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      Computing the Local Field Potential (LFP) from Integrate-and-Fire Network Models

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          Abstract

          Leaky integrate-and-fire (LIF) network models are commonly used to study how the spiking dynamics of neural networks changes with stimuli, tasks or dynamic network states. However, neurophysiological studies in vivo often rather measure the mass activity of neuronal microcircuits with the local field potential (LFP). Given that LFPs are generated by spatially separated currents across the neuronal membrane, they cannot be computed directly from quantities defined in models of point-like LIF neurons. Here, we explore the best approximation for predicting the LFP based on standard output from point-neuron LIF networks. To search for this best “LFP proxy”, we compared LFP predictions from candidate proxies based on LIF network output (e.g, firing rates, membrane potentials, synaptic currents) with “ground-truth” LFP obtained when the LIF network synaptic input currents were injected into an analogous three-dimensional (3D) network model of multi-compartmental neurons with realistic morphology, spatial distributions of somata and synapses. We found that a specific fixed linear combination of the LIF synaptic currents provided an accurate LFP proxy, accounting for most of the variance of the LFP time course observed in the 3D network for all recording locations. This proxy performed well over a broad set of conditions, including substantial variations of the neuronal morphologies. Our results provide a simple formula for estimating the time course of the LFP from LIF network simulations in cases where a single pyramidal population dominates the LFP generation, and thereby facilitate quantitative comparison between computational models and experimental LFP recordings in vivo.

          Author Summary

          Leaky integrate-and-fire (LIF) networks are often used to model neural network activity. The spike trains they produce, however, cannot be directly compared to the local field potentials (LFPs) that are measured by low-pass filtering the potential recorded from extracellular electrodes. This is because LFPs are generated by neurons with spatial extensions, while LIF networks typically consist of point neurons. In order to still be able to approximately predict LFPs from LIF network simulations, we here explore simple proxies for computing LFPs based on standard output from LIF network simulations. Predictions from the various LFP proxies were compared with “ground-truth” LFPs computed by means of well-established volume conduction theory where synaptic currents corresponding to the LIF network simulation were injected into populations of multi-compartmental neurons with realistic morphologies. We found that a simple weighted sum of the LIF synaptic currents with a single universally applicable set of weights excellently capture the time course of the LFP signal when the LFP predominantly is generated by a single population of pyramidal cells. Our study therefore provides a simple formula by which the LFP signal can be estimated directly from the LIF network activity, providing a missing quantitative link between simple neural models and LFP measures in vivo.

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          Simple model of spiking neurons.

          A model is presented that reproduces spiking and bursting behavior of known types of cortical neurons. The model combines the biologically plausibility of Hodgkin-Huxley-type dynamics and the computational efficiency of integrate-and-fire neurons. Using this model, one can simulate tens of thousands of spiking cortical neurons in real time (1 ms resolution) using a desktop PC.
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            In this paper we present an approach to the identification of nonlinear input-state-output systems. By using a bilinear approximation to the dynamics of interactions among states, the parameters of the implicit causal model reduce to three sets. These comprise (1) parameters that mediate the influence of extrinsic inputs on the states, (2) parameters that mediate intrinsic coupling among the states, and (3) [bilinear] parameters that allow the inputs to modulate that coupling. Identification proceeds in a Bayesian framework given known, deterministic inputs and the observed responses of the system. We developed this approach for the analysis of effective connectivity using experimentally designed inputs and fMRI responses. In this context, the coupling parameters correspond to effective connectivity and the bilinear parameters reflect the changes in connectivity induced by inputs. The ensuing framework allows one to characterise fMRI experiments, conceptually, as an experimental manipulation of integration among brain regions (by contextual or trial-free inputs, like time or attentional set) that is revealed using evoked responses (to perturbations or trial-bound inputs, like stimuli). As with previous analyses of effective connectivity, the focus is on experimentally induced changes in coupling (cf., psychophysiologic interactions). However, unlike previous approaches in neuroimaging, the causal model ascribes responses to designed deterministic inputs, as opposed to treating inputs as unknown and stochastic.
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              Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons.

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              The dynamics of networks of sparsely connected excitatory and inhibitory integrate-and-fire neurons are studied analytically. The analysis reveals a rich repertoire of states, including synchronous states in which neurons fire regularly; asynchronous states with stationary global activity and very irregular individual cell activity; and states in which the global activity oscillates but individual cells fire irregularly, typically at rates lower than the global oscillation frequency. The network can switch between these states, provided the external frequency, or the balance between excitation and inhibition, is varied. Two types of network oscillations are observed. In the fast oscillatory state, the network frequency is almost fully controlled by the synaptic time scale. In the slow oscillatory state, the network frequency depends mostly on the membrane time constant. Finite size effects in the asynchronous state are also discussed.
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                Author and article information

                Contributors
                Role: Editor
                Journal
                PLoS Comput Biol
                PLoS Comput. Biol
                plos
                ploscomp
                PLoS Computational Biology
                Public Library of Science (San Francisco, CA USA )
                1553-734X
                1553-7358
                14 December 2015
                December 2015
                : 11
                : 12
                Affiliations
                [1 ]The Biorobotics Institute, Scuola Superiore Sant’Anna, Pontedera, Pisa, Italy
                [2 ]Neural Computation Laboratory, Center for Neuroscience and Cognitive Systems @UniTn, Istituto Italiano di Tecnologia, Rovereto, Italy
                [3 ]Department of Neuroscience and Pharmacology, University of Copenhagen, Copenhagen, Denmark
                [4 ]Department of Computational Biology, School of Computer Science and Communication, Royal Institute of Technology–KTH, Stockholm, Sweden
                [5 ]Ernst Strüngmann Institute (ESI) for Neuroscience in Cooperation with Max Planck Society, Frankfurt/Main, Germany
                [6 ]Institute of Clinical Neuroanatomy, Goethe University Frankfurt, Frankfurt/Main, Germany
                [7 ]Frankfurt Institute for Advanced Studies (FIAS), Frankfurt/Main, Germany
                [8 ]Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, Ås, Norway
                [9 ]Department of Physics, University of Oslo, Oslo, Norway
                University College London, UNITED KINGDOM
                Author notes

                The authors have declared that no competing interests exist.

                Conceived and designed the experiments: AM HL HC AL SP GTE. Performed the experiments: AM HL HC. Analyzed the data: AM. Contributed reagents/materials/analysis tools: AM HL HC. Wrote the paper: AM HL HC AL SP GTE.

                Article
                PCOMPBIOL-D-15-00871
                10.1371/journal.pcbi.1004584
                4682791
                26657024
                116b2415-a9d4-4aa4-8efb-6466ed67dbff
                © 2015 Mazzoni et al

                This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited

                Page count
                Figures: 10, Tables: 5, Pages: 38
                Product
                Funding
                AM was supported by the NEBIAS European project (EUFP7-ICT-611687), by the PRIN/HandBot Italian project (CUP: B81J12002680008; prot.: 20102YF2RY), by the SI-CODE Project of the FP7 of the European Commission (FET-Open, Grant FP7-284553) and by the Italian Ministry of Foreign Affairs and International Cooperation, Directorate General for Country Promotion (Economy, Culture and Science)—Unit for Scientific and Technological Cooperation, via the Italy-Sweden bilateral research project on "Brain network mechanisms for integration of natural tactile input patterns". HL was supported by The Danish Council for Independent Research and FP7 Marie Curie Actions – COFUND (grant id: DFF – 1330-00226), EU grant 269921 (BrainScaleS) and The Dynamical Systems Interdisciplinary Network, University of Copenhagen. HC was supported by BMBF grant 01GQ1406 (Bernstein Award 2013). SP was supported by the SI-CODE Project of the FP7 of the European Commission (FET-Open, Grant FP7-284553), and the Autonomous Province of Trento (“Grandi Progetti 2012,” the “Characterizing and Improving Brain Mechanisms of Attention–ATTEND” project). GTE was supported by the Research Council of Norway (ISP-Physics), EU grants 269921 (BrainScaleS) and 604102 (Human Brain Project). Computer time on supercomputer Stallo was provided by Research Council of Norway (Notur). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
                Categories
                Research Article
                Custom metadata
                The Python codes used to generate LFP from artificial morphologies injected with LIF spike dynamics are available on the LFPy official site ( http://lfpy.github.io/). Current based and conductance based LIF model source codes are identical to those already available on the ModelDB sharing repository ( http://senselab.med.yale.edu/ModelDB/ShowModel.asp?model=152539) with accession number 152539. Data generated by both networks are available to all and are stored on a public repository here: http://dx.doi.org/10.5061/dryad.j5r51

                Quantitative & Systems biology
                Quantitative & Systems biology

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