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

Spiking network scaling that approximately preserves mean firing rates and covariances.A Diagram illustrating the network and indicating the parameters that are adjusted. B Excitatory in-degrees K′, relative inhibitory synaptic weight g′, and relative number of inhibitory neurons γ′ versus scaling factor N1/N. The dashed vertical line indicates the limit below which the scaling fails. C Total number of neurons Ntotal = (1+γ′)N1 and total number of synapses Nsyn = (1+γ′)2K′ N1 versus scaling factor. D Rates of external excitatory and inhibitory Poisson inputs necessary for keeping firing rates constant. Average firing rates are between 23.1 and 23.5 spikes/s for both excitatory and inhibitory populations and all network sizes. E Integrated covariances, corresponding to zero-frequency components in the Fourier domain. Crosses: simulation results, dots: theoretical predictions. F Average covariance between excitatory-inhibitory neuron pairs for different network sizes. The dashed curve indicates the theoretical prediction for N = 10,000. Each network was simulated for 100 s.
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pcbi.1004490.g007: Spiking network scaling that approximately preserves mean firing rates and covariances.A Diagram illustrating the network and indicating the parameters that are adjusted. B Excitatory in-degrees K′, relative inhibitory synaptic weight g′, and relative number of inhibitory neurons γ′ versus scaling factor N1/N. The dashed vertical line indicates the limit below which the scaling fails. C Total number of neurons Ntotal = (1+γ′)N1 and total number of synapses Nsyn = (1+γ′)2K′ N1 versus scaling factor. D Rates of external excitatory and inhibitory Poisson inputs necessary for keeping firing rates constant. Average firing rates are between 23.1 and 23.5 spikes/s for both excitatory and inhibitory populations and all network sizes. E Integrated covariances, corresponding to zero-frequency components in the Fourier domain. Crosses: simulation results, dots: theoretical predictions. F Average covariance between excitatory-inhibitory neuron pairs for different network sizes. The dashed curve indicates the theoretical prediction for N = 10,000. Each network was simulated for 100 s.

Mentions: When the populations receive statistically identical external inputs, we have a1 = a2 = r, since the internal inputs are also equal. Fig 7 illustrates the network scaling for the choice w′ = w. Results are shown as a function of the relative size N1/N of the excitatory population. External drive is provided at each network size to keep the mean and standard deviation of the total inputs to each neuron at the level indicated. The mean is supplied as a constant current input, while the variability is afforded by Poisson inputs according to Eqs (17) and (18) (Fig 7D). It is seen that the transformations (Fig 7B) are able to reduce both the total numbers of neurons and the total number of synapses (Fig 7C) while approximately preserving covariance sizes and shapes (Fig 7E,7F). Small fluctuations in the theoretical predictions in Fig 7E are due to the discreteness of numbers of neurons and synapses, and deviations of the effective inhibitory weight from the linear approximation gw. The fact that the theoretical prediction in Fig 7F misses the small dips around t = 0 may be due to the approximation of the autocorrelations by delta functions, eliminating the relative refractoriness due to the reset. The numbers of neurons and synapses increase again below some N1/N, and diverge as g′ becomes zero. This limits the scalability despite the additional freedom provided by the symmetry.


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)

Spiking network scaling that approximately preserves mean firing rates and covariances.A Diagram illustrating the network and indicating the parameters that are adjusted. B Excitatory in-degrees K′, relative inhibitory synaptic weight g′, and relative number of inhibitory neurons γ′ versus scaling factor N1/N. The dashed vertical line indicates the limit below which the scaling fails. C Total number of neurons Ntotal = (1+γ′)N1 and total number of synapses Nsyn = (1+γ′)2K′ N1 versus scaling factor. D Rates of external excitatory and inhibitory Poisson inputs necessary for keeping firing rates constant. Average firing rates are between 23.1 and 23.5 spikes/s for both excitatory and inhibitory populations and all network sizes. E Integrated covariances, corresponding to zero-frequency components in the Fourier domain. Crosses: simulation results, dots: theoretical predictions. F Average covariance between excitatory-inhibitory neuron pairs for different network sizes. The dashed curve indicates the theoretical prediction for N = 10,000. Each network was simulated for 100 s.
© Copyright Policy
Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC4556689&req=5

pcbi.1004490.g007: Spiking network scaling that approximately preserves mean firing rates and covariances.A Diagram illustrating the network and indicating the parameters that are adjusted. B Excitatory in-degrees K′, relative inhibitory synaptic weight g′, and relative number of inhibitory neurons γ′ versus scaling factor N1/N. The dashed vertical line indicates the limit below which the scaling fails. C Total number of neurons Ntotal = (1+γ′)N1 and total number of synapses Nsyn = (1+γ′)2K′ N1 versus scaling factor. D Rates of external excitatory and inhibitory Poisson inputs necessary for keeping firing rates constant. Average firing rates are between 23.1 and 23.5 spikes/s for both excitatory and inhibitory populations and all network sizes. E Integrated covariances, corresponding to zero-frequency components in the Fourier domain. Crosses: simulation results, dots: theoretical predictions. F Average covariance between excitatory-inhibitory neuron pairs for different network sizes. The dashed curve indicates the theoretical prediction for N = 10,000. Each network was simulated for 100 s.
Mentions: When the populations receive statistically identical external inputs, we have a1 = a2 = r, since the internal inputs are also equal. Fig 7 illustrates the network scaling for the choice w′ = w. Results are shown as a function of the relative size N1/N of the excitatory population. External drive is provided at each network size to keep the mean and standard deviation of the total inputs to each neuron at the level indicated. The mean is supplied as a constant current input, while the variability is afforded by Poisson inputs according to Eqs (17) and (18) (Fig 7D). It is seen that the transformations (Fig 7B) are able to reduce both the total numbers of neurons and the total number of synapses (Fig 7C) while approximately preserving covariance sizes and shapes (Fig 7E,7F). Small fluctuations in the theoretical predictions in Fig 7E are due to the discreteness of numbers of neurons and synapses, and deviations of the effective inhibitory weight from the linear approximation gw. The fact that the theoretical prediction in Fig 7F misses the small dips around t = 0 may be due to the approximation of the autocorrelations by delta functions, eliminating the relative refractoriness due to the reset. The numbers of neurons and synapses increase again below some N1/N, and diverge as g′ becomes zero. This limits the scalability despite the additional freedom provided by the symmetry.

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