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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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(a) Diagram of a communication-efficiency morphospace for toy-networks; the location of a toy-network within this morphospace of abstract networks can be associated with specific aspects of network structure that favour two distinct communication schemes: diffusion-based communication (Ediff) and routing-based communication (Erout). (b) Geometric example of Pareto optimality: the area of three circles represents three objectives to be maximized; circles are constrained to be contained within an equilateral triangle and cannot overlap with each other. There are several solutions to the problem; the top triangle shows a solution that could be improved by increasing the area of the blue circle; thus, it is not Pareto-optimal. The two bottom triangles show solutions in which the area of none of the circles can be increased without having to decrease the area of another circle; therefore, the solutions are Pareto-optimal [45]. (c) Example of a Pareto front, where three objective functions are to be maximized. All the points in the plot represent feasible solutions; however, only the red points belong to the Pareto front. (d) Rewiring rule: the weights of the edges {i, l} and {j, k} are randomly selected, provided that the total wiring cost is preserved. (e) Matrix of Euclidean distances (left side) and interpolated fibre lengths (right side) between all pairs of nodes of the LAU1 dataset. The colour map on the human cortex images represents the fibre lengths after interpolation between node 300 (whose location is indicated with a white circle) and all other nodes of the LAU1 network.
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RSTB20130530F1: (a) Diagram of a communication-efficiency morphospace for toy-networks; the location of a toy-network within this morphospace of abstract networks can be associated with specific aspects of network structure that favour two distinct communication schemes: diffusion-based communication (Ediff) and routing-based communication (Erout). (b) Geometric example of Pareto optimality: the area of three circles represents three objectives to be maximized; circles are constrained to be contained within an equilateral triangle and cannot overlap with each other. There are several solutions to the problem; the top triangle shows a solution that could be improved by increasing the area of the blue circle; thus, it is not Pareto-optimal. The two bottom triangles show solutions in which the area of none of the circles can be increased without having to decrease the area of another circle; therefore, the solutions are Pareto-optimal [45]. (c) Example of a Pareto front, where three objective functions are to be maximized. All the points in the plot represent feasible solutions; however, only the red points belong to the Pareto front. (d) Rewiring rule: the weights of the edges {i, l} and {j, k} are randomly selected, provided that the total wiring cost is preserved. (e) Matrix of Euclidean distances (left side) and interpolated fibre lengths (right side) between all pairs of nodes of the LAU1 dataset. The colour map on the human cortex images represents the fibre lengths after interpolation between node 300 (whose location is indicated with a white circle) and all other nodes of the LAU1 network.

Mentions: The concept of morphospace originated in the context of evolutionary biology [44]. It provides a framework to map all the possible biological forms that can result by varying the parameter values of a geometrical or mathematical model of form. Most importantly, it allows the identification of forms that have been produced in nature and forms that have not. The parameters of a model of form define the axes or dimensions of a morphospace, where different locations within each dimension specify the parameter values and are associated with a particular biological form. Here, we extend the concept of morphospace to encompass network structure; hence, the dimensions of a network morphospace are given by network structural measures and positions within the morphospace correspond to characteristic aspects of network topology (figure 1a). Specifically, in this paper we define a three-dimensional morphospace with axes given by Ediff, Erout and CN; therefore, networks are placed within the morphospace according to the previously defined measures.Figure 1.


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)

(a) Diagram of a communication-efficiency morphospace for toy-networks; the location of a toy-network within this morphospace of abstract networks can be associated with specific aspects of network structure that favour two distinct communication schemes: diffusion-based communication (Ediff) and routing-based communication (Erout). (b) Geometric example of Pareto optimality: the area of three circles represents three objectives to be maximized; circles are constrained to be contained within an equilateral triangle and cannot overlap with each other. There are several solutions to the problem; the top triangle shows a solution that could be improved by increasing the area of the blue circle; thus, it is not Pareto-optimal. The two bottom triangles show solutions in which the area of none of the circles can be increased without having to decrease the area of another circle; therefore, the solutions are Pareto-optimal [45]. (c) Example of a Pareto front, where three objective functions are to be maximized. All the points in the plot represent feasible solutions; however, only the red points belong to the Pareto front. (d) Rewiring rule: the weights of the edges {i, l} and {j, k} are randomly selected, provided that the total wiring cost is preserved. (e) Matrix of Euclidean distances (left side) and interpolated fibre lengths (right side) between all pairs of nodes of the LAU1 dataset. The colour map on the human cortex images represents the fibre lengths after interpolation between node 300 (whose location is indicated with a white circle) and all other nodes of the LAU1 network.
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Related In: Results  -  Collection

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RSTB20130530F1: (a) Diagram of a communication-efficiency morphospace for toy-networks; the location of a toy-network within this morphospace of abstract networks can be associated with specific aspects of network structure that favour two distinct communication schemes: diffusion-based communication (Ediff) and routing-based communication (Erout). (b) Geometric example of Pareto optimality: the area of three circles represents three objectives to be maximized; circles are constrained to be contained within an equilateral triangle and cannot overlap with each other. There are several solutions to the problem; the top triangle shows a solution that could be improved by increasing the area of the blue circle; thus, it is not Pareto-optimal. The two bottom triangles show solutions in which the area of none of the circles can be increased without having to decrease the area of another circle; therefore, the solutions are Pareto-optimal [45]. (c) Example of a Pareto front, where three objective functions are to be maximized. All the points in the plot represent feasible solutions; however, only the red points belong to the Pareto front. (d) Rewiring rule: the weights of the edges {i, l} and {j, k} are randomly selected, provided that the total wiring cost is preserved. (e) Matrix of Euclidean distances (left side) and interpolated fibre lengths (right side) between all pairs of nodes of the LAU1 dataset. The colour map on the human cortex images represents the fibre lengths after interpolation between node 300 (whose location is indicated with a white circle) and all other nodes of the LAU1 network.
Mentions: The concept of morphospace originated in the context of evolutionary biology [44]. It provides a framework to map all the possible biological forms that can result by varying the parameter values of a geometrical or mathematical model of form. Most importantly, it allows the identification of forms that have been produced in nature and forms that have not. The parameters of a model of form define the axes or dimensions of a morphospace, where different locations within each dimension specify the parameter values and are associated with a particular biological form. Here, we extend the concept of morphospace to encompass network structure; hence, the dimensions of a network morphospace are given by network structural measures and positions within the morphospace correspond to characteristic aspects of network topology (figure 1a). Specifically, in this paper we define a three-dimensional morphospace with axes given by Ediff, Erout and CN; therefore, networks are placed within the morphospace according to the previously defined measures.Figure 1.

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