Dynamics of spin systems with randomly asymmetric bonds: Ising spins and Glauber dynamics

Escaping Kinetic Traps Using Nonreciprocal Interactions

Kinetic traps are a notorious problem in equilibrium statistical mechanics, where temperature quenches ultimately fail to bring the system to low energy configurations. Using multifarious self-assembly as a model system, we introduce a mechanism to escape kinetic traps by utilizing nonreciprocal interactions between components. Introducing nonequilibrium effects offered by broken action-reaction symmetry in the system pushes the trajectory of the system out of arrested dynamics. The dynamics of the model is studied using tools from the physics of interfaces and defects. Our proposal can find applications in self-assembly, glassy systems, and systems with arrested dynamics to facilitate escape from local minima in rough energy landscapes.

Does the brain behave like a (complex) network? I. Dynamics

Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.

Machine learning at the mesoscale: A computation-dissipation bottleneck

The cost of information processing in physical systems calls for a trade-off between performance and energetic expenditure. Here we formulate and study a computation-dissipation bottleneck in mesoscopic systems used as input-output devices. Using both real data sets and synthetic tasks, we show how nonequilibrium leads to enhanced performance. Our framework sheds light on a crucial compromise between information compression, input-output computation and dynamic irreversibility induced by nonreciprocal interactions.

Nonequilibrium thermodynamics of the asymmetric Sherrington-Kirkpatrick model

Most natural systems operate far from equilibrium, displaying time-asymmetric, irreversible dynamics characterized by a positive entropy production while exchanging energy and matter with the environment. Although stochastic thermodynamics underpins the irreversible dynamics of small systems, the nonequilibrium thermodynamics of larger, more complex systems remains unexplored. Here, we investigate the asymmetric Sherrington-Kirkpatrick model with synchronous and asynchronous updates as a prototypical example of large-scale nonequilibrium processes. Using a path integral method, we calculate a generating functional over trajectories, obtaining exact solutions of the order parameters, path entropy, and steady-state entropy production of infinitely large networks. Entropy production peaks at critical order-disorder phase transitions, but is significantly larger for quasi-deterministic disordered dynamics. Consequently, entropy production can increase under distinct scenarios, requiring multiple thermodynamic quantities to describe the system accurately. These results contribute to developing an exact analytical theory of the nonequilibrium thermodynamics of large-scale physical and biological systems and their phase transitions.

Modelling time-varying interactions in complex systems: the Score Driven Kinetic Ising Model

A common issue when analyzing real-world complex systems is that the interactions between their elements often change over time. Here we propose a new modeling approach for time-varying interactions generalising the well-known Kinetic Ising Model, a minimalistic pairwise constant interactions model which has found applications in several scientific disciplines. Keeping arbitrary choices of dynamics to a minimum and seeking information theoretical optimality, the Score-Driven methodology allows to extract from data and interpret the presence of temporal patterns describing time-varying interactions. We identify a parameter whose value at a given time can be directly associated with the local predictability of the dynamics and we introduce a method to dynamically learn its value from the data, without specifying parametrically the system's dynamics. We extend our framework to disentangle different sources (e.g. endogenous vs exogenous) of predictability in real time, and show how our methodology applies to a variety of complex systems such as financial markets, temporal (social) networks, and neuronal populations.

From random point processes to hierarchical cavity master equations for stochastic dynamics of disordered systems in random graphs: Ising models and epidemics

We start from the theory of random point processes to derive n-point coupled master equations describing the continuous dynamics of discrete variables in random graphs. These equations constitute a hierarchical set of approximations that generalize and improve the cavity master equation (CME), a recently obtained closure for the usual master equation representing the dynamics. Our derivation clarifies some of the hypotheses and approximations that originally led to the CME, considered now as the first order of a more general technique. We tested the scheme in the dynamics of three models defined over diluted graphs: the Ising ferromagnet, the Viana-Bray spin-glass, and the susceptible-infectious-susceptible model for epidemics. In the first two, the equations perform similarly to the best-known approaches in literature. In the latter, they outperform the well-known pair quenched mean-field approximation.

Overcoming the complexity barrier of the dynamic message-passing method in networks with fat-tailed degree distributions

The dynamic cavity method provides the most efficient way to evaluate probabilities of dynamic trajectories in systems of stochastic units with unidirectional sparse interactions. It is closely related to sum-product algorithms widely used to compute marginal functions from complicated global functions of many variables, with applications in disordered systems, combinatorial optimization, and computer science. However, the complexity of the cavity approach grows exponentially with the in-degrees of the interacting units, which creates a defacto barrier for the successful analysis of systems with fat-tailed in-degree distributions. In this paper, we present a dynamic programming algorithm that overcomes this barrier by reducing the computational complexity in the in-degrees from exponential to quadratic, whenever couplings are chosen randomly from (or can be approximated in terms of) discrete, possibly unit-dependent, sets of equidistant values. As a case study, we analyze the dynamics of a random Boolean network with a fat-tailed degree distribution and fully asymmetric binary ±J couplings, and we use the power of the algorithm to unlock the noise-dependent heterogeneity of stationary node activation patterns in such a system.

Persistent individual bias in a voter model with quenched disorder

Many theoretical studies of the voter model (or variations thereupon) involve order parameters that are population-averaged. While enlightening, such quantities may obscure important statistical features that are only apparent on the level of the individual. In this work, we ask which factors contribute to a single voter maintaining a long-term statistical bias for one opinion over the other in the face of social influence. To this end, a modified version of the network voter model is proposed, which also incorporates quenched disorder in the interaction strengths between individuals and the possibility of antagonistic relationships. We find that a sparse interaction network and heterogeneity in interaction strengths give rise to the possibility of arbitrarily long-lived individual biases, even when there is no population-averaged bias for one opinion over the other. This is demonstrated by calculating the eigenvalue spectrum of the weighted network Laplacian using the theory of sparse random matrices.

Analysis of temporal correlation in heart rate variability through maximum entropy principle in a minimal pairwise glassy model

In this work we apply statistical mechanics tools to infer cardiac pathologies over a sample of M patients whose heart rate variability has been recorded via 24 h Holter device and that are divided in different classes according to their clinical status (providing a repository of labelled data). Considering the set of inter-beat interval sequences \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\mathbf {r}(i) \} = \{ r_1(i), r_2(i), \ldots , \}$$\end{document}{r(i)}={r1(i),r2(i),…,}, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\ldots ,M$$\end{document}i=1,…,M, we estimate their probability distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(\mathbf {r})$$\end{document}P(r) exploiting the maximum entropy principle. By setting constraints on the first and on the second moment we obtain an effective pairwise \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(r_n,r_m)$$\end{document}(rn,rm) model, whose parameters are shown to depend on the clinical status of the patient. In order to check this framework, we generate synthetic data from our model and we show that their distribution is in excellent agreement with the one obtained from experimental data. Further, our model can be related to a one-dimensional spin-glass with quenched long-range couplings decaying with the spin–spin distance as a power-law. This allows us to speculate that the 1/f noise typical of heart-rate variability may stem from the interplay between the parasympathetic and orthosympathetic systems.

Inference of the kinetic Ising model with heterogeneous missing data

We consider the problem of inferring a causality structure from multiple binary time series by using the kinetic Ising model in datasets where a fraction of observations is missing. Inspired by recent work on mean field methods for the inference of the model with hidden spins, we develop a pseudo-expectation-maximization algorithm that is able to work even in conditions of severe data sparsity. The methodology relies on the Martin-Siggia-Rose path integral method with second-order saddle-point solution to make it possible to approximate the log-likelihood in polynomial time, giving as output an estimate of the couplings matrix and of the missing observations. We also propose a recursive version of the algorithm, where at every iteration some missing values are substituted by their maximum-likelihood estimate, showing that the method can be used together with sparsification schemes such as lasso regularization or decimation. We test the performance of the algorithm on synthetic data and find interesting properties regarding the dependency on heterogeneity of the observation frequency of spins and when some of the hypotheses that are necessary to the saddle-point approximation are violated, such as the small couplings limit and the assumption of statistical independence between couplings.

Dynamics and orientation selectivity in a cortical model of rodent V1 with excess bidirectional connections

Recent experiments have revealed fine structure in cortical microcircuitry. In particular, bidirectional connections are more prevalent than expected by chance. Whether this fine structure affects cortical dynamics and function has not yet been studied. Here we investigate the effects of excess bidirectionality in a strongly recurrent network model of rodent V1. We show that reciprocal connections have only a very weak effect on orientation selectivity. We find that excess reciprocity between inhibitory neurons slows down the dynamics and strongly increases the Fano factor, while for reciprocal connections between excitatory and inhibitory neurons it has the opposite effect. In contrast, excess bidirectionality within the excitatory population has a minor effect on the neuronal dynamics. These results can be explained by an effective delayed neuronal self-coupling which stems from the fine structure. Our work suggests that excess bidirectionality between inhibitory neurons decreases the efficiency of feature encoding in cortex by reducing the signal to noise ratio. On the other hand it implies that the experimentally observed strong reciprocity between excitatory and inhibitory neurons improves the feature encoding.

Ubiquity of collective irregular dynamics in balanced networks of spiking neurons

We revisit the dynamics of a prototypical model of balanced activity in networks of spiking neurons. A detailed investigation of the thermodynamic limit for fixed density of connections (massive coupling) shows that, when inhibition prevails, the asymptotic regime is not asynchronous but rather characterized by a self-sustained irregular, macroscopic (collective) dynamics. So long as the connectivity is massive, this regime is found in many different setups: leaky as well as quadratic integrate-and-fire neurons; large and small coupling strength; and weak and strong external currents.

Pattern Storage, Bifurcations, and Groupwise Correlation Structure of an Exactly Solvable Asymmetric Neural Network Model

Despite their biological plausibility, neural network models with asymmetric weights are rarely solved analytically, and closed-form solutions are available only in some limiting cases or in some mean-field approximations. We found exact analytical solutions of an asymmetric spin model of neural networks with arbitrary size without resorting to any approximation, and we comprehensively studied its dynamical and statistical properties. The network had discrete time evolution equations and binary firing rates, and it could be driven by noise with any distribution. We found analytical expressions of the conditional and stationary joint probability distributions of the membrane potentials and the firing rates. By manipulating the conditional probability distribution of the firing rates, we extend to stochastic networks the associating learning rule previously introduced by Personnaz and coworkers. The new learning rule allowed the safe storage, under the presence of noise, of point and cyclic attractors, with useful implications for content-addressable memories. Furthermore, we studied the bifurcation structure of the network dynamics in the zero-noise limit. We analytically derived examples of the codimension 1 and codimension 2 bifurcation diagrams of the network, which describe how the neuronal dynamics changes with the external stimuli. This showed that the network may undergo transitions among multistable regimes, oscillatory behavior elicited by asymmetric synaptic connections, and various forms of spontaneous symmetry breaking. We also calculated analytically groupwise correlations of neural activity in the network in the stationary regime. This revealed neuronal regimes where, statistically, the membrane potentials and the firing rates are either synchronous or asynchronous. Our results are valid for networks with any number of neurons, although our equations can be realistically solved only for small networks. For completeness, we also derived the network equations in the thermodynamic limit of infinite network size and we analytically studied their local bifurcations. All the analytical results were extensively validated by numerical simulations.

Path-integral methods for analyzing the effects of fluctuations in stochastic hybrid neural networks

We consider applications of path-integral methods to the analysis of a stochastic hybrid model representing a network of synaptically coupled spiking neuronal populations. The state of each local population is described in terms of two stochastic variables, a continuous synaptic variable and a discrete activity variable. The synaptic variables evolve according to piecewise-deterministic dynamics describing, at the population level, synapses driven by spiking activity. The dynamical equations for the synaptic currents are only valid between jumps in spiking activity, and the latter are described by a jump Markov process whose transition rates depend on the synaptic variables. We assume a separation of time scales between fast spiking dynamics with time constant [Formula: see text] and slower synaptic dynamics with time constant τ. This naturally introduces a small positive parameter [Formula: see text], which can be used to develop various asymptotic expansions of the corresponding path-integral representation of the stochastic dynamics. First, we derive a variational principle for maximum-likelihood paths of escape from a metastable state (large deviations in the small noise limit [Formula: see text]). We then show how the path integral provides an efficient method for obtaining a diffusion approximation of the hybrid system for small ϵ. The resulting Langevin equation can be used to analyze the effects of fluctuations within the basin of attraction of a metastable state, that is, ignoring the effects of large deviations. We illustrate this by using the Langevin approximation to analyze the effects of intrinsic noise on pattern formation in a spatially structured hybrid network. In particular, we show how noise enlarges the parameter regime over which patterns occur, in an analogous fashion to PDEs. Finally, we carry out a [Formula: see text]-loop expansion of the path integral, and use this to derive corrections to voltage-based mean-field equations, analogous to the modified activity-based equations generated from a neural master equation.

Dynamic mean-field and cavity methods for diluted Ising systems

We compare dynamic mean-field and dynamic cavity methods to describe the stationary states of dilute kinetic Ising models. We compute dynamic mean-field theory by expanding in interaction strength to third order, and we compare to the exact dynamic mean-field theory for fully asymmetric networks. We show that in diluted networks, the dynamic cavity method generally predicts magnetizations of individual spins better than both first-order ("naive") and second-order ("TAP") dynamic mean-field theory.

Statistical physics of pairwise probability models

Statistical models for describing the probability distribution over the states of biological systems are commonly used for dimensional reduction. Among these models, pairwise models are very attractive in part because they can be fit using a reasonable amount of data: knowledge of the mean values and correlations between pairs of elements in the system is sufficient. Not surprisingly, then, using pairwise models for studying neural data has been the focus of many studies in recent years. In this paper, we describe how tools from statistical physics can be employed for studying and using pairwise models. We build on our previous work on the subject and study the relation between different methods for fitting these models and evaluating their quality. In particular, using data from simulated cortical networks we study how the quality of various approximate methods for inferring the parameters in a pairwise model depends on the time bin chosen for binning the data. We also study the effect of the size of the time bin on the model quality itself, again using simulated data. We show that using finer time bins increases the quality of the pairwise model. We offer new ways of deriving the expressions reported in our previous work for assessing the quality of pairwise models.

Learning of spatiotemporal patterns in Ising-spin neural networks: analysis of storage capacity by path integral methods

We encode periodic spatiotemporal patterns in Ising-spin neural networks, using the simple learning rule inspired by the spike-timing-dependent synaptic plasticity. It is then found that periodically oscillating spin neurons successfully reproduce phase differences of the encoded periodic patterns. The storage capacity of this associative memory neural network is enhanced with an adequate level of asymmetry in synapse connections. To understand the properties of these nonequilibrium retrieval states of the neural network, we carry out an analysis based on a path integral method. The relation of a dynamic crosstalk term to time-persistent oscillation of a correlation function well explains the enhancement of the storage capacity in spite of our approximation on nonpersistent terms. We investigate the accuracy of this approximation further by detailed comparison with numerical simulations.

A constructive mean-field analysis of multi-population neural networks with random synaptic weights and stochastic inputs

We deal with the problem of bridging the gap between two scales in neuronal modeling. At the first (microscopic) scale, neurons are considered individually and their behavior described by stochastic differential equations that govern the time variations of their membrane potentials. They are coupled by synaptic connections acting on their resulting activity, a nonlinear function of their membrane potential. At the second (mesoscopic) scale, interacting populations of neurons are described individually by similar equations. The equations describing the dynamical and the stationary mean-field behaviors are considered as functional equations on a set of stochastic processes. Using this new point of view allows us to prove that these equations are well-posed on any finite time interval and to provide a constructive method for effectively computing their unique solution. This method is proved to converge to the unique solution and we characterize its complexity and convergence rate. We also provide partial results for the stationary problem on infinite time intervals. These results shed some new light on such neural mass models as the one of Jansen and Rit (1995): their dynamics appears as a coarse approximation of the much richer dynamics that emerges from our analysis. Our numerical experiments confirm that the framework we propose and the numerical methods we derive from it provide a new and powerful tool for the exploration of neural behaviors at different scales.

Visual recognition and inference using dynamic overcomplete sparse learning

We present a hierarchical architecture and learning algorithm for visual recognition and other visual inference tasks such as imagination, reconstruction of occluded images, and expectation-driven segmentation. Using properties of biological vision for guidance, we posit a stochastic generative world model and from it develop a simplified world model (SWM) based on a tractable variational approximation that is designed to enforce sparse coding. Recent developments in computational methods for learning overcomplete representations (Lewicki & Sejnowski, 2000; Teh, Welling, Osindero, & Hinton, 2003) suggest that overcompleteness can be useful for visual tasks, and we use an overcomplete dictionary learning algorithm (Kreutz-Delgado, et al., 2003) as a preprocessing stage to produce accurate, sparse codings of images. Inference is performed by constructing a dynamic multilayer network with feedforward, feedback, and lateral connections, which is trained to approximate the SWM. Learning is done with a variant of the back-propagation-through-time algorithm, which encourages convergence to desired states within a fixed number of iterations. Vision tasks require large networks, and to make learning efficient, we take advantage of the sparsity of each layer to update only a small subset of elements in a large weight matrix at each iteration. Experiments on a set of rotated objects demonstrate various types of visual inference and show that increasing the degree of overcompleteness improves recognition performance in difficult scenes with occluded objects in clutter.

Signal buffering in random networks of spiking neurons: microscopic versus macroscopic phenomena

In randomly connected networks of pulse-coupled elements a time-dependent input signal can be buffered over a short time. We studied the signal buffering properties in simulated networks as a function of the networks' state, characterized by both the Lyapunov exponent of the microscopic dynamics and the macroscopic activity derived from mean-field theory. If all network elements receive the same signal, signal buffering over delays comparable to the intrinsic time constant of the network elements can be explained by macroscopic properties and works best at the phase transition to chaos. However, if only 20% of the network units receive a common time-dependent signal, signal buffering properties improve and can no longer be attributed to the macroscopic dynamics.

The dynamic neural filter: a binary model of spatiotemporal coding

We describe and discuss the properties of a binary neural network that can serve as a dynamic neural filter (DNF), which maps regions of input space into spatiotemporal sequences of neuronal activity. Both deterministic and stochastic dynamics are studied, allowing the investigation of the stability of spatiotemporal sequences under noisy conditions. We define a measure of the coding capacity of a DNF and develop an algorithm for constructing a DNF that can serve as a source of given codes. On the basis of this algorithm, we suggest using a minimal DNF capable of generating observed sequences as a measure of complexity of spatiotemporal data. This measure is applied to experimental observations in the locust olfactory system, whose reverberating local field potential provides a natural temporal scale allowing the use of a binary DNF. For random synaptic matrices, a DNF can generate very large cycles, thus becoming an efficient tool for producing spatiotemporal codes. The latter can be stabilized by applying to the parameters of the DNF a learning algorithm with suitable margins.

Dynamical working memory and timed responses: the role of reverberating loops in the olivo-cerebellar system

This article explores dynamical properties of the olivo-cerebellar system that arise from the specific wiring of inferior olive (IO), cerebellar cortex, and deep cerebellar nuclei (DCN). We show that the irregularity observed in the firing pattern of the IO neurons is not necessarily produced by noise but can instead be the result of a purely deterministic network effect. We propose that this effect can serve as a dynamical working memory or as a neuronal clock with a characteristic timescale of about 100 ms that is determined by the slow calcium dynamics of IO and DCN neurons. This concept provides a novel explanation of how the cerebellum can solve timing tasks on a timescale that is two orders of magnitude longer than the millisecond timescale usually attributed to neuronal dynamics. One of the key ingredients of our model is the observation that due to postinhibitory rebound, DCN neurons can be driven by GABAergic ("inhibitory") input from cerebellar Purkinje cells. Topographic projections from the DCN to the IO form a closed reverberating loop with an overall synaptic transmission delay of about 100 ms that is in resonance with the intrinsic oscillatory properties of the inferior olive. We use a simple time-discrete model based on McCulloch-Pitts neurons in order to investigate in a first step some of the fundamental properties of a network with delayed reverberating projections. The macroscopic behavior is analyzed by means of a mean-field approximation. Numerical simulations, however, show that the microscopic dynamics has a surprisingly rich structure that does not show up in a mean-field description. We have thus performed extensive numerical experiments in order to quantify the ability of the network to serve as a dynamical working memory and its vulnerability by noise. In a second step, we develop a more realistic conductance-based network model of the inferior olive consisting of about 20 multicompartment neurons that are coupled by gap junctions and receive excitatory and inhibitory synaptic input via AMPA and GABAergic synapses. The simulations show that results for the time-discrete model hold true in a time-continuous description.

Dynamic ultrametricity in spin glasses

We investigate the dynamics of spin glasses from the "rheological" point of view, in which aging is suppressed by the action of small, nonconservative forces. The different features can be expressed in terms of the scaling of relaxation times with the magnitude of the driving force, which plays the role of the critical parameter. Stated in these terms, ultrametricity loses much of its mystery and can be checked rather easily. This approach also seems a natural starting point to investigate what would be the real-space structures underlying the hierarchy of time scales. We study in detail the appearance of this many-scale behavior in a mean-field model, in which dynamic ultrametricity is clearly present. A similar analysis is performed on numerical results obtained for a three-dimensional spin glass: In that case, our results are compatible with either that dynamic ultrametricity is absent or that it develops so slowly that even in experimental time-windows it is still hardly observable.

Retrieval properties of a Hopfield model with random asymmetric interactions

The process of pattern retrieval in a Hopfield model in which a random antisymmetric component is added to the otherwise symmetric synaptic matrix is studied by computer simulations. The introduction of the anti-symmetric component is found to increase the fraction of random inputs that converge to the memory states. However, the size of the basin of attraction of a memory state does not show any significant change when asymmetry is introduced in the synaptic matrix. We show that this is due to the fact that the spurious fixed points, which are destabilized by the introduction of asymmetry, have very small basins of attraction. The convergence time to spurious fixed-point attractors increases faster than that for the memory states as the asymmetry parameter is increased. The possibility of convergence to spurious fixed points is greatly reduced if a suitable upper limit is set for the convergence time. This prescription works better if the synaptic matrix has an antisymmetric component.