Parameters of stochastic diffusion processes estimated from observations of first-hitting times: Application to the leaky integrate-and-fire neuronal model

A theoretical model has to stand the test against the real world to be of any practical use. The first step is to identify parameters in the model estimated from experimental data. In many applications where renewal point data are available, models of first-hitting times of underlying diffusion processes arise. Despite the seemingly simplicity of the model, the problem of how to estimate parameters of the underlying stochastic process has resisted solution. The few attempts have either been unreliable, difficult to implement, or only valid in subsets of the relevant parameter space. Here we present an estimation method that overcomes these difficulties, is computationally easy and fast to implement, and also works surprisingly well on small data sets. The method is illustrated on simulated and experimental data. Two common neuronal models—the Ornstein-Uhlenbeck and Feller models—are investigated.

Figure 1

Densities of estimates for simulated data based on 1000 estimates. Upper panels: estimates of $\alpha$. Lower panels: estimates of $\beta$. Solid black lines: sample sizes of 500. Dashed black lines: sample sizes of 100. Solid gray lines: sample sizes of 50. Dashed gray lines: sample sizes of 10. Vertical lines: true values used in the simulations: (A), (E) $(\alpha ,\beta )=(0.8,1)$; (B), (F) $(\alpha ,\beta )=(1,0.1)$; (C), (G) $(\alpha ,\beta )=(2,0.1)$; (D), (H) $(\alpha ,\beta )=(2,1)$. Note different scales.

Figure 2

Auditory neurons. Comparison of the (normalized) right-hand side of Eq. (12) (solid curves) to the (normalized) empirical equation (13) (dashed curves), calculated with estimated parameters. Vertical axes can be interpreted as a cumulative probability; horizontal axes are ISIs (s). (A) Spontaneous record: $\widehat{\alpha }=0.85$, $\widehat{\beta }=0.09$. (B) Stimulated record: $\widehat{\alpha }=4.8$, $\widehat{\beta }=0.63$. Note different time axes.

Figure 3

Olfactory neurons. (A), (B), (C) Neuron that fits the models. (D), (E), (F) Neuron that did not fit the models. (A), (D) OU model, normalized comparison of the right-hand side of Eq. (12) (solid lines) to the empirical equation (13) (dashed lines), calculated with estimated parameters. Vertical axes can be interpreted as a cumulative probability; horizontal axes are ISIs (s). (B), (E) Likewise for the Feller model. (C), (F) Corresponding spike trains consisting of 100 spikes. Note different time axes.