Limits...
Balancing robustness against the dangers of multiple attractors in a Hopfield-type model of biological attractors.

Anafi RC, Bates JH - PLoS ONE (2010)

Bottom Line: By progressively removing the links of fully connected Hopfield nets, we found that a designated attractor of the nets could still be supported until only slightly more than 1 link per node remained.Furthermore, with more than about twice the minimum of links, the net became increasingly able to support a second attractor.We speculate that homeostatic biological networks may have evolved to assume a degree of connectivity that balances robustness and agility against the dangers of becoming trapped in an abnormal attractor.

View Article: PubMed Central - PubMed

Affiliation: Division of Sleep Medicine, University of Pennsylvania, Philadelphia, Pennsylvania, United States of America.

ABSTRACT

Background: Many chronic human diseases are of unclear origin, and persist long beyond any known insult or instigating factor. These diseases may represent a structurally normal biologic network that has become trapped within the basin of an abnormal attractor.

Methodology/principal findings: We used the Hopfield net as the archetypical example of a dynamic biological network. By progressively removing the links of fully connected Hopfield nets, we found that a designated attractor of the nets could still be supported until only slightly more than 1 link per node remained. As the number of links approached this minimum value, the rate of convergence to this attractor from an arbitrary starting state increased dramatically. Furthermore, with more than about twice the minimum of links, the net became increasingly able to support a second attractor.

Conclusions/significance: We speculate that homeostatic biological networks may have evolved to assume a degree of connectivity that balances robustness and agility against the dangers of becoming trapped in an abnormal attractor.

Show MeSH
Fraction of times that networks converged to a second orthogonal attractor versus the number of links per node in the network, for nets of N = 50, 100, 200 and 400 nodes.The minimal set of links previously determined to support the first designated attractor in each net were not affected during the link pruning process.
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pone-0014413-g004: Fraction of times that networks converged to a second orthogonal attractor versus the number of links per node in the network, for nets of N = 50, 100, 200 and 400 nodes.The minimal set of links previously determined to support the first designated attractor in each net were not affected during the link pruning process.

Mentions: We searched for the second attractor point by adding several thousand links back to individual examples of fully pruned networks (such as those shown in Fig. 1), with the weights of these new links adjusted so that the network supported both the original designated attractor and a second designated attractor that was orthogonal to the first. We then randomly removed these new links without touching the minimal set of links previously found necessary to support the first attractor. At each step in this process, the network was launched from 200 random initial configurations, all of which were required to converge to one or other of the two designated attractors before the links were permanently deleted. As the new links were pruned, the fraction of the runs that converged to the second attractor decreased (Fig. 4). These results also scaled with network size so that, regardless of N, the chance of convergence to the second attractor became extremely small when the network was reduced to approximately 3N connections. When fewer than about 2 connections per node remained, the second designated attractor was no longer supported as a fixed point of the system. There thus appears to be a range of networks connectivities between 1 to 2 connections per node that can support a single designated attractor but not a second one. This degree of sparsity, however, comes at the cost of both network fragility and speed, as Fig. 3 indicates a sharp decline in convergence speed as connectivity falls below 3 connections per node.


Balancing robustness against the dangers of multiple attractors in a Hopfield-type model of biological attractors.

Anafi RC, Bates JH - PLoS ONE (2010)

Fraction of times that networks converged to a second orthogonal attractor versus the number of links per node in the network, for nets of N = 50, 100, 200 and 400 nodes.The minimal set of links previously determined to support the first designated attractor in each net were not affected during the link pruning process.
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC3008716&req=5

pone-0014413-g004: Fraction of times that networks converged to a second orthogonal attractor versus the number of links per node in the network, for nets of N = 50, 100, 200 and 400 nodes.The minimal set of links previously determined to support the first designated attractor in each net were not affected during the link pruning process.
Mentions: We searched for the second attractor point by adding several thousand links back to individual examples of fully pruned networks (such as those shown in Fig. 1), with the weights of these new links adjusted so that the network supported both the original designated attractor and a second designated attractor that was orthogonal to the first. We then randomly removed these new links without touching the minimal set of links previously found necessary to support the first attractor. At each step in this process, the network was launched from 200 random initial configurations, all of which were required to converge to one or other of the two designated attractors before the links were permanently deleted. As the new links were pruned, the fraction of the runs that converged to the second attractor decreased (Fig. 4). These results also scaled with network size so that, regardless of N, the chance of convergence to the second attractor became extremely small when the network was reduced to approximately 3N connections. When fewer than about 2 connections per node remained, the second designated attractor was no longer supported as a fixed point of the system. There thus appears to be a range of networks connectivities between 1 to 2 connections per node that can support a single designated attractor but not a second one. This degree of sparsity, however, comes at the cost of both network fragility and speed, as Fig. 3 indicates a sharp decline in convergence speed as connectivity falls below 3 connections per node.

Bottom Line: By progressively removing the links of fully connected Hopfield nets, we found that a designated attractor of the nets could still be supported until only slightly more than 1 link per node remained.Furthermore, with more than about twice the minimum of links, the net became increasingly able to support a second attractor.We speculate that homeostatic biological networks may have evolved to assume a degree of connectivity that balances robustness and agility against the dangers of becoming trapped in an abnormal attractor.

View Article: PubMed Central - PubMed

Affiliation: Division of Sleep Medicine, University of Pennsylvania, Philadelphia, Pennsylvania, United States of America.

ABSTRACT

Background: Many chronic human diseases are of unclear origin, and persist long beyond any known insult or instigating factor. These diseases may represent a structurally normal biologic network that has become trapped within the basin of an abnormal attractor.

Methodology/principal findings: We used the Hopfield net as the archetypical example of a dynamic biological network. By progressively removing the links of fully connected Hopfield nets, we found that a designated attractor of the nets could still be supported until only slightly more than 1 link per node remained. As the number of links approached this minimum value, the rate of convergence to this attractor from an arbitrary starting state increased dramatically. Furthermore, with more than about twice the minimum of links, the net became increasingly able to support a second attractor.

Conclusions/significance: We speculate that homeostatic biological networks may have evolved to assume a degree of connectivity that balances robustness and agility against the dangers of becoming trapped in an abnormal attractor.

Show MeSH