The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.Ĭompeting interests: The authors have declared that no competing interests exist.ĭue to their extended branching in both dendritic and axonal fields many classes of neurons are not electrically compact objects, in that the membrane voltage varies significantly throughout their spatial structure. EP/L015374/1, CDT in Mathematics for Real-World Systems, to RPG. 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.ĭata Availability: All relevant data are within the manuscript and its Supporting Information files.įunding: We acknowledge funding from the Engineering and Physical Sciences Research Council funding under Grant No. Received: JAccepted: JanuPublished: April 20, 2020Ĭopyright: © 2020 Gowers et al. The generality of these results suggests they will provide a mathematical framework for future studies that might require the structure of neurons to be taken into account, such as the effect of electrical fields or multiple synaptic input streams that target distinct spatial domains of cortical pyramidal cells.Ĭitation: Gowers RP, Timofeeva Y, Richardson MJE (2020) Low-rate firing limit for neurons with axon, soma and dendrites driven by spatially distributed stochastic synapses. Though the models are simple, these preliminary results show that it is possible to obtain useful formulae that capture the effects of spatially distributed synaptic drive. ![]() We illustrate this approach using basic neuronal morphologies that capture the fundamentals of neuronal structure. Here we show that in a physiologically relevant, low-firing-rate regime, an approximate level-crossing approach can be used to provide an estimate for the neuronal firing rate even when the dendrites and axons are included. This has been largely due to the mathematical complexity of including the effects of spatially distributed synaptic input. However, until relatively recently the majority of the mathematical formulae describing how fluctuating synaptic drive triggers action potentials have been applicable only for small neurons with the dendritic and axonal structure ignored. The dynamics of how these signals are integrated and how they ultimately trigger outgoing spikes have been modelled extensively since the late 1960s. The resulting synaptic input causes voltage fluctuations throughout their structure that evolve in space and time. In an active neuronal network, neurons receive vast numbers of incoming synaptic pulses throughout their dendrites and cell body that each exhibit significant variability in amplitude and arrival time. Neurons are extended cells with multiple branching dendrites, a cell body and an axon. The combination of simplicity and generality promises a framework that can be built upon to incorporate increasing levels of biophysical detail and extend beyond the low-rate firing limit treated in this paper. The approach necessitates only calculating the mean and variances of the non-thresholded voltage and its rate of change in neuronal structures subject to spatio-temporal synaptic fluctuations. ![]() Even for simple models, the analytical approximations derived demonstrate a surprising richness including: independence of the firing rate to the electrotonic length for certain models, but with a form distinct to the point-like leaky integrate-and-fire model a non-monotonic dependence of the firing rate on the number of dendrites receiving synaptic drive a significant effect of the axonal and somatic load on the firing rate and the role that the trigger position on the axon for spike initiation has on firing properties. Here we demonstrate that an extension of the level-crossing method of Rice, previously used for compact cells, provides a general framework for approximating the firing rate of neurons with spatial structure. ![]() Calculation of the complex flow of electrical activity driven by stochastic spatio-temporal synaptic input streams in these structures has presented a significant analytical challenge. These approaches have yielded considerable insight into how single-cell properties affect network activity however, many neuronal classes, such as cortical pyramidal cells, are electrically extended objects. Since the 1960s, the majority of approaches have treated neurons as being electrically compact and therefore isopotential. Analytical forms for neuronal firing rates are important theoretical tools for the analysis of network states.
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