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Table of contents for Handbook of dynamic system modeling / editor, Paul A. Fishwick.
A Nature Research Journal. Many dynamical systems, both natural and artificial, are stimulated by time dependent external signals, somehow processing the information contained therein. We demonstrate how to quantify the different modes in which information can be processed by such systems and combine them to define the computational capacity of a dynamical system. This is bounded by the number of linearly independent state variables of the dynamical system, equaling it if the system obeys the fading memory condition. It can be interpreted as the total number of linearly independent functions of its stimuli the system can compute.
Our theory combines concepts from machine learning reservoir computing , system modeling, stochastic processes, and functional analysis. We illustrate our theory by numerical simulations for the logistic map, a recurrent neural network, and a two-dimensional reaction diffusion system, uncovering universal trade-offs between the non-linearity of the computation and the system's short-term memory.
Many dynamical systems, both natural and artificial, are driven by external input signals and can be seen as performing non-trivial, real-time computations of these signals. The aim of this work is to study the online information processing that takes place within these dynamical systems.
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The importance of this question is underlined by the multitude of examples from nature and engineering which fall into this category. Recent insight in robotics has demonstrated that the control of animal and robot locomotion can greatly be simplified by exploiting the physical body's response to its environment 10 , Additional systems to which our theory could apply are population dynamics responding to, e.
All of these exemplary systems display completely different physical implementations. We present a general framework that allows to compare the computational properties of a broad class of dynamical systems. We are able to characterize the information processing that takes place within these systems in a way which is independent of their physical realization, using a quantitive measure.
It is normalized appropriately such that completely different systems can be compared and it allows us to characterize different computational regimes linear vs. Some initial steps in this direction are provided by the linear memory capacity introduced in 12 later theoretically extended to linear systems in discrete time 13 and continuous time systems 14 , and using Fischer information Although it has been argued that short-term memory of a neural system is crucial for the brain to perform useful computation on sensory input streams 15 , 16 , it is our belief that the general paradigm underlying the brain's computation cannot solely rely on maximizing linear memory.
The neural circuits have to compute complex non-linear and spatio-temporal functions of the inputs. Prior models focusing on linear memory capacity in essence assumed the dynamic system to only implement memory, while the complex non-linear mapping is off-loaded to an unspecified read-out mechanism.
Handbook of Dynamic System Modeling
We propose to endow the dynamic system with all the required computational capacity, and use only simple linear read-out functions. The capacity measures we introduce are therefore of great interest since they quantify all the information processing that takes place within the dynamical system, and don't introduce an artificial separation between linear and non-linear information processing.
One of the startling results of this work with potentially far reaching consequences is that all dynamical systems, provided they obey the condition of having a fading memory 17 and have linearly independent internal variables, have in principle the same total normalized capacity to process information. The ability to carry out useful information processing is therefore an almost universal characteristic of dynamical systems.
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This result provides theoretical justification for the widely used paradigm of reservoir computing with a linear readout layer 2 , 3 , 4 ; indeed it confirms that such systems can be universal for computation of time invariant functions with fading memory. We illustrate the versatility of our approach by using numerical simulations on three very different dynamical systems: the logistic map, a recurrent neural circuit, and a two-dimensional reaction diffusion system.
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These examples allow us to uncover the potentially universal trade-off between memory depth and non-linear computations performed by the dynamic system itself. We also discuss how the influence of noise decreases the computation capacity of a dynamical system. The dynamical systems we consider are observed in discrete time. The set J may be finite or enumerably or continuously infinite. Typical examples for the set U include a discrete set, or subsets of such as the unit interval or the whole real line.
When studying a dynamical system, we do not necessarily have access to all the dynamical variables x t. This corresponds to experimental situations where many variables may not be measurable, and is also the case if the number of dynamical variables J is infinite. Note that we use the term internal variables throughout the paper for easy of notation.
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The theory however also holds if we cannot directly observe the internal states, but only have access to them via N observation functions. The system evolves according to an evolution equation of the form: with T j the mapping from input and previous state to the j 'th current state.
We denote a dynamical system with the evolution law eq. In order to evaluate the performance of the system in a standardized way and to make analytical derivations tractable, we will take the inputs u t to be independent and identically drawn from some probability distribution p u.
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It should be noted that in many, if not all, real world situations the inputs are not i. Most of the theory should also hold for the non-i.
However our aim is to characterize the dynamical system itself. By taking the inputs to be i. The distribution p u can be chosen according to what is most relevant for the dynamical system under study. In order to study this dynamical system experimentally, we proceed as follows.
This to ensure that potential initial transients have died out. Next, we drive the system for T timesteps with inputs from the same distribution. During this later phase we record both the values of the inputs u t and of the state variables x t which will be used to get insight into how information is processed by the dynamical system. A central quantity in our analysis will be the capacity to reconstruct the function z from the state of a dynamical system using a linear estimator.
We introduce this notion as follows. Consider a linear estimator constructed from the observed data We measure the quality of the estimator by the Mean Square Error MSE : where here and throughout we denote by a subscript T quantities that are generated from the measured data. We are interested in the estimator which minimizes the MSE.
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The Capacity for the dynamical system X to reconstruct the function z is where. The notion of capacity can also be interpreted from the point of view of computation. In this approach, one wishes to use the dynamical system to compute the function z u —h , and to this end one uses, following the ideas introduced in reservoir computing 2 , 3 , 4 , the simple linear readout eq. The capacity C [ X , z ] measures how successful the dynamical system is at computing z. The main difference with the notation of 12 is the treatment of a possible bias term in eq. In what follows we do not introduce such a bias term, as this makes the statement of the results simpler.
We discuss the relation between the two approaches in the supplementary material. Note that C T [ X , z ] depends on a single set of recorded data. The computed value of the capacity is therefore affected by statistical fluctuations: if one repeats the experiment one will get a slightly different value. This must be taken into account, particularly when interpreting very small values of the capacity C T.
We discuss this issue in the supplementary material. The capacity is based on simple linear estimators.
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