Note on “ Characterization of subthreshold voltage fluctuations in neuronal membranes ”

Different analytic expressions for the membrane potential distribution of membranes subject to synaptic noise have been proposed, and can be very helpful to analyze experimental data. However, all of these expressions are either a pproximations or limit cases, and it is not clear how they compare, and which expression sho uld be used in a given situation. In this note, we provide a comparison of the diffe rent approximations available, with an aim to delineate which expression is most suitable fo r analyzing experimental data. Synaptic noise can be modeled by fluctuating conductances de scrib d by Ornstein-Uhlenbeck stochastic processes (Destexhe, Rudolph, Fellous, & Sejnowski, 20 01). This system was investigated by using stochastic calculus to obtain analytic expressions f r the steady-state membrane potential (V m) distribution (Rudolph & Destexhe, 2003; 2005). Analytic ex pressions can also be obtained for the moments of the underlying three-dimensional Fokker-Planc k equation (FPE) (Richardson, 2004), or by considering this equation under different limit cases (L indner & Longtin, 2006). One of the greatest promises of such analytic expressions is that they can be used to deduce the characteristics of conductance fluctuations from intracellular recordings in vivo (Rudolph et al., 2004; 2005). A recent article (Lindner & Longtin, 2006) provided an in-de pth analysis of some of these expressions, as well as different analytically-exact limit c ases. One of the conclusions of this analysis was that the original expression provided by Rudolph & Deste xhe (2003) was derived using steps that were incorrect for colored noise, and that the expression ob tained matches numerical simulations only for restricted ranges of parameters. The latter conclusion was in agreement with the analysis provided in Rudolph & Destexhe (2005). Another conclusion was that th e “extended expression” proposed by Rudolph & Destexhe (2005), although providing an excellent fit to Vm distributions in general, does not match for some parameter values and in particular, it doe s n t agree with the analytically-exact static-noise limit. This extended expression is therefore not an exact solution of the system either. Since several analytic expressions were provided for the st eady-state Vm distribution (Rudolph & Destexhe, 2003; Richardson, 2004; Rudolph & Destexhe, 2005 ; Lindner & Longtin, 2006), and since all of these expressions are either approximations or limit cases, it is not clear how they compare and which expression should be used in a given situation. In part icular, it is unclear which expression should be used to analyze experimental recordings. In the pr esent note, we attempt to answer these questions by clarifying a number of points about some of the p revious expressions, and by providing a detailed comparison of the different expressions availab le in the literature. First, we would like to clarify a number of misleading statem ents we made in the original article (Rudolph & Destexhe, 2003), and which may lead to confusion. The goal of this paper was to obtain an analytic expression for the steady-state V m distribution of membranes subject to conductance-based colored noise sources. To obtain this, we considered the ful l system under at → ∞ limit. In this limit, we noted that the noise time constants become infinitesimall y small compared to the time over which the system is considered, and this property allowed us to tre at he system as for white noise. Our 2 main assumption was that this procedure would allow us to obt ain the correct steady-state properties like the Vm distribution. Our approach was to obtain a simplified FPE whi ch gives the same steadystate solutions as the FPE describing the full system. These assumptions were stated in the Results of Rudolph & Destexhe (2003), but were not clearly stated in the Abstract and Discussion, and it could be understood that we claimed to provide an FPE valid for the f ull system. We clarify here that the treatment followed in that paper did not intend to describe t he full system, but was only restricted to steady-state solutions. Unlike the original expression (Rudolph & Destexhe, 2003), which matches only for a restricted range of parameters, the extended expression (Rudolph & Des texhe, 2005) matches for several orders of magnitude of the parameters (see also supplementary i fo mation of Rudolph & Destexhe, 2005). Why the extended expression matches so well, althoug h it is not an exact solution of the system (Lindner & Longtin, 2006), is presently unknown. It i s not due to the presence of boundary conditions, which could compensate for mismatches “by c han e”. Simulations with and without boundary conditions gave equally good fits for the parameter s considered here (see NEURON code in supplementary information). Our interpretation (Rudolph & Destexhe, 2005) is that the t → ∞ limit altered the spectral structure of the stochastic process (fi ltering), and one can recover a better spectral structure by following the same approximation for a system t hat is solvable (e.g., that of Richardson, 2004) and correct it accordingly. Thus, as also found by Lind ner & Longtin (2006), the extended expression is a very good approximation of the steady-state Vm distribution. Other expressions have been proposed under different approximations (Richardson , 2004) or limit cases (Lindner & Longtin, 2006) and also match well the simulations for the applicable range of parameters. Since different expressions were proposed corresponding t o different approximations (Rudolph & Destexhe, 2003, 2005; Richardson, 2004; Lindner & Longtin, 2006), we investigated which expression must be used in practical situations. We have considere d an extended range of parameters and tested all expressions by running the model for 10,000 rando mly-selected values within this parameter space. The results of this procedure are shown in Fig. 1AD. The smallest error between analytic expressions and numerical simulations was found for the ext ended expression of Rudolph & Destexhe (2005), followed by Gaussian approximations of the same aut hors and that of Richardson (2004). The fourth best approximation was the static-noise limit by Lin dner & Longtin (2006). By scanning only within physiologically-relevant values based on conducta n e measurements in cats in vivo (Rudolph et al., 2005), the same ranking was observed (Fig. 1E), with e ven more drastic differences (up to 95%; see supplementary information). Manual examination o f the different parameter sets where the extended expression was not the best estimate revealed that this happened when both time constants were slow (“slow synapses”; decay time constants >50 ms). Indeed, performing parameter scans restricted to this region of parameters showed that the extend ed expression, while still providing good fits to the simulations, ranked first for less than 30% of the ca ses, while the static-noise limit was the best estimate for almost 50% of parameter sets (Fig. 1F; see d etails in supplementary information). Scanning parameters within a wider range of values includin g fast/slow synapses and weak/strong conductances showed that the extended expression was still the best estimate (about 47%), followed by the static-noise limit (37%; see supplementary informat ion).