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Prüser A, Rosmej S, Engel A. Nature of the Volcano Transition in the Fully Disordered Kuramoto Model. PHYSICAL REVIEW LETTERS 2024; 132:187201. [PMID: 38759180 DOI: 10.1103/physrevlett.132.187201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/13/2023] [Accepted: 03/29/2024] [Indexed: 05/19/2024]
Abstract
Randomly coupled phase oscillators may synchronize into disordered patterns of collective motion. We analyze this transition in a large, fully connected Kuramoto model with symmetric but otherwise independent random interactions. Using the dynamical cavity method, we reduce the dynamics to a stochastic single-oscillator problem with self-consistent correlation and response functions that we study analytically and numerically. We clarify the nature of the volcano transition and elucidate its relation to the existence of an oscillator glass phase.
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Affiliation(s)
- Axel Prüser
- Carl von Ossietzky University Oldenburg, Institut für Physik, D26111 Oldenburg, Germany
| | - Sebastian Rosmej
- Carl von Ossietzky University Oldenburg, Institut für Physik, D26111 Oldenburg, Germany
| | - Andreas Engel
- Carl von Ossietzky University Oldenburg, Institut für Physik, D26111 Oldenburg, Germany
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2
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Garcia ER, Crumpton MJ, Galla T. Niche overlap and Hopfield-like interactions in generalized random Lotka-Volterra systems. Phys Rev E 2023; 108:034120. [PMID: 37849207 DOI: 10.1103/physreve.108.034120] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/27/2023] [Accepted: 07/01/2023] [Indexed: 10/19/2023]
Abstract
We study communities emerging from generalized random Lotka-Volterra dynamics with a large number of species with interactions determined by the degree of niche overlap. Each species is endowed with a number of traits, and competition between pairs of species increases with their similarity in trait space. This leads to a model with random Hopfield-like interactions. We use tools from the theory of disordered systems, notably dynamic mean-field theory, to characterize the statistics of the resulting communities at stable fixed points and determine analytically when stability breaks down. Two distinct types of transition are identified in this way, both marked by diverging abundances but differing in the behavior of the integrated response function. At fixed points only a fraction of the initial pool of species survives. We numerically study the eigenvalue spectra of the interaction matrix between extant species. We find evidence that the two types of dynamical transition are, respectively, associated with the bulk spectrum or an outlier eigenvalue crossing into the right half of the complex plane.
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Affiliation(s)
- Enrique Rozas Garcia
- Department of Physics, Gothenburg University, 41296 Gothenburg, Sweden
- Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
| | - Mark J Crumpton
- Department of Mathematics, King's College London, London WC2R 2LS, United Kingdom
- Department of Physics and Astronomy, School of Natural Sciences, The University of Manchester, Manchester M13 9PL, United Kingdom
| | - Tobias Galla
- Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
- Department of Physics and Astronomy, School of Natural Sciences, The University of Manchester, Manchester M13 9PL, United Kingdom
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Mignacco F, Urbani P, Zdeborová L. Stochasticity helps to navigate rough landscapes: comparing gradient-descent-based algorithms in the phase retrieval problem. MACHINE LEARNING: SCIENCE AND TECHNOLOGY 2021. [DOI: 10.1088/2632-2153/ac0615] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022] Open
Abstract
Abstract
In this paper we investigate how gradient-based algorithms such as gradient descent (GD), (multi-pass) stochastic GD, its persistent variant, and the Langevin algorithm navigate non-convex loss-landscapes and which of them is able to reach the best generalization error at limited sample complexity. We consider the loss landscape of the high-dimensional phase retrieval problem as a prototypical highly non-convex example. We observe that for phase retrieval the stochastic variants of GD are able to reach perfect generalization for regions of control parameters where the GD algorithm is not. We apply dynamical mean-field theory from statistical physics to characterize analytically the full trajectories of these algorithms in their continuous-time limit, with a warm start, and for large system sizes. We further unveil several intriguing properties of the landscape and the algorithms such as that the GD can obtain better generalization properties from less informed initializations.
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Sidhom L, Galla T. Ecological communities from random generalized Lotka-Volterra dynamics with nonlinear feedback. Phys Rev E 2020; 101:032101. [PMID: 32289927 DOI: 10.1103/physreve.101.032101] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2019] [Accepted: 01/28/2020] [Indexed: 12/11/2022]
Abstract
We investigate the outcome of generalized Lotka-Volterra dynamics of ecological communities with random interaction coefficients and nonlinear feedback. We show in simulations that the saturation of nonlinear feedback stabilizes the dynamics. This is confirmed in an analytical generating-functional approach to generalized Lotka-Volterra equations with piecewise linear saturating response. For such systems we are able to derive self-consistent relations governing the stable fixed-point phase and to carry out a linear stability analysis to predict the onset of unstable behavior. We investigate in detail the combined effects of the mean, variance, and covariance of the random interaction coefficients, and the saturation value of the nonlinear response. We find that stability and diversity increases with the introduction of nonlinear feedback, where decreasing the saturation value has a similar effect to decreasing the covariance. We also find cooperation to no longer have a detrimental effect on stability with nonlinear feedback, and the order parameters mean abundance and diversity to be less dependent on the symmetry of interactions with stronger saturation.
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Affiliation(s)
- Laura Sidhom
- Theoretical Physics, Department of Physics and Astronomy, School of Natural Sciences, The University of Manchester, Manchester M13 9PL, United Kingdom
| | - Tobias Galla
- Theoretical Physics, Department of Physics and Astronomy, School of Natural Sciences, The University of Manchester, Manchester M13 9PL, United Kingdom and Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
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Çakmak B, Opper M. Memory-free dynamics for the Thouless-Anderson-Palmer equations of Ising models with arbitrary rotation-invariant ensembles of random coupling matrices. Phys Rev E 2019; 99:062140. [PMID: 31330731 DOI: 10.1103/physreve.99.062140] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/07/2019] [Indexed: 11/07/2022]
Abstract
We propose an iterative algorithm for solving the Thouless-Anderson-Palmer equations of Ising models with arbitrary rotation-invariant (random) coupling matrices. In the thermodynamic limit, we prove by means of the dynamical functional method that the proposed algorithm converges when the so-called de Almeida Thouless criterion is fulfilled. Moreover, we give exact analytical expressions for the rate of the convergence.
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Affiliation(s)
- Burak Çakmak
- Artificial Intelligence Group, Technische Universität Berlin, 10587 Berlin, Germany
| | - Manfred Opper
- Artificial Intelligence Group, Technische Universität Berlin, 10587 Berlin, Germany
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6
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Pena RFO, Vellmer S, Bernardi D, Roque AC, Lindner B. Self-Consistent Scheme for Spike-Train Power Spectra in Heterogeneous Sparse Networks. Front Comput Neurosci 2018; 12:9. [PMID: 29551968 PMCID: PMC5840464 DOI: 10.3389/fncom.2018.00009] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/29/2017] [Accepted: 02/07/2018] [Indexed: 11/13/2022] Open
Abstract
Recurrent networks of spiking neurons can be in an asynchronous state characterized by low or absent cross-correlations and spike statistics which resemble those of cortical neurons. Although spatial correlations are negligible in this state, neurons can show pronounced temporal correlations in their spike trains that can be quantified by the autocorrelation function or the spike-train power spectrum. Depending on cellular and network parameters, correlations display diverse patterns (ranging from simple refractory-period effects and stochastic oscillations to slow fluctuations) and it is generally not well-understood how these dependencies come about. Previous work has explored how the single-cell correlations in a homogeneous network (excitatory and inhibitory integrate-and-fire neurons with nearly balanced mean recurrent input) can be determined numerically from an iterative single-neuron simulation. Such a scheme is based on the fact that every neuron is driven by the network noise (i.e., the input currents from all its presynaptic partners) but also contributes to the network noise, leading to a self-consistency condition for the input and output spectra. Here we first extend this scheme to homogeneous networks with strong recurrent inhibition and a synaptic filter, in which instabilities of the previous scheme are avoided by an averaging procedure. We then extend the scheme to heterogeneous networks in which (i) different neural subpopulations (e.g., excitatory and inhibitory neurons) have different cellular or connectivity parameters; (ii) the number and strength of the input connections are random (Erdős-Rényi topology) and thus different among neurons. In all heterogeneous cases, neurons are lumped in different classes each of which is represented by a single neuron in the iterative scheme; in addition, we make a Gaussian approximation of the input current to the neuron. These approximations seem to be justified over a broad range of parameters as indicated by comparison with simulation results of large recurrent networks. Our method can help to elucidate how network heterogeneity shapes the asynchronous state in recurrent neural networks.
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Affiliation(s)
- Rodrigo F O Pena
- Laboratório de Sistemas Neurais, Department of Physics, School of Philosophy, Sciences and Letters of Ribeirão Preto, University of São Paulo, São Paulo, Brazil
| | - Sebastian Vellmer
- Theory of Complex Systems and Neurophysics, Bernstein Center for Computational Neuroscience, Berlin, Germany.,Department of Physics, Humboldt Universität zu Berlin, Berlin, Germany
| | - Davide Bernardi
- Theory of Complex Systems and Neurophysics, Bernstein Center for Computational Neuroscience, Berlin, Germany.,Department of Physics, Humboldt Universität zu Berlin, Berlin, Germany
| | - Antonio C Roque
- Laboratório de Sistemas Neurais, Department of Physics, School of Philosophy, Sciences and Letters of Ribeirão Preto, University of São Paulo, São Paulo, Brazil
| | - Benjamin Lindner
- Theory of Complex Systems and Neurophysics, Bernstein Center for Computational Neuroscience, Berlin, Germany.,Department of Physics, Humboldt Universität zu Berlin, Berlin, Germany
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Advani M, Bunin G, Mehta P. Statistical physics of community ecology: a cavity solution to MacArthur's consumer resource model. JOURNAL OF STATISTICAL MECHANICS (ONLINE) 2018; 2018:033406. [PMID: 30636966 PMCID: PMC6329381 DOI: 10.1088/1742-5468/aab04e] [Citation(s) in RCA: 29] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/21/2023]
Abstract
A central question in ecology is to understand the ecological processes that shape community structure. Niche-based theories have emphasized the important role played by competition for maintaining species diversity. Many of these insights have been derived using MacArthur's consumer resource model (MCRM) or its generalizations. Most theoretical work on the MCRM has focused on small ecosystems with a few species and resources. However theoretical insights derived from small ecosystems many not scale up large ecosystems with many resources and species because large systems with many interacting components often display new emergent behaviors that cannot be understood or deduced from analyzing smaller systems. To address these shortcomings, we develop a statistical physics inspired cavity method to analyze MCRM when both the number of species and the number of resources is large. Unlike previous work in this limit, our theory addresses resource dynamics and resource depletion and demonstrates that species generically and consistently perturb their environments and significantly modify available ecological niches. We show how our cavity approach naturally generalizes niche theory to large ecosystems by accounting for the effect of collective phenomena on species invasion and ecological stability. Our theory suggests that such phenomena are a generic feature of large, natural ecosystems and must be taken into account when analyzing and interpreting community structure. It also highlights the important role that statistical-physics inspired approaches can play in furthering our understanding of ecology.
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Affiliation(s)
| | - Guy Bunin
- Technion-Israel Institute of Technology, Haifa, 3200003, Israel
| | - Pankaj Mehta
- Dept. of Physics, Boston University, Boston, MA 02215
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8
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Abstract
The inclusion of a macroscopic adaptive threshold is studied for the retrieval dynamics of both layered feedforward and fully connected neural network models with synaptic noise. These two types of architectures require a different method to be solved numerically. In both cases it is shown that, if the threshold is chosen appropriately as a function of the cross-talk noise and of the activity of the stored patterns, adapting itself automatically in the course of the recall process, an autonomous functioning of the network is guaranteed. This self-control mechanism considerably improves the quality of retrieval, in particular the storage capacity, the basins of attraction and the mutual information content.
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Affiliation(s)
- D Bollé
- Institute for Theoretical Physics, Katholieke Universiteit Leuven, Celestijnenlaan 200 D, B-3001, Leuven, Belgium.
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9
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Bergmann UM, Kühn R, Stamatescu IO. Learning with incomplete information in the committee machine. BIOLOGICAL CYBERNETICS 2009; 101:401-410. [PMID: 19888596 DOI: 10.1007/s00422-009-0345-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/01/2009] [Accepted: 10/23/2009] [Indexed: 05/28/2023]
Abstract
We study the problem of learning with incomplete information in a student-teacher setup for the committee machine. The learning algorithm combines unsupervised Hebbian learning of a series of associations with a delayed reinforcement step, in which the set of previously learnt associations is partly and indiscriminately unlearnt, to an extent that depends on the success rate of the student on these previously learnt associations. The relevant learning parameter lambda represents the strength of Hebbian learning. A coarse-grained analysis of the system yields a set of differential equations for overlaps of student and teacher weight vectors, whose solutions provide a complete description of the learning behavior. It reveals complicated dynamics showing that perfect generalization can be obtained if the learning parameter exceeds a threshold lambda ( c ), and if the initial value of the overlap between student and teacher weights is non-zero. In case of convergence, the generalization error exhibits a power law decay as a function of the number of examples used in training, with an exponent that depends on the parameter lambda. An investigation of the system flow in a subspace with broken permutation symmetry between hidden units reveals a bifurcation point lambda* above which perfect generalization does not depend on initial conditions. Finally, we demonstrate that cases of a complexity mismatch between student and teacher are optimally resolved in the sense that an over-complex student can emulate a less complex teacher rule, while an under-complex student reaches a state which realizes the minimal generalization error compatible with the complexity mismatch.
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Affiliation(s)
- Urs M Bergmann
- Institut für Theoretische Physik, Universität Heidelberg, Heidelberg, Germany.
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Yoshino Y, Galla T, Tokita K. Rank abundance relations in evolutionary dynamics of random replicators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:031924. [PMID: 18851082 DOI: 10.1103/physreve.78.031924] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2008] [Revised: 07/09/2008] [Indexed: 05/26/2023]
Abstract
We present a nonequilibrium statistical mechanics description of rank abundance relations (RAR) in random community models of ecology. Specifically, we study a multispecies replicator system with quenched random interaction matrices. We here consider symmetric interactions as well as asymmetric and antisymmetric cases. RARs are obtained analytically via a generating functional analysis, describing fixed-point states of the system in terms of a small set of order parameters, and in dependence on the symmetry or otherwise of interactions and on the productivity of the community. Our work is an extension of Tokita [Phys. Rev. Lett. 93, 178102 (2004)], where the case of symmetric interactions was considered within an equilibrium setup. The species abundance distribution in our model come out as truncated normal distributions or transformations thereof and, in some case, are similar to left-skewed distributions observed in ecology. We also discuss the interaction structure of the resulting food-web of stable species at stationarity, cases of heterogeneous cooperation pressures as well as effects of finite system size and of higher-order interactions.
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Affiliation(s)
- Yoshimi Yoshino
- Graduate School of Science and Cybermedia Center, Osaka University, Toyonaka, Osaka 560-0043, Japan.
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11
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Kühn R, Stamatescu IO. Learning with incomplete information and the mathematical structure behind it. BIOLOGICAL CYBERNETICS 2007; 97:99-112. [PMID: 17534648 DOI: 10.1007/s00422-007-0162-4] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/04/2007] [Accepted: 05/02/2007] [Indexed: 05/15/2023]
Abstract
We investigate the problem of learning with incomplete information as exemplified by learning with delayed reinforcement. We study a two phase learning scenario in which a phase of Hebbian associative learning based on momentary internal representations is supplemented by an 'unlearning' phase depending on a graded reinforcement signal. The reinforcement signal quantifies the success-rate globally for a number of learning steps in phase one, and 'unlearning' is indiscriminate with respect to associations learnt in that phase. Learning according to this model is studied via simulations and analytically within a student-teacher scenario for both single layer networks and, for a committee machine. Success and speed of learning depend on the ratio lambda of the learning rates used for the associative Hebbian learning phase and for the unlearning-correction in response to the reinforcement signal, respectively. Asymptotically perfect generalization is possible only, if this ratio exceeds a critical value lambda( c ), in which case the generalization error exhibits a power law decay with the number of examples seen by the student, with an exponent that depends in a non-universal manner on the parameter lambda. We find these features to be robust against a wide spectrum of modifications of microscopic modelling details. Two illustrative applications-one of a robot learning to navigate a field containing obstacles, and the problem of identifying a specific component in a collection of stimuli-are also provided.
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Affiliation(s)
- Reimer Kühn
- Department of Mathematics, King's College, London, UK.
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Anand K, Kühn R. Phase transitions in operational risk. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:016111. [PMID: 17358228 DOI: 10.1103/physreve.75.016111] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2006] [Indexed: 05/14/2023]
Abstract
In this paper we explore the functional correlation approach to operational risk. We consider networks with heterogeneous a priori conditional and unconditional failure probability. In the limit of sparse connectivity, self-consistent expressions for the dynamical evolution of order parameters are obtained. Under equilibrium conditions, expressions for the stationary states are also obtained. Consequences of the analytical theory developed are analyzed using phase diagrams. We find coexistence of operational and nonoperational phases, much as in liquid-gas systems. Such systems are susceptible to discontinuous phase transitions from the operational to nonoperational phase via catastrophic breakdown. We find this feature to be robust against variation of the microscopic modeling assumptions.
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Affiliation(s)
- Kartik Anand
- Department of Mathematics, King's College London, Strand, London WC2R 2LS, United Kingdom.
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Lerchner A, Sterner G, Hertz J, Ahmadi M. Mean field theory for a balanced hypercolumn model of orientation selectivity in primary visual cortex. NETWORK (BRISTOL, ENGLAND) 2006; 17:131-50. [PMID: 16818394 DOI: 10.1080/09548980500444933] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/10/2023]
Abstract
We present a complete mean field theory for a balanced state of a simple model of an orientation hypercolumn, with a numerical procedure for solving the mean-field equations quantitatively. With our treatment, one can determine self-consistently both the firing rates and the firing correlations, without being restricted to specific neuron models. Here, we solve the mean-field equations numerically for integrate-and-fire neurons. Several known key properties of orientation selective cortical neurons emerge naturally from the description: Irregular firing with statistics close to - but not restricted to - Poisson statistics; an almost linear gain function (firing frequency as a function of stimulus contrast) of the neurons within the network; and a contrast-invariant tuning width of the neuronal firing. We find that the irregularity in firing depends sensitively on synaptic strengths. If the Fano factor is considerably larger (smaller) than 1 at some stimulus orientation, then it is also larger (resp. maller) than 1 for all other stimulus orientations that elicit firing. We also find that the tuning of the noise in the input current is the same as the tuning of the external input, while that for the mean input current depends on both the external input and the intracortical connectivity.
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Lerchner A, Ursta C, Hertz J, Ahmadi M, Ruffiot P, Enemark S. Response Variability in Balanced Cortical Networks. Neural Comput 2006. [DOI: 10.1162/neco.2006.18.3.634] [Citation(s) in RCA: 37] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/04/2022]
Abstract
We study the spike statistics of neurons in a network with dynamically balanced excitation and inhibition. Our model, intended to represent a generic cortical column, comprises randomly connected excitatory and inhibitory leaky integrate-and-fire neurons, driven by excitatory input from an external population. The high connectivity permits a mean field description in which synaptic currents can be treated as gaussian noise, the mean and autocorrelation function of which are calculated self-consistently from the firing statistics of single model neurons. Within this description, a wide range of Fano factors is possible. We find that the irregularity of spike trains is controlled mainly by the strength of the synapses relative to the difference between the firing threshold and the postfiring reset level of the membrane potential. For moderately strong synapses, we find spike statistics very similar to those observed in primary visual cortex.
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Affiliation(s)
- Alexander Lerchner
- Current address: Laboratory of Neuropsychology, NIMH, NIH, Bethesda, MD 20893, USA
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Lerchner A, Ahmadi M, Hertz J. High-conductance states in a mean-field cortical network model. Neurocomputing 2004. [DOI: 10.1016/j.neucom.2004.01.149] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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Bollé D, Blanco JB, Verbeiren T. The signal-to-noise analysis of the Little–Hopfield model revisited. ACTA ACUST UNITED AC 2004. [DOI: 10.1088/0305-4470/37/6/001] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Hertz J, Richmond B, Nilsen K. Anomalous response variability in a balanced cortical network model. Neurocomputing 2003. [DOI: 10.1016/s0925-2312(02)00775-0] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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Luo P, Wong KYM. Dynamical and stationary properties of on-line learning from finite training sets. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:011906. [PMID: 12636531 DOI: 10.1103/physreve.67.011906] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/29/2002] [Indexed: 05/24/2023]
Abstract
The dynamical and stationary properties of on-line learning from finite training sets are analyzed by using the cavity method. For large input dimensions, we derive equations for the macroscopic parameters, namely, the student-teacher correlation, the student-student autocorrelation and the learning force fluctuation. This enables us to provide analytical solutions to Adaline learning as a benchmark. Theoretical predictions of training errors in transient and stationary states are obtained by a Monte Carlo sampling procedure. Generalization and training errors are found to agree with simulations. The physical origin of the critical learning rate is presented. Comparison with batch learning is discussed throughout the paper.
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Affiliation(s)
- Peixun Luo
- Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China.
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21
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Kłos J, Kobe S. Time decay of the remanent magnetization in the +/-J spin glass model at T=0. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:066111. [PMID: 11415177 DOI: 10.1103/physreve.63.066111] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2000] [Indexed: 05/23/2023]
Abstract
Using the zero-temperature Metropolis dynamics, the time decay of the remanent magnetization in the +/-J Edward-Anderson spin glass model with a uniform random distribution of ferromagnetic and antiferromagnetic interactions has been investigated. Starting from the saturation, the magnetization per spin m reveals a slow decrease with time, which can be approximated by a power law: m(t)=m(infinity)+(t/a(0))(a(1)), a(1)<0. Moreover, its relaxation does not lead it into one of the ground states, and therefore the system is trapped in metastable isoenergetic microstates remaining magnetized. Such behavior is discussed in terms of a random walk that the system performs on its available configuration space.
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Affiliation(s)
- J Kłos
- Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany
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Heimel JA, Coolen AC. Generating functional analysis of the dynamics of the batch minority game with random external information. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:056121. [PMID: 11414975 DOI: 10.1103/physreve.63.056121] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/04/2000] [Indexed: 05/23/2023]
Abstract
We study the dynamics of the batch minority game, with random external information, using generating functional techniques introduced by De Dominicis. The relevant control parameter in this model is the ratio alpha=p/N of the number p of possible values for the external information over the number N of trading agents. In the limit N-->infinity we calculate the location alphac of the phase transition (signaling the onset of anomalous response), and solve the statics for alpha>alphac exactly. The temporal correlations in global market fluctuations turn out not to decay to zero for infinitely widely separated times. For alpha<alphac the stationary state is shown to be nonunique. For alpha-->0 we analyze our equations in leading order in alpha, and find asymptotic solutions with diverging volatility sigma=O(alpha(-1/2)) (as regularly observed in simulations), but also asymptotic solutions with vanishing volatility sigma=O(alpha(1/2)). The former, however, are shown to emerge only if the agents' initial strategy valuations are below a specific critical value.
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Affiliation(s)
- J A Heimel
- Department of Mathematics, King's College London, The Strand, London WC2R 2LS, United Kingdom
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Fulvi Mari C. Random networks of spiking neurons: instability in the Xenopus tadpole moto-neural pattern. PHYSICAL REVIEW LETTERS 2000; 85:210-213. [PMID: 10991196 DOI: 10.1103/physrevlett.85.210] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/06/1999] [Indexed: 05/23/2023]
Abstract
A large network of integrate-and-fire neurons is studied analytically when the synaptic weights are independently randomly distributed according to a Gaussian distribution with arbitrary mean and variance. The relevant order parameters are identified, and it is shown that such network is statistically equivalent to an ensemble of independent integrate-and-fire neurons with each input signal given by the sum of a self-interaction deterministic term and a Gaussian colored noise. The model is able to reproduce the quasisynchronous oscillations, and the dropout of their frequency, of the central nervous system neurons of the swimming Xenopus tadpole. Predictions from the model are proposed for future experiments.
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Affiliation(s)
- C Fulvi Mari
- Nonlinear and Complex Systems Group, Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, United Kingdom.
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Nutzel K, Krey U. Subtle dynamic behaviour of finite-size Sherrington-Kirkpatrick spin glasses with nonsymmetric couplings. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/26/14/001] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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