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Hidden Order: The Geometry of Complex Patterns
Salvatore Torquato gives a public lecture in the Simons Foundation's 2026 Randomness series, connecting the geometry of complex systems with random structures and applications across mathematics, computer science, physics and chemistry.
Interacting spiral wave patterns underlie complex brain dynamics and are related to cognitive processing
The large-scale activity of the human brain exhibits rich and complex patterns, but the spatiotemporal dynamics of these patterns and their functional roles in cognition remain unclear. Here by characterizing moment-by-moment fluctuations of human cortical functional magnetic resonance imaging signals, we show that spiral-like, rotational wave patterns (brain spirals) are widespread during both resting and cognitive task states. These brain spirals propagate across the cortex while rotating around their phase singularity centres, giving rise to spatiotemporal activity dynamics with non-stationary features. The properties of these brain spirals, such as their rotational directions and locations, are task relevant and can be used to classify different cognitive tasks. We also demonstrate that multiple, interacting brain spirals are involved in coordinating the correlated activations and de-activations of distributed functional regions; this mechanism enables flexible reconfiguration of task-driven activity flow between bottom-up and top-down directions during cognitive processing. Our findings suggest that brain spirals organize complex spatiotemporal dynamics of the human brain and have functional correlates to cognitive processing.
Abstraction doesn't happen all at once (despite what some models of concept learning suggest)
In the past few years, there has been growing evidence that the basic ability for relational generalization starts in early infancy, with 3-month-olds seeming to learn relational abstractions with little training. Further, work with toddlers seem to suggest that relational generalizations are no more difficult than those based on objects, and they can readily consider both simultaneously. Likewise, causal learning research with adults suggests that people infer causal relationships at multiple levels of abstraction simultaneously as they learn about novel causal systems. These findings all appear counter to theories of concept learning that posit when concepts are first learned they tend to be concrete (tied to specific contexts and features) and abstraction proceeds incrementally as learners encounter more examples. The current talk will not question the veracity of any of these findings but will present several others from my and others’ research on relational learning that suggests that when the perceptual or conceptual content becomes more complex, patterns of incremental abstraction re-emerge. Further, the specific contexts and task parameters that support or hinder abstraction reveal the underlying cognitive processes. I will then consider whether the models that posit simultaneous, immediate learning at multiple levels of abstraction can accommodate these more complex patterns.