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

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.Author Summary: Neural networks have two basic components: their structural elements (neurons and synapses), and the dynamics of these constituents. The so-called effective connectivity combines both components to yield a measure of the actual influence of physical connections. Previous work showed effective connectivity to determine correlations, which quantify the co-activation of different neurons. Conversely, methods for estimating network structure from correlations have been developed. We here extend the range of networks for which the mapping between effective connectivity and correlations can be shown to be one-to-one, and clarify the conditions under which this equivalence holds. These findings apply to a class of networks that is often used, with some variations, to model the activity of cerebral cortex. Since the numbers of neurons and synapses in real mammalian brains are vast, such models tend to be reduced in size for simulation purposes. However, our findings imply that if we wish to retain the original dynamics including correlations, effective connectivity needs to be unchanged, from which we derive scaling laws for synaptic strengths and external inputs, and fundamental limits on the reducibility of network size. The work points to the importance of considering networks with realistic numbers of neurons and synapses.