Inferring network connectivity using kinetic Ising models

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Inference of kinetic Ising model on sparse graphs

Based on dynamical cavity method, we propose an approach to the inference of kinetic Ising model, which asks to reconstruct couplings and external fields from given time-dependent output of original system. Our approach gives an exact result on tree graphs and a good approximation on sparse graphs, it can be seen as an extension of Belief Propagation inference of static Ising model to kinetic Ising model. While existing mean field methods to the kinetic Ising inference e.g., na\" ive mean-field, TAP equation and simply mean-field, use approximations which calculate magnetizations and correlations at time $t$ from statistics of data at time $t-1$, dynamical cavity method can use statistics of data at times earlier than $t-1$ to capture more correlations at different time steps. Extensive numerical experiments show that our inference method is superior to existing mean-field approaches on diluted networks.Comment: 9 pages, 3 figures, comments are welcom

The Effect of Nonstationarity on Models Inferred from Neural Data

Neurons subject to a common non-stationary input may exhibit a correlated firing behavior. Correlations in the statistics of neural spike trains also arise as the effect of interaction between neurons. Here we show that these two situations can be distinguished, with machine learning techniques, provided the data are rich enough. In order to do this, we study the problem of inferring a kinetic Ising model, stationary or nonstationary, from the available data. We apply the inference procedure to two data sets: one from salamander retinal ganglion cells and the other from a realistic computational cortical network model. We show that many aspects of the concerted activity of the salamander retinal neurons can be traced simply to the external input. A model of non-interacting neurons subject to a non-stationary external field outperforms a model with stationary input with couplings between neurons, even accounting for the differences in the number of model parameters. When couplings are added to the non-stationary model, for the retinal data, little is gained: the inferred couplings are generally not significant. Likewise, the distribution of the sizes of sets of neurons that spike simultaneously and the frequency of spike patterns as function of their rank (Zipf plots) are well-explained by an independent-neuron model with time-dependent external input, and adding connections to such a model does not offer significant improvement. For the cortical model data, robust couplings, well correlated with the real connections, can be inferred using the non-stationary model. Adding connections to this model slightly improves the agreement with the data for the probability of synchronous spikes but hardly affects the Zipf plot.Comment: version in press in J Stat Mec

Effect of coupling asymmetry on mean-field solutions of direct and inverse Sherrington-Kirkpatrick model

We study how the degree of symmetry in the couplings influences the performance of three mean field methods used for solving the direct and inverse problems for generalized Sherrington-Kirkpatrick models. In this context, the direct problem is predicting the potentially time-varying magnetizations. The three theories include the first and second order Plefka expansions, referred to as naive mean field (nMF) and TAP, respectively, and a mean field theory which is exact for fully asymmetric couplings. We call the last of these simply MF theory. We show that for the direct problem, nMF performs worse than the other two approximations, TAP outperforms MF when the coupling matrix is nearly symmetric, while MF works better when it is strongly asymmetric. For the inverse problem, MF performs better than both TAP and nMF, although an ad hoc adjustment of TAP can make it comparable to MF. For high temperatures the performance of TAP and MF approach each other

Bump formation in a binary attractor neural network

This paper investigates the conditions for the formation of local bumps in the activity of binary attractor neural networks with spatially dependent connectivity. We show that these formations are observed when asymmetry between the activity during the retrieval and learning is imposed. Analytical approximation for the order parameters is derived. The corresponding phase diagram shows a relatively large and stable region, where this effect is observed, although the critical storage and the information capacities drastically decrease inside that region. We demonstrate that the stability of the network, when starting from the bump formation, is larger than the stability when starting even from the whole pattern. Finally, we show a very good agreement between the analytical results and the simulations performed for different topologies of the network.Comment: about 14 page

Stimulus-dependent maximum entropy models of neural population codes

Neural populations encode information about their stimulus in a collective fashion, by joint activity patterns of spiking and silence. A full account of this mapping from stimulus to neural activity is given by the conditional probability distribution over neural codewords given the sensory input. To be able to infer a model for this distribution from large-scale neural recordings, we introduce a stimulus-dependent maximum entropy (SDME) model---a minimal extension of the canonical linear-nonlinear model of a single neuron, to a pairwise-coupled neural population. The model is able to capture the single-cell response properties as well as the correlations in neural spiking due to shared stimulus and due to effective neuron-to-neuron connections. Here we show that in a population of 100 retinal ganglion cells in the salamander retina responding to temporal white-noise stimuli, dependencies between cells play an important encoding role. As a result, the SDME model gives a more accurate account of single cell responses and in particular outperforms uncoupled models in reproducing the distributions of codewords emitted in response to a stimulus. We show how the SDME model, in conjunction with static maximum entropy models of population vocabulary, can be used to estimate information-theoretic quantities like surprise and information transmission in a neural population.Comment: 11 pages, 7 figure

U.S. stock market interaction network as learned by the Boltzmann Machine

We study historical dynamics of joint equilibrium distribution of stock returns in the U.S. stock market using the Boltzmann distribution model being parametrized by external fields and pairwise couplings. Within Boltzmann learning framework for statistical inference, we analyze historical behavior of the parameters inferred using exact and approximate learning algorithms. Since the model and inference methods require use of binary variables, effect of this mapping of continuous returns to the discrete domain is studied. The presented analysis shows that binarization preserves market correlation structure. Properties of distributions of external fields and couplings as well as industry sector clustering structure are studied for different historical dates and moving window sizes. We found that a heavy positive tail in the distribution of couplings is responsible for the sparse market clustering structure. We also show that discrepancies between the model parameters might be used as a precursor of financial instabilities.Comment: 15 pages, 17 figures, 1 tabl

Beyond inverse Ising model: structure of the analytical solution for a class of inverse problems

I consider the problem of deriving couplings of a statistical model from measured correlations, a task which generalizes the well-known inverse Ising problem. After reminding that such problem can be mapped on the one of expressing the entropy of a system as a function of its corresponding observables, I show the conditions under which this can be done without resorting to iterative algorithms. I find that inverse problems are local (the inverse Fisher information is sparse) whenever the corresponding models have a factorized form, and the entropy can be split in a sum of small cluster contributions. I illustrate these ideas through two examples (the Ising model on a tree and the one-dimensional periodic chain with arbitrary order interaction) and support the results with numerical simulations. The extension of these methods to more general scenarios is finally discussed.Comment: 15 pages, 6 figure

Generalized mean field approximation for parallel dynamics of the Ising model

The dynamics of the non-equilibrium Ising model with parallel updates is investigated using a generalized mean field approximation that incorporates multiple two-site correlations at any two time steps, which can be obtained recursively. The proposed method shows significant improvement in predicting local system properties compared to other mean field approximation techniques, particularly in systems with symmetric interactions. Results are also evaluated against those obtained from Monte Carlo simulations. The method is also employed to obtain parameter values for the kinetic inverse Ising modeling problem, where couplings and local field values of a fully connected spin system are inferred from data. © 2014 IOP Publishing Ltd and SISSA Medialab srl

Statistical pairwise interaction model of stock market

Financial markets are a classical example of complex systems as they comprise many interacting stocks. As such, we can obtain a surprisingly good description of their structure by making the rough simplification of binary daily returns. Spin glass models have been applied and gave some valuable results but at the price of restrictive assumptions on the market dynamics or others are agent-based models with rules designed in order to recover some empirical behaviours. Here we show that the pairwise model is actually a statistically consistent model with observed first and second moments of the stocks orientation without making such restrictive assumptions. This is done with an approach based only on empirical data of price returns. Our data analysis of six major indices suggests that the actual interaction structure may be thought as an Ising model on a complex network with interaction strengths scaling as the inverse of the system size. This has potentially important implications since many properties of such a model are already known and some techniques of the spin glass theory can be straightforwardly applied. Typical behaviours, as multiple equilibria or metastable states, different characteristic time scales, spatial patterns, order-disorder, could find an explanation in this picture.Comment: 11 pages, 8 figure