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Biological Cybernetics

, Volume 108, Issue 4, pp 475–493 | Cite as

Estimating latency from inhibitory input

  • Marie Levakova
  • Susanne Ditlevsen
  • Petr Lansky
Original Paper

Abstract

Stimulus response latency is the time period between the presentation of a stimulus and the occurrence of a change in the neural firing evoked by the stimulation. The response latency has been explored and estimation methods proposed mostly for excitatory stimuli, which means that the neuron reacts to the stimulus by an increase in the firing rate. We focus on the estimation of the response latency in the case of inhibitory stimuli. Models used in this paper represent two different descriptions of response latency. We consider either the latency to be constant across trials or to be a random variable. In the case of random latency, special attention is given to models with selective interaction. The aim is to propose methods for estimation of the latency or the parameters of its distribution. Parameters are estimated by four different methods: method of moments, maximum-likelihood method, a method comparing an empirical and a theoretical cumulative distribution function and a method based on the Laplace transform of a probability density function. All four methods are applied on simulated data and compared.

Keywords

Response latency Selective interaction Neuronal firing Inhibition Maximum likelihood  Laplace transform 

1 Introduction

In the nervous system, information is transmitted through the firing of action potentials (spikes) by neurons. The time course of the action potentials themselves varies very little and probably carries no information. We therefore consider the output of the neuron as a sequence of point events, which is called a spike train. Spike trains appear to be stochastic and are thus modeled as realizations of stochastic point processes. Experimentally measured quantities are time intervals between consecutive spikes, so-called interspike intervals (ISIs).

Traditionally, it has been believed that most of the relevant information is contained in the firing rate of the neuron. However, behavioral experiments show that reaction times on some stimuli are often rather short and it is not possible to evaluate the firing rate in such a short time window (Rullen et al. 1998, 2005). This suggests that the firing rate cannot be the only form of neural code and the exact timing of spikes plays a role. This is called temporal coding. The temporal coding can be affected by response latency (Bonnasse-Gahot and Nadal 2012; Gautrais and Thorpe 1997), which is in general defined as a time-lag between the stimulus onset and the evoked modulation in neural activity. For example, results of Chase and Young (2007) show that first-spike latency codes could be a feasible mechanism for information transfer.

When stimulated, the neuronal response is in most cases characterized by an increase in firing rate. However, sometimes a change of conditions is followed by an apparent decrease of the neuronal activity. For example, inhibitory response is a common phenomenon and has been observed, e.g., in the olfactory system of many animals. Krofczik et al. (2009) report that about 12 % of odors in their experiment evoked inhibitory response in lateral projection neurons of the honeybee. Moreover, the suppression of neural activity was so rapid that not a single response action potential was elicited when stimulating with a mixture of odors. Similar responses are not exceptional in olfactory receptor and cortical neurons of the frog Rana ridibunda (Rospars et al. 2000; Duchamp-Viret et al. 1996) and in primary olfactory centers of the moth Manduca sexta (Reisenman et al. 2008).

Existence of the so-called spontaneous activity makes it impossible to measure the latency exactly, rendering estimation difficult. Methods of estimation based on records of the entire spike train obtained in \(n\) independent trials have been presented (Baker and Gerstein 2001; Friedman and Priebe 1998; Commenges et al. 1986). The disadvantage of these methods is that they are designed primarily for excitatory stimuli. Although it is possible to apply them also when the stimulus is inhibitory, the estimates are often less precise. Different approaches assuming that the reaction to the stimulus is of short duration are proposed by Pawlas et al. (2010) and Tamborrino et al. (2012, 2013), where only observations up to the first spike after the stimulus onset are used for estimation. However, also these methods are build on models where an excitatory stimulus is explicitly assumed and the models cannot easily include inhibitory stimuli. It seems that the problem of latency estimation of inhibitory stimuli has been somewhat neglected. Here, we deal specifically with this situation. Therefore, whenever evoked activity is referred to, it is assumed that the firing activity is lower compared to the spontaneous activity.

In this paper, only the times of the first spikes following the stimulus onset from repeated trials instead of entire spike trains are used for estimation. This approach is especially suitable for inhibitory stimuli because a response can consist of a few long ISIs only. The usual approach of estimating the firing rate over an extended time window and detecting the change point might be inappropriate for responses of a few spikes and with a nonstationary firing rate.

Our approach is based on parametric models of spike trains. The key assumption is the Poissonian character of the spontaneous activity. It is generally agreed on that firing of real neurons is not Poissonian, in many cases, it is not even renewal as has been evidenced in a number of papers, e.g., in a review by Farkhooi et al. (2009). Nevertheless, the Poisson assumption is often used because it makes calculations less difficult, and it is often an acceptable approximation during spontaneous activity. Although a more precise description is appropriate, it would lead to major inconveniences when trying to handle the problem mathematically. Moreover, the Poisson assumption is in this case not so strong because it is required only locally since proposed models are constructed so that the assumption concerns only the ISI containing the stimulus onset and only prior to the beginning of the response. In addition, many elaborated mathematical models support this approximation as adequate. The statistical properties of neuronal firing of the classical Hodgkin–Huxley model with a stochastic input fluctuation was analyzed by Chow and White (1996). It was shown that the spontaneous activity arising from channel fluctuations is well described by the Poisson model. Also, the firing of the leaky integrate-and-fire model without input current (but with stochastic fluctuations of the membrane potential) is described by the Poisson spiking model (the so-called subthreshold regime) (Ditlevsen and Lansky 2005). In general, applications of the Poisson model to neuronal data are numerous.

The ISIs evoked by a stimulus presentation are often assumed gamma distributed. This distribution serves as a typical example in theoretical studies on neuronal firing (Dorval 2008; Kang and Amari 2008; Miura et al. 2006; Nawrot et al. 2008; Shimokawa and Shinomoto 2009), where the point process of the spike times is often called a gamma renewal process. Furthermore, the distribution was often checked in experimental studies, e.g., Hentall (2000), Mandl (1993) and McKeegan (2002).

The response latency is treated as a random variable. This approach can be justified by the fact that noise of all kinds (e.g. synaptic, membrane, or channel noise) has an influence on the actual latency. For example, the impact of channel noise on latency variability was studied theoretically by Wainrib et al. (2010). They investigated the Morris-Lecar model with a finite number of channels, which leads to random latency, whose asymptotic distribution is derived there as well. The approach assuming random latency was employed previously by Nawrot et al. (2003) who proposed a method for elimination of response latency variability. We discuss the special case of constant latency as well.

Two alternatives of the basic model are considered, and their stochastic properties are investigated. Characteristics of the distribution of the time between the stimulus onset and the first spike following it, like the probability density function (pdf), its Laplace transform and the cumulative distribution function (cdf), are derived and moments (mean and variance) are calculated. Then, four estimation methods of the mean latency are proposed. The first one is nonparametric, and only assumes that the latency is constant across trials. It is based on comparison between the theoretical and the empirical cdf. The remaining estimators are parametric, thus assuming specific models and distributions. The second estimator is obtained by the method of moments where knowledge of the mean and variance is crucial. This can be particularly useful when moments are available, but explicit expressions for the distribution are not. The two remaining methods are the maximum-likelihood method, which employs the pdf, and the method based on its Laplace transform. Normally, the maximum-likelihood method is the preferred method of choice, when available, but in our setting, the usual regularity conditions are not fulfilled, since the likelihood function is discontinuous in the parameter of interest; the mean latency. Thus, it is not obvious that it will behave better than other estimators, and the usual asymptotic tools to evaluate the quality of the estimators based on the Hessian, are not available. Moreover, in some of the considered models, the likelihood function is not available, and other methods are necessary.

All estimating routines were implemented in the free statistical software R (see R Core Team 2013) and can be found in the supplementary material of the paper.

2 Character of experimental data

Data are obtained from \(n\) trials under identical experimental conditions. In each trial, the stimulus is presented and the resulting spike train is recorded during a time period, which spans from a time instant preceding the stimulus onset to a time instant after the stimulus onset. Before the stimulation, the activity of the neuron is spontaneous and it fires irregularly, but with some stable firing rate. The stimulus is applied at a fixed time, denoted by \(t_s\), in each trial. During some unknown period of time \(\varTheta \), variable or fixed, after \(t_s\), the activity of the neuron remains spontaneous and is not influenced by the stimulation. This lag is called the response latency. After that the spiking activity changes. Henceforth, we assume that the evoked activity is lower than the spontaneous, since we focus on inhibitory stimuli. We model the observed spike train by two different random point processes. The first one describes the spontaneous activity, and the second characterizes evoked activity. The observed spike train is a realization of the first spontaneous process up to time \(t_\mathrm{s} + \varTheta \), and of the second evoked process after time \(t_\mathrm{s} + \varTheta \).

Estimation methods presented here require generally only knowledge of the time interval from \(t_\mathrm{s}\) to the first following spike. In addition, some methods also allow to use measurements of an entire ISI after \(t_\mathrm{s}\), i.e. the time between the first and the second spike after \(t_\mathrm{s}\), which improves the estimates. Nevertheless, spike times subsequent to the first spike are not necessarily required, which is an advantage if the response lasts for a short time only. The time between \(t_\mathrm{s}\) and the first following spike is a random variable denoted by \(T\), its realizations from \(n\) independent trials are \(\{t_1,t_2,\ldots ,t_n\}\). Observations \(t_i\) can be divided into two subgroups with respect to \(\varTheta \). An observation \(t_i\) can be shorter than \(\varTheta \), which means that the first spike after \(t_\mathrm{s}\) belongs to the spontaneous activity, or it is longer than \(\varTheta \) and the first observed spike after \(t_\mathrm{s}\) is influenced by the stimulus. The time between the first and the second spike after \(t_\mathrm{s}\) is a random variable \(X\) and its realizations are denoted by \(\{x_1,x_2,\ldots ,x_n\}\).

Two different concepts of latency are considered. Either the latency \(\varTheta \) is supposed to be a constant, thus it is fixed in all trials. In that case, it is denoted by lower case, \(\theta \). Or the latency is assumed to be a random variable, which is denoted by upper case, \(\varTheta \), and so the exact latency differs across trials. Then, the mean value of \(\varTheta \) is of interest. In order to estimate the latency, it is necessary to develop a probabilistic model of an after-stimulus spike train and to know its properties.

3 General models of an after-stimulus spike train

We distinguish between two types of observations \(t_i\) according to whether they are influenced by the stimulation or not. The time from \(t_\mathrm{s}\) to the occurrence of the first spontaneous spike (due to the spontaneous random point process) is denoted by \(W\) and its pdf by \(f_W(t)\). The time from \(t_\mathrm{s}\) to the first evoked spike (due to the evoked random point process) is denoted by \(R\) and its pdf by \(f_R(t)\). If \(W > \varTheta \), \(W\) is not observable since it is suppressed by the stimulation. In general, \(T\) satisfies
$$\begin{aligned} T={\left\{ \begin{array}{ll} W &{} W \le \varTheta \\ R &{} W > \varTheta . \end{array}\right. } \end{aligned}$$
(1)
This is illustrated in Fig. 1. It is supposed that the spontaneous activity forms a Poisson process with fixed intensity \(\lambda \). Then, \(f_W(t)\) is exponential with parameter \(\lambda \). For the first evoked spike, we assume two models.
Fig. 1

Illustration of the general model of an after-stimulus spike train. Spikes are indicated with dots. At time \(t_\mathrm{s}\) a stimulus is applied. The observed spike train is determined by the spontaneous activity until \(t_\mathrm{s}+\theta \), then it is determined by the evoked activity. Realizations \(w\) and \(r\) denote the time from \(t_\mathrm{s}\) to the first event of the spontaneous and the evoked activity, respectively. a If \(w \le \theta \), the time to the first observed spike after \(t_\mathrm{s}\) is equal to \(w\); \(t=w\). b If \(w > \theta \), the first spike after the stimulus is the first event of the evoked activity; \(t=r\). In both cases, the first complete ISI after \(t_\mathrm{s}\) is denoted by \(x\)

Model A The random variable \(R\) is given by the sum \(R = \varTheta + U\), where \(U\) is a nonnegative random variable with pdf \(f_U(t)\). Moreover, \(W\) and \(U\) are independent, hence
$$\begin{aligned} T_\mathrm{A} = {\left\{ \begin{array}{ll} W &{} W \le \varTheta \\ \varTheta + U &{} W > \varTheta . \end{array}\right. } \end{aligned}$$
(2)
This model assumes that at time \(t_\mathrm{s}+\varTheta \) the stimulus causes a change in the distribution of the observed point process.
Model B The random variable \(R\) is given by the sum \(R = W + U\), where \(U\) is a nonnegative random variable with pdf \(f_U(t)\), again independent of \(W\). Then \(T\) is given by
$$\begin{aligned} T_\mathrm{B} = {\left\{ \begin{array}{ll} W &{} W \le \varTheta \\ W + U &{} W > \varTheta . \end{array}\right. } \end{aligned}$$
(3)
This model assumes that the stimulus delays the first spike after time \(\varTheta \). A schematic description of \(T\) in Models A and B is shown in Fig. 2.
Fig. 2

Schematic description of an after-stimulus spike train for Model A (\(\mathbf{a}\ T_{A}\ \hbox {if}\ W \le \Theta , \mathbf{b}\ T_{A}\ \hbox {if}\ W > \Theta \)) and for Model B (\(\mathbf{c}\ T_{B}\ \hbox {if}\ W \le \Theta , \mathbf{d}\ T_{B}\ \hbox {if}\ W > \theta \)). In Model A, \(r\) is given as the sum of \(\theta \) and \(u\). In Model B, \(r\) is the sum of \(w\) and \(u\)

To study \(\varTheta \), we first derive the conditional distribution of \(T\) and its moments for a given value \(\theta \) of \(\varTheta \).

3.1 Conditional distribution of \(T\)

Model A The conditional pdf of \(T_\mathrm{A}\), given that it is smaller than \(t_\mathrm{s}+\theta \), is \(f_{T_\mathrm{A}|\,\theta }\,(t\,|\,W\le \theta ) = \lambda \mathrm{e}^{-\lambda t}/ (1-\mathrm{e}^{-\lambda \theta })\) for \(t \in [0,\theta ]\). If \(W > \theta \), i.e. the first spike occurs after \(t_\mathrm{s}+\theta \), the conditional pdf of \(T_\mathrm{A}\) is \(f_{T_\mathrm{A}|\,\theta }\,(t\,|\,W>\theta ) = f_R(t) = f_U(t-\theta )\) for \(t \in (\theta ,\infty )\). Using the formula for the total probability, we obtain the conditional pdf of \(T_\mathrm{A}\)
$$\begin{aligned} f_{T_\mathrm{A}|\,\theta }(t) = {\left\{ \begin{array}{ll} \lambda \mathrm{e}^{-\lambda t} &{} t \in [0,\theta ] \\ \mathrm{e}^{-\lambda \theta } f_U(t-\theta ) &{} t \in (\theta ,\infty ). \end{array}\right. } \end{aligned}$$
(4)
The conditional cdf, Laplace transform, mean and variance of \(T_\mathrm{A}\) are given in “Appendix 1”, Eqs. (27)–(30).
Model B The conditional pdf \(f_{T_\mathrm{B}|\,\theta }(t\,|\,W\le \theta )\) is equal to \(f_{T_\mathrm{A}|\,\theta }(t\,|\,W\le \theta )\). For \(T_\mathrm{B} > \theta \), \(T_\mathrm{B}\) is the sum of \(W>\theta \) and \(U\). Therefore, \(f_{T_\mathrm{B}|\,\theta }(t \,|\, W > \theta ) = \mathrm{e}^{\lambda \theta } \int _\theta ^t \lambda \mathrm{e}^{-\lambda y}f_U(t-y)\,\mathrm{d}y\) for \(t \in (\theta ,\infty )\). Altogether, the pdf of \(T_\mathrm{B}\) is
$$\begin{aligned} f_{T_\mathrm{B}|\,\theta }(t) = {\left\{ \begin{array}{ll} \lambda \mathrm{e}^{-\lambda t} &{} t \in [0,\theta ] \\ \int _\theta ^t \lambda \mathrm{e}^{-\lambda y}f_U(t-y)\,\mathrm{d}y &{} t \in (\theta ,\infty ) \end{array}\right. } \end{aligned}$$
(5)
The conditional cdf, Laplace transform, mean and variance of \(T_\mathrm{B}\) are given in “Appendix 1”, Eqs. (34)–(37).
Note that for both models, \(\theta \) satisfies
$$\begin{aligned} \theta = \inf \{t>0: F_{T|\,\theta }(t)\ne F_W(t)\}. \end{aligned}$$
(6)
This property is later exploited for the estimation of latency.

3.2 Unconditional distribution of \(T\)

Let \(\varTheta \) be a random variable with pdf \(f_{\varTheta }(\theta )\). Henceforward, the latency is assumed to be exponentially distributed with mean \(\theta ^*\), i.e., \(f_{\varTheta }(\theta ) = \exp (-\theta /\theta ^*)/\theta ^*\). This assumption is very strict and will be discussed later. The unconditional pdf of \(T\) is \(f_T(t)=\int _0^\infty f_{T|\,\theta }(t)f_{\varTheta }(\theta )\,\mathrm{d}\theta \). The pdf for Model A is
$$\begin{aligned} f_{T_\mathrm{A}}(t) = \mathrm{e}^{-\left( \lambda +\frac{1}{\theta ^*}\right) t}\left[ \lambda + \frac{1}{\theta ^*}\int \limits _0^t \mathrm{e}^{\left( \lambda +\frac{1}{\theta ^*}\right) u}f_U(u)\,\mathrm{d}u \right] \end{aligned}$$
(7)
and for Model B it is
$$\begin{aligned} f_{T_\mathrm{B}}(t)&= \lambda \mathrm{e}^{-(\lambda +\frac{1}{\theta ^*})t} + \nonumber \\&+ \frac{1}{\theta ^*}\int \limits _0^t\mathrm{e}^{-\frac{\theta }{\theta ^*}} \left( \int \limits _\theta ^t \lambda \mathrm{e}^{-\lambda y} f_U(t-y)\,\mathrm{d}y \right) \mathrm{d}\theta . \end{aligned}$$
(8)
In “Appendix 1”, Eqs. (31)–(33) and (38)–(40), the cdfs, Laplace transforms, means and variances are given.

4 Examples of models

In this Section, particular examples are considered by assuming specific distributions for the evoked activity \(U\). First, we assume that the latency is constant across trials and equal to \(\theta ^*\). Thus, \(\mathrm{Pr}(\varTheta =\theta ^*)=1\). The distribution of \(T\) is found using the formulas derived in Sect. 3.1 with \(\theta =\theta ^*\). Constant \(\theta ^*\) plays the role of a parameter of the distribution of \(T\).

Model 1: Model A with constant latency and exponentially distributed \(U\). Let \(U\) follow an exponential distribution with rate parameter \(\kappa \) in Model A. We assume \(\kappa <\lambda \) to describe the inhibitory response. The complete spike train is a realization of a Poisson process with non-constant firing rate \(\lambda (t)\) given by
$$\begin{aligned} \lambda (t)={\left\{ \begin{array}{ll} \lambda &{} t \le t_\mathrm{s} + \theta ^*\\ \kappa &{} t > t_\mathrm{s} +\theta ^*. \end{array}\right. } \end{aligned}$$
(9)
The pdf of \(T\) can be derived from (4) with \(f_U(t)=\kappa \mathrm{e}^{-\kappa t}\),
$$\begin{aligned} f_{T_{M1}}(t) = {\left\{ \begin{array}{ll} \lambda \mathrm{e}^{-\lambda t} &{} t \in [0,\theta ^*] \\ \kappa \mathrm{e}^{-\lambda \theta ^*-\kappa (t-\theta ^*)} &{} t \in (\theta ^*,\infty ) \end{array}\right. }\!. \end{aligned}$$
(10)
The model is illustrated in Fig. 3 and the pdf of \(T_{M1}\) in Fig. 5a. The Laplace transform, mean and variance of (10) can be found in “Appendix 2”, Eqs. (41)–(43).
Fig. 3

Schematic description of a spike train in Model 1. The spike train before \(t_\mathrm{s} + \theta ^*\) (vertical dashed line) is a realization of a Poisson process with parameter \(\lambda \), thus ISIs are exponentially distributed with rate \(\lambda \). The spike train after \(t_\mathrm{s} + \theta ^*\) is a realization of a Poisson process with parameter \(\kappa \), hence ISIs are exponential with rate \(\kappa \)

Model 2: Model B with constant latency and gamma distributed \(U\). Let \(U\) in Model B follow a gamma distribution with shape parameter \(k>0\) and rate parameter \(\lambda \), which is the same value as the parameter of the distribution of \(W\). For integer \(k\), this model postulates that the spontaneous activity is determined by a Poisson process with parameter \(\lambda \), and that during the response only every \((k+1)\)th event of the process is observable, whereas the remaining events are eliminated. The model is illustrated in Fig. 4.
Fig. 4

Schematic description of a spike train in Model 2. The whole spike train is generated by a Poisson process with parameter \(\lambda \). Before \(t_\mathrm{s} + \theta ^*\) every event of the process generates a spike. After \(t_\mathrm{s}+\theta ^*\) only every \((k+1)\)th event of the process produces a spike and the previous \(k\) events are supressed and therefore unobservable. Thus, in the first part of the spike train, ISIs are exponentially distributed with parameter \(\lambda \), in the second part ISIs follow a gamma distribution with shape parameter \(k+1\) and rate parameter \(\lambda \)

The pdf of \(T\), obtained by inserting the gamma density \(f_U(t)=\lambda ^{k}t^{k-1}\mathrm{e}^{-\lambda t}/\Gamma (k)\) into (5), is
$$\begin{aligned} f_{T_{M2}}(t) = {\left\{ \begin{array}{ll} \lambda \text {e}^{-\lambda t} &{} t \in [0,\theta ^{*}] \\ \frac{\lambda ^{k+1}}{\Gamma (k+1)}(t-\theta ^{*})^{k}\text {e}^{-\lambda t} &{} t \in (\theta ^{*},\infty ). \end{array}\right. } \end{aligned}$$
(11)
It is illustrated in Fig. 5b. The Laplace transform, mean and variance of (11) can be found in “Appendix 2”, Eqs. (44)–(46).
Fig. 5

Probability density functions of \(T\) in Model 1 and 2. a The pdf of \(T_{M1}\) in Model 1 with \(\theta ^*=1\), \(\lambda =1\) and \(\kappa =0.1\) (solid), \(\kappa =0.3\) (dotted), \(\kappa =0.5\) (dashed line). b The pdf of \(T_{M2}\) in Model 2 with \(\theta ^*=1\), \(\lambda =1\) and \(k=1\) (solid), \(k=2\) (dotted), \(k=3\) (dashed line)

We now relax the assumption that the latency is constant across trials and let it be a realization of an exponentially distributed random variable \(\varTheta \) with mean \(\theta ^*\), different in each trial. The formulas from Sect. 3.2 can be applied.

Model 3: Model A with exponentially distributed latency and exponentially distributed \(U\). We generalize Model 1 to non-constant latency. Using (7), the pdf of \(T\) is
$$\begin{aligned} f_{T_{M3}}(t) = \lambda \mathrm{e}^{-(\lambda +1/\theta ^*)t} + \frac{\kappa \left( \mathrm{e}^{-\kappa t} - \mathrm{e}^{-(\lambda +1/\theta ^*)t}\right) }{(\lambda -\kappa )\theta ^*+1} \end{aligned}$$
(12)
The Laplace transform, mean and variance of (12) can be found in “Appendix 2”, Eqs. (47)–(49).

Model 4: Model B with exponentially distributed latency and gamma distributed \(U\). We generalize Model 2 to non-constant latency. The pdf is not available. The Laplace transform, mean and variance can be found in “Appendix 2”, Eqs. (50)–(52).

4.1 Model with selective interaction during the response

Models of selective interaction are presented and discussed in detail, e.g., by Fienberg (1974). The basic idea is that there exist two series of pulses, which form the input to the neuron. The pulses of the first sequence, representing excitatory pulses, are able to evoke a spike. The pulses of the second sequence, representing inhibitory pulses, cause that some excitatory pulses are canceled. The model in its simplest form can be described by the following assumptions:
  1. 1.
    The input to the neuron consists of pulses generated by two independent renewal processes, one excitatory and the other inhibitory.
    1. (a)

      An interval between two subsequent excitatory pulses is a random variable \(X_E\) with pdf \(f_E(t)\).

       
    2. (b)

      An interval between inhibitory pulses is a random variable \(X_I\) with pdf \(f_I(t)\). For simplicity, in this paper, it is assumed that inhibitory pulses form a Poisson process with mean \(\theta ^*\), i.e. \(f_I(t) = \exp (- t/\theta ^*)/\theta ^*\).

       
     
  2. 2.

    Whenever one or more inhibitory pulses occur, the effect of the next excitatory pulse is eliminated.

     
  3. 3.

    A spike is observed whenever an excitatory pulse occurs, unless it is deleted by a preceding inhibitory pulse.

     
We suppose that in absence of stimulation the neuronal input consists of excitatory pulses only. The stimulation at time \(t_\mathrm{s}\) initiates a series of inhibitory pulses. In this framework, the time from \(t_\mathrm{s}\) to the first inhibitory pulse can be considered to be the latency, hence this random variable is denoted by \(\varTheta \). An example of a spike train generated by such a process is in Fig. 6.
Fig. 6

Spike train generated by a process with selective interaction. Spontaneous activity is determined entirely by the excitatory pulses coming before \(t_\mathrm{s}\) (on the first line). After \(t_\mathrm{s}\) the inhibitory pulses start to come. They are depicted on the second line. Whenever an inhibitory pulse occurs, it causes deletion of the next excitatory pulse. The resulting spike train can be seen on the third line

First, we study the properties of the ISIs during the response period. Let an ISI in the response period be denoted by \(X\) with pdf \(f_X(t)\). In Fienberg (1974), the Laplace transform of \(f_X(t)\) is derived under the assumption that the inhibitory process is a Poisson process,
$$\begin{aligned} \widehat{f_X}(s) = \frac{\widehat{f_E}(s+1/\theta ^*)}{1+\widehat{f_E}(s+1/\theta ^*)-\widehat{f_E}(s)}. \end{aligned}$$
(13)
The mean of \(X\), which can be calculated from (13), is
$$\begin{aligned} \mathbb {E}(X) = \frac{\mathbb {E}(X_E)}{\widehat{f_E}(1/\theta ^*)}. \end{aligned}$$
(14)
Nevertheless, the variable of interest is \(T\). The procedure used to derive the Laplace transform of \(f_T(t)\) is analogous to the derivation of \(\widehat{f_X}(s)\). Assume that the realization of \(T\) lies in the interval \((t,t+\Delta t)\). Two different scenarios are possible. Either the first excitatory pulse arrives at time \(t\) after \(t_\mathrm{s}\) with no intervening inhibitory pulses before, or the first excitatory pulse arrives at time \(\tau \) after \(t_\mathrm{s}\) (\(\tau <t\)), but it is preceded by an arrival of one or more inhibitory pulses. Thus we have
$$\begin{aligned} f_T(t) = \! f^+_E(t)\mathrm{e}^{-t/\theta ^*} + \int \limits _0^t f^+_E(\tau )(1-\mathrm{e}^{-\tau /\theta ^*})f_X(t - \tau )\,\mathrm{d}\tau , \end{aligned}$$
(15)
where \(f^+_E(t)\) is the pdf of the forward recurrence time of the excitatory process. It holds for renewal processes that \(f^+_E(t) =\left( 1\!-\!\int _0^t f_E(\tau )\,\mathrm{d}\tau \right) /\int _0^\infty tf_E(t)\,\mathrm{d}t\) and the Laplace transform is therefore \(\widehat{f^+_E}(s) =(1 - \widehat{f_E}(s))/(s \mathbb {E}(X_E))\). It should be stressed that formula (15) is valid only if inhibitory pulses are generated by a Poisson process with rate parameter \(1/\theta ^*\). Taking the Laplace transform of (15) yields
$$\begin{aligned} \widehat{f_T}(s) = \widehat{f^+_E}(s + 1/\theta ^*\!) + \widehat{f_X}(s) \left[ \widehat{f^+_E}(s) - \widehat{f^+_E}(s + 1/\theta ^*)\right] . \end{aligned}$$
(16)
After inserting (13) into (16) we get
$$\begin{aligned} \widehat{f_T}(s)&= \widehat{f^+_E}(s+1/\theta ^*) \nonumber \\&\quad + \frac{\widehat{f_E}(s+1/\theta ^*)\left[ \widehat{f^+_E}(s) - \widehat{f^+_E}(s+1/\theta ^*)\right] }{1-\widehat{f_E}(s) + \widehat{f_E}(s+1/\theta ^*)}. \end{aligned}$$
(17)
The mean of \(T\) can be calculated from (17),
$$\begin{aligned} \mathbb {E}(T) = \frac{\mathbb {E}\left( X_E^2 \right) }{2 \mathbb {E}(X_E)} + \frac{\mathbb {E}(X_E) +\theta ^*\left[ \widehat{f_E}(1/\theta ^*)-1\right] }{\widehat{f_E}(1/\theta ^*)}. \end{aligned}$$
(18)
An example of a model with selective interaction is presented below.

Model 5: Model with selective interaction and excitatory pulses as a Gamma process. Suppose that excitatory pulses form a renewal process, where ISIs follow a gamma distribution with shape parameter \(k\) and rate parameter \(\lambda \). The pdf of \(T\) can only be found in the special case where \(k=1\). The Laplace transforms and means of \(X\) and \(T\), as well as the pdf of \(T\) for \(k=1\) can be found in “Appendix 2, Eqs. (53)–(57).

The distributions of \(X_{M5}\) and \(T_{M5}\) are illustrated by histograms of simulated observations in Figs. 7 and 8.
Fig. 7

Histograms of simulated observations of \(X_{M5}\) for two choices of \(k, \lambda \) and \(\theta ^*\). a \(k=5, \lambda =5, \theta ^*=1\), b \(k=20, \lambda =20, \theta ^*=1\). For each histogram \(100,000\) simulated observations of \(X_{M5}\) were generated

Fig. 8

Histograms of simulated observations of \(T_{M5}\) for two choices of \(k, \lambda \) and \(\theta ^*\). a \(k=5, \lambda =5, \theta ^*=1\), b \(k=20, \lambda =20, \theta ^*=1\). For each histogram \(100,000\) simulated observations of \(T_{M5}\) were generated

A brief overview of Models 1–5 is given in Table 1.
Table 1

Overview of proposed models

 

Class of models

Latency

Spontaneous activity

Evoked activity

Model 1

Model A

Constant

\(W \sim Exp(\lambda )\)

\(R = \varTheta + U\)

\(\varTheta =\theta ^*\)

\(U \sim Exp(\kappa )\)

Model 2

Model B

Constant

\(W \sim Exp(\lambda )\)

\(R = W + U\)

\(\varTheta =\theta ^*\)

\(U \sim Gamma(k,\lambda )\)

Model 3

Model A

Random

\(W \sim Exp(\lambda )\)

\(R = \varTheta + U\)

\(\varTheta \sim Exp(1/\theta ^*)\)

\(U \sim Exp(\kappa )\)

Model 4

Model B

Random

\(W \sim Exp(\lambda )\)

\(R = W + U\)

\(\varTheta \sim Exp(1/\theta ^*)\)

\(U \sim Gamma(k,\lambda )\)

Model 5

Model with selective interaction

Random

Gamma process

Inhibition by Poisson process

\(\varTheta \sim Exp(1/\theta ^*)\)

\(X_E \sim Gamma(k,\lambda )\)

\(X_I \sim Exp(1/\theta ^*)\)

5 Estimation of latency

The aim of this paper was the estimation of \(\theta ^*\), which represents the exact latency in models with constant latency and the mean latency in models with random latency. Four estimation methods are proposed: a nonparametric method based on the cdf of \(T\), the method of moments, maximum-likelihood estimation and a method based on the Laplace transform of the pdf of \(T\). We focus on estimation of \(\theta ^*\), although there are other parameters. We assume that \(\lambda \) is known. This may not be completely true, but it is possible to estimate it from the record of the spontaneous activity before the stimulus onset. We do not deal with this issue here since it is out of scope of this paper. Estimation of the spontaneous firing rate under Poisson, renewal and stationarity assumptions is discussed e.g. in Tamborrino et al. (2012). Other parameters of the evoked activity, namely \(\kappa \) in Model 1 and 3 and \(k\) in Model 2 and 4, are unknown, and it can be necessary to estimate them too. All estimating routines were implemented in the free statistical software R (see R Core Team 2013) and can be found in the supplementary material of the paper.

5.1 Estimators of \(\theta ^*\) based on cumulative distribution functions

This nonparametric estimator has the advantage of being nearly assumption-free. The only assumptions are the Poissonian character of the spontaneous activity (or any other distribution, as long as it is known) and that the latency is constant across trials. The price one pays is loss of efficiency in the sense of larger variance on the estimators since no information on the specific model is used. This approach was originally proposed by Tamborrino et al. (2012) for the case of excitatory stimulus. The idea is to compare the empirical cumulative distribution function (ecdf), denoted by \(\widehat{F}_{T}(t)\), obtained from observations \(t_i\) of \(T\) in \(n\) independent trials, to the theoretical cdf of \(W\), denoted by \(F_W(t)\). We have \(F_W(t) = 1-\mathrm{e}^{-\lambda t}\). Note that it is straightforward to use any other distribution function for the spontaneous activity.

For constant latency, \(\theta ^* = \inf \{t>0: F_{T}(t)\ne F_W(t)\}\), see (6). To identify this point, the difference \(F_W(t)-\widehat{F}_{T}(t)\) of the two cdfs can be employed. This function has the following approximate properties:
  1. 1.

    On the interval \([0, \theta ]\), \(F_W(t) - \widehat{F}_{T}(t)\) oscillates near zero.

     
  2. 2.

    For \(t \in (\theta , t_{(n)})\), where \(t_{(n)}\) is the maximal observation of \(T, F_W(t)\) is greater than \(F_{T}(t)\) (because of the lower firing rate during the response period) and thus \(F_W(t) - \widehat{F}_{T}(t)\) tends to be positive.

     
  3. 3.

    For \(t \ge t_{(n)}\) the ecdf \(\widehat{F}_{T}(t)\) is equal to \(1\), while \(F_W(t)\) approaches 1 in the limit. Thus, their distance \(F_W(t) - \widehat{F}_{T}(t)\) is negative and converges to 0.

     
The method aims at detecting the point, which divides the oscillating and the positive part of \(F_W(t) - \widehat{F}_{T}(t)\). A simple way is to find the maximal \(t\) at which \(F_W(t) - \widehat{F}_{T}(t)\) is equal to zero and from which the difference \(F_W(t) - \widehat{F}_{T}(t)\) is strictly positive up to time \(t_{(n)}\). The resulting estimator is
$$\begin{aligned} \hat{\theta }^*_{\mathrm{{ECDF}},1} = \max \{t \in [0,t_{(n)}):F_W(t)-\widehat{F}_{T}(t)\le 0\}. \end{aligned}$$
(19)
A more sophisticated way to distinguish random fluctuations of \(F_W(t)-\widehat{F}_{T}(t)\) around zero from a systematic departure was proposed by Tamborrino et al. (2012). The difference \(F_W(t)-\widehat{F}_{T}(t)\) is compared with its standard deviation under the assumption that \(t\le \theta ^*\), and therefore, it does not depend on \(R\). The variance \(\sigma ^2(t)\) is
$$\begin{aligned} \sigma ^2(t)&= \mathrm{Var}\left( F_W(t)-\widehat{F}_{T}(t)\right) = \mathrm{Var}\left( \widehat{F}_{T}(t)\right) \\&= \mathrm{Var}\left( \frac{1}{n}\sum _{i=1}^n \mathbb {1}_{\{t_i\le t\}}\right) = \frac{1}{n}F_W(t)(1-F_W(t)) \end{aligned}$$
and the estimator is
$$\begin{aligned} \hat{\theta }^*_{\mathrm{{ECDF}},2} = \max \{t \in [0,\tilde{t}]:F_W(t)-\widehat{F}_{T}(t)\le \sigma (t)\}, \end{aligned}$$
(20)
where \(\tilde{t}=\arg \max _{t \in [0,t_{(n)}]} (F_W(t)-\widehat{F}_{T}(t))\). The estimation is illustrated in Fig. 9. An improvement might be obtained if \(\sigma (t)\) is multiplied by a suitable constant or some function of time to reduce the bias in \(\hat{\theta }^*_{\mathrm{{ECDF}},2}\) (see results of the simulation study in Sect. 6). This is not further explored here.
Fig. 9

Determination of \(\hat{\theta }^*_{\mathrm{{ECDF}}}\). The difference between \(F_W(t)\) and \(\widehat{F}_T(t)\) is calculated. The true latency is marked by the dashed line and the estimate by the solid line. a \(\hat{\theta }^*\) is the maximum \(t\) at which \(F_W(t)-\hat{F}_T(t)=0\) and such that for \(t\in (\hat{\theta }, t_{(n)})\) the difference \(F_W(t)-\hat{F}_T(t)\) is strictly positive (area under the curve on this interval is in gray). b The point \(\tilde{t}\), which maximizes \(F_W(t)-\hat{F}_T(t)\), is found. The maximum \(t\) smaller than \(\tilde{t}\), at which \(F_W(t)-\hat{F}_T(t)=\sigma (t)\) is the estimate \(\hat{\theta }^*_{\mathrm{{ECDF}},2}\)

Note that it is straightforward to adapt the estimator to a response, where a priori it is unknown if it is excitatory or inhibitory, as well as use it as a test of whether the stimulus has any effect on the neuronal activity at all.

5.2 Estimators obtained by the method of moments

This is a parametric estimator, which is useful when the likelihood is not available. A moment estimator of \(\theta ^*\) is obtained as the solution to the moment equations. The system of equations must have as many moment equations as the number of unknown parameters. All models considered in this paper have at most two unknown parameters, therefore we solve the following system
$$\begin{aligned} \mathbb {E}(T) = \bar{t}; \quad \mathrm{Var}(T) = s^2, \end{aligned}$$
(21)
where \(\bar{t}\) is the average of observations \(\{t_1,t_2,\ldots ,t_n\}\) and \(s^2\) is their sample variance. In “Appendix 3”, we deal with the question of finding a solution to these equations for the example models introduced in Sect. 4.

5.3 Maximum-likelihood estimators

When the pdf of \(T\) is available, we can perform maximum-likelihood estimation. Under standard regularity conditions, it is the most efficient estimator, meaning that the asymptotic variance of the estimator is smallest among all non-biased estimators. The regularity conditions are not met here, though, since the likelihood function is not differentiable. Nevertheless, we shall later see in the simulation study that it still outperforms the other estimators. Because the likelihood functions for models with constant latency have some typical features, such that maximization of the likelihood function is done differently from the models with random latency and selective interaction, it is described in more detail.

5.3.1 Models with constant latency

First, we discuss maximization of the log-likelihood function generally for Model A and B without assuming any particular distribution of \(U\). The pdfs of \(T_\mathrm{A}\) and \(T_\mathrm{B}\) are given in (4) and (5) and the log-likelihood functions \(l(\theta ^*)\) are given in “Appendix 4”, Eqs. (63) and (64). To maximize \(l(\theta ^*)\), we put the derivative with respect to \(\theta ^*\) equal to zero. However, \(l(\theta ^*)\) is discontinuous at \(\theta ^*=t_i\) for all \(i=1, \ldots , n\). It is therefore necessary to find all local maxima on the intervals \((0,t_{(1)}), (t_{(1)},t_{(2)}), \ldots , (t_{(n)},\infty )\), where \(t_{(1)}<t_{(2)}<\cdots <t_{(n)}\) are ordered observations of \(T\). Moreover, the global maximum could be achieved on the boundary of any of these intervals, therefore the one-sided limits for \(\theta \rightarrow t_i^-\) and \(\theta \rightarrow t_i^+\) must be evaluated.

For Model A, the following condition is fulfilled at stationary points
$$\begin{aligned} \sum _{t_{(j)} \ge t_{(i)}} \frac{f_U'(t_{(j)}-\theta ^*)}{f_U(t_{(j)}-\theta ^*)} = -\lambda (n-i+1), \quad i=1,\ldots ,n. \end{aligned}$$
(22)
For Model B, it can be shown that the likelihood function is decreasing on intervals \([0,t_{(1)}), [t_{(i-1)},t_{(i)}), i=2,\ldots ,n\) and is constant on the interval \([t_{(n)},\infty )\). This implies that it is sufficient to evaluate the likelihood function at points \(\theta ^*=t_i\) only. This follows from the shape of the contribution of the \(i\)th observation \(t_i\) to the total log-likelihood,
$$\begin{aligned} l^B_i(\theta ^*;t_i) = {\left\{ \begin{array}{ll} \ln \int _{\theta ^*}^{t_i} \lambda \mathrm{e}^{-\lambda y} f_U(t_i-y)\,\mathrm{d}y &{} \theta ^* < t_i \\ \ln \lambda - \lambda t_i &{} \theta ^* \ge t_i. \end{array}\right. } \end{aligned}$$
(23)
For \(\theta ^* \ge t_i, l^B_i(\theta ^*;t_i)\) is a constant independent of \(\theta ^*\). For \(\theta ^* < t_i, l^B_i(\theta ^*;t_i) \) is an integral of a positive function over an interval depending on \(\theta ^*\). If \(\theta ^*\) increases, the domain of integration is smaller and the integral decreases. Thus, the total log-likelihood is a sum of functions \(l^B_i, i=1,\ldots ,n\), which are decreasing and constant piecewise and where at least for one \(i\) the function \(l^B_i\) is decreasing on any interval \([0,t_{(1)})\), \([t_{(i-1)},t_{(i)}), i=2,\ldots ,n\). Thus, the total log-likelihood is decreasing on given intervals, and constant on \([t_{(n)},\infty )\), because all \(l^B_i(\theta ^*;t_i)\) are constant there.
The log-likelihood functions \(l^{M1}(\theta ^* \! ,\kappa )\) and \(l^{M2}(\theta ^* \! ,k)\) for Model 1 and 2 are given in “Appendix 4”, Eqs. (66) and (67). They are illustrated in Fig. 10.
Fig. 10

Cross-section of the log-likelihood functions a in Model 1 for a fixed value of \(\kappa ~(l^{M1}(\theta ^{*};\cdot ))\), c in Model 2 for a fixed value of \(k~(l^{M2}(\theta ^{*};\cdot ))\). The log-likelihood functions are decreasing piecewise with respect to \(\theta ^*\) for \(\theta ^* < t_{(n)}\). At \(\theta ^*=t_i\), \(i=1, \ldots , n\), discontinuities occur and for \(\theta ^* > t_{(n)}\) the log-likelihood functions are constant. Dotted lines correspond to points \(\theta ^*=t_i\), \(i=1,\ldots ,10\). The true latency is marked by the dashed line. Log-likelihood functions and the procedure of finding their maximum b in Model 1 (\(l^{M1}(\theta ^{*};\kappa )\)), d in Model 2 (\(l^{M2}(\theta ^{*};k)\)). First, log-likelihood functions are explored at every \(\theta ^*=t_i\), then local maxima with respect to \(\kappa \) or \(k\) are found (white points). Finally, the greatest local maximum (the black point) is determined as the global maximum and corresponding values of \(\theta ^*\) (black solid lines) are maximum-likelihood estimates. The true values of \(\theta ^*\), \(\kappa \) and \(k\) are illustrated by dashed lines. The log-likelihood functions were constructed given 10 observations of \(T\)

If \(\lambda > \kappa \), which corresponds to an inhibitory response, \(l^{M1}\) is decreasing on intervals \([0,t_{(1)}), [t_{(i-1)},t_{(i)}), i=2,\ldots ,n\) and has no stationary points within these intervals. Therefore, we only need to examine \(l^{M1}\) at points \(\theta ^*=t_i\). The same is true for Model 2 as it is a special case of Model B. However, in both models, there is another unknown parameter, \(\kappa \) or \(k\), respectively, so we look for a maximum of a two-dimensional function. Therefore, for every \(\theta ^*=t_i\), we must find the maximum with respect to \(\kappa \) (in Model 1) or \(k\) (in Model 2) first. Then, we determine which of the local maxima is the global one.

The \(\kappa \), which maximizes \(l^{M1}\) for fixed \(\theta ^*\), is
$$\begin{aligned} \kappa = \frac{n-n_{\theta ^*}}{\sum _{t_i> \theta ^*} (t_i - \theta ^*)}, \end{aligned}$$
(24)
where \(n_{\theta ^*}\) is the number of observations \(t_i \le \theta ^*\). For \(\theta ^* \ge t_{(n)}\), \(\kappa \) in (24) is not defined, because \(l^{M1}\) is constant here. For Model 2, local maxima with respect to \(k\) must be found numerically.

5.3.2 Models with random latency

The log-likelihood function for Model 3 is given in “Appendix 4”, Eq. (67). The estimator is obtained by numerical maximization of (67) with respect to \(\theta ^*\) and \(\kappa \).

For Model 4, the pdf of \(T_{M4}\) and thus the likelihood function are not available in a closed form.

5.3.3 Model with selective interaction

In Model 5, it is difficult to find the inverse Laplace transform of \(\widehat{f_T}(t)\), and the maximum-likelihood estimator cannot be calculated in general. However, if \(k=1\), the pdf of \(T\) is available [see (57)] and the log-likelihood function is given in “Appendix 4”, Eq. (68). The estimator of \(\theta ^*\) is obtained by direct numerical maximization of (68).

We can get more precise estimates by doubling the number of observations, using observations \(t_i\) and \(x_i\) together. Because observations of \(T\) and \(X\) come from the same distribution, they can all be inserted into (68).

5.4 Estimators based on the Laplace transform

The empirical moment generating function was first used in estimation problems by Quandt and Ramsey (1978) and later by Epps and Pulley (1985). Parameter estimation in particular distributions has been discussed (Koutrouvelis and Canavos 1997; Koutrouvelis et al. 2005; Koutrouvelis and Meintainis 2002). We implement this method for estimation of \(\theta ^*\), but we employ the Laplace transform of \(f_T(t)\) instead of the moment generating function. This modification makes no difference.

The method is justified by the one-to-one relationship between the pdf of \(T\) and its Laplace transform, and hence, the distribution is fully characterized by the Laplace transform. The Laplace transform of \(f_T(t)\) is the mean of the exponential transformation of \(T\), \(\widehat{f_T}(s) = \mathbb {E}\left( \mathrm{e}^{-sT} \right) \) and can thus be approximated by its empirical counterpart obtained from data,
$$\begin{aligned} \widetilde{f_T}(s) = \frac{1}{n}\sum _{i=1}^n \mathrm{e}^{-st_i}. \end{aligned}$$
(25)
A particular case is exploited in Ditlevsen and Lansky (2005) and Ditlevsen and Lansky (2006) for the Ornstein–Uhlenbeck and Feller Leaky Integrate-and-Fire Models, where explicit expressions are obtained for specific values of \(s\). The main idea is to compare the empirically estimated Laplace transform with the theoretical \(\widehat{f_T}(s)\) evaluated at suitably chosen points \(s\) and find \(\theta ^*\) such that some difference is minimal. The following nonlinear regression equation is used
$$\begin{aligned} \widetilde{f_T}(s_i) = \widehat{f_T}(s_i;\theta ^*) + \varepsilon _i, \end{aligned}$$
(26)
where \(s_1,s_2,\ldots ,s_q\) is a chosen grid of points. The unknown parameters are estimated by ordinary least squares. The quality of the estimator depends on the choice of grid points \(s_i\); nevertheless, it is not generally clear how the optimal grid should be constructed, though some suggestions for some special distributions exist (Koutrouvelis et al. 2005).
All four estimation methods and their properties are summarized in Table 2. R script file, which contains functions implementing all presented estimation methods, is provided in the supplementary material.
Table 2

Summary of estimation methods

Estimation method

Description

Advantages

Disadvantages

Method based on cdfs

Returns \(\theta ^*\) at which the empirical cdf differs significantly from the exponential cdf with rate \(\lambda \)

No assumptions about evoked activity

Less accurate estimatesApplicable only if latency is constant across trials

Method of moments

Returns parameters for which the theoretical mean (and variance) are equal to their empirical counterparts

Applicable for all models with known momentsCan be extended to take into account subsequent ISIs

Needs particular assumptions about evoked activity

Maximum-likelihood method

Returns parameters for which the probability that data were generated under the given model is maximal

Has often good asymptotic properties such as efficiency and consistency, even when regularity conditions are not met

Needs particular assumptions about evoked activity Pdf of the data must be known

Laplace method

Returns parameters for which the deviation of the theoretical Laplace transform of the pdf from the empirically estimated Laplace transform is minimal

Applicable for all models with known Laplace transform

Needs particular assumptions about evoked activity Accuracy strongly influenced by the choice of the grid points

A common disadvantage is that all methods are based on particular assumptions on spontaneous activity

6 Simulation studies and numerical results

The performance of the estimators introduced in Sect. 5 is examined for Model 1–5 by simulations. First \(W\) and \(R\) and in the case of models with random latency also \(\varTheta \) are simulated. Then, Eq. (1) provides a realization of \(T\). We used a sample size of \(n=100\), i.e., the number of trials under identical conditions. For every combination of parameter values, \(N=1,000\) samples were simulated and therefore \(1{,}000\) different estimates of the mean latency were computed.

6.1 Parameter setting and results

To compare estimators, we use relative mean error (RME) to evaluate the bias, and relative mean square error (RMSE), incorporating both the variance and the bias. They are defined by
$$\begin{aligned} \mathrm{RME}(\hat{\theta }^*)&= \frac{1}{N}\sum _{i=1}^N\left( \frac{\hat{\theta }^*_i-\theta ^*}{\theta ^*}\right) \end{aligned}$$
and
$$\begin{aligned} \mathrm{RMSE}(\hat{\theta }^*) = \frac{1}{N}\sum _{i=1}^N\left( \frac{\hat{\theta }^*_i-\theta ^*}{\theta ^*}\right) ^2, \end{aligned}$$
where \(N = 1{,}000\). Parameter values used in the simulations were:
  • \(\lambda = 1\), \(\quad \kappa \in \{0.1, 0.25\}\), \(\quad k \in \{3,9\}\)

  • \(\theta ^*\) varying from \(0.1\) to \(1.5\) in steps of \(0.1\)

For the estimator based on the Laplace transform, the grid points \(s_i\) used in the regression were \(q=50\) equidistant points between 0.0002 and 0.01 (\(s_{i+1}-s_i=0.0002\)). This grid was chosen to give an acceptable precision of estimates, evaluated from simulations. A finer grid did not improve estimates much, whereas a coarser grid significantly deteriorated estimates.
Results for models with constant latency (Model 1 and 2) Results are provided in Fig. 11a–d. The nonparametric estimators \(\hat{\theta }^*_{\mathrm{{ECDF}},1}\) and \(\hat{\theta }^*_{\mathrm{{ECDF}},2}\) perform quite well for all values of \(\theta ^*\), but they suffer from the largest bias and the variance is also large. Therefore, the results are shown separately in Fig. 11a, b with broader scales. The moment and Laplace estimator give similar results, the variance of the Laplace estimator seems to be slightly smaller. A disadvantage of these estimators is that the RMSE for small \(\theta ^*\) is very large. Moreover, a considerable number of estimates of \(\theta ^*\) are negative (details about this phenomenon are given below). It is apparent that the maximum-likelihood estimator has smaller variance than the other estimators, since its RMSEs are generally the smallest. In conclusion, for latencies in the range used in the simulations, the maximum-likelihood estimator seems to be the best estimator.
Fig. 11

Results of the simulation study for models with constant and random latency. a RMEs and b RMSEs of nonparametric estimates \(\theta ^*_{\mathrm{{ECDF}},1}\) and \(\theta ^*_{\mathrm{{ECDF}},2}\) in Model 1 and 2. c RMEs and d RMSEs of parametric estimates obtained by the method of moments, maximum-likelihood method and method based on the Laplace transform of the pdf of \(T\) in Model 1 and 2. e RMEs and f RMSEs of estimates obtained by the method of moments, maximum-likelihood method and method based on the Laplace transform of the pdf of T in Model 3 and 4. Maximum-likelihood estimation is not possible for Model 4

Results for models with random latency (Model 3 and 4) Results can be seen in Fig. 11e, f. Again, the moment estimator and the Laplace estimator give similar results with large errors for small values of mean latency. The Laplace method leads to slightly better estimates. As expected, maximum-likelihood estimates are best.

Results for model with selective interaction (Model 5)

Results can be seen in Fig. 12. The moment estimator and the Laplace estimator give almost identical results. Using observations of \(X\) together with observations of \(T\) leads to significant improvement of estimates. If \(k=1\), the maximum-likelihood estimator tends to be most precise again.
Fig. 12

Results of the simulation study for models with selective interaction. a RMEs and b RMSEs of estimates in Model 5 obtained by the method of moments based on observations of \(T\), based on observations of \(X\) and based on observations of \(T\) and \(X\) together, the maximum-likelihood method based only on observations of \(T\) and based on observations of \(T\) and \(X\) together and the method based on the Laplace transform of the pdf of \(T\). Maximum-likelihood estimation is not possible if \(k>1\)

Identifiability of parameters Estimates of \(\theta ^*\) are required to be positive. Only the estimator based on ecdf and the maximum-likelihood estimator for models with constant latency are constructed so that they ensure non-negativity of estimates. Estimates obtained by other methods can become negative. In our simulations, estimates in models with selective interaction were all positive. Maximum-likelihood estimates in all models were almost always positive, problems with identifiability arose only in Model 3 for very small \(\theta ^*\) (\(\theta ^*=\lambda /5\) and \(\theta ^*=\lambda /10\)), where at most \(1\,\%\) of estimates were negative. However, a large proportion of negative estimates occurred among moment estimates and estimates obtained from the Laplace transform. The worst results were obtained for Model 1 and 3 (both are special cases of Model A) for small \(\theta ^*\).

Exact proportions of negative estimates for \(\theta ^*\) in the range \([0.1,0.6]\) are shown in Fig. 13. If \(\theta ^*\) is larger, estimates become negative very rarely.
Fig. 13

Proportions of negative estimates obtained a for Model 1 and 2, b for Model 3 and 4 (second row). Only those estimators where negative estimates occurred are shown

6.2 Conclusion

Based on the results from the simulation study, we suggest the following ranking of the estimators according to their accuracy. Overall, the best method is maximum likelihood. It is succeeded by the method of moments and the Laplace method, which are of a similar quality, whereas the cdf-method gives the poorest results. However, there are many exceptions to these rules. The maximum-likelihood method performs best if only RMSE is taken into consideration. Nevertheless, if the bias is to be minimized, then it is outperformed for very short latencies (\(\theta \le \lambda /5\), approximately) by the method of moments and the Laplace method in all models except Model 3. The method of moments and the Laplace method represent basic approaches to inhibitory latency estimation, because they are not as limited in their applicability to particular models, since e.g. moments are often available even if the distribution is not. The method of moments is better than the Laplace method for the model with selective interaction if there is a good reason to assume that the response is long enough and satisfies the renewal assumption and observations of subsequent intervals are available. On the other hand, the Laplace method avoids the problem with potential unidentifiability of parameters with higher probability. Finally, the method based on cdfs is the best choice if one is not willing to assume a parametric model for the evoked activity.

7 Discussion

Although the notation used in this paper mimics the notation used by Pawlas et al. (2010), the methods for latency estimation presented there cannot be applied on inhibitory response. The main reason is that the variable \(R\) has in Pawlas et al. (2010) the meaning of both the time to the first evoked spike, as well as the so-called first-spike latency. Therefore, the general aim of the methods proposed there is to distinguish observations of \(W\) and \(R\). When observations of \(W\) are excluded, statistical properties of the variable \(R\) are estimated. On the contrary, here we have observations of \(W\) and \(R\) as well, but the variable of interest is \(\varTheta \), which is not measured directly and has only indirect influence on the observations obtained in the experiment.

The models presented in the papers by Tamborrino et al. (2012) and Tamborrino et al. (2013) are more similar to our models, but the difference is that \(T = \min \{W,R\}\) if \(W>\varTheta \). Thus, estimation methods based on the assumptions about evoked activity cannot be used for our model. The only exception is the estimator based on comparison of cumulative distribution functions, because it uses only the assumptions about spontaneous and evoked activity being different.

The simulation study shows that the maximum-likelihood method, when available, is the best choice, which is in agreement with our expectations. However, the method of moments and the Laplace method can be applied for all considered models while the maximum-likelihood method cannot. The nonparametric estimators are biased, \(\hat{\theta }^*_{\mathrm{{ECDF}},1}\) underestimates and \(\hat{\theta }^*_{\mathrm{{ECDF}},2}\) overestimates \(\theta ^*\). Since these two estimators differ only in the threshold used for detecting the beginning of the response (\(0\) and \(\sigma (t)\), respectively), it suggests that a less biased estimator (with smaller RME for each \(\theta ^*\)) could be obtained if the difference \(F_W(t)-\widehat{F}_T(t)\) is compared with \(\alpha \sigma (t)\), where \(\alpha \) is a suitable constant in \((0,1)\). Remember that this nonparametric method only assumes Poissonian spontaneous activity and constant latency.

It was shown that estimators of \(\theta ^*\) for models with selective interaction, which are based on observations of both \(T\) and \(X\), are much better for arbitrary choice of the mean latency \(\theta ^*\). On the other hand, the method of estimation using only observations of \(T\) can have an advantage over estimators using observations of \(T\) and \(X\), if the assumption that the spontaneous activity as well as the evoked activity are given by renewal processes, is not satisfied. They can also be better when the response period is relatively short, and the corresponding part of the spike train consists mainly of a few spikes. Then, it is not certain that the first ISI after the stimulus belongs entirely to the response period. However, it is clear that any estimation method based on measurements of \(T\) alone would fail, whenever the true latency is so long that all observations of \(T\) (or a considerable amount of them) are shorter than \(\varTheta \). In that case, observations of the subsequent ISIs are necessary for estimation.

The assumptions of the presented models could be relaxed at the cost of less explicit estimators and larger computational costs, e.g. by allowing for more general distribution families for the spontaneous activity. The sensitivity on the Poisson assumption should be tested, either on real data or on simulated data, where this assumption is violated by design. The assumption that the latency follows an exponential distribution is very simplistic and more realistic models could be considered. The justification of this restriction is that the exponential distribution enables explicit calculations and to obtain the Laplace transform of the pdf of \(T\) in a manageable form. For other distributions, numerical methods would be required. In particular, eq. (15) is no longer valid. Another shortcoming is that stationarity across trials is implicitly assumed, whereas experimental data are usually more complicated, e.g. because of adaptation and plasticity. On the other hand, the assumption of reproducibility is intrinsic to statistical methods and nearly always implicitly assumed by experimentalists when preparing post-stimulus time histograms.

Some of the presented models could also be used for excitatory responses, namely Model A and its special cases Model 1 and Model 3. They can describe inhibitory as well as excitatory response, if the distribution of \(U\) is appropriately chosen, e.g. \(U \sim Exp(\kappa )\), where \(\kappa > \lambda \). In that case, all estimation methods could be applied with only little alterations. The method based on cdfs would require to work with the difference \(\hat{F}_T(t) - F_W(t)\) instead of \(F_W(t) - \hat{F}_T(t)\). In fact, this method was originally proposed this way for excitatory response (Tamborrino et al. 2012). The method of moments and the Laplace method would not be influenced by this change. The maximum-likelihood method is also unchanged, the only difference concerns the likelihood function in Model 1 (with constant latency), which is piecewise decreasing for inhibitory response and piecewise increasing for excitatory response. Nevertheless, this change has no impact on the determination of estimates.

Notes

Acknowledgments

M.L. and P.L. were supported by the Grant Agency of the Czech Republic, project P304/12/G069, and by RVO:67985823. S.D. was supported by the Danish Council for Independent Research | Natural Sciences. The work is part of the Dynamical Systems Interdisciplinary Network, University of Copenhagen.

Supplementary material

422_2014_614_MOESM1_ESM.txt (22 kb)
Supplementary material 1 (txt 22 KB)

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Marie Levakova
    • 1
    • 3
  • Susanne Ditlevsen
    • 2
  • Petr Lansky
    • 3
  1. 1.Department of Mathematics and Statistics, Faculty of ScienceMasaryk UniversityBrnoCzech Republic
  2. 2.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark
  3. 3.Institute of PhysiologyAcademy of Sciences of the Czech RepublicPrague 4Czech Republic

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