#### J.-P. Nadal: abstracts of papers published since 1992

Chronological order: most recent papers at the end.
.

## Information capacity of a perceptron

#### Brunel N., Nadal J.-P., and Toulouse G.

We study the information storage capacity of a simple perceptron in the error regime. For random unbiased patterns the geometrical analysis gives a logarithmic dependence of the information content in the asymptotic limit.
In that regime the statistical physics approach, when used at the simplest level of replica theory, does not give satisfactory results. However for perceptrons with finite stability, the information content can be simply calculated with statistical physics methods in a region above the critical storage level, for biased as well as for unbiased patterns.

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## A network model of the coupling of ion channels with secondary messenger in cell signaling

#### Plouraboué F., Atlan H., Weisbuch G. and Nadal J.-P.

We demonstrate that formal neural networks techniques allow to build the simplest models compatible with a limited but systematic set of experimental data. The experimental system under study is the growth of mouse macrophage like cell lines under the combined influence of two ion channels, the growth factor receptor and adenylate cyclase. We conclude that 3 components out of 4 can be described by linear multithreshold automata. The remaining component behavior being non-monotonous necessitate the introduction of a fifth hidden variable, or of non-linear interactions.

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## Duality Between Learning Machines: A Bridge Between Supervised and Unsupervised Learning

#### Jean-Pierre Nadal and Nestor Parga

We exhibit a duality between two perceptrons that allows us to compare the theoretical analysis of supervised and unsupervised learning tasks - more exactly of parameter estimation and encoding tasks. The first perceptron has one output and is asked to learn a classification of p patterns. The second (dual) perceptron has p outputs and is asked to transmit as much information as possible on a distribution of inputs. We show in particular that the maximum information that can be stored in the couplings for the supervised learning task is equal to the maximum information that can be transmitted by the dual perceptron.

Neural Computation, Vol. 6, Issue 3 (May 1994), pages 491-508.
(preprint1992.ps.gz, preprint1992.pdf).

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## Relevant parameters for a class of sequence-retrieving neural networks

#### Lefèvre O. and Nadal J.-P.

Reconsidering a recently introduced model of sequence-retrieving neural network, we introduce appropriate analogues of the well-known stabilities and show how these, together with two coupling parameters $\lambda$ and $\vartheta$, entirely control the dynamics in the case of strong dilution. The model is exactly solved and phase diagrams are drawn for two different choices of the synaptic matrices; they reveal a rich structure. We then briefly speculate as to the role of these parameters within a more general framework.

Journal de Physique I (France), Volume 3 Nb. 6 (1993) pages 1303-1328
abstract and full paper available on EDP Sciences web site, and from HAL.

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## Information processing by a perceptron in an unsupervised learning task

#### Jean-Pierre Nadal and Nestor Parga

We study the ability of a simple neural network (a perceptron architecture, no hidden units, binary outputs) to process information in the context of an unsupervised learning task. The network is asked to provide the best possible neural representation of a given input distribution, according to some criterion taken from Information Theory. We compare various optimization criteria that have been proposed : maximum information transmission, minimum redundancy and closeness to factorial code.
We show that for the perceptron one can compute the maximal information that the code (the output neural representation) can convey about the input. We show that one can use Statistical Mechanics techniques, such as the replica techniques, to compute the typical mutual information between input and output distributions. More precisely, for a Gaussian input source with a given correlation matrix, we compute the typical mutual information when the couplings are chosen randomly. We determine the correlations between the synaptic couplings which maximize the gain of information. We analyse the results in the case of a one dimensional receptive field.

Network: Computation in Neural Systems, Volume 4, Number 3 (August 1993), pages 295-312
IOP, this journal has moved to Taylor & Francis/informa healthcare.
(preprint .ps.gz, .pdf).

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## Optimal unsupervised learning

#### Tim L. H. Watkin and Jean-Pierre Nadal

We introduce an inferential approach to unsupervised learning which allows us to define an optimal learning strategy. Applying these ideas to a simple, previously studied model, we show that it is impossible to detect structure in data until a critical number of examples have been presented-an effect which will be observed in all problems with certain underlying symmetries. Thereafter, the advantage of optimal learning over previously studied learning algorithms depends critically upon the distribution of patterns; optimal learning may be exponentially faster. Models with more subtle correlations are harder to analyse, but in a simple limit of one such problem we calculate exactly the efficacy of an algorithm similar to some used in practice, and compare it to that of the optimal prescription.

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## Nonlinear neurons in the low-noise limit: a factorial code maximizes information transfer

#### Jean-Pierre Nadal and Nestor Parga

We investigate the consequences of maximizing information transfer in a simple neural network (one input layer, one output layer), focussing on the case of non linear transfer functions. We assume that both receptive fields (synaptic efficacies) and transfer functions can be adapted to the environment.

The main result is that, for bounded and invertible transfer functions, in the case of a vanishing additive output noise, and no input noise, maximization of information (Linsker's infomax principle) leads to a factorial code - hence to the same solution as required by the redundancy reduction principle of Barlow, or, in the signal processing language, to Independent Component Analysis (ICA).

We show also that this result is valid for linear, more generally unbounded, transfer functions, provided optimization is performed under an additive constraint, that is which can be written as a sum of terms, each one being specific to one output neuron. Finally we study the effect of a non zero input noise. We find that, at first order in the input noise, assumed to be small as compared to the - small - output noise, the above results are still valid, provided the output noise is uncorrelated from one neuron to the other.

Network: Computation in Neural Systems, Volume 5, Number 4 (November 1994), pages 565-581
IOP, this journal has moved to Taylor & Francis/informa healthcare.
(preprint.ps.gz, preprint.pdf).

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## Trio learning: a new strategy for building hybrid trees

#### d'Alché-Buc F., Zwierski D. and Nadal J.-P.

Neural trees are constructive algorithms which build decision trees whose nodes are binary neurons. We propose a new learning scheme, "trio-learning", which leads to a significant reduction in the tree complexity. In the trio strategy, each node of the tree is optimized by taking into account the knowledge that it will be followed by two son nodes. Moreover, trio-learning can be used to build hybrid trees, with internal nodes and terminal nodes of different nature, for solving any standard task (e.g. classification, regression, density estimation). Promising results on a handwritten character classification are presented.

Int. Journ. of Neur. Syst. Volume 5, Issue 4 (December 1994) pages 259-274.
(abstract and full paper from IJNS web site)

This work has been performed at Laboratoires d'Electronique Philips S.A.S. (LEP), Limeil-Brévannes, France. Other publications with LEP: click here, and see below.

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## Sparsely coded neural networks

#### Claude Meunier and Jean-Pierre Nadal

Review paper on sparse coding in (auto)associative memories - in particular in Willshaw et al (1969) and Hopfield (1982) type models.

In The Handbook of Brain Theory and Neural Networks, Arbib M. A. Ed. (MIT Press, 1995) pp. 899-901

Related works:

• Associative memory: on the (puzzling) sparse coding limit
J.-P. Nadal, Journal of Physics A: Mathematical and General, Vol. 24 (1991) pp. 1093-1101
(abstract and full paper available on IoP electronic journals web site).

• Information storage in sparsely-coded memory nets
J.-P. Nadal and G. Toulouse, Network: Computation in Neural Systems, Vol. 1 (1990) pp. 61-74
abstract and full paper on the review web site. NB: initially published by IOP, this journal has moved to Taylor & Francis/informa healthcare.

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## Maximization of mutual information in a linear noisy network: a detailed study

#### Campa A., Del Giudice P., Parga N. and Nadal J.-P.

We consider a linear, one-layer feedforward neural network performing a coding task. The goal of the network is to provide a statistical neural representation that convey as much information as possible on the input stimuli in noisy conditions. We determine the family of synaptic couplings that maximizes the mutual information between input and output distribution. Optimization is performed under different constraints on the synaptic efficacies. We analyze the dependence of the solutions on input and output noises. This work goes beyond previous studies of the same problem in that:
(i) we perform a detailed stability analysis in order to find the global maxima of the mutual information;
(ii) we examine the properties of the optimal synaptic configurations under different constraints;
(iii) we do not assume translational invariance of the input data, as it is usually done when input are assumed to be visual stimuli.

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## Asymptotic performance of a constructive algorithm

#### Florence d'Alché-Buc and Jean-Pierre Nadal

We present a numerical study of a neural tree learning algorithm, the trio-learning strategy. We study the behaviour of the algorithm as a function of the size of the training set (figure). The results show that a limited number of examples can be used to estimate both the network performance and the network complexity that would result from running the algorithm on a large data set.

Neural Processing Letters 2(2):1-4 (March 1995). (TOC of the issue on H. G. Schuster web site)
(preprint.ps.gz, preprint.pdf)

This work has been performed at Laboratoires d'Electronique Philips S.A.S. (LEP), Limeil-Brévannes, France. Other publications with LEP: click here.

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## A formal approach to market organization: choice functions, mean field approximation and maximum entropy principle

#### Nadal J.-P., Weisbuch G., Chenevez O. and Kirman A.

We present a formal, although simple, approach to the modeling of a buyer behavior in the type of markets studied in Weisbuch, Kirman and Herreiner, 1995. We compare possible buyer's choice functions, such as linear or logit function. We study the resulting behaviour, showing that they depend on some convexity properties of the choice function. Our results make use of standard Statistical Physics concepts and techniques. In particular we use the "mean field approximation" to derive the long term behaviour of buyers, and we show that the standard "logit" choice function can be justified from a general optimization principle, leading to an exploration-exploitation compromise.

In Advances in Self-Organization and Evolutionary Economics, J. Lesourne and A. Orléan Eds. (Economica, London, 1998), pp. 149-159.
(preprint1996.ps.gz, preprint1996.pdf).

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## Redundancy Reduction and Independent Component Analysis: Conditions on Cumulants and Adaptive Approaches

#### Jean-Pierre Nadal and Nestor Parga

In the context of both sensory coding and signal processing, building factorized codes has been shown to be an efficient strategy. In a wide variety of situations, the signal to be processed is a linear mixture of statistically independent sources. Building a factorized code is then equivalent to performing blind source separation. Thanks to the linear structure of the data, this can be done, in the language of signal processing, by finding an appropriate linear filter, or equivalently, in the language of neural modeling, by using a simple feedforward neural network.
In this article, we discuss several aspects of the source separation problem. We give simple conditions on the network output that, if satisfied, guarantee that source separation has been obtained. Then we study adaptive approaches, in particular those based on redundancy reduction and maximization of mutual information. We show how the resulting updating rules are related to the BCM theory of synaptic plasticity. Eventually we briefly discuss extensions to the case of nonlinear mixtures. Throughout this article, we take care to put into perspective our work with other studies on source separation and redundancy reduction. In particular we review algebraic solutions, pointing out their simplicity but also their drawbacks.

Neural Computation, Vol. 9, Issue 7 (October 1997), pages 1421-1456
(preprint.pdf)

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## Information processing by a noisy binary channel

#### Korutcheva E., Parga N. and Nadal J.-P.

We study the information processing properties of a binary channel receiving data from a gaussian source. A systematic comparison with linear processing is done. A remarkable property of the binary sytem is that, as the ratio $\alpha$ between the number of output and input units increases, binary processing becomes equivalent to linear processing with a quantization output noise that depends on $\alpha$. In this regime , that holds up to $O( \alpha^{-4})$ , information processing occurs as if populations of $\alpha$ binary units cooperate to represent one $\alpha$-bit output unit. Unsupervised learning of a noisy environment by optimization of the parameters of the binary channel is also considered.

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## Self-Similarity Properties of Natural Images Resemble Those of Turbulent Flows

#### Antonio Turiel, Germ´n Mato, Néstor Parga and Jean-Pierre Nadal

We show that the statistics of an edge type variable in natural images exhibits self-similarity properties which resemble those of local energy dissipation in turbulent flows. Our results show that self-similarity and extended self-similarity hold remarkably for the statistics of the local edge variance, and that the very same models can be used to predict all of the associated exponents. These results suggest using natural images as a laboratory for testing more elaborate scaling models of interest for the statistical description of turbulent flows. The properties we have exhibited are relevant for the modeling of the early visual system: They should be included in models designed for the prediction of receptive fields.
[© 1998 by The American Physical Society]

Physical Review Letters, Volume 80, Issue 5 (February 2, 1998) pp. 1098-1101
(preprint.ps.gz, preprint.pdf) (paper from PRL on line)

For more recent works on this subject, see below and the web site of Antonio Turiel.

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## Nonlinear feedforward networks with stochastic outputs: infomax implies redundancy reduction

#### Jean-Pierre Nadal, Nicolas Brunel and Nestor Parga

Independent Component Analysis (ICA), and in particular Blind Source Separation (BSS), can be obtained from the maximization of mutual information, as first shown in Nadal and Parga 1994. The practical interest of this information theoretic based cost function was then demonstrated in several BSS applications (see e.g. Bell and Sejnowski 1995, ICA at CNL).
In the present paper the main result of Nadal and Parga 1994 is extended to the case of stochastic outputs. More precisely, we prove that maximization of mutual information between the output and the input of a feedforward neural network leads to full redundancy reduction under the following sufficient conditions:
(1) the input signal is a (possibly nonlinear) invertible mixture of independent components; (2) there is no input noise; (3) the activity of each output neuron is a (possibly) stochastic variable with a probability distribution depending on the stimulus through a deterministic function of the inputs; both the probability distributions and the functions can be different from neuron to neuron; (4) optimization of the mutual information is performed over all these deterministic functions.

Network: Computation in Neural Systems, Volume 9, Number 2 (May 1998) pages 207-217
Initially published by IOP, this journal has moved to Taylor & Francis/informa healthcare.
(preprint.ps.gz, preprint.pdf)

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## Mutual information, Fisher information and population coding

#### Nicolas Brunel and Jean-Pierre Nadal

In the context of parameter estimation and model selection, it is only quite recently that a direct link between the Fisher information and information theoretic quantities has been exhibited. We give an interpretation of this link within the standard framework of information theory. We show that in the context of population coding, the mutual information between the activity of a large array of neurons and a stimulus to which the neurons are tuned is naturally related to the Fisher information.
In the light of this result we consider the optimization of the tuning curves parameters in the case of neurons responding to a stimulus represented by an angular variable.

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## Unsupervised and supervised learning: the mutual information between parameters and observations

#### Didier Herschkowitz and Jean-Pierre Nadal

Recent works in parameter estimation and neural coding have demonstrated that optimal performance are related to the mutual information between parameters and data. In this paper:

We study the mutual information between parameter and data for a family of supervised and unsupervised learning tasks. The parameter is a possibly, but not necessarily, high-dimensional vector. We derive exact bounds and asymptotic behaviors for the mutual information as a function of the data size and of some properties of the probability of the data given the parameter. We compare these exact results with the predictions of replica calculations. We briefly discuss the universal properties of the mutual information as a function of data size.
[© 1999 The American Physical Society]

Phys. Rev. E Volume 59, issue 3 (March 1, 1999), pp. 3344-3360.
(preprint.ps.gz, preprint.pdf) (abstract and paper from Phys. Rev. E online).

Short version presented at NIPS*98:
Unsupervised clustering: the mutual information between parameters and observations

in Advances in Neural Information Processing Systems 11, M. S. Kearns, S. A. Solla, D. A. Cohn, eds., MIT Press 1999, pp. 232-238.

Related works at LPS :
D. Herschkowitz and M. Opper, "Retarded Learning: Rigorous Results from Statistical Mechanics", Phys.Rev.Lett. 86, 2174 (2001).

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## Blind Source Separation with Time Dependent Mixtures

#### Nestor Parga and Jean-Pierre Nadal

We address the problem of blind source separation in the case of a time dependent mixture matrix. For a slowly and smoothly varying mixture matrix, we propose a systematic expansion which leads to a practical algebraic solution when stationary and ergodic properties hold for the sources.
[© 2000 Elsevier Science B. V.]

Signal Processing, volume 80 issue 10 (October 2000) pp. 2187-2194.
(preprint.ps.gz, preprint.pdf) (abstract and full text from Elsevier Science Direct)

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## Analysis of geophysical time series and information theory: Independent Component Analysis

#### Filipe Aires, Alain Chédin and Jean-Pierre Nadal

Résumé :
Dans le but d'identifier les causes physiques de la variabilité d'un système dynamique, la communauté géophysique utilise de façon intensive les techniques statistiques d'extraction de composantes. Un algorithme récemment développé, fondé sur la théorie de l'information, est introduit dans ce travail : l'analyse en composantes indépendantes (ACI). Cette technique présente deux avantages majeurs sur les techniques classiques. Premièrement, elle a pour but d'extraire des composantes statistiquement indépendantes, là où les techniques classiques cherchent uniquement la décorrélation. Deuxièmement, l'hypothèse linéaire pour le mélange des composantes n'est pas requise. Cette nouvelle technique est présenté dans le contexte de l'analyse de séries temporelles géophysiques. L'algorithme ACI est appliqué à l'étude de la variabilité de la température de surface de l'océan (TSO) tropical, avec une attention particulière pour l'analyse des liens entre le phénomène El Niño/Southern Oscillation (Enso) et la variabilité de la TSO Atlantique.
(© 1999 - Académie des Sciences/ Éditions Scientifiques et Médicales Elsevier SAS)

Les Comptes rendus de l'Académie des Sciences (CRAS), Geoscience (Série IIa, Sciences de la terre et des planètes), vol. 328 num. 9 (1999) pp. 569-575.

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## Independent component analysis of multivariate time series. Application to the tropical SST variability

#### Filipe Aires, Alain Chédin and Jean-Pierre Nadal

With the aim of identifying the physical causes of variability of a given dynamical system, the geophysical community has made an extensive use of classical component extraction techniques such as principal component analysis (PCA) or rotational techniques (RT). We introduce a recently developed algorithm based on information theory: independent component analysis (ICA). This new technique presents two major advantages over classical methods. First, it aims at extracting statistically independent components where classical techniques search for decorrelated components (i.e., a weaker constraint). Second, the linear hypothesis for the mixture of components is not required. In this paper, after having briefly summarized the essentials of classical techniques, we present the new method in the context of geophysical time series analysis. We then illustrate the ICA algorithm by applying it to the study of the variability of the tropical sea surface temperature (SST), with a particular emphasis on the analysis of the links between El Niño Southern Oscillation (ENSO) and Atlantic SST variability. The new algorithm appears to be particularly efficient in describing the complexity of the phenomena and their various sources of variability in space and time.
(© 2000 by the American Geophysical Union)

Journal of Geophysical Research - Atmospheres, Vol. 105 , No. D13 , p. 17,437 (2000).

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## Meet, Discuss and Segregate!

#### Gerard Weisbuch, Guillaume Deffuant, Frederic Amblard, Jean-Pierre Nadal

We present a model of opinion dynamics in which agents adjust continuous opinions as a result of random binary encounters whenever their difference in opinion is below a given threshold. High thresholds yield convergence of opinions toward an average opinion, whereas low thresholds result in several opinion clusters. The model is further generalized to network interactions, threshold heterogeneity, adaptive thresholds, and binary strings of opinions.

Complexity Vol 7:3 (2002) pp 55-63 (abstract and paper from Wiley InterScience)

Preprint Nov. 2001, "Interacting Agents and Continuous Opinions Dynamics" cond-mat/0111494
also as Santa Fe working paper #01-11-072 and on RePEc and HAL open archives.

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## Rigorous Bounds to Retarded Learning

#### Arnaud Buhot, Mirta B. Gordon, Jean-Pierre Nadal

We show that the lower bound to the critical fraction of data needed to infer (learn) the orientation of the anisotropy axis of a probability distribution, determined by Herschkowitz and Opper [Phys.Rev.Lett. 86, 2174 (2001)], is not always valid. If there is some structure in the data along the anisotropy axis, their analysis is incorrect, and learning is possible with much less data points.
[© 2002 by The American Physical Society]

Comment, Physical Review Letters 88, 099801 (2002)
(preprint: cond-mat/0201256)
This article has been selected for the February 15, 2002 issue of the Virtual Journal of Biological Physics Research published by the American Institute of Physics and the American Physical Society.

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## Orientational minimal redundancy wavelets: from edge detection to perception

#### Antonio Turiel, Jean-Pierre Nadal and Nestor Parga

Natural images are complex but very structured objects and, in spite of its com- plexity, the sensory areas in the neocortex in mammals are able to devise learned strategies to encode them endciently. How is this goal achieved? In this paper, we will discuss the multiscaling approach, which has been recently used to derive a redundancy reducing wavelet basis. This kind of representation can be statistically learned from the data and is optimally adapted for image coding; besides, it presents some remarkable features found in the visual pathway. We will show that the introduction of oriented wavelets is necessary to provide a complete description, which stresses the role of the wavelets as edge detectors.

Vision Research Vol. 43:9 (2003) pp. 1061-1079.
(preprint.pdf)
For related works, see above and the web site of Antonio Turiel.

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## Is there something out there? Infering space from sensorimotor dependencies

#### David Philipona, Kevin O'Regan and Jean-Pierre Nadal

This letter suggests that in biological organisms, the perceived structure of reality, in particular the notions of body, environment, space, object, and attribute, could be a consequence of an effort on the part of brains to account for the dependency between their inputs and their outputs in terms of a small number of parameters. To validate this idea, a procedure is demonstrated whereby the brain of an organism with arbitrary input and output connectivity can deduce the dimensionality of the rigid group of the space underlying its input-output relationship, that is the dimension of what the organism will call physical space.

Neural Computation Vol 15:9 (Sept. 2003) pp 2029-2049.
(selected as sample article of the Sept. 2003 issue)
(preprint.ps.gz - preprint.pdf)

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## Identifying genes from up-down properties of microarray expression series

#### Karen Willbrand, Francois Radvanyi, Jean-Pierre Nadal, Jean-Paul Thiery, and Thomas M. A. Fink

Motivation: We consider any collection of microarrays that can be ordered to form a progression, as a function of time, or severity of disease, or dose of a stimulant, for example. By plotting the expression level of each gene as a function of time, or severity, or dose, we form an expression series, or curve, for each gene. While most of these curves will exhibit random fluctuations, some will contain pattern, and it is these genes which are most likely associated with the independent variable.
Results: We introduce a method of identifying pattern and hence genes in microarray expression curves without knowing what kind of pattern to look for. Key to our approach is the sequence of ups and downs formed by pairs of consecutive data points in each curve. As a benchmark, we blindly identified yeast cell cycles genes without selecting for periodic or any other anticipated behaviour.

Bioinformatics 2005, Oct. 15; 21(20):3859-3864
Supplementary information can be found at http://www.lps.ens.fr/~willbran/up-down/

Related publications by K. Willbrand (LPS ENS) and Th. Fink (Inst. Curie): see here

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## Optimal Information Storage and the Distribution of Synaptic Weights: Perceptron versus Purkinje Cell

#### Nicolas Brunel, Vincent Hakim, Philippe Isope, Jean-Pierre Nadal and Boris Barbour

It is widely believed that synaptic modifications under lie learning and memory. However, few studies have examined what can be deduced about the learning process from the distribution of synaptic weights. We analyze the perceptron, a prototypical feedforward neural network, and obtain the optimal synaptic weight distribution for a perceptron with excitatory synapses. It contains more than 50% silent synapses, and this fraction increases with storage reliability: silent synapses are therefore a necessary byproduct of optimizing learning and reliability. Exploiting the classical analogy between the perceptron and the cerebellar Purkinje cell, we fitted the optimal weight distribution to that measured for granule cell-Purkinje cell synapses. The two distributions agreed well, suggesting that the Purkinje cell can learn up to 5 kilobytes of information in the form of 40,000 input-output associations.

Neuron, Vol 43, 745-757, 2 September 2004
(with supplemental data, see here).

Related work here.

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## Pre-attentive segmentation of oriented textures

#### Ingrid Machecler and Jean-Pierre Nadal

The interpretation of geophysical data, such as images of subsurface rocks (seismic data, borehole scans), requires one in particular to perform an elaborate segmentation analysis on strongly textured, anisotropic, and not necessarily brightness-contrasted images. In this paper we explore the possibility of deriving new segmentation algorithms from recent advances in the neural modelling of pre-attentive segmentation in human vision. More specifically we consider a neural model proposed by Zhaoping Li. First, we reproduce some specific results obtained by Zhaoping Li on simple artificial and real images sharing some textural characteristics with geophysical data. Next, from the analysis of the model behaviour, we propose an image processing workflow depending on the textural characteristics and on the type of segmentation (contour enhancement or texture edge detection) one is interested in. With this algorithm one gets promising results: from the computation of a single attribute one extracts the oriented textured feature boundaries without prior classification.

J. Geophys. Eng. 1 (2004) 312-326.
Published online 22 November 2004 - Print publication: Issue 4 (10 December 2004)
Currently freely available online as one of the featured articles of the review.

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## Heterogeneity and feedback in an agent-based market model

#### François Ghoulmié, Rama Cont and Jean-Pierre Nadal

We propose an agent-based model of a single-asset financial market, described in terms of a small number of parameters, which generates price returns with statistical properties similar to the stylized facts observed in financial time series. Our agent-based model generically leads to the absence of autocorrelation in returns, self-sustaining excess volatility, mean-reverting volatility, volatility clustering and endogenous bursts of market activity non-attributable to external noise. The parsimonious structure of the model allows the identification of feedback and heterogeneity as the key mechanisms leading to these effects.

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## Seller's dilemma due to social interactions between customers

In this paper, we consider a discrete choice model where heterogeneous agents are subject to mutual influences. We explore some consequences on the market's behaviour, in the simplest case of a uniform willingness to pay distribution. We exhibit a first-order phase transition in the profit optimization by the monopolist: if the social influence is strong enough, there is a regime where, if the mean willingness to pay increases, or if the production costs decrease, the optimal solution for the monopolist jumps from a solution with a high price and a small number of buyers, to a solution with a low price and a large number of buyers. Depending on the path of prices adjustments by the monopolist, simulations show hysteretic effects on the fraction of buyers.

Physica A, Volume 356, Issues 2-4 , 15 October 2005, Pages 628-640
(available online 13 June 2005).
preprint.pdf

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## Multiple equilibria in a monopoly market with heterogeneous agents and externalities

We explore the effects of social influence in a simple market model in which a large number of agents face a binary choice: to buy/not to buy a single unit of a product at a price posted by a single seller (monopoly market). We consider the case of positive externalities: an agent is more willing to buy if other agents make the same decision. We consider two special cases of heterogeneity in the individuals' decision rules, corresponding in the literature to the Random Utility Models of Thurstone, and of McFadden and Manski. In the first one the heterogeneity fluctuates with time, leading to a standard model in Physics: the Ising model at finite temperature (known as annealed disorder) in a uniform external field. In the second approach the heterogeneity among agents is fixed; in Physics this is a particular case of quenched disorder models known as random field Ising model, at zero temperature. We study analytically the equilibrium properties of the market in the limiting case where each agent is influenced by all the others (the mean field limit), and we illustrate some dynamic properties of these models making use of numerical simulations in an Agent based Computational Economics approach.

Quantitative Finance Vol.5, No. 6, December 2005, 557-568.
(preprint.pdf - preliminary version: condmat 0311096)

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## The acquisition of allophonic rules: statistical learning with linguistic constraints

#### Sharon Peperkamp, Rozenn Le Calvez, Jean-Pierre Nadal and Emmanuel Dupoux

Phonological rules relate surface phonetic word forms to abstract underlying forms that are stored in the lexicon. Infants must thus acquire these rules in order to infer the abstract representation of words. We implement a statistical learning algorithm for the acquisition of one type of rule, namely allophony, which introduces context-sensitive phonetic variants of phonemes. This algorithm is based on the observation that different realizations of a single phoneme typically do not appear in the same contexts (ideally, they have complementary distributions). In particular, it measures the discrepancies in context probabilities for each pair of phonetic segments. In Experiment 1, we test the algorithm.s performances on a pseudo-language and show that it is robust to statistical noise due to sampling and coding errors, and to non-systematic rule application. In Experiment 2, we show that a natural corpus of semiphonetically transcribed child-directed speech in French presents a very large number of near-complementary distributions that do not correspond to existing allophonic rules. These spurious allophonic rules can be eliminated by a linguistically motivated filtering mechanism based on a phonetic representation of segments. We discuss the role of a priori linguistic knowledge in the statistical learning of phonology.

Cognition, Volume 101, Issue 3, October 2006, Pages B31-B41
(preprint Oct. 2005 - paper.pdf)

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## Discrete Choices under Social Influence: Generic Properties

#### Mirta B. Gordon, Jean-Pierre Nadal, Denis Phan and Viktoriya Semeshenko

We consider a model of socially interacting individuals that make a binary choice in a context of positive additive endogenous externalities. It encompasses as particular cases several models from the sociology and economics literature. We extend previous results to the case of a general distribution of idiosyncratic preferences, called here Idiosyncratic Willingnesses to Pay (IWP).
Positive additive externalities yield a family of inverse demand curves that include the classical downward sloping ones but also new ones with non constant convexity. When $j$, the ratio of the social influene strength to the standard deviation of the IWP distribution, is small enough, the inverse demand is a classical monotonic (decreasing) function of the adoption rate. Even if the IWP distribution is mono-modal, there is a critical value of $j$ above which the inverse demand is non monotonic, decreasing for small and high adoption rates, but increasing within some intermediate range. Depending on the price there are thus either one or two equilibria.
Beyond this first result, we exhibit the {\em generic} properties of the boundaries limiting the regions where the system presents different types of equilibria (unique or multiple). These properties are shown to depend {\em only} on qualitative features of the IWP distribution: modality (number of maxima), smoothness and type of support (compact or infinite). The main results are summarized as {\em phase diagrams} in the space of the model parameters, on which the regions of multiple equilibria are precisely delimited.

Mathematical Models and Methods in Applied Sciences (M3AS), Volume: 19, Supplementary Issue 1(2009) pp. 1441-1481 (DOI: 10.1142/S0218202509003887)
Preprint arXiv:0704.2333v1 [physics.soc-ph], and HAL-SHS or RePEc (Research Papers in Economics) open archives, March 2007

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## Collective states in social systems with interacting learning agents

#### Viktoriya Semeshenko, Mirta B. Gordon and Jean-Pierre Nadal

We consider a social system of interacting heterogeneous agents with learning abilities, a model close to Random Field Ising Models, where the random field corresponds to the idiosyncratic willingness to pay. Given a fixed price, agents decide repeatedly whether to buy or not a unit of a good, so as to maximize their expected utilities. We show that the equilibrium reached by the system depends on the nature of the information agents use to estimate their expected utilities.

Physica A: Statistical Mechanics and its Applications Volume 387, Issues 19-20, August 2008, Pages 4903-4916 ( online 10 April 2008 ).
Preprint arXiv:0704.2324v1 [physics.soc-ph], April 2007.

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## Crime and punishment: the economic burden of impunity

#### Mirta B. Gordon, J. Roberto Iglesias, Viktoriya Semeshenko and Jean-Pierre Nadal

Crime is an economically relevant activity. It may represent a mechanism of wealth distribution but also a social and economic burden because of the interference with regular legal activities and the cost of the law enforcement system. Sometimes it may be less costly for the society to allow for some level of criminality. However, a drawback of such a policy is that it may lead to a high increase of criminal activity, that may become hard to reduce later on. Here we investigate the level of law enforcement required to keep crime within acceptable limits. A sharp phase transition is observed as a function of the probability of punishment. We also analyze other consequences of criminality as the growth of the economy, the inequality in the wealth distribution (the Gini coefficient) and other relevant quantities under different scenarios of criminal activity and probabilities of apprehension.

Eur. Phys. J. B 68, 133-144 (2009) DOI: 10.1140/epjb/e2009-00066-x
Preprint arXiv:0710.3751 Oct. 2007.

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## What can we learn from synaptic weight distributions?

#### Boris Barbour, Nicolas Brunel, Vincent Hakim and Jean-Pierre Nadal

Much research effort into synaptic plasticity has been motivated by the idea that modifications of synaptic weights (or strengths or efficacies) underlie learning and memory. Here, we examine the possibility of exploiting the statistics of experimentally measured synaptic weights to deduce information about the learning process. Analysing distributions of synaptic weights requires a theoretical framework to interpret the experimental measurements, but the results can be unexpectedly powerful, yielding strong constraints on possible learning theories as well as information that is difficult to obtain by other means, such as the information storage capacity of a cell. We review the available experimental and theoretical techniques as well as important open issues.

Trends in Neurosciences, Volume 30, Issue 12, December 2007, Pages 622-629
online November 5th, 2007, on TINS (Elsevier - ScienceDirect) web site.

Related work here.

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## Neural Coding of Categories:Information Efficiency and Optimal Population Codes

#### Laurent Bonnasse-Gahot and Jean-Pierre Nadal

This paper deals with the analytical study of coding a discrete set of categories by a large assembly of neurons. We consider population coding schemes, which can also be seen as instances of exemplar models proposed in the literature to account for phenomena in the psychophysics of categorization. We quantify the coding efficiency by the mutual information between the set of categories and the neural code, and we characterize the properties of the most efficient codes, considering different regimes corresponding essentially to different signal-to-noise ratio. One main outcome is to find that, in a high signal-to-noise ratio limit, the Fisher information at the population level should be the greatest between categories, which is achieved by having many cells with the stimulus-discriminating parts (steepest slope) of their tuning curves placed in the transition regions between categories in stimulus space. We show that these properties are in good agreement with both psychophysical data -- from different domains such as object recognition and speech perception --, and with the neurophysiology of the inferotemporal cortex in the monkey, a cortex area known to be specifically involved in classification tasks.

Journal of Computational Neuroscience 25:1 August 2008 pp. 169-187 (DOI: 10.1007/s10827-007-0071-5)
online 31 January 2008 - preprint.pdf (May 2007).

Related works, same authors:

• "Perception of categories: from coding efficiency to reaction times", here.
• "From Exemplar Theory to Population Coding and Back - An Ideal Observer Approach" (preprint.pdf), proceedings of the workshop "Exemplar-Based Models of Language Acquisition and Use", Dublin, 2007.

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## Pricing Strategies of Goods with Externalities

#### Mirta B. Gordon, Jean-Pierre Nadal, Denis Phan and Viktoriya Semeshenko

This paper summarizes the effects of social influences in a monopoly market with heterogeneous agents. The market equilibria are presented in the limiting case of global influence. Considering static profit maximization there may exist two different regimes: to sell either to a large fraction of customers at a low price, or to a small fraction of them at a higher price. This arises for numerous mono-modal distributions of idiosyncratic willingness to pay if the social influence is strong enough. The seller's optimal strategy switches from one regime to the other at parameter values where the demand has two different Nash equilibria; but the strategy of posting low prices to attract large fractions of buyers may fail due to a lack of coordination.

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## Phase diagram of a Schelling segregation model

The collective behavior in a variant of Schelling's segregation model is characterized with methods borrowed from statistical physics, in a context where their relevance was not conspicuous. A measure of segregation based on cluster geometry is defined and several quantities analogous to those used to describe physical lattice models at equilibrium are introduced. This physical approach allows to distinguish quantitatively several regimes and to characterize the transitions between them, leading to the building of a phase diagram. Some of the transitions evoke empirical sudden ethnic turnovers. We also establish links with 'spin-1' models in physics. Our approach provides generic tools to analyze the dynamics of other socio-economic systems.

The European Physical Journal B - Condensed Matter and Complex Systems (EPJB) Volume 70:2 (2009) pp. 293-304 (DOI: 10.1140/epjb/e2009-00234-0)
online 8 July 2009 (full text on EPJB site) - preprint arXiv:0903.4694, March 2009.

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## Cycles of cooperation and free-riding in social systems

#### Yiping P. Ma, Sebastian Gonçalves, Sylvain Mignot, Jean-Pierre Nadal and Mirta B. Gordon

Basic evidences on non-profit making and other forms of benevolent-based organizations reveal a rough partition of members between some {\em pure consumers} of the public good (free-riders) and {\em benevolent} individuals (cooperators). We study the relationship between the community size and the level of cooperation in a simple model where the utility of joining the community is proportional to its size. We assume an idiosyncratic willingness to join the community ; cooperation bears a fixed cost while free-riding bears a (moral) idiosyncratic cost proportional to the fraction of cooperators. We show that the system presents two types of equilibria: fixed points (Nash equilibria) with a mixture of cooperators and free-riders and cycles where the size of the community, as well as the proportion of cooperators and free-riders, vary periodically.

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## Disentangling collective trends from local dynamics

#### Marc Barthelemy, Jean-Pierre Nadal and Henri Berestycki

A single social phenomenon (such as crime, unemployment or birth rate) can be observed through temporal series corresponding to units at different levels (cities, regions, countries...). Units at a given local level may follow a collective trend imposed by external conditions, but also may display fluctuations of purely local origin. The local behavior is usually computed as the difference between the local data and a global average (e.g. a national average), a view point which can be very misleading. In this article, we propose a method for separating the local dynamics from the global trend in a collection of correlated time series. We take an independent component analysis approach in which we do not assume a small average local contribution in contrast with previously proposed methods. We first test our method on financial time series for which various data analysis tools have already been used. For the S&P500 stocks, our method is able to identify two classes of stocks with marked different behaviors: the followers' (stocks driven by the collective trend), and the leaders' (stocks for which local fluctuations dominate). Furthermore, as a byproduct contributing to its validation, the method also allows to classify stocks in several groups consistent with industrials sectors. We then consider crime rate series, a domain where the separation between global and local policies is still a major subject of debate. We apply our method to the states in the US and the regions in France. In the case of the US data, we observe large fluctuations in the transition period of mid-70's during which crime rates increased significantly, whereas since the 80's, the state crime rates are governed by external factors, and the importance of local specificities being decreasing.

Proceedings of the National Academy of Sciences (PNAS)
online April 12, 2010, doi: 10.1073/pnas.0910259107
Preprint arXiv:0909.1490 September 2009.
Echo in the nonacademic press: Where local policy matters, 16 April 2010, in Emerging Health Threats Forum (a not-for-profit Community Interest Company, established with support from the UK's Health Protection Agency).

## Schelling segregation in an open city: a kinetically constrained Blume-Emery-Griffiths spin-1 system

In the 70s Schelling introduced a multiagent model to describe the segregation dynamics that may occur with individuals having only weak preferences for "similar" neighbors. Recently variants of this model have been discussed, in particular, with emphasis on the links with statistical physics models. Whereas these models consider a fixed number of agents moving on a lattice, here, we present a version allowing for exchanges with an external reservoir of agents. The density of agents is controlled by a parameter which can be viewed as measuring the attractiveness of the city lattice. This model is directly related to the zero-temperature dynamics of the Blume-Emery-Griffiths spin-1 model, with kinetic constraints. With a varying vacancy density, the dynamics with agents making deterministic decisions leads to a variety of "phases" whose main features are the characteristics of the interfaces between clusters of agents of different types. The domains of existence of each type of interface are obtained analytically as well as numerically. These interfaces may completely isolate the agents leading to another type of segregation as compared to what is observed in the original Schelling model, and we discuss its possible socioeconomic correlates.

Phys. Rev. E 81, 066120 (2010)
Preprint arXiv:1002.3758, February 2010.

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## Perception of categories: from coding efficiency to reaction times

#### Laurent Bonnasse-Gahot and Jean-Pierre Nadal

Reaction-times in perceptual tasks are the subject of many experimental and theoretical studies. With the neural decision making process as main focus, most of these works concern discrete (typically binary) choice tasks, implying the identification of the stimulus as an exemplar of a category. Here we address issues specific to the perception of categories (e.g. vowels, familiar faces, ...), making a clear distinction between identifying a category (an element of a discrete set) and estimating a continuous parameter (such as a direction). We exhibit a link between optimal Bayesian decoding and coding efficiency, the latter being measured by the mutual information between the discrete category set and the neural activity. We characterize the properties of the best estimator of the likelihood of the category, when this estimator takes its inputs from a large population of stimulus-specific coding cells. Adopting the diffusion-to-bound approach to model the decisional process, this allows to relate analytically the bias and variance of the diffusion process underlying decision making to macroscopic quantities that are behaviorally measurable. A major consequence is the existence of a quantitative link between reaction times and discrimination accuracy. The resulting analytical expression of mean reaction times during an identification task accounts for empirical facts, both qualitatively (e.g. more time is needed to identify a category from a stimulus at the boundary compared to a stimulus lying within a category), and quantitatively (working on published experimental data on phoneme identification tasks).

Preprint arXiv:1102.4749, Feb. 2011. Supporting information here.
Brain Research, Volume 1434, 24 January 2012, Pages 47-61.

Related works, same authors:

• "Neural Coding of Categories: Information Efficiency and Optimal Population Codes", J. of Comput. Neuroscience 2008, here.
• "From Exemplar Theory to Population Coding and Back - An Ideal Observer Approach" (paper here), proceedings of the workshop "Exemplar-Based Models of Language Acquisition and Use", Dublin, 2007.

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## Modeling urban housing market dynamics: can the socio-spatial segregation preserve some social diversity?

#### Laetitia Gauvin, Annick Vignes and Jean-Pierre Nadal

Addressing issues in social diversity, we introduce a model of housing transactions between agents heterogeneous in their willingness to pay. A key assumption is that agents preferences for a place depend on both an intrinsic attractiveness and on the social characteristics of its neighborhood. The stationary space distribution of income is analytically and numerically characterized. The main results are that socio-spatial segregation occurs whenever the social influence is strong enough, but even so, some social diversity is preserved at most locations. Comparing with the Parisian housing market, the results reproduce general trends concerning the price distribution and the income spatial segregation.

Journal of Economic Dynamics and Control (JEDC), ), Volume 37, Issue 7, July 2013, Pages 1300-1321
doi:10.1016/j.jedc.2013.03.001.
Preprint arXiv:1012.2606.
Extended abstract in the ICCS2011 online proceedings.

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## Between order and disorder: a 'weak law' on recent electoral behavior among urban voters?

#### Christian Borghesi, Jean Chiche and Jean-Pierre Nadal

A new viewpoint on electoral involvement is proposed from the study of the statistics of the proportions of abstentionists, blank and null, and votes according to list of choices, in a large number of national elections in different countries. Considering 11 countries without compulsory voting (Austria, Canada, Czech Republic, France, Germany, Italy, Mexico, Poland, Romania, Spain and Switzerland), a stylized fact emerges for the most populated cities when one computes the entropy associated to the three ratios, which we call the entropy of civic involvement of the electorate. The distribution of this entropy (over all elections and countries) appears to be sharply peaked near a common value. This almost common value is typically shared since the 1970's by electorates of the most populated municipalities, and this despite the wide disparities between voting systems and types of elections. Performing different statistical analyses, we notably show that this stylized fact reveals particular correlations between the blank/null votes and abstentionists ratios. We suggest that the existence of this hidden regularity, which we propose to coin as a weak law on recent electoral behavior among urban voters', reveals an emerging collective behavioral norm characteristic of urban citizen voting behavior in modern democracies. Analyzing exceptions to the rule provide insights into the conditions under which this normative behavior can be expected to occur.

PLoS ONE 7(7): e39916. doi:10.1371/journal.pone.0039916 (published: July 25, 2012) - Preprint arXiv:1202.6307.

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## Storage of correlated patterns in standard and bistable Purkinje cell models

#### Claudia Clopath, Jean-Pierre Nadal and Nicolas Brunel

The cerebellum has long been considered to undergo supervised learning, with climbing fibers acting as a 'teaching' or 'error' signal. Purkinje cells (PCs), the sole output of the cerebellar cortex, have been considered as analogs of perceptrons storing input/output associations. In support of this hypothesis, a recent study found that the distribution of synaptic weights of a perceptron at maximal capacity is in striking agreement with experimental data in adult rats. However, the calculation was performed using random uncorrelated inputs and outputs. This is a clearly unrealistic assumption since sensory in- puts and motor outputs carry a substantial degree of temporal correlations. In this paper, we consider a binary output neuron with a large number of inputs, which is required to store associations between temporally correlated sequences of binary inputs and outputs, modelled as Markov chains. Storage capacity is found to increase with both input and output correlations, and diverges in the limit where both go to unity. We also investigate the capacity of a bistable output unit, since PCs have been shown to be bistable in some experimental conditions. Bistability is shown to enhance storage capacity whenever the output correlation is stronger than the input correlation. Distribution of synaptic weights at maximal capacity is shown to be independent on correlations, and is also unaffected by the presence of bistability.

PLoS Computational Biology, 8(4): e1002448 (2012) doi:10.1371/journal.pcbi.1002448s

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## Entanglement between Demand and Supply in Markets with Bandwagon Goods

#### Mirta B. Gordon, Jean-Pierre Nadal, Denis Phan and Viktoriya Semeshenko

Whenever customers'choices (e.g. to buy or not a given good) depend on others choices (cases coined 'positive externalities' or 'bandwagon effect' in the economic literature), the demand may be multiply valued: for a same posted price, there is either a small number of buyers, or a large one -- in which case one says that the customers coordinate. This leads to a dilemma for the seller: should he sell at a high price, targeting a small number of buyers, or at low price targeting a large number of buyers? In this paper we show that the interaction between demand and supply is even more complex than expected, leading to what we call the curse of coordination: the pricing strategy for the seller which aimed at maximizing his profit corresponds to posting a price which, not only assumes that the customers will coordinate, but also lies very near the critical price value at which such high demand no more exists. This is obtained by the detailed mathematical analysis of a particular model formally related to the Random Field Ising Model and to a model introduced in social sciences by T. C. Schelling in the 70's.

Journal of Statistical Physics: Volume 151, Issue 3 (2013), Page 494-522 doi:10.1007/s10955-012-0660-1
Online 13 Dec. 2012 - Preprint arXiv:1209.1321

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## A model of riots dynamics: Shocks, diffusion and thresholds

#### Henri Berestycki, Jean-Pierre Nadal and Nancy Rodríguez

We introduce and analyze several variants of a system of differential equations which model the dynamics of social outbursts, such as riots. The systems involve the coupling of an explicit variable representing the intensity of rioting activity and an underlying (implicit) field of social tension. Our models include the effects of exogenous and endogenous factors as well as various propagation mechanisms. From numerical and mathematical analysis of these models we show that the assumptions made on how different locations influence one another and how the tension in the system disperses play a major role on the qualitative behavior of bursts of social unrest. Furthermore, we analyze here various properties of these systems, such as the existence of traveling wave solutions, and formulate some new open mathematical problems which arise from our work.

Networks and Heterogeneous Media (NHM), Vol. 10, Nber 3, pp. 443-475, September 2015 (online July 2015)
doi:10.3934/nhm.2015.10.443 (abstract and paper on NHM web site).
Preprint: arXiv:1502.04725

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## Représentation du langage et modèles d'évolution linguistique : la grammaticalisation comme perspective

#### Quentin Feltgen, Benjamin Fagard & Jean-Pierre Nadal

Though numerous numerical studies have investigated language change, grammaticalization and diachronic phenomena of language renewal have been left aside, or so it seems. We argue that previous models, dedicated to other purposes, make representational choices that cannot easily account for this type of phenomenon. In this paper we propose a new framework, aiming to depict linguistic renewal through numerical simulations. We illustrate it with a specific implementation which brings to light the phenomenon of semantic bleaching.

Traitement Automatique du Langage (TAL), 2014 Volume 55 Num. 3, pp. 47-71
Paper
online May 2016 (article in French).
Related works, same authors (in English):
"Modeling Language Change: The Pitfall of Grammaticalization", book chapter, in "Language in Complexity: The Emerging Meaning", Springer 2016, pp. 49-72
,
and paper below.

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## Cerebellar learning using perturbations

#### G. Bouvier, C. Clopath, C. Bimbard, J.-P. Nadal, N. Brunel, V. Hakim and B. Barbour

The cerebellum aids the learning and execution of fast coordinated movements, with acquired information being stored by plasticity of parallel fibre--Purkinje cell synapses. According to the current consensus, erroneously active parallel fibre synapses are depressed by complex spikes arising as climbing fibres signal movement errors. However, this theory cannot solve the credit assignment problem of using the limited information from a global movement evaluation to optimise behaviour by guiding the plasticity in numerous neurones. We identify the possible implementation of an algorithm solving this problem, whereby spontaneous complex spikes perturb ongoing movements, create an eligibility trace for plasticity and signal resulting error changes to guide plasticity. These error changes are extracted by adaptively cancelling the average error. This framework, stochastic gradient descent with estimated global errors, generates specific predictions for synaptic plasticity rules that contradict the current consensus. However, in vitro plasticity experiments under physiological conditions verified our predictions, highlighting the sensitivity of plasticity studies to unphysiological conditions. Using numerical and analytical approaches we demonstrate the convergence and estimate the capacity of learning in our implementation. Finally, a similar mechanism may operate during optimisation of action sequences by the basal ganglia, where dopamine could both initiate movements and signal rewards, analogously to the dual perturbation and correction role of the climbing fibre outlined here.

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## Epidemiological modeling of the 2005 French riots: a spreading wave and the role of contagion

#### Laurent Bonnasse-Gahot, Henri Berestycki, Marie-Aude Depuiset, Mirta B. Gordon, Sebastian Roché, Nancy Rodríguez and Jean-Pierre Nadal

As a large-scale instance of dramatic collective behavior, the 2005 French riots started in a poor suburb of Paris, then spread in all of France, lasting about three weeks. Remarkably, although there were no displacements of rioters, the riot activity did traveled. Daily national police data to which we had access have allowed us to take advantage of this natural experiment to explore the dynamics of riot propagation. Here we show that an epidemic-like model, with less than 10 free parameters and a single sociological variable characterizing neighborhood deprivation, accounts quantitatively for the full spatio-temporal dynamics of the riots. This is the first time that such data-driven modeling involving contagion both within and between cities (through geographic proximity or media) at the scale of a country is performed. Moreover, we give a precise mathematical characterization to the expression `wave of riots'', and provide a visualization of the propagation around Paris, exhibiting the wave in a way not described before. The remarkable agreement between model and data demonstrates that geographic proximity played a major role in the riot propagation, even though information was readily available everywhere through media. Finally, we argue that our approach gives a general framework for the modeling of spontaneous collective uprisings.

Submitted - arXiv preprint http://arxiv.org/abs/1701.07479 (Jan. 25, 2017).

Supporting information, videos: see here.

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## Frequency patterns of semantic change: Corpus-based evidence of a near-critical dynamics in language change

#### Quentin Feltgen, Benjamin Fagard and Jean-Pierre Nadal

It is generally believed that, when a linguistic item acquires a new meaning, its overall frequency of use in the language rises with time with an S-shaped growth curve. Yet, this claim has only been supported by a limited number of case studies. In this paper, we provide the first corpus-based quantitative confirmation of the genericity of the S-curve in language change. Moreover, we uncover another generic pattern, a latency phase of variable duration preceding the S-growth, during which the frequency of use of the semantically expanding word remains low and more or less constant. We also propose a usage-based model of language change supported by cognitive considerations, which predicts that both phases, the latency and the fast S-growth, take place. The driving mechanism is a stochastic dynamics, a random walk in the space of frequency of use. The underlying deterministic dynamics highlights the role of a control parameter, the strength of the cognitive impetus governing the onset of change, which tunes the system at the vicinity of a saddle-node bifurcation. In the neighborhood of the critical point, the latency phase corresponds to the diffusion time over the critical region, and the S-growth to the fast convergence that follows. The duration of the two phases is computed as specific first passage times of the random walk process, leading to distributions that fit well the ones extracted from our dataset. We argue that our results are not specific to the studied corpus, but apply to semantic change in general.

Submitted. arXiv preprint https://arxiv.org/abs/1703.00203 (Mar. 1st, 2017).
Related works, same authors: here (in French) and here.

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