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.

#### Simulation data

binary/asymm/dist_Kin_scale: data on E-I network of Ginzburg neurons for Fig. 1C. binary/asymm/mcculloch_pitts: data on binary E-I network with population-specific connection probability for Fig. 5. binary/symm: data on binary E-I network with population-independent connection probability, scaled to preserve zero-lag covariances, for Fig. 6. spiking/E_I_LIF_network: data on scaling of network of leaky integrate-and-fire neurons with population-independent connection probability, preserving correlations, for Fig. 7. spiking/inhibitory_LIF_network: data on single-population network of inhibitory leaky integrate-and-fire neurons, for Fig. 2. spiking/microcircuit_param_scan: data on scaling of microcircuit model of Potjans & Diesmann (Cereb Cortex, 2014) with and without regard for covariances, for Figs. 3 and 4.

data.tar.gz

#### Python scripts for analysis and plotting

fig1.py, fig2.py, fig3.py, fig4.py, fig5.py, fig6.py, fig7.py: reproduce the figures of the paper using the data and additional figure files in this repository. binary_network_asymm.py, binary_network_symm.py, g_gamma_K_scaling.py, ginzburg_network_dist_Kin_scale.py, inhibitory_LIF_network.py: NEST simulation scripts for random recurrent networks. parameters_binary_asymm.py, parameters_binary_symm.py, parameters_ginzburg.py, parameters_inhibitory_LIF_network.py: corresponding parameter files. hambach_run_g_gamma_K_scaling.py, hambach_run_inhibitory_LIF_network.py, hambach_run_sim_dist_Kin_scale.py: scripts for running the corresponding simulations on a cluster. h5py_wrapper.py: for storing and loading h5 files. plotfuncs.py, plot_utils.py, ps_manip.py: general plotting scripts. ginzburg_theory.py, ginzburg_theory_distKin.py, hawkesnet.py, siegert.py: contain functions for computing covariances and related quantities. microcircuit_analysis.py: functions for loading and analyzing data from simulations of the Potjans and Diesmann (Cereb Cortex, 2014) microcircuit model. microcircuit: folder containing NEST simulation scripts for the Potjans and Diesmann microcircuit model.

py.tar.gz

#### Accessory figure files

fig_scaling_framework.json: configuration file for Fig. 1. relevant_scales.eps, why_scaling.eps, scaling_as_subsampling.eps: diagrams included in Fig. 1. redistributing_synapes.eps: diagram included in Fig. 6. spiking_network_scaling_diagram.eps: diagram included in Fig. 7.

figures.tar.gz

#### Simulation data - submission 2

binary/asymm/dist_Kin_scale: data on E-I network of Ginzburg neurons for Fig. 1C.
binary/asymm/mcculloch_pitts: data on binary E-I network with population-specific connection probability for Fig. 3.
binary/symm: data on binary E-I network with population-independent connection probability, scaled to preserve zero-lag covariances, for Fig. 6.
spiking/E_I_LIF_network: data on scaling of network of leaky integrate-and-fire neurons with population-independent connection probability, preserving correlations, for Fig. 7.
spiking/inhibitory_LIF_network: data on single-population network of inhibitory leaky integrate-and-fire neurons, for Fig. 2.
spiking/LIF_network_rate_comparison: data on E-I network of leaky integrate-and-fire neurons with different firing rates, for Fig. 5.
spiking/microcircuit: data on scaling of microcircuit model of Potjans & Diesmann (Cereb Cortex, 2014), for Fig. 4.

data2.tar.gz

#### Python scripts for analysis and plotting - submission 2

fig1.py, fig2.py, fig3.py, fig4.py, fig5.py, fig6.py, fig7.py: reproduce the figures of the paper using the data and additional figure files in this repository.
binary_network_asymm.py, binary_network_symm.py, g_gamma_K_scaling.py, ginzburg_network_dist_Kin_scale.py, inhibitory_LIF_network.py, LIF_network_rate_comparison.py: NEST simulation scripts for random recurrent networks.
parameters_binary_asymm.py, parameters_binary_symm.py, parameters_g_gamma_K_scaling.py, parameters_ginzburg.py, parameters_inhibitory_LIF_network.py, parameters_LIF_network_rate_comparison.py: corresponding parameter files.
hambach_run_g_gamma_K_scaling.py, hambach_run_inhibitory_LIF_network.py, hambach_run_LIF_network_rate_comparison.py, hambach_run_sim_dist_Kin_scale.py: scripts for running the corresponding simulations on a cluster.
h5py_wrapper.py: for storing and loading h5 files.
plotfuncs.py, plot_utils.py, ps_manip.py: general plotting scripts.
corr_hawkes_like.py, ginzburg_theory.py, ginzburg_theory_distKin.py, hawkesnet.py, siegert.py: contain functions for computing covariances and related quantities. microcircuit: folder containing NEST simulation scripts for the Potjans and Diesmann microcircuit model.

py2.tar.gz

#### Accessory figure files - submission 2

fig_scaling_framework.json: configuration file for Fig. 1.
relevant_scales.eps, why_scaling.eps, scaling_as_subsampling.eps: diagrams included in Fig. 1.
redistributing_synapes.eps: diagram included in Fig. 6.
spiking_network_scaling_diagram.eps: diagram included in Fig. 7.

figures2.tar.gz