Cable theory

Classical cable theory uses mathematical models to calculate the electric current (and accompanying voltage) along passive neurites, particularly the dendrites that receive synaptic inputs at different sites and times. Estimates are made by modeling dendrites and axons as cylinders composed of segments with capacitances c_m and resistances r_m combined in parallel (see Fig. 1). The capacitance of a neuronal fiber comes about because electrostatic forces are acting through the very thin lipid bilayer (see Figure 2). The resistance in series along the fiber r_l is due to the axoplasm's significant resistance to movement of electric charge. History Cable theory in computational neuroscience has roots leading back to the 1850s, when Professor William Thomson (later known as Lord Kelvin) began developing mathematical models of signal decay in submarine (underwater) telegraphic cables. The model

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

Benjamin Torben-Nielsen, Willem Anna Mark Wybo

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.

Springer Berlin Heidelberg2013

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

Daniele Boccalini, Marc-Oliver Gewaltig, Benjamin Torben-Nielsen, Willem Anna Mark Wybo

We prove that when a class of partial differential equations, generalized from the cable equation, is defined on tree graphs and the inputs are restricted to a spatially discrete, well chosen set of points, the Green's function (GF) formalism can be rewritten to scale asO(n) with the number n of inputs locations, contrary to the previously reported O(n(2)) scaling. We show that the linear scaling can be combined with an expansion of the remaining kernels as sums of exponentials to allow efficient simulations of equations from the aforementioned class. We furthermore validate this simulation paradigm on models of nerve cells and explore its relation with more traditional finite difference approaches. Situations in which a gain in computational performance is expected are discussed.

Massachusetts Institute of Technology Press2015

High-speed mechano-active multielectrode array for investigating rapid stretch effects on cardiac tissue

Etienne de Coulon, Christian Dellenbach, Matthias Imboden, Alexandre Poulin, Stephan Rohr, Samuel Rosset, Herbert Shea

Systematic investigations of the effects of mechano-electric coupling (MEC) on cellular cardiac electrophysiology lack experimental systems suitable to subject tissues to in-vivo like strain patterns while simultaneously reporting changes in electrical activation. Here, we describe a self-contained motor-less device (mechano-active multielectrode-array, MaMEA) that permits the assessment of impulse conduction along bioengineered strands of cardiac tissue in response to dynamic strain cycles. The device is based on polydimethylsiloxane(PDMS) cell culture substrates patterned with dielectric actuators (DEAs) and compliantgold ion-implanted extracellular electrodes. The DEAs induce uniaxial stretch and compression in defined regions of the PDMS substrate at selectable amplitudes and with rates up to 18 s−1. Conduction along cardiomyocyte strands was found to depend linearly on static strain according to cable theory while, unexpectedly, being completely independent on strain rates. Parallel operation of multiple MaMEAs provides for systematic high-throughput investigations of MEC during spatially patterned mechanical perturbations mimicking in-vivo conditions.

2019

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