Computational Convergence of the Path Integral for Real Dendritic Morphologies
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Computational Convergence of the Path Integral for Real Dendritic Morphologies Quentin Caudron · Simon R. Donnelly · Samuel P.C. Brand · Yulia Timofeeva Received: 19 June 2012 / Accepted: 11 September 2012 / Published online: 22 November 2012 © 2012 Q. Caudron et al.; licensee Springer. 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.
Abstract Neurons are characterised by a morphological structure unique amongst biological cells, the core of which is the dendritic tree. The vast number of dendritic geometries, combined with heterogeneous properties of the cell membrane, continue to challenge scientists in predicting neuronal input-output relationships, even in the case of sub-threshold dendritic currents. The Green’s function obtained for a given dendritic geometry provides this functional relationship for passive or quasi-active dendrites and can be constructed by a sum-over-trips approach based on a path integral formalism. In this paper, we introduce a number of efficient algorithms for realisation of the sum-over-trips framework and investigate the convergence of these algorithms on different dendritic geometries. We demonstrate that the convergence of the trip sampling methods strongly depends on dendritic morphology as well as the biophysical properties of the cell membrane. For real morphologies, the number of trips to guarantee a small convergence error might become very large and strongly affect computational efficiency. As an alternative, we introduce a highly-efficient matrix method which can be applied to arbitrary branching structures. Keywords Dendrites · Path integral · Sum-over-trips · Morphology · Dendritic computation Q. Caudron () · S.P.C. Brand · Y. Timofeeva Centre for Complexity Science, University of Warwick, Coventry, CV4 7AL, UK e-mail: [email protected] Q. Caudron · Y. Timofeeva Department of Computer Science, University of Warwick, Coventry, CV4 7AL, UK S.R. Donnelly Doctoral Training Centre in Neuroinformatics and Computational Neuroscience, University of Edinburgh, Edinburgh, EH8 9AB, UK S.P.C. Brand Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
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1 Introduction Discovered more than a century ago by Santiago Ramón y Cajal [1], dendrites form the vast majority of the surface area of a neuron, with the dendritic trees of some motoneurons representing up to 97% of total neuronal surface area and 75% of the total neuronal volume [2]. These complex branching structures are responsible for transferring electrical activity between synapses and the soma. As technology evolved, interest in dendrites began to gather momentum, with the invention of sharp micropipette electrodes in the early 1950s allowing intracellular recordings to be made. It was the breakthrough work of Wilfrid Rall [3] on the applic
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