Multi-scale account of the network structure of macaque visual cortex

We estimate the number of neurons in each area and population in three steps:

The connection probabilities of the microcircuit model (Potjans and Diesmann 2014, Table 5) form the basis for the local circuit of each area. They provide an \(8\times 8\) matrix of population-specific connection probabilities that was compiled from anatomical and electrophysiological studies (with large contributions from Binzegger et al. 2004; Thomson and Lamy 2007). We adapt this circuit to all 32 areas by preserving the relative indegrees between local projections which leads to area-specific connection probabilities. To determine the fraction of type I and II (i.e., within-area) synapses for each area, we use retrograde tracing data from Markov et al. (2011) consisting of fractions of labeled neurons (FLN) per area as a result of injections into one area at a time. The measured FLN thus determine the numbers of source neurons for each projection. The fraction intrinsic to the injected area, \(\hbox {FLN}_{\mathrm {i}}\), is approximately equal for all nine areas where this fraction was determined, with a mean of 0.79. We assume that source neurons on average establish the same number of synapses in a given target area, independent of their location (inside or outside the given area). Combining this with the area-specific total numbers of synapses leads us to the total numbers of local synapses, which we distribute as further detailed in the “Results” section.

We treat all cortico-cortical connections as originating and terminating in the \(1\,\mathrm {mm}^{2}\) patches, ignoring their spatial divergence and convergence. We determine whether a pair of areas is connected based on the union of all connections reported in the FV91 scheme in the CoCoMac database (Stephan et al. 2001; Bakker et al. 2012; Suzuki and Amaral 1994a; Felleman and Van Essen 1991; Rockland and Pandya 1979; Barnes and Pandya 1992) (see Supplementary Sec. “Processing of CoCoMac data” for details) and all connections reported by Markov et al. (2014a). Numbers of synapses between areas are determined on the basis of the retrograde tracing data from Markov et al. (2014a). The data consist of fractions of labeled neurons \(\hbox {FLN}_{AB}=\hbox {NLN}_{AB}/\sum _{B^{\prime }}\hbox {NLN}_{AB^{\prime }}\) (analogous to the intrinsic fraction of labeled neurons \(\hbox {FLN}_{\mathrm {i}}\)), with \(\hbox {NLN}_{AB}\) the number of labeled neurons in area B upon injection in area A. To translate the data into numbers of synapses, we assume, similarly to the assumption on intrinsically versus extrinsically labeled neurons, that a neuron projecting to a target area establishes the same number of synapses regardless of the source area it is located in. Markov et al. (2014a) used a parcellation scheme called M132 which is also available as a cortical surface, both in native and in F99 space, a standard macaque cortical surface included with Caret (Van Essen et al. 2001). For each injection, we identify the corresponding area in the FV91 parcellation (Supplementary Table S4) by registering the coordinates of the injection site to the F99 atlas available via the Scalable Brain Atlas (Bakker et al. 2015). There are data for 11 visual areas in the FV91 scheme with repeat injections in six areas, for which we take the arithmetic mean. To map data on the source side from M132 to FV91, we count the number of overlapping triangles on the F99 surface between any given pair of regions and distribute the FLN proportionally to the amount of overlap, using the F99 region overlap tool at the CoCoMac site (http://​cocomac.​g-node.​org). To fill in gaps in the FLN data, we exploit the exponential decay of connection density with inter-areal distance (Ercsey-Ravasz et al. 2013). Supplementary Table S5 lists all distance values, which we compute as the median of the distances between all vertex pairs of the two areas in their surface representation in F99 space.

Layer-specific tracing results from the CoCoMac database (Stephan et al. 2001; Bakker et al. 2012) and Markov et al. 2014b help us determine the distribution of connections across source and target layers. On the source side, the laminar projection pattern can be expressed as the fraction of supragranular labeled neurons (SLN) in retrograde tracing experiments (Markov et al. 2014b). To map the SLN from the M132 to the FV91 scheme, we use the exact coordinates of the injections to determine the corresponding target area A in the FV91 parcellation, and for each pair of areas we take the mean SLN across injections. At this point, the source areas are still in the M132 parcellation. To map the source areas from M132 to FV91, we weight the SLN by the overlap \(c_{B,\beta }\) between area \(\beta\) in the former (M132) and area B in the latter (FV91) scheme and the FLN,This weighting with the FLN reflects the fact that denser connections more strongly determine the overall distribution of labeled neurons across supra- and infragranular layers. We estimate missing values based on a sigmoidal fit of SLN versus the logarithmized ratio of overall cell densities of the two areas (Fig. 5a). This is similar to the relation between SLN and the hierarchical level differences found by Markov et al. (2014b), although there, the hierarchical ordering of areas was obtained using the SLN data in the first place. With this approach, the goodness of fit is difficult to evaluate, because some degrees of freedom are used up to determine the hierarchy itself. A relationship between laminar patterns and log ratios of neuron densities was suggested by Beul et al. (2017). Following Markov et al. (2014b), we use a beta-binomial model, assuming the numbers of labeled neurons in the source areas to sample from a beta-binomial distribution (e.g., Weisstein 2005). This distribution arises as a combination of a binomial distribution with probability p of supragranular labeling in a given area, and a beta distribution of p across areas with dispersion parameter \(\phi\). With the probit link function g (e.g. McCulloch et al. 2008), the measured \(\hbox {SLN}_{AB}\) relates to the log ratio \(\ell _{AB}\) of overall neuron densities for each pair of areas aswhere \(\{a_{0},a_{1}\}\) are scalar fit parameters. We perform this fit in the original scheme (M132) under the assumption that mapping cell densities between schemes introduces fewer errors than mapping SLN would. For mapping the cell densities to M132 we again employ the overlap tool of CoCoMac (see above) and compute the cell density of each area in the M132 scheme as a weighted average over the associated FV91 areas. For areas with identical names in both schemes, we simply take the neuron density from the FV91 scheme. We compute the SLN fit in R (R Core Team 2015) with the betabin function of the aod package (Lesnoff and Lancelot 2012). In contrast to Markov et al. (2014b), who exclude certain areas when fitting SLN versus hierarchical distance in view of ambiguous hierarchical relations, we take all data points into account to obtain a simple and uniform rule. We also tested a logit link function and found nearly identical results (Supplementary Fig. S1C).

As a further step, we combine SLN with tracing results from CoCoMac (Felleman and Van Essen 1991; Barnes and Pandya 1992; Suzuki and Amaral 1994b; Morel and Bullier 1990; Perkel et al. 1986; Seltzer and Pandya 1994). The data sets complement each other: SLN provides quantitative information on laminar patterns of outgoing projections for about one quarter of the connected areas, distinguishing only between supra- and infragranular layers. CoCoMac has values for all six layers (which we denote by \(\alpha (\nu )\)), but limited to a qualitative strength ranging from 0 (absent) to 3 (strong) which we take to represent numbers of synapses in orders of magnitude (for further details see Supplementary Sec. “Processing of CoCoMac data”). On the target side, we determine the pattern of target layers \(P_{\mathrm {t}}\) from anterograde tracer studies in CoCoMac (Jones et al. 1978; Rockland and Pandya 1979; Morel and Bullier 1990; Webster et al. 1991; Felleman and Van Essen 1991; Barnes and Pandya 1992; Distler et al. 1993; Suzuki and Amaral 1994b; Webster et al. 1994) if available (29% coverage), and otherwise determine it from the source pattern, as further described in the “Results” section.

Anterograde tracing experiments characterize target patterns of projections in terms of the locations of the synapses, whereas the layer that forwards incoming input depends on the location of the cell body. Therefore, to characterize polysynaptic pathways, it is necessary to bridge the descriptions in terms of cell body and synapse locations. To this end, we relate synapse to target cell body locations using the following cat V1 data from Binzegger et al. (2004), which are listed in the table in Fig. 9 of Izhikevich and Edelman (2008): first, the probability \(\mathcal {P}(s_{\mathrm {cc}}|c_{\mathrm {B}}\bigcap s\in v)\) for a synapse in layer v on a cell of type \(c_{\mathrm {B}}\) (e.g., a pyramidal cell with soma in L5) to be of cortico-cortical origin (20th column in the table in Fig. 9 of Izhikevich and Edelman 2008); second, the relative occurrence \({\mathcal {P}}(c_{\mathrm {B}})\) of the cell type \(c_{\mathrm {B}}\) (first column); and third, the total numbers of synapses \(N^{\mathrm {syn}}(v,c_{\mathrm {B}})\) in layer v onto individual cells of the given type (second column). The latter do not equal the numbers of synapses onto the neurons in our network; we rather use them as auxiliary quantities in the calculation. Binzegger et al. (2004) distinguish 17 different cell types, which we map to the 8 cortical populations considered in our network based on the laminar position of their cell body and their excitatory or inhibitory nature (Supplementary Table S10). To transform the data of Binzegger et al. (2004) from cat to macaque, we adjust the occurrence of each cell type \(c_{\mathrm {B}}\) associated with a population i according to the different relative population sizes in cat and macaque V1. For this, we compute the occurrence of population i in macaque V1, \(\mathcal {P}_{i,\mathrm {V1}}=N_{i,\mathrm {V1}}/N_{\mathrm {total,}V1}\) and divide it by the sum of occurrences of all cell types associated with population i in cat. The occurrence of \(c_{\mathrm {B}}\) is then multiplied by this factor: \({\mathcal {P}}(c_{\mathrm {B}})\rightarrow \mathcal {\mathcal {\mathcal {P}}}(c_{\mathrm {B}})\cdot \mathcal {P}_{i,\mathrm {V1}}/\sum _{c_{\mathrm {B}}^{\prime }\in i}\mathcal {P}(c_{B}^{\prime })\). The “Results” section details the corresponding derivation.

Cortico-cortical feedback connections preferentially target excitatory rather than inhibitory neurons, i.e., a disproportionately high number of synapses is formed onto excitatory neurons (Johnson and Burkhalter 1996; Anderson et al. 2011). We choose a fraction of 93% of connections targeting excitatory neurons, as an average over experimental values ranging between 87 and 98%.

Intra-areal synapses originating outside the \(1\,\mathrm {mm}^{2}\) patches (type II) and those coming from outside vision-related cortex, that is, non-visual and subcortical inputs (type IV) are external inputs for our purposes. While they do not form an intrinsic part of the system under consideration, these inputs provide a more comprehensive picture of the network and are relevant for investigations of the network dynamics. Therefore, we estimate these inputs for completeness. As further explained in the results, we can estimate the numbers of type II synapses to some extent from local connectivity profiles, but the available data do not allow us to faithfully determine the contribution from remote intra-area connectivity (patchy connections) for all areas. Furthermore, quantitative area-specific data on non-visual and subcortical inputs are highly incomplete. For these reasons, we jointly describe the type II and type IV synapses using a simple scheme: for each area, we compute the total number of external synapses as the difference between the total number of synapses (determined from the volume density of synapses) and those of type I and III and distribute these such that all neurons in the given area have the same indegree for external sources. Supplementary Table S6 lists the resulting external indegrees. Overall, external inputs amount to approximately 32% of the total inputs to each neuron in the network.

We investigate the community structure of the area-level network with the map equation method (Rosvall et al. 2009). In this clustering algorithm, an agent performs random walks between graph nodes according to their degree of connectivity and a certain probability of jumping to a random network node. We choose the probability for a certain target node to be selected to be proportional to the outdegree of the connection, and \(p=0.15\) as the probability of a random jump. The algorithm detects clusters in the graph by minimizing the length of a binary description of the network using a Huffman code. To assess the quality of the clustering, we compute a modularity measure which extends a measure for unweighted, directed networks (Leicht and Newman 2008) to weighted networks, analogous to Newman (2004),where \(\mathcal {K}_{AB}^{\mathrm {out}}\) and \(\mathcal {K}_{AB}\) respectively are the matrices of relative outdegrees and indegrees, \(m=\sum _{A,B}\mathcal {K}_{AB}^{\mathrm {out}}\) and \(\delta _{\mathcal {C}_{A},\mathcal {C}_{B}}=1\) if areas A and B are in the same cluster and 0 otherwise. \(Q=0\) reflects equal connectivity within and between clusters, while \(Q=1\) corresponds to connectivity exclusively within clusters.

To study paths in the network, we construct the weighted and directed graph of the network connectivity at the population level. While this graph only contains anatomical information, to identify the paths that are most relevant for activity propagation we take into account (1) the relative weight of inhibitory compared to excitatory synapses; and (2) the near-criticality of the brain (Robinson et al. 2014; Priesemann et al. 2014). Following Potjans and Diesmann (2014), we define the synaptic weight \(J_{\mathcal {E}}=0.15\,\mathrm {mV}\) for excitatory connections and \(J_{\mathcal {I}}=- \,4J_{\mathcal {E}}\) for inhibitory connections. We then construct a weight matrix G with elements \(g_{ij}=K_{ij}\cdot |J|\) where J is chosen depending on whether the source population is excitatory or inhibitory. This matrix is transformed into an approximate gain matrix by scaling the matrix by a factor representing the susceptibility of the target populations, i.e., the change in output activity for a unit change in input. For simplicity, we assume this susceptibility to be identical across populations. To reflect the near-criticality of the brain, we choose it to be equal the reciprocal of the maximal real part of the eigenvalues of G: \(G^{\prime }=G/\left[ {\mathrm {Re}}(\lambda )\right] _{\mathrm {max}}\), so that the maximal real part of the eigenvalues of the resulting matrix is \(\left[ {\mathrm {Re}}(\lambda ^{\prime })\right] _{\mathrm {max}}=1\). This scaling is relevant because it modulates the relative strengths of direct and indirect paths: a larger value of \(\left[ {\mathrm {Re}}(\lambda ^{\prime })\right] _{\mathrm {max}}\) increases the relative weighting of indirect paths. The weight of the edge from population j to i is then defined as \(g_{ij}^{\prime }\). The distance between two nodes in the graph is defined as the logarithm of the reciprocal of the weight, \(d_{ij}={\mathrm {log}}(1/w_{ij})\), so that summing the distances reflects a multiplication of the corresponding weights. We find the shortest paths between any two nodes of the graph using the Bellman–Ford algorithm (Shimbel 1955; Ford 1956; Bellman 1958). This algorithm finds the shortest paths emanating from vertex i on a graph with N vertices in an iterative manner: it starts by assigning an infinite path length to all other nodes k of the graph. Then, it loops through all edges (j, k) of the graph, tests if the path length \(p_{ij}\) plus the distance of the edge \(d_{jk}\) is smaller than the currently stored path length \(p_{ik}\), and, if so, assigns \(p_{ik}\rightarrow p_{ij}+d_{jk}\). By repeating this \(N-1\) times for all edges, the algorithm considers paths of increasing length at every iteration and ultimately finds the shortest paths between each pair of vertices. In contrast to Dijkstra’s algorithm, it can deal with edges with negative distance values.

We here derive the number of neurons in each population of the network from measured and estimated neuron densities, laminar thicknesses, and proportions of excitatory and inhibitory neurons. Overall neuron density and L4 neuron density increase with architectural type (Fig. 2a). Since we assign neuron densities to areas with missing data according to their architectural types, the same trends are present throughout the model. Total cortical thickness decreases with increasing logarithmized overall neuron density, \(\mathrm {log}\,\rho\), providing thickness estimates for the 18 areas not included in the empirical data set (Fig. 2b). The ratio of L4 thickness and total cortical thickness increases with \(\mathrm {log}\,\rho\), which predicts the relative L4 thickness for areas with missing data (Fig. 2c). Since the relative thicknesses of the other layers show no notable change with \(\mathrm {log}\,\rho\), we fill in missing values using the mean of the known data for these quantities and then normalize the sum of the relative thicknesses to 1 (Supplementary Table S7).

For deriving the local connectivity, as an intermediate step we require the full surface areas (Supplementary Table S8) to sample the tails of the Gaussian connectivity profiles, and not just the \(1\,\mathrm {mm}^{2}\) patches. For this purpose, we approximate each brain area as a flat disk with radius R and surface area \(S(R)=\pi R^{2}\), so that the number of neurons in population i of area A iswhere \(v_{i}\) denotes the layer of population i, \(D_{A,v_{i}}\) the thickness of layer \(v_{i}\), and \(\mathcal {E},\,\mathcal {I}\) the pool of excitatory and inhibitory populations, respectively. Supplementary Table S9 gives the population sizes corresponding to the \(1\,\mathrm {mm}^{2}\) area size we consider.

Each neuron receives synapses of four different origins (Fig. 3a): those originating inside the \(1\,\mathrm {mm}^{2}\) microcircuit (type I), the remaining intra-areal synapses (type II), cortico-cortical synapses from other vision-related areas (type III), and synapses from outside vision-related cortex (type IV).

For combined local and long-range connections, we assume a constant volume density of synapses across areas (Harrison et al. 2002). Experimental values for the average indegree in monkey V1 vary between 2300 (O’Kusky and Colonnier 1982) and 5600 (Cragg 1967) synapses per neuron. We take the average (3950) as representative for V1, resulting in a synapse density of \(\rho _{\mathrm {syn}}=8.3\times 10^{8}\;\frac{\text {synapses}}{\text {mm}^{3}}\), not far from the value of \(6.3\times 10^{8}\;\frac{\text {synapses}}{\text {mm}^{3}}\) measured in rat somatosensory cortex (Markram et al. 2015). This constant synapse density diversifies the areas in terms of their connectivity due to their different neuron densities. Combined with decreasing cell density along the gradient of architectural types (Fig. 2a), the constant synapse density leads to an increase in the average indegree of neurons in low-type areas compared to high-type areas (Fig. 3b). Primary visual cortex V1 has the lowest average indegree of \(\sim\)3950 synapses per neuron while neurons in the area with the lowest architectural type, TH, receive on average \(\sim\)14,000 synapses. The average indegree does not strictly increase with the overall neuron density due to differences in the area-specific laminar composition. Averaged across all areas, a neuron in the network receives approximately 9800 synapses.

We base the fraction of intra-areal synapses (types I + II) on fractions of labeled neurons intrinsic to the injected area (\(\hbox {FLN}_{\mathrm {i}}\)) in retrograde tracing experiments by Markov et al. (2011). Since the reported values are approximately constant across injected areas, we use the mean value of 0.79 for all 32 areas of the network. This leads us to the assumption that 79% of the synapses to a neuron are intra-areal (types I and II combined) and the remaining 21% stem from sources outside of the areas (types III and IV combined).

In the following two subsections, we explain the integration of the different data sources to yield the area- and population-specific numbers of synapses of types I and III.

We test if the network follows known organizing principles by analyzing the community structure in the weighted and directed graph of area-level connectivity. The map equation method (Rosvall et al. 2009) applied on the outdegrees reveals six clusters (Fig. 7). We test the significance of the corresponding modularity \(Q=0.38\) by comparing with 1000 surrogate networks conserving the total outdegree of each area by shuffling its targets. This yields \(Q=-0.03\pm 0.03\), indicating the significance of the clustering. The anatomical community structure shows a correspondence with known functional groupings. Two large clusters comprise ventral stream areas along with parahippocampal areas TH and TF, and dorsal stream areas, respectively. The grouping of areas TF and TH with ventral stream areas is reasonable in view of the involvement of these parahippocampal areas in object and spatial memory processes (Bachevalier and Nemanic 2008; Nemanic et al. 2004) and was also obtained for the binary connection matrix of Felleman and Van Essen (1991) containing about half of the connections present in our weighted connectivity matrix (Hilgetag et al. 2000). The polysensory dorsal stream areas STPp and STPa have strong recurrent connections and are thus grouped outside of the large dorsal stream cluster. Ventral areas VOT and PITd are grouped with early visual area VP and dorsal area MSTd. Early visual areas V1 and V2 form a separate cluster, as do the two frontal areas FEF and 46. Nonetheless, the clusters are heavily interconnected (Fig. 7). The basic separation into ventral and dorsal clusters matches that found for the binary connection matrix of Felleman and Van Essen (1991) (Jouve et al. 1998; Hilgetag et al. 2000), but there are also important differences. For instance, our clustering groups area 7a with the dorsal instead of the ventral stream, better matching the scheme described by Nassi and Callaway (2009), and early visual areas V1 and V2 as well as frontal areas 46 and FEF are placed in separate clusters, respectively, in line with the differential functional properties of these areas and their non-unique association with the dorsal and ventral streams. The community structure is robust against excluding a small percentage (\(\sim\)10%) of the experimental data of Markov et al. (2014a) that underlie the fit of the exponential relation between connection densities and inter-areal distance and estimating the connection densities of these connections from the fit (Supplementary Fig. S4). Furthermore, the community structure is robust against adding a random fluctuation to the estimated FLN on the order of the spread of the experimental data around the fit in Fig. 4a (Supplementary Fig. S4).

To investigate the implications of the derived connectivity for the communication between areas, we detect the shortest paths between pairs of areas in the network (see “Materials and methods”). The shortest paths between cortical areas follow distinct patterns depending on the structural and hierarchical relation between the areas. The laminar patterns of the shortest paths between directly connected areas depend on the hierarchical relation between the areas (Fig. 8a). In the feedforward direction, shortest paths predominantly start in 2/3E and end in 4E, as expected from the layer-specific connectivity (Fig. 5). Lateral shortest paths similarly mostly originate in 2/3E and terminate in 4E. Feedback paths, on the other hand, mostly start in the infragranular layers 5 and 6 and target neurons in layers 4 and 5.

Taking all pairs of areas into account regardless of whether they are directly connected, a similar picture emerges (Fig. 8b). Since SLN are only available for directly connected areas, we here group pairs of areas according to the difference between their architectural types. We call pathways from structurally differentiated to less differentiated areas ‘high-to-low-type’, those in the opposite direction ‘low-to-high-type’ and those between structurally similar areas ‘horizontal’. High-to-low-type pathways as well as horizontal pathways follow the 23E\(\rightarrow\)4E pattern. Low-to-high-type pathways, on the other hand, are more uniformly distributed with most paths starting in 5E or 6E, and ending in 4E or 5E. These observations consider only the first and last populations of the entire path. However, 45% of the shortest paths take a detour via one or multiple intermediate areas. Even if the two areas are directly connected, the direct connection is not the shortest (strongest) path in 10% of the cases. In intermediate areas, the shortest paths involve one or two populations. From high-type to low-type areas, these intra-area paths are mostly from 4E to 2/3E (Fig. 8c), in line with the start-end pattern shown in Fig. 8b, but a substantial fraction passes through 2/3E and 5E only. Indirect, horizontal paths mostly involve a relay via 5E, and to a lesser extent 2/3E and the 4E\(\rightarrow\)2/3E pattern. Similarly, connections from low-type to high-type areas are mostly forwarded by the 5E population only. These results suggest that directionally distinct paths in the population-level connectivity open up communication channels for specifically targeted cortico-cortical communication across sets of areas. We test the robustness of these findings against altering the SLN thresholds used for the hierarchical categorization of connections and against pruning of the SLN data underlying the network construction, including the sigmoidal fit (Fig. 5a). We found no qualitative variations in the paths between areas, meaning that our findings are independent of moderate variations in the underlying data and heuristics (Supplementary Fig. S5 and Supplementary Fig. S6). Furthermore, the laminar patterns of shortest paths remain qualitatively unchanged when only connections with available experimental SLN data are included in the analysis (Supplementary Fig. S7).