ARTICLE IN PRESS Neural Networks ( ) – Contents lists available at ScienceDirect Neural Networks journal homepage: www.elsevier.com/locate/neunet 2009 Special Issue Understanding neurodynamical systems via Fuzzy Symbolic Dynamics Krzysztof Dobosz a , Włodzisław Duch b,∗ a Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland b Department of Informatics, Nicolaus Copernicus University, Toruń, Poland article info Article history: Received 11 February 2009 Received in revised form 23 November 2009 Accepted 10 December 2009 Keywords: Symbolic dynamics Neurodynamical system Visualization of multidimensional time series abstract Neurodynamical systems are characterized by a large number of signal streams, measuring activity of individual neurons, local field potentials, aggregated electrical (EEG) or magnetic potentials (MEG), oxygen use (fMRI) or activity of simulated neurons. Various basis set decomposition techniques are used to analyze such signals, trying to discover components that carry meaningful information, but these techniques tell us little about the global activity of the whole system. A novel technique called Fuzzy Symbolic Dynamics (FSD) is introduced to help in understanding of the multidimensional dynamical system’s behavior. It is based on a fuzzy partitioning of the signal space that defines a non-linear mapping of the system’s trajectory to the low-dimensional space of membership function activations. This allows for visualization of the trajectory showing various aspects of observed signals that may be difficult to discover looking at individual components, or to notice otherwise. FSD mapping can be applied to raw signals, transformed signals (for example, ICA components), or to signals defined in the time–frequency domain. To illustrate the method two FSD visualizations are presented: a model system with artificial radial oscillatory sources, and the output layer (50 neurons) of Respiratory Rhythm Generator (RRG) composed of 300 spiking neurons. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction Neuroimaging data and simulated neurodynamical systems are characterized by multiple streams of non-stationary data, and thus may be represented only in high-dimensional signal spaces. For example, functional magnetic resonance imaging (fMRI) provides thousands of streams corresponding to the changing activity of voxels, with sampling rate of a few hertz, and electroencephalographic (EEG) recordings hundreds of streams with sampling frequency of hundreds of hertz. High data volumes that quickly change in time make such signals very hard to understand. Popular signal processing techniques remove artifacts by various filtering techniques, involving waveform analysis, morphological analysis, decomposition of data streams into meaningful components using Fourier or Wavelet Transforms, Principal and Independent Component Analysis (PCA, ICA), etc. (Rangayyan, 2001; Sanei & Chambers, 2008). Interesting events are then searched for using processed signal components, with time–frequency–intensity maps calculated for each component. Such techniques are very useful, but do not show global prop- erties of processes in the high-dimensional signal spaces. Simu- lation of complex dynamics is usually described in terms of at- ∗ Corresponding author. E-mail address: wduch@is.umk.pl (W. Duch). tractors, but precise characterization of their basins and possi- ble transitions between them is rarely attempted. A mapping that separates interesting segments of the trajectory could help to categorize such events. Global analysis is needed to character- ize different types of system’s behavior, see how attractors trap dynamics, notice partial synchronization and desynchronization events, or filter the high frequency noise. For many applications (including brain–computer interfaces) a snapshot of the whole tra- jectory helping to understand its main characteristics, would be very useful. This is the goal of our paper, presenting a global ap- proach to the high-dimensional signal analysis (to focus attention we shall talk about neurodynamics, although any dynamical sys- tem can be analyzed in this way). Two inspirations have been important in the development of our approach. First, an observation that different brain areas probably ‘‘understand’’ and collaborate with each other by filtering the main properties of their large-scale activity, reacting to specific activations that may be roughly characterized in a symbolic way. The second inspiration comes from the successes of the symbolic dynamics (Hao & Zheng, 1998) in understanding and simplifying the description of dynamical systems. Symbolic dynamics may be used as an approximation to brain processes if hard partitioning of the activity of various brain regions is done and labeled by a finite set of symbols. However, such a discretization may for most applications be either too rough or require too many symbols to be useful. The Fuzzy Symbolic Dynamics (FSD) introduced in 0893-6080/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.neunet.2009.12.005 Please cite this article in press as: Dobosz, K., & Duch, W. Understanding neurodynamical systems via Fuzzy Symbolic Dynamics. Neural Networks (2009), doi:10.1016/j.neunet.2009.12.005