Scalability of asynchronous networks is limited by one-to-one mapping between effective connectivity and correlations

TL;DR

This paper demonstrates that the effective connectivity and correlation structure in asynchronous networks are fundamentally linked, limiting the extent to which such networks can be scaled without altering their dynamics.

Contribution

It reveals the one-to-one relationship between effective connectivity and correlations and derives conditions for preserving network dynamics during scaling.

Findings

Effective connectivity determines the correlation structure in networks.

Scaling networks requires careful adjustment of synaptic weights.

There are fundamental limits to network reducibility based on external input variance.

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.