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.
Highlights
While many aspects of brain dynamics and function remain unexplored, the numbers of neurons and synapses in a given volume are well known, and as such constitute basic parameters that should be taken seriously
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 second-order statistics are to be maintained
The so-called effective connectivity combines both components to yield a measure of the actual influence of physical connections
Summary
While many aspects of brain dynamics and function remain unexplored, the numbers of neurons and synapses in a given volume are well known, and as such constitute basic parameters that should be taken seriously. Computational capacity ranges from a few tens of millions of synapses on laptop or desktop computers, or on dedicated hardware when fully exploited [4, 5], to 1012 − 1013 synapses on supercomputers [6]. This upper limit is still about two orders of magnitude below the full human brain, underlining the need for downscaling in computational modeling. Any brain model that approximates a fraction of the recurrent connections as external inputs is in some sense downscaled: the missing interactions need to be absorbed into the network and input parameters in order to obtain the appropriate statistics. The implications of such scaling are usually not investigated
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