Empirical data only afford a partial view of a system’s spatiotemporal organization, observed dynamics being restricted to certain (often linear) domains of phase space. Dynamical modeling provides a simplified but more comprehensive view: it extends the boundaries of empirical data, exposes continuity between qualitatively different regimes, shows the paths leading from one regime to another, attempts to reveal the entire parameter space of the underlying dynamics, and may ultimately lead to uncovering first principles (Kelso, 1995; 2012; Kelso at al., 2009; in press). A complementary aspect of our approach combines extensive experimental observations with theoretical models. This continuous dialog between experiments and models (and between experimental and theoretical scientists on our team) is a key feature of our three-way plan, which adds theoretical modeling to assess the impact of specific properties or parameters such as coupling strength and directionality on dyadic interaction (de Guzman et al., 2010; Kelso, 2012; Kelso at al., in press).

The Haken-Kelso-Bunz (HKB) model

The Haken-Kelso-Bunz (HKB) model was proposed in that spirit: theoretical modeling of a system of (nonlinearly) coupled nonlinear oscillators reproduced essential properties of biological coordination (e.g. different forms of phase synchrony, instability, phase transitions) and predicted others (critical slowing, fluctuation enhancement, hysteresis, etc., Riley, et al., 2011 for recent review). With over 1200 citations, HKB is probably the best known and most extensively tested quantitative model in human movement behavior (Fuchs & Jirsa, 2008; Haken, Kelso & Bunz, 1985). More importantly, the HKB model also encompasses coordination at neural, behavioral and social scales thus offering a powerful tool for integrating different levels of brain and behavior (Kelso, 1995; 2012; Kelso, et al., in press). The extended version of the HKB (Kelso et al., 1990) investigated how the coordination dynamics is modulated when the interacting systems have different intrinsic dynamics (e.g. frequency of oscillation).

Representation of the potential function for the extended HKB model. Black balls symbolize stable coordinated behaviors and white balls correspond to unstable behavioral states.

Integrating such ‘broken-symmetry’ into the model provided new insight into the phenomenon of metastability which has been proposed as a fundamental principle of brain and behavior (Bressler and Kelso, 2001; Freeman & Holmes, 2005; Friston, 1997; Kelso, 1995; 2012; Kelso & Tognoli, 2007; Perez Velazquez & Wennberg, 2004; Rabinovitch et al., 2008; Sporns, 2010; Tognoli & Kelso, 2009; Werner, 2007).

The Excitator model

In line with the foregoing theoretical research, a further advance was to create the mathematical conditions for discrete behaviors to arise from the continuous dynamics of the system’s self-sustained oscillators, the so-called “excitator” model (Jirsa & Kelso 2005). Although it seems intuitive that continuous behavior arises from the juxtaposition of discrete actions, nature sometimes appears to proceed the other way around, using basic building blocks with self-sustained dynamics such as central pattern generators to produce the sophisticated functional mechanics of the neocortex (Yuste et al., 2005; Kelso, 2009).

Different theoretical realizations of the intrinsic dynamic of the excitator model.

The excitator model (Jirsa & Kelso 2005) shares many properties of excitable two-dimensional dynamical systems in neurobiology but is especially relevant here because it provides a tool to study both continuous and discrete dynamics. More than generalizing the HKB model, it provides an entry point to the understanding of a variety of new phenomena such as false starts (Fink, Kelso & Jirsa, 2009) and the geometry of phase space trajectories (Jirsa & Kelso, 2005).

The Virtual Partner Interaction (VPI)

A further advance made in the previous funding period was the introduction of Virtual Partner Interaction or VPI (Kelso at al., 2009), a hybrid model/experiment paradigm analog to the “dynamic clamp” of cellular neuroscience (Prinz et al., 2004). VPI consists of a human partner and its mathematical mirror (“virtual partner”), who are reciprocally coupled via the HKB equations of coordination dynamics. Human and virtual partners are provided coordination tasks to jointly accomplish and behavioral self-organization is studied (Kelso at al., 2009; de Guzman et al., 2009). The experimenter can tune parameters to assess the impact of specific properties on coupling strength and directionality (de Guzman et al., 2010; Kelso, 2012; Kelso at al., in press). The VPI technique offers access to a broad range of on-line and reciprocal interactions, including regions of the dynamical state space that cannot be easily explored during live human-human interactions. This extended parameter range opens up the possibility of systematically driving social interaction; it has already led to the discovery of novel coordination behaviors, stable patterns of social interaction, never seen before because of their location in orphaned regions of the parameter space. Because VPI left human subjects with a strong sense of agency and intentionality from their virtual partner, this work has also led to expanded enquiry on agency and perception of cooperation (Drever et al., 2012).

Neurocomputational model of social interaction

Finally, in order to integrate measurements from multiple levels of description into a single dynamical account (Kelso, et al., in press; Tognoli et al., 2010), we extend our approach to develop neurocomputational models of social behavior. Earlier on, we have shown theoretically how bimanual behavior relies on brain architecture, especially inter-hemispheric anatomical connectivity (Banerjee & Jirsa, 2007; Banerjee et al., 2012; Jirsa, Fuchs & Kelso, 1998). This allowed for fruitful quantitative predictions about the relationship between brain and behavior (Kelso et al., Nature, 1998). More recently, we turned the same logic to the question of social as opposed to individual behavior (Dumas et al. 2012). A novel neurocomputational model was developed to understand the relationship between brain architecture (including interbrain structural symmetries) and social interactions. The resulting social network was built from a pair of real human connectomes. Those connectomes were obtained by parcellation of brain regions combined with diffusion tensor imaging (DTI) datasets. Brain areas were mapped to self-sustained oscillators, which were coupled neurally within brain (connectomes) and between brains. At rest or combined with functional simulation, such networks further our understanding of how structure and dynamics are intertwined within and between human brains, and how they relate to social behavior. The method has already shown that the anatomical connectivity of the human brain enhances similarities of the neural dynamics and facilitates the creation of sensorimotor coupling between individuals (Dumas et al. 2012). This computational social neuroscience approach leads to insights and specific predictions about the neurobiological mechanisms underlying social behavior, including the link between structural anomalies and self-other dysfunction in pathologies such as autism and schizophrenia (Dumas, 2011).