Scalability of Asynchronous Networks Is Limited by One-to-One Mapping between Effective Connectivity and Correlations.
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.
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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 |
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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. |
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.
No MeSH data available.