Weakly Connected Oscillatory Networks for Dynamic Pattern Recognition Fernando Corinto, Michele Bonnin, and Marco Gilli Dept. of Electronics, Politecnico di Torino, Torino, Italy E-mails: fernando.corinto@polito.it, michele.bonnin@polito.it, marco.gilli@polito.it Abstract— Recent studies on the thalamo-cortical system have shown that weakly connected oscillatory networks (WCNs) ex- hibit associative properties and can be exploited for dynamic pattern recognition. In this manuscript we focus on WCNs, composed of oscillators that admit of a Lur’e like description and are organized in such a way that they communicate one another, through a common medium. The main dynamic features are investigated by exploiting the phase deviation equation (i.e. the equation that describes the phase deviation due to the weak coupling). Furthermore, by using a simple learning algorithm, the phase-deviation equation is designed in such a way that given sets of patterns can be stored and recalled. In particular, two models of WCNs associative and dynamic memories are provided. I. I Nonlinear oscillatory networks have been widely used in Bi- ology, Physics and Engineering for modeling complex space- time phenomena [1]. The most significant and worth studying property of oscillatory systems is synchronization, either when they are coupled with other oscillators or when they are subject to an external driving signal. In this work we focus on weakly connected oscillatory net- works that represent bio-inspired architectures for information and image processing. Recent studies in neuroscience have shown that some significant features of the visual systems, like the binding problem [2], can be investigated, by exploiting nonlinear dynamic network models [3]. Some studies on the thalamo-cortical system have suggested new architectures for neurocomputers, that consist of coupled arrays of oscillators, with a periodic and/or complex dynamic behavior (including the possibility of chaos) [4], [5]. In particular, it has been shown that nonlinear oscillatory networks can behave as Hopfield neural networks, whose attractors are limit cycles instead of equilibrium points [4], [5]. The mathematical model of a weakly connected oscilla- tory network (WCN) consists of a large system of coupled nonlinear ordinary differential equations (ODEs), that may exhibit a rich spatio-temporal dynamics, including several attractors and bifurcation phenomena [6]. For this reason WCN dynamics has been mainly investigated through time-domain numerical simulation. Recently some spectral techniques have been applied to space-invariant networks, in order to charac- terize some space-time phenomena (see [6] and in particular [7], [8]). However the proposed methods are not suitable for characterizing the global dynamic behavior of complex networks, that exhibit a large number of attractors. The global dynamic behavior of WCNs can be investigated through the phase deviation equation [5], i.e. the equation that describes the evolution of the phase deviations, due to the weak coupling. We have employed this method for investigating one- dimensional weakly connected networks, composed by third order oscillators (Chua’s circuits) [9]. In this manuscript we consider WCNs composed by oscil- lators that admit of a Lur’e like description and organized in such a way that each oscillator communicates with the others through a central system (called master cell). As shown in [4] and [10], such networks can be employed as oscillatory associative memories because there is a one to one correspondence between the equilibrium points of the phase deviation equation and the limit cycles of the WCN. Firstly, we derive an accurate analytic expression of the phase deviation equation, by extending the technique presented in [9]. Then we show that the equilibrium points of the phase deviation equation can be designed through a simple learning rule in order to retrieve a given set of stored pattern. In particular, we presents two WCN models such that: (a) the outputs of the oscillators are only in-phase or anti-phase; (b) the outputs of the oscillators are not in-phase or anti-phase. II. W C N We consider weakly connected networks (WCNs) [5], hav- ing the star topology [10], that originates from the bio-inspired architecture proposed in [4]. All the cells are connected to a central complex cell O 0 (called master cell) in the shape of a star and communicate each other only through the central system. Let us assume that each cell O i is a dynamical system of order m described by the following system of nonlinear ordinary differential equations (ODEs) (1 ≤ i ≤ n): ˙ X i = F i (X i ), X = [X T 1 , ... X T n ] T (1) where X i ∈ R m represents the state vector of each cell, F i : R m → R m and T denotes transposition. The cells O i (1 ≤ i ≤ n) interact only through the master cell that supplies the signal G i (X 0 , X ) to each cell, where G i : R m×(n+1) → R m and X T 0 is the state vector of the master cell whose dynamics is described by ˙ X 0 = F 0 (X 0 ) with X 0 ∈ R m . Star topology WCNs, composed by n cells and one master cell, are then described by ˙ X i = F i (X i ) + ε G i (X 0 , X ) where 0 ≤ i ≤ n and ε represents a small parameter that guarantees a weak connection among the cells and G 0 (X 0 , X ) = 0. 1-4244-0437-1/06/$20.00 ©2006 IEEE. 61 Authorized licensed use limited to: Rensselaer Polytechnic Institute. Downloaded on February 4, 2010 at 12:18 from IEEE Xplore. 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