Limits...
Using Pareto optimality to explore the topology and dynamics of the human connectome.

Avena-Koenigsberger A, Goñi J, Betzel RF, van den Heuvel MP, Griffa A, Hagmann P, Thiran JP, Sporns O - Philos. Trans. R. Soc. Lond., B, Biol. Sci. (2014)

Bottom Line: This architecture embodies an inherently complex relationship between connection topology, the spatial arrangement of network elements, and the resulting network cost and functional performance.Using a multi-objective evolutionary approach that approximates a Pareto-optimal set within the morphospace, we investigate the capacity of anatomical brain networks to evolve towards topologies that exhibit optimal information processing features while preserving network cost.This approach allows us to investigate network topologies that emerge under specific selection pressures, thus providing some insight into the selectional forces that may have shaped the network architecture of existing human brains.

View Article: PubMed Central - PubMed

Affiliation: Department of Psychological and Brain Sciences, Indiana University, Bloomington, IN, USA.

ABSTRACT
Graph theory has provided a key mathematical framework to analyse the architecture of human brain networks. This architecture embodies an inherently complex relationship between connection topology, the spatial arrangement of network elements, and the resulting network cost and functional performance. An exploration of these interacting factors and driving forces may reveal salient network features that are critically important for shaping and constraining the brain's topological organization and its evolvability. Several studies have pointed to an economic balance between network cost and network efficiency with networks organized in an 'economical' small-world favouring high communication efficiency at a low wiring cost. In this study, we define and explore a network morphospace in order to characterize different aspects of communication efficiency in human brain networks. Using a multi-objective evolutionary approach that approximates a Pareto-optimal set within the morphospace, we investigate the capacity of anatomical brain networks to evolve towards topologies that exhibit optimal information processing features while preserving network cost. This approach allows us to investigate network topologies that emerge under specific selection pressures, thus providing some insight into the selectional forces that may have shaped the network architecture of existing human brains.

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Population of proximal network elements of (a) LAU1, (b) LAU2 and (c) UTR networks, respectively. Every blue dot shows the location of a network that was created by applying three rewiring steps to the respective empirical network. Grey dots show the two-dimensional projections onto the distinct planes of the morphospace. Red dots at the origin of the arrows indicate the projections in each two-dimensional plane of the three-dimensional coordinates (1, 1, 1), which correspond to the location of the empirical networks within the respective morphospace. Arrows point towards the preferred direction in which proximal networks are located in each plane.
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RSTB20130530F3: Population of proximal network elements of (a) LAU1, (b) LAU2 and (c) UTR networks, respectively. Every blue dot shows the location of a network that was created by applying three rewiring steps to the respective empirical network. Grey dots show the two-dimensional projections onto the distinct planes of the morphospace. Red dots at the origin of the arrows indicate the projections in each two-dimensional plane of the three-dimensional coordinates (1, 1, 1), which correspond to the location of the empirical networks within the respective morphospace. Arrows point towards the preferred direction in which proximal networks are located in each plane.

Mentions: The second aim of our work is to characterize the effects of small perturbations on the structure of brain networks. To do so, we generated three populations of 10 000 minimally rewired network variants, each created by carrying out three rewiring steps on each of the three empirical networks, LAU1, LAU2 and UTR. We used three rewiring steps because it is the minimum number of rewiring steps that allows us to distinguish numerical differences in all three dimensions of the morphospace. In this way, we explore the proximal morphospace, that is, the space that contains the closest neighbouring network elements of each empirical network. Interestingly, in all three datasets we found that 99% of the neighbouring networks are in a region of the morphospace defined by Ediff > 1, i.e. most networks have higher values of Ediff, compared with their corresponding empirical networks. This suggests that the three empirical networks are located very close to a (local) Ediff minimum and that the region of morphospace defined by Ediff ≤ 1 is difficult to access, given the topological constraints imposed by the rewiring algorithm (see §2d,e). The proportion of networks contained in the region Erout > 1 and CN > 1 varied across datasets: 35.26%, 85.14% and 75.22% of the populations extracted from the LAU1, LAU2 and UTR datasets, respectively, were in the region Erout > 1; 28.21%, 16.07% and 27.25% of the populations (LAU1, LAU2 and UTR, respectively) were in the region CN > 1; finally, 9.66%, 14.59% and 21.92% of the populations were in the region {Ediff > 1, Erout > 1, CN > 1} of the respective morphospace (LAU1, LAU2 and UTR). Figure 3 shows the distributions of proximal networks embedded in the three morphospaces corresponding to each dataset. The shape of the region occupied by these networks shows that the accessibility of the sub-region of morphospace surrounding the coordinates (Ediff, Erout, CN) = (1, 1, 1) is not uniform and that there are ‘preferred’ directions along each axis in which networks are located.Figure 3.


Using Pareto optimality to explore the topology and dynamics of the human connectome.

Avena-Koenigsberger A, Goñi J, Betzel RF, van den Heuvel MP, Griffa A, Hagmann P, Thiran JP, Sporns O - Philos. Trans. R. Soc. Lond., B, Biol. Sci. (2014)

Population of proximal network elements of (a) LAU1, (b) LAU2 and (c) UTR networks, respectively. Every blue dot shows the location of a network that was created by applying three rewiring steps to the respective empirical network. Grey dots show the two-dimensional projections onto the distinct planes of the morphospace. Red dots at the origin of the arrows indicate the projections in each two-dimensional plane of the three-dimensional coordinates (1, 1, 1), which correspond to the location of the empirical networks within the respective morphospace. Arrows point towards the preferred direction in which proximal networks are located in each plane.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4150305&req=5

RSTB20130530F3: Population of proximal network elements of (a) LAU1, (b) LAU2 and (c) UTR networks, respectively. Every blue dot shows the location of a network that was created by applying three rewiring steps to the respective empirical network. Grey dots show the two-dimensional projections onto the distinct planes of the morphospace. Red dots at the origin of the arrows indicate the projections in each two-dimensional plane of the three-dimensional coordinates (1, 1, 1), which correspond to the location of the empirical networks within the respective morphospace. Arrows point towards the preferred direction in which proximal networks are located in each plane.
Mentions: The second aim of our work is to characterize the effects of small perturbations on the structure of brain networks. To do so, we generated three populations of 10 000 minimally rewired network variants, each created by carrying out three rewiring steps on each of the three empirical networks, LAU1, LAU2 and UTR. We used three rewiring steps because it is the minimum number of rewiring steps that allows us to distinguish numerical differences in all three dimensions of the morphospace. In this way, we explore the proximal morphospace, that is, the space that contains the closest neighbouring network elements of each empirical network. Interestingly, in all three datasets we found that 99% of the neighbouring networks are in a region of the morphospace defined by Ediff > 1, i.e. most networks have higher values of Ediff, compared with their corresponding empirical networks. This suggests that the three empirical networks are located very close to a (local) Ediff minimum and that the region of morphospace defined by Ediff ≤ 1 is difficult to access, given the topological constraints imposed by the rewiring algorithm (see §2d,e). The proportion of networks contained in the region Erout > 1 and CN > 1 varied across datasets: 35.26%, 85.14% and 75.22% of the populations extracted from the LAU1, LAU2 and UTR datasets, respectively, were in the region Erout > 1; 28.21%, 16.07% and 27.25% of the populations (LAU1, LAU2 and UTR, respectively) were in the region CN > 1; finally, 9.66%, 14.59% and 21.92% of the populations were in the region {Ediff > 1, Erout > 1, CN > 1} of the respective morphospace (LAU1, LAU2 and UTR). Figure 3 shows the distributions of proximal networks embedded in the three morphospaces corresponding to each dataset. The shape of the region occupied by these networks shows that the accessibility of the sub-region of morphospace surrounding the coordinates (Ediff, Erout, CN) = (1, 1, 1) is not uniform and that there are ‘preferred’ directions along each axis in which networks are located.Figure 3.

Bottom Line: This architecture embodies an inherently complex relationship between connection topology, the spatial arrangement of network elements, and the resulting network cost and functional performance.Using a multi-objective evolutionary approach that approximates a Pareto-optimal set within the morphospace, we investigate the capacity of anatomical brain networks to evolve towards topologies that exhibit optimal information processing features while preserving network cost.This approach allows us to investigate network topologies that emerge under specific selection pressures, thus providing some insight into the selectional forces that may have shaped the network architecture of existing human brains.

View Article: PubMed Central - PubMed

Affiliation: Department of Psychological and Brain Sciences, Indiana University, Bloomington, IN, USA.

ABSTRACT
Graph theory has provided a key mathematical framework to analyse the architecture of human brain networks. This architecture embodies an inherently complex relationship between connection topology, the spatial arrangement of network elements, and the resulting network cost and functional performance. An exploration of these interacting factors and driving forces may reveal salient network features that are critically important for shaping and constraining the brain's topological organization and its evolvability. Several studies have pointed to an economic balance between network cost and network efficiency with networks organized in an 'economical' small-world favouring high communication efficiency at a low wiring cost. In this study, we define and explore a network morphospace in order to characterize different aspects of communication efficiency in human brain networks. Using a multi-objective evolutionary approach that approximates a Pareto-optimal set within the morphospace, we investigate the capacity of anatomical brain networks to evolve towards topologies that exhibit optimal information processing features while preserving network cost. This approach allows us to investigate network topologies that emerge under specific selection pressures, thus providing some insight into the selectional forces that may have shaped the network architecture of existing human brains.

Show MeSH