The Green’s function formalism as a bridge between single- and multi-compartmental modeling

Abstract

Neurons are spatially extended structures that receive and process inputs on their dendrites. It is generally accepted that neuronal computations arise from the active integration of synaptic inputs along a dendrite between the input location and the location of spike generation in the axon initial segment. However, many application such as simulations of brain networks use point-neurons—neurons without a morphological component—as computational units to keep the conceptual complexity and computational costs low. Inevitably, these applications thus omit a fundamental property of neuronal computation. In this work, we present an approach to model an artificial synapse that mimics dendritic processing without the need to explicitly simulate dendritic dynamics. The model synapse employs an analytic solution for the cable equation to compute the neuron’s membrane potential following dendritic inputs. Green’s function formalism is used to derive the closed version of the cable equation. We show that by using this synapse model, point-neurons can achieve results that were previously limited to the realms of multi-compartmental models. Moreover, a computational advantage is achieved when only a small number of simulated synapses impinge on a morphologically elaborate neuron. Opportunities and limitations are discussed.

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Notes

  1. 1.

    Note that formally, the operator \(\hat{L}_d\) depends on \(x\) explicitly in a discontinuous way: for \(0<x<L_d\hbox {: }\hat{L}_d(x) = \frac{\pi a_d^2}{r_a}\frac{\mathrm{\partial }^2 }{\mathrm{\partial }x^2} - 2\pi a_d g_m - 2\pi a_d c_m \frac{\mathrm{\partial }}{\mathrm{\partial }t}\), and for \(x=0\): \(\hat{L}_d(x=0) = \sum _{d=1}^N \frac{\pi a_d^2}{r_a}\frac{\mathrm{\partial }}{\mathrm{\partial }x} - G_{\text {som}} - C_{\text {som}} \frac{\mathrm{\partial }}{\mathrm{\partial }t}\).

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Acknowledgments

We thank Marc-Oliver Gewaltig for comments on the manuscript and Moritz Deger for helpful discussion. This work was supported by the BrainScaleS EU FET-proactive FP7 grant.

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Correspondence to Willem A. M. Wybo.

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Wybo, W.A.M., Stiefel, K.M. & Torben-Nielsen, B. The Green’s function formalism as a bridge between single- and multi-compartmental modeling. Biol Cybern 107, 685–694 (2013). https://doi.org/10.1007/s00422-013-0568-0

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Keywords

  • Morphological simplification
  • Cable theory
  • Interacting synapses
  • Green’s function formalism
  • Transfer functions