Detection of dependence patterns with delay

Chevallier, Julien; Laloë, Thomas

doi:10.1002/bimj.201400235

Cited by 2 publications

(3 citation statements)

References 43 publications

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“…strap procedures to more than two neurons. Indeed, delayed coincidence counts have already been introduced in this case in (Chevallier & Laloë, 2015) and similar bootstrap procedures have been developed for more than two real valued variables in the precursor work of (Romano, 1989). Figure 4: Schematic view of the three bootstrap procedures.…”

Section: Discussionmentioning

confidence: 99%

Surrogate Data Methods Based on a Shuffling of the Trials for Synchrony Detection: The Centering Issue

Albert¹,

Bouret²,

Fromont

et al. 2016

Neural Computation

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We investigate several distribution-free dependence detection procedures, all based on a shuffling of the trials, from a statistical point of view. The mathematical justification of such procedures lies in the bootstrap principle and its approximation properties. In particular, we show that such a shuffling has mainly to be done on centered quantities-that is, quantities with zero mean under independence-to construct correct p-values, meaning that the corresponding tests control their false positive (FP) rate. Thanks to this study, we introduce a method, named permutation UE, which consists of a multiple testing procedure based on permutation of experimental trials and delayed coincidence count. Each involved single test of this procedure achieves the prescribed level, so that the corresponding multiple testing procedure controls the false discovery rate (FDR), and this with as few assumptions as possible on the underneath distribution, except independence and identical distribution across trials. The mathematical meaning of this assumption is discussed, and it is in particular argued that it does not mean what is commonly referred in neuroscience to as cross-trials stationarity. Some simulations show, moreover, that permutation UE outperforms the trial-shuffling of Pipa and Grün ( 2003 ) and the MTGAUE method of Tuleau-Malot et al. ( 2014 ) in terms of single levels and FDR, for a comparable amount of false negatives. Application to real data is also provided.

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Section: Discussionmentioning

confidence: 99%

Surrogate Data Methods Based on a Shuffling of the Trials for Synchrony Detection: The Centering Issue

Albert¹,

Bouret²,

Fromont

et al. 2016

Neural Computation

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show abstract

“…These random functions are F 0 -measurable and they satisfy the first two lines of (10) thanks to the independence of the H ij 's and the N j − 's. Hence, one can consider the intensity given by (12) with such a choice of F to represent the contribution of the processes (N i − ) i=1,...,n to the dynamics after time 0. In this example, the F ij 's are obviously dependent from the N j − 's.…”

Section: Ii1 Parameters Of the Modelmentioning

confidence: 99%

“…where (S t− ) t≥0 is the predictable age process associated with N and δ is a parameter corresponding to the time length of the strict refractory period of a neuron. This sounds as an alternative to the strategy used in [12]. Therein, refractory periods are described by choosing, in the standard formulation of Hawkes processes, strongly negative selfinteraction functions at a very short range.…”

Section: Introductionmentioning

confidence: 99%

Mean-field limit of generalized Hawkes processes

Chevallier

2017

Stochastic Processes and their Applications

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101

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We generalize multivariate Hawkes processes mainly by including a dependence with respect to the age of the process, i.e. the delay since the last point.Within this class, we investigate the limit behaviour, when n goes to infinity, of a system of n mean-field interacting age-dependent Hawkes processes. We prove that such a system can be approximated by independent and identically distributed age dependent point processes interacting with their own mean intensity. This result generalizes the study performed by Delattre, Fournier and Hoffmann in [17].In continuity with [11], the second goal of this paper is to give a proper link between these generalized Hawkes processes as microscopic models of individual neurons and the age-structured system of partial differential equations introduced by Pakdaman, Perthame and Salort in [42] as macroscopic model of neurons.Keywords: Hawkes process, mean-field approximation, interacting particle systems, renewal equation, neural network Mathematical Subject Classification: 60G55, 60F05, 60G57, 92B20 I IntroductionIn the recent years, the self-exciting point process known as the Hawkes process [29] has been used in very diverse areas. First introduced to model earthquake replicas [32] or [41] (ETAS model), it has been used in criminology to model burglary [39], in genomic data analysis to model occurrences of genes [27,50], in social networks analysis to model viewing or popularity [3,14], as well as in finance [1,2]. We refer to [35] or [56] for more extensive reviews on applications of Hawkes processes. A univariate (nonlinear) Hawkes process is a point process N admitting a stochastic intensity of the formwhere Φ : R → R + is called the intensity function, h : R + → R is called the selfinteraction function (also called exciting function or kernel function in the literature) and N (dz) denotes the point measure associated with N . We refer to [53,54,55,57,58] for recent papers dealing with nonlinear Hawkes process. Such a form of the intensity is motivated by practical cases where all the previous points of the process may impact the rate of appearance of a new point. The influence of the past points is formulated in terms of the delay between those past occurrences and the present time, through the weight function h. In the natural framework where h is nonnegative and Φ increasing, this choice of interaction models an excitatory phenomenon: * Corresponding author: e-mail: julien.chevallier@unice.fr 1 each time the process has a jump, it excites itself in the sense that it increases its intensity and thus the probability of finding a new point. A classical case is the linear Hawkes process for which h is non-negative and Φ(x) = µ + x where µ is a positive constant called the spontaneous rate. Note however that Hawkes processes can also describe inhibitory phenomena. For example, the function h may take negative values, Φ being the positive part modulo the spontaneous rate µ, i.e. Φ(x) = max(0, µ + x).Multivariate Hawkes processes consist of multivariate point processes...

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Detection of dependence patterns with delay

Cited by 2 publications

References 43 publications

Surrogate Data Methods Based on a Shuffling of the Trials for Synchrony Detection: The Centering Issue

Surrogate Data Methods Based on a Shuffling of the Trials for Synchrony Detection: The Centering Issue

Mean-field limit of generalized Hawkes processes

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