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

Towards a Calculus of Echo State Networks

Reservoir computing is a recent trend in neural networks which uses the dynamical perturbations on the phase space of a system to compute a desired target function. We present how one can formulate an expectation of system performance in a simple class of reservoir computing called echo state networks. In contrast with previous theoretical frameworks, which only reveal an upper bound on the total memory in the system, we analytically calculate the entire memory curve as a function of the structure of the system and the properties of the input and the target function. We demonstrate the precision of our framework by validating its result for a wide range of system sizes and spectral radii. Our analytical calculation agrees with numerical simulations. To the best of our knowledge this work presents the first exact analytical characterization of the memory curve in echo state networks

Memristor models for machine learning

In the quest for alternatives to traditional CMOS, it is being suggested that digital computing efficiency and power can be improved by matching the precision to the application. Many applications do not need the high precision that is being used today. In particular, large gains in area- and power efficiency could be achieved by dedicated analog realizations of approximate computing engines. In this work, we explore the use of memristor networks for analog approximate computation, based on a machine learning framework called reservoir computing. Most experimental investigations on the dynamics of memristors focus on their nonvolatile behavior. Hence, the volatility that is present in the developed technologies is usually unwanted and it is not included in simulation models. In contrast, in reservoir computing, volatility is not only desirable but necessary. Therefore, in this work, we propose two different ways to incorporate it into memristor simulation models. The first is an extension of Strukov's model and the second is an equivalent Wiener model approximation. We analyze and compare the dynamical properties of these models and discuss their implications for the memory and the nonlinear processing capacity of memristor networks. Our results indicate that device variability, increasingly causing problems in traditional computer design, is an asset in the context of reservoir computing. We conclude that, although both models could lead to useful memristor based reservoir computing systems, their computational performance will differ. Therefore, experimental modeling research is required for the development of accurate volatile memristor models.Comment: 4 figures, no tables. Submitted to neural computatio

Exploiting short-term memory in soft body dynamics as a computational resource

Soft materials are not only highly deformable but they also possess rich and diverse body dynamics. Soft body dynamics exhibit a variety of properties, including nonlinearity, elasticity, and potentially infinitely many degrees of freedom. Here we demonstrate that such soft body dynamics can be employed to conduct certain types of computation. Using body dynamics generated from a soft silicone arm, we show that they can be exploited to emulate functions that require memory and to embed robust closed-loop control into the arm. Our results suggest that soft body dynamics have a short-term memory and can serve as a computational resource. This finding paves the way toward exploiting passive body dynamics for control of a large class of underactuated systems.Comment: 22 pages, 11 figures; email address correcte

Product Reservoir Computing: Time-Series Computation with Multiplicative Neurons

Echo state networks (ESN), a type of reservoir computing (RC) architecture, are efficient and accurate artificial neural systems for time series processing and learning. An ESN consists of a core of recurrent neural networks, called a reservoir, with a small number of tunable parameters to generate a high-dimensional representation of an input, and a readout layer which is easily trained using regression to produce a desired output from the reservoir states. Certain computational tasks involve real-time calculation of high-order time correlations, which requires nonlinear transformation either in the reservoir or the readout layer. Traditional ESN employs a reservoir with sigmoid or tanh function neurons. In contrast, some types of biological neurons obey response curves that can be described as a product unit rather than a sum and threshold. Inspired by this class of neurons, we introduce a RC architecture with a reservoir of product nodes for time series computation. We find that the product RC shows many properties of standard ESN such as short-term memory and nonlinear capacity. On standard benchmarks for chaotic prediction tasks, the product RC maintains the performance of a standard nonlinear ESN while being more amenable to mathematical analysis. Our study provides evidence that such networks are powerful in highly nonlinear tasks owing to high-order statistics generated by the recurrent product node reservoir