Limits...
Scalability of Asynchronous Networks Is Limited by One-to-One Mapping between Effective Connectivity and Correlations.

van Albada SJ, Helias M, Diesmann M - PLoS Comput. Biol. (2015)

Bottom Line: The one-to-one correspondence between effective connectivity and the temporal structure of pairwise averaged correlations implies that network scalings should preserve the effective connectivity if pairwise averaged correlations are to be held constant.Changes in effective connectivity can even push a network from a linearly stable to an unstable, oscillatory regime and vice versa.Our results therefore show that the reducibility of asynchronous networks is fundamentally limited.

View Article: PubMed Central - PubMed

Affiliation: Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA BRAIN Institute I, Jülich Research Centre, Jülich, Germany.

ABSTRACT
Network models are routinely downscaled compared to nature in terms of numbers of nodes or edges because of a lack of computational resources, often without explicit mention of the limitations this entails. While reliable methods have long existed to adjust parameters such that the first-order statistics of network dynamics are conserved, here we show that limitations already arise if also second-order statistics are to be maintained. The temporal structure of pairwise averaged correlations in the activity of recurrent networks is determined by the effective population-level connectivity. We first show that in general the converse is also true and explicitly mention degenerate cases when this one-to-one relationship does not hold. The one-to-one correspondence between effective connectivity and the temporal structure of pairwise averaged correlations implies that network scalings should preserve the effective connectivity if pairwise averaged correlations are to be held constant. Changes in effective connectivity can even push a network from a linearly stable to an unstable, oscillatory regime and vice versa. On this basis, we derive conditions for the preservation of both mean population-averaged activities and pairwise averaged correlations under a change in numbers of neurons or synapses in the asynchronous regime typical of cortical networks. We find that mean activities and correlation structure can be maintained by an appropriate scaling of the synaptic weights, but only over a range of numbers of synapses that is limited by the variance of external inputs to the network. Our results therefore show that the reducibility of asynchronous networks is fundamentally limited.

No MeSH data available.


Related in: MedlinePlus

Binary network scaling that approximately preserves both mean activities and zero-lag covariances.A Increased covariances due to reduced network size can be countered by a change in the relative inhibitory synaptic weight combined with a redistribution of the synapses so that a fraction comes from outside the network. Adjusting a combination of the threshold and external drive restores the working point. B Scaling parameters versus relative network size for an example network. Since γ = 1 in this example, the scaling only works down to g = 1 (indicated by the horizontal and vertical dashed lines): Lower values of g only allow a silent or fully active network as steady states. C, E The mean activities are well preserved both by the conventional scaling in Eq (1) with an appropriate adjustment of θ (panel C), and by the method proposed here (panel E). D, F Conventional scaling increases the magnitude of zero-lag covariances in simulated data (panel D), while the proposed method preserves them (panel F). Dark colors: full-scale network. Light colors: downscaled network. Crosses and dots indicate zero-lag correlations in the full-scale and downscaled networks, respectively.
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pcbi.1004490.g006: Binary network scaling that approximately preserves both mean activities and zero-lag covariances.A Increased covariances due to reduced network size can be countered by a change in the relative inhibitory synaptic weight combined with a redistribution of the synapses so that a fraction comes from outside the network. Adjusting a combination of the threshold and external drive restores the working point. B Scaling parameters versus relative network size for an example network. Since γ = 1 in this example, the scaling only works down to g = 1 (indicated by the horizontal and vertical dashed lines): Lower values of g only allow a silent or fully active network as steady states. C, E The mean activities are well preserved both by the conventional scaling in Eq (1) with an appropriate adjustment of θ (panel C), and by the method proposed here (panel E). D, F Conventional scaling increases the magnitude of zero-lag covariances in simulated data (panel D), while the proposed method preserves them (panel F). Dark colors: full-scale network. Light colors: downscaled network. Crosses and dots indicate zero-lag correlations in the full-scale and downscaled networks, respectively.

Mentions: We use Eq (22) to perform a more sophisticated downscaling (cf. Fig 6). Let the new size of the excitatory population be N′. Eq (22) shows that the covariances can only be preserved when a combination of We, γ, and g is adjusted. We take γ constant, and apply the transformationWe→fWe;g→g′.(27)Solving Eq (22) for f and g′ yields (cf. Fig 6B)f=aceeN′+γcii2(cee-cii)We[(aN′+cee)(aN+γcii)-γ4(cee+cii)2](28)g′=cee(cee-cii)-2aN′ciiγcii(cee-cii)+2aN′cee.(29)The change in We can be captured by K → fK as long as the working point (μ, σ) is maintained. This intuitively corresponds to a redistribution of the synapses so that a fraction f comes from inside the network, and 1 − f from outside (cf. Fig 6A). However, the external drive does not have the same mean and variance as the internal inputs, since it needs to make up for the change in g. The external input can be modeled as a Gaussian noise with parametersμext=KJ(1-γg)⟨n⟩-fKJ(1-γg′)⟨n⟩(30)σext2=KJ2(1+γg2)a-fKJ2(1+γg′2)a,(31)independent for each neuron.


Scalability of Asynchronous Networks Is Limited by One-to-One Mapping between Effective Connectivity and Correlations.

van Albada SJ, Helias M, Diesmann M - PLoS Comput. Biol. (2015)

Binary network scaling that approximately preserves both mean activities and zero-lag covariances.A Increased covariances due to reduced network size can be countered by a change in the relative inhibitory synaptic weight combined with a redistribution of the synapses so that a fraction comes from outside the network. Adjusting a combination of the threshold and external drive restores the working point. B Scaling parameters versus relative network size for an example network. Since γ = 1 in this example, the scaling only works down to g = 1 (indicated by the horizontal and vertical dashed lines): Lower values of g only allow a silent or fully active network as steady states. C, E The mean activities are well preserved both by the conventional scaling in Eq (1) with an appropriate adjustment of θ (panel C), and by the method proposed here (panel E). D, F Conventional scaling increases the magnitude of zero-lag covariances in simulated data (panel D), while the proposed method preserves them (panel F). Dark colors: full-scale network. Light colors: downscaled network. Crosses and dots indicate zero-lag correlations in the full-scale and downscaled networks, respectively.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4556689&req=5

pcbi.1004490.g006: Binary network scaling that approximately preserves both mean activities and zero-lag covariances.A Increased covariances due to reduced network size can be countered by a change in the relative inhibitory synaptic weight combined with a redistribution of the synapses so that a fraction comes from outside the network. Adjusting a combination of the threshold and external drive restores the working point. B Scaling parameters versus relative network size for an example network. Since γ = 1 in this example, the scaling only works down to g = 1 (indicated by the horizontal and vertical dashed lines): Lower values of g only allow a silent or fully active network as steady states. C, E The mean activities are well preserved both by the conventional scaling in Eq (1) with an appropriate adjustment of θ (panel C), and by the method proposed here (panel E). D, F Conventional scaling increases the magnitude of zero-lag covariances in simulated data (panel D), while the proposed method preserves them (panel F). Dark colors: full-scale network. Light colors: downscaled network. Crosses and dots indicate zero-lag correlations in the full-scale and downscaled networks, respectively.
Mentions: We use Eq (22) to perform a more sophisticated downscaling (cf. Fig 6). Let the new size of the excitatory population be N′. Eq (22) shows that the covariances can only be preserved when a combination of We, γ, and g is adjusted. We take γ constant, and apply the transformationWe→fWe;g→g′.(27)Solving Eq (22) for f and g′ yields (cf. Fig 6B)f=aceeN′+γcii2(cee-cii)We[(aN′+cee)(aN+γcii)-γ4(cee+cii)2](28)g′=cee(cee-cii)-2aN′ciiγcii(cee-cii)+2aN′cee.(29)The change in We can be captured by K → fK as long as the working point (μ, σ) is maintained. This intuitively corresponds to a redistribution of the synapses so that a fraction f comes from inside the network, and 1 − f from outside (cf. Fig 6A). However, the external drive does not have the same mean and variance as the internal inputs, since it needs to make up for the change in g. The external input can be modeled as a Gaussian noise with parametersμext=KJ(1-γg)⟨n⟩-fKJ(1-γg′)⟨n⟩(30)σext2=KJ2(1+γg2)a-fKJ2(1+γg′2)a,(31)independent for each neuron.

Bottom Line: The one-to-one correspondence between effective connectivity and the temporal structure of pairwise averaged correlations implies that network scalings should preserve the effective connectivity if pairwise averaged correlations are to be held constant.Changes in effective connectivity can even push a network from a linearly stable to an unstable, oscillatory regime and vice versa.Our results therefore show that the reducibility of asynchronous networks is fundamentally limited.

View Article: PubMed Central - PubMed

Affiliation: Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA BRAIN Institute I, Jülich Research Centre, Jülich, Germany.

ABSTRACT
Network models are routinely downscaled compared to nature in terms of numbers of nodes or edges because of a lack of computational resources, often without explicit mention of the limitations this entails. While reliable methods have long existed to adjust parameters such that the first-order statistics of network dynamics are conserved, here we show that limitations already arise if also second-order statistics are to be maintained. The temporal structure of pairwise averaged correlations in the activity of recurrent networks is determined by the effective population-level connectivity. We first show that in general the converse is also true and explicitly mention degenerate cases when this one-to-one relationship does not hold. The one-to-one correspondence between effective connectivity and the temporal structure of pairwise averaged correlations implies that network scalings should preserve the effective connectivity if pairwise averaged correlations are to be held constant. Changes in effective connectivity can even push a network from a linearly stable to an unstable, oscillatory regime and vice versa. On this basis, we derive conditions for the preservation of both mean population-averaged activities and pairwise averaged correlations under a change in numbers of neurons or synapses in the asynchronous regime typical of cortical networks. We find that mean activities and correlation structure can be maintained by an appropriate scaling of the synaptic weights, but only over a range of numbers of synapses that is limited by the variance of external inputs to the network. Our results therefore show that the reducibility of asynchronous networks is fundamentally limited.

No MeSH data available.


Related in: MedlinePlus