Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model

The stochastic Ornstein-Uhlenbeck neuronal model is studied, and estimators of the model input parameters, depending on the firing regime of the process, are derived. Closed expressions for the Laplace transforms of the first two moments of the normalized first-passage time through a constant boundary in the suprathreshold regime are derived, which is used to define moment estimators. In the subthreshold regime, the exponentiality of the first-passage time is utilized to characterize the input parameters. In the threshold regime and for the Wiener process approximation, analytic expressions for the first-passage-time density are used to derive the maximum-likelihood estimators of the parameters. The methods are illustrated on simulated data under different conditions, including misspecification of the intrinsic parameters of the model. Finally, known approximations of the first-passage-time moments are improved.

Figure 1

Realizations of $X_t$ (membrane potential against time, arbitrary units). When the threshold $S$ is reached at time $T$, both process and time are reset to 0. The dashed line $\mu \tau$ is the asymptotic depolarization. Upper panel: suprathreshold regime. Lower panel: subthreshold regime.

Figure 2

Gaussian quantile plots (empirical versus theoretical quantiles) for the estimators (23, 24) from the 1000 artificial data sets simulated with $\mu =1.5\mathrm{mV}∕\mathrm{msec}$, $\sigma =1\mathrm{mV}∕\sqrt{\mathrm{msec}}$, $\tau =10\mathrm{msec}$, and $S=10\mathrm{mV}$. The straight lines pass through the first and third quartiles; the horizontal lines are set at the parameter values used in the simulations.

Figure 3

Mean from 100 realizations of the estimators (23, 24) when $S$ and $\tau$ are misspecified. Each data set consists of 100 simulated ISI’s. Left panel: estimates of $\mu$ versus correct values. For each value of $\mu$, $\sigma =0.1\mathrm{mV}∕\sqrt{\mathrm{msec}}$ (left points), $\sigma =1.0\mathrm{mV}∕\sqrt{\mathrm{msec}}$ (middle points), and $\sigma =1.5\mathrm{mV}∕\sqrt{\mathrm{msec}}$ (right points). Right panel: estimates of $\sigma$ versus correct values. For each value of $\sigma$, $\mu =1.0\mathrm{mV}∕\mathrm{msec}$ (left points), $\mu =1.5\mathrm{mV}∕\mathrm{msec}$ (middle points), and $\mu =2.0\mathrm{mV}∕\mathrm{msec}$ (right points). Also estimator (16) is plotted in the right panel when $S$ and $\tau$ are well specified (points above plotting region are missing). The dashed lines are the identity lines, indicating the best possible estimates.

Figure 4

Estimation assuming the full model and the Wiener approximation, respectively, on data sets simulated from the full model for different time constants. Mean from 100 realizations. Left panel: estimates of $\mu$. Right panel: estimates of $\sigma$. Points above plotting region of estimator (16) are missing (right panel). The dashed lines are the identity lines, indicating the best possible estimates.