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.

A Sparse Reformulation of the Green’s Function Formalism Allows Efficient Simulations of Morphological Neuron Models

“…Thus the choice of the SGF formalism over the finite difference approximation depends on the neural system at hand. In some systems, such as bipolar neurons used in auditory coincidence detection (Agmon-Snir et al, 1998;Wybo et al, 2013), inputs occur only at a small set of locations, and the cells' computation is performed by linear membrane properties. Other systems, such as myelinated axons, have non-linear membrane currents that are concentrated at a discrete set of locations -the nodes of Ranvier, with stretches of myelinated fiber in between that behave approximately linear.…”

“…It is immediately clear that the computational cost of the SGF formalism is far lower than the computational cost of the GF formalism (Wybo et al, 2013) from the number of required kernels alone (Fig 2D). However, the comparison of computational cost with the canonical finite difference approaches requires a more careful discussion.…”

“…In this section we prove formally that a sparse reformulation of equation 2exists. Fourier transforming this equation gives the GF formalism in the Fourier domain (Butz and Cowan, 1974;Wybo et al, 2013), :…”

“…Often however, stretches of neural fiber behave approximately linear (Schoen et al, 2012), and one is not interested in the explicit voltage at all locations, but only at a specific output location. For this reason we proposed the idea of using the Green's function formalism to simulate neuron models that receive inputs at a limited number of locations (Wybo et al, 2013). Given a number of n inputs, the voltage at an output location x i can be written in this formalism as:…”

Electrical Compartmentalization in Neurons

“…To derive NETs, we rely on the Green's function (GF) (Koch, 1998;Wybo et al, 2013Wybo et al, , 2015 Zðx; x 0 ; tÞ. The GF is a function of three variables: two locations x and x 0 along the dendritic arborization and a temporal variable t. We compute the GF in an exponential basis:…”