P101 How random connectivity shapes fluctuations of finite-size neural populations

Nils E. Greven*1, 2, Jonas Ranft3, Tilo Schwalger1,2

1Department of Mathematics, Technische Universität Berlin, Berlin, Germany
2Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany
3Institut de Biologie de l’ENS, Ecole normale supérieure, PSL University, CNRS, Paris.

*Email: greven@math.tu-berlin.de
Introduction

A fundamental problem in computational neuroscience is to understand the variability of neural population dynamics and their response to stimuli in the brain [1,2]. Mean-field models have proven useful to study the mechanisms underlying neural variability in spiking neural networks, however, previous models that describe fluctuations typically assume either infinitely large network sizesN[3] or all-to-all connectivity [4] assumptions that seem unrealistic for cortical populations. To gain insight into the case of both finite network size and non-full connectivity together, we derive here a nonlinear stochastic mean-field model for a network of spiking Poisson neurons with quenched random connectivity.

Methods
We treat the quenched disorder of the connectivity by an annealed approximation [3] that leads to a simpler fully connected network with additional independent noise in the neurons. This annealed network enables a reduction to a low-dimensional closed system of coupled Langevin equations (MF2) for the mean and variance of the neuronal membrane potentials.We comparethe theory ofthis mesoscopic modelto simulations of the underlying microscopic model.An additional comparison toprevious mesoscopic models(MF1)that neglected the recurrent noise effect caused by quenched disorderallowsto investigateand analytically understand theeffects of taking quenched random connectivityand finite network-sizeinto account.

Results
In comparison, the novel mesoscopic model MF2 well describes the fluctuations and nonlinearities of finite-size neuronal populations and outperforms MF1. This effect can be analytically understood as a softening of the effective nonlinearityof the population transfer function (Fig 1A). The mesoscopic theory predicts a large effect of the connection probability (Fig 1B) and stimulus strength on the variance of the population firing rate (Fig 1C, D) that MF1 cannot sufficiently explain.

Discussion
In conclusion, our mesoscopic theory elucidates how disordered connectivity shapes nonlinear dynamics and fluctuations of neural populations at the mesoscopic scale and showcases a useful mean-field method to treat non-full connectivity in finite-size, spiking neural networks.In the paper presented here, we investigated the effect of quenched randomness on finite networks of Poisson neurons. As an extensionwe can analyzethe annealed approximation for networksof Integrate and-fire neuronswith reset.




Figure 1. A) Population transfer function F for MF2 (always blue) is flatter than for MF1 (always yellow) resulting in different fixed points =intersection with black B) MF2 captures dependence on connection probability p for the variance of the population firing rate r, MF1 is p-independent C,D) Variance of r for different external drive μ is massively different for MF1 vs MF2 and different network sizes
Acknowledgements
We are grateful to Jakob Stubenrauch for useful comments on the manuscript.
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[2] G. Hennequin, Y. Ahmadian, D. B. Rubin, M. Lengyel, K. D. Miller, Neuron 98, 846 (2018).

[3] N. Brunel, V. Hakim, Neural Comput. 11, 1621 (1999).

[4] T. Schwalger, M. Deger, W. Gerstner, PLoS Comput. Biol. 13, e1005507 (2017).