Modelling the emergence of whisker barrels
Abstract
Brain development relies on an interplay between genetic specification and selforganization. Striking examples of this relationship can be found in the somatosensory brainstem, thalamus, and cortex of rats and mice, where the arrangement of the facial whiskers is preserved in the arrangement of cell aggregates to form precise somatotopic maps. We show in simulation how realistic whisker maps can selforganize, by assuming that information is exchanged between adjacent cells only, under the guidance of gene expression gradients. The resulting model provides a simple account of how patterns of gene expression can constrain spontaneous pattern formation to faithfully reproduce functional maps in subsequent brain structures.
eLife digest
How does the brain wire itself up? One possibility is that a precise genetic blueprint tells every brain cell explicitly how it should be connected to other cells. Another option is that complex patterns emerge from relatively simple interactions between growing cells, which are more loosely controlled by genetic instruction.
The barrel cortex in the brains of rats and mice features one of the most distinctive wiring patterns. There, cylindrical clusters of cells – or barrels – are arranged in a pattern that closely matches the arrangement of the whiskers on the face. Neurons in a barrel become active when the corresponding whisker is stimulated. This precise mapping between individual whiskers and their brain counterparts makes the whiskerbarrel system ideal for studying brain wiring.
Guidance fields are a way the brain can create cell networks with wiring patterns like the barrels. In this case, genetic instructions help to create gradients of proteins across the brain. These help the axons that connect neurons together to grow in the right direction, by navigating towards regions of higher or lower concentrations. A large number of guidance fields could map out a set of centrepoint locations for axons to grow towards, ensuring the correct barrel arrangement. However, there are too few known guidance fields to explain how the barrel cortex could form by this kind of genetic instruction alone.
Here, James et al. tried to find a mechanism that could create the structure of the barrel cortex, relying only on two simple guidance fields. Indeed, two guidance fields should be enough to form a coordinate system on the surface of the cortex. In particular, it was examined whether the cortical barrel map could reliably selforganize without a full genetic blueprint prespecifying the barrel centrepoints in the cortex.
To do so, James et al. leveraged a mathematical model to create computer simulations; these showed that only two guidance fields are required to reproduce the map. However, this was only the case if axons related to different whiskers competed strongly for space while making connections, causing them to concentrate into whiskerspecific clusters. The simulations also revealed that the target tissue does not need to specify centrepoints if, instead, the origin tissue directs how strongly the axons should respond to the guidance fields. So this model describes a simple way that specific structures can be copied across the central nervous system.
Understanding the way the barrel cortex is set up could help to grasp how healthy brains develop, how brain development differs in certain neurodevelopmental disorders, and how brain wiring reorganizes itself in different contexts, for example after a stroke. Computational models also have the potential to reduce the amount of animal experimentation required to understand how brains are wired, and to cast light on how brain wiring is shaped by evolution.
Introduction
Spatial patterns in neural connectivity provide clues about the constraints under which brains evolve and develop (Purves et al., 1992). Perhaps the most distinctive pattern can be found in the barrel cortex of many rodent species (Woolsey and Van der Loos, 1970). The barrels are identifiable soon after birth in layer 4 of primary somatosensory cortex as dense clusters of thalamocortical axons, which are enclosed by borders a few neurons thick from postnatal day 3 (Erzurumlu and Gaspar, 2012). In the plane tangential to the cortical surface the barrels constitute a somatotopic map of the whiskers, with cells within adjacent barrels responding most strongly and quickly to deflection of adjacent whiskers (ArmstrongJames et al., 1992). Barrel patterning reflects subcortical whisker maps comprising cell aggregates called barrelettes in the brainstem and barreloids in the thalamus (Ma, 1991; Van Der Loos, 1976).
Barrel formation requires afferent input from whisker stimulation and thalamic calcium waves (AntónBolaños et al., 2019), and depends on a complex network of axon guidance molecules such as ephrinA5 and A7 and adhesion molecules such as cadherin6 and 8 (Vanderhaeghen et al., 2000; Miller et al., 2006). This network is orchestrated by interactions between morphogens Fgf8 and Fgf17 and transcription factors Emx2, Pax6, Sp8, and Couptf1 (Shimogori and Grove, 2005; Bishop et al., 2000), which are expressed in gradients spanning the cortical sheet that mark orthogonal axes and can be manipulated to stretch, shrink, shift, and even duplicate barrels (Assimacopoulos et al., 2012).
The barrel boundaries form a Voronoi tessellation (Senft and Woolsey, 1991; Figure 1A), suggesting that barreloid topology is preserved in the projection of thalamocortical axons into the cortex, and that a barrel forms by lateral axon branching from an initial centrepoint that ceases upon contact with axons branching from adjacent centres. However, the assumption of prearranged centrepoints is difficult to resolve with the observation that axons arrive in the cortical plate as an undifferentiated bundle, prior to barreloid formation (Agmon et al., 1993). In mice, axons from the trigeminal ganglion arrive in the principal division of the trigeminal nucleus (PrV) at E12, then axons from the PrV arrive in the ventroposteromedial nucleus of the thalamus (VPM) at E17, then axons from the VPM arrive in the cortical plate at E18/P0. Distinct whiskerrelated clusters then become apparent in the PrV at P0P1, in the VPM at P2P3, and in the cortex at P3P5 (Erzurumlu and Gaspar, 2012; Sehara and Kawasaki, 2011).
Alternatively, reactiondiffusion dynamics could generate a Voronoi tessellation without prearranged centres, by amplifying characteristic modes in a noisy initial distribution of axon branches, as a net effect of shortrange cooperative and longrange competitive interactions. Accordingly, the barrel pattern would be determined by the relative strength of these interactions and by the shape of the cortical field boundary. However, intrinsic cortical dynamics alone cannot account for the topographic correspondence between thalamic and cortical domains, the irregular sizes and specific arrangement of the barrels in rows and arcs, or the influence of gene expression gradients.
The centrepoint and reactiondiffusion models are not mutually exclusive. Preorganized centres could bias reactiondiffusion processes to generate specific arrangements more reliably, and mechanisms of lateral axon branching may constitute the tension between cooperation and competition required for selforganization. However, proof that barrel patterning can emerge from an undifferentiated bundle of axons, based only on local interactions, would show that a separate stage and/or extrinsic mechanism for preorganizing thalamocortical connections need not be assumed. To this end, we ask whether barrel maps can emerge in a system with reactiondiffusion dynamics, under the guidance of signalling gradients, and in the absence of predefined centres.
Models
Karbowski and Ermentrout, 2004 developed a reactiondiffusion style model of how extrinsic signalling gradients can constrain the emergence of distinct fields from intrinsic cortical dynamics. Their model defines how the fraction of occupied synapses ${c}_{i}(x,t)$ and the density of axon branches ${a}_{i}(x,t)$ interact at time t, along a 1D anteriorposterior axis x, for N thalamocortical projections indexed by i. The model was derived from the assumption that the rates at which a_{i} and c_{i} grow are reciprocally coupled. Extending the original 1D model to simulate arealization on a 2D cortical sheet, we use ${a}_{i}(\mathbf{\mathbf{x}},t)$ and ${c}_{i}(\mathbf{\mathbf{x}},t)$, and model synaptogenesis as
Accordingly, where the total fraction of synaptic connections sums to one, connections decay at rate α. Otherwise, ${c}_{i}(\mathbf{\mathbf{x}},t)$ increases nonlinearly ($k>1$) with the density of axon branching. Axon branching is modelled as
The first term on the right describes the divergence (indicated by $\mathrm{\nabla}\cdot$) of the quantity in parentheses, which is referred to as the ‘flux’ of axonal branching. The flux represents diffusion across the cortical sheet, at rate D, and the influence of M molecular signalling fields, $\rho (\mathbf{\mathbf{x}})$. The influence of a given field (indexed by j) on a given thalamic projection (indexed by i), is determined by ${\gamma}_{i,j}$, which may be positive or negative in order that axons may branch in the direction of either higher or lower concentrations. Note that computing the divergence in simulation requires cells on the cortical sheet to communicate with immediately adjacent cells only (see Methods). Here ${\chi}_{i}=0$ is a placeholder. The second term on the right represents the coupling between axon branching and synaptogenesis, and an assumption that the spatial distribution of synaptic density across the cortical sheet is broadly homogeneous. As such, the quantity c_{i} can be thought of as the connection density.
Results
First, we verified that all results established by Karbowski and Ermentrout, 2004 for a 1D axis could be reproduced using our extension to a 2D cortical sheet. Using an elliptical domain, S, with $M=3$ offset guidance gradients aligned to the longer axis, $N=5$ thalamocortical projections gave rise to five distinct cortical fields at locations that preserved the topographic ordering defined by the original γ values. However, we found that specifying N ordered areas required $M\approx (N+1)/2$ signalling fields. This is because localization of axon densities occurs only when projections are influenced by interactions with two or more signalling gradients that encourage migration in opposing directions. As the number of guidance fields is unlikely to approach the number of individual barrels, modifications to the model were required.
We reasoned that an arbitrary number of distinct field locations may be determined by a minimum of two guidance gradients, if the concentration of the projection densities is influenced by competition between projections, and if a projection that interacts more strongly with a given guidance gradient migrates further in the direction of that gradient. Accordingly, projections that interact most strongly with a given guidance gradient would come to occupy cortical locations at which that field has extreme values, leaving adjacent locations available to be occupied by projections with the next strongest interactions, and so forth. This would in principle allow the relative locations of the fields to be specified by the relative values of the interaction parameters, γ, and hence for a topological map in the cortex to be specified by a spatial ordering of the γ values at the level of the thalamus.
Such dynamics are quite unlike those described by classic chemospecificity models (Sperry, 1963), which essentially assume centrepoints by specifying conditions in the target tissue that instruct preidentified afferents to stop growing. Consider, for example, that when simulated in isolation from oneanother, all projections in the model described would simply migrate to the extrema of the cortical guidance fields.
Testing this reasoning required increasing the strength of the competition between simulated thalamocortical projections for cortical territory, by increasing the tendency for each projection to compete for cortical space in which to branch and make connections. The major modification required was thus to introduce into the model an additional source of competition between thalamic projections. The term in parentheses in Equation 1 represents competition between thalamocortical projections for a limited availability of cortical connections. To introduce competition also in terms of axon branching, whilst ensuring that a_{i} is conserved over time, we redefined
This term contributes to the flux of axonal branching as an additional source of diffusion, scaled by $\u03f5$, which reduces the branching density for a given projection where the branches of other projections are dense. Note that this operation is local to individual afferent projections.
In addition, the model we have outlined requires that molecular guidance gradients in the cortex are complemented by graded values of the interaction strengths, γ, at the level of the thalamus. While the precise mechanisms by which thalamic and cortical gradients interact during development have not been fully characterised, the presence of complementary thalamic and cortical guidance gradients has been well established experimentally. In particular, the EphA4 receptor and its ligand ephrinA5 are distributed in complementary gradients in the somatosensory thalamus and cortex (Vanderhaeghen et al., 2000; Miller et al., 2006). Cells originating in VPM express high levels of EphA receptors and project to the lateral part of S1, which expresses low levels of ephrinA5, and cells originating in the VPL express low levels of EphA receptors and project to the medial part of S1, which expresses high levels of ephrinA5 (see Gao et al., 1998; Dufour et al., 2003; Vanderhaeghen and Polleux, 2004; Speer and Chapman, 2005; Torii et al., 2013). We assume that such patterning arises because the relative strengths of interaction with guidance molecules (e.g., ephrinA5) in the cortex are correlated with the relative concentrations of complementary molecules (e.g., EphA4) in the thalamus, and thus with thalamic position along the axis to which their gradients are aligned.
For simplicity, the two simulated thalamic interaction gradients, as well as the two cortical guidance gradients, were initially chosen to be linear and orthogonal. Hence a given pair of γ values corresponds to the coordinate of a barreloid centre in the VPM. Coordinates, in a reference plane defined by the anteriorposterior and mediallateral axes, were estimated from Figure 5d of Haidarliu and Ahissar, 2001, and scaled such that $\gamma \in \pm 2$. Note that this scaling is arbitrary because according to the model the coordinates provide relative position information only.
A cortical boundary enclosing barrels for 41 macrovibrissae was traced from a cytochrome oxidase stain from Zheng et al., 2001 (using original data kindly supplied by the authors), and Equations 1–3 were solved for $N=41$ projections on the resulting domain, S, using $M=2$ linear signalling gradients aligned with the anteriorposterior and mediallateral axes. These gradients are shown with the barrel field boundary in Figure 1A for clarity, though like ephrinA5 they may be thought of as extending across the cortical hemisphere (Miller et al., 2006). Simulations were stepped through 30000 iterations of Equations 1–3 ($\delta t=0.0001$).
Across a wide range of parameter values, random initial conditions (a uniform random distribution for $a(\mathbf{\mathbf{x}},0)\in (0.2,0.4)$, $c(\mathbf{\mathbf{x}},0)=0$) eventually yielded a clear Voronoilike tessellation of topographically organized thalamocortical projections, confirming that barrel maps can selforganize in the absence of prespecified centre points. The organization is apparent in a plot of the identity of the projection for which the connection density is maximal at each simulated cortical location, as shown in Figure 1C. Parameter values for this example simulation (see also Figure 1—video 1) were obtained by conducting a full parameter sweep and choosing a combination ($\alpha =3.6$, $\beta =16.67$, $k=3$, $D=0.5$, $\u03f5=1.2$) that scored well against the following three measures.
First, we used an algorithm introduced by Honda to measure the discrepancy of each barrel shape from a Dirichlet domain shape (Honda, 1983). Low overall values of this Hondaδ metric obtained from simulated barrels indicate a close correspondence of the simulated barrel field with a Voronoi tessellation, and thus with a biological barrel field (for mice $\delta \approx 0.054$, Senft and Woolsey, 1991, and our analysis of data from Zheng et al., 2001 indicates that the value for rats is similar). For the tessellation that is overlaid on the real barrel field in Figure 1A, $\delta =0.025$, and a reduction in δ in the example simulation over time confirmed that an equivalent ‘good’ Voronoi pattern can emerge within ≈ 20000 iterations (Figure 1D, red circles). Second, we devised a pattern difference measure that is sensitive to deviations in the component shapes and overall topographic registration between two tessellations, η, and we used this measure to compare the simulated barrel fields to the real barrel field from which the boundary shape applied to the simulation was obtained (see Methods for details). A similar reduction in η in the development of the example simulation confirmed that the shapes and arrangement of emergent connection fields came to match those of the real barrel field by around 20000 iterations (Figure 1D, black squares). Third, we measured the connection selectivity, ω, at each location on the cortical sheet, as the connection density of the most dense projection divided by the sum over all projection densities. The overall connection selectivity increased as the barrel map selforganized in the example simulation (Figure 1D, grey hexagons), and the selectivity became concentrated in regions overlapping with the emergent barrel centres (Figure 1E).
Against these three metrics we are also able to characterise the robustness of selforganization to the model parameters, and to investigate the sensitivity of the model to variation in its inputs. Figure 2A shows values of δ, η and ω obtained after 30000 iterations, from 216 independent simulations, each representing a unique combination of the model parameters D, $\u03f5$, and the ratio $\alpha /\beta $. First, we observe that selforganization is highly robust to the ratio $\alpha /\beta $, across five orders of magnitude, with respect to all three metrics. Second, the most strongly Voronoiconforming patterns (low Hondaδ) were generated by simulations in which the diffusion constant D and the strength of competition $\u03f5$ were high. Third, strongest overall connection selectivities, ω, were obtained for lower values of D. Fourth, variation in the pattern difference metric, η, indicated that the alignment between real and simulated patterns was greatest for intermediate rates of diffusion, $D\approx 0.5$. Together these results indicate that when competition is strong, the rate of diffusion determines a tradeoff such that fields emerge to be barrelshaped when diffusion is fast and they emerge to be more selectively innervated when diffusion is slow.
The parameters of the example simulation are indicated in Figure 2A using an asterisk. In Figure 2B, we also present examples of alternative patterns that emerge for different choices of D. Decreasing the rate of diffusion may be considered equivalent to increasing the overall size of the domain, S. Hence, insights into barrel development in species with a larger representation of the vibrissae, which do not have barrel fields, may be gained by studying pattern formation when D is small. In this context, it is interesting to note that for small D, the organization is predicted to be topological but highly irregular, with a general expansion in the territory occupied by the central versus peripheral domains that would presumably manifest as an absence of identifiable barrel fields (Figure 2Biii).
Next we conducted a sensitivity analysis to determine the extent to which the quality of the pattern (after $t=30000$ iterations) is affected by perturbations to (i) the magnitude and offset of the noise applied to a_{i} at $t=0$; (ii) noise applied to the interaction parameters, ${\gamma}_{i,j}$; (iii) noise (at various length scales) applied to the guidance fields; and (iv) the magnitude and orientation of one cortical guidance field relative to the other (Figure 1A).
Using the parameters of the example simulation (Figure 1C) we established baseline mean and standard deviations from ten independent simulations with initial uniform random values for $a(\mathbf{\mathbf{x}},0)\in (0.2,0.4)$, to be $\delta =0.089\pm 0.004$, $\eta =0.2108\pm 0.002$, and $\omega =0.2165\pm 0.0001$. Repeating with the variation in the initial noise doubled ($a(\mathbf{\mathbf{x}},0)\in (0.1,0.5)$), or removed altogether ($a(\mathbf{\mathbf{x}},0)=0.3$), generated distributions of δ, η, and ω that were not statistically different, as established using paired twosample ttests. Adding noise to the interaction parameters (γ) affected neither the Hondaδ or the connection selectivity measures substantially (see Figure 3A), and an increase in the pattern difference reflected an increase in the occurrence of topological defects only when perturbations became so large as to cause the ordering of γ values from neighbouring thalamic sites to be switched (see example map Figure 3Bi). Adding noise to the cortical guidance field values, ${\rho}_{1}(\mathbf{\mathbf{x}})$ and ${\rho}_{2}(\mathbf{\mathbf{x}})$, disrupted pattern formation only for high levels of noise applied at short length scales, which manifested as nonstraight edges at the domain boundaries (Figure 3Bii). Varying the slope of one linear gradient ${\rho}_{1}(\mathbf{\mathbf{x}})$ while keeping that of the other constant caused elongation of the emergent domains along the corresponding axis (Figure 3Biii), while pattern formation was not strongly influenced by relaxing the assumption that the gradients of the two cortical guidance fields are orthogonal (Figure 3Biv). Overall, the sensitivity analysis revealed that selforganization of barrellike fields in the model is highly robust to a wide range of sources of perturbation.
To further investigate the interplay of genes intrinsic to the developing neocortex and extrinsic factors such as thalamocortical input, we simulated two well known experimental manipulations of barrel development. First, we simulated a seminal barrel duplication paradigm (Shimogori and Grove, 2005; Assimacopoulos et al., 2012) in which the growth factor Fgf8, which is normally expressed at the anterior end of the cortical subplate from around E9.5 (Crossley and Martin, 1995), is ectopically expressed (by electroporation) also at the posterior pole. We assume that this results in a mirror of the primary barrel cortex boundary along the rostrocaudal axis (Assimacopoulos et al., 2012) and a mirroring of the anteriorposterior guidance gradient ${\rho}_{1}$ at the border between them (Figure 4A). The result after 30000 iterations, and otherwise using the parameters of the example simulation, was two mirrorsymmetrical barrel fields comprising $2N$ barrels (Figure 4B), consistent with the outcome of the original experiments.
Finally, to investigate the response of the model to environmental manipulation, we simulated a whisker deprivation experiment. In a critical period comprising the first postnatal days, removal of the whiskers by electrocauterization, plucking, or trimming leads to observable changes in brain structures, including the barrel field (Jeanmonod et al., 1981). Amongst other changes, deprivation of individual whiskers leads to smaller barrels (Kossut, 1992). We simulated trimming of the individual whisker C3 during the critical period by reducing the competitiveness of the C3 thalamocortical projection, $\u03f5$. As a result, the corresponding field size was smaller (Figure 3C), and the size of the fields representing the neighbours of C3 increased in size. A reduction in area to 65%, comparable to that induced by Kossut, 1992, was obtained in simulation when $\u03f5$ was reduced to 86% of its default value (Figure 3D), and the C3 barrel disappeared altogether when $\u03f5$ was less than half of its original value.
Although individual whisker trimming reduces barrel size, if an entire row is trimmed, barrel sizes for the trimmed row are not obviously changed (Land and Simons, 1985). We investigated with a simulation in which we varied $\u03f5$ for all of the row C projections. Although the barrels which formed for row C did show some reduction in area (Figure 4D), this reduction was small when compared with the individually trimmed C3 simulation. If the effect of trimming any whisker is to reduce its $\u03f5$ (the competitiveness of its projection) to 86% of its original value, then the model predicts that the cortical barrels for the trimmed C row will retain 91% of their area on average.
Together with the results of simulated misexpression, the consistency of the simulated whisker trimming results with those of the original studies demonstrates how the model can be used to investigate the contribution of intrinsic and extrinsic factors to the development of cortical fields.
Discussion
The present results suggest that the key requirements for the emergence of realistic barrel patterning are (i) at each cortical location thalamocortical projections compete for a limited number of available synaptic connections (Equations 1–2), (ii) at each location the branching rate of a given projection is reduced by the density of other projections (Equation 3), and (iii) the branch density of each projection is conserved over time.
The emergence of barrels in simulation required competition between thalamic projections in terms of synaptic connectivity and also competition in terms of cortical space, as represented by χ, with an implicit requirement for a self/other identifier amongst projections. This latter form of competition may account for the absence of barrels in rodents with larger brains, such as capybara, for which competition for space is presumably weaker (Woolsey et al., 1975). Hence, irrespective of whether barrels are necessary for adaptive whisker function, the emergence of somatotopically ordered modular structures may be an inevitable consequence of local competition for cortical territory driven by input from an array of discrete sensory organs (Purves et al., 1992).
In reality, the Voronoi tessellation is extended by scores of smaller barrels alongside the Erow barrels, which represent the microvibrissae, and presumably form via the same competitive processes. Enforcing here the same boundary condition as used to represent the true edges of the barrel field was necessary to ensure the stability of the simulation, though we acknowledge that this region of the boundary was enforced primarily to keep the number of simulated projections, and hence the overall computational complexity of the simulation, manageable (simulating an extra projection introduces 13030 new dynamical variables).
It is important to emphasize that the formulation of the model is entirely local, insofar as simulation requires no information to be communicated from a given cortical grid cell to any but those immediately adjacent (via diffusion). Hence the simulations demonstrate how a selforganizing system, constrained by genetically specified guidance cues and by the shape of the cortical field boundary, can faithfully reproduce an arrangement of cell aggregates in one neural structure as a topographic map in another.
Moreover, the present results confirm that somatotopic map formation does not require the prespecification of centrepoints by as yet undetermined additional developmental mechanisms.
Methods
We concentrated on the representation of the fortyone macrovibrissae that constitute a given barrel field, because their thalamic and cortical correlates are easily identifiable and consistently located, excluding the five rhinal whiskers as their cortical representation is isolated from the main barrel field. We excluded the representation of the microvibrissae to limit the overall complexity of the simulations.
The cortical sheet was modelled as a two dimensional hexagonal lattice, which simplifies the computation of the 2D Laplacian. Within a boundary traced around the edge of a rat barrel field (Figure 1A) we set the hextohex distance d to 0.03 mm, which resulted in a lattice containing 6515 hexes for the simulations shown in Figure 1A,C and D and 12739 hexes for the Fgf8 misexpression study shown in Figure 3. Each hex contained 82 timedependent variables: 41 branching densities (a_{i}) and 41 connection densities (c_{i}). The rate of change of each of the timedependent variables (Equations 1 and 2) was computed using a fourthorder RungeKutta method.
The most involved part of this computation is to find the divergence of the flux of axonal branching, ${\mathbf{\mathbf{J}}}_{i}(\mathbf{\mathbf{x}},t)$, the term in parentheses in Equation 2:
where ${\widehat{a}}_{i}\equiv {\sum}_{j\ne i}^{N}{a}_{j}$. Note that the sum of the guidance gradients is timeindependent and define ${\mathbf{\mathbf{g}}}_{i}(\mathbf{\mathbf{x}})\equiv {\sum}_{j=1}^{M}{\gamma}_{i,j}\nabla {\rho}_{j}(\mathbf{\mathbf{x}})$. Because the divergence operator is distributive, Equation 4 can be expanded using vector calculus identities (dropping references to $\mathbf{\mathbf{x}}$ and t for clarity):
Applying the vector calculus product rule identity yields
which has five elements to compute: (i) $D\mathrm{\nabla}\cdot \mathrm{\nabla}{a}_{i}$ (the Laplacian of a_{i}); (ii) a timeindependent modulator of a_{i} (because $\mathrm{\nabla}\cdot {\mathbf{g}}_{i}$ is a timeindependent static field); (iii) the scalar product of the static vector field ${\mathbf{\mathbf{g}}}_{i}$ and the gradient of a_{i}; (iv) the Laplacian of ${\widehat{a}}_{i}$; and (v) a term involving the gradients of a_{i} and ${\widehat{a}}_{i}$. Each of the divergences can be simplified by means of Gauss’s Theorem following Lee et al., 2014.
The computation of the mean value of the Laplacian across one hexagon of area $\mathrm{\Omega}=\frac{\sqrt{3}}{2}{d}^{2}$, located at position ${\mathbf{\mathbf{p}}}_{0}$, with neighbours at positions ${\mathbf{\mathbf{p}}}_{1}$–${\mathbf{\mathbf{p}}}_{6}$ is
(7) $\begin{array}{ll}{\displaystyle \u27e8D\mathrm{\nabla}\cdot \mathrm{\nabla}{a}_{i}({\mathbf{p}}_{0},t)\u27e9=\frac{D}{\mathrm{\Omega}}{\u222f}_{\mathrm{\Omega}}\mathrm{\nabla}\cdot \mathrm{\nabla}{a}_{i}(\mathbf{x},t)\phantom{\rule{negativethinmathspace}{0ex}}\mathrm{d}\mathrm{\Omega}}& {\displaystyle =\frac{D}{\mathrm{\Omega}}\oint \frac{\mathrm{\partial}{a}_{i}}{\mathrm{\partial}\hat{\mathbf{n}}}\phantom{\rule{negativethinmathspace}{0ex}}\mathrm{d}\gamma}\\ & {\displaystyle \approx \frac{D}{\mathrm{\Omega}}\sum _{j=1}^{6}\frac{\mathrm{\partial}{a}_{i}({\mathbf{p}}_{j})}{\mathrm{\partial}\hat{\mathbf{n}}}{}_{\mathrm{m}\mathrm{i}\mathrm{d}}v}\\ & {\displaystyle =\frac{2D}{\sqrt{3}{d}^{2}}\sum _{j=1}^{6}\frac{{a}_{i}({\mathbf{p}}_{j}){a}_{i}({\mathbf{p}}_{0})}{d}\frac{d}{\sqrt{3}}}\\ & {\displaystyle =\frac{2D}{3{d}^{2}}\sum _{j=1}^{6}({a}_{i}({\mathbf{p}}_{j}){a}_{i}({\mathbf{p}}_{0})),}\end{array}$where $v=d/\sqrt{3}$ is the length of each edge of the hexagon and $\mathrm{d}\gamma $ is an infinitesimally small distance along its perimeter.
The computation of the second term in Equation 6, $\u27e8{a}_{i}({\mathbf{p}}_{0},t)\mathrm{\nabla}\cdot {\mathbf{g}}_{i}({\mathbf{p}}_{0})\u27e9$, can be written out similarly:
(8) $\begin{array}{ll}{\displaystyle \frac{1}{\mathrm{\Omega}}{\u222f}_{\mathrm{\Omega}}{a}_{i}\mathrm{\nabla}\cdot {\mathbf{g}}_{i}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\mathrm{d}\mathrm{\Omega}}& {\displaystyle =\frac{{a}_{i}({\mathbf{p}}_{0},t)}{\mathrm{\Omega}}\oint {\mathbf{g}}_{i}\cdot \phantom{\rule{negativethinmathspace}{0ex}}\mathrm{d}\hat{\mathbf{n}}}\\ & {\displaystyle \approx \frac{{a}_{i}({\mathbf{p}}_{0},t)}{\mathrm{\Omega}}\sum _{j=1}^{6}\frac{{\mathbf{g}}_{i}({\mathbf{p}}_{j})+{\mathbf{g}}_{i}({\mathbf{p}}_{0})}{2}\cdot \hat{\mathbf{n}}\phantom{\rule{thickmathspace}{0ex}}v}\\ & {\displaystyle =\frac{2{a}_{i}({\mathbf{p}}_{0},t)v}{\sqrt{3}{d}^{2}}\sum _{j=1}^{6}[\frac{{g}_{i}^{x}({\mathbf{p}}_{j})+{g}_{i}^{x}({\mathbf{p}}_{0})}{2}\cdot \hat{\mathbf{n}}+\frac{{g}_{i}^{y}({\mathbf{p}}_{j})+{g}_{i}^{y}({\mathbf{p}}_{0})}{2}\cdot \hat{\mathbf{n}}]}\\ {\displaystyle \Rightarrow \u27e8{a}_{i}({\mathbf{p}}_{0},t)\mathrm{\nabla}\cdot \mathbf{g}({\mathbf{p}}_{0})\u27e9}& {\displaystyle \approx \frac{{a}_{i}({\mathbf{p}}_{0},t)}{3d}\sum _{j=1}^{6}[({g}_{i}^{x}({\mathbf{p}}_{j})+{g}_{i}^{x}({\mathbf{p}}_{0}))\mathrm{cos}(\frac{\pi}{3}(j1))+({g}_{i}^{y}({\mathbf{p}}_{j})+{g}_{i}^{y}({\mathbf{p}}_{0}))\mathrm{sin}(\frac{\pi}{3}(j1))],}\end{array}$where ${g}_{i}^{x}$ and ${g}_{i}^{y}$ are the Cartesian components of ${\mathbf{\mathbf{g}}}_{i}$. Both this last expression, and the final expression of Equation 7 can be computed locally, by summing over values of the nearest neighbours.
The middle term in Equation 6 is the scalar product of two vector fields which is straightforward to compute from their Cartesian components.
The same method used to compute $\mathrm{\nabla}\cdot \mathrm{\nabla}{a}_{i}$ in term (i) is used to compute $\mathrm{\nabla}\cdot \mathrm{\nabla}{\hat{a}}_{i}$.
The final term is the scalar product of the two vector fields $\nabla {a}_{i}$ and $\nabla {\widehat{a}}_{i}$.
By separating the computation of Equation 4 into parts (i)–(v), the noflux boundary condition,
can be fulfilled. On the boundary, the contribution to $\mathbf{\mathbf{J}}$ resulting from the first term of Equation 6 can be fixed to 0 by the ‘ghost cell method’ in which, during the evaluation of (i), a hex outside the boundary containing the same value as the hex inside the boundary is imagined to exist such that the flux of $\mathbf{\mathbf{J}}$ across the boundary is 0. Then, ${\mathbf{\mathbf{g}}}_{i}(\mathbf{\mathbf{x}})$ can be tailored so that it, and its normal derivative, approach 0 at the boundary, ensuring that the second and third terms of Equation 6 also contribute nothing to $\mathbf{\mathbf{J}}$. This is achieved by multiplying ${\mathbf{\mathbf{g}}}_{i}(\mathbf{\mathbf{x}})$ by a sharp logistic function of the distance, d_{b}, from $\mathbf{\mathbf{x}}$ to the boundary, of the form $1/[1+\mathrm{exp}(100({d}_{f}{d}_{b}))]$, where ${d}_{f}=0.1$ mm $\approx 3d$ is the boundary falloff distance.
The pattern difference metric, η, incorporates information about the differences in the areas of simulated and experimentally determined (real) barrels (${\mathcal{A}}_{i}^{\mathrm{sim}}$ and ${\mathcal{A}}_{i}^{\mathrm{exp}}$), as well as information contained in the ‘adjacency vector’ for each barrel, ${\mathcal{V}}_{i}$, the jth element of which is the length of the border between barrel i and j. For a wellformed barrel pattern, ${\mathcal{V}}_{i}$ is a sparse vector. A dimensionless quantity can be obtained from the scalar product of the simulated and experimental adjacency vectors: $\frac{1}{N}\sum _{i}\frac{{\mathcal{\mathcal{V}}}_{i}^{\mathrm{s}\mathrm{i}\mathrm{m}}}{{b}_{i}^{\mathrm{s}\mathrm{i}\mathrm{m}}}\text{}\cdot \text{}\frac{{\mathcal{\mathcal{V}}}_{i}^{\mathrm{e}\mathrm{x}\mathrm{p}}}{{b}_{i}^{\mathrm{e}\mathrm{x}\mathrm{p}}}$, where for example ${b}_{i}^{\mathrm{exp}}$, is the total length of the border around real barrel i. This quantity is small when the fields that form in simulation have dissimilar neighbour relations to those of the real barrels (e.g., Figure 1C, $t=1000$), and maximal for a precise topological map ($t=10000$). A second comparison considers the mean magnitude of the difference between the simulated and experimental vectors: $\frac{1}{N}{\sum}_{i}\parallel {\mathcal{V}}_{i}^{\mathrm{sim}}{\mathcal{V}}_{i}^{\mathrm{exp}}\parallel $. This tends to 0 for a perfect match and can separate patterns with straight boundaries from those with ‘noisy edges’ (as in Figure 3Bii). We combined these terms into a single metric:
which has units of mm^{3}.
The code required to reproduce these results is available at https://github.com/ABRGModels/BarrelEmerge/tree/eLife (James, 2020, copy archived at https://github.com/elifesciencespublications/BarrelEmerge). The computations described in (i)–(v) may be found in the class method RD_James_comp2::compute_divJ() which calculates term1, term2, term3, term1_1 and term1_2, respectively. BarrelEmerge depends on the software library morphologica (RRID:SCR_018813), which must also be compiled.
Data availability
All data and code required to reproduce the results in this work are available in the public repository https://github.com/ABRGModels/BarrelEmerge/tree/eLife (copy archived at https://github.com/elifesciencespublications/BarrelEmerge).
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Article and author information
Author details
Funding
James S. McDonnell Foundation (220020516)
 Sebastian S James
 Leah A Krubitzer
 Stuart P Wilson
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
The authors thank Jason Berwick at the University of Sheffield for advice and for access to the rat barrel stains used to construct Figure 1A. This work was supported by a Collaborative Activity Award, Cortical Plasticity Within and Across Lifetimes, from the James S McDonnell Foundation (grant 220020516).
Copyright
© 2020, James et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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