GoodnessofFit Tests and Nonparametric Adaptive Estimation for Spike Train Analysis
 Patricia ReynaudBouret^{1}Email author,
 Vincent Rivoirard^{2},
 Franck Grammont^{1} and
 Christine TuleauMalot^{1}
https://doi.org/10.1186/2190856743
© P. ReynaudBouret et al.; licensee Springer 2014
Received: 15 February 2013
Accepted: 4 November 2013
Published: 17 April 2014
Abstract
When dealing with classical spike train analysis, the practitioner often performs goodnessoffit tests to test whether the observed process is a Poisson process, for instance, or if it obeys another type of probabilistic model (Yana et al. in Biophys. J. 46(3):323–330, 1984; Brown et al. in Neural Comput. 14(2):325–346, 2002; Pouzat and Chaffiol in Technical report, http://arxiv.org/abs/arXiv:0909.2785, 2009). In doing so, there is a fundamental plugin step, where the parameters of the supposed underlying model are estimated. The aim of this article is to show that plugin has sometimes very undesirable effects. We propose a new method based on subsampling to deal with those plugin issues in the case of the Kolmogorov–Smirnov test of uniformity. The method relies on the plugin of good estimates of the underlying model that have to be consistent with a controlled rate of convergence. Some nonparametric estimates satisfying those constraints in the Poisson or in the Hawkes framework are highlighted. Moreover, they share adaptive properties that are useful from a practical point of view. We show the performance of those methods on simulated data. We also provide a complete analysis with these tools on single unit activity recorded on a monkey during a sensorymotor task.
Electronic Supplementary Material
The online version of this article (doi:10.1186/2190856743) contains supplementary material.
1 Introduction
In neuroscience, the action potentials (spikes) are the main components for the realtime information processing in the brain. Moreover, it is possible to record in vivo several neurons and to have access to simultaneous spike trains. The duration of each spike is very small, about one millisecond. Moreover, the number and the position of each spike fluctuate from one trial to another trial. It is consequently quite natural to assimilate a spike to a random event. Therefore, in this article, we mathematically model spike trains as realvalued point processes that have been deeply described and studied for a long time in the literature (see [4] for a review) and often used in neuroscience (see, for instance, [2] and the references therein). However, except in very particular tests of independence (see, for instance, [5, 6]), it is most of the time necessary to describe spike trains as realizations of particular stochastic processes.
Most of the analyses start by answering a standard basic question. Is the process an homogeneous Poisson process or not? See, for instance, [7–9]. Indeed, for this simple model, extensively used in neuroscience, there is only one parameter to infer, namely the firing rate. The study of firing rates in neuroscience has lead to significative advances in the understanding of the coding of the direction of movements [10] for instance. But most of the time, spikes trains are more complex than homogeneous Poisson processes. Various studies have exhibited different kinds of correlations between some motor, sensory, or cognitive events in a behaving animal and a variation of the firing rate of specific neurons, before, during or after this event [11, 12]. In particular, such data cannot be stationary. So, constraints on the previous model are relaxed and processes can be assumed to be inhomogeneous Poisson processes. In this setting, the firing rate is now timedependent and is modeled by a function $\lambda (\cdot )$, which is the intensity of the inhomogeneous Poisson process (see [8, 9]). Several studies have also established statistical evidence of dependence between the occurrences of the spikes of several neurons (see [5, 6, 13–15]) or even within a given neuron. In this case, standard homogeneous or inhomogeneous Poisson processes cannot be used and models based on univariate or multivariate Hawkes processes or variations of them are quite natural to capture dependence of spikes occurrences [16–21]. Hawkes processes, extensively described and discussed later on, generalize homogeneous Poisson processes by using functions quantifying interactions between spikes. These functions are called interaction functions. Such interaction functions are used in neuroscience to model excitation and inhibition phenomena [22].
Whatever the chosen model, this model has to be tested before any other inference based on this model. A plugin step to infer unknown parameters is most of the time unavoidable to perform these tests. More precisely, for general models on point processes, the main ingredient consists in transforming the data so that the time changed process becomes a homogeneous Poisson process, fact which can be easily tested. However, the parameters of the transformation are usually unknown and are replaced by estimates. This plugin trick has been widely popularized since [23]. It is widely used in neuroscience since [1] (see also the textbook of Tuckwell [24], [3], or [2]). The main goal of this article is to precisely show that the plugin step may sometimes lead to undesirable effects and to propose the subsampling as a reasonable and quite universal solution. We focus here on the Kolmogorov–Smirnov (K.S.) test of uniformity. Indeed this K.S. test is usually considered as one of the three main tests on the firstorder statistics that can be done to test the homogeneous Poisson hypothesis (see [1] and the references therein). More generally, the K.S. test (see [25] for its first use up to our knowledge) is one of the main omnibus tests [26], meaning that it is effective against a wide class of alternatives. However, it is known that a plugin has to be taken with care for this test (see [27] for some brief discussion of this point). By using aggregated or cumulated tests, we propose 5 tests based on subsampling as goodnessoffit tests, for which plugin issues are solved. Note that, in neuroscience, plugin problems have already been emphasized for other types of tests, namely the independence tests [22].
 1.
They are obtained by completely datadriven procedures that can be used even by neophytes in nonparametric statistics.
 2.
They achieve optimal convergence rates.
 3.
They do not assume light tails or any shape (exponential, unimodal, etc.) about the underlying function.
 4.
They adapt to the smoothness of the underlying function.
Furthermore, the developed strategies considerably extend the procedures proposed by [7, 30]. In particular, new datadriven kernel rules are introduced to estimate the intensity of inhomogeneous Poisson processes. We also derive a lassotype estimate for recovering interaction functions of multivariate Hawkes processes when observing n trials. Some new interpretations of the estimate and connections with classical tools of the neuroscience literature such as joint peristimulus time histograms (JPSTH) and cross correlograms are also proposed. Theoretical results are established by using the oracle approach (see later).
The article is organized as follows. We first explain how subsampling can overcome the issues raised by plugin for goodnessoffit tests for the special case of the K.S. test. Then we extensively discuss adaptive nonparametric estimation and its advantages with respect to parametric estimation. This is illustrated on Poisson or Hawkes processes and a wide range of nonparametric methods are proposed. Finally, some simulations have been performed and real data sets coming from the recordings of a sensorymotor task (that can be found in [15], for instance) are analyzed thanks to these new methods. Most of the analysis has been performed with the software R. We refer to [7] for a complete list of its advantages.
The function $\Lambda (\cdot )$ is therefore continuous nondecreasing. This is also the timetransformation on which the timerescaling theorem is based [2]. In the sequel, ${X}_{p}\underset{p\to \mathrm{\infty}}{\overset{\mathbb{P}}{\to}}0$ means that the sequence ${X}_{p}$ converges in probability toward 0 when p tends to infinity; ${X}_{p}\underset{p\to \mathrm{\infty}}{\overset{\mathcal{L}}{\to}}X$ means that the distribution of ${X}_{p}$ tends to the one of X when p tends to infinity.
2 GoodnessofFit Tests: The Plugin Drawback and Subsampling as a Possible Universal Solution
Once spike trains have been obtained and sorted, neurophysiologists often perform a very basic data analysis, which consists in testing several features such as stationarity for instance among other statistical inferences [7]. Following Ventura et al. [8], the first step of a “good practice” is usually to test whether the observed spike train is homogeneous Poisson or not. But it is usually admitted that real spike trains cannot be that simple and this hypothesis is most of the time rejected. To explain the rejection, the next step, still following [8], is to impute it to a lack of stationarity or to something more complex. It means that we have to test whether the process is an inhomogeneous Poisson process or not. For this purpose, one uses the timerescaling theorem (see [32] but also [4, 31]) under the hypothesis that the process is a Poisson process with deterministic intensity $\lambda (\cdot )$. Its associated compensator $\Lambda (\cdot )$ is in this case deterministic as well. The timerescaling theorem, in its simplest version, states therefore that if N is a Poisson process with intensity $\lambda (\cdot )$, observed on $[0,{T}_{\mathrm{max}}]$, then $\mathcal{N}=\{X=\Lambda (T):T\in N\}$ is an homogeneous Poisson process on $[0,\Lambda ({T}_{\mathrm{max}})]$ with intensity 1, fact which can be tested by practitioners. However, there is a misspecification in the method since $\lambda (\cdot )$ is unknown. The most popular and widely used method in neuroscience consists in plugging an estimate $\stackrel{\u02c6}{\lambda}(\cdot )$ in [8]. As explained in the Introduction, we first illustrate the drawbacks of noncautious plugins for goodnessoffit tests on the K.S. test, which has already been observed by [27]. We then propose a remedy to overcome these drawbacks based on subsampling.
2.1 Elementary Situation for Illustration
Under ${H}_{0}$, if ${F}_{0}$ is continuous, the distribution of ${\mathit{KS}}_{n}$ is known and it does not depend on ${F}_{0}$, so it can be tabulated [27]. For any $\alpha \in (0,1)$, let ${k}_{n,1\alpha}$ be the $1\alpha $ quantile of this distribution. The classical (without plugin) K.S. test consists in rejecting ${H}_{0}$ whenever ${\mathit{KS}}_{n}>{k}_{n,1\alpha}$ and this test is of exact level α. Note also that when n tends to ∞, the random variable $\sqrt{n}{\mathit{KS}}_{n}$ tends in distribution to a tabulated distribution (see [33]). As a consequence, if ${\tilde{k}}_{1\alpha}$ is the $1\alpha $ quantile of , $\sqrt{n}{k}_{n,1\alpha}$ tends to ${\tilde{k}}_{1\alpha}$ and the approximation is valid as soon as $n>45$ [34] (see also Durbin’s modification in [27]).
 (i)
Estimate λ by $\stackrel{\u02c6}{\lambda}=1/\overline{X}$, where $\overline{X}$ is the empirical mean of the ${X}_{i}$’s: $\overline{X}={n}^{1}{\sum}_{i=1}^{n}{X}_{i}$.
 (ii)
Plug in the estimate $\stackrel{\u02c6}{\lambda}$ and estimate ${F}_{0}$ by $u\to \stackrel{\u02c6}{F}(u)=1exp(\stackrel{\u02c6}{\lambda}u)$.
 (iii)
Form the K.S. statistic (1) by replacing ${F}_{0}$ by $\stackrel{\u02c6}{F}$. This leads to ${\mathit{KS}}^{(1)}$.
 (iv)
Reject ${H}_{0}$ whenever ${\mathit{KS}}^{(1)}>{k}_{n,1\alpha}$.
 (i)
Estimate λ by $\tilde{\lambda}=1/\overline{\overline{X}}$, where $\overline{\overline{X}}$ is the empirical mean of the first half of the ${X}_{i}$’s: $\overline{\overline{X}}=2/n{\sum}_{i=1}^{n/2}{X}_{i}$.
 (ii)
Plug in the estimate $\tilde{\lambda}$ and estimate ${F}_{0}$ by $u\to \tilde{F}(u)=1exp(\tilde{\lambda}u)$.
 (iii)
Form the K.S. statistic (1) by replacing ${F}_{0}$ by $\tilde{F}$, but also by replacing ${F}_{n}$ by the empirical cumulative distribution function only based on ${X}_{n/2+1},\dots ,{X}_{n}$. This leads to ${\mathit{KS}}^{(2)}$.
 (iv)
Reject ${H}_{0}$ whenever ${\mathit{KS}}^{(2)}>{k}_{n/2,1\alpha}$.
The pvalues of this test are represented on Fig. 1. Surprisingly, the distribution of the pvalues shows that the resulting test is not conservative enough. Indeed, the test will reject ${H}_{0}$ more than required and this procedure is even worse than the first strategy. Therefore, we turn toward a much more universal strategy, subsampling, thanks to the following result (see the Additional File 1 for the proof).
Technical arguments of Additional File 1 prove that the previous test is of exact level α asymptotically. More importantly, in practice this conclusion remains true even for relatively small values of n as shown in Fig. 1 illustrated with $n=40$. Even if this test is not as powerful as the one described in [27], it has the main advantage to be almost universal. It can be adapted to most of parametric situations, since the use of subsampling makes the condition (2) quite easy to fulfill.
We want now to adapt this method to the more general scheme of goodnessoffit tests for counting processes. From now on and whatever the situation, p will always correspond to the size of a subsample, i.e., a positive integer much smaller than n the total number of observations.
2.2 Aggregated Test of ${H}_{0}$: “The Observed Processes Are i.i.d. Poisson Processes”
To fix notation, we consider in the sequel that we observe n i.i.d. trials. Consequently, we have access to ${N}_{1},\dots ,{N}_{n}$, n i.i.d. point processes observed on $[0,{T}_{\mathrm{max}}]$ representing the n i.i.d. spike trains of a fixed recorded neuron during ${T}_{\mathrm{max}}$ seconds.
It is not possible to assess on just one realization whether a point process is a (non necessarily homogeneous) Poisson process or not since the variations of the repartition of the points between different parts of one trial can either be due to nonstationarity or to more complex dependency structures that cannot be studied on just one run.
Since F is unknown in our present situation, one has to estimate it, which leads to exactly the same plugin problem as before. Fortunately, we are able to prove the following result.
In Additional File 1, we prove that the previous test is of asymptotic level α. Note that Condition (5) can be demanding and rejection can be due to nonfulfillment of this condition. For instance, estimates $\stackrel{\u02c6}{\lambda}$ based on parametric estimates on a prescribed parametric model (such as maximum likelihood estimates for instance, see [8]) fulfill (5) if the prescribed model is true, but cannot fulfill this condition if the prescribed parametric model is not true. Hence, using parametric estimates in this setting lead to test both ${H}_{0}$ and “the prescribed parametric model is correct,” which is not satisfying. Therefore, it is natural to make no parametric assumption on the underlying model for $\lambda (\cdot )$ and to try to fulfill (5) by using nonparametric estimation.
Finally, as already observed by [8], aggregation can dilute the dependencies between the points. Therefore, Tests 2 and 3 cannot be really powerful as we will see later.
2.3 Cumulated GoodnessofFit Tests
Those general models are usually described through their conditional intensity $\lambda (\cdot )$, which represents the probability of occurrence of a point at time t given the past before t. So, defining a model through its conditional intensity is the easiest way to model the dependence between points. For instance, when we assume the conditional intensity λ to be a deterministic function f, we are assuming independence with respect to the past. This is equivalent to assuming that the process is an inhomogeneous Poisson process with intensity $\lambda =f$. Therefore, testing ${H}_{0}$: “the observed processes are i.i.d. with conditional intensity $\lambda =f$ and unknown deterministic function f” is equivalent to testing ${H}_{0}$: “the observed processes are i.i.d. inhomogeneous Poisson processes.”
where μ is a positive real parameter and h a non negative integrable function with support in ${\mathbb{R}}_{+}^{\ast}$ and with $f=(\mu ,h)$. For instance, if the function h is supported by the interval $(0,2]$, then the probability of occurrence at time t randomly depends on the occurrences of the process on $[t2,t)$. Testing whether the process is a classical selfexciting Hawkes or not can be rephrased as testing whether the process has conditional intensity given by the form ${\lambda}_{f}$ defined in (6), with unknown f. Other famous examples in biomedical areas such as the multiplicative Aalen intensity or the Cox model can be found in [29].
As in the previous subsection, we use the timerescaling theorem but in a deeper way. Remember that the general timerescaling theorem [2] states that for any point process N on $[0,{T}_{\mathrm{max}}]$ with compensator $\Lambda (\cdot )$, the point process $\mathcal{N}=\{X=\Lambda (T):T\in N\}$ is a Poisson process with intensity 1 on $[0,\Lambda ({T}_{\mathrm{max}})]$. Therefore, it is more interesting to cumulate the processes after timerescaling than in the usual time space $[0,{T}_{\mathrm{max}}]$. For general conditional intensity models, $\Lambda (\cdot )$ is random. Therefore the state space $[0,\Lambda ({T}_{\mathrm{max}})]$ is also random in general. Moreover, when we are dealing with p i.i.d. processes ${N}_{1},\dots ,{N}_{p}$, each ${N}_{i}$ has a different compensator ${\Lambda}_{i}(\cdot )$ which depends on the history of the i th trial. So except in the Poisson case where $\Lambda (\cdot )$ is deterministic, we do not apply the same transformation to all the points. We finally have to deal with p processes ${\mathcal{N}}_{i}=\{X={\Lambda}_{i}(T):T\in {N}_{i}\}$ that are Poisson processes of intensity 1, and whose occurrences lie in $[0,{\Lambda}_{i}({T}_{\mathrm{max}})]$. Even if the ${\Lambda}_{i}({T}_{\mathrm{max}})$ are i.i.d., they are not equal in general.
This leads to two main remarks. First, it is not possible to aggregate in general the timetransformed processes since we would aggregate processes with different lengths (see Fig. 2). Therefore, Tests 2 and 3 cannot be transferred to the most general case straightforwardly. However, one can cumulate those processes as done in Fig. 2 and this even if the intervals have different lengths. The resulting process ${\mathcal{N}}^{c,p}$ is therefore a Poisson process with intensity 1 on the random interval $\mathcal{I}=[0,{\sum}_{i=1}^{p}{\Lambda}_{i}({T}_{\mathrm{max}})]$ (see also Additional File 1 for a more precise formula and a proof of this statement). The second remark consists in noting that ${\sum}_{i=1}^{p}{\Lambda}_{i}({T}_{\mathrm{max}})$ being a random quantity, it is not true in general that conditionally to the total number of points in ℐ, the points of ${\mathcal{N}}^{c,p}$ behave like an i.i.d. uniform sample, and in the sequel we shall need to restrict ourselves to an interval of the form $[0,p\theta ]$ with a deterministic bound pθ, which is with high probability, smaller than ${\sum}_{i=1}^{p}{\Lambda}_{i}({T}_{\mathrm{max}})$.
Besides we have to deal with estimation of unknown transformations ${\Lambda}_{i}(\cdot )$. For this purpose, we introduce estimates of the type $t\to {\stackrel{\u02c6}{\Lambda}}_{i}(t)={\int}_{0}^{t}{\stackrel{\u02c6}{\lambda}}_{i}(u)du$, where ${\stackrel{\u02c6}{\lambda}}_{i}(\cdot )$ estimates ${\lambda}_{i}(\cdot )$, the conditional intensity of the i th process ${N}_{i}$. We obtain a cumulate process ${\stackrel{\u02c6}{\mathcal{N}}}^{c,p}$ built from the ${\stackrel{\u02c6}{\Lambda}}_{i}(\cdot )$’s. We have the following equivalent to Proposition 2.
This test, as a special case of Test 4, is also of exact asymptotic level α as soon as $\Lambda ({T}_{\mathrm{max}})>\theta $. Tests 4 and 5 are more powerful to detect dependencies or to reject the Poisson assumption than Tests 2 or 3, as we will see later.
As for Test 3, and for exactly the same reasons, we want to find nonparametric estimates satisfying (8) or (9). We provide in the next section powerful tools to deal with this problem and theoretical guarantees of performance of these estimates.
3 Nonparametric and Adaptive Estimation
3.1 Why Is Adaptive Estimation Useful?
Nonparametric estimation, and in particular nonparametric estimation of Poisson process intensity, is at the root of most of the data analyses performed on spike trains. Indeed, peristimulus time histograms (PSTH) [36] are usually the first graphical representations of an experiment. Those histograms have usually a fixed length for each interval (typically 10 ms) and are quite noisy from a statistical point of view (see, for instance, the representations of [8]). Therefore, there have been several attempts to provide smoother estimates, either by kernel estimates (see, for instance, [30]) or by projection on an orthonormal basis (see, for instance, [7] for the use of splines). These methods provide a first illustration of the data with as few assumptions as possible on the underlying “true” firing rate. They are originally not linked at all to any statistical or probabilistic models and constitute descriptive statistics. In particular, no parametric assumption on the underlying intensity is made at this step, the parametric model and its associated maximum likelihood estimator (MLE) being given in a second time once the shape of the curve is guessed [8]. Because of this lack of parametric assumption, those estimates seem to be the best candidates at first glance for the estimate $\stackrel{\u02c6}{\lambda}$ that needs to be plugged in Tests 3 or 5.
However, the problem of the convergence rate remains. In all these methods, there is a tuning parameter that needs to be chosen: it is the length of the interval for histograms, the bandwidth in kernel rules or the number of coefficients in the orthonormal expansion. The problem of the choice of this parameter has first been tackled very roughly in the neuroscience literature by choosing a fixed value. On the real data presented here or on the ones in [8], it was usually considered that a bandwidth of 50 or 100 ms was a good choice. However, such a very rough choice cannot guarantee a convergence rate when n goes to ∞. Indeed let us look more closely at the kernel estimate.
In this setting, this choice can be applied to Tests 3 and 5, since the Markov inequality implies that (5) or (9) are satisfied with $p(n)={n}^{\delta}$ and $\delta <(2r)/(2r+1)$. The choice $r=1$ gives $\delta <2/3$ and $r=2$ gives $\delta <4/5$.
Of course, in practice the choice of the bandwidth is capital. Since the smoothness of λ is unknown, the practitioner cannot use the previous choice. Furthermore, guessing the order of magnitude of $h(n)$ is not enough to achieve good performance since the leading constant plays an essential role. Hence, the theoretical considerations developed before do not solve the practical problem. Several directions have been proposed to overcome this problem. One of the most famous ones consists in using leaveoneout or other crossvalidation methods [30, 38]: among a finite family of fixed bandwidths, such methods choose the best one in an asymptotic setting. However, to our knowledge, nothing can be said when the family of bandwidths is not fixed and some bandwidths tend to 0 with n. It is not clear at all that the resulting estimate achieves a prescribed rate and, therefore, it cannot be used for the proposed tests in particular. Other methods based on the rule of the thumb (and variations of it) have been proposed in the density or the Poisson setting [8, 39], and in this case the resulting bandwidth is of the form $h(n)=C{n}^{1/5}$ for various possible choices of the constant C. Generally, those choices lead to poor results as noted by [8] (see also our the simulation study).
Adaptive estimation [37] aims at tuning in a datadriven way the unknown parameters of those methods (kernels, histograms, etc.) such that the resulting estimate has good practical performance and a guaranteed convergence rate. The adaptive estimates are usually mathematically proved to achieve the best possible rate of convergence and this even if the regularity is unknown. Moreover, they do not depend on any restrictive assumption such as, for instance, some parametric assumption. The only assumption lies in the underlying probabilistic model (for instance, one assumes that the processes are inhomogeneous Poisson processes). Their reconstructions are therefore much more trustworthy than other methods for which those extra assumptions may not be fulfilled. As a conclusion, adaptive estimates constitute ideal candidates to be plugged in Tests 3, 4, or 5.
The main aim of next subsections is therefore to present adaptive estimates in the Poisson or in the Hawkes model that will have these good properties.
3.2 Adaptive Estimation for Poisson Processes
3.2.1 Kernel Estimates
As mentioned previously, the Poisson setting is very close to the density setting. In the density setting, the main adaptive method for choosing a bandwidth is the Lepski method, which has been recently updated to the multidimensional framework and to deal with the problem of choosing the leading constant of the bandwidth. Due to Goldenshluger and Lepski [40], it is referred in the sequel as the GL method. We propose here to adapt this method to the Poisson setting in the following way and to prove its adaptive properties.
Note that in (12), ${\parallel K\parallel}_{2}^{2}{N}^{a,n}([0,{T}_{\mathrm{max}}])/({n}^{2}h)$ is an unbiased estimate of the variance term in (11) and therefore the previous criterion mimics the biasvariance decomposition of the risk of ${\stackrel{\u02c6}{\lambda}}_{n}^{{K}_{h}}$ up to some multiplicative constant. Once K, ℋ and η are chosen, we obtain a turnkey procedure. The following theoretical result justifies our procedure.
where ${C}_{1}$ is a constant depending on ${\parallel K\parallel}_{1}$ and η and ${C}_{2}$ is a constant depending on δ, η, ${\parallel K\parallel}_{2}$, ${\parallel K\parallel}_{1}$, ${\parallel \lambda \parallel}_{1}$, and ${\parallel \lambda \parallel}_{\mathrm{\infty}}$.
which is the optimal rate of convergence over such spaces. This rate is achieved, even if we do not know in advance the regularity r of λ, which is from a theoretical point of view the main improvement with respect to the theory described in the previous subsection.
If K is the Gaussian kernel, then ${\parallel K\parallel}_{1}=1$ and ${\parallel K\parallel}_{2}={2}^{1/2}{\pi}^{1/4}$. Moreover, ${K}_{h}\star {K}_{{h}^{\prime}}={K}_{\sqrt{{h}^{2}+{{h}^{\prime}}^{2}}}$ and a straightforward computation shows that explicit formula for ${\parallel {\stackrel{\u02c6}{\lambda}}_{n}^{h,{h}^{\prime}}{\stackrel{\u02c6}{\lambda}}_{n}^{{K}_{{h}^{\prime}}}\parallel}_{2}$ are also available. It is consequently very easy to implement the method, the computational cost being almost of the same order as crossvalidation. We will see in the simulation study that this practical choice is also quite robust.
3.2.2 Histograms
In the Poisson setup, there are several ways to select datadriven partitions that lead to adaptive histogram estimates. For instance, one can use model selection as in [41]. Model selection can either select a regular partition or an irregular partition on a grid. When regular partitions are considered, the resulting estimator satisfies an oracle inequality similar to the oracle inequality established in Theorem 2 for the kernel rule combined with the GL method. Indeed the bin for the histograms plays exactly the same role as the kernel bandwidth. Therefore, it leads to similar theoretical performance, except that the histograms cannot become smooth enough to guarantee an optimal convergence rate for regular intensities (namely $r>1$). Therefore, the choice of regular partitions is probably not the best one and one may prefer the GL method. The datadriven choice of the partition becomes much more interesting when the partition is not forced to be regular. Indeed irregular partitions can capture a fast increase of the firing rate followed very quickly by a fast decrease at some particular moment of the experiment, without leading to too noisy estimates as the classical PSTH, since over smoother periods, the length of the interval can be much larger. However, the method of [41] is too time consuming to be really considered in practice. Another possible direction is the context of Markov modulated Poisson processes [42], where the algorithms are also quite time consuming without ensuring any adaptive property in terms of convergence rate (despite some possible interpretation with respect to hidden Markov processes).
In [43], it has been proved that a slight modification of this estimate satisfies an oracle inequality in the same spirit as Theorem 2. It also generalizes this estimate by considering general biorthogonal bases instead of the Haar basis, leading to smooth estimates (see [43, 44]). In this case, for a slight modification of the threshold, the resulting estimate has the same convergence rates as the kernel estimate combined with the GL method, up to some logarithmic term, as soon $\gamma >1$. The choice $\gamma <1$ has been shown to lead to bad convergence rates and the choice $\gamma =1$ has been shown to work well on extensive simulations in both [43, 44]. This method is easily implementable leading to very fast algorithms that are in particular faster than algorithms based on the GL method.
3.2.3 More Sophisticated Procedures
Thresholding rules and irregular partitions overcome a drawback of kernel estimates that suffer from a lack of spatial adaptivity on the time axis. Several attempts have been proposed to build more local choices of the bandwidth (see [30] for instance), but to our knowledge no mathematical proof of this spatial adaptation has been established, whereas histograms and in particular the previous Haar thresholding estimator can adapt the length of the bin to the heterogeneity of the data. But the resulting estimator is not smooth at all. As explained, we can consider a smoother wavelet basis, but this extension does not completely address the issue.
The best alternative, to our knowledge, when the support of $\lambda (\cdot )$ is known and bounded (here $[0,{T}_{\mathrm{max}}]$) and when $\lambda (\cdot )$ does not vanish for a significant period of time, is due to Willett and Nowak [45]. Their method is quite intricate to describe. Informally, a penalized loglikelihood criterion is used to select a piecewise polynomial. Both the partition and the degree of each polynomial on each interval of the partition are free (on a very refined grid of resolution). Willett and Nowak have proved that such an estimator achieves optimal rates of convergence for various classes of regularity and in an adaptive way. From a practical point of view, a dyadic tree algorithm is used. Its complexity is much smaller than a full model selection method on the same piecewise polynomial family of models. It is a bit more complex than a thresholding algorithm, but there exist a program (FreeDegree) in Matlab interfaced with C which makes its use in practice quite easy. For a more complete description of the method, we refer to [45]. Note that in practice because of its adaptive properties, this estimator is able to be piecewise constant when the true intensity is piecewise constant but also very smooth (with high degree for the polynomials) when the underlying intensity is smooth and when the number of points is sufficient. It is also able to be spatially adaptive, the underlying datadriven partition being irregular. In the sequel, we denote this method ${\stackrel{\u02c6}{\lambda}}_{n}^{\mathrm{WN}}$.
3.3 Adaptive Estimation for Hawkes Processes
If inhomogeneous Poisson processes can model nonstationary data, they are not appropriate to model dependencies between points. However, several studies have established potential dependence of spike occurrences for different neurons. This has been detected via descriptive statistics, via independence tests for a given fixed model or via modelfree independence tests based on permutations (also called trialsshuffling) [5, 6, 13, 15, 22, 46].
One simple model of dependency is the multivariate Hawkes process, which is the point process equivalent to the autoregressive model. It has first been introduced by Hawkes [47], as a selfexciting point process, that is useful in particular in seismology (see, for instance, [23]). It has also been used to model positions of motifs along the DNA molecule [48, 49]. In neuroscience, it explicitly appears in the 1980s with [19] and is close in spirit to [50, 51], with the additional advantage of modeling potential feedback between the neurons.
In (17), the ${\nu}^{(m)}$’s are positive parameters representing the spontaneous firing rates and the ${h}_{\ell}^{(m)}$’s are the interaction functions and have support included into ${\mathbb{R}}_{+}^{\ast}$. More precisely, before the first occurrence of the multivariate process, the ${N}^{(m)}$’s behave like homogeneous Poisson processes with constant intensities ${\nu}^{(m)}$. The first occurrence (and the next ones) affects all the processes by increasing or decreasing the conditional intensity via the interaction functions ${h}_{\ell}^{(m)}$’s. For instance, if ${h}_{\ell}^{(m)}$ takes large positive values in the neighborhood of the delay d and is null elsewhere, then after the delay d of one occurrence of ${N}^{(\ell )}$, the probability to have a new occurrence of ${N}^{(m)}$ will significantly increase: The process ${N}^{(\ell )}$ excites the process ${N}^{(m)}$. On the contrary, if ${h}_{\ell}^{(m)}$ is negative around d, then after the delay d of one occurrence of ${N}^{(\ell )}$, the probability to have a new occurrence of ${N}^{(m)}$ will significantly decrease: The process ${N}^{(\ell )}$ inhibits the process ${N}^{(m)}$. Note in particular that the functions ${h}_{m}^{(m)}$’s model selfinteractions.
The Hawkes process as described above cannot really model nonstationary data. Indeed, when t grows (and under conditions on the interaction functions), the process converges quite quickly toward an equilibrium, which is stationary (see, for instance, [52, 53], and the references therein). If these conditions are not satisfied, the number of points in the process grows too fast to be a realistic model for spike trains anyway. Hence, Hawkes processes as defined in (17) cannot model nonstationary data, but can model dependent data.
where it is assumed that the interaction functions are bounded with support in $[0,A]$ with ${T}_{1}>A$.
Inference for Hawkes models based on the likelihood can be found in the literature, in particular, for parametric models [23, 49]. However, in neuroscience, for flexibility, the used parametric models are based on a large number of parameters. Therefore, they require several thousand spikes per neuron to be observed in a stationary way to achieve good estimation [19]. Classical model selection based on AIC and BIC criteria has also been used to select the number of knots for the spline estimate [21, 48, 54]. However, these criteria do not adapt well to irregular functions. This is the reason why alternative nonparametric adaptive inference has recently been developed in such models. The univariate case ($M=1$) has been studied in [55], where rates of convergence depending on the underlying regularity of the selfinteraction function have been derived. We can also mention the alternatives proposed in [20, 56] but no theoretical validation is provided in those works.
A multivariate approach, valid for very general counting processes including Hawkes processes and based on ${\ell}_{1}$ penalties, has been recently developed in [57]. Based on minimization of convex criteria, its computational cost is more reasonable than procedures proposed in [55] and it is also proved to satisfy oracle inequalities. We shall detail this method in the case of Hawkes processes and with piecewise constant estimates of the underlying interaction functions.
In the next section that can be skipped at first reading, we describe the method in a technical way. Then we give heuristic arguments to understand more deeply the presented method (see also [58] for a quicker view on this estimate). In particular, the method does not rely on the likelihood, but on a leastsquare contrast, which can be reinterpreted in terms of JPSTH [59].
3.3.1 Intensity Candidates and LeastSquare Contrast on One Trial
which is called the leastsquare contrast. This expression is observable and can be minimized if f is parameterized by a fixed number of parameters.
Hence, by (23), proposing ${\psi}_{t}^{(m)}(f)$ as a candidate for the intensity ${\lambda}^{(m)}$ of ${N}^{(m)}$ amounts to proposing a linear combination of instantaneous counts with delay to model the probability of the next occurrence of a point in ${N}^{(m)}$.
showing that the estimate given in (24) should be a convenient preliminary estimate.
3.3.2 LeastSquare Estimates on Several Trials and Connections with JPSTH and CrossCorrelograms
JPSTH and cross correlograms have been used for a long time in neuroscience, without links with any model. The formula (26), for the leastsquare estimate, shows the link between those descriptive statistics (more precisely the ${\overline{\mathbf{n}}}_{m,\ell}$’s) and the parameters of the Hawkes model. To recover the parameters, we need, in particular, to inverse the matrix $({\sum}_{i=1}^{n}{\mathbf{G}}^{(i)})$. This matrix quantifies for instance the following situation. Assume that $M=3$ and that the interaction functions ${h}_{2}^{(1)}$ and ${h}_{3}^{(2)}$ are large on $[0,\delta ]$ and null elsewhere. We also assume that all the other interaction functions are null. In this situation, ${\overline{\mathbf{n}}}_{1,3}$ (or at least its first coordinate) will be large even if there is no direct interaction from ${N}_{3}$ on ${N}_{1}$. The matrix $({\sum}_{i=1}^{n}{\mathbf{G}}^{(i)})$ cumulates all these features (and also the fixed effect due to the spontaneous parameter, which needs to be subtracted) and inverting it enables us to find an estimate of the true interactions. See also [58] for a more visual transcription.
Note, however, that even if many coefficients are null as in the above described situation, due to the random noise, the estimates ${\stackrel{\u02c6}{\mathbf{a}}}^{(m)}$ have nonzero coordinates almost surely. Therefore, it is difficult to interpret the resulting estimate in terms of functional connectivity graph [58]. Moreover, if we wish to capture all the features, it is preferable to take A large and δ small. Therefore, the number of parameters of the model, depending on $K=A{\delta}^{1}$, increases. With a small number of trials n and a small interval $[{T}_{1},{T}_{2}]$, the leastsquare estimate is doomed to be quite poor as the MLE [19].
To remedy these problems, we now consider ${\ell}_{1}$ penalization to find a nonparametric estimate with adaptive properties and prescribed convergence rate.
3.3.3 Lasso Estimate
Since the criterion (27) is convex, the minimization problem can be performed quite easily. The function $f\in \mathcal{H}$ associated with $\tilde{\mathbf{a}}$ is denoted ${\stackrel{\u02c6}{f}}^{\mathrm{B}}$, in reference to the Bernstein inequality that governs the shape of the weights (see [57]).
Because the penalty term added to the leastsquare criterion is a weighted ${\ell}_{1}$norm, the resulting estimate is sparse and many coordinates in ${\tilde{\mathbf{a}}}^{(m)}$ will be null (see [60] for the seminal paper on Lasso methods). This estimate and much more general forms have been studied quite intensively in [57]. In Additional File 3, we prove an oracle inequality for a slight modification of the present estimate, whose exact form can also be found in [58].
that constitutes a compromise, as usual, between a bias term and a variance term. Minimizing the bias gives the best linear approximation of λ of the form $\psi (f)$ and this even if λ is not of the form $\psi (f)$. In this sense, it applies in particular to Hawkes processes with selfinhibition (i.e., negative ${h}_{m}^{(m)}$’s), which models refractory periods [22] and for which $f\to \lambda ={(\psi (f))}_{+}$ is not linear anymore. Finally, (29) leads to a control of the lefthand side of (8) adapted to the context of this section. Under further technical assumptions, we can then prove that Test 4 can be applied. We refer the reader to [57] for more details that are omitted here to avoid too tedious technical aspects.
The last point already developed in [57] is that Lasso estimates are most of the time biased in practice. To overcome this problem, a two step procedure is proposed. It consists in finding the non zero coefficients of ${\stackrel{\u02c6}{f}}^{\mathrm{B}}$ and performing a classical leastsquare estimate on this support. We denote this twostep estimate ${\stackrel{\u02c6}{f}}^{\mathrm{BO}}$.
4 Practical Performance
4.1 Description of the Data
4.1.1 Real Data
The data used here are a small subset of already partially published data in previous experimental studies [15, 22, 61, 62]. These data were collected on a 5yearold male rhesus monkey who was trained to perform a delayed multidirectional pointing task. The animal sat in a primate chair in front of a vertical panel on which seven touchsensitive lightemitting diodes were mounted, one in the center and six placed equidistantly (60 degrees apart) on a circle around it. The monkey had to initiate a trial by touching and then holding with the left hand the central target. After a delay of 500 ms, the preparatory signal (PS) was presented by illuminating one of the six peripheral targets in green. After a delay of either 600 or 1200 ms, selected at random with various probability, it turned red, serving as the response signal and pointing target. During the first part of the delay, the probability for the response signal to occur at $500+600\phantom{\rule{0.3em}{0ex}}\text{ms}=1.1\phantom{\rule{0.3em}{0ex}}\text{s}$ was 0.3. Once this moment passed without signal occurrence, the conditional probability for the signal to occur at $500+600+600\phantom{\rule{0.3em}{0ex}}\text{ms}=1.7\phantom{\rule{0.3em}{0ex}}\text{s}$ changed to 1. The monkey was rewarded by a drop of juice after each correct trial, i.e., a trial for which the monkey touches the correct target at the correct moment.
Signals recorded from up to seven independently moving microelectrodes (quartz insulated platinum–tungsten electrodes, impedance: 2–5 MO at 1000 Hz) were amplified and bandpass filtered from 300 Hz to 10 kHz. Single unit activity was obtained by performing an online discrimination of spikes on each electrode. Spikes were firstly selected by taking into account their amplitude using an online window discriminator with highpass and lowpass filters. In cases where spikes were not discriminable due to their amplitude only, the electrode was moved until the signals were sufficiently distinct to be discriminable on this basis. Although offline spike sorting was available, it was not used in this study. Indeed, beyond the reservations that one may have concerning the variable quality of the output of such software, the use of clean original electrophysiological signals makes safer the more specific study of precise neuronal synchronization. Neuronal data along with behavioral events (occurrences of signals and performance of the animal) were stored on a PC for offline analysis with a time resolution of 1 kHz.
Repartition of number of trials on the real data sets
Direction  Total  

1  2  3  4  5  6  
Data Set A  28  31  30  35  28  25  177 
Data Set B  23  24  26  18  30  20  141 
Therefore, n, the total number of trials will be close to 200 if one aggregates over all the directions or will belong to the interval $[20,35]$ if one considers the trials according to the directions. Those trials are assumed to be i.i.d. This assumption is more reasonable if one considers trials for a fixed given direction.
4.1.2 Simulated Data
To assess the performance of our procedure, simulated data for which the underlying model is known have also been simulated. Three different data sets have been simulated, with the thinning method [63]:

(SHomPoi) Spikes are distributed according to an homogeneous Poisson processes of intensity 20 Hz on $[0,2]$ s.

(SInPoi) Spikes are distributed according to an inhomogeneous Poisson processes with piecewise continuous intensity on $[0,2]$ s given by$t\to \lambda (t)=\sum _{i=1}^{3}[{g}_{i}+{h}_{i}{e}^{4\ast {(t{c}_{i})}^{2}/({r}_{i}^{2}{(t{c}_{i})}^{2})}]{\mathbf{1}}_{t\in [{c}_{i}{r}_{i};{c}_{i}+{r}_{i})},$
with $g=[5,30,0]$, $h=[12.5,15,12.5]$, $c=[0.375,1.25,1.825]$, and $r=[0.375,0.5,0.125]$.

(SHaw) Two spike trains are simulated according to a bivariate Hawkes process observed on $[0,2]$. Each process is respectively denoted ${N}^{(1)}$ and ${N}^{(2)}$. Their intensities are given by (17) with spontaneous parameters ${\nu}^{(1)}={\nu}^{(2)}=20$ Hz and interaction functions ${h}_{1}^{(1)}={h}_{2}^{(2)}=20\times {\mathbf{1}}_{[0,0.005]}$, ${h}_{2}^{(1)}=60\times {\mathbf{1}}_{[0,0.01]}$ and ${h}_{1}^{(2)}=0$.
Each time a n i.i.d. sample is drawn.
The several treatments have been done in R except Willett and Nowak’s estimate (WN) for which Matlab has been used.
4.2 Results
4.2.1 Checking the Homogeneous Poisson Assumption
pvalues of the chisquare test of the Poissonian distribution for the number of spikes per trial. The following code is used: ∘ corresponds to a pvalue of the test by upper values in $[{10}^{2},{10}^{1})$, ▵ to a pvalue of the test by upper values in $[{10}^{3},{10}^{2})$, ▵▵ to a pvalue of the test by upper value in $[{10}^{4},{10}^{3})$, $\mathrm{\u25b5}\mathrm{\u25b5}\mathrm{\u25b5}$ to a pvalue of the test by upper value in $(\mathrm{\infty},{10}^{4})$. The signs are filled in black if the pvalues correspond to rejection of a Benjamini and Hochberg (BH) multiple test method [64] either on the simulated data (left part of the table) or on Data Sets A and B (right part of the table)
n  Directions  Pooled  

40  200  1  2  3  4  5  6  
SHomPoi  N1A  ∘  ∘  •  ▲  ▲▲▲  
SInPoi  N2A  ▲▲  ∘  ∘  ▲▲▲  
SHaw (${N}^{(1)}$)  N1B  ∘  ∘  ∘  ▲▲▲  
SHaw (${N}^{(2)}$)  ∘  N2B  ∘  •  ▲▲▲ 
pvalues of the classical K.S. test of uniformity on the spikes, all trials being aggregated. Same codes as in Table 2. Note that none of the pvalues were close enough to 1 to force a rejection by the test by lower values (i.e., rejection when the test statistic is smaller than ${k}_{n,\alpha}$, which corresponds to pvalues of the test by upper values larger than $1\alpha $)
n  Directions  Pooled  

40  200  1  2  3  4  5  6  
SHomPoi  N1A  ▲▲▲  ▲▲▲  ▲▲▲  •  ▲▲▲  ▲▲  ▲▲▲  
SInPoi  ▲▲▲  ▲▲▲  N2A  ▲▲▲  ▲▲▲  ▲▲▲  ▲▲▲  ▲▲▲  ▲▲▲  ▲▲▲ 
SHaw (${N}^{(1)}$)  N1B  ▲▲▲  ▲▲▲  ▲▲▲  ▲▲▲  ▲▲▲  ▲▲▲  ▲▲▲  
SHaw (${N}^{(2)}$)  N2B  ▲▲▲  ▲▲▲  ▲▲▲  ▲▲▲  ▲▲▲  ▲  ▲▲▲ 
pvalues of Test 1 on the ISI, with a subsample size $\lfloor {[{n}_{\mathrm{tot}}^{\mathrm{ISI}}]}^{2/3}\rfloor $, where ${n}_{\mathrm{tot}}^{\mathrm{ISI}}$ is the total number of ISI that have been observed in all the trials. Same codes as in Table 2 for the pvalues. Note that none of the pvalues were close enough to 1 to force a rejection by the test by lower values (i.e., rejection when the test statistic is smaller than ${k}_{n,\alpha}$, which corresponds to pvalues of the test by upper values larger than $1\alpha $)
n  Directions  Pooled  

40  200  1  2  3  4  5  6  
SHomPoi  N1A  ▲▲▲  ▲▲▲  ▲  •  ∘  ▲▲▲  
SInPoi  ▲▲  ▲▲▲  N2A  ▲▲▲  ▲▲▲  ▲▲▲  ▲  ▲▲▲  ▲▲▲  ▲▲▲ 
SHaw (${N}^{(1)}$)  ∘  ▲  N1B  ▲▲▲  ▲▲▲  ▲▲▲  •  ▲▲▲  ▲▲▲  ▲▲▲ 
SHaw (${N}^{(2)}$)  ▲  ▲▲▲  N2B  •  ▲▲▲  ▲  ∘  ▲▲  ▲  ▲▲▲ 
4.2.2 Checking the Inhomogeneous Poisson Assumption
First of all, ${\stackrel{\u02c6}{\lambda}}_{n}^{{\mathrm{SW}}_{h}}$ is clearly the worst choice, as expected for such a rough kernel. Figure 5a shows the reconstruction of a constant intensity. Kernel estimates with Gaussian kernels are oscillating; the thumb rule bandwidth ${h}^{\ast}$ is larger than $\stackrel{\u02c6}{h}$, the GL bandwidth. The fixed bandwidth $h=0.05$ is the smallest one and is quite inadequate in this setting. The adaptive Haar thresholding rule ${\stackrel{\u02c6}{\lambda}}_{n}^{\mathrm{Th}}$ is much better in this case. The WN method is able to reconstruct perfectly the flat line. For (SInPoi), the intensity has large jumps and smooth bumps (Figs. 5b and 5c). For such an irregular intensity and for a small number of trials (Fig. 5b), the thresholding estimate and the WN method are both able to recover the jumps perfectly but the smooth bumps are estimated by a piecewise constant function. The Gaussian kernels are better for the estimation of the bumps, but of course, they cannot detect the jumps. In this respect, the GL bandwidth is the best, whereas ${\stackrel{\u02c6}{\lambda}}_{n}^{{K}_{{h}^{\ast}}}$ and ${\stackrel{\u02c6}{\lambda}}_{n}^{{K}_{h}}$ are too smooth. For a large number of trials (Fig. 5c), the thresholding estimate is a bit refined but clearly suffers from a lack of smoothness. Unlike ${\stackrel{\u02c6}{\lambda}}_{n}^{{K}_{{h}^{\ast}}}$ and ${\stackrel{\u02c6}{\lambda}}_{n}^{{K}_{h}}$, ${\stackrel{\u02c6}{\lambda}}_{n}^{\mathrm{GL}}$ is reconstructing all the three bumps. The WN method is reconstructing more accurately the jumps despite some important boundary artefacts. It also gives smoother reconstructions for the bumps. In conclusion, the GL method clearly gives a bandwidth choice that adapts to high irregularity of the intensity with respect to other choices, whereas the thresholding estimate, which leads to an adaptive histogram, is more spatially adaptive despite its lack of smoothness. Up to boundary effects, the WN methodology seems to be the most accurate, since it adapts to the regularity of the underlying intensity. Note, however, that on an interval with a few number of points, this method provides a piecewise constant reconstruction, even if the underlying intensity is smooth, because this choice is more robust. This conclusion is also coherent with two previous and more extensive studies (see [43, 44]).
Now let us apply the different proposed tests. First, Test 2 has been applied on the simulated data sets with $n=40$ and $p=\lfloor {n}^{2/3}\rfloor $ and nothing was declared significant. However, the pvalues corresponding to (SHaw) are abnormally large. To take this into account, we have performed also a variant of Test 2 where one rejects if the same test statistic is now smaller than ${k}_{n,\alpha}$. This test consists therefore in rejecting the Poisson hypothesis when both estimated distributions are too close. As Test 2, this test is also asymptotically of level α, by application of Proposition 2. Actually, all the tests presented here can be said to reject “by upper values” and have therefore a version “by lower values.” On each set (simulated or not) of Test 2 pvalues (by upper and lower values), one can perform a Benjamini and Hochberg procedure with FDR 5 % [64]. It declares both processes in (SHaw) as nonPoissonian in the family of simulated data (pvalues in $[{10}^{3},{10}^{2})$). Due to the high variability of the reconstructions on Data Sets A and B, which depend on the considered direction, it was not possible to pool the data together and, therefore, the corresponding tests have been performed direction by direction. However, Tests 2 by upper or lower values do not detect anything on Data Sets A and B.
pvalues of Test 3 on Data Sets A and B, with a subsample size $\lfloor {n}^{2/3}\rfloor $, where n is the number of trials. Same codes as Table 2 for the pvalues of the test by upper values. For the pvalues of the test by lower values, same codes except that ∘ becomes □ and ▵ becomes ▽
Directions  

1  2  3  4  5  6  
N1A  Th  □  
GL  
WN  
N2A  Th  
GL  
WN  
N1B  Th  ▵  ∘  
GL  ▲▲  ▲▲▲  ▲▲▲  ▲▲  ▲  
WN  ▵  
N2B  Th  ▵  ∘  □  
GL  ∘  ∘  ∘  ∘  
WN  ∘ 
4.2.3 Checking the Hawkes Assumption
5 Conclusion
When using the timerescaling theorem to assess whether an observed spike train obeys a certain probabilistic model (e.g., Poisson, Hawkes, etc.), a plugin step is currently performed [1–3, 8, 24]. If this plugin step is done without care the resulting test may be much too conservative leading to poor detections (see Fig. 1). We propose here to use the subsampling as an almost universal solution when dealing with Kolmogorov–Smirnov tests of uniformity, such a universal solution being completely new. The main requirement is to have access to an estimate of the underlying intensity, whose rate of convergence is known (see, for instance, (5)).
In classical previous works such as [8, 23], parametric estimates such as MLE over a prescribed parametric model are used as plugin estimates, which are converging toward the true intensity when the underlying parametric model is true. Therefore, when this parametric plugin is used inside omnibus tests (and in particular the ones presented here), one has to be careful that we are not only testing the probabilistic assumption, but also the fact that the intensity belongs to the parametric model. To overcome this problem, we advertise for the use of nonparametric estimates and more precisely to adaptive estimates, for which rates of convergence are known under lighter assumptions on the intensity than a prescribed parametric assumption. Moreover, those adaptive methods have very good practical performance making them also very good in practice for estimation.
On some simulated data and on some real data sets, we have shown that our method performs very well. However, there are still two main directions in which our work need to be pursued in order to provide a more complete answer on real data sets. First of all, the KS test of exponentiality on the ISI can also be performed instead of the KS test of uniformity [1]. If the method with subsampling can be adapted to this case, we have presently no guarantee that this test would have a controlled level. Indeed we have no equivalent of Proposition 2 or Theorem 1 for the ISI repartition. The second main drawback is that we are clearly able to reject (at least on the presented real data sets) both homogeneous and inhomogeneous Poisson assumptions. We are also able to test whether the processes are Hawkes or not, which in particular takes into account refractory periods and dependence between several spike trains. However, the Hawkes model reflects stationary features, and cannot model nonstationary data. Therefore, we would need a more general model, which includes the dependence as in the Hawkes model presented here, but also the non stationarity as in the inhomogeneous Poisson process. At the present moment, models reflecting both are not compatible with a full agnostic approach where no assumption is made on the underlying functions, the estimation problem being not completely identifiable [16]. A first step will be therefore to provide a trade off between estimation capacity and not too restrictive assumptions on the process itself, with respect to real spike train data.
Electronic Supplementary Material
If ${({a}_{n})}_{n\ge 0}$ and ${({b}_{n})}_{n\ge 0}$ are two sequences, $a(n)\asymp b(n)$ means that there exists two positive constants ${c}_{1}$, ${c}_{2}$ such that for all $n\ge 0$, ${c}_{1}{a}_{n}\le {b}_{n}\le {c}_{2}{a}_{n}$.
Declarations
Acknowledgements
We are especially thankful to Alexa Riehle, leader of the Laboratory in which the data used in this article were previously collected. The authors also wish to thank F. Picard for fruitful discussions during several steps of this work. This research is partly supported by the French Agence Nationale de la Recherche (ANR 2011 BS01 010 01 projet Calibration) and by the PEPS BMI 20122013 Estimation of dependence graphs for thalamocortical neurons and multivariate Hawkes processes.
Authors’ Affiliations
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