Abstract
Computational models offer a unique tool for understanding the network-dynamical mechanisms which mediate between physiological and biophysical properties, and behavioral function. A traditional challenge in computational neuroscience is, however, that simple neuronal models which can be studied analytically fail to reproduce the diversity of electrophysiological behaviors seen in real neurons, while detailed neuronal models which do reproduce such diversity are intractable analytically and computationally expensive. A number of intermediate models have been proposed whose aim is to capture the diversity of firing behaviors and spike times of real neurons while entailing the simplest possible mathematical description. One such model is the exponential integrate-and-fire neuron with spike rate adaptation (aEIF) which consists of two differential equations for the membrane potential (V) and an adaptation current (w). Despite its simplicity, it can reproduce a wide variety of physiologically observed spiking patterns, can be fit to physiological recordings quantitatively, and, once done so, is able to predict spike times on traces not used for model fitting. Here we compute the steady-state firing rate of aEIF in the presence of Gaussian synaptic noise, using two approaches. The first approach is based on the 2-dimensional Fokker-Planck equation that describes the (V,w)-probability distribution, which is solved using an expansion in the ratio between the time constants of the two variables. The second is based on the firing rate of the EIF model, which is averaged over the distribution of the w variable. These analytically derived closed-form expressions were tested on simulations from a large variety of model cells quantitatively fitted to in vitro electrophysiological recordings from pyramidal cells and interneurons. Theoretical predictions closely agreed with the firing rate of the simulated cells fed with in-vivo-like synaptic noise.
Highlights
In recent years there has been an increasing push toward neurobiologically highly realistic large-scale network models that incorporate a lot of anatomical and physiological detail (Traub et al, 1988; Whittington et al, 2000; Traub et al, 2005; Markram, 2006; Izhikevich and Edelman, 2008; Lansner, 2009; Lundqvist et al, 2010)
Neural network dynamics has been identified as a crucial link between more basic genetic, molecular, and physiological factors on the one hand, and cognitive function and behavior on the other (e.g., Balaguer-Ballester et al, 2011; Mante et al, 2013, and citations therein), and has been described as a point of convergence for various pathophysiological and psychiatric mechanisms (Durstewitz and Seamans, 2012; Mitchell et al, 2012)
In the following, we first derive the 2-dimensional Fokker-Planck equation for the EIF model with spike-triggered adaptation. This is a special case of the AdEx model, a simple 2-dimensional neuron model which combines the exponential integrate-and-fire (EIF) neuron (Fourcaud–Trocmé et al, 2003) with a second differential equation for an adaptation current (Benda and Herz, 2003; Izhikevich, 2003; Brette and Gerstner, 2005; Naud et al, 2008; Hertäg et al, 2012)
Summary
In recent years there has been an increasing push toward neurobiologically highly realistic large-scale network models that incorporate a lot of anatomical and physiological detail (Traub et al, 1988; Whittington et al, 2000; Traub et al, 2005; Markram, 2006; Izhikevich and Edelman, 2008; Lansner, 2009; Lundqvist et al, 2010). Neurobiologically detailed models reach computational and mathematical limits very fast: Their large number of parameters, stiff nonlinearities, and very high dimensionality make fitting to physiological data a very tedious ad-hoc process, numerical simulations very time-consuming, and prevent a deep understanding of the underlying dynamical mechanisms For these reasons, mean-field theories (MFT) have been developed for networks of simpler single cell models such as the leaky integrate-and-fire neuron (LIF) These analytical expressions should provide an essential building block for the theoretical analysis of large networks composed of physiologically realistic elements, and improved tools for a mechanistic understanding of the relations between physiologically recorded network activity and behavior
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