Theory and Practice of Computing with Excitable Dynamics

Reservoir computing (RC) is a promising paradigm for time series processing. In this paradigm, the desired output is computed by combining measurements of an excitable system that responds to time-dependent exogenous stimuli. The excitable system is called a reservoir and measurements of its state are combined using a readout layer to produce a target output. The power of RC is attributed to an emergent short-term memory in dynamical systems and has been analyzed mathematically for both linear and nonlinear dynamical systems. The theory of RC treats only the macroscopic properties of the reservoir, without reference to the underlying medium it is made of. As a result, RC is particularly attractive for building computational devices using emerging technologies whose structure is not exactly controllable, such as self-assembled nanoscale circuits. RC has lacked a formal framework for performance analysis and prediction that goes beyond memory properties. To provide such a framework, here a mathematical theory of memory and information processing in ordered and disordered linear dynamical systems is developed. This theory analyzes the optimal readout layer for a given task. The focus of the theory is a standard model of RC, the echo state network (ESN). An ESN consists of a fixed recurrent neural network that is driven by an external signal. The dynamics of the network is then combined linearly with readout weights to produce the desired output. The readout weights are calculated using linear regression. Using an analysis of regression equations, the readout weights can be calculated using only the statistical properties of the reservoir dynamics, the input signal, and the desired output. The readout layer weights can be calculated from a priori knowledge of the desired function to be computed and the weight matrix of the reservoir. This formulation explicitly depends on the input weights, the reservoir weights, and the statistics of the target function. This formulation is used to bound the expected error of the system for a given target function. The effects of input-output correlation and complex network structure in the reservoir on the computational performance of the system have been mathematically characterized. Far from the chaotic regime, ordered linear networks exhibit a homogeneous decay of memory in different dimensions, which keeps the input history coherent. As disorder is introduced in the structure of the network, memory decay becomes inhomogeneous along different dimensions causing decoherence in the input history, and degradation in task-solving performance. Close to the chaotic regime, the ordered systems show loss of temporal information in the input history, and therefore inability to solve tasks. However, by introducing disorder and therefore heterogeneous decay of memory the temporal information of input history is preserved and the task-solving performance is recovered. Thus for systems at the edge of chaos, disordered structure may enhance temporal information processing. Although the current framework only applies to linear systems, in principle it can be used to describe the properties of physical reservoir computing, e.g., photonic RC using short coherence-length light

Statistical Models for Co-occurrence Data

Modeling and predicting co-occurrences of events is a fundamental problem of unsupervised learning. In this contribution we develop a statistical framework for analyzing co-occurrence data in a general setting where elementary observations are joint occurrences of pairs of abstract objects from two finite sets. The main challenge for statistical models in this context is to overcome the inherent data sparseness and to estimate the probabilities for pairs which were rarely observed or even unobserved in a given sample set. Moreover, it is often of considerable interest to extract grouping structure or to find a hierarchical data organization. A novel family of mixture models is proposed which explain the observed data by a finite number of shared aspects or clusters. This provides a common framework for statistical inference and structure discovery and also includes several recently proposed models as special cases. Adopting the maximum likelihood principle, EM algorithms are derived to fit the model parameters. We develop improved versions of EM which largely avoid overfitting problems and overcome the inherent locality of EM--based optimization. Among the broad variety of possible applications, e.g., in information retrieval, natural language processing, data mining, and computer vision, we have chosen document retrieval, the statistical analysis of noun/adjective co-occurrence and the unsupervised segmentation of textured images to test and evaluate the proposed algorithms

The Kuramoto model in complex networks

181 pages, 48 figures. In Press, Accepted Manuscript, Physics Reports 2015 Acknowledgments We are indebted with B. Sonnenschein, E. R. dos Santos, P. Schultz, C. Grabow, M. Ha and C. Choi for insightful and helpful discussions. T.P. acknowledges FAPESP (No. 2012/22160-7 and No. 2015/02486-3) and IRTG 1740. P.J. thanks founding from the China Scholarship Council (CSC). F.A.R. acknowledges CNPq (Grant No. 305940/2010-4) and FAPESP (Grants No. 2011/50761-2 and No. 2013/26416-9) for financial support. J.K. would like to acknowledge IRTG 1740 (DFG and FAPESP).Peer reviewedPreprin

Mutual information in random Boolean models of regulatory networks

The amount of mutual information contained in time series of two elements gives a measure of how well their activities are coordinated. In a large, complex network of interacting elements, such as a genetic regulatory network within a cell, the average of the mutual information over all pairs is a global measure of how well the system can coordinate its internal dynamics. We study this average pairwise mutual information in random Boolean networks (RBNs) as a function of the distribution of Boolean rules implemented at each element, assuming that the links in the network are randomly placed. Efficient numerical methods for calculating show that as the number of network nodes N approaches infinity, the quantity N exhibits a discontinuity at parameter values corresponding to critical RBNs. For finite systems it peaks near the critical value, but slightly in the disordered regime for typical parameter variations. The source of high values of N is the indirect correlations between pairs of elements from different long chains with a common starting point. The contribution from pairs that are directly linked approaches zero for critical networks and peaks deep in the disordered regime.Comment: 11 pages, 6 figures; Minor revisions for clarity and figure format, one reference adde

Complex and Adaptive Dynamical Systems: A Primer

An thorough introduction is given at an introductory level to the field of quantitative complex system science, with special emphasis on emergence in dynamical systems based on network topologies. Subjects treated include graph theory and small-world networks, a generic introduction to the concepts of dynamical system theory, random Boolean networks, cellular automata and self-organized criticality, the statistical modeling of Darwinian evolution, synchronization phenomena and an introduction to the theory of cognitive systems. It inludes chapter on Graph Theory and Small-World Networks, Chaos, Bifurcations and Diffusion, Complexity and Information Theory, Random Boolean Networks, Cellular Automata and Self-Organized Criticality, Darwinian evolution, Hypercycles and Game Theory, Synchronization Phenomena and Elements of Cognitive System Theory.Comment: unformatted version of the textbook; published in Springer, Complexity Series (2008, second edition 2010

An Initial Framework Assessing the Safety of Complex Systems

Trabajo presentado en la Conference on Complex Systems, celebrada online del 7 al 11 de diciembre de 2020.Atmospheric blocking events, that is large-scale nearly stationary atmospheric pressure patterns, are often associated with extreme weather in the mid-latitudes, such as heat waves and cold spells which have significant consequences on ecosystems, human health and economy. The high impact of blocking events has motivated numerous studies. However, there is not yet a comprehensive theory explaining their onset, maintenance and decay and their numerical prediction remains a challenge. In recent years, a number of studies have successfully employed complex network descriptions of fluid transport to characterize dynamical patterns in geophysical flows. The aim of the current work is to investigate the potential of so called Lagrangian flow networks for the detection and perhaps forecasting of atmospheric blocking events. The network is constructed by associating nodes to regions of the atmosphere and establishing links based on the flux of material between these nodes during a given time interval. One can then use effective tools and metrics developed in the context of graph theory to explore the atmospheric flow properties. In particular, Ser-Giacomi et al. [1] showed how optimal paths in a Lagrangian flow network highlight distinctive circulation patterns associated with atmospheric blocking events. We extend these results by studying the behavior of selected network measures (such as degree, entropy and harmonic closeness centrality)at the onset of and during blocking situations, demonstrating their ability to trace the spatio-temporal characteristics of these events.This research was conducted as part of the CAFE (Climate Advanced Forecasting of sub-seasonal Extremes) Innovative Training Network which has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 813844

Interpreting multi-stable behaviour in input-driven recurrent neural networks

Recurrent neural networks (RNNs) are computational models inspired by the brain. Although RNNs stand out as state-of-the-art machine learning models to solve challenging tasks as speech recognition, handwriting recognition, language translation, and others, they are plagued by the so-called vanishing/exploding gradient issue. This prevents us from training RNNs with the aim of learning long term dependencies in sequential data. Moreover, a problem of interpretability affects these models, known as the ``black-box issue'' of RNNs. We attempt to open the black box by developing a mechanistic interpretation of errors occurring during the computation. We do this from a dynamical system theory perspective, specifically building on the notion of Excitable Network Attractors. Our methodology is effective at least for those tasks where a number of attractors and a switching pattern between them must be learned. RNNs can be seen as massively large nonlinear dynamical systems driven by external inputs. When it comes to analytically investigate RNNs, often in the literature the input-driven property is neglected or dropped in favour of tight constraints on the input driving the dynamics, which do not match the reality of RNN applications. Trying to bridge this gap, we framed RNNs dynamics driven by generic input sequences in the context of nonautonomous dynamical system theory. This brought us to enquire deeply into a fundamental principle established for RNNs known as the echo state property (ESP). In particular, we argue that input-driven RNNs can be reliable computational models even without satisfying the classical ESP formulation. We prove a sort of input-driven fixed point theorem and exploit it to (i) demonstrate the existence and uniqueness of a global attracting solution for strongly (in amplitude) input-driven RNNs, (ii) deduce the existence of multiple responses for certain input signals which can be reliably exploited for computational purposes, and (iii) study the stability of attracting solutions w.r.t. input sequences. Finally, we highlight the active role of the input in determining qualitative changes in the RNN dynamics, e.g. the number of stable responses, in contrast to commonly known qualitative changes due to variations of model parameters