1
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Bardella G, Franchini S, Pan L, Balzan R, Ramawat S, Brunamonti E, Pani P, Ferraina S. Neural Activity in Quarks Language: Lattice Field Theory for a Network of Real Neurons. ENTROPY (BASEL, SWITZERLAND) 2024; 26:495. [PMID: 38920504 PMCID: PMC11203154 DOI: 10.3390/e26060495] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/25/2024] [Revised: 05/28/2024] [Accepted: 05/30/2024] [Indexed: 06/27/2024]
Abstract
Brain-computer interfaces have seen extraordinary surges in developments in recent years, and a significant discrepancy now exists between the abundance of available data and the limited headway made in achieving a unified theoretical framework. This discrepancy becomes particularly pronounced when examining the collective neural activity at the micro and meso scale, where a coherent formalization that adequately describes neural interactions is still lacking. Here, we introduce a mathematical framework to analyze systems of natural neurons and interpret the related empirical observations in terms of lattice field theory, an established paradigm from theoretical particle physics and statistical mechanics. Our methods are tailored to interpret data from chronic neural interfaces, especially spike rasters from measurements of single neuron activity, and generalize the maximum entropy model for neural networks so that the time evolution of the system is also taken into account. This is obtained by bridging particle physics and neuroscience, paving the way for particle physics-inspired models of the neocortex.
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Affiliation(s)
- Giampiero Bardella
- Department of Physiology and Pharmacology, Sapienza University of Rome, Piazzale Aldo Moro 5, 00185 Roma, Italy (E.B.); (P.P.); (S.F.)
| | - Simone Franchini
- Department of Physiology and Pharmacology, Sapienza University of Rome, Piazzale Aldo Moro 5, 00185 Roma, Italy (E.B.); (P.P.); (S.F.)
| | - Liming Pan
- School of Cyber Science and Technology, University of Science and Technology of China, Hefei 230026, China;
| | - Riccardo Balzan
- Laboratoire de Chimie et Biochimie Pharmacologiques et Toxicologiques, UMR 8601, UFR Biomédicale et des Sciences de Base, Université Paris Descartes-CNRS, PRES Paris Sorbonne Cité, 75006 Paris, France;
| | - Surabhi Ramawat
- Department of Physiology and Pharmacology, Sapienza University of Rome, Piazzale Aldo Moro 5, 00185 Roma, Italy (E.B.); (P.P.); (S.F.)
| | - Emiliano Brunamonti
- Department of Physiology and Pharmacology, Sapienza University of Rome, Piazzale Aldo Moro 5, 00185 Roma, Italy (E.B.); (P.P.); (S.F.)
| | - Pierpaolo Pani
- Department of Physiology and Pharmacology, Sapienza University of Rome, Piazzale Aldo Moro 5, 00185 Roma, Italy (E.B.); (P.P.); (S.F.)
| | - Stefano Ferraina
- Department of Physiology and Pharmacology, Sapienza University of Rome, Piazzale Aldo Moro 5, 00185 Roma, Italy (E.B.); (P.P.); (S.F.)
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2
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Hutt A, Trotter D, Pariz A, Valiante TA, Lefebvre J. Diversity-induced trivialization and resilience of neural dynamics. CHAOS (WOODBURY, N.Y.) 2024; 34:013147. [PMID: 38285722 DOI: 10.1063/5.0165773] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2023] [Accepted: 01/01/2024] [Indexed: 01/31/2024]
Abstract
Heterogeneity is omnipresent across all living systems. Diversity enriches the dynamical repertoire of these systems but remains challenging to reconcile with their manifest robustness and dynamical persistence over time, a fundamental feature called resilience. To better understand the mechanism underlying resilience in neural circuits, we considered a nonlinear network model, extracting the relationship between excitability heterogeneity and resilience. To measure resilience, we quantified the number of stationary states of this network, and how they are affected by various control parameters. We analyzed both analytically and numerically gradient and non-gradient systems modeled as non-linear sparse neural networks evolving over long time scales. Our analysis shows that neuronal heterogeneity quenches the number of stationary states while decreasing the susceptibility to bifurcations: a phenomenon known as trivialization. Heterogeneity was found to implement a homeostatic control mechanism enhancing network resilience to changes in network size and connection probability by quenching the system's dynamic volatility.
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Affiliation(s)
- Axel Hutt
- MLMS, MIMESIS, Université de Strasbourg, CNRS, Inria, ICube, 67000 Strasbourg, France
| | - Daniel Trotter
- Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
- Krembil Brain Institute, University Health Network, Toronto, Ontario M5T 0S8, Canada
| | - Aref Pariz
- Krembil Brain Institute, University Health Network, Toronto, Ontario M5T 0S8, Canada
- Department of Biology, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
| | - Taufik A Valiante
- Krembil Brain Institute, University Health Network, Toronto, Ontario M5T 0S8, Canada
- Department of Electrical and Computer Engineering, Institute of Medical Science, Institute of Biomedical Engineering, Division of Neurosurgery, Department of Surgery, CRANIA (Center for Advancing Neurotechnological Innovation to Application), Max Planck-University of Toronto Center for Neural Science and Technology, University of Toronto, Toronto, Ontario M5S 3G8, Canada
| | - Jérémie Lefebvre
- Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
- Krembil Brain Institute, University Health Network, Toronto, Ontario M5T 0S8, Canada
- Department of Biology, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
- Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada
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3
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Liu C, Wu ZX, Wang CY, Yang HX, Guan JY. Double resonance induced by group coupling with quenched disorder. CHAOS (WOODBURY, N.Y.) 2023; 33:013114. [PMID: 36725631 DOI: 10.1063/5.0132107] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/26/2022] [Accepted: 12/13/2022] [Indexed: 06/18/2023]
Abstract
Results show that the astrocytes can not only listen to the talk of large assemble of neurons but also give advice to the conversations and are significant sources of heterogeneous couplings as well. In the present work, we focus on such regulation character of astrocytes and explore the role of heterogeneous couplings among interacted neuron-astrocyte components in a signal response. We consider reduced dynamics in which the listening and advising processes of astrocytes are mapped into the form of group coupling, where the couplings are normally distributed. In both globally coupled overdamped bistable oscillators and an excitable FitzHugh-Nagumo (FHN) neuron model, we numerically and analytically demonstrate that two types of bell-shaped collective response curves can be obtained as the ensemble coupling strength or the heterogeneity of group coupling rise, respectively, which can be seen as a new type of double resonance. Furthermore, through the bifurcation analysis, we verify that these resonant signal responses stem from the competition between dispersion and aggregation induced by heterogeneous group and positive pairwise couplings, respectively. Our results contribute to a better understanding of the signal propagation in coupled systems with quenched disorder.
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Affiliation(s)
- Cong Liu
- Lanzhou Center for Theoretical Physics and Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Zhi-Xi Wu
- Lanzhou Center for Theoretical Physics and Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Chong-Yang Wang
- Institute of Computational Physics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Han-Xin Yang
- Department of Physics, Fuzhou University, Fuzhou 350108, People's Republic of China
| | - Jian-Yue Guan
- Lanzhou Center for Theoretical Physics and Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
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4
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Peng X, Lin W. Complex Dynamics of Noise-Perturbed Excitatory-Inhibitory Neural Networks With Intra-Correlative and Inter-Independent Connections. Front Physiol 2022; 13:915511. [PMID: 35812336 PMCID: PMC9263264 DOI: 10.3389/fphys.2022.915511] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/08/2022] [Accepted: 05/09/2022] [Indexed: 11/24/2022] Open
Abstract
Real neural system usually contains two types of neurons, i.e., excitatory neurons and inhibitory ones. Analytical and numerical interpretation of dynamics induced by different types of interactions among the neurons of two types is beneficial to understanding those physiological functions of the brain. Here, we articulate a model of noise-perturbed random neural networks containing both excitatory and inhibitory (E&I) populations. Particularly, both intra-correlatively and inter-independently connected neurons in two populations are taken into account, which is different from the most existing E&I models only considering the independently-connected neurons. By employing the typical mean-field theory, we obtain an equivalent system of two dimensions with an input of stationary Gaussian process. Investigating the stationary autocorrelation functions along the obtained system, we analytically find the parameters’ conditions under which the synchronized behaviors between the two populations are sufficiently emergent. Taking the maximal Lyapunov exponent as an index, we also find different critical values of the coupling strength coefficients for the chaotic excitatory neurons and for the chaotic inhibitory ones. Interestingly, we reveal that the noise is able to suppress chaotic dynamics of the random neural networks having neurons in two populations, while an appropriate amount of correlation coefficient in intra-coupling strengths can enhance chaos occurrence. Finally, we also detect a previously-reported phenomenon where the parameters region corresponds to neither linearly stable nor chaotic dynamics; however, the size of the region area crucially depends on the populations’ parameters.
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Affiliation(s)
- Xiaoxiao Peng
- Shanghai Center for Mathematical Sciences, School of Mathematical Sciences, and LMNS, Fudan University, Shanghai, China
- Research Institute of Intelligent Complex Systemsand Center for Computational Systems Biology, Fudan University, Shanghai, China
- *Correspondence: Xiaoxiao Peng, ; Wei Lin,
| | - Wei Lin
- Shanghai Center for Mathematical Sciences, School of Mathematical Sciences, and LMNS, Fudan University, Shanghai, China
- Research Institute of Intelligent Complex Systemsand Center for Computational Systems Biology, Fudan University, Shanghai, China
- State Key Laboratory of Medical Neurobiology, MOE Frontiers Center for Brain Science, and Institutes of Brain Science, Fudan University, Shanghai, China
- *Correspondence: Xiaoxiao Peng, ; Wei Lin,
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5
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Aminzare Z, Srivastava V. Stochastic synchronization in nonlinear network systems driven by intrinsic and coupling noise. BIOLOGICAL CYBERNETICS 2022; 116:147-162. [PMID: 35441346 DOI: 10.1007/s00422-022-00928-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/01/2021] [Accepted: 03/07/2022] [Indexed: 06/14/2023]
Abstract
In this paper, we consider a noisy network of nonlinear systems in the sense that each system is driven by two sources of state-dependent noise: (1) an intrinsic noise that can be generated by the environment or any internal fluctuations and (2) a noisy coupling which is generated by interactions with other systems. Our goal is to understand the effect of noise and coupling on synchronization behaviors of such networks. First, we assume that all the systems are driven by a common noise and show how a common noise can be detrimental or beneficial for network synchronization behavior. Then, we assume that the systems are driven by independent noise and study network approximate synchronization behavior. We numerically illustrate our results using the example of coupled Van der Pol oscillators.
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Affiliation(s)
- Zahra Aminzare
- Department of Mathematics, University of Iowa, Iowa City, IA, USA.
| | - Vaibhav Srivastava
- Electrical and Computer Engineering, Michigan State University, East Lansing, MI, USA
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6
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Khelfa B, Korbmacher R, Schadschneider A, Tordeux A. Heterogeneity-induced lane and band formation in self-driven particle systems. Sci Rep 2022; 12:4768. [PMID: 35306506 PMCID: PMC8934355 DOI: 10.1038/s41598-022-08649-4] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2021] [Accepted: 03/08/2022] [Indexed: 01/23/2023] Open
Abstract
The collective motion of interacting self-driven particles describes many types of coordinated dynamics and self-organisation. Prominent examples are alignment or lane formation which can be observed alongside other ordered structures and nonuniform patterns. In this article, we investigate the effects of different types of heterogeneity in a two-species self-driven particle system. We show that heterogeneity can generically initiate segregation in the motion and identify two heterogeneity mechanisms. Longitudinal lanes parallel to the direction of motion emerge when the heterogeneity statically lies in the agent characteristics (quenched disorder). While transverse bands orthogonal to the motion direction arise from dynamic heterogeneity in the interactions (annealed disorder). In both cases, non-linear transitions occur as the heterogeneity increases, from disorder to ordered states with lane or band patterns. These generic features are observed for a first and a second order motion model and different characteristic parameters related to particle speed and size. Simulation results show that the collective dynamics occur in relatively short time intervals, persist stationary, and are partly robust against random perturbations.
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Affiliation(s)
- Basma Khelfa
- School for Mechanical Engineering and Safety Engineering, University of Wuppertal, Wuppertal, Germany
| | - Raphael Korbmacher
- School for Mechanical Engineering and Safety Engineering, University of Wuppertal, Wuppertal, Germany
| | | | - Antoine Tordeux
- School for Mechanical Engineering and Safety Engineering, University of Wuppertal, Wuppertal, Germany.
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7
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Mambuca AM, Cammarota C, Neri I. Dynamical systems on large networks with predator-prey interactions are stable and exhibit oscillations. Phys Rev E 2022; 105:014305. [PMID: 35193197 DOI: 10.1103/physreve.105.014305] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/29/2021] [Accepted: 12/12/2021] [Indexed: 06/14/2023]
Abstract
We analyze the stability of linear dynamical systems defined on sparse, random graphs with predator-prey, competitive, and mutualistic interactions. These systems are aimed at modeling the stability of fixed points in large systems defined on complex networks, such as ecosystems consisting of a large number of species that interact through a food web. We develop an exact theory for the spectral distribution and the leading eigenvalue of the corresponding sparse Jacobian matrices. This theory reveals that the nature of local interactions has a strong influence on a system's stability. We show that, in general, linear dynamical systems defined on random graphs with a prescribed degree distribution of unbounded support are unstable if they are large enough, implying a tradeoff between stability and diversity. Remarkably, in contrast to the generic case, antagonistic systems that contain only interactions of the predator-prey type can be stable in the infinite size limit. This feature for antagonistic systems is accompanied by a peculiar oscillatory behavior of the dynamical response of the system after a perturbation, when the mean degree of the graph is small enough. Moreover, for antagonistic systems we also find that there exist a dynamical phase transition and critical mean degree above which the response becomes nonoscillatory.
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Affiliation(s)
| | - Chiara Cammarota
- Department of Mathematics, King's College London, Strand, London, WC2R 2LS, United Kingdom
- Dipartimento di Fisica, Sapienza Università di Roma, P. le A. Moro 5, 00185 Rome, Italy
| | - Izaak Neri
- Department of Mathematics, King's College London, Strand, London, WC2R 2LS, United Kingdom
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8
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Nwogugu MIC. Complex systems and ‘‘
Spatio ‐Temporal Anti‐Compliance Coordination
’’ In cyber‐physical networks: A critique of the
Hipster Effect
, bankruptcy prediction and alternative risk premia. COGNITIVE COMPUTATION AND SYSTEMS 2021. [DOI: 10.1049/ccs2.12029] [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] Open
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9
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Fasoli D, Panzeri S. Stationary-State Statistics of a Binary Neural Network Model with Quenched Disorder. ENTROPY (BASEL, SWITZERLAND) 2019; 21:e21070630. [PMID: 33267344 PMCID: PMC7515124 DOI: 10.3390/e21070630] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 05/20/2019] [Revised: 06/17/2019] [Accepted: 06/23/2019] [Indexed: 06/12/2023]
Abstract
In this paper, we study the statistical properties of the stationary firing-rate states of a neural network model with quenched disorder. The model has arbitrary size, discrete-time evolution equations and binary firing rates, while the topology and the strength of the synaptic connections are randomly generated from known, generally arbitrary, probability distributions. We derived semi-analytical expressions of the occurrence probability of the stationary states and the mean multistability diagram of the model, in terms of the distribution of the synaptic connections and of the external stimuli to the network. Our calculations rely on the probability distribution of the bifurcation points of the stationary states with respect to the external stimuli, calculated in terms of the permanent of special matrices using extreme value theory. While our semi-analytical expressions are exact for any size of the network and for any distribution of the synaptic connections, we focus our study on networks made of several populations, that we term "statistically homogeneous" to indicate that the probability distribution of their connections depends only on the pre- and post-synaptic population indexes, and not on the individual synaptic pair indexes. In this specific case, we calculated analytically the permanent, obtaining a compact formula that outperforms of several orders of magnitude the Balasubramanian-Bax-Franklin-Glynn algorithm. To conclude, by applying the Fisher-Tippett-Gnedenko theorem, we derived asymptotic expressions of the stationary-state statistics of multi-population networks in the large-network-size limit, in terms of the Gumbel (double exponential) distribution. We also provide a Python implementation of our formulas and some examples of the results generated by the code.
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10
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Large deviations for randomly connected neural networks: I. Spatially extended systems. ADV APPL PROBAB 2018. [DOI: 10.1017/apr.2018.42] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
Abstract
In a series of two papers, we investigate the large deviations and asymptotic behavior of stochastic models of brain neural networks with random interaction coefficients. In this first paper, we take into account the spatial structure of the brain and consider first the presence of interaction delays that depend on the distance between cells and then the Gaussian random interaction amplitude with a mean and variance that depend on the position of the neurons and scale as the inverse of the network size. We show that the empirical measure satisfies a large deviations principle with a good rate function reaching its minimum at a unique spatially extended probability measure. This result implies an averaged convergence of the empirical measure and a propagation of chaos. The limit is characterized through a complex non-Markovian implicit equation in which the network interaction term is replaced by a nonlocal Gaussian process with a mean and covariance that depend on the statistics of the solution over the whole neural field.
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11
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Fasoli D, Cattani A, Panzeri S. Transitions between asynchronous and synchronous states: a theory of correlations in small neural circuits. J Comput Neurosci 2017; 44:25-43. [PMID: 29124505 PMCID: PMC5770155 DOI: 10.1007/s10827-017-0667-3] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/11/2017] [Revised: 09/04/2017] [Accepted: 10/10/2017] [Indexed: 12/11/2022]
Abstract
The study of correlations in neural circuits of different size, from the small size of cortical microcolumns to the large-scale organization of distributed networks studied with functional imaging, is a topic of central importance to systems neuroscience. However, a theory that explains how the parameters of mesoscopic networks composed of a few tens of neurons affect the underlying correlation structure is still missing. Here we consider a theory that can be applied to networks of arbitrary size with multiple populations of homogeneous fully-connected neurons, and we focus its analysis to a case of two populations of small size. We combine the analysis of local bifurcations of the dynamics of these networks with the analytical calculation of their cross-correlations. We study the correlation structure in different regimes, showing that a variation of the external stimuli causes the network to switch from asynchronous states, characterized by weak correlation and low variability, to synchronous states characterized by strong correlations and wide temporal fluctuations. We show that asynchronous states are generated by strong stimuli, while synchronous states occur through critical slowing down when the stimulus moves the network close to a local bifurcation. In particular, strongly positive correlations occur at the saddle-node and Andronov-Hopf bifurcations of the network, while strongly negative correlations occur when the network undergoes a spontaneous symmetry-breaking at the branching-point bifurcations. These results show how the correlation structure of firing-rate network models is strongly modulated by the external stimuli, even keeping the anatomical connections fixed. These results also suggest an effective mechanism through which biological networks may dynamically modulate the encoding and integration of sensory information.
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Affiliation(s)
- Diego Fasoli
- Laboratory of Neural Computation, Center for Neuroscience and Cognitive Systems @UniTn, Istituto Italiano di Tecnologia, 38068, Rovereto, Italy.
- Center for Brain and Cognition, Computational Neuroscience Group, Universitat Pompeu Fabra, 08002, Barcelona, Spain.
| | - Anna Cattani
- Laboratory of Neural Computation, Center for Neuroscience and Cognitive Systems @UniTn, Istituto Italiano di Tecnologia, 38068, Rovereto, Italy
- Department of Biomedical and Clinical Sciences "L. Sacco", University of Milan, Milan, Italy
| | - Stefano Panzeri
- Laboratory of Neural Computation, Center for Neuroscience and Cognitive Systems @UniTn, Istituto Italiano di Tecnologia, 38068, Rovereto, Italy
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12
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Ly C, Marsat G. Variable synaptic strengths controls the firing rate distribution in feedforward neural networks. J Comput Neurosci 2017; 44:75-95. [DOI: 10.1007/s10827-017-0670-8] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/26/2017] [Revised: 10/18/2017] [Accepted: 10/19/2017] [Indexed: 12/27/2022]
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13
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Barreiro AK, Kutz JN, Shlizerman E. Symmetries Constrain Dynamics in a Family of Balanced Neural Networks. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2017; 7:10. [PMID: 29019105 PMCID: PMC5635020 DOI: 10.1186/s13408-017-0052-6] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 04/12/2017] [Accepted: 09/19/2017] [Indexed: 06/07/2023]
Abstract
We examine a family of random firing-rate neural networks in which we enforce the neurobiological constraint of Dale's Law-each neuron makes either excitatory or inhibitory connections onto its post-synaptic targets. We find that this constrained system may be described as a perturbation from a system with nontrivial symmetries. We analyze the symmetric system using the tools of equivariant bifurcation theory and demonstrate that the symmetry-implied structures remain evident in the perturbed system. In comparison, spectral characteristics of the network coupling matrix are relatively uninformative about the behavior of the constrained system.
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Affiliation(s)
- Andrea K. Barreiro
- Department of Mathematics, Southern Methodist University, POB 750156, Dallas, TX 75275 USA
| | - J. Nathan Kutz
- Department of Applied Mathematics, University of Washington, Box 353925, Seattle, WA 98195-3925 USA
| | - Eli Shlizerman
- Department of Applied Mathematics, University of Washington, Box 353925, Seattle, WA 98195-3925 USA
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14
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Nicola W, Tripp B, Scott M. Obtaining Arbitrary Prescribed Mean Field Dynamics for Recurrently Coupled Networks of Type-I Spiking Neurons with Analytically Determined Weights. Front Comput Neurosci 2016; 10:15. [PMID: 26973503 PMCID: PMC4770054 DOI: 10.3389/fncom.2016.00015] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/17/2015] [Accepted: 02/05/2016] [Indexed: 11/20/2022] Open
Abstract
A fundamental question in computational neuroscience is how to connect a network of spiking neurons to produce desired macroscopic or mean field dynamics. One possible approach is through the Neural Engineering Framework (NEF). The NEF approach requires quantities called decoders which are solved through an optimization problem requiring large matrix inversion. Here, we show how a decoder can be obtained analytically for type I and certain type II firing rates as a function of the heterogeneity of its associated neuron. These decoders generate approximants for functions that converge to the desired function in mean-squared error like 1/N, where N is the number of neurons in the network. We refer to these decoders as scale-invariant decoders due to their structure. These decoders generate weights for a network of neurons through the NEF formula for weights. These weights force the spiking network to have arbitrary and prescribed mean field dynamics. The weights generated with scale-invariant decoders all lie on low dimensional hypersurfaces asymptotically. We demonstrate the applicability of these scale-invariant decoders and weight surfaces by constructing networks of spiking theta neurons that replicate the dynamics of various well known dynamical systems such as the neural integrator, Van der Pol system and the Lorenz system. As these decoders are analytically determined and non-unique, the weights are also analytically determined and non-unique. We discuss the implications for measured weights of neuronal networks.
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Affiliation(s)
- Wilten Nicola
- Department of Applied Mathematics, University of WaterlooWaterloo, ON, Canada
| | - Bryan Tripp
- Department of Systems Design Engineering, University of WaterlooWaterloo, ON, Canada
- Center for Theoretical Neuroscience, University of WaterlooWaterloo, ON, Canada
| | - Matthew Scott
- Department of Applied Mathematics, University of WaterlooWaterloo, ON, Canada
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15
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Allahverdyan AE, Steeg GV, Galstyan A. Memory-induced mechanism for self-sustaining activity in networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:062824. [PMID: 26764761 DOI: 10.1103/physreve.92.062824] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/20/2015] [Indexed: 06/05/2023]
Abstract
We study a mechanism of activity sustaining on networks inspired by a well-known model of neuronal dynamics. Our primary focus is the emergence of self-sustaining collective activity patterns, where no single node can stay active by itself, but the activity provided initially is sustained within the collective of interacting agents. In contrast to existing models of self-sustaining activity that are caused by (long) loops present in the network, here we focus on treelike structures and examine activation mechanisms that are due to temporal memory of the nodes. This approach is motivated by applications in social media, where long network loops are rare or absent. Our results suggest that under a weak behavioral noise, the nodes robustly split into several clusters, with partial synchronization of nodes within each cluster. We also study the randomly weighted version of the models where the nodes are allowed to change their connection strength (this can model attention redistribution) and show that it does facilitate the self-sustained activity.
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Affiliation(s)
- A E Allahverdyan
- Yerevan Physics Institute, Alikhanian Brothers Street 2, Yerevan 375036, Armenia
| | - G Ver Steeg
- USC Information Sciences Institute, 4676 Admiralty Way, Marina del Rey, California 90292, USA
| | - A Galstyan
- USC Information Sciences Institute, 4676 Admiralty Way, Marina del Rey, California 90292, USA
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16
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Firing rate dynamics in recurrent spiking neural networks with intrinsic and network heterogeneity. J Comput Neurosci 2015; 39:311-27. [DOI: 10.1007/s10827-015-0578-0] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/07/2015] [Revised: 07/06/2015] [Accepted: 09/23/2015] [Indexed: 11/25/2022]
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17
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Wainrib G, Galtier M. Regular graphs maximize the variability of random neural networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:032802. [PMID: 26465523 DOI: 10.1103/physreve.92.032802] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/17/2014] [Indexed: 06/05/2023]
Abstract
In this work we study the dynamics of systems composed of numerous interacting elements interconnected through a random weighted directed graph, such as models of random neural networks. We develop an original theoretical approach based on a combination of a classical mean-field theory originally developed in the context of dynamical spin-glass models, and the heterogeneous mean-field theory developed to study epidemic propagation on graphs. Our main result is that, surprisingly, increasing the variance of the in-degree distribution does not result in a more variable dynamical behavior, but on the contrary that the most variable behaviors are obtained in the regular graph setting. We further study how the dynamical complexity of the attractors is influenced by the statistical properties of the in-degree distribution.
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Affiliation(s)
- Gilles Wainrib
- Ecole Normale Supérieure, Département d'Informatique, équipe DATA, Paris, France
| | - Mathieu Galtier
- European Institute for Theoretical Neuroscience, Paris, France
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Sonnenschein B, Schimansky-Geier L. Approximate solution to the stochastic Kuramoto model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:052111. [PMID: 24329218 DOI: 10.1103/physreve.88.052111] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/25/2013] [Indexed: 06/03/2023]
Abstract
We study Kuramoto phase oscillators with temporal fluctuations in the frequencies. The infinite-dimensional system can be reduced in a Gaussian approximation to two first-order differential equations. This yields a solution for the time-dependent order parameter, which characterizes the synchronization between the oscillators. The known critical coupling strength is exactly recovered by the Gaussian theory. Extensive numerical experiments further show that the analytical results are very accurate below and sufficiently above the critical value. We obtain the asymptotic order parameter in closed form, which suggests a tighter upper bound for the corresponding scaling. As a last point, we elaborate the Gaussian approximation in complex networks with distributed degrees.
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Affiliation(s)
- Bernard Sonnenschein
- Department of Physics, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489 Berlin, Germany and Bernstein Center for Computational Neuroscience Berlin, Philippstrasse 13, 10115 Berlin, Germany
| | - Lutz Schimansky-Geier
- Department of Physics, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489 Berlin, Germany and Bernstein Center for Computational Neuroscience Berlin, Philippstrasse 13, 10115 Berlin, Germany
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García del Molino LC, Pakdaman K, Touboul J, Wainrib G. Synchronization in random balanced networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:042824. [PMID: 24229242 DOI: 10.1103/physreve.88.042824] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/18/2013] [Indexed: 06/02/2023]
Abstract
Characterizing the influence of network properties on the global emerging behavior of interacting elements constitutes a central question in many areas, from physical to social sciences. In this article we study a primary model of disordered neuronal networks with excitatory-inhibitory structure and balance constraints. We show how the interplay between structure and disorder in the connectivity leads to a universal transition from trivial to synchronized stationary or periodic states. This transition cannot be explained only through the analysis of the spectral density of the connectivity matrix. We provide a low-dimensional approximation that shows the role of both the structure and disorder in the dynamics.
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Wainrib G, Touboul J. Topological and dynamical complexity of random neural networks. PHYSICAL REVIEW LETTERS 2013; 110:118101. [PMID: 25166580 DOI: 10.1103/physrevlett.110.118101] [Citation(s) in RCA: 45] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2012] [Indexed: 06/03/2023]
Abstract
Random neural networks are dynamical descriptions of randomly interconnected neural units. These show a phase transition to chaos as a disorder parameter is increased. The microscopic mechanisms underlying this phase transition are unknown and, similar to spin glasses, shall be fundamentally related to the behavior of the system. In this Letter, we investigate the explosion of complexity arising near that phase transition. We show that the mean number of equilibria undergoes a sharp transition from one equilibrium to a very large number scaling exponentially with the dimension on the system. Near criticality, we compute the exponential rate of divergence, called topological complexity. Strikingly, we show that it behaves exactly as the maximal Lyapunov exponent, a classical measure of dynamical complexity. This relationship unravels a microscopic mechanism leading to chaos which we further demonstrate on a simpler disordered system, suggesting a deep and underexplored link between topological and dynamical complexity.
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Affiliation(s)
- Gilles Wainrib
- LAGA, Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), 99 avenue J.B. Clément, F-93430 Villetaneuse, France
| | - Jonathan Touboul
- The Mathematical Neuroscience Laboratory, CIRB/Collège de France (CNRS UMR 7241, INSERM U1050, UPMC ED 158, MEMOLIFE PSL*), 11, place Marcelin Berthelot, 75005 Paris, France and BANG Laboratory, INRIA Paris-Rocquencourt, Domaine de Voluceau, 78153 Le Chesnay, France
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