measured growth of postcrossing axons since a recent study

(Wang et al., 2007) showed that *in vivo* suppression of spontaneous electrical activity in mouse callosal axons decreased rates of axon outgrowth on the postcrossing but not the precrossing side of the callosum. Treatment with blockers to IP3 receptors slowed axon outgrowth of postcrossing axons but their normal rates were restored after washout of the blocking reagent. However, blocking Ca**2**+ releasefrom stores had no effect on guidance of callosal axons whose trajectories (measured from images obtained at the beginning and end of the 3-h experiments) after crossing the midline were similar to those of untreated controls. In contrast, pharmacologically blocking Ca**2**+ entry through TRP channels not only slowed axon outgrowth but also severely misrouted postcrossing callosal axons, which turned prematurely toward the cortical plate or turned ventrally toward the septum. We verified the involvement of calcium activity by blocking TRP channels and then measuring effects on Ca**2**+ transients in individual postcrossing callosal axons. As shown in **Figure 8I**, blocking TRP channels attenuated frequencies of Ca**2**+ transients to about 3 h compared with 12 h for controls, showing directly that the growth and guidance defects resulted from decreased Ca**2**+ activity. Furthermore, as we found in dissociated cultures, calcium signaling by store release or by Ca**2**+ entry through TRP channels respectively mediates axon outgrowth or both axon outgrowth and guidance in the corpus callosum.

Although these results demonstrate the requirement for calcium signaling in outgrowth and guidance of callosal axons, it was important to show the specific involvement of Wnt5a signaling. *In vivo* knockout mice lacking the Wnt5a Ryk receptor showed guidance errors in callosal axons only after they crossed the midline (Keeble et al., 2006). This is consistent with the finding that mouse callosal axons do not respond to Wnt5a until the age when they cross the midline (E 16) and begin to express the Ryk receptor (Keeble et al., 2006). However, these experiments did not address signaling mechanisms downstream of Ryk receptors. Therefore, we knocked down the Ryk receptor by electroporation of Ryk siRNA in a small number of axons and then analyzed effects on DsRed2 labeled callosal axons in living cortical slices (Hutchins et al., 2011). Ryk knockdown slowed the growth rate of postcrossing callosal axons to half their normal growth rate, but had no effect on precrossing axon growth rates, consistent with Ryk expression on callosal axons only after they cross the midline (Keeble et al., 2006). Importantly, Ryk knockdown also caused severe guidance errors in about a third of labeled postcrossing axons that deviated inappropriately toward the septum or prematurely turned toward the overlying cortex. Since we found previously (Li et al., 2009) that knocking down the Ryk receptor reduced the proportion of neurons expressing Ca**2**+ transients in response to Wnt5a, we asked whether growth and guidance defects in callosal axons in which Ryk was knocked down resulted from interference with Wnt5a evoked calcium signaling. To address this question we coelectroporated the calcium biosensor GCaMP2 with Ryk siRNA to measure calcium activity in neurons in which RyK/Wnt signaling was disrupted. In such neurons frequencies of Ca**2**+ transients were reduced fourfold in postcrossing axons. Remarkably, in growth cones projecting aberrantly toward the septum, Ca**2**+ transients were undetectable. Taken together, these results suggest that callosal axon growth and guidance errors caused by Ryk knockdown result from attenuation of calcium activity. Since CaMKII is also a component of the Wnt/calcium signaling pathway (Li et al., 2009), we investigated its role in callosal axon growth and guidance by cortical transfection of plasmids encoding an inhibitory

CaMKII protein (Tang and Kalil, 2005). For postcrossing but not precrossing axons, this treatment slowed axon outgrowth by about half and caused dramatic axon guidance errors in which some axons projected aberrantly while others looped backward in the corpus callosum. Do these defects represent a failure of axon repulsion mechanisms mediated by Wnt5a? Using turning assays in which dissociated neurons were exposed to Wnt5a gradients, we found that neurons transfected with the CaMKII inhibitor were unable to increase their growth rates or show repulsive turning. These results suggest that CaMKII, downstream of Wnt/calcium signaling, is necessary for the outgrowth and repulsive guidance mechanisms by which Wnt5a regulates development of callosal axons.

Taken together these results, summarized in a cortical slice model of the developing corpus callosum (**Figure 9**), show that Wnt/calcium signaling pathways are essential for regulating callosal axon outgrowth and guidance. Nevertheless, additional growth and guidance factors are known to regulate development of the corpus callosum (Lindwall et al., 2007). For example, in addition toWnt5a, sources of netrins, semaphorins, and Slit 2 surround the corpus callosum and their role in callosal axon guidance across the midline has been well characterized (Shu and Richards, 2001; Shu et al., 2003; Niquille et al., 2009; Piper et al., 2009). Although direct evidence for netrins in guiding callosal axons to the midline is lacking, callosal axons express the netrin-1 receptor DCC and netrin-1 knockout mice lack all forebrain commissural projections (Serafini et al., 1996). Slits are chemorepellent proteins expressed in glial structures adjacent to the callosal midline in a pattern similar to that of Wnt5a. Knockout mice lacking the Slit receptor Robo 1 display aberrant projections of callosal axons that are misdirected at the midline, suggesting that Slits may channel callosal axons into the correct pathway (Lindwall et al., 2007). Mutant mice whose axons are unable to bind semaphorins also show callosal malformations. However, our finding that inhibiting calcium signaling only affected growth of axons after but not before the midline suggests the involvement of Wnt5a signaling because cortical axons do not respond to Wnt5a until the age at which they cross the midline (Keeble et al., 2006). Although Slit 2 affects both pre- and post-crossing axons (Shu and Richards, 2001; Shu et al., 2003) our results show that interfering withWnt/calcium/CaMKII signaling only affects postcrossing axon outgrowth.

#### **MICROTUBULE DYNAMICS REGULATE Wnt5a EVOKED AXON OUTGROWTH AND GUIDANCE**

Growth cone behaviors are regulated by reorganization and dynamics of the actin and microtubule (MT) cytoskeleton (Dent and Gertler, 2003; Lowery and Van Vactor, 2009), which can be modulated by extracellular guidance cues (Luo, 2002; Huber et al., 2003; Kalil and Dent, 2005). Stable microtubules occupy the central region of the growth cone while dynamic MTs that grow and shrink in dynamic instability actively explore the periphery (Dent et al., 1999; Dent and Kalil, 2001; Schaefer et al., 2002, 2008; Lowery and Van Vactor, 2009). Inhibiting MT dynamics can abolish growth cone turning and localized MT stabilization and de-stabilization can induce attractive turning and repulsion (Buck and Zheng, 2002), suggesting an instructive role for MTs in axon guidance. Wnt signaling increases MT stability and Wnt3a

the contralateral callosum have high frequencies of calcium transients evoked by Wnt5a signaling, which accelerates rates of axon outgrowth. In contrast, ipsilateral axons not yet responsive to Wnt5a, have low levels of calcium activity and thus grow more slowly. LV, lateral ventricle; CP,

release of calcium from intracellular stores and calcium influx through the plasma membrane. These calcium transients activate CaMKII to accelerate axon extension and repel growth cones away from the glial wedge toward the contralateral cortex. (Reprinted from Hutchins et al., 2011).

and Wnt7a specifically decrease axon length and increase growth cone size of DRG and cerebellar mossy fiber terminals (Lucas and Salinas, 1997; Hall et al., 2000) by reorganization of MTs into stable loops (Ciani and Salinas, 2005; Salinas, 2007; Purro et al., 2008). This occurs through changes in MT directionality across the growth cone rather than toward the leading edge (Purro et al., 2008). In contrast we showed that Wnt5a increases cortical axon outgrowth and induces growth cone repulsion (Li et al., 2009) but the cytoskeletal mechanisms are unknown. Our preliminary results (Li et al., unpublished observations) suggest that reorganization of dynamic MTs plays a major role in increased axon outgrowth and repulsive growth cone turning induced by Wnt5a.

We first confirmed that dynamic MTs are required for the growth promoting and guidance effects of Wnt5a by applying nocodazole, an inhibitor of MT dynamics (Dent and Kalil, 2001), which prevented Wnt5a induced increases in axon outgrowth and repulsive turning in Wnt5a gradients. To determine the relationship between directionality of MTs and the direction of growth cone advance induced by Wnt5a, we used high resolution fluorescence (total internal reflection fluorescence, TIRF) microscopy to image growth cones of early postnatal cortical neurons transfected with an EGFP-EB3 construct to label dynamic MT plus ends and DsRed2 to label the cytoplasm. EB3 is a +TIP protein that binds to the growing plus ends of MTs (Nakagawa et al., 2000) and EGFP-EB3 has been used to image MT dynamics in neurons (Stepanova et al., 2003). Dynamic growing MT tips were visualized as EB3 "comets"in cortical growth cones before and after bath application of Wnt5a. Wnt5a oriented dynamic MTs in the direction of axon elongation, suggesting that increased axon outgrowth by Wnt5a occurs by reorientation of dynamic MTs. We then applied Wnt5a as a gradient to investigate the possible redistribution of dynamic MTs in growth cones during repulsive turning behaviors. Preliminary results (**Figure 10**) showed that before application of Wnt5a

**FIGURE 10 |Wnt5a gradients induce redistribution and reorganization of dynamic microtubules (MTs) in growth cones. (A)** Schematic of measurement of the distribution of EB3 comets on MT plus ends in the presence of a pipette gradient of Wnt5a locally applied at the right side of the growth cone. Individual EB3 comets are detected by fluorescence (green dots) and their center of intensity (cross) is determined. **(B)** Localization of the center of intensity of EB3 comets evoked by a Wnt5a gradient in 10 growth cones in independent experiments including the growth cone shown in **(C,D)**. Negative numbers indicate EB3 localization to the side of the growth cone farthest from the Wnt5a source. Change of the center of intensity = −0.35 ± 0.14μm (p < 0.05, Wilcoxon signed rank test). **(C,D)** Composite images of the same growth cone showing redistribution of EB3 comets over time 30 min before and during 30 min of application of the Wnt5a gradient (n = 601 images before Wnt5a and 601 images during Wnt5a application). The maximum intensity was computed for each pixel from the entire image stack by using algorithms in Metamorph. Using these values stacks of all images were superimposed and pixels assigned values in pseudocolor to indicate fluorescence intensity.

EB3 comets were randomly distributed throughout the growth cone but during 30 min exposure of growth cones to Wnt5a gradients EB3 comets redistributed to the far side of the growth cone away from the Wnt5a source. These results suggest that in Wnt5a gradients asymmetric redistribution of dynamic MTs may be a first step in repulsive turning.

Gradients of guidance cues can evoke asymmetries in calcium activity (Gomez and Zheng, 2006) that are typically highest on the side of the growth cone facing a source of either an attractive or repulsive guidance cue gradient. Thus it was surprising to find that Wnt5a can evoke Ca**2**+ transients localized to the far side of the growth cone facing away from the Wnt5a source. The coincidence of MT and Ca**2**+ asymmetries led us to investigate a possible relationship between asymmetries in Ca**2**+ transients and the redistribution of dynamic MTs in growth cones exposed to Wnt5a gradients. One candidate link is the MT associated protein tau which binds to MTs to increase their stability (Dehmelt and Halpain, 2005). CaMKII phosphorylates tau at the MT binding site (Litersky et al., 1996; Sironi et al., 1998). This leads to detachment of tau from MTs and thus decreased MT stability and increased MT dynamics (Biernat et al., 1993). Furthermore, Wnt5a is known to activate CaMKII (Kuhl et al., 2000). Together these findings led us to investigate tau as a possible link between calcium signaling and dynamic MTs. To determine whether tau is required for the growth promoting effects of Wnt5a on cortical axons, we knocked down tau with siRNAs which prevented Wnt5a from increasing rates of cortical axon outgrowth. Importantly, this treatment also prevented growth cone repulsion in Wnt5a gradients. We then co-transfected cortical neurons with EGFP-EB3 and tau siRNA, and imaged dynamic MTs by following EB3 comets in growth cones. Remarkably, we found that tau knockdown prevented the redistribution of dynamic MTs toward the far side of the growth cone facing away from theWnt5a gradient. As shown in **Figure 11** dynamic MTs remain randomly distributed throughout

the growth cones as in untreated controls. Since calcium activity is required for the growth and guidance effects of Wnt5a on cortical axons *in vitro* (Li et al., 2009) and within the corpus callosum (Hutchins et al., 2011) these results suggest a relationship, possibly mediated by tau, between Wnt5a evoked calcium signaling and the redistribution of dynamic MTs during Wnt5a evoked cortical growth cone repulsion.

#### **FUTURE DIRECTIONS**

This review has emphasized the central role of intracellular calcium signaling in transducing the effects of guidance molecules on growth and guidance of cortical axons during development. However, many questions remain unanswered. First, the mechanisms by which guidance cues evoke calcium transients localized to small regions of axons and growth cones are not well understood. Second, how competitive mechanisms, mediated by calcium signaling, favor the growth of one cortical axon process over another requires further study. Third, the relationship between calcium activity and cytoskeletal dynamics leading to cortical axon branching, outgrowth and guidance remains obscure. Fourth, effects of guidance cues on cytoskeletal reorganization leading to specific growth cone behaviors are only beginning to be addressed. Finally, it is not always clear that axon growth and guidance mechanisms identified in tissue culture are applicable to development *in vivo*. Thus a major challenge will be to elucidate how axons of neurons in the complex *in vivo* environment of the cerebral cortex can integrate multiple extracellular guidance cues and intracellular signaling pathways necessary for appropriate axon outgrowth, guidance, and branching during wiring of cortical circuits.

#### **ACKNOWLEDGMENTS**

The work from our laboratory was funded by National Institutes of Health Grant NSO14428 and grants from the Whitehall

**FIGURE 11 | The microtubule associated protein tau is required forWnt5a induced redistribution of dynamic MTs in growth cones. (A,B)** Composite images showing the redistribution of EB3 comets in the same growth cone over time 30 min before and during 30 min of application of the Wnt5a gradient (n = 361 images before Wnt5a and 601 images during Wnt5a application). The maximum intensity was computed for each pixel from the entire image stack by using algorithms in Metamorph. Using these values stacks of all images were superimposed and pixels assigned values in pseudocolor to indicate fluorescence intensity. Wnt5a was locally applied as a

gradient at the right side of the growth cone. In this growth cone tau was knocked down with siRNA, which prevented the asymmetric redistribution of EB3 comets during exposure of the growth cone to Wnt5a gradients. **(C)** Localization of the center of intensity of EB3 comets evoked by Wnt5a gradients in four growth cones in independent experiments including the growth cone shown in **(A,B)**. Quantitative analysis shows that Wnt5a fails to induce redistribution of EB3 comets to the far side of the growth cone after tau knockdown with siRNA (change in center of intensity not different from 0. p > 0.05, Wilcoxon signed rank test).

Hutchins). Bruce Ian Hutchins is currently supported by the Intramural Research Program of NINDS and the Pharmacology

Foundation and the University of Wisconsin Graduate School (to Katherine Kalil), a Herman Shapiro fellowship (to Li Li), and National Research Service Award GM801642 (to Bruce Ian

**REFERENCES**

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in the rostral CNS. *J. Neurosci.* 22, 8563–8573.

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**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 20 June 2011; accepted: 08 September 2011; published online: 28 September 2011.*

*Citation: Kalil K, Li L and Hutchins BI (2011) Signaling mechanisms in cortical axon growth, guidance, and branching. Front. Neuroanat. 5:62. doi: 10.3389/fnana.2011.00062*

*Copyright © 2011 Kalil, Li and Hutchins. This is an open-access article subject to a non-exclusive license between the authors and Frontiers Media SA, which permits use, distribution and reproduction in other forums, providedthe original authors and source are credited and other Frontiers conditions are complied with.*

## Wiring of divergent networks in the central auditory system

#### *Charles C. Lee1\*†, Amar U. Kishan2† and Jeffery A.Winer3*

1 Department of Comparative Biomedical Sciences, School of Veterinary Medicine, Louisiana State University, Baton Rouge, LA, USA

2 Harvard Medical School, Boston, MA, USA

3 Division of Neurobiology, Department of Molecular and Cell Biology, University of California, Berkeley, CA, USA

#### *Edited by:*

Julian Budd, University of Sussex, UK

*Reviewed by:* Sarah L. Pallas, Georgia State University, USA

Hisayuki Ojima, Tokyo Medical and Dental University, Japan

#### *\*Correspondence:*

Charles C. Lee, Department of Comparative Biomedical Sciences, School of Veterinary Medicine, Louisiana State University, Skip Bertman Drive, Baton Rouge, LA 70803, USA. e-mail<EMAIL_ADDRESS>
†Charles C. Lee and Amar U. Kishan have contributed equally to this work. Divergent axonal projections are found throughout the central auditory system. Here, we evaluate these branched projections in terms of their types, distribution, and putative physiological roles. In general, three patterns of axon collateralization are found: intricate local branching, long-distance collaterals, and branched axons (BAs) involved in feedback-control loops. Local collaterals in the auditory cortex may be involved in local processing and modulation of neuronal firing, while long-range collaterals are optimized for wide-dissemination of information. Rarely do axons branch to both ascending and descending targets. Branched projections to two or more widely separated nuclei or areas are numerically sparse but widespread. Finally, branching to contralateral targets is evident at multiple levels of the auditory pathway and may enhance binaural computations for sound localization. These patterns of axonal branching are comparable to those observed in other modalities. We conclude that the operations served by BAs are area- and nucleusspecific and may complement the divergent unbranched projections of local neuronal populations.

**Keywords: branched axon, auditory system, collaterals, cortical, thalamocortical, brainstem**

#### **INTRODUCTION**

A cardinal feature of axons is their divergent projections, which range from sparse branching in the thalamic input to different auditory cortex (AC) areas (Morel and Imig, 1987; Lee et al., 2004a; Kishan et al., 2008) to the many collaterals and thousand of boutons of single spiral Ia cochlear ganglion axons (Brown, 1981). Branched axons (BAs) are present throughout the auditory system (Fekete et al., 1984; Willard and Martin, 1984; Ojima, 1994; Hazama et al., 2004; Coomes et al., 2005; Kimura et al., 2005; Lee and Winer, 2008a,b,c) and can take many forms, from local (Brown et al., 1988a,b) to very distant (Hashikawa et al., 1995; Cetas et al., 1999; Huang and Winer, 2000), presumably allowing neurons to synchronize remote events or form multiple feature-specific representations.

Different patterns of axonal branching prevail at different levels of the auditory system (**Figures 1**–**3**). For instance, branching between different nuclei is common in the pathways to and from the medial nucleus of the trapezoid body (MTB; Morest, 1968; Spirou et al., 1990; Kuwabara and Zook, 1991, 1992; Kuwabara et al., 1991; Smith et al., 1991), while thalamocortical axons rarely project to different cortical fields, such as the primary auditory cortex (AI) and the anterior auditory field (AAF; Morel and Imig, 1987; Lee et al., 2004a,b). Other axons have both descending and ascending projections, e.g., from MTB cell axons projecting to the cochlear nucleus (CoN) and the inferior colliculus (IC), <1% project to both (Schofield, 1994).

In discussing the wide variety of branching patterns present in the auditory system, it is imperative to acknowledge that various methods allow the detection of different patterns of axonal branching, and that these different methods have inherent limitations in terms of the conclusions that can be drawn from their use. Thus, we review the technical considerations inherent in assessing axonal branching. An especially important caveat to establish at the outset, however, is that dual retrograde injections can only ascertain axonal branching to the specific regions within the nuclei injected; conclusions cannot be drawn about other forms of axonal branching from these studies. Nonetheless, the use of dual retrograde tracing has been useful in formulating hypotheses about neural function.

**Abbreviations:** AAF, anterior auditory field; AC, auditory cortex; AI, primary auditory area; AII, second auditory cortex; AVCN, anteroventral cochlear nucleus; BA, branched axon; CF, characteristic frequency; CN, central nucleus of the IC; CoN, cochlear nucleus; CT, corticothalamic; CTβ, cholera toxin β fragment; CTβG, cholera toxin β fragment, gold conjugated; DCoN, dorsal CoN; DL, double-labeled neuron; DLL, dorsal nucleus of the lateral lemniscus; DlP, dorsolateral periolivary area; DmP, DMPO, dorsomedial periolivary area; DZ, dorsal auditory zone; ED, dorsal posterior ectosylvian area; EE, excitatory–excitatory band; EI, excitatory–inhibitory response band; EI, intermediate posterior ectosylvian area; EV, ventral posterior ectosylvian area; IC, inferior colliculus; II, V, VI, auditory cortex layers; In, insular cortex; IL, intermediate nucleus of the lateral lemniscus; LA, lateral amygdaloid nucleus; La, lateral nucleus of the IC; LOC,lateral olivocochlear neurons; LSO,lateral superior olive; LT, LTB, lateral nucleus of the trapezoid body; MG, medial geniculate body; MGBd, MGd, dorsal division of the MG; MGBv, MGv, ventral division of the MGv; MGm, medial division of the MG; MOC, medial olivocochlear system; MSO, medial superior olive; MTB, medial nucleus of the trapezoid body; NA, nucleus angularis; NL, nucleus laminaris; NM, nucleus magnocellularis; P, posterior auditory area; PDL, percentage of double-labeled neurons; PIN, posterior intralaminar nucleus; PON, periolivary nuclei; PRh, perirhinal area; PVCN, posteroventral cochlear nucleus; RC, radiate multipolar cell; RP, rostral pole of the MG; SC, superior colliculus; SOC, superior olivary complex; SPN, superior paraolivary nucleus; TC, thalamocortical; Te, temporal cortex; Te3, third area of temporal cortex; TRN, thalamic reticular nucleus; VCN, ventral CoN; Ve, ventral auditory area; VIII, auditory nerve; VNLL, VL, ventral nucleus of the lateral lemniscus; VPO, ventral periolivary nucleus; VTB, ventral nucleus of the trapezoid body.

Although the functional implications of BAs are numerous (Morest, 1968; Kuwabara et al., 1991; Ojima et al., 1991, 1992; Li and Mizuno, 1997a,b; Kuwabara and Zook, 1999; Ye et al., 2000; Mulders and Robertson, 2002, 2003; Mulders et al., 2007), we are treating the function of BAs from the perspective of general organizational principles.

In the first two sections (see Branched Axons in the Auditory Cortical System, Branched Axons in the Auditory Brainstem and Midbrain), we review the existence, magnitude, and possible functions of BAs in the auditory cortex and thalamus as compared with those at earlier levels of the auditory system. These initial sections review the specifics of axonal branching in the auditory system, which the general reader may wish to skim in favor of the final sections (see Technical Considerations, Thematic Perspective, Alternatives to Collateralization in theAuditory Cortex,Collaterals in Other Modalities, and Summary), where we examine principles of axonal branching and evaluate the technical difficulties inherent in detecting BAs.

#### **BRANCHED AXONS IN THE AUDITORY CORTICAL SYSTEM THALAMOCORTICAL SYSTEM**

All regions of the auditory cortex (AC) receive an input from the thalamus (Lee andWiner, 2008a). The principal source of auditory thalamocortical (TC) input, the medial geniculate body (MG), has tonotopic ventral (MGv) and rostral pole (RP) divisions, and non-tonotopic dorsal (MGd) and medial (MGm) divisions, which project in varying degrees to each of the 13 auditory cortical (AC) areas in the cat (Huang and Winer, 2000). Although focal regions within a thalamic nucleus can project broadly to multiple cortical areas based on anterograde tracing studies (Huang and Winer, 2000), axonal divergence of single neurons beyond a few millimeters is quite rare based on retrograde double labeling studies (Kishan et al., 2008). Thus, axonal branching in the auditory thalamocortical system is highly local, but with unique topographical features.

One of these features is the patchy distribution of TC BAs, which extend over 300–500μm in layers IIIb and IV of the primate AC core (Hashikawa et al., 1995). In the lateral and posteromedial auditory cortical areas, larger (1000–1500 μm) patches arise from the MG anterodorsal and/or posterodorsal nuclei. In the rabbit, TC BAs form patches 1–2 mm apart in AI layers III and IV, with tangential layer I BAs up to 7 mm long (Cetas et al., 1999; **Figure 2A**). In the cat, similar patches are seen in AI, AAF, ventral, and the posterior AC (P) following injections of anterograde tracers into the MGv (Huang and Winer, 2000). More divergence occurs after similar MGd and MGm deposits, though not explicitly from BAs. Thick MGm axons in AC layer Ia project laterally across wide expanses, and vertical branches in layers II, IVb, and Va have fewer lateral BAs (Huang and Winer, 2000). Axons in layer IIIb also have many local BAs shorter than those in layers Ia and VIb.

The patchy distribution of MG afferents in AC may correlate with parvalbumin immunoreactivity and perhaps with modules of broadly and narrowly tuned neurons (Read et al., 2008) or binaural excitatory–excitatory/inhibitory (EE, EI) modules, though physiological–anatomical studies suggest that EE and EI columns are not linked by BAs (Middlebrooks and Zook, 1983). Similar patchy distributions in AC areas ostensibly lacking a binaural columnar arrangement imply that BAs are unrelated to binaurality. Intraareal BAs linking EE or EI columns are also sparse (Middlebrooks and Zook, 1983, but see Brandner and Redies, 1990).

Another canonical feature of the primary auditory cortical areas is the orderly spatial arrangement of neurons according to characteristic frequency (CF), i.e., tonotopy. A question that naturally arises is whether TC BAs contribute to the creation of the multiple AC CF maps (Morel and Imig, 1987) from the two representations in the MG (Imig and Morel, 1985a,b, 1985a,b)? Based on retrograde studies where different tracers are placed into matched isofrequency loci in different primary cortical areas, few double-labeled thalamic neurons are found (Morel and Imig,

1987), with some differences among the MGv and RP (Lee et al., 2004a; Kishan et al., 2008). Due to the paucity of double labeling in such studies, it appears that TC BAs do not create multiple CF maps in these areas.

Finally, the MG and intralaminar nuclei also project widely to non-auditory cortex. Thalamic BAs targeting both the lateral amygdaloid nucleus and the perirhinal or primary AC could influence autonomic and affective responses to auditory and multisensory stimuli (Namura et al.,1997). BAs may link some intralaminar nuclei with the dorsal (and, less so) ventral perirhinal cortex, and rarely arise from MGd/m neurons, though up to 17% of MGm cells project to perirhinal cortex and to the lateral amygdaloid nucleus (**Figure 2B**; **Table 1**). Although MGd cells project to both the frontal cortex and primary/non-primary AC, these originate from unbranched sources (Kurokawa and Saito, 1995). Thus, these TC parts of the auditory and motor pathways are segregated, despite extensive interdigitation of the projection cells. Overall, the few studies and diversity of relevant pathways make it difficult to specify the role of BAs in TC projections to non-auditory cortex.

#### **CORTICOCORTICAL SYSTEM**

Every area of the auditory cortex receives extensive input from local intrinsic cortical connections and extrinsic connections from other cortical areas in both hemispheres (Winer and Lee, 2007; Lee and Winer, 2008b,c), which provide ∼95% of the total input to an area (Lee et al., 2004a; Lee and Winer, 2011). As with the thalamocortical system, anterograde, axon-filling, and retrograde studies each provide complementary evidence about BAs in the corticocortical system.

On a local level, neurons in the auditory cortex branch within an area to create extensive divergent laminar circuits. In particular, layer II and III pyramidal cell axons branch proximally and distally to the cell body (Ojima et al., 1991; **Figure 2C**), forming an axonal network that extends across layers I–V, with two-tofive thick collaterals in layer III or V in addition to the main axon descending to the white matter for other cortical targets (Ojima et al., 1992). The horizontal branches in layer III or V run parallel to the pia for 500–2500 μm and emit, at a few distant points, local plexuses of secondary branches extending to upper and lower layers. This collateralization as a whole forms a columnar terminal field in layers I through V with a branchsparse gap in layer IV (**Figure 2C**). Each neuron has a number of vertical branches distributing around its cell body, forming a columnar terminal field, which is similar to that formed at distant points. Some non-projecting pyramidal neurons have thick, bifurcated axons with recurrent oblique or horizontal BAs; the latter extend 1–2 mm in layer V, and oblique branches project heavily in layers II–IV, with weaker input to layers I or II. Such cells may interact with those producing the synchronized oscillations arising in layer V (Silva et al., 1991). Several long-range dorsoventrally oriented BAs may link or segregate AI isofrequency loci in the cat (Read et al., 2001). Alternatively, they may synchronize cells with similar CF response properties, analogously to pyramidal neurons in visual cortex (Gray and Singer, 1989; Gray et al., 1989). Perhaps TC BAs complement these rich, local periodic projections.

As with the thalamocortical system, branched corticocortical projections that link similar CF regions are sparse, comprising <1% of AI and AAF cells projecting to matched CF regions (Lee et al., 2004a), although earlier studies using anterograde methods found extensive interconnections among matched CF regions (Imig and Reale,1980), perhaps accountedfor by neuronal populations that project in an unbranched manner to matched CF regions in different areas. Thus, long-range cortical BAs may be more rare than axon filling studies suggest. This implies that cortical BAs do not contribute significantly to spectral maps in different AC areas and illustrates afundamental difference between the auditory forebrain and the brainstem, where axons subdivide profusely to innervate many different targets (Irvine, 1986). Intrinsic intraareal BAs across frequency laminae are also rare (Kishan et al., 2008), but may be more prevalent along an isofrequency contour.

Commissural AI axons may also target disparate areas, with homo- and hetero-topic terminal sites; a dual retrograde study found that some rat BAs target both sites (Rüttgers et al., 1990). However, <1% of AI or AAF neurons project commissurally to frequency-matched loci in both fields, and <4% of non-primary (AII, Te, and In) neurons project to two loci in their contralateral counterparts (Kishan et al., 2008).

#### **CORTICOFUGAL PROJECTIONS**

The auditory corticofugal system targets many thalamic,midbrain, and brainstem nuclei (Winer, 2006). Of these, the corticothalamic (CT) system is massive, with each major MG division receiving input from four or more AC areas (Winer et al., 2001). Two types of terminals arise from AI: small endings from thin axons of layer VI pyramidal neurons and large boutons from thick axons of layer V pyramidal neurons (Ojima, 1994; Winer et al., 1999; Llano and Sherman, 2008). Layer VI CT neurons typically project in a feedback manner to the thalamic nucleus from which they receives their major TC input, while layer V CT neurons project in a feedforward manner to a higher order thalamic nucleus (Winer et al., 2001; Sherman and Guillery, 2006).

Layer V CT pyramidal cell targets include MGm, MGd, and ventrolateral MGv, with thick horizontal BAs occurring in cortical layers V and VI forming heterogeneous en passant and spine-like boutons, and thin vertical axons ending above layer IV (Ojima et al., 1992), and with no BAs to the contralateral AI (Wong and Kelly, 1981), reserving collateralization to the ipsilateral AC. BAs crossing the cortical CF axis may enhance inhibition at other CFs, while those parallel to the isofrequency contours could have local roles (Ojima et al., 1991; Song et al., 2006).

Layer VI CT neurons branch extensively in both thalamus and cortex. Some layer VI CT cells have recurrent branches in cortical layerVI, then ascend to layers III and IV,where their processes form a dense plexus. In the thalamus, thin fiber BAs form dorsoventrally elongated bands parallel to MGv CF laminae (Rouiller and de Ribaupierre, 1990). Layer VI CT cells may activate local columnar neurons, while layer V CT neurons target more remote columns at the same or different CF. In addition, anterograde tracer deposits at separate frequency loci in the cat label terminals segregated in the MG, suggesting that microtopography complements BAs (Takayanagi and Ojima, 2006).

Corticothalamic projections include BAs to the thalamic reticular nucleus (TRN; Lam and Sherman, 2010). Layer V or VI axons traverse the TRN (Hazama et al., 2004); forming elongated slabs; these may be BAs of cells targeting the MGv. High- and low-CF loci in rat primary and non-primary AC areas converge in the MGv and target different TRN regions (Kimura et al., 2005). The TRN has inhibitory input to much of the MG, and some TRN neurons project to both the ventrolateral MGv and MGd, or to both the MGv *pars ovoidea* and MGm (Crabtree, 1998). This branching pattern might enable two AC tonotopic areas to convergently excite one MG region via direct CT projections, while divergently inhibiting separate MG regions via indirect reticulothalamic projections (Kimura et al., 2005). The AC also targets the midbrain, medulla, and striatum (Winer, 2006), and these corticofugal cells may also have intracortical BAs. Layer V corticostriatal neurons have vertical and short-range horizontal BAs. The vertical BAs form a dense network of terminal arbors in layers III and IV, perhaps reinforcing supragranular, reciprocal connections between AC CF loci projecting to similar striatal targets.

The corticocollicular system is also a rich substrate for axonal branching (Winer et al., 1998; Winer, 2006). Rat corticocollicular cells project to the caudal striatum (Moriizumi and Hattori, 1991b), and some corticofugal cells target the superior olivary complex (SOC) and IC, or the IC and the CoN, via BAs (Doucet et al., 2002, 2003). Some corticocollicular cells send BAs to the nucleus of the brachium of the IC (Saldaña et al.,1996). Retrograde experiments indicate that ∼5% of layer V neurons project to both IC (Willard and Martin, 1984; Coomes et al., 2005). Almost half of contralaterally projecting corticocollicular cells project bilaterally. Given the conservative estimates provided by retrograde tracers, all contralaterally projecting cells may target both ICs (Coomes et al., 2005), though no neurons appear to have BAs targeting both the IC and MG (Wong and Kelly, 1981).

#### **BRANCHED AXONS IN THE AUDITORY BRAINSTEM AND MIDBRAIN**

#### **BRAINSTEM PROJECTIONS**

Now, we consider the axonal branching patterns observed in the auditory brainstem and midbrain, in comparison with those of the auditory cortical systems described previously. Do similar branching patterns and principles apply across multiple stages of the auditory pathway? The numerous connections among brainstem

and midbrain nuclei might suggest different patterns of axonal branching exist at these stages. As noted in morphological studies, auditory BAs begin in the periphery (Lorente de Nó, 1981). At the earliest levels, type I auditory nerve fibers branch extensively in the CoN (Fekete et al., 1984). One main branch targets the ventral cochlear nucleus (VCoN) and the other ends in the dorsal cochlear

nucleus (DCoN). Near this bifurcation, the parent trunk has few collaterals at low CFs, while axons at higher CFs have more numerous and complex axonal branches. Descending axons have 14–30 collaterals and, in the DCoN, the main trunk often makes parallel branches ending within 100μm. Many BAs end in simple, en passant swellings, and others terminate diffusely in the neuropil. BAs have regional morphologic variations, e.g., in the posteroventral cochlear nucleus, some have en passant swellings, while in the central part of the nucleus, fibers with a CF >4 kHz have many BAs that extend for hundreds of microns. These are parallel to octopus cell primary dendrites and could enhance the sharpness of tuning near the intensity threshold and broaden tuning at higher intensities. The ascending branch has 4–16 collaterals and ends in calyces of Held. These collaterals form complex, endbulb-like endings or en passant swellings and often remain within 100 μm of the parent branch, though one-third end in the anteroventral CoN. Small branches from high- and low-CF fiber may create heterotopic high frequency response zones in the VCoN (Fekete et al., 1984).

Cochlear nucleus afferents also branch. VCoN neurons send branches to matching frequency loci in the cat IC and contralateral DCoN (Adams, 1983a). Planar and radiate multipolar cells (Tand D-stellate cells, respectively) in the anterior VCoN branch to the DCoN and posterior VCoN, mainly to the multipolar cell area (Oertel et al., 1990). Radiate multipolar cells project to both the ipsilateral DCoN and the contralateral CoN (Doucet and Ryugo, 2006). Up to half the cells projecting to CoN also target the thalamic ventrobasal complex and may provide information about head and body position useful in sound localization or for somatic sensory–auditory interactions (Li and Mizuno, 1997a).

A prominent CoN target is the contralateral MTB (Morest, 1968),whose principal cells provide glycinergic input to the ipsilateral lateral superior olive (LSO) for interaural intensity difference computations (Smith et al., 1998). CoN projections form calyces of Held endings onMTB principal cells (Smith et al.,1991) and collateralize ipsilateral to the CoN of origin, targeting the lateral nucleus of the trapezoid body (LTB), posterior periolivary nucleus, or ventrolateral periolivary nucleus and end in large terminal swellings of variable shapes (**Figure 3B**; Spirou et al., 1990). En passant swellings are rare.

Most CoN BAs are precalycine. These traverse the MTB and ventral nucleus of the trapezoid body (VTB) toward the lateral lemniscus, forming branches in the anterolateral periolivary nucleus, the rostral LTB, and the VTB. Some fibers form collaterals at their branch point near the abducens nerve root, and branch sparsely before ending in the nucleus paragigantocellularis lateralis. Other precalycine collaterals target the dorsomedial and ventral periolivary nuclei and branch repeatedly within it (Kuwabara et al., 1991). About 40% of ipsilateral calyciferous branches end axosomatically in the ventral periolivary nucleus (VPO), 20% in the LTB and LSO, and 7% near the MTB in an area associated with the medial olivocochlear system (MOC). All axons have extensive BAs within the MTB, perhaps contributing to lateral inhibition. Other BAs end diffusely in the adjacent periolivary nuclei, the LTB, and the LSO, and 25% reach the lateral lemniscus (**Figure 3A**). Of the calycine collaterals, all terminate 20–80 μm from their origin in varicosities. Thus, ascending input to the MTB reaches parts of the ipsilateral lateral and medial olivocochlear system and diverse contralateral brain stem nuclei. MOC BAs to the CoN often converge with type II auditory nerve fiber endings (Benson and Brown, 2004), and areas targeted by such axons also project to the MOC, forming another prospective feedback-gain loop (Ye et al., 2000).

Perhaps unsurprisingly for brainstem projections, MTB axons are also collateralized (**Figure 3A**; Morest, 1968; Kuwabara et al., 1991). Principal cell axons send 2–6 BAs to the periolivary nuclei, superior paraolivary nucleus (SPN; the rodent homolog of the cat dorsomedial periolivary nucleus), and the VTB. Half of these axons also branch to the medial superior olive (MSO), and 25% branch to the lateral lemniscus. Recurrent MTB collaterals are also seen. The main axon often ends in a cascade of terminal BAs in the LSO; sometimes forming 1–2 thick perpendicular branches and then arborizing in the neuropil. MTB branches to the MSO are tonotopically organized (Smith et al., 1998).

Many brain stem neurons sample both the outputs of the MTB as well as collaterals bifurcatingfrom input to theMTB, perhapsfor monitoring or instructing gain control (Morest, 1968; Kuwabara et al., 1991). LSO-projecting neurons from the LTB also have collaterals to MSO (except in big brown bats), which, like MTB BAs, have axosomatic input on bipolar cells (Kuwabara and Zook, 1992). These inhibitory inputs may complement excitatory CoN afferents, perhaps preceding excitatory inputs because the contralateral calyciferous axons are much thicker than the CoN axons directly projecting to the contralateral MSO. Cell filling experiments in gerbil brain stem slices demonstrate that the MSO input to the SPN is highly branched, with >40% of thick, ascending MSO axons having one or more short BAs from their main trunk that ramify sparsely in the SPN (Kuwabara and Zook, 1999).

Not all brain stem projections have BAs. While some CoN efferent axons in the guinea pig target both CoN-projecting and IC-projecting cells in the SPN, their BAs may not be extensive (Schofield, 1995). Further, <1% of MTB neurons project to both the IC and CoN ipsilaterally, contralaterally, or have one ipsilateral and one contralateral target (Schofield, 1994).

#### **PROJECTIONS OF THE INFERIOR COLLICULUS**

The IC is the midbrain target for auditory input arising from earlier brainstem sources, e.g., the CoN, SOC, lateral lemniscal nuclei, AC, and many other non-auditory structures. The tonotopic central nucleus of the IC (CN) contains narrowly tuned neurons,while the cells in the dorsal cortex and lateral cortex (La) have broader frequency-tuning and multisensory properties. The IC projects to the MG, CoN, SOC, dorsal column nuclei, superior colliculi (SC), and other nuclei (for review see Winer and Schreiner, 2005).

The projection from the ventral nucleus of the lateral lemniscus to the CN has few BAs to different high- and low-frequency regions in the rat CN (Merchán and Berbel, 1996).

Such tonotopic precision is implicit in the narrow frequency tuning of anteroventral CoN cells (Bourk et al., 1981). In addition, in the rat lateral lemniscal nuclei, no neurons project to both the IC and the SC, or to both SCs, though cells in the dorsal nucleus of the lateral lemniscus may project to the SC deep layers for acoustic motor reflexes and head orientation (Tanaka et al., 1985).

The proportion of brainstem afferents that target both ICs via BAs may be species specific. In the cat IC, only 2% of LSO olivocollicular neurons project to both IC, while surprisingly, in the opossum, 20–25% of LSO olivocollicular neurons and almost all MSO olivocollicular cells project to both (Willard and Martin, 1984). Similar work in the guinea pig finds no branched projections in the LSO, MSO, or VCoN, but in the DCoN, 68% of ipsilateral

#### **Table 1 | Retrograde studies of auditory branched projections.**

#### **Table 1 | Continued**

#### **Table 1 | Continued**

#### **Table 1 | Continued**

1Originally counted as being part of the suprageniculate nucleus, which is considered as part of the MGd; c, contralateral; i, ipsilateral; 3H-BSA, tritiated bovine serum albumin; AAF, anterior auditory field; AC, auditory cortex; AI, primary auditory area; AII, second auditory cortex; Bb, bisbenzimide; BSA, bovine serum albumin; CoN, cochlear nucleus; CN, central nucleus of the IC; CTβ, β subunit of cholera toxin; CTβG, gold conjugate of CTβ; Cu, cuneate nucleus; DCoN, dorsal CN; DLPO, dorsolateral periolivary nucleus; DNLL, dorsal nucleus of the LL; DY, diamidino yellow; EB, Evans blue; EE, excitatory–excitatory band; EI, excitatory–inhibitory response band; FB, fast blue; FD, fluorescein–dextran; FG, fluorogold; FR, fluororuby; Fr1, frontal cortex; Gr, gracile nucleus; HRP, horseradish peroxidase; IC, inferior colliculus; La, lateral nucleus of the IC; In, insular cortex; LL, lateral lemniscus; LTB, lateral nucleus of the trapezoid body; LOC, lateral olivocochlear neurons; LSO, lateral superior olive; MG, medial geniculate body; MGd, dorsal division of the MG; MGm, medial division of the MG; MGv, ventral division of the MGv; MTB, medial nucleus of the trapezoid body; MOC, medial olivocochlear system; MSO, medial superior olive; NY, nuclear yellow; PI, propidium iodide; PIN, posterior intralaminar nucleus; PN, pontine nuclei; Pr, primulin; RC-multipolar, radiate multipolar cells projecting contralaterally; RP, rostral pole of the MG; SC, superior colliculus; SOC, superior olivary complex; SPFm, medial portion of the subparafascicular nucleus; SPFp, posterior portion of the subparafascicular nucleus; STN, spinal trigeminal nucleus; TB, true blue; Te, temporal cortex; Te1, primary auditory area; Te3, non-auditory temporal cortex; TMRDA, tetramethylrhodamine–dextran amine; TRN, thalamic reticular nucleus; VCN, ventral CN; VNLL, ventral nucleus of the LL.

IC-projecting cells have BAs to the contralateral IC (**Figure 3C**; Schofield and Cant, 1996b). Compared with the corticofugal system (see above), in both the guinea pig and the opossum, ∼6% of AC neurons project bilaterally to the IC.

Branched brainstem projections to the IC and other targets are also rare. In the SPN, ∼2% of IC-projecting cells branch to the CoN (**Figure 3D**; Schofield, 1991). Similarly, in the guinea pig SOC, only 1% of IC-projecting neurons send axons to the CoN (**Figure 3E**; Schofield, 2002). These few branched projections originate in the ventral periolivary region, including the anteroventral periolivary nucleus and the VTB, but no cells project to both targets contralaterally, or to one ipsilaterally – and the other contralaterally. In addition, some non-auditory afferents also have BAs (Moriizumi and Hattori, 1991a,b; Li and Mizuno, 1997a,b).

Within the IC itself, local connections are highly collateralized as revealed by intracellular filling studies in the cat (Oliver et al., 1991). These intrinsic BAs sometimes parallel the dendrites, extending for hundreds of microns (as in the CoN), while other IC neurons have non-oriented CN BAs. This diversity suggests extensive IC computational roles for local BAs and interneurons (Oliver et al., 1991). Axons of these cells extend toward the brachium of the IC, and many likely project to the MG (Winer et al., 1996). Some of these tectothalamic neurons are inhibitory (Winer et al., 1996; Peruzzi et al., 1997; Bartlett and Smith, 2002; Lee and Sherman, 2010), providing a source of feedforward inhibition that is unique to the auditory system.

However, most long-range IC projections have few BAs. Few colliculobulbar neurons target both CoNs in the guinea pig (Schofield, 2001) and rat (Okoyama et al., 2006). Instead, the IC may exert descending divergent influence disynaptically through contact with cells that projecting bilaterally to the CoN, particularly in the VTB and anteroventral periolivary nucleus (Schofield and Cant, 1999).  IC neurons branching to the MG and the contralateral IC also target the contralateral CoN, and comprise 1–10% of all tectothalamic cells (González-Hernández et al., 1991; Okoyama et al., 2006). Similarly, few axons target both the contralateral IC and the SOC or CoN.

#### **TECHNICAL CONSIDERATIONS**

Many approaches have been used to characterize BAs. Dual retrograde tract tracing (Hayes and Rustioni, 1979; Kuypers et al., 1980; Jones, 1983) can provide a profile of BA projections, as the many labeled cells permit quantitative analyses (**Table 1**). However, these studies presume equivalent uptake affinity, injection site size and efficacy, visualization methods, the interaction of damage with uptake, and transport rate (Schofield et al., 2007). To label significant numbers of cells, sufficiently large deposits can complicate the collection of quantitative data. Thus, for example, injections restricted to a single binaural response bands may be too small to label sufficient cells to provide reliable statistically appropriate estimates of double-labeled cells (DLs; Kishan et al., 2008).

Negative results are also problematic. Few DLs suggest that the injected regions do not receive BAs, though other areas might, or that the tracers were neither equivalent spatially nor equally likely to be transported. If BAs are oriented selectively, and the injections are not aligned appropriately, DL estimates would be spurious. Finally, dual retrograde tracing methods are limited since BAs to only a few sites can be detected, even if multiple targets are present. Thus, dual retrograde tracing likely underestimates the divergence of axonal projections.

In comparison, focal anterograde injections may overestimate the degree of single axon divergence by labeling fibersof-passage or filling closely apposed neurons that project to separate loci. However, both anterograde and axon filling studies can demonstrate recurrent, local, and distant BAs. Some BAs are too near their source to be detected reliably by retrograde means (Winer, 1986), and anterograde or filling approaches do not require a precise or systematic injection orientation to reveal them. Anterograde studies may not reveal the full range of targets since incomplete filling of fine or long processes or insufficient transport time may confound estimates.

Intracellular filling studies are highly constrained by sample size (Fekete et al., 1984; Ojima et al., 1991). While a few axons may have collaterals (or, alternatively, lack branches), it is uncertain whether these are representative. As in anterograde studies, incomplete staining or insufficient transport time can constrain firm conclusions or population values except when many axons can be filled and their targets visualized (Brown, 1981; Humphrey et al., 1985).

#### **THEMATIC PERSPECTIVE**

Branched axons are common in the auditory cortical system, as well as in the auditory midbrain and brainstem. However, several general principles are evident from a comparison across processing levels. First, most axons branch according to one of three patterns: intricate local BAs, long-distance collaterals, and BAs involved in feedback-control loops. Second, cells rarely project to both ascending and descending targets, suggesting that these streams are well segregated and that descending projections play specific roles rather than merely feedback or modulatory ones (Guinan, 2006; Winer, 2006). Third, some neurons have both ascending and contralateral targets, e.g., CoN neurons projecting to the IC and the contralateral CoN (Adams, 1983b), and IC neurons targeting the MG and the contralateral IC (González-Hernández et al., 1991). Most projections are contra- or ipsi-lateral because of the acoustic chiasm (Glendenning and Masterton, 1983); thus, these BAs may enhance binaural computations for sound localization or otherwise modulate ascending input from an ear with contralateral influence. This may not pertain to descending projections since corticothalamic neurons are not commissural (Wong and Kelly, 1981). Fourth, bilateral projection neurons are part of at least the corticofugal and olivocochlear streams, with ∼5% of corticocollicular neurons projecting to both IC (Willard and Martin, 1984; Coomes and Schofield, 2004), and a similar proportion of MOC cells targeting both cochleae (Thompson and Thompson, 1986; Robertson et al., 1987a,1987a,b; Aschoff and Ostwald, 1988). Such bilaterally projecting neurons in ascending pathways are differentially distributed in various nuclei.

#### **ALTERNATIVES TO COLLATERALIZATION IN THE AUDITORY CORTEX**

In the auditory cortex, one might predict that BAs would be an ideal way to create multiple independent representations of frequency, aurality, amplitopy, or other dimensions required for computation (Ehret, 1997). It is somewhat unexpected that BAs to matched frequency regions are comparatively rare, especially in the forebrain (Lee et al., 2004a), where the emergence of multiple CF maps (Reale and Imig, 1980) suggest that they might be more common.

A robust alternative mechanism is provided by heterotopic projections that arise from interleaved thalamic and cortical neurons situated in close proximity and serving presumably similar physiologic roles but whose targets are separated widely (Lee et al., 2004b; Lee and Winer, 2005). Three obvious advantages accrue to this arrangement. First, precise branching to multiple targets is unnecessary, and neurons that target multiple cortical areas can migrate as a group and assemble their connectivity with comparative ease relative to the precision required by multiple branches

that must terminate in exact register in different targets. Second, and perhaps most critically, heterotopic arrangements enable easy coordination of activity across large spatial territories, a prospectively problematic issue when coordinating diverse spatiotemporal patterns across vast expanses of brain (Lee et al., 2004b; Winer et al., 2004). Third, they provide a simple mechanism enabling the precise coordination of discharge patterns among resident thalamic or cortical neurons, either via local circuit neurons or, in their absence (Winer and Larue, 1996), via the BAs of excitatory neurons.

A second alternative is that the terminal plexus of many axons is highly divergent, and can span wide arrays, as in the TC axons in visual (Ferster and LeVay, 1978), somatic sensory (Landry and Deschênes, 1981), and auditory (Velenovsky et al., 2003) cortex. Such axons engage large areas and could readily initiate or sustain parallel intracortical and corticocortical modularity (DeFelipe et al., 1986) in networks larger than the comparatively finer scale of interneuronal projections (Kisvárday et al., 1994). The complexity of these axons belies point-to-point models of connectivity (Brandner and Redies, 1990).

#### **COLLATERALS IN OTHER MODALITIES**

Comparable, and perhaps even more extensive, collaterals systems exist in other modalities. The complexity of the subcortical auditory pathwayfrustrates direct comparisons with the visual, somatic sensory, or autonomic systems. Nonetheless, some comparisons can be drawn. For example, primary phrenic afferents send BAs to different spinal cord laminae (Goshgarian and Roubal, 1986), as do Ia muscle spindle (Brown and Fyffe, 1978), and Ib Golgi tendon organ (Brown and Fyffe, 1978) afferents. Many cuneate nucleus inputs are collateralized (Weinberg et al., 1990), resembling type I ganglion cell axons near the CoN. Retinofugal fibers to the lateral geniculate nucleus (LGN) ramify within the LGN (Conley and Fitzpatrick, 1989), resembling type I ganglion axons within the CoN.

Forebrain connections are compared more readily. The visual TC system may have more interareal BAs and intraareal BAs to matched functional domains than the somatic sensory or auditory systems. Retinotopically matched deposits in areas 17 and 18 double label 3–16% of neurons in the LGN A lamina (Bullier, 1984; Birnbacher and Albus, 1987; Salin et al., 1989), while matched somatotopic injections (SI) in the primary and secondary somatosensory areas only double label 2.3% of cells (Fisher et al., 1983).

Horizontal BAs are also present in all modalities. In the visual system, extensive lateral collaterals, similar to those seen in AI link loci with similar functional properties (Gilbert and Wiesel, 1979; Michalski et al., 1983; LeVay, 1988). There are also horizontal connections in higher-level areas such as the macaque inferior temporal cortex (Tanigawa et al., 2005), and long-range horizontal collaterals from SI pyramidal cells may target neurons in other fields (DeFelipe et al., 1986).

As in AI, some rat SI CT cells have local collaterals to neurons in the same column, while others project remotely (Zhang and Deschênes, 1997). Mirroring the absence of AI corticofugal BAs to diverse targets, <2% of SI cells have BAs to the corticostriatal, corticorubral, corticopontine, and corticospinal pathways (Akintunde and Buxton, 1992). In the somatic sensory (Bourassa et al., 1995) and visual (Bourassa and Deschênes, 1995) systems, the CT fibers arising from axons of layer V neurons in V1 or SI were collaterals of corticotectal or corticopontine axons, unlike the auditory CT system. This suggests modality specific rules for BAs, whose ontogeny and functional specificity remain for further investigation.

#### **SUMMARY**

The floridness of axonal branching throughout the central auditory system, and other modalities, is indicative of the functional importance of divergent processing in sensory systems. Such branching ranges across scales, from intrinsic branches that modulate firing in local circuits, to long-range collaterals that widely disseminate information. Yet, it remains an open question whether BAs as a wiring principle is more efficient from an ontological and developmental standpoint, compared with the targeting of separate loci by unbranched neuronal ensembles. In addition, the degree to which separate branches have similar synaptic properties and efficacy in terms of transmitting auditory information remains to be investigated. Indeed, widely varying synaptic properties at separate axonal branches would have profound effects on the divergent dissemination of auditory information. Thus, defining the functional role of axonal divergence will require a convergence of future theory and experiments.

#### **ACKNOWLEDGMENTS**

This work is dedicated to the late Jeffery A. Winer, who served as a mentor and inspirational figure for innumerable students, colleagues, and friends, and without whom this work would not have reached fruition. The void left by his absence testifies to the breadth and depth of his scholarship, friendship, and humanity. Jeff, we miss you greatly. We also thank David T. Larue for helpful discussions and assistance with the figures. This work was supported by NIH Grants R03 DC 11361 (Charles C. Lee) and R01 DC 2319 (Jeffery A. Winer).

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Winer, J. A., and Schreiner, C. E. (2005). "The central auditory system: a functional analysis," in *The Inferior Colliculus*, eds J. A. Winer and C. E. Schreiner (New York: Springer-Verlag), 1–68.

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**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 12 May 2011; accepted: 09 July 2011; published online: 28 July 2011.*

*Citation: Lee CC, Kishan AU and Winer JA (2011) Wiring of divergent networks in the central auditory system. Front. Neuroanat. 5:46. doi: 10.3389/fnana.2011.00046*

*Copyright © 2011 Lee, Kishan andWiner. This is an open-access article subject to a non-exclusive license between the authors and Frontiers Media SA, which permits use, distribution and reproduction in other forums, providedthe original authors and source are credited and other Frontiers conditions are complied with.*

## How the cortex gets its folds: an inside-out, connectivity-driven model for the scaling of mammalian cortical folding

#### *Bruno Mota1,2 and Suzana Herculano-Houzel 1,2\**

1 Instituto de Ciências Biomédicas, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil 2 Instituto Nacional de Neurociência Translacional, São Paulo, Brazil

#### *Edited by:*

Julian Budd, University of Sussex, UK

#### *Reviewed by:*

Charles F. Stevens, The Salk Institute for Biological Studies, USA David C. Van Essen, Washington University in St. Louis, USA

#### *\*Correspondence:*

Suzana Herculano-Houzel, Instituto de Ciências Biomédicas, Universidade Federal do Rio de Janeiro, Av. Brigadeiro Trompowski, s/n, Ilha do Fundão, Rio de Janeiro, Rio de Janeiro 21941-590, Brazil. e-mail<EMAIL_ADDRESS>
Larger mammalian cerebral cortices tend to have increasingly folded surfaces, often considered to result from the lateral expansion of the gray matter (GM), which, in a volume constrained by the cranium, causes mechanical compression that is relieved by inward folding of the white matter (WM), or to result from differential expansion of cortical layers. Across species, thinner cortices, presumably more pliable, would offer less resistance and hence become more folded than thicker cortices of a same size. However, such models do not acknowledge evidence in favor of a tension-based pull onto the GM from the inside, holding it in place even when the constraint imposed by the cranium is removed. Here we propose a testable, quantitative model of cortical folding driven by tension along the length of axons in the WM that assumes that connections through the WM are formed early in development, at the same time as the GM becomes folded, and considers that axonal connections through the WM generate tension that leads to inward folding of the WM surface, which pulls the GM surface inward. As an important necessary simplifying hypothesis, we assume that axons leaving or entering the WM do so approximately perpendicularly to the WM–GM interface. Cortical folding is thus driven by WM connectivity, and is a function of the fraction of cortical neurons connected through theWM, the average length, and the average cross-sectional area of the axons in the WM. Our model predicts that the different scaling of cortical folding across mammalian orders corresponds to different combinations of scaling of connectivity, axonal cross-sectional area, and tension along WM axons, instead of being a simple function of the number of GM neurons. Our model also explains variations in average cortical thickness as a result of the factors that lead to cortical folding, rather than as a determinant of folding; predicts that for a same tension, folding increases with connectivity through the WM and increased axonal cross-section; and that, for a same number of neurons, higher connectivity through the WM leads to a higher degree of folding as well as an on average thinner GM across species.

 Already in the late

eighteenth century, Franz Gall acknowledged that cortical folding, and thus the non-isometrical expansion of the cerebral cortical gray matter (GM), allowed a faster increase in number of neurons than would be granted by increased cranial volume alone if the cortical surface expanded isometrically, or simply as a balloon would (Gross, 1999). Later, His (1874) considered that cerebral shape could be explained by unequal growth, competing volume demands, and resulting tension of different brain structures.

Cortical folding has since been considered to increase with brain size as the GM expands laterally and supposedly pushes inward the underlying white matter (WM; Le Gros Clark, 1945), specially given the constraint imposed by the skull (Welker, 1990). Because the distribution of neurons beneath the cortical surface has traditionally been considered to be constant across species (Rockel et al., 1980), cortical folding would thus be a direct function of the number of neurons in the cortex. This is in line with the

**Abbreviations:** *a*, average cross-section area of myelinated axons in WM; *A*E, Exposed surface area of cerebral cortex; *A*T, Total surface area of cerebral cortex; *A*W, White–gray matter interface surface area; C, Cortical estimated computational capacity; *F*G =*A*T/*A*E, Cortical folding index; *F*W =*A*I/4π*R*W2, white matter folding index; GM, cerebral cortical gray matter; *l*, average length of myelinated axons in WM; *L* = l *n*·NN, total length of myelinated axons in WM; MG, mass of cerebral cortex GM; *M*W, mass of cerebral cortex subcortical WM; *n*, GM connectivity through the WM, defined as the fraction of neurons in GM with myelinated axon in WM; *N*, Number of neurons in GM; *O*, number of other (non-neuronal) cells in WM; *R*W(3*V* W/4π)1/3, WM "radius" (or, more properly, length scale); *T* =*V* G/*A*T, average thickness of cerebral cortex GM; *V*, volume of cerebral cortex (GM +WM); *V* G, volume of cerebral cortex gray matter (GM); *V* W, volume of cerebral cortex white matter (WM); WM, subcortical white matter.

usual expectation that elephants and large cetaceans, with larger brains than humans, have larger numbers of cortical neurons in their more folded cortices than humans do.

Contrary to these traditional views, however, we have recently shown that cortical size is not a uniform function of the number of cortical neurons across mammals. Using the isotropic fractionator to determine numbers of brain neurons (Herculano-Houzel and Lent, 2005), we find that cortical mass increases much faster in rodents than in primates as the cerebral cortex gains neurons across species (Herculano-Houzel et al., 2006, 2007, 2011; Gabi et al., 2010), which is due to an increase in average neuronal cell size in larger-brained rodents while there is barely any increase in average neuronal cell size in larger-brained primates (Herculano-Houzel, 2011). As a result, neuronal densities are larger in primate cortices than in rodent cortices of a comparable size (Herculano-Houzel, 2011). Moreover, we recently showed that the number of neurons beneath the cortical surface is not constant across primate species as previously thought (Herculano-Houzel et al., 2008), and is also not constant across rodents, actually decreasing in larger brains (Ventura, Mota, and Herculano-Houzel, to be submitted), which means that there is not a single relationship that applies between cortical surface area and number of cortical neurons across mammals. The combination of these findings leaves one no longer any reason to expect cortical folding to be a simple, homogeneous function of increased numbers of cortical neurons across mammals. For instance, although large cetacean and artiodactyla brains have more folded cerebral cortices than similar sized primate brains, they probably have not more but rather far fewer neurons in the cortex than primates, due to their very low neuronal densities (Tower, 1954; Herculano-Houzel, 2009; and our unpublished observations).

This is not to say, however, that cortical folding is not driven by a shared, conserved mechanism across mammalian species. In fact, our recent work on the scaling of the subcortical WM in primates as a function of their numbers of cerebral cortical neurons unexpectedly led us to realize that cortical folding can be universally predicted as a function of not simply the total number of cortical neurons, but of the number of cortical neurons that are connected through the WM compounded with the average caliber of their axons in the WM and the tension in these axons (Herculano-Houzel et al., 2010). Cortical folding, according to this view, would not be driven by the GM, but rather primarily by tension in the WM.

A qualitative, tension-based theory of cortical morphogenesis was first proposed by Van Essen (1997), taking into consideration the patterns of connectivity between cortical areas. His connectivity-driven hypothesis for the placement of cortical folds accounts for the consistent formation of convolutions in a speciesspecific pattern. It does not, however, explain the increased cortical folding that accompanies increasing cortical size across species; Van Essen himself still resorted to a GM driven mechanism to account for that. There are, however, a number of other evidences against a GM driven mechanism of cortical folding. Partial removal of the skull during development does not have a dramatic effect on the fissure pattern, and lesion experiments suggest that cortical folding is not primarily dependent on a disproportionate growth between cortical and subcortical structures (reviewed in

Kaas, 2009). Thus, the primary source of fissure formation must be sought in factors within the cortex itself – or underneath it.

Based on our findings on the scaling of cortical connectivity and WM volume in primates (Herculano-Houzel et al., 2010), we proposed in that paper an extension of Van Essen's qualitative tension-based theory of cortical folding to explain quantitatively how increased folding accompanies increasing cortical size across primate species. According to our model, rather than *driving* the folding of the WM surface, the folding of the external surface of the GM *results* from folding of the WM surface, which, in turn, results from increased tension within the WM due to increased numbers of axons composing the WM depending on their physical properties of caliber and tension. Our model is quantitative; acknowledges that the cortex scales differently in size across mammalian orders as different power functions of its number of neurons; is therefore applicable, in principle, to all mammalian species; and makes easily testable predictions for all of them. The following is a description of the model, its assumptions, and a discussion of its implications and predictions, and how they can be tested.

#### **ASSUMPTIONS AND RATIONALE**

So far, we have found the size of the different brain structures, the numbers of cells that compose them, and their average densities, and therefore average cell size, to be parameters related to one another by power functions (Herculano-Houzel et al., 2006, 2007, 2011; Sarko et al., 2009; Gabi et al., 2010). Generically, one should expect brain and cortical allometric scaling rules that are valid over several orders of magnitude across species within a particular mammalian order to take the form of power laws (i.e., relating measurable quantities "*x*" and "*y*" by *y* = *kxa*). This is because relations which are expected to remain valid over many orders of magnitude should not be given in terms of parameters that specify a particular size scale, that is, they should be scale-independent. As can be easily demonstrated, a one-variable function is scale-independent if and only if it is a power law. We therefore expect to find the scaling rules that determine cortical folding to also be scale-invariant, and therefore power laws.

Even more importantly,we assume that the number of neuronal cells is the mainfree parameter that coordinates the scaling of every other quantity of interest, measurable, or estimated. There are several reasons for this assumption. First, neurons, rather than glia, are the first cells to populate the developing brain in large numbers, and their connectivity begins to be established at the same time as convolutions begin to form, even before the final neuronal complement is in place (Goldman-Rakic, 1980; Goldman-Rakic and Rakic, 1984). Therefore, and in contrast to most earlier studies on brain allometry that implicitly or explicitly regarded the number of neurons as a consequence of brain size, we believe instead that any biologically plausible model of brain allometry must consider brain size, in all its aspects, to be a *consequence* of its number of neurons, according to scaling laws that may vary across different phyla. Second, we have found that, in contrast to the order- and structure-specific neuronal scaling rules, the scaling of different brain structures seems to occur as a universally shared function of their numbers of glial cells, both across orders and

structures (Herculano-Houzel, 2011). This result, combined with the late onset of gliogenesis in post-natal development (Sauvageot and Stiles, 2002), suggests that we can assume that cortical composition is determined essentially by the number of neurons and their average mass (which is itself closely related to the number of neurons by an order-specific power law); after neurogenesis, nearly invariant glia will then infiltrate the intraneuronal space in proportion to the total neuronal mass (Herculano-Houzel, 2011). The development of an adult mammalian cortex can then be viewed as a process whereby total numbers of neurons, numbers of neurons connected through the WM, their size (which includes the soma, all dendrites, and an axon of a particular caliber), and cortical folding vary in lockstep, over an invariant background of glia.

Our model thus assumes that all parameters related to cortical scaling and folding can be described as power functions of the total number of cortical neurons. Note that the fact that general allometric rules exist for cortical morphology in each order, expressible as power laws of their number of neurons, does not mean that the latter is the only significant degree of freedom in brain or cortical evolution. Rather, the power laws tell us that any other significant degrees of freedom must be present either at a substructure level, thus being erased by measurements that average over the entire structure, or at the microscopic level of detailed connectivity, which is not accessible to our methods, but also not relevant to the model at hand.

Our model, as presented for cross-species comparisons, considers total cortical volumes and areas, and average values of neuronal density and cortical thickness for the whole cortex, in line with the empirical studies that generated the numerical data used here (Herculano-Houzel et al., 2006, 2007, 2010, 2011; Sarko et al., 2009). Neuronal density is now known to vary across the cortical surface within primate species (Collins et al., 2010), and it is known that cortical thickness and gyrification also vary across the cortex (Zilles et al., 1988; Toro et al., 2008). However, comparative studies on the scaling of cortical gyrification traditionally analyze whole-brain patterns (Hofman, 1985; Pillay and Manger, 2007), and attempts to understand the scaling of cortical gyrification have similarly been directed toward whole-brain comparisons. We therefore developed our model based on average values for whole cortex that can be compared across orders, but predict that the scaling rules proposed here to govern gyrification at the level of the whole cortex might also be applicable at the local level across cortical areas.

White matter is largely composed of axons connecting neurons in the GM, mostly with each other but also with subcortical structures, along with the glial cells that support their function. The volume of each axon is simply its cross-sectional area multiplied by its length. If there is no significant correlation between these two latter quantities (which can be proven mathematically that will be the case when axon bundle volume is constrained and average signal propagation time is minimized), then the total axonal volume is the product of average axon cross-section area *a,* the average axonal length *l*, and the total number of axons present in the WM. We can assume further that the volume of the intra-axonal space, including in particular the myelin sheath and the myelinating oligodendrocytes, is proportional to axonal volume, given the experimental support for a linear relation between axon diameter and myelin sheath diameter (Sadahiro et al., 2000). Using the common assumption of a linear relation between total number of oligodendrocytes and the total axon length (Barres and Raff, 1994, 1999), then the total volume of the WM can be written to scale with the product of the total axon length (or total number of oligodendrocytes) and average axonal cross-sectional area.

We also assume that WM axons intersect the surface of the WM–GM interface in a perpendicular direction. This is a simplifying assumption, since it can observed from direct imagery that multiple fiber orientations can be present very close to each other even at the WM–GM interface. However, we believe on theoretical grounds that this is a reasonable if imperfect approximation of a somewhat more complex anatomical reality. Indeed, axons typically cross this interface in parallel bundles (Mori et al., 2002), which are the most volume-economical way of building such surface; in a growing cortex the combination of axonal longitudinal tension,WM hydrostatic pressure, and WM–GM interface surface tension should align most bundles very close to perpendicularly to the interface.We must make this assumption because we are unfortunately aware of no systematic studies of the distribution axonal incidence angles in the literature, although published diffusion tensor imaging tracing studies show a clear (but unquantified) preference for perpendicular angles of incidence (for instance, Mori et al., 2002). A systematic variation in the average incidence angle across species would alter our results somewhat, but not appreciably except for a very large range of values1.

Finally, our model assumes that connections through the WM are formed early in development, at the same time as the GM becomes folded, an assumption that is supported by experimental evidence (reviewed in Welker, 1990); that most axons in mammalian cortical WM are myelinated (Olivares et al., 2001); and that most of the WM volume amounts to myelinated axons (Wang et al., 2008), thus neglecting the contribution to the volume of (small) non-myelinated fibers in the WM.

#### **THE MODEL**

We consider that the surface of the WM–GM interface, with total area*A*W,is crossed nearly perpendicularly by most axons leaving or entering the WM, of an average cross-sectional area *a*, and which, together with the ensheathing glial cells, comprise the entirety of the WM surface. *A*W can thus be quantified as the product of the number of cortical neurons,*N*; the fraction *n* of these neurons that are connected through the WM; and their cross-sectional area, *a* (**Figure 1**). Thus,

*AW* ∼ γ−1*nNa*,

where γ is the average cosine of the incidence angle of axons at the WM–GM interface. The value of γ is of course 1 in the simplified

1 A deviation from perpendicular incidence would be reflected on a proportional discrepancy in the estimation of axonal lengths; but only a systematic variation (with *N*) of incidence angle would alter the calculated power law coefficients. Unfortunately we cannot at this point rule out or numerically constraint such variation, and must therefore recognize this fact as a limitation of our model as it stands.

case of perfectly orthogonal incidence, and presumably close to 1 in reality. The WM volume *V* W is the sum total of the volumes of all fibers and is thus equal to one half of the product of *A*W and the average axonal length in the WM, *l*, such that

sending or receiving axons (of an average cross-sectional area a) through it.

*VW* ∼ *nNal*.

Note that the average value of axonal length is given by *l* = *2V* w/γ*A*w. A direct measurement of γ would provide us with a direct measurement of *l* for each species. Assuming γ = 1, then *l* can be obtained from the existing measurements of *V* w and *A*w. Strictly speaking, these quantities are only lower bounds on the average axonal lengths; if however γ ≈ 1 as we postulate, they can be taken as good approximations of their actual values.

If the WM scales under tension, the cubic root of its volume should increase more slowly than the square root of its surface area, leading to deformation of the latter, that is, to folding of the GM– WM surface. To quantify the extent of WM folding, we define a folding index *F*W, which is the ratio between the actual WM surface, *A*W, and the exposed surface expected from its volume (9π/2)1/3 *V* W2/3.

*F*W = (2/9π) 1/3 (*n Na*) - (*nNa l*) 2/3

Thus, a *F*W value of exactly one implies a spherical WM, and larger values imply more convoluted forms. Importantly, notice that it is not necessary to model the cerebrum as a sphere for the 2/3 scaling relationship between its surface area and volume to hold; a volume of any shape that scaled isometrically would have the same scaling relationship of 2/3. In this case of isometric growth, which would ensue if the WM did not scale under tension, then we would expect *F*W to be invariant as function of *N*.

Now, considering that *a*, *n*, and *l* are themselves related to *N* as power functions such that *a* ∼ *N*α, *n* ∼ *Nc* , and *l* ∼ *N*λ, the relationships above can be entirely rewritten as power functions of *N*:

$$\begin{aligned} A\_W &\sim N^C N \ N^\alpha \sim N^{c+1+\alpha} \\ V\_W &\sim N^C N \ N^\alpha N^\lambda \sim N^{c+1+\alpha+\lambda} \\ F\_W &\sim N^{(1+c+\alpha-2\lambda)/3} \end{aligned}$$

AW and the average axonal length in the white matter, l.

Note that if we took into account a systematic variation of the incidence angle of fibers at the GM–WM interface as a power law of *N*, we would have to introduce a (non-zero) new coefficient at the expression for *A*w. There is unfortunately currently no experimental way of estimating the value of such coefficient. We have assumed throughout that it is small enough to be disregarded, but should it prove to be otherwise we will have to recalculate the other coefficients accordingly, and revisit the conclusion obtained.

Simultaneously,for cortices with average GM thickness T much smaller than the cortical characteristic length so that the internal and external areas of GM scale linearly (that is, *A*G ∝ *A*W), T can be defined as simply the ratio between the volume of the GM,*V* G, and area of the GM–WM interface, *A*W. Given that *V* G scales as a power function of *N*, with *Nv*, then

$$T \sim N^r / N^{c+1+\alpha}$$

and therefore cortical thickness T scales with *Nt* , such that

$$t = \nu - \varepsilon - 1 - \alpha.$$

Instead of *v*, we can use a more biologically meaningful parameter *d*, which is the exponent relating neuronal density D in the GM to *N*, given that *V* G = *N*·D−1 and that D ∼ *Nd*. Because *V* G ∼ *N*1−*d* , then *v* = 1 − *d*, and the equation above becomes

$$t = -d - c - \alpha.$$

Average cortical thickness, therefore, scales a function of a GMrelated variable (the scaling of neuronal density with *N*), and two WM-related variables that, together with *N*, determine WM folding (the scaling of connectivity and of axonal cross-sectional area with *N*). Remarkably, cortical thickness is therefore not itself a function of *N*, but rather of how exponents *d*, *c*, and α are interrelated.

Finally, the extent of GM folding, *F*G, can be expressed as the ratio between its actual surface, AG (which can be written as proportional to *V* G/T, or *N*1−d−t ), and the exposed surface expected from the total volume (*V* G +*V* W)2/3:

$$F\_G = N^{1-d-t}/(N^{-d} + N^{c+1+\alpha+\lambda})^{2/3}$$

Note that although this equation is not an exact power law, it can in practice be well approximated by one since*V* W and*V* G scale in fairly similar ways with *N*. Another way to express *F*G, now as an exact power law, is by writing the total volume *V*T =*V* G +*V* W itself as a measurable power function of *N*, varying with *Nz* . In this case, *F*G becomes

$$F\_G = N^{1-d-t} / (N^Z)^{2/3}$$

Returning to the first equation of *F*G, and recalling that *t* = −*d*−*c*−α, GM folding is thus a combined function of the number of cortical neurons; the fraction of these neurons that are connected through the WM; and the average cross-section area of the axons in the WM. Further, the thickness of the GM is thus a consequence of some of the same parameters that determine how the cortex folds, and not a determinant of it. A schematic of the model is depicted in **Figure 2**.

#### **PREDICTIONS FROM THE MODEL**

Useful mathematical models are those that lead to a number of testable predictions. This is one major advantage of our model: it allows us to derive not only testable qualitative insights on the scaling of cortical folding, but also quantitative predictions that can be tested experimentally.

#### **QUALITATIVE PREDICTIONS ABOUT CORTICAL FOLDING**

Our model predicts that the folding of the GM is related to the folding of the WM, and the scaling of the former across species depends on the scaling of the latter. We predict that WM folding scales across mammalian species with the number of cortical neurons; the fraction of these neurons that are interconnected through the WM; the average length of the myelinated fibers in the WM; and their average cross-sectional area. GM folding then scales depending, additionally, on the scaling of the GM thickness, which in turn is determined by the scaling of neuronal density

fundamental parameters determining cortical folding, probably determined genetically, and which we postulate to vary alometrically with N: the fraction of cortical neurons connected through the gray matter (n), the average cross-sectional area of the axons in the white matter (a), the average neuronal density in the gray matter (D, which is approximately proportional to the inverse of average neuronal cell

determine the folding of the white matter surface (FW). On top of AW, the gray matter becomes organized depending on the average size of its neurons, which, combined to a and n, determine cortical thickness, T. The degree of folding of the gray matter, F G, is thus a consequence of the folding of the white matter, which is in turn dependent on how the parameters determining cortical connectivity (c, a, and l) scale with N.

in the GM (besides the scaling of connectivity and average axonal cross-sectional area in theWM). **Figure 3** illustrates how the interplay across the scaling of these parameters determines cortical morphology and folding.

One remarkable characteristic of our model is that, in principle, it applies universally across mammalian orders (and therefore describes the scaling of cortical folding universally), even though relationships such as those among folding index, cortical thickness, and cortical size are different across orders (Pillay and Manger,

2007). In fact, we can predict that these relationships will be different across orders depending on the particular defining exponents that apply to each order; order-specific characteristics of the scaling of cortical folding will result from combinations of these exponents. Thus, it is conceivable that cortical folding increases in larger brains with no change in connectivity and no change in the average cross-sectional area of the axons in the WM; with decreasing connectivity and increasing average cross-sectional area of the WM; and so forth.

Indeed, one of the strengths of the model is that one can predict how the different scaling exponents will be related. Because folding of the WM scales with *N*(1+*c*+α−2λ)/3, then larger cortices in a mammalian order will only be increasingly folded when (1 + *c* + α − 2λ)/3 > 0. This condition imposes constraints on the combinations of values of *c*, α, and λ that will lead to increased folding in larger cortices (that is, as *N* increases).

For instance, supposing that cortical scaling occurs in an order with a fixed λ of 0.5, then *c* + α > 0, which implies that if folding increases in larger cortices, then any decrease in connectivity (that is, a negative value of *c*), or even a constant connectivity (that is, *c* = 0) is necessarily surpassed by a positive scaling of average axonal cross-section in the WM. In these circumstances, the thickness of the GM, which scales with *t* = –*d*−*c*−α, will depend on the scaling of neuronal density minus the positive sum of *c* + α; as a consequence, the number of neurons underneath a given cortical surface will not be constant (a condition that is only met when *t* = −*d*).

By the same token, if cortical scaling occurs in an order with increasing folding in the presence of fixed connectivity (that is, with *c* = 0), then necessarily α > 2λ − 1. In these circumstances, the average axonal cross-sectional area in the WM will increase in larger-brained cortices for any value of λ ≥ 0.5. Consequently, the thickness of the GM will scale depending on the value of –*d*−α.

Likewise, it is possible for cortical scaling to occur with increased cortical folding within an order with fixed connectivity and unchanging average axonal cross-sectional area in the WM, as long as λ < 0.5. In these conditions, it can be predicted that the thickness of the GM will scale depending on the scaling of neuronal density alone, and thus occur indeed with a constant number of neurons beneath the cortical surface, as intended in some models (Rockel et al., 1980).

#### **CORTICAL THICKNESS**

Thinner mammalian cortices are usually found to be more folded than thicker cortices of a similar size (Hofman, 1985; Pillay and Manger, 2007). This finding has been attributed to thinner cortices being supposedly more pliable than thicker cortices, which would render the former less resistant to being folded (Pillay and Manger, 2007). In contrast, our model predicts that cortical thickness is actually determined by two WM-related factors that also determine the degree of cortical folding (connectivity and average axonal cross-sectional area in the WM), and a third, GM-related variable (neuronal density). Remarkably, the scaling of cortical thickness is therefore not simply a function of *N*, but rather of how *d*, *c*, and α are interrelated; if neuronal density, connectivity, and average axonal cross-sectional area are unchanging in a mammalian order, then larger cortices, with larger *N*, would be expected to have an unchanging thickness as well. In another scenario, even if neuronal densities in the GM were still constant across species in an order, cortical thickness would still increase with *N* as long as *c* + α < 0, which would be the case, for instance, if connectivity decreased (*c* < 0) and average axonal cross-sectional area did not change (α = 0).

#### **CORTICAL UNIFORMITY**

Remarkably, the relationship between cortical thickness, connectivity, neuronal density, and axonal cross-sectional area predicts that a uniform number of neurons underneath a cortical surface area will only be found across species (Rockel et al., 1980) when a very specific condition is met. Given that *V* G =*A*W *T* = *N*/*D*, then the ratio *N*/*A*W will only be constant when the product *D*·*T* is constant. Written as a function of *N*, and remembering that *D* varies with *Nd*, and *T* with *N* −*d*−*c*−α, then this condition is met only when *d* + (−*d*−*c*−α) = 0, that is, when the sum of the exponents *c* and α is zero, or, alternatively, when cortical connectivity through theWM remains constant and simultaneously the average axonal cross-section in the WM does not scale (that is, both *c* and α are zero). In all other cases, the number of neurons underneath a cortical surface will scale with a non-zero combination of *d* and *t*. Notice that this prediction is valid both for the scaling of the entire cerebral cortex and for different cortical areas.

#### **SCALING OF WM WITH GM**

Because *V* G can be defined as the product of *A*W and *T*, and therefore varies with *Nc*+1+α *Nt* , or *Nc*+1+α+*t* , then the ratio *V* W/*V* G varies with *Nc*+1+α+λ/*Nc*+1+α+*t* = *N*λ − *t* , that is, depending on which is steeper: the scaling of axonal length reduction (faster with smaller values of *l*) or of cortical thickening (faster with larger value of *t*) as a function of *N*. Ratios larger than 0 mean that axons shorten relative to isometry more *slowly* than the cortex thickens as it gains neurons. The implication here is that the ratio*V* W/*V* G will be constant across mammalian species when λ = *t* and that it will increase when λ > *t*; and may even scale similarly across orders when they share a similar relationship between λ and *t*. Note that a similar scaling of the *V* W/*V* G relationship across orders does not imply similar values of λ or of *t*, but only a similar difference between them across orders. The increase in the ratio *V* W/*V* G in larger cortices (Zhang and Sejnowski, 2000) can thus be predicted to be a consequence of a slower minimization of average axonal length relative to isometry than the thickening of the GM as a function of *N*.

#### **PROPAGATION TIME**

Another way of thinking about *F*W is to express it in terms of the average axon length *l* and the WM characteristic length *R*W (defined as the radius of a sphere with volume *V* W). This relationship can be written as *F*W = 2*R*W/3*l*, where the WM folding index is simply proportional to the ratio between the characteristic size of the WM and the average axonal length. Thus, the more axonal tension curtails the growth of *l*, the greater the *F*W and the more convoluted the WM becomes. Compared to a smooth WM surface, folding the WM results in axons having to travel shorter distances to connect GM neurons. A more folded WM will have shorter axons, as a fraction of its characteristic size.

Since the whole purpose of the axons in WM is to transmit signals, it makes sense to quantify how well and quickly they do it, in terms of the scaling rules obtained above. It is well known that an action potential impulse propagates along a myelinated axon in a time proportional to the axon length and inversely proportional to the square root of axon cross-sectional area. The average impulse propagation time should then be given by *t* ¯ α *l*/*a*1/2.

Since *l* = 2*VW/A*W ∝ *N*λ,*A*W ∼ *Nc*+1+α+λ, and*V* W ∼ *Nc*+1+α+λ, we can combine the three equations to describe how the average propagation time scales with *N*:

$$\begin{aligned} \overline{t} &\propto \frac{l}{a^{1/2}} \propto \frac{2V\text{W}}{A\_\text{W}N^{\alpha/2}} \propto N^{\text{tr}}\\ \mathfrak{r} &= \lambda - \frac{\alpha}{2} \end{aligned}$$

From the equation above, it is clear that, with all else being equal, increasing axonal thickness would result in smaller propagation times. However, if all axons in the WM were to grow thicker by the same factor, then a tightly packed WM would also have to expand to accommodate the extra volume. But a largerWM would mean that GM neurons would be further apart, and axons would have to be longer to connect them.

For the sake of argument, let us considering an isometric doubling of all the lengths and diameters (i.e., a fourfold increase in average cross-sectional areas) of all axons in the WM, such that the overall shape of the WM does not change. According to the formula given above, the average axon impulse propagation time in this isometrically scaled up WM would also be unchanged. Signals have to propagate further due to the doubling of *l*, but the propagation speed is proportionally faster due to the quadrupling of *a*. Thus, as far as average propagation times in the WM are concerned, there is no difference between scaled up or scaled down isometric versions of the WM. In contrast, the hypometric scaling of axonal length in the WM has the obvious consequence of decreasing propagation time: the smaller the value of *l*, the smaller the increase in axonal cross-sectional area required to maintain a constant average propagation time as the cortex gains neurons interconnected through the WM.

With the realization that average propagation time scales with *N* raised to an exponent of λ − α/2, then the folding of the WM can be written to scale as

$$F\_W \sim N^{(1+c+\alpha-2\lambda)/3} \sim N^{(1+c-2\tau)/3}$$

Thus, theWM folding index can be expressed in such a way as to depend on only two quantities: The total number of axons in WM (which scales with *Nc*+1), and the inverse of the square of the average axonal signal propagation time (which scales as *N*λ−α/2). This means that cortices in which connectivity through the WM and average propagation time scale similarly will have folding indices that also scale similarly. This also means that, for a given value of *c*, a faster upscaling of WM folding will be accompanied by a slower increase in propagation times. Increasing cortical folding in larger brains is thus associated with the advantage of a diminished increase in the average propagation time that would otherwise be expected if the WM grew isometrically.

#### **COMPUTATIONAL CAPACITY**

Signal propagation times tell us how fast a cortex computes information, but not how effectively. To quantify computational capacity in a simple way, consider a simple neuronal circuit composed of a few neurons connected by axons passing through the WM. A discrete"operation"in such a circuit consists of a set number of signals being passed back and forth among the neurons (for instance in response to a specific external input). A typical such operation is memory retrieval: The circuit receives as input an incomplete pattern that is a partial match to a stored pattern. After a few cycles over its feedback loops, the circuit's output eventually converges to the stored pattern. Clearly, each such computational cycle (which can be as simple as two neurons with reciprocal connections) is completed in the time it takes for a signal to propagate along the axons of its interconnected, constituting neurons. Thus, in general terms, and assuming that all propagation times in the circuit scale up similarly, then the time needed to perform one operation in this circuit is proportional to the average propagation time along its axons; conversely, the number of operations it can perform per unit time is inversely proportional to the average impulse propagation time of the circuit's axons. In this case, *C*, the number of operations involving WM fibers (i.e., non-local operations) that a cortex would be able to perform per unit time, a very simple proxy for its overall computational capacity, is then proportional to the number of "circuits," or the number of axons, and inversely proportional to the average propagation time.

$$C \propto \frac{nN}{\mathfrak{r}} = \frac{O^{\frac{1}{2}} A\_{\mathbb{W}}^2}{V\_{\mathbb{W}}^{\frac{2}{2}}} \propto N^{\theta}$$

$$\theta = 1 + c - \mathfrak{r} = 1 + c - \lambda + \frac{\alpha}{2}$$

Note that, like the WM folding index, the scaling of *C* depends only on the number of axons crossing the WM, and on the coefficient τ for the scaling of average propagation time.

2

We can also define a volumetric computational efficiency, that is, computational capacity per unit of WM cortical volume, *C*ef = *C*/*V* W,

$$\begin{aligned} C^{\text{ef}} \propto \frac{nN}{V\_{\text{W}}\pi} &= \frac{N\_O^{\frac{1}{2}} A\_{\text{W}}^2}{V\_{\text{W}}^{\frac{2}{2}}} \propto N^{\varepsilon},\\ \varepsilon &= -2\lambda - \frac{\alpha}{2}. \end{aligned}$$

The computational efficiency of the WM is thus predicted to scale with *N* −2λ − (α/2). Note that *Cef* is highly dependent on the scaling of axonal length, but only weakly so on the scaling of axonal cross-sectional area.

#### **WHEN SHOULD THE CORTEX FOLD?**

If the WM scales isometrically, that is, without folding, then the average axonal length *l* will scale with *V* W1/3, or with *A*W1/2. If on the contrary the WM scales under tension, with a smaller increase in*V* W than expected, that implies that *l* is scaling more slowly than expected. Assuming that *A*W ∼ *nNa* and that *a*, *n*, and *l* are themselves related to *N* as power functions such that *a* ∼ *N*α, *n* ∼ *Nc* , and *l* ∼ *N*λ, then, in the case of isometric scaling of *V* W,

$$l \sim N^{\hat{\lambda}} \sim (n \text{ N } a)^{1/2} \sim N^{(\epsilon + 1 + \alpha)/2}$$

so λ = (*c* + 1 + α)/2 if the WM scales isometrically.

In all scenarios where λ < (*c* + 1 + α)/2, the WM will become convoluted as it increases in size, and one can therefore expect the cortical GM to also become more and more folded at the same time. Qualitatively, this means that, due to increasing WM folding, axons do not grow in direct proportion to the characteristic length of the brain (*R*W), but rather much more slowly; with cortical growth, connections become relatively shorter, as if axons resisted being "stretched" to accommodate more and more connections through the WM. As we have seen, this results in average propagation times increasing more slowly with the addition of more axons in WM.

Finally, in these scenarios where λ < (*c* + 1 + α)/2 and the WM becomes increasingly folded in larger cortices, it can also be predicted that τ < (*c* + 1)/2 (by substituting τ = λ − α/2 in the equation), which implies that there is an upper limit to the scaling exponent of propagation time in the WM determined by the connectivity fraction. This leads to the interesting realization that decreasing the fraction of cortical neurons connected through the WM (that is, more negative values of *c*) minimizes the upscaling of propagation times in larger cortices.

#### **TESTING THE MODEL**

There are two main ways in which our model can be tested: by designing experiments to address the prediction that cortical connectivity through the WM affects the establishment of cortical folds; and by testing the quantitative relationships predicted by the model.While we have not yet had the opportunity to design experiments specifically to test this model, earlier experiments showing that cortical folding is altered after disrupting cortical connectivity but not after partial removal of the skull during development (reviewed in Kaas, 2009) do support our proposal that cortical folding is driven by pulling on the inside of the cortex, rather than from the cortex pushing inward.

Testing the numerical relationships predicted by the current model requires quantifying, across different mammalian species, their numbers of neurons in the GM, numbers of other cells in the WM; obtaining surface and volume measurements for the GM and WM; and determining, from the scaling across these parameters, whether the exponents thus calculated match the predictions from the model. Here we discuss how the scaling exponents can be obtained; in the next section, we will address how well the model's predictions are matched by experimental data available so far from rodents and primates.

#### **MEASURING** *l* **AND λ**

We showed above how, given that *l* = 2*V* W/*A*W ∝ *N*λ, the allometric exponent λ for scaling of *l* with *N* can be determined experimentally from the relationship between the ratio between the measureable values *V* W/*A*W and *N*. Additionally, the same relationship also allows the average value of *l* to be estimated for each species, as the quotient 2*V* W/*A*W. The factor of 2 is due to the fact that we expect the vast majority of fibers in the WM to be cortico-cortical (Braitenberg and Schüz, 1998) and thus cross the WM–GM interface twice, coming in and out of the WM as they connect different areas of the cortical GM. Notice that because any deviation of the true factor will be in the direction of being smaller than 2, the value of *l* thus determined can be considered an upper limit of the average axon length in the WM for that species; moreover, deviations from this value of 2 do not affect the exponent, λ.

#### **DETERMINING** *c* **AND α**

Experimentally, one can obtain the sum *c* + α from plotting AW as a function of *N*, but not the individual values of these exponents. A simple, well-fundamented assumption, however, allows the exponents *c* and α to be determined empirically using the isotropic fractionator (Herculano-Houzel and Lent, 2005) to count the number of non-neuronal (other) cells in WM, *O*, most of which are oligodendrocytes myelinating neighboring axons. According to Barres and Raff (1994, 1999), it is reasonable to assume that the total length *L* of all axons in WM, given by *L* = *n*·*N*·*l*, is proportional to the total number of oligodendrocytes found in the WM. Thus, *O* ∼ *L*.

 Because the power law relating *O* to *N*, *O* ∼ *N*ω, can be determined empirically (Herculano-Houzel et al., 2010), *V* W may be rewritten as

### *VW* ∼ *N*ω*N*α.

This is a power function that allows the exponent α to be determined, given that ω is known. The exponent *c* can next be calculated simply from the scaling relationship above between *A*W and *N*, now that has been determined.

#### **TESTING THE PREDICTIONS**

Once the values of exponents λ, *c*, and α are obtained from the measurements of *V* W, *A*W, *N*, and *O*, it becomes possible to predict if the WM should become increasingly folded in larger brains from the comparison between λ and (*c* + 1 + α)/2. If these quantities are equal, then the WM should scale isometrically. If, on the other hand, λ < (*c* + 1 + α)/2, the WM will become convoluted as it increases in size, and one can therefore expect the cortical GM to also become more and more folded at the same time.

The actual folding of the WM, *F*W, is easily determined experimentally for each species as being proportional to *A*W/*V* W2/3; the exponent of the experimentally obtained scaling of *F*W with *N* can thus be compared to the predicted exponent given by *F*W ∼ *N*(1+*c*+α−2λ)/3.

Next, once the average thickness of the GM is determined for each species as the ratio *V* G/*A*W, the exponent *t* of its scaling with *N* can be determined experimentally, as can the exponent *d* of the scaling relationship between neuronal density and number of cortical neurons. Once these exponents are available, it can be tested whether, as predicted, *t* = −*d*−*c*−α; and whether the extent of GM folding, *F*G, expressed as the ratio AG/(*V* G +*V* W)2/3, scales as predicted, with

$$F\_{\mathcal{G}} = N^{1-d-t} / \left( N^{-d} + N^{C+1+\alpha+\lambda} \right)^{2/3}$$

#### **EXPERIMENTAL FINDINGS**

This model has so far been applied to a primate dataset containing 11 species, including humans (Herculano-Houzel et al., 2010), and more recently to a dataset of 5 rodent species (Ventura, Mota, and Herculano-Houzel, to be submitted). Experimentally, we find that the average cross-section area remains nearly invariant in primates as a function of *N*, while it increases sharply with *N* across rodents. This is qualitatively very similar to what happens with average neuron size in both orders: It increases significantly with *N* in rodents, but increases very slowly with *N* across primate species (Herculano-Houzel,2011), suggesting there is a connection between average neuron size and axon caliber.

As for the fraction of GM neurons projecting axons through the WM, we find that it decreases with *N* in both orders, but at a rate that is more pronounced in primates than in rodents. As a result, the WM becomes increasingly folded in larger primate brains, but less rapidly in larger rodent brains.

Using the experimental method described above to determine the values of the scaling exponents that appear in the model, we obtained, in rodents and in primates, the values listed in **Table 1**.

In both orders, *l* scales sublinearly with *N*, which is a significant finding given that any increase in average axon length implies an increase in volumetric and propagation time costs. However, while the rodent λ closely matches the value of 0.689 expected in case of isometric scaling of *V* W of λ = (*c* + 1 + α)/2, the primate λ = 0.207 is significantly smaller than the value of 0.436 expected in the case of isometric scaling of theWM. This is a strong indication that in primates, the increase in distance between interconnected cortical regions is minimized by effective shortening of the axons, as would be expected to happen if they grew under longitudinal tension (Van Essen, 1997).

As a consequence of these exponents, the folding of the WM is predicted to scale as the cortex gains neurons with *N*0.153 in primates, which matches precisely the observed scaling of the WM calculated as *A*W/*V* W2/3; and is predicted to scale with *N*0, that is, not to increase in larger rodents cortices. Notice that this

**Table 1 | Experimentally determined exponents for, respectively, the scaling of average axonal length in theWM (***l***), average axonal cross-sectional area in theWM (***a***), fraction of GM neurons connected through theWM (***n***), average GM thickness (***t***), and neuronal density in the GM (***d***) as power functions of the number of cortical neurons (***N***).**

Note that the values in the table value slightly from the values reported in Herculano-Houzel et al. (2010) because here they are estimated for values of VW, which are not available for human brains in our dataset, while in that study, they were estimated for values of MW, which are available for humans, although AW is not. Because of the lack of all values for human brains, we chose to recalculate the exponents here for only those species for which VW and AW are available. \*Neuronal density in the GM was not found to scale with N in the original study where λ, α, c, and t were estimated (Herculano-Houzel et al., 2010), but in a later review of a large dataset was found to scale with d = −0.168 (Herculano-Houzel, 2011). Again, we chose to report the value that applies to the current dataset, for consistency.

prediction apparently contradicts the finding that large rodent cortices, such as those of the agouti and capybara, are indeed folded. We believe, however, that the apparent failure of the model to predict the folding of large rodent cortices is due to the fact that in our sample, three of the five species (mouse, rat, and guinea pig) are small-brained and practically lissencephalic. Thus, rodent cortices seem to scale without becoming folded only up to a certain point, beyond which larger cortices do become increasingly folded. This is actually circumstantial evidence in favor of the push–pull model that we propose, in which the WM only begins to fold once the traction that it exerts upon the GM exceeds the resistance of the latter to becoming folded inward.

#### **NUMBER OF CORTICAL NEURONS CONNECTED THROUGH THE WM**

Our model predicts that GM connectivity *n* (the fraction of GM neurons that sends an axon through the WM) decreases as the GM gains neurons, in a manner that we estimate in primates as *n* ∝ *N* −0.212. Although we do not dispose of estimates of the actual number of cortical neurons connected through the WM, it is illuminating to consider the following exercise scenario. Supposing, for the sake of argument, that 50% of all cortical neurons were connected through the WM in the marmoset, then a scaling of *n* with *N* −0.212 would imply that in a monkey-sized cortex with 10 times more neurons than the marmoset, WM connectivity would fall to 10−0.212 = 0.61 × 50% = 30% of all neurons; and a human-sized cortex with about 100 times more neurons than a marmoset would have only 19% of its neurons interconnected through the WM.

Note that decreased connectivity occurs in the face of an increased total number of axons in the WM, which is proportional to *n*·*N*, or *N*1+*c*. In the exercise scenario above, the total number of axons in the WM would increase from about 122 million in the marmoset, to 510 million in the macaque, to 3.0 billion in the human cortex. Larger primate cortices, therefore, increase in size proportionally to *N*1 neurons in the GM, of which a number proportional to *N*0.788 send axons into the WM.

#### **A CONSTANT NUMBER OF NEURONS BENEATH THE CORTICAL SURFACE?**

As observed above, our model predicts that the ratio *N*/AG will only be constant, as assumed in several models (e.g., Prothero, 1997; Zhang and Sejnowski, 2000), under particular circumstances, when *d* = −*t*, that is, when both *c* or α are zero, or so is their sum. Indeed, we have recently found that the ratio *N*/AG (which is similar to the ratio*N*/*A*W),farfrom being constant across primate species, varies threefold across primate species, and actually scales with variations in neuronal density across the species (Herculano-Houzel et al., 2008).

In primates, the predicted value of *t* = 0.127 agrees nicely with a measured value of 0.109 ± 0.025 SE. In rodents, the predicted value of *t* = 0.262 is well below the measured value of 0.427 ± 0.048, possibly because our current estimate of *d* for the GM is not as accurate as in primates for methodological reasons (Ventura, Mota, and Herculano-Houzel, to be submitted). In both cases, however, cortical thickness clearly does not scale as the inverse of neuronal density (because either *c* or α or both are significantly different from zero, and non-canceling), thus explaining why the number of neurons underneath a given cortical surface does not remain constant as cortical size varies (Herculano-Houzel et al., 2008; Ventura, Mota, and Herculano-Houzel, to be submitted).

#### **SCALING OF** *V* **W/***V* **G**