We define a mathematical formalism based on the concept of an ''open dynamical system'' and show how it can be used to model embodied cognition. This formalism extends classical dynamical systems theory by distinguishing a ''total system'' (which models an agent in an environment) and an ''agent system'' (which models an agent by itself), and it includes tools for analyzing the collections of overlapping paths that occur in an embedded agent's state space. To illustrate the way this formalism can be applied, several neural network models are embedded in a simple model environment. Such phenomena as masking, perceptual ambiguity, and priming are then observed. We also use this formalism to reinterpret examples from the embodiment literature, arguing that it provides for a more thorough analysis of the relevant phenomena.

Aron Gurwitsch's theory of the structure and dynamics of consciousness has much to offer contemporary theorizing about consciousness and its basis in the embodied brain. On Gurwitsch's account, as we develop it, the field of consciousness has a variable sized focus or "theme" of attention surrounded by a structured periphery of inattentional contents. As the field evolves, its contents change their status, sometimes smoothly, sometimes abruptly. Inner thoughts, a sense of one's body, and the physical environment are dominant field contents. These ideas can be linked with (and help unify) contemporary theories about the neural correlates of consciousness, inattention, the small world structure of the brain, meta-stable dynamics, embodied cognition, and predictive coding in the brain.

Traditionally different approaches to the study of cognition have been viewed as competing explanatory frameworks. An alternative view, explanatory pluralism, regards different approaches to the study of cognition as complementary ways of studying the same phenomenon, at specific temporal and spatial scales, using appropriate methodological tools. Explanatory pluralism has been often described abstractly, but has rarely been applied to concrete cases. We present a case study of explanatory pluralism. We discuss three separate ways of studying the same phenomenon: a perceptual decision-making task (Bahrami et al., 2010), where pairs of subjects share information to jointly individuate an oddball stimulus among a set of distractors. Each approach analyzed the same corpus but targeted different units of analysis at different levels of description: decision-making at the behavioral level, confidence sharing at the linguistic level, and acoustic energy at the physical level. We discuss the utility of explanatory pluralism for describing this complex, multiscale phenomenon, show ways in which this case study sheds new light on the concept of pluralism, and highlight good practices to critically assess and complement approaches.

Historically cognition was understood as the result of processes occurring solely in the brain. Recently, however, cognitive scientists and philosophers studying "embodied" or "situated" cognition have begun emphasizing the role of the body and environment in which brains are situated, i.e. they view the brain as an "open system". However, these theorists frequently rely on dynamical systems which are traditionally viewed as closed systems. We address this tension by extending the framework of dynamical systems theory. We show how structures which appear in the state space of an embodied agent differ from those that appear in closed systems, and we show how these structures can be used to model representational processes in embodied agents. We focus on neural networks as models of embodied cognition.

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