Identification of noisy response latency

Massimiliano Tamborrino, Susanne Ditlevsen, and Petr Lansky

Phys. Rev. E 86, 021128 – Published 23 August 2012

Abstract

In many physical systems there is a time delay before an applied input (stimulation) has an impact on the output (response), and the quantification of this delay is of paramount interest. If the response can only be observed on top of an indistinguishable background signal, the estimation can be highly unreliable, unless the background signal is accounted for in the analysis. In fact, if the background signal is ignored, however small it is compared to the response and however large the delay is, the estimate of the time delay will go to zero for any reasonable estimator when increasing the number of observations. Here we propose a unified concept of response latency identification in event data corrupted by a background signal. It is done in the context of information transfer within a neural system, more specifically on spike trains from single neurons. The estimators are compared on simulated data and the most suitable for specific situations are recommended.

Received 22 March 2012

DOI:https://doi.org/10.1103/PhysRevE.86.021128

Authors & Affiliations

Massimiliano Tamborrino^* and Susanne Ditlevsen^†

Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK 2100 Copenhagen, Denmark

Petr Lansky^‡

Institute of Physiology, Academy of Sciences of the Czech Republic Videnska 1083, 142 20 Prague 4, Czech Republic

^*mt@math.ku.dk
^†susanne@math.ku.dk
^‡lansky@biomed.cas.cz

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Vol. 86, Iss. 2 — August 2012

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Images

Figure 1
Schematic description of the single experimental trial. Spikes are indicated with dots. At time 0 the measurements start and at time $t_{s}$ a stimulus is applied. $W$ ( $R$ ) denotes the time to the first spontaneous (evoked) spike after $t_{s}$ . For an observer, they cannot be distinguished. Here $T$ represents the time to the first spike after the stimulus onset and is measured. In the presence of an absolute response latency $θ$ , the response latency $R$ is given by $θ + Z$ and no evoked spikes can occur in $[t_{s}, t_{s} + θ]$ . In contrast, spontaneous spikes might occur in that time interval. The relative response latency $Z$ denotes the time to the first evoked spike after $t_{s} + θ$ , the spontaneous ISIs are denoted by $X$ , and the first spike after 0 is denoted by $W_{0}$ . Finally, $W^{-}$ corresponds to the backward recurrence time, defined as the time to the last spontaneous spike before $t_{s}$ .Reuse & Permissions
Figure 2
Dependence of $R_{ME}$ $(\hat{p})$ and $R_{MSE}$ $(\hat{p})$ (average over 10000 simulations) on the number of observations and the absolute response latency when $W$ is exponential with rate $λ = 1 s^{- 1}$ . Top panels: $Z$ is exponential with rate $ω = 10 s^{- 1}$ . Bottom panels: $Z$ is gamma with $α = 0.05 s$ and $β = 2$ . In both cases $E [Z] = 0.1 s$ . Left panels: Different values of $n$ for fixed $θ = 0.2 s$ , with $p \approx 0.26$ . Right panels: Different values of $θ$ for fixed $n = 50$ . Here $p$ varies between 0.14 and 0.39. Also shown are the estimators of $p$ under the renewal assumption ${\hat{p}}_{a}$ (solid line), the stationarity assumption ${\hat{p}}_{b}$ (dashed line), and the parametric assumption ${\hat{p}}_{c}$ (dot-dashed line), given by Eq. (17).Reuse & Permissions
Figure 3
Dependence of $R_{ME}$ $(\hat{θ})$ (average over 10000 simulations) on the number of observations and the absolute response latency when $W$ is exponential with rate $λ = 1 s^{- 1}$ for estimators with errors less than 10 $%$ . Top panels: $Z$ is exponential with rate $ω = 10 s^{- 1}$ . Bottom panels: $Z$ is gamma with $α = 0.05 s$ and $β = 2$ . In both cases $E [Z] = 0.1 s$ . Left panels: Different values of $n$ and $θ = 0.2 s$ . Right panels: Different values of $θ$ and $n = 50$ . The following estimators are shown: the $p$ estimator ${\hat{θ}}_{2; c}$ under the parametric assumption (solid line); the CDF estimators ${\hat{θ}}_{3}$ under the renewal assumption (dot-dashed line), the stationarity assumption (dotted line), or the parametric assumption (long-dashed line); the MLE ${\hat{θ}}_{4}$ (circles); the ME ${\hat{θ}}_{5}$ (crosses) (only in the top panels); and the misspecified estimator ${\hat{θ}}_{6}$ (gray circles) (only in the bottom panels). The estimators ${\hat{θ}}_{2}$ under the renewal and stationarity assumptions ${\hat{θ}}_{2; a}$ and ${\hat{θ}}_{2; b}$ are not reported since they are almost identical to ${\hat{θ}}_{2; c}$ .Reuse & Permissions
Figure 4
Dependence of $R_{MSE}$ $(\hat{θ})$ (average over 10000 simulations) on the number of observations and the absolute response latency when $W$ is exponential with rate $λ = 1 s^{- 1}$ for estimators with errors less than 10 $%$ . Top panels: $Z$ is exponential with rate $ω = 10 s^{- 1}$ . Bottom panels: $Z$ is gamma with $α = 0.05 s$ and $β = 2$ . Left panels: Different values of $n$ and $θ = 0.2 s$ . Right panels: Different values of $θ$ and $n = 50$ . The following estimators are shown: the $p$ estimator ${\hat{θ}}_{2; c}$ under the parametric assumption (solid line); the CDF estimators ${\hat{θ}}_{3}$ under the renewal assumption (dot-dashed line), the stationarity assumption (dotted line), or the parametric assumption (long-dashed line); the MLE ${\hat{θ}}_{4}$ (circles); the ME ${\hat{θ}}_{5}$ (crosses) (only in the top panels); and the misspecified estimator ${\hat{θ}}_{6}$ (gray circles) (only in the bottom panels). The estimators ${\hat{θ}}_{2}$ under the renewal and stationarity assumptions ${\hat{θ}}_{2; a}$ and ${\hat{θ}}_{2; b}$ are not reported since they are almost identical to ${\hat{θ}}_{2; c}$ .Reuse & Permissions
Figure 5
Percentage of repetitions out of 10000 that do not fulfill condition (25). For these data sets, the ME ${\hat{θ}}_{5}$ cannot be evaluated. Here $W$ is exponential with rate $λ = 1 s^{- 1}$ ; $Z$ is exponential with rate $ω = 10 s^{- 1}$ (solid line) or $Z$ is gamma with $α = 0.05 s$ and $β = 2$ (dashed line). Left panels: Different values of $n$ for fixed $θ = 0.2 s$ . Right panels: Different values of $θ$ for fixed $n = 50$ .Reuse & Permissions

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