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Huang L, Khabou M. Nonlinear Poisson autoregression and nonlinear Hawkes processes. Stoch Process Their Appl 2023. [DOI: 10.1016/j.spa.2023.03.015] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 04/04/2023]
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2
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Mascart C, Scarella G, Reynaud-Bouret P, Muzy A. Scalability of Large Neural Network Simulations via Activity Tracking With Time Asynchrony and Procedural Connectivity. Neural Comput 2022; 34:1915-1943. [PMID: 35896155 DOI: 10.1162/neco_a_01524] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/15/2021] [Accepted: 04/27/2022] [Indexed: 11/04/2022]
Abstract
We present a new algorithm to efficiently simulate random models of large neural networks satisfying the property of time asynchrony. The model parameters (average firing rate, number of neurons, synaptic connection probability, and postsynaptic duration) are of the order of magnitude of a small mammalian brain or of human brain areas. Through the use of activity tracking and procedural connectivity (dynamical regeneration of synapses), computational and memory complexities of this algorithm are proved to be theoretically linear with the number of neurons. These results are experimentally validated by sequential simulations of millions of neurons and billions of synapses running in a few minutes using a single thread of an equivalent desktop computer.
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Affiliation(s)
| | - Gilles Scarella
- Université Côte d'Azur, CNRS, I3S, France.,Université Côte d'Azur, CNRS, LJAD, 06103 Nice, France
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3
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Limit theorems for Hawkes processes including inhibition. Stoch Process Their Appl 2022. [DOI: 10.1016/j.spa.2022.04.002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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4
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Neuronal Network Inference and Membrane Potential Model using Multivariate Hawkes Processes. J Neurosci Methods 2022; 372:109550. [PMID: 35247493 DOI: 10.1016/j.jneumeth.2022.109550] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2021] [Revised: 02/22/2022] [Accepted: 02/26/2022] [Indexed: 11/21/2022]
Abstract
BACKGROUND In this work, we propose to catch the complexity of the membrane potential's dynamic of a motoneuron between its spikes, taking into account the spikes from other neurons around. Our approach relies on two types of data: extracellular recordings of multiple spikes trains and intracellular recordings of the membrane potential of a central neuron. NEW METHOD We provide a unified framework and a complete pipeline to analyze neuronal activity from data extraction to statistical inference. To the best of our knowledge, this is the first time that a Hawkes-diffusion model is investigated on such complex data. The first step of the proposed procedure is to select a subnetwork of neurons impacting the central neuron using a multivariate Hawkes process. Then we infer a jump-diffusion dynamic in which jumps are driven from a Hawkes process, the occurrences of which correspond to the spike trains of the aforementioned subset of neurons that interact with the central neuron. RESULTS From the Hawkes estimation step we recover a small connectivity graph which contains the central neuron, and we show that taking into account this information improves the inference of membrane potential through the proposed jump-diffusion model. A goodness of fit test is applied to validate the relevance of the Hawkes model in such context. COMPARISON WITH EXISTING METHODS We compare an empirical inference method and two sparse estimation procedures based on the Hawkes assumption for the reconstruction of the connectivity graph using the spike-trains. Then, the Hawkes-diffusion model is competed with the simple diffusion in terms of best fit to describe the behavior of the membrane potential of a central neuron surrounded by a network. CONCLUSIONS The present method takes advantage of both spike trains and membrane potential to understand the behavior of a fixed neuron. The entire code has been developed and is freely available on GitHub.
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Erny X, Löcherbach E, Loukianova D. Mean field limits for interacting Hawkes processes in a diffusive regime. BERNOULLI 2022. [DOI: 10.3150/21-bej1335] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
- Xavier Erny
- Université Paris-Saclay, CNRS, Univ Evry, Laboratoire de Mathématiques et Modélisation d’Evry, 91037, Evry, France
| | - Eva Löcherbach
- Statistique, Analyse et Modélisation Multidisciplinaire, Université Paris 1 Panthéon-Sorbonne, EA 4543 et FR FP2M 2036 CNRS, France
| | - Dasha Loukianova
- Université Paris-Saclay, CNRS, Univ Evry, Laboratoire de Mathématiques et Modélisation d’Evry, 91037, Evry, France
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Chagny G, Channarond A, Hoang VH, Roche A. Adaptive nonparametric estimation of a component density in a two-class mixture model. J Stat Plan Inference 2022. [DOI: 10.1016/j.jspi.2021.05.004] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
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Diffusion approximation of multi-class Hawkes processes: Theoretical and numerical analysis. ADV APPL PROBAB 2021. [DOI: 10.1017/apr.2020.73] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
AbstractOscillatory systems of interacting Hawkes processes with Erlang memory kernels were introduced by Ditlevsen and Löcherbach (Stoch. Process. Appl., 2017). They are piecewise deterministic Markov processes (PDMP) and can be approximated by a stochastic diffusion. In this paper, first, a strong error bound between the PDMP and the diffusion is proved. Second, moment bounds for the resulting diffusion are derived. Third, approximation schemes for the diffusion, based on the numerical splitting approach, are proposed. These schemes are proved to converge with mean-square order 1 and to preserve the properties of the diffusion, in particular the hypoellipticity, the ergodicity, and the moment bounds. Finally, the PDMP and the diffusion are compared through numerical experiments, where the PDMP is simulated with an adapted thinning procedure.
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Linear Response of General Observables in Spiking Neuronal Network Models. ENTROPY 2021; 23:e23020155. [PMID: 33514033 PMCID: PMC7911777 DOI: 10.3390/e23020155] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 12/10/2020] [Revised: 01/20/2021] [Accepted: 01/21/2021] [Indexed: 11/17/2022]
Abstract
We establish a general linear response relation for spiking neuronal networks, based on chains with unbounded memory. This relation allow us to predict the influence of a weak amplitude time dependent external stimuli on spatio-temporal spike correlations, from the spontaneous statistics (without stimulus) in a general context where the memory in spike dynamics can extend arbitrarily far in the past. Using this approach, we show how the linear response is explicitly related to the collective effect of the stimuli, intrinsic neuronal dynamics, and network connectivity on spike train statistics. We illustrate our results with numerical simulations performed over a discrete time integrate and fire model.
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Erny X. A convergence criterion for systems of point processes from the convergence of their stochastic intensities. ELECTRONIC COMMUNICATIONS IN PROBABILITY 2021. [DOI: 10.1214/21-ecp372] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
- Xavier Erny
- Université Paris-Saclay, CNRS, Univ Evry, Laboratoire de Mathématiques et Modélisation d’Evry, 91037, Evry, France
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Cofré R, Maldonado C, Cessac B. Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics. ENTROPY (BASEL, SWITZERLAND) 2020; 22:E1330. [PMID: 33266513 PMCID: PMC7712217 DOI: 10.3390/e22111330] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 10/09/2020] [Revised: 11/13/2020] [Accepted: 11/15/2020] [Indexed: 12/04/2022]
Abstract
The Thermodynamic Formalism provides a rigorous mathematical framework for studying quantitative and qualitative aspects of dynamical systems. At its core, there is a variational principle that corresponds, in its simplest form, to the Maximum Entropy principle. It is used as a statistical inference procedure to represent, by specific probability measures (Gibbs measures), the collective behaviour of complex systems. This framework has found applications in different domains of science. In particular, it has been fruitful and influential in neurosciences. In this article, we review how the Thermodynamic Formalism can be exploited in the field of theoretical neuroscience, as a conceptual and operational tool, in order to link the dynamics of interacting neurons and the statistics of action potentials from either experimental data or mathematical models. We comment on perspectives and open problems in theoretical neuroscience that could be addressed within this formalism.
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Affiliation(s)
- Rodrigo Cofré
- CIMFAV-Ingemat, Facultad de Ingeniería, Universidad de Valparaíso, Valparaíso 2340000, Chile
| | - Cesar Maldonado
- IPICYT/División de Matemáticas Aplicadas, San Luis Potosí 78216, Mexico;
| | - Bruno Cessac
- Inria Biovision team and Neuromod Institute, Université Côte d’Azur, 06901 CEDEX Inria, France;
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Donnet S, Rivoirard V, Rousseau J. Nonparametric Bayesian estimation for multivariate Hawkes processes. Ann Stat 2020. [DOI: 10.1214/19-aos1903] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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13
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Liu C. Statistical inference for a partially observed interacting system of Hawkes processes. Stoch Process Their Appl 2020. [DOI: 10.1016/j.spa.2020.04.003] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
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Raad MB, Ditlevsen S, Löcherbach E. Stability and mean-field limits of age dependent Hawkes processes. ANNALES DE L'INSTITUT HENRI POINCARÉ, PROBABILITÉS ET STATISTIQUES 2020. [DOI: 10.1214/19-aihp1023] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Lauw HW, Wong RCW, Ntoulas A, Lim EP, Ng SK, Pan SJ. BRUNCH: Branching Structure Inference of Hybrid Multivariate Hawkes Processes with Application to Social Media. ADVANCES IN KNOWLEDGE DISCOVERY AND DATA MINING 2020. [PMCID: PMC7206163 DOI: 10.1007/978-3-030-47426-3_43] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
Abstract
Multivariate Hawkes processes (MHPs) are a class of point processes where an arrival in one dimension can affect the future arrivals in all dimensions. Existing MHPs are associated with homogeneouslink functions. However, in reality, different dimensions may exhibit different temporal characteristics. In this paper, we augment MHPs by incorporating heterogeneouslink functions, referred to as hybrid MHPs, to capture the temporal characteristics in different dimensions. Since the branching structure can be utilized to equivalently represent MHPs, we propose a novel model called BRUNCH via intensity-driven Chinese Restaurant Processes (intCRP) to identify the optimal branching structure of hybrid MHPs. Furthermore, we relax the constraint on the shapes of triggering kernels in MHPs. We develop a Monte Carlo-based inference algorithm called MEDIA to infer the branching structure. Experiments on real-world datasets demonstrate the superior performance of BRUNCH and its usefulness in social media applications.
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Affiliation(s)
- Hady W. Lauw
- School of Information Systems, Singapore Management University, Singapore, Singapore
| | - Raymond Chi-Wing Wong
- Department of Computer Science and Engineering, Hong Kong University of Science and Technology, Hong Kong, Hong Kong
| | - Alexandros Ntoulas
- Department of Informatics and Telecommunications, National and Kapodistrian University of Athens, Athens, Greece
| | - Ee-Peng Lim
- School of Information Systems, Singapore Management University, Singapore, Singapore
| | - See-Kiong Ng
- Institute of Data Science, National University of Singapore, Singapore, Singapore
| | - Sinno Jialin Pan
- School of Computer Science and Engineering, Nanyang Technological University, Singapore, Singapore
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Cessac B. Linear response in neuronal networks: From neurons dynamics to collective response. CHAOS (WOODBURY, N.Y.) 2019; 29:103105. [PMID: 31675822 DOI: 10.1063/1.5111803] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2019] [Accepted: 09/11/2019] [Indexed: 06/10/2023]
Abstract
We review two examples where the linear response of a neuronal network submitted to an external stimulus can be derived explicitly, including network parameters dependence. This is done in a statistical physicslike approach where one associates, to the spontaneous dynamics of the model, a natural notion of Gibbs distribution inherited from ergodic theory or stochastic processes. These two examples are the Amari-Wilson-Cowan model [S. Amari, Syst. Man Cybernet. SMC-2, 643-657 (1972); H. R. Wilson and J. D. Cowan, Biophys. J. 12, 1-24 (1972)] and a conductance based Integrate and Fire model [M. Rudolph and A. Destexhe, Neural Comput. 18, 2146-2210 (2006); M. Rudolph and A. Destexhe, Neurocomputing 70(10-12), 1966-1969 (2007)].
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Affiliation(s)
- Bruno Cessac
- Université Côte d'Azur, Inria, Biovision team, Sophia-Antipolis, France
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18
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Chevallier J, Duarte A, Löcherbach E, Ost G. Mean field limits for nonlinear spatially extended Hawkes processes with exponential memory kernels. Stoch Process Their Appl 2019. [DOI: 10.1016/j.spa.2018.02.007] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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20
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León JR, Samson A. Hypoelliptic stochastic FitzHugh–Nagumo neuronal model: Mixing, up-crossing and estimation of the spike rate. ANN APPL PROBAB 2018. [DOI: 10.1214/17-aap1355] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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21
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Ćmiel B, Szkutnik Z, Wojdyła J. Asymptotic confidence bands in the Spektor-Lord-Willis problem via kernel estimation of intensity derivative. Electron J Stat 2018. [DOI: 10.1214/18-ejs1391] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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22
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23
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24
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Cadre B, Massiot G, Truquet L. Nonparametric tests for Cox processes. J Stat Plan Inference 2017. [DOI: 10.1016/j.jspi.2016.12.001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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Abstract
We consider a size-structured model describing a population of cells proliferating by division. Each cell contain a quantity of toxicity which grows linearly according to a constant growth rateα. At division, the cells divide at a constant rateRand share their content between the two daughter cells into fractionsΓand 1 −ΓwhereΓhas a symmetric densityhon [ 0,1 ], since the daughter cells are exchangeable. We describe the cell population by a random measure and observe the cells on the time interval [ 0,T] with fixedT. We address here the problem of estimating the division kernelh(or fragmentation kernel) when the division tree is completely observed. An adaptive estimator ofhis constructed based on a kernel functionKwith a fully data-driven bandwidth selection method. We obtain an oracle inequality and an exponential convergence rate, for which optimality is considered.
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Onaga T, Shinomoto S. Emergence of event cascades in inhomogeneous networks. Sci Rep 2016; 6:33321. [PMID: 27625183 PMCID: PMC5022041 DOI: 10.1038/srep33321] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/05/2016] [Accepted: 08/24/2016] [Indexed: 11/09/2022] Open
Abstract
There is a commonality among contagious diseases, tweets, and neuronal firings that past events facilitate the future occurrence of events. The spread of events has been extensively studied such that the systems exhibit catastrophic chain reactions if the interaction represented by the ratio of reproduction exceeds unity; however, their subthreshold states are not fully understood. Here, we report that these systems are possessed by nonstationary cascades of event-occurrences already in the subthreshold regime. Event cascades can be harmful in some contexts, when the peak-demand causes vaccine shortages, heavy traffic on communication lines, but may be beneficial in other contexts, such that spontaneous activity in neural networks may be used to generate motion or store memory. Thus it is important to comprehend the mechanism by which such cascades appear, and consider controlling a system to tame or facilitate fluctuations in the event-occurrences. The critical interaction for the emergence of cascades depends greatly on the network structure in which individuals are connected. We demonstrate that we can predict whether cascades may emerge, given information about the interactions between individuals. Furthermore, we develop a method of reallocating connections among individuals so that event cascades may be either impeded or impelled in a network.
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Affiliation(s)
- Tomokatsu Onaga
- Department of Physics, Kyoto University, Kyoto 606-8502, Japan
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Jovanović S, Rotter S. Interplay between Graph Topology and Correlations of Third Order in Spiking Neuronal Networks. PLoS Comput Biol 2016; 12:e1004963. [PMID: 27271768 PMCID: PMC4894630 DOI: 10.1371/journal.pcbi.1004963] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/26/2016] [Accepted: 05/02/2016] [Indexed: 01/06/2023] Open
Abstract
The study of processes evolving on networks has recently become a very popular research field, not only because of the rich mathematical theory that underpins it, but also because of its many possible applications, a number of them in the field of biology. Indeed, molecular signaling pathways, gene regulation, predator-prey interactions and the communication between neurons in the brain can be seen as examples of networks with complex dynamics. The properties of such dynamics depend largely on the topology of the underlying network graph. In this work, we want to answer the following question: Knowing network connectivity, what can be said about the level of third-order correlations that will characterize the network dynamics? We consider a linear point process as a model for pulse-coded, or spiking activity in a neuronal network. Using recent results from theory of such processes, we study third-order correlations between spike trains in such a system and explain which features of the network graph (i.e. which topological motifs) are responsible for their emergence. Comparing two different models of network topology—random networks of Erdős-Rényi type and networks with highly interconnected hubs—we find that, in random networks, the average measure of third-order correlations does not depend on the local connectivity properties, but rather on global parameters, such as the connection probability. This, however, ceases to be the case in networks with a geometric out-degree distribution, where topological specificities have a strong impact on average correlations. Many biological phenomena can be viewed as dynamical processes on a graph. Understanding coordinated activity of nodes in such a network is of some importance, as it helps to characterize the behavior of the complex system. Of course, the topology of a network plays a pivotal role in determining the level of coordination among its different vertices. In particular, correlations between triplets of events (here: action potentials generated by neurons) have recently garnered some interest in the theoretical neuroscience community. In this paper, we present a decomposition of an average measure of third-order coordinated activity of neurons in a spiking neuronal network in terms of the relevant topological motifs present in the underlying graph. We study different network topologies and show, in particular, that the presence of certain tree motifs in the synaptic connectivity graph greatly affects the strength of third-order correlations between spike trains of different neurons.
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Affiliation(s)
- Stojan Jovanović
- Bernstein Center Freiburg & Faculty of Biology, University of Freiburg, Freiburg, Germany
- CB, CSC, KTH Royal Institute of Technology, Stockholm, Sweden
- * E-mail:
| | - Stefan Rotter
- Bernstein Center Freiburg & Faculty of Biology, University of Freiburg, Freiburg, Germany
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Delattre S, Fournier N. Statistical inference versus mean field limit for Hawkes processes. Electron J Stat 2016. [DOI: 10.1214/16-ejs1142] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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31
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Albert M, Bouret Y, Fromont M, Reynaud-Bouret P. Bootstrap and permutation tests of independence for point processes. Ann Stat 2015. [DOI: 10.1214/15-aos1351] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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32
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Chevallier J, Laloë T. Detection of dependence patterns with delay. Biom J 2015; 57:1110-30. [DOI: 10.1002/bimj.201400235] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/03/2014] [Revised: 06/17/2015] [Accepted: 07/09/2015] [Indexed: 11/12/2022]
Affiliation(s)
- Julien Chevallier
- Laboratoire de Mathématiques J.A. Dieudonné; UMR 7351 CNRS, Université de Nice Sophia Antipolis; 06108 Nice Cedex 02 France
| | - Thomas Laloë
- Laboratoire de Mathématiques J.A. Dieudonné; UMR 7351 CNRS, Université de Nice Sophia Antipolis; 06108 Nice Cedex 02 France
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Mastromatteo I, Bacry E, Muzy JF. Linear processes in high dimensions: Phase space and critical properties. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:042142. [PMID: 25974473 DOI: 10.1103/physreve.91.042142] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/18/2014] [Indexed: 06/04/2023]
Abstract
In this work we investigate the generic properties of a stochastic linear model in the regime of high dimensionality. We consider in particular the vector autoregressive (VAR) model and the multivariate Hawkes process. We analyze both deterministic and random versions of these models, showing the existence of a stable phase and an unstable phase. We find that along the transition region separating the two regimes the correlations of the process decay slowly, and we characterize the conditions under which these slow correlations are expected to become power laws. We check our findings with numerical simulations showing remarkable agreement with our predictions. We finally argue that real systems with a strong degree of self-interaction are naturally characterized by this type of slow relaxation of the correlations.
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Affiliation(s)
- Iacopo Mastromatteo
- Centre de Mathématiques Appliquées, CNRS, École Polytechnique, UMR 7641, 91128 Palaiseau, France
| | - Emmanuel Bacry
- Centre de Mathématiques Appliquées, CNRS, École Polytechnique, UMR 7641, 91128 Palaiseau, France
| | - Jean-François Muzy
- Laboratoire Sciences Pour l'Environnement, CNRS, Université de Corse, UMR 6134, 20250 Corté, France
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