Fig. 1
From: Inferring and validating mechanistic models of neural microcircuits based on spike-train data
Estimation results for background input statistics using synthetic data. a Example membrane voltage time series with indicated spike times from a leaky I&F neuron (Eqs. (5–7)) together with membrane voltage and ISI histograms, and associated probability densities pV, pISI calculated using the Fokker–Planck equation (see Methods sections “Method 1: conditioned spike time likelihood” and “Method 2: derived spike rate model”). Parameter estimates are shown in (b). Dots denote the values of the conditioned spike time likelihoods, i.e., the factors in Eq. (1), which are points of pISI here. Method 1 was used for parameter estimation. b Log-likelihood subtracted from the maximal value as a function of the input mean μ and standard deviation σ, based on 400 spikes from the example in (a), with true and estimated parameter values indicated. c Maximal log-likelihood across μ and σ as a function of fixed τm, subtracted from the maximal value across τm which is attained at the indicated value. d Mean and central 50% (i.e., 25–75th percentile) of relative errors between estimated and true parameter values for μ and σ as a function of number of spikes K. Insets: empirical density of parameter estimates with true values indicated for K = 100 and K = 400. e top: spike rate and ISI coefficient of variation (CV) calculated analytically using the Fokker–Planck equation (lines; see Methods sections “Method 1: conditioned spike time likelihood” and “Method 2: derived spike rate model”) and empirically using numerical simulations (dots) as a function of μ for different values of σ; bottom: standard deviation of estimates for μ and σ according to the theoretical bound given by the Cramer–Rao inequality (lines; see Methods section “Calculation of the Cramer–Rao bound”) and empirical values from simulations (dots) for K = 400. In a, b, d, e τm was set to the true value
