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Privault N, Thieullen M. Closed-form modeling of neuronal spike train statistics using multivariate Hawkes cumulants. Phys Rev E 2022; 106:054410. [PMID: 36559454 DOI: 10.1103/physreve.106.054410] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/14/2022] [Accepted: 10/27/2022] [Indexed: 06/17/2023]
Abstract
We derive exact analytical expressions for the cumulants of any orders of neuronal membrane potentials driven by spike trains in a multivariate Hawkes process model with excitation and inhibition. Such expressions can be used for the prediction and sensitivity analysis of the statistical behavior of the model over time and to estimate the probability densities of neuronal membrane potentials using Gram-Charlier expansions. Our results are shown to provide a better alternative to Monte Carlo estimates via stochastic simulations and computer codes based on combinatorial recursions are included.
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Affiliation(s)
- Nicolas Privault
- Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore and LPSM - UMR 8001, Sorbonne Université, 4 Place Jussieu, 75252 Paris, France
| | - Michèle Thieullen
- Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore and LPSM - UMR 8001, Sorbonne Université, 4 Place Jussieu, 75252 Paris, France
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2
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Miehl C, Onasch S, Festa D, Gjorgjieva J. Formation and computational implications of assemblies in neural circuits. J Physiol 2022. [PMID: 36068723 DOI: 10.1113/jp282750] [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: 05/20/2022] [Accepted: 08/22/2022] [Indexed: 11/08/2022] Open
Abstract
In the brain, patterns of neural activity represent sensory information and store it in non-random synaptic connectivity. A prominent theoretical hypothesis states that assemblies, groups of neurons that are strongly connected to each other, are the key computational units underlying perception and memory formation. Compatible with these hypothesised assemblies, experiments have revealed groups of neurons that display synchronous activity, either spontaneously or upon stimulus presentation, and exhibit behavioural relevance. While it remains unclear how assemblies form in the brain, theoretical work has vastly contributed to the understanding of various interacting mechanisms in this process. Here, we review the recent theoretical literature on assembly formation by categorising the involved mechanisms into four components: synaptic plasticity, symmetry breaking, competition and stability. We highlight different approaches and assumptions behind assembly formation and discuss recent ideas of assemblies as the key computational unit in the brain. Abstract figure legend Assembly Formation. Assemblies are groups of strongly connected neurons formed by the interaction of multiple mechanisms and with vast computational implications. Four interacting components are thought to drive assembly formation: synaptic plasticity, symmetry breaking, competition and stability. This article is protected by copyright. All rights reserved.
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Affiliation(s)
- Christoph Miehl
- Computation in Neural Circuits, Max Planck Institute for Brain Research, 60438, Frankfurt, Germany.,School of Life Sciences, Technical University of Munich, 85354, Freising, Germany
| | - Sebastian Onasch
- Computation in Neural Circuits, Max Planck Institute for Brain Research, 60438, Frankfurt, Germany.,School of Life Sciences, Technical University of Munich, 85354, Freising, Germany
| | - Dylan Festa
- Computation in Neural Circuits, Max Planck Institute for Brain Research, 60438, Frankfurt, Germany.,School of Life Sciences, Technical University of Munich, 85354, Freising, Germany
| | - Julijana Gjorgjieva
- Computation in Neural Circuits, Max Planck Institute for Brain Research, 60438, Frankfurt, Germany.,School of Life Sciences, Technical University of Munich, 85354, Freising, Germany
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3
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Mean-field limits for non-linear Hawkes processes with excitation and inhibition. Stoch Process Their Appl 2022. [DOI: 10.1016/j.spa.2022.07.006] [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: 11/23/2022]
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4
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5
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Fast multi-resolution segmentation for nonstationary Hawkes process using cumulants. INTERNATIONAL JOURNAL OF DATA SCIENCE AND ANALYTICS 2020. [DOI: 10.1007/s41060-020-00223-3] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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6
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Montangie L, Miehl C, Gjorgjieva J. Autonomous emergence of connectivity assemblies via spike triplet interactions. PLoS Comput Biol 2020; 16:e1007835. [PMID: 32384081 PMCID: PMC7239496 DOI: 10.1371/journal.pcbi.1007835] [Citation(s) in RCA: 19] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/20/2019] [Revised: 05/20/2020] [Accepted: 03/31/2020] [Indexed: 01/08/2023] Open
Abstract
Non-random connectivity can emerge without structured external input driven by activity-dependent mechanisms of synaptic plasticity based on precise spiking patterns. Here we analyze the emergence of global structures in recurrent networks based on a triplet model of spike timing dependent plasticity (STDP), which depends on the interactions of three precisely-timed spikes, and can describe plasticity experiments with varying spike frequency better than the classical pair-based STDP rule. We derive synaptic changes arising from correlations up to third-order and describe them as the sum of structural motifs, which determine how any spike in the network influences a given synaptic connection through possible connectivity paths. This motif expansion framework reveals novel structural motifs under the triplet STDP rule, which support the formation of bidirectional connections and ultimately the spontaneous emergence of global network structure in the form of self-connected groups of neurons, or assemblies. We propose that under triplet STDP assembly structure can emerge without the need for externally patterned inputs or assuming a symmetric pair-based STDP rule common in previous studies. The emergence of non-random network structure under triplet STDP occurs through internally-generated higher-order correlations, which are ubiquitous in natural stimuli and neuronal spiking activity, and important for coding. We further demonstrate how neuromodulatory mechanisms that modulate the shape of the triplet STDP rule or the synaptic transmission function differentially promote structural motifs underlying the emergence of assemblies, and quantify the differences using graph theoretic measures. Emergent non-random connectivity structures in different brain regions are tightly related to specific patterns of neural activity and support diverse brain functions. For instance, self-connected groups of neurons, known as assemblies, have been proposed to represent functional units in brain circuits and can emerge even without patterned external instruction. Here we investigate the emergence of non-random connectivity in recurrent networks using a particular plasticity rule, triplet STDP, which relies on the interaction of spike triplets and can capture higher-order statistical dependencies in neural activity. We derive the evolution of the synaptic strengths in the network and explore the conditions for the self-organization of connectivity into assemblies. We demonstrate key differences of the triplet STDP rule compared to the classical pair-based rule in terms of how assemblies are formed, including the realistic asymmetric shape and influence of novel connectivity motifs on network plasticity driven by higher-order correlations. Assembly formation depends on the specific shape of the STDP window and synaptic transmission function, pointing towards an important role of neuromodulatory signals on formation of intrinsically generated assemblies.
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Affiliation(s)
- Lisandro Montangie
- Computation in Neural Circuits Group, Max Planck Institute for Brain Research, Frankfurt, Germany
| | - Christoph Miehl
- Computation in Neural Circuits Group, Max Planck Institute for Brain Research, Frankfurt, Germany
- Technical University of Munich, School of Life Sciences, Freising, Germany
| | - Julijana Gjorgjieva
- Computation in Neural Circuits Group, Max Planck Institute for Brain Research, Frankfurt, Germany
- Technical University of Munich, School of Life Sciences, Freising, Germany
- * E-mail:
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7
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Kass RE, Amari SI, Arai K, Brown EN, Diekman CO, Diesmann M, Doiron B, Eden UT, Fairhall AL, Fiddyment GM, Fukai T, Grün S, Harrison MT, Helias M, Nakahara H, Teramae JN, Thomas PJ, Reimers M, Rodu J, Rotstein HG, Shea-Brown E, Shimazaki H, Shinomoto S, Yu BM, Kramer MA. Computational Neuroscience: Mathematical and Statistical Perspectives. ANNUAL REVIEW OF STATISTICS AND ITS APPLICATION 2018; 5:183-214. [PMID: 30976604 PMCID: PMC6454918 DOI: 10.1146/annurev-statistics-041715-033733] [Citation(s) in RCA: 22] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/06/2023]
Abstract
Mathematical and statistical models have played important roles in neuroscience, especially by describing the electrical activity of neurons recorded individually, or collectively across large networks. As the field moves forward rapidly, new challenges are emerging. For maximal effectiveness, those working to advance computational neuroscience will need to appreciate and exploit the complementary strengths of mechanistic theory and the statistical paradigm.
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Affiliation(s)
- Robert E Kass
- Carnegie Mellon University, Pittsburgh, PA, USA, 15213;
| | - Shun-Ichi Amari
- RIKEN Brain Science Institute, Wako, Saitama Prefecture, Japan, 351-0198
| | | | - Emery N Brown
- Massachusetts Institute of Technology, Cambridge, MA, USA, 02139
- Harvard Medical School, Boston, MA, USA, 02115
| | | | - Markus Diesmann
- Jülich Research Centre, Jülich, Germany, 52428
- RWTH Aachen University, Aachen, Germany, 52062
| | - Brent Doiron
- University of Pittsburgh, Pittsburgh, PA, USA, 15260
| | - Uri T Eden
- Boston University, Boston, MA, USA, 02215
| | | | | | - Tomoki Fukai
- RIKEN Brain Science Institute, Wako, Saitama Prefecture, Japan, 351-0198
| | - Sonja Grün
- Jülich Research Centre, Jülich, Germany, 52428
- RWTH Aachen University, Aachen, Germany, 52062
| | | | - Moritz Helias
- Jülich Research Centre, Jülich, Germany, 52428
- RWTH Aachen University, Aachen, Germany, 52062
| | - Hiroyuki Nakahara
- RIKEN Brain Science Institute, Wako, Saitama Prefecture, Japan, 351-0198
| | | | - Peter J Thomas
- Case Western Reserve University, Cleveland, OH, USA, 44106
| | - Mark Reimers
- Michigan State University, East Lansing, MI, USA, 48824
| | - Jordan Rodu
- Carnegie Mellon University, Pittsburgh, PA, USA, 15213;
| | | | | | - Hideaki Shimazaki
- Honda Research Institute Japan, Wako, Saitama Prefecture, Japan, 351-0188
- Kyoto University, Kyoto, Kyoto Prefecture, Japan, 606-8502
| | | | - Byron M Yu
- Carnegie Mellon University, Pittsburgh, PA, USA, 15213;
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Abstract
We expand the theory of Hawkes processes to the nonstationary case, in which the mutually exciting point processes receive time-dependent inputs. We derive an analytical expression for the time-dependent correlations, which can be applied to networks with arbitrary connectivity, and inputs with arbitrary statistics. The expression shows how the network correlations are determined by the interplay between the network topology, the transfer functions relating units within the network, and the pattern and statistics of the external inputs. We illustrate the correlation structure using several examples in which neural network dynamics are modeled as a Hawkes process. In particular, we focus on the interplay between internally and externally generated oscillations and their signatures in the spike and rate correlation functions.
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Affiliation(s)
- Neta Ravid Tannenbaum
- Edmond and Lily Safra Center for Brain Sciences, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel
| | - Yoram Burak
- Edmond and Lily Safra Center for Brain Sciences, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel and Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel
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Rostami V, Porta Mana P, Grün S, Helias M. Bistability, non-ergodicity, and inhibition in pairwise maximum-entropy models. PLoS Comput Biol 2017; 13:e1005762. [PMID: 28968396 PMCID: PMC5645158 DOI: 10.1371/journal.pcbi.1005762] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/06/2016] [Revised: 10/17/2017] [Accepted: 09/05/2017] [Indexed: 11/30/2022] Open
Abstract
Pairwise maximum-entropy models have been used in neuroscience to predict the activity of neuronal populations, given only the time-averaged correlations of the neuron activities. This paper provides evidence that the pairwise model, applied to experimental recordings, would produce a bimodal distribution for the population-averaged activity, and for some population sizes the second mode would peak at high activities, that experimentally would be equivalent to 90% of the neuron population active within time-windows of few milliseconds. Several problems are connected with this bimodality: 1. The presence of the high-activity mode is unrealistic in view of observed neuronal activity and on neurobiological grounds. 2. Boltzmann learning becomes non-ergodic, hence the pairwise maximum-entropy distribution cannot be found: in fact, Boltzmann learning would produce an incorrect distribution; similarly, common variants of mean-field approximations also produce an incorrect distribution. 3. The Glauber dynamics associated with the model is unrealistically bistable and cannot be used to generate realistic surrogate data. This bimodality problem is first demonstrated for an experimental dataset from 159 neurons in the motor cortex of macaque monkey. Evidence is then provided that this problem affects typical neural recordings of population sizes of a couple of hundreds or more neurons. The cause of the bimodality problem is identified as the inability of standard maximum-entropy distributions with a uniform reference measure to model neuronal inhibition. To eliminate this problem a modified maximum-entropy model is presented, which reflects a basic effect of inhibition in the form of a simple but non-uniform reference measure. This model does not lead to unrealistic bimodalities, can be found with Boltzmann learning, and has an associated Glauber dynamics which incorporates a minimal asymmetric inhibition. Networks of interacting units are ubiquitous in various fields of biology; e.g. gene regulatory networks, neuronal networks, social structures. If a limited set of observables is accessible, maximum-entropy models provide a way to construct a statistical model for such networks, under particular assumptions. The pairwise maximum-entropy model only uses the first two moments among those observables, and can be interpreted as a network with only pairwise interactions. If correlations are on average positive, we here show that the maximum entropy distribution tends to become bimodal. In the application to neuronal activity this is a problem, because the bimodality is an artefact of the statistical model and not observed in real data. This problem could also affect other fields in biology. We here explain under which conditions bimodality arises and present a solution to the problem by introducing a collective negative feedback, corresponding to a modified maximum-entropy model. This result may point to the existence of a homeostatic mechanism active in the system that is not part of our set of observable units.
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Affiliation(s)
- Vahid Rostami
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA BRAIN Institute I, Jülich Research Centre, Jülich, Germany
- * E-mail:
| | - PierGianLuca Porta Mana
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA BRAIN Institute I, Jülich Research Centre, Jülich, Germany
| | - Sonja Grün
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA BRAIN Institute I, Jülich Research Centre, Jülich, Germany
- Theoretical Systems Neurobiology, RWTH Aachen University, Aachen, Germany
| | - Moritz Helias
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA BRAIN Institute I, Jülich Research Centre, Jülich, Germany
- Department of Physics, Faculty 1, RWTH Aachen University, Aachen, Germany
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10
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Ocker GK, Hu Y, Buice MA, Doiron B, Josić K, Rosenbaum R, Shea-Brown E. From the statistics of connectivity to the statistics of spike times in neuronal networks. Curr Opin Neurobiol 2017; 46:109-119. [PMID: 28863386 DOI: 10.1016/j.conb.2017.07.011] [Citation(s) in RCA: 28] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/08/2017] [Revised: 07/21/2017] [Accepted: 07/27/2017] [Indexed: 10/19/2022]
Abstract
An essential step toward understanding neural circuits is linking their structure and their dynamics. In general, this relationship can be almost arbitrarily complex. Recent theoretical work has, however, begun to identify some broad principles underlying collective spiking activity in neural circuits. The first is that local features of network connectivity can be surprisingly effective in predicting global statistics of activity across a network. The second is that, for the important case of large networks with excitatory-inhibitory balance, correlated spiking persists or vanishes depending on the spatial scales of recurrent and feedforward connectivity. We close by showing how these ideas, together with plasticity rules, can help to close the loop between network structure and activity statistics.
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Affiliation(s)
| | - Yu Hu
- Center for Brain Science, Harvard University, United States
| | - Michael A Buice
- Allen Institute for Brain Science, United States; Department of Applied Mathematics, University of Washington, United States
| | - Brent Doiron
- Department of Mathematics, University of Pittsburgh, United States; Center for the Neural Basis of Cognition, Pittsburgh, United States
| | - Krešimir Josić
- Department of Mathematics, University of Houston, United States; Department of Biology and Biochemistry, University of Houston, United States; Department of BioSciences, Rice University, United States
| | - Robert Rosenbaum
- Department of Mathematics, University of Notre Dame, United States
| | - Eric Shea-Brown
- Allen Institute for Brain Science, United States; Department of Applied Mathematics, University of Washington, United States; Department of Physiology and Biophysics, and University of Washington Institute for Neuroengineering, United States.
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11
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Abstract
Recent experimental advances are producing an avalanche of data on both neural connectivity and neural activity. To take full advantage of these two emerging datasets we need a framework that links them, revealing how collective neural activity arises from the structure of neural connectivity and intrinsic neural dynamics. This problem of structure-driven activity has drawn major interest in computational neuroscience. Existing methods for relating activity and architecture in spiking networks rely on linearizing activity around a central operating point and thus fail to capture the nonlinear responses of individual neurons that are the hallmark of neural information processing. Here, we overcome this limitation and present a new relationship between connectivity and activity in networks of nonlinear spiking neurons by developing a diagrammatic fluctuation expansion based on statistical field theory. We explicitly show how recurrent network structure produces pairwise and higher-order correlated activity, and how nonlinearities impact the networks’ spiking activity. Our findings open new avenues to investigating how single-neuron nonlinearities—including those of different cell types—combine with connectivity to shape population activity and function. Neuronal networks, like many biological systems, exhibit variable activity. This activity is shaped by both the underlying biology of the component neurons and the structure of their interactions. How can we combine knowledge of these two things—that is, models of individual neurons and of their interactions—to predict the statistics of single- and multi-neuron activity? Current approaches rely on linearizing neural activity around a stationary state. In the face of neural nonlinearities, however, these linear methods can fail to predict spiking statistics and even fail to correctly predict whether activity is stable or pathological. Here, we show how to calculate any spike train cumulant in a broad class of models, while systematically accounting for nonlinear effects. We then study a fundamental effect of nonlinear input-rate transfer–coupling between different orders of spiking statistic–and how this depends on single-neuron and network properties.
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Affiliation(s)
- Gabriel Koch Ocker
- Allen Institute for Brain Science, Seattle, Washington, United States of America
| | - Krešimir Josić
- Department of Mathematics and Department of Biology and Biochemistry, University of Houston, Houston, Texas, United States of America
- Department of BioSciences, Rice University, Houston, Texas, United States of America
| | - Eric Shea-Brown
- Allen Institute for Brain Science, Seattle, Washington, United States of America
- Department of Applied Mathematics, University of Washington, Seattle, Washington, United States of America
- Department of Physiology and Biophysics, and UW Institute of Neuroengineering, University of Washington, Seattle, Washington, United States of America
| | - Michael A. Buice
- Allen Institute for Brain Science, Seattle, Washington, United States of America
- Department of Applied Mathematics, University of Washington, Seattle, Washington, United States of America
- * E-mail:
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12
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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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Sanlı C, Lambiotte R. Local Variation of Hashtag Spike Trains and Popularity in Twitter. PLoS One 2015; 10:e0131704. [PMID: 26161650 PMCID: PMC4498919 DOI: 10.1371/journal.pone.0131704] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2015] [Accepted: 06/04/2015] [Indexed: 11/19/2022] Open
Abstract
We draw a parallel between hashtag time series and neuron spike trains. In each case, the process presents complex dynamic patterns including temporal correlations, burstiness, and all other types of nonstationarity. We propose the adoption of the so-called local variation in order to uncover salient dynamical properties, while properly detrending for the time-dependent features of a signal. The methodology is tested on both real and randomized hashtag spike trains, and identifies that popular hashtags present regular and so less bursty behavior, suggesting its potential use for predicting online popularity in social media.
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Affiliation(s)
- Ceyda Sanlı
- CompleXity and Networks, naXys, Department of Mathematics, University of Namur, Namur, Belgium
- * E-mail:
| | - Renaud Lambiotte
- CompleXity and Networks, naXys, Department of Mathematics, University of Namur, Namur, Belgium
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