Are the input parameters of white noise driven integrate and fire neurons uniquely determined by rate and CV?

Abstract

Integrate and fire (IF) neurons have found widespread applications in computational neuroscience. Particularly important are stochastic versions of these models where the driving consists of a synaptic input modeled as white Gaussian noise with mean $\mu$ and noise intensity $D$. Different IF models have been proposed, the firing statistics of which depends nontrivially on the input parameters $\mu$ and $D$. In order to compare these models among each other, one must first specify the correspondence between their parameters. This can be done by determining which set of parameters $(\mu ,D)$ of each model is associated with a given set of basic firing statistics as, for instance, the firing rate and the coefficient of variation (CV) of the interspike interval (ISI). However, it is not clear a priori whether for a given firing rate and CV there is only one unique choice of input parameters for each model. Here we review the dependence of rate and CV on input parameters for the perfect, leaky, and quadratic IF neuron models and show analytically that indeed in these three models the firing rate and the CV uniquely determine the input parameters.

Introduction

Stochastic integrate and fire (IF) neurons constitute an important tool in theoretical neuroscience, having been used to address a number of relevant biological problems. For instance, different variants of these models have been employed in the debate on the high variability of the interspike interval (ISI) observed for cortical neurons (Softky and Koch, 1993; Gutkin and Ermentrout, 1998). Other problems in which stochastic IF models have been applied include the response to fast signals (Brunel et al., 2001; Lindner and Schimansky-Geier, 2001; Fourcaud-Trocmé et al., 2003; Naundorf et al., 2005), asynchronous spiking in recurrent networks (Brunel, 2000), and oscillations of firing activity in systems with spatially correlated noisy driving (Doiron et al., 2003, Doiron et al., 2004; Lindner et al., 2005b).

IF neurons can be classified according to the nonlinearities that govern their subthreshold dynamics. Three simple and important variants are the perfect integrate and fire (PIF), leaky integrate and fire (LIF), and quadratic integrate and fire (QIF) models, in which the subthreshold dynamics is described by a constant, a linear, and a quadratic voltage dependence, respectively. Noisy inputs with different degrees of biological realism have been considered for these models. One simple choice is a white Gaussian input current, corresponding to the so-called diffusion approximation of synaptic spike train input (Holden, 1976; Ricciardi, 1977; Tuckwell, 1989). The PIF with white noise drive (also referred to as the random walk model of neural firing) was first considered by Gerstein and Mandelbrot (1964). The LIF with a white Gaussian input current has been studied by Johannesma (1968) and afterwards by many other authors (see Holden, 1976; Ricciardi, 1977; Tuckwell, 1989; Burkitt, 2006 and references therein); it is also referred to as Ornstein–Uhlenbeck neuron (Lánský and Rospars, 1995). White noise driving in the QIF was studied by Gutkin and Ermentrout (1998) and Lindner et al. (2003).

The firing statistics of the various IF models may depend sensitively on the specific nonlinearity in the respective model, and so it is not clear at the first glance which model is capable of reproducing which features of the firing statistics of real neurons. For instance, it has been shown that LIF and QIF display rather different phase shifts if driven by a periodic stimulus (Fourcaud-Trocmé et al., 2003). Further, the LIF with periodic stimulation can transmit signals of very high frequencies if they are encoded in the noise intensity (Lindner and Schimansky-Geier, 2001), while the QIF cannot (Naundorf et al., 2005). Last but not least, when the strength of the input noise is varied, the LIF can show coherence resonance (CR) (Pakdaman et al., 2001; Lindner et al., 2002) whereas the PIF and the QIF do not display CR (Lindner et al., 2003) (see also below). A thorough comparison of different stochastic IF neurons and the roles of their respective nonlinearities constitutes therefore an interesting and largely open problem.

If one wants to compare two specific IF models one should tune their input parameters (mean and intensity of the input fluctuations) such that the basic firing statistics is the same. A simple choice for the basic firing statistics is the firing rate, quantifying the intensity of the spike train, and the ISI's coefficient of variation (CV), quantifying the irregularity of the spike train. Setting both models in this way in the same firing regime (e.g. low spike rate and high ISI variability), one can then ask for higher statistics (e.g. the power spectrum of the spike train) or for their response characteristics (e.g. to weak periodic stimulation or step currents). The idea for such a tuning tacitly assumes that there is for each of the models at most one input parameter set yielding a desired firing statistics, for instance, rate and CV. At a closer look, however, this is not evident at all: are the input parameters for a white noise driven IF model indeed uniquely determined if we prescribe certain values of the rate and the CV? This is the question that we address in this paper and we will answer it for the three IF models mentioned above, namely, PIF, LIF, and QIF neurons.

The question discussed here is also related to the problem of parameter estimation for an IF model from experimental data, which has been subject of several studies (see Lánský and Ditlevsen, 2008 and references therein). In one approach, the model parameters are inferred from subthreshold membrane measurements (for a recent reference see Badel et al., 2008). In another approach, model parameters are estimated using solely the ISI statistics (Tuckwell and Richter, 1978; Inoue et al., 1995; Rauch et al., 2003; Camera et al., 2004; Shinomoto et al., 1999; Ditlevsen and Lansky, 2005, Ditlevsen and Lansky, 2007; Mullowney and Iyengar, 2008). The latter approach is of importance, since often subthreshold data are not available. Most of these studies consider the LIF model. To the best of our knowledge, the problem of the uniqueness of input parameters has been addressed only by Kostal et al. (2007), where it is mentioned without a proof that uniqueness of the input parameters given fixed rate and CV holds for the LIF.

In this paper, we show analytically that rate and CV uniquely determine the input parameters for the three models. We first introduce the models and the statistics studied here (Section 2). In the main part of the paper (3 PIF neuron, 4 LIF neuron, 5 QIF neuron), we briefly review the rate and CV as functions of the input parameters and then prove the uniqueness of the relation between these parameters and the former statistics. We give an outlook to generalizations of the considered problem in Section 6.