Investigating Echo-State Networks Dynamics by Means of Recurrence Analysis

Biological computation through recurrence

One of the defining features of living systems is their adaptability to changing environmental conditions. This requires organisms to extract temporal and spatial features of their environment, and use that information to compute the appropriate response. In the last two decades, a growing body of work, mainly coming from the machine learning and computational neuroscience fields, has shown that such complex information processing can be performed by recurrent networks. Temporal computations arise in these networks through the interplay between the external stimuli and the network's internal state. In this article we review our current understanding of how recurrent networks can be used by biological systems, from cells to brains, for complex information processing. Rather than focusing on sophisticated, artificial recurrent architectures such as long short-term memory (LSTM) networks, here we concentrate on simpler network structures and learning algorithms that can be expected to have been found by evolution. We also review studies showing evidence of naturally occurring recurrent networks in living organisms. Lastly, we discuss some relevant evolutionary aspects concerning the emergence of this natural computation paradigm.

Excitatory/inhibitory balance emerges as a key factor for RBN performance, overriding attractor dynamics

Reservoir computing provides a time and cost-efficient alternative to traditional learning methods. Critical regimes, known as the "edge of chaos," have been found to optimize computational performance in binary neural networks. However, little attention has been devoted to studying reservoir-to-reservoir variability when investigating the link between connectivity, dynamics, and performance. As physical reservoir computers become more prevalent, developing a systematic approach to network design is crucial. In this article, we examine Random Boolean Networks (RBNs) and demonstrate that specific distribution parameters can lead to diverse dynamics near critical points. We identify distinct dynamical attractors and quantify their statistics, revealing that most reservoirs possess a dominant attractor. We then evaluate performance in two challenging tasks, memorization and prediction, and find that a positive excitatory balance produces a critical point with higher memory performance. In comparison, a negative inhibitory balance delivers another critical point with better prediction performance. Interestingly, we show that the intrinsic attractor dynamics have little influence on performance in either case.

Utilizing variable auto encoder-based TDO optimization algorithm for predicting loneliness from electrocardiogram signals

Several seniors and a substantial part of the general population are living in social isolation. This frequently occurs in vulnerability, isolation, and depression, which then have a poor impact on other health-related factors. A number of health problems, including a higher risk of cardio problems, are brought on by social isolation and loneliness. Electrocardiogram (ECG) usage for mental condition recognition enables accurate determination of a person's internal representation. The electrocardiogram (ECG) signals can be thoroughly analyzed to uncover hidden data that may be helpful for the precise identification of cardiac problems. ECG time-series information typically have great dimensions and complicated componentry. Using relevant information to guide training is among the main achievements of this type of learning. An ECG signal plays a significant part in the individual body's ability to manage behavior. Furthermore, loneliness identification is crucial since it has the worse effect on the circumstances that afflict persons. This study suggested an approach for detecting loneliness from an ECG signal to use a variable auto encoder-based optimization algorithm for ESN technique. The suggested approach consists of three phases for identifying a person's loneliness. Firstly, undecimated discrete wavelet transform is used to preprocess the acquired ECG data. Next, further characteristics are extracted from the precompiled signals using a variable auto encoder. For the precise categorization of loneliness in the ECG signal, a metaheuristic optimized ESN is, therefore, presented. The outcomes of the tests demonstrate that the suggested system with suitable ECG representations produces improved accuracy as well as performance.

Reservoir Dynamic Interpretability for Time Series Prediction: A Permutation Entropy View

An echo state network (ESN) is an efficient recurrent neural network (RNN) that is widely used in time series prediction tasks due to its simplicity and low training cost. However, the "black-box" nature of reservoirs hinders the development of ESN. Although a large number of studies have concentrated on reservoir interpretability, the perspective of reservoir modeling is relatively single, and the relationship between reservoir richness and reservoir projection capacity has not been effectively established. To tackle this problem, a novel reservoir interpretability framework based on permutation entropy (PE) theory is proposed in this paper. In structure, this framework consists of reservoir state extraction, PE modeling, and PE analysis. Based on these, the instantaneous reservoir states and neuronal time-varying states are extracted, which are followed by phase space reconstruction, sorting, and entropy calculation. Firstly, the obtained instantaneous state entropy (ISE) and global state entropy (GSE) can measure reservoir richness for interpreting good reservoir projection capacity. On the other hand, the multiscale complexity-entropy analysis of global and neuron-level reservoir states is performed to reveal more detailed dynamics. Finally, the relationships between ESN performance and reservoir dynamic are investigated via Pearson correlation, considering different prediction steps and time scales. Experimental evaluations on several benchmarks and real-world datasets demonstrate the effectiveness and superiority of the proposed reservoir interpretability framework.

Replication in Energy Markets: Use and Misuse of Chaos Tools

As pointed out by many researchers, replication plays a key role in the credibility of applied sciences and the confidence in all research findings. With regard, in particular, to energy finance and economics, replication papers are rare, probably because they are hampered by inaccessible data, but their aim is crucial. We consider two ways to avoid misleading results on the ostensible chaoticity of price series. The first one is represented by the proper mathematical definition of chaos and the related theoretical background, while the latter is represented by the hybrid approach that we propose here-i.e., consisting of considering the dynamical system underlying the price time series as a deterministic system with noise. We find that both chaotic and stochastic features coexist in the energy commodity markets, although the misuse of some tests in the established practice in the literature may say otherwise.

Integer Echo State Networks: Efficient Reservoir Computing for Digital Hardware

We propose an approximation of echo state networks (ESNs) that can be efficiently implemented on digital hardware based on the mathematics of hyperdimensional computing. The reservoir of the proposed integer ESN (intESN) is a vector containing only n -bits integers (where is normally sufficient for a satisfactory performance). The recurrent matrix multiplication is replaced with an efficient cyclic shift operation. The proposed intESN approach is verified with typical tasks in reservoir computing: memorizing of a sequence of inputs, classifying time series, and learning dynamic processes. Such architecture results in dramatic improvements in memory footprint and computational efficiency, with minimal performance loss. The experiments on a field-programmable gate array confirm that the proposed intESN approach is much more energy efficient than the conventional ESN.

Synthesizing Mesh Deformation Sequences With Bidirectional LSTM

Synthesizing realistic 3D mesh deformation sequences is a challenging but important task in computer animation. To achieve this, researchers have long been focusing on shape analysis to develop new interpolation and extrapolation techniques. However, such techniques have limited learning capabilities and therefore often produce unrealistic deformation. Although there are already networks defined on individual meshes, deep architectures that operate directly on mesh sequences with temporal information remain unexplored due to the following major barriers: irregular mesh connectivity, rich temporal information, and varied deformation. To address these issues, we utilize convolutional neural networks defined on triangular meshes along with a shape deformation representation to extract useful features, followed by long short-term memory (LSTM) that iteratively processes the features. To fully respect the bidirectional nature of actions, we propose a new share-weight bidirectional scheme to better synthesize deformations. An extensive evaluation shows that our approach outperforms existing methods in sequence generation, both qualitatively and quantitatively.

On the Post Hoc Explainability of Optimized Self-Organizing Reservoir Network for Action Recognition

This work proposes a novel unsupervised self-organizing network, called the Self-Organizing Convolutional Echo State Network (SO-ConvESN), for learning node centroids and interconnectivity maps compatible with the deterministic initialization of Echo State Network (ESN) input and reservoir weights, in the context of human action recognition (HAR). To ensure stability and echo state property in the reservoir, Recurrent Plots (RPs) and Recurrence Quantification Analysis (RQA) techniques are exploited for explainability and characterization of the reservoir dynamics and hence tuning ESN hyperparameters. The optimized self-organizing reservoirs are cascaded with a Convolutional Neural Network (CNN) to ensure that the activation of internal echo state representations (ESRs) echoes similar topological qualities and temporal features of the input time-series, and the CNN efficiently learns the dynamics and multiscale temporal features from the ESRs for action recognition. The hyperparameter optimization (HPO) algorithms are additionally adopted to optimize the CNN stage in SO-ConvESN. Experimental results on the HAR problem using several publicly available 3D-skeleton-based action datasets demonstrate the showcasing of the RPs and RQA technique in examining the explainability of reservoir dynamics for designing stable self-organizing reservoirs and the usefulness of implementing HPOs in SO-ConvESN for the HAR task. The proposed SO-ConvESN exhibits competitive recognition accuracy.

Learn to synchronize, synchronize to learn

In recent years, the artificial intelligence community has seen a continuous interest in research aimed at investigating dynamical aspects of both training procedures and machine learning models. Of particular interest among recurrent neural networks, we have the Reservoir Computing (RC) paradigm characterized by conceptual simplicity and a fast training scheme. Yet, the guiding principles under which RC operates are only partially understood. In this work, we analyze the role played by Generalized Synchronization (GS) when training a RC to solve a generic task. In particular, we show how GS allows the reservoir to correctly encode the system generating the input signal into its dynamics. We also discuss necessary and sufficient conditions for the learning to be feasible in this approach. Moreover, we explore the role that ergodicity plays in this process, showing how its presence allows the learning outcome to apply to multiple input trajectories. Finally, we show that satisfaction of the GS can be measured by means of the mutual false nearest neighbors index, which makes effective to practitioners theoretical derivations.

Reservoir Computing Approaches for Representation and Classification of Multivariate Time Series

Classification of multivariate time series (MTS) has been tackled with a large variety of methodologies and applied to a wide range of scenarios. Reservoir computing (RC) provides efficient tools to generate a vectorial, fixed-size representation of the MTS that can be further processed by standard classifiers. Despite their unrivaled training speed, MTS classifiers based on a standard RC architecture fail to achieve the same accuracy of fully trainable neural networks. In this article, we introduce the reservoir model space, an unsupervised approach based on RC to learn vectorial representations of MTS. Each MTS is encoded within the parameters of a linear model trained to predict a low-dimensional embedding of the reservoir dynamics. Compared with other RC methods, our model space yields better representations and attains comparable computational performance due to an intermediate dimensionality reduction procedure. As a second contribution, we propose a modular RC framework for MTS classification, with an associated open-source Python library. The framework provides different modules to seamlessly implement advanced RC architectures. The architectures are compared with other MTS classifiers, including deep learning models and time series kernels. Results obtained on the benchmark and real-world MTS data sets show that RC classifiers are dramatically faster and, when implemented using our proposed representation, also achieve superior classification accuracy.

Non-linear analysis of the heart rate variability in characterization of manic and euthymic phases of bipolar disorder

BACKGROUND

- Bipolar Disorder (BD) has been associated with autonomic nervous system (ANS) dysregulation, with a consequent increase in mortality. Recent work highlights the non-linear analysis of ANS function. Our objective was to compare ANS modulation using recurrence plots (RP) and symbolic analysis (SA) in manic and euthymic phases of BD to controls.

METHODS

- Eighteen male patients (33.1 ± 12.0 years) were assessed during mania and at discharge in the euthymic phase compared and to a healthy group matched by age (33.9 ± 10.8 years). Electrocardiographic series (1000 RR intervals, at rest, in supine position) were captured using Polar Advantage RS800CX equipment and Heart Rate Variability (HRV) was analysed using RP and SA. Statistical analysis was performed using ANOVA with Tukey's post-test. The threshold for statistical significance was set at P < 0.05 and Cohen's d effect size was also quantified considering d > 0.8 as an important effect. The study was registered into the Clinical Trials Registration (ClinicalTrials.gov: NCT01272518).

RESULTS

Manic group presented significantly higher linearity before treatment (P<0.05) compared to controls considering RP variables. Cohen's d values had a large effect size ranging from 0.888 to 1.227. In the manic phase, SA showed predominance of the sympathetic component (OV%) with reduction of the parasympathetic component (2LV% and 2UV%) with reversion post treatment including higher Shannon Entropy (SE) indicating higher complexity.

LIMITATIONS

- short follow-up (1 month) and small number of patients.

CONCLUSIONS

- Non-linear analyzes may be used as supplementary tools for understanding autonomic function in BD during mania and after drug treatment.

Recurrence Quantification Analysis for Human Activity Recognition

Human Activity Recognition (HAR) is a central unit to understand and predict human behavior. HAR has been used to estimate the levels of a sedentary, monitor lifestyle habits, track the levels of people's health, or build a recommendation system. Many researchers have utilized the inertial measurement unit as an input tool to explore the HAR land. The recurrence plot (RP) technique recently has its applications diverse in various areas. From the recurrence plot, a machine-auto or hand-crafted approach can be used to extract feature vectors. While the machine-auto based approach has been reported in the literature, the latter hand-crafted based method has not. For that reason, this paper evaluated and demonstrated the feasibility of utilizing Recurrence Quantification Analysis (RQA), which was a typical hand-crafted method from RP, to classify human activities. A Linear Discriminant Analysis classifier yielded a 95.08% accuracy, which belonged in the top accuracy reported in the literature. Compare to the machine-auto or end-to-end approach, RQA is a far less complicated and more lean system that should be further analyzed in a HAR application.

Evolutionary aspects of reservoir computing

Reservoir computing (RC) is a powerful computational paradigm that allows high versatility with cheap learning. While other artificial intelligence approaches need exhaustive resources to specify their inner workings, RC is based on a reservoir with highly nonlinear dynamics that does not require a fine tuning of its parts. These dynamics project input signals into high-dimensional spaces, where training linear readouts to extract input features is vastly simplified. Thus, inexpensive learning provides very powerful tools for decision-making, controlling dynamical systems, classification, etc. RC also facilitates solving multiple tasks in parallel, resulting in a high throughput. Existing literature focuses on applications in artificial intelligence and neuroscience. We review this literature from an evolutionary perspective. RC's versatility makes it a great candidate to solve outstanding problems in biology, which raises relevant questions. Is RC as abundant in nature as its advantages should imply? Has it evolved? Once evolved, can it be easily sustained? Under what circumstances? (In other words, is RC an evolutionarily stable computing paradigm?) To tackle these issues, we introduce a conceptual morphospace that would map computational selective pressures that could select for or against RC and other computing paradigms. This guides a speculative discussion about the questions above and allows us to propose a solid research line that brings together computation and evolution with RC as test model of the proposed hypotheses. This article is part of the theme issue 'Liquid brains, solid brains: How distributed cognitive architectures process information'.

Spatio-Temporal Interpolated Echo State Network for Meteorological Series Prediction

Spatio-temporal series prediction has attracted increasing attention in the field of meteorology in recent years. The spatial and temporal joint effect makes predictions challenging. Most of the existing spatio-temporal prediction models are computationally complicated. To develop an accurate but easy-to-implement spatio-temporal prediction model, this paper designs a novel spatio-temporal prediction model based on echo state networks. For real-world observed meteorological data with randomness and large changes, we use a cubic spline method to bridge the gaps between the neighboring points, which results in a pleasingly smooth series. The interpolated series is later input into the spatio-temporal echo state networks, in which the spatial coefficients are computed by the elastic-net algorithm. This approach offers automatic selection and continuous shrinkage of the spatial variables. The proposed model provides an intuitive but effective approach to address the interaction of spatial and temporal effects. To demonstrate the practicality of the proposed model, we apply it to predict two real-world datasets: monthly precipitation series and daily air quality index series. Experimental results demonstrate that the proposed model achieves a normalized root-mean-square error of approximately 0.250 on both datasets. Similar results are achieved on the long short-term memory model, but the computation time of our proposed model is considerably shorter. It can be inferred that our proposed neural network model has advantages on predicting meteorological series over other models.

Sensitive Finite-State Computations Using a Distributed Network With a Noisy Network Attractor

We exhibit a class of smooth continuous-state neural-inspired networks composed of simple nonlinear elements that can be made to function as a finite-state computational machine. We give an explicit construction of arbitrary finite-state virtual machines in the spatiotemporal dynamics of the network. The dynamics of the functional network can be completely characterized as a "noisy network attractor" in phase space operating in either an "excitable" or a "free-running" regime, respectively, corresponding to excitable or heteroclinic connections between states. The regime depends on the sign of an "excitability parameter." Viewing the network as a nonlinear stochastic differential equation where a deterministic (signal) and/or a stochastic (noise) input is applied to any element, we explore the influence of the signal-to-noise ratio on the error rate of the computations. The free-running regime is extremely sensitive to inputs: arbitrarily small amplitude perturbations can be used to perform computations with the system as long as the input dominates the noise. We find a counter-intuitive regime where increasing noise amplitude can lead to more, rather than less, accurate computation. We suggest that noisy network attractors will be useful for understanding neural networks that reliably and sensitively perform finite-state computations in a noisy environment.

Determination of the Edge of Criticality in Echo State Networks Through Fisher Information Maximization

It is a widely accepted fact that the computational capability of recurrent neural networks (RNNs) is maximized on the so-called "edge of criticality." Once the network operates in this configuration, it performs efficiently on a specific application both in terms of: 1) low prediction error and 2) high short-term memory capacity. Since the behavior of recurrent networks is strongly influenced by the particular input signal driving the dynamics, a universal, application-independent method for determining the edge of criticality is still missing. In this paper, we aim at addressing this issue by proposing a theoretically motivated, unsupervised method based on Fisher information for determining the edge of criticality in RNNs. It is proved that Fisher information is maximized for (finite-size) systems operating in such critical regions. However, Fisher information is notoriously difficult to compute and requires the analytic form of the probability density function ruling the system behavior. This paper takes advantage of a recently developed nonparametric estimator of the Fisher information matrix and provides a method to determine the critical region of echo state networks (ESNs), a particular class of recurrent networks. The considered control parameters, which indirectly affect the ESN performance, are explored to identify those configurations lying on the edge of criticality and, as such, maximizing Fisher information and computational performance. Experimental results on benchmarks and real-world data demonstrate the effectiveness of the proposed method.

Multiplex visibility graphs to investigate recurrent neural network dynamics

A recurrent neural network (RNN) is a universal approximator of dynamical systems, whose performance often depends on sensitive hyperparameters. Tuning them properly may be difficult and, typically, based on a trial-and-error approach. In this work, we adopt a graph-based framework to interpret and characterize internal dynamics of a class of RNNs called echo state networks (ESNs). We design principled unsupervised methods to derive hyperparameters configurations yielding maximal ESN performance, expressed in terms of prediction error and memory capacity. In particular, we propose to model time series generated by each neuron activations with a horizontal visibility graph, whose topological properties have been shown to be related to the underlying system dynamics. Successively, horizontal visibility graphs associated with all neurons become layers of a larger structure called a multiplex. We show that topological properties of such a multiplex reflect important features of ESN dynamics that can be used to guide the tuning of its hyperparamers. Results obtained on several benchmarks and a real-world dataset of telephone call data records show the effectiveness of the proposed methods.