Functional Architectures and Hierarchies of Time Scales Dionysios Perdikis1, Marmaduke Woodman1,2, and Viktor Jirsa1 1
Theoretical Neuroscience Group, Institute of Movement Sciences, UMR CNRS 6233 & University of Mediterranean, 163, Av. de Luminy, F-13288, Marseille cedex 09, France 2 Center for Complex Systems and Brain Sciences, Florida Atlantic University, 777 Glades Rd, Florida 33431, USA
[email protected]
Abstract. Dynamical system theory offers approaches towards cognitive modeling and computation inspired by self-organization and pattern formation in open systems operating far from thermodynamical equilibrium. In this spirit we propose a functional architecture for the emergence of complex functions such as sequential motor behaviors. We model elementary functions as Structured Flows on Manifolds (SFM) that provide an unambiguous deterministic description of the functional dynamics, while still remaining compatible with the intrinsically low dimensionality of elementary behaviors. Pattern competition processes (operating on a hierarchy of time scales) provide the means to compose complex functions out of simpler constituent ones. Our underlying hypothesis is that complex functions can be decomposed in functional modes (simpler building blocks). Simulations of generating cursive handwriting provide proof of concept and suggest exciting avenues towards extending the current framework to other human functions including learning and language. Keywords: cognitive architectures, function, phase flow, self-organization, pattern formation, non-linear dynamics.
1 Introduction It is a common assumption in cognitive modeling and Artificial Intelligence that human function is constituted in a structured manner (even if that structure is as complex as irregular behavior [1,2] or of a purely statistical nature [3,4]. Complex functions are often characterized by the repetition of invariant patterns such as in dancing, musical performance and language related functions including speech, writing and typing. In all these cases, dancing figures, musical notes, elementary conceptual schemas, phonemes, graphemes or keyboard pressings combine to form complex sequences in time. To understand the organization of a complex sequence, the two traditional approaches, symbolic dynamics and connectionism, define functional units as well as a set of operations on them, commonly referred to as “computation”. In symbolic dynamics information is represented explicitly as organized symbols and computation takes the form of syntactic rules combining them [5]. In connectionist models on the other hand, patterns of activation, distributed across the nodes of large networks, allow for parallel computations [6]. Moreover, hybrid cognitive architectures involving symbolic knowledge representation with connectionist learning algorithms K. Diamantaras, W. Duch, L.S. Iliadis (Eds.): ICANN 2010, Part II, LNCS 6353, pp. 353 – 361, 2010. © Springer-Verlag Berlin Heidelberg 2010
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D. Perdikis, M. Woodman, and V. Jirsa
have also been proposed [7]. It is a common theme, however, that in all but a few cases [2,8,9], functional units are considered as static patterns or “states”. Even when a dynamical process is modeled and dealt with in real time (see the broad literature of recurrent neural networks [10,11,12]), it is eventually broken into a succession of discrete states (encoding past context) and treated as such.
Fig. 1. The word “flow” is generated by the functional architecture. From top to bottom: (a) workspace output: the trajectory on the plane x-y, (b) time series of the slow sequential dynamics ξ1-4 (different line styles are used to distinguish among the four functional modes used), (c) functional dynamics: the trajectory in the 3-dimensional phase (state) space spanned by ux, uy and uz, and the respective time series, and (d) the control signal I.
In contrast our here proposed approach is based upon the time structure of complex human function. We propose that complex functional sequences can be decomposed in a dynamical repertoire of functional modes, namely elementary processes (as opposed to static states) that play the role of functional units. Additional processes operating in different characteristic time scales use the latter ones as building blocks of more complex functions. Thus, computation emerges in a self-organized manner. The ensemble of subsystems, acting upon two different time scales, defines a functional architecture allowing for novel forms of biologically inspired computation. Potential applications of the here proposed functional architecture may be found in motor control where complex movements are composed of elementary building blocks called motor programs [13] or movement primitives [14,15]. Production and comprehension of speech perception in terms of syllables [16,17] or language in terms of basic conceptual schemas [18,19] may serve as another promising field for application. Here we illustrate the basic functional principles in the context of
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sequential motor behavior because movements constitute dynamical phenomena easily open to observation. In the following we present the mathematical framework of Structured Flows on Manifolds and then introduce the functional architecture in section 2. In section 3 we demonstrate an application in cursive handwriting as a proof of concept providing the respective computer simulations. Finally, in section 4 we discuss several potential extensions with respect to neural coding, learning, embodied intelligence and analog biological inspired computation. 1.1 Structured Flows on Manifolds (SFMs) We model functional modes as Structured Flows on Manifolds (SFM) that have only recently been proposed [20,21] as a general framework for understanding functional dynamics. Such is accomplished through a fast adiabatic contraction from an inherently high dimensional space to a functionally relevant subset of the phase (state) space, the so-called manifold. On the manifold a phase flow is prescribed and a dynamics evolves for the duration of a specific functional mode. This behavior is mathematically expressed as:
({u } ,{s }) u + μ f ({u } ,{s }) , = − s + h ( {u } ,{ s } )
ui = − g s j
i
j
j
i
i
j
i
j
(1)
u ∈ \ N , s ∈ \M , N M where the value of the so called “smallness” parameter μ is constrained as in 0
