Instituto de Física de São Carlos, Universidade de São Paulo, Caixa Postal 369, São Carlos, SP, 13560-970, Brazil

School of Computing Science, Newcastle University, Claremont Tower, Newcastle upon Tyne, NE1 7RU, UK

Institute of Neuroscience, Henry Wellcome Building for Neuroecology, Newcastle University, Framlington Place, Newcastle upon Tyne, NE2 4HH, UK

Jacobs University Bremen, School of Engineering and Science, Campus Ring 6, 28759 Bremen, Germany

Boston University, Sargent College, Department of Health Sciences, 635 Commonwealth Ave, Boston, MA 02215, USA

Abstract

Background

The organization of the connectivity between mammalian cortical areas has become a major subject of study, because of its important role in scaffolding the macroscopic aspects of animal behavior and intelligence. In this study we present a computational reconstruction approach to the problem of network organization, by considering the topological and spatial features of each area in the primate cerebral cortex as subsidy for the reconstruction of the global cortical network connectivity. Starting with all areas being disconnected, pairs of areas with similar sets of features are linked together, in an attempt to recover the original network structure.

Results

Inferring primate cortical connectivity from the properties of the nodes, remarkably good reconstructions of the global network organization could be obtained, with the topological features allowing slightly superior accuracy to the spatial ones. Analogous reconstruction attempts for the

Conclusion

The close relationship between area-based features and global connectivity may hint on developmental rules and constraints for cortical networks. Particularly, differences between the predictions from topological and spatial properties, together with the poorer recovery resulting from spatial properties, indicate that the organization of cortical networks is not entirely determined by spatial constraints.

Background

Scientific-technological advances over the last decades have produced ever-increasing experimental knowledge about brain organization and dynamics. In particular, modern anatomical techniques have provided extensive data on the interconnections of cerebral cortical areas in the brains of animals such as the cat or rat, or non-human primates such as the rhesus monkey. The intricate, non-random connectivity of cortical brain regions mediates the diverse and flexible sensory, cognitive and behavioral functions of the mammalian brain. However, the topological organization of these networks

A fundamental open problem in systems neuroscience is the relationship between specialized features of local nodes, such as areas of the cerebral cortex, and the global interaction and integration of these nodes in the neural networks. One aspect of this relationship concerns the question from which features of the local nodes structural connectivity between them might be predicted.

We address this question with the help of network analysis approaches

It is important to note that the problem of network reconstruction from topological features is in a sense circular. Such features are derived from the complete connectivity of the network, so global connectivity may be inferred by taking itself into account. However, this is by no means a trivial task. For instance, guessing which nodes are specifically interconnected, based on measurements such as their degree or clustering coefficient, is almost invariably an impossible task. The exercise of trying to reconstruct the connections from a collection of topological measurements therefore provides an interesting new way to look at specific properties and structural organization of a complex network. For instance, in case the connectivity could be reasonably guessed from the node degree correlations, this would provide a key insight about its underlying organization.

We consider topological as well as spatial parameters, as biological networks, and brain networks in particular, are embedded in space. It is an interesting question to ask how the topological and spatial organization of these networks relates to each other. In particular, how do the topological and spatial features of individual nodes relate to the connectivity and layout of the whole network? Answers on these questions may inform current theories on the evolution and development of complex biological networks.

The Methods section of this article presents the adopted topological and spatial features and describes the reconstruction methodology based on similarity between sets of features. The analysis was applied to primate cortical brain connectivity (2,402 connections among 95 cortical areas of the Macaque monkey). In order to provide a comparative case, we also describe the application of the same methodology to

Analysis of

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Results

Overview and community analysis

Figure

Two-dimensional projection of the network of 95 cortical area-nodes at the centers of mass of the cortical regions, obtained through principal component analysis and for the complete set of their existing interconnections

Two-dimensional projection of the network of 95 cortical area-nodes at the centers of mass of the cortical regions, obtained through principal component analysis and for the complete set of their existing interconnections.

Figure

Spatial distribution of network connections

Spatial distribution of network connections. The histograms of Euclidean distances between all pairs of edges (a), number of existing edges with respective distance (b) and the ratio between (b) and (a).

Given a network, it is often the case that a subset of its nodes is more interconnected with one another than with the remainder of the network. Such a subset of nodes, together with the respective interconnections, is called a

The two identified communities represented in terms of the center of mass of each cortical region

The two identified communities represented in terms of the center of mass of each cortical region. Communities 1 and 2 are identified respectively by '×' and '*'.

Communities 1 and 2 were of comparable size and included _{1 }= 44 and _{2 }= 51 nodes, respectively, and _{1 }= 1326 and _{2 }= 1280 directed edges. The clustering coefficients obtained for the two identified communities were found to be equal to 0.52 and 0.68, respectively. This might be explained by the higher number of connections within communities. Whereas the global edge density of the network was 0.17, the densities within the communities were 0.50 and 0.66.

Topological characterization

Now we focus our attention on the analysis of the local node properties and connectivity of these two communities. Figure

Histograms of node degree, clustering coefficient, average distance, average topological distance, and matching index for community 1 (left-hand side column) and community 2 (right-hand side column)

Histograms of node degree, clustering coefficient, average distance, average topological distance, and matching index for community 1 (left-hand side column) and community 2 (right-hand side column). The asterisks identify the average of each measure. The histogram distributions of the two communities differed markedly for the node degree, however, the average degrees of the communities were very similar for all measures.

Spatial characterization

Figure

Histograms of local density, coefficient of variation, and area obtained for the two communities

Histograms of local density, coefficient of variation, and area obtained for the two communities. The three Cartesian coordinates are not shown because they are not invariant with position and rotation and can be readily inferred from the projection in Figure 3. The asterisks show the average of each measure. Although the average of the measurements for each respective community tends to be similar, the distributions in all histograms are markedly different.

An important issue to be considered while adopting several measurements is the quantification of possible relationships between them, which can be indicated by the Pearson correlation coefficients for all pairwise combinations of measures. The Pearson coefficients calculated independently for the topological and spatial measurements are given in Tables

Correlations of topological node-based measures.

1.00

---

---

---

0.36**

1.00

---

---

-0.46**

-0.28**

1.00

---

0.94**

0.34**

0.46**

1.00

Pearson correlation coefficients were obtained for all pairs of considered topological measurements (** indicates that correlation is significant at the 0.01 level, two-tailed).

Correlations of spatial node-based measures.

1.00

---

---

0.22*

1.00

---

-0.10

0.11

1.00

Pearson correlation coefficients were obtained for all pairs of considered spatial measurements (* indicates that correlation is significant at the 0.05 level, two-tailed).

Except for the strong correlation between the node degree and matching index, all other pairs of measurements were unremarkable, supporting the complementariness of the adopted sets of features.

Comparison between original and reconstructed networks

Table

Expected ratios of correct ones and zeroes, and respective geometrical average.

**Network**

_{1 }=_{1}

_{0 }= _{0}

**Cortical community 1**

0.68

0.32

0.46

**Cortical community 2**

0.49

0.51

0.50

Theoretically expected average ratios of correct ones and zeroes, as well as their respective geometrical averages, obtained for the two cortical communities.

We performed an exhaustive search while taking into account all 1-by-1, 2-by-2 and 3-by-3 combinations of each of the two types of considered measurements for a whole sequence of threshold values _{1 }and _{2 }with respect to the two main cortical communities considered in this work. It is clear from this table that the best synthesized networks were obtained by the matching index for the first community and the combination of (clustering coefficient, matching index) for the second community.

Network reconstruction from individual and combined topological node measures.

Community1

Community2

**Measurements**

1

0.797

0.647

2

0.563

0.577

3

0.539

0.611

4

**
**

0.664

1, 2

0.637

0.706

1, 3

0.770

0.670

1, 4

0.802

0.663

2, 3

0.532

0.638

2, 4

0.704

**
**

3, 4

0.782

0.678

1, 2, 3

0.688

0.694

1, 2, 4

0.782

0.720

1, 3, 4

0.800

0.678

2, 3, 4

0.689

0.703

Geometrical averages of the connectivity estimation obtained for the two communities while considering the 14 combinations of measurements listed in the first column. The best combinations for communities 1 and 2 were respectively the matching index (4) and the pair of measurements involving the clustering coefficient (2) and matching index (4). The configurations leading to the best matches have been emphasized.

Figure

Adjacency matrices of the original communities (a,b) and those of the respectively most similar reconstructed networks (c,d) considering topological features

Adjacency matrices of the original communities (a,b) and those of the respectively most similar reconstructed networks (c,d) considering topological features. The reconstructions in this figure correspond to the highlighted configurations in Table 4. Black points indicate presence of connections.

The qualities of the reconstructions obtained by considering the 4 spatial features are given in Table

Network reconstruction from individual and combined spatial node measures.

Community1

Community2

**Measurements**

5

**
**

0.612

6

0.471

0.587

7

0.498

0.554

8

0.503

0.552

5, 6

0.602

0.635

5, 7

0.654

**
**

5, 8

0.503

0.554

6, 7

0.451

0.621

6, 8

0.503

0.554

7, 8

0.504

0.555

5, 6, 7

0.574

0.638

5, 6, 8

0.503

0.554

5, 7, 8

0.504

0.556

6, 7, 8

0.504

0.554

Geometrical averages of connectivity estimation obtained for the two cortical communities while considering the 14 combinations of measurements listed in the first column. The best combinations for communities 1 and 2 were respectively the local density (5) and the pair of measurements including the local density (5) and cortical area (7). The configurations leading to the best matches have been emphasized.

In order to investigate how the combinations of topological and spatial features perform with respect to the network reconstruction, we also considered hybrid combinations between the two topological (i.e. clustering coefficient and matching index) and the two spatial (i.e. local density and area size) features which were found to produce the best results in Table

Network reconstruction from combinations of topological and spatial node measures.

Community1

Community2

**Measurements**

2

0.563

0.577

4

**
**

0.664

5

0.727

0.612

7

0.498

0.584

2, 4

0.710

0.750

2, 5

0.590

0.650

2, 7

0.495

0.634

4, 5

0.790

0.685

4, 7

0.753

0.727

5, 7

0.654

0.643

2, 4, 5

0.753

0.721

2, 4, 7

0.688

**
**

2, 5, 7

0.595

0.673

4, 5, 7

0.753

0.708

Geometrical averages of connectivity estimation obtained for the two cortical communities while considering the 14 combinations of measurements listed in the first column. The best combinations for communities 1 and 2 were respectively the local density (5) and the pair of measurements including the local density (5) and cortical area (7). The configurations leading to the best matches have been emphasized.

Figure

Adjacency matrices obtained for the two networks constructed based on the 4 spatial features

Adjacency matrices obtained for the two networks constructed based on the 4 spatial features. Original communities (a,b) and the respectively most similar reconstructed networks (c,d). Black points indicate presence of connections.

Interestingly, a comparison between the adjacency matrices in Figure

It is quite surprising that such good reconstructions of the original matrices could be obtained by considering relatively simple topological and spatial features. Table

Comparison of network reconstructions with random benchmarks.

**Network**

** Hamm. Dist**.

**
R
**

**
R
**

**
Random
**

**Cortex/Comm. 1**

**Features reconstr**.

1486

0.793

0.831

0.46

**0.810**

**Cortex/Comm. 1**

**Distance reconstr**.

711

0.664

0.792

0.46

**0.725**

**Cortex/Comm. 1**

**Mixed feats. reconstr**.

1486

0.793

0.831

0.46

**0.810**

**Cortex/Comm. 2**

**Features reconstr**.

898

0.641

0.921

0.50

**0.750**

**Cortex/Comm. 2**

**Distance reconstr**.

764

0.513

0.807

0.50

**0.643**

**Cortex/Comm. 2**

**Mixed feats. reconstr**.

520

0.954

0.628

0.50

**0.775**

Overview of measurements comparing the original and reconstructed networks: ratio of overall matches, percentage of correct zeros, percentage of correct ones, and geometrical averages between the two latter percentages expected for random comparisons and obtained from the considered experiments. It is apparent that the topological and spatial reconstructions of the two cortical communities have quality substantially superior to the random reference.

Predicting unknown connections

Connections which have not yet been tested in tract tracing studies were so far treated as absent in this study. This is due to the fact that only one of the three compilations contributing to the present dataset distinguished between absent and unknown connections. For this compilation

Confirmation or mismatch of connections, and prediction of unknown connections in a reconstructed submatrix of the cortical network

Confirmation or mismatch of connections, and prediction of unknown connections in a reconstructed submatrix of the cortical network. Data for the reconstruction of this 31 × 31 graph was based on ref [9]. Green fields denote confirmed existing (1) and absent (0) connections, respectively, whereas red fields indicate a mismatch between the original and the shown reconstructed connectivity (either by inserting connections into the matrix or removing them from the original). Yellow fields highlight connections that were predicted to exist (1) or to be absent (0) by the reconstruction approach and whose status was previously not known.

We also explored the impact of the potential existence of the currently unknown connections, by creating two additional simulated versions of the 31 × 31 area subgraph matrix, in which (a) all unknown connections were assumed to exist ('full' version), (b) 31% of the unknown connections were assumed to exist (this reflects the average edge density in cortical networks, 'relative' version). Reasonable reconstructions were obtained in all these three cases, as demonstrated by the respective Hamming distances and geometrical average errors (Table

Potential impact of unknown data.

Version of matrix

Hamming error

Geometrical error

Original

307

0.655

Relative

356

0.612

Full

278

0.731

Assessment of reconstructions for a 31 × 31 subgraph, based on data from ref. [9], which either was considered in its 'original' state, with all currently absent connections assumed to exist ('full'), or with about 1/3 of the absent connections assumed to exist ('relative').

Discussion

We have explored the role of local topological and spatial features in determining cortical connectivity. Topological features had been analyzed before

In general, a small number of local features is sufficient for predicting connections between regions. In the case of the topological features, the matching index represented the most effective individual feature for reconstruction of both communities, while the best selection for community 2 also required the clustering coefficient. This result substantiates the particular role of this feature for cortical organization

Concerning single features for the prediction of connections, topological features led to a better estimation than spatial features. This may be partly explained by the fact that topological node features by their definition are indirectly linked to global network organization, as mentioned previously. It is, however, surprising that the 'purest' spatial parameter (parameter 8: area coordinates, which expresses the proximity between areas) did not result in a strong prediction for connectivity, as spatial distance has been previously put forward as an important factor in primate cortical connectivity

Since previous tract tracing studies have focused on the visual cortex, there might exist additional connections mainly within and between motor, auditory, and somatosensory cortices. As demonstrated for a smaller subgraph of the primate cortical network, our reconstruction approach could be used to guide future experimental studies, by deriving hypotheses about currently unknown projections which would be expected to exist or be absent. The analysis of different versions of this subgraph, with varying proportions of unknown connections assumed to exist, also demonstrated that the principal conclusions of this study do not depend on the number of currently unknown connections which may be discovered in the future.

An earlier analysis of the relationship between the surface size of cortical areas and the number of projections they send or receive found no significant correlation between these parameters

For the feature analysis we transformed unidirectional projections into bidirectional connections. This resulted in 3,044 directed edges compared to the original 2,402 directed edges. This step was necessary as the reconstruction based on spatial distance depends on the Euclidean distance which is symmetric in both directions. It may be an interesting task for the future to repeat the topological analyses based on unidirectional measures.

The observed relationships between local node properties and global connectivity may hint on developmental rules. As the reconstruction approach worked well for the primate network, but not for the neuronal connections in

Although the connectivity in non-human primates such as the macaque monkey is relatively well known, there is still only little information available about human connectivity. New methods such as diffusion tensor imaging

Conclusion

The reconstruction of neural connectivity from local node properties offers insights into constraints of network organization. In particular, it suggests that neuronal networks in

Methods

Neural network data

We analyzed the organization of 2,402 projections among 95 cortical areas and sub-areas of the primate (Macaque monkey) brain. The connectivity data were retrieved from CoCoMac (

For comparison, we also analyzed two-dimensional spatial representations of the rostral neuronal network (131 neurons, 764 connections) of the nematode

The cortical as well as the

Graph-theoretical representation

The connections between cortical regions can be represented and understood as a graph, eg, _{non }of the adjacency matrix

Because we also have information about the spatial position of each cortical region, it is possible to construct a _{non }and _{non}(_{non}(_{non }and _{non }to be comparable with the symmetric spatial distance matrix

Network characterization indices

The following 8 node-based measurements (4 topological and 4 spatial) were considered in the analysis.

Feature 1 (Topological) – Node degree

This simple but informative measurement quantifies the number of edges attached to a node. In the case of non-directed networks, the node degree of node

Note that the node degree provides a direct measurement of the degree in which the specific node is connected to the rest of the network.

Feature 2 (Topological) – Clustering coefficient

Given a subset

where _{i }is the set containing the immediate neighbors of _{i}) is the number of edges between such neighbors, and |_{i}| is the number of elements in the set _{i }≤ 1.

Feature 3 (Topological) – Average shortest path distance

Given any two nodes _{i,j }between the two nodes _{i}. Note that all nodes in the cortical network are connected.

Feature 4 (Topological) – Matching index

This measurement, introduced in _{i }∧ _{j}| is the number of common projections that occur in nodes _{i }∨ _{j}|. The matching index is then calculated as:

A low matching index value indicates that the nodes have diverging input and output and are linked to substantially different parts of the network. As with the shortest path, the matching indices are averaged for all nodes.

Feature 5 (Spatial) – Local density

It is often the case with point distributions (as the centers of mass of the cortical areas) that the number of points per unit area varies along the space. In such cases, it is interesting to consider the _{i }of neighboring points contained in a sphere of small radius

The quantity _{i}(

Feature 6 (Spatial) – Coefficient of variation of the nearest distances

Given a reference point

Feature 7 (Spatial) – Area size of each cortical region

This measurement corresponds simply to the area size of the two-dimensional surface of each cortical region. The surface area was measured directly within three-dimensional space; that means, we did not use a flattened two-dimensional map to estimate the surface extent of a cortical region.

Feature 8 (Spatial) – Cartesian coordinates of the cortical areas center of mass

These features, considered together for simplicity's sake, correspond to the

Table

Node-based characterization measures used for network reconstruction.

MEASUREMENTS

IDENTIFICATION NUMBER

**Node degree**

**1**

**Clustering coefficient**

**2**

**Avg. shortest path distance**

**3**

**Matching index**

**4**

The table lists node-based measurements for network reconstruction by considering topological (in bold) and spatial (in italics) properties of the cortical data.

Network reconstruction from node features

Hypothetical cortical networks were created by assessing the pairwise similarity of nodes with respect to each of the eight features. Undirected links were created between nodes, if their similarity exceeded a threshold. In order to avoid the need to specify this threshold, we considered a sequence of equally spaced thresholds during the reconstruction and took as result the threshold leading to the best results (i.e., best recovery of the original connectivity).

The topological and spatial context around each node

Now it is possible to use the methodology suggested in _{i}(

where

Network comparisons

Because the reconstructed networks are fully congruent with the original data, in the sense that they have the same number of nodes and each node refer to the same cortical region, it is possible to obtain a simple and effective measurement of the difference between the original network

where _{G }and _{F }are the adjacency matrices of the original and reconstructed matrices, and

However, this measure, the Hamming distance, provides a biased quantification of the similarity between any two matrices in case the number of zeros and ones is significantly different. For instance, in case a matrix contains few ones and many zeros, its Hamming distance to a null matrix (all entries equal to zero) will be very small. In order to provide a more balanced overall measurement of the similarity between two adjacency matrices _{1}) and correct zeros (_{0}). More specifically, in case matrix _{0 }zeroes and _{1 }ones, and matrix _{0 }zeroes coinciding with the zeroes of _{1 }coinciding ones, we define _{1 }= _{1}/_{1 }and _{0 }= _{0}/_{0}. The two matrices will be maximally similar in case _{1 }= _{0 }= 1.

It is possible to obtain a random reference for the comparison between any two adjacency matrices _{1 }= _{1}/^{2 }and the ratio of zeroes be _{0 }= _{0}/^{2}. It can be shown that the average expected ratios of correct ones and zeros while comparing matrix _{1 }of ones are given as _{1 }=_{1 }and _{0 }= _{0}.

These comparisons with the original connectivity and random benchmarks were applied to all adjacency matrices reconstructed from individual and combined node features.

Authors' contributions

The initial proposal of checking the relationship between spatial position and connectivity was suggested by CCH and MK, while LDFC proposed the methodology of network reconstructions. LDFC performed all experimental simulations and analyses, except the determination of the correlation statistical tests, performed by CCH. The discussion and interpretation of the results, as well as the paper writing, was performed jointly by the three authors.

Acknowledgements

Luciano da F. Costa thanks FAPESP (05/00587-5) and CNPq (308231/03-1) for sponsorship. Marcus Kaiser acknowledges support from EPSRC (EP/E002331/1).