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A simple mechanism for higher-order correlations in integrate-and-fire neurons

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Recent work [1] shows that common input gives rise to higher-order correlations in the Dichotomized Gaussian neuron model. Here we study a homogeneous population of integrate-and-fire neurons receiving correlated input. Each neuron receives an independent white noise input and all neurons receive a common Gaussian input. To quantify the contributions of higher-order correlations we use a maximum entropy model. The model with interactions up to second order (i.e. pairwise correlations) is known as the Ising model. The Kullbach-Leibler divergence between the Ising model and the model with interactions of all orders allows us to quantitatively describe the presence of higher-order correlations.

We observe from numerical simulations that for low firing rates, the Kullbach-Leibler divergence grows with increasing correlation i.e. strength of the common input (Figure 1A). For population size N=100, the Ising model predicts a vastly different distribution of spike outputs (Figures 1B,C).

Figure 1
figure1

A, KL-divergence grows with increasing correlation between the neurons. B, Distribution of spike outputs from numerical simulation of LIF neurons. C, Predicted distribution of spike outputs from Ising model.

For a leaky IF or exponential IF neuron receiving an input signal identical in all trials, and a background noise independent from trial to trial, it is possible to explicitly calculate the linear response function [2, 3]. We use this linear filter to compute instantaneous firing probabilities for the N cells in our setup. This gives us a theoretical basis for our central finding that strong higher-order correlations arise naturally in integrate and fire cells receiving common inputs.

References

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    Macke JH, Opper M, Bethge M: Common input explains higher-order correlations and entropy in a simple model of neural population activity. Phys Rev Letters. 2011, 106:

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    Ostojic S, Brunel N: From spiking neuron models to linear-nonlinear models. PLOS Computational Biology. 2011, 7 (1):

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    Richardson MJE: Firing rate response of linear and nonlinear integrate and fire neurons to modulated current-based and conductance-based synaptic drive. Phys Rev E. 2007, 76:

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Acknowledgements

This work was funded in part by the Burroughs Wellcome Fund Scientific Interfaces Program.

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Correspondence to David A Leen.

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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Leen, D.A., Shea-Brown, E. A simple mechanism for higher-order correlations in integrate-and-fire neurons. BMC Neurosci 13, P45 (2012) doi:10.1186/1471-2202-13-S1-P45

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Keywords

  • Firing Rate
  • Ising Model
  • Neuron Model
  • Pairwise Correlation
  • Linear Filter