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Individual differences in brain functional networks may be related to complex personal identifiers, including health, age, and ability. Dynamic network theory has been used to identify properties of dynamic brain function from fMRI data, but the majority of analyses and findings remain at the level of the group. Here, we apply hypergraph analysis, a method from dynamic network theory, to quantify individual differences in brain functional dynamics. Using a summary metric derived from the hypergraph formalism-hypergraph cardinality-we investigate individual variations in two separate, complementary data sets. The first data set ("multi-task") consists of 77 individuals engaging in four consecutive cognitive tasks. We observe that hypergraph cardinality exhibits variation across individuals while remaining consistent within individuals between tasks; moreover, the analysis of one of the memory tasks revealed a marginally significant correspondence between hypergraph cardinality and age. This finding motivated a similar analysis of the second data set ("age-memory"), in which 95 individuals, aged 18-75, performed a memory task with a similar structure to the multi-task memory task. With the increased age range in the age-memory data set, the correlation between hypergraph cardinality and age correspondence becomes significant. We discuss these results in the context of the well-known finding linking age with network structure, and suggest that hypergraph analysis should serve as a useful tool in furthering our understanding of the dynamic network structure of the brain., Competing Interests: The authors have declared that no competing interests exist.

Activity in the human brain moves between diverse functional states to meet the demands of our dynamic environment, but fundamental principles guiding these transitions remain poorly understood. Here, we capitalize on recent advances in network science to analyze patterns of functional interactions between brain regions. We use dynamic network representations to probe the landscape of brain reconfigurations that accompany task performance both within and between four cognitive states: a task-free resting state, an attention-demanding state, and two memory-demanding states. Using the formalism of hypergraphs, we identify the presence of groups of functional interactions that fluctuate coherently in strength over time both within (task-specific) and across (task-general) brain states. In contrast to prior emphases on the complexity of many dyadic (region-to-region) relationships, these results demonstrate that brain adaptability can be described by common processes that drive the dynamic integration of cognitive systems. Moreover, our results establish the hypergraph as an effective measure for understanding functional brain dynamics, which may also prove useful in examining cross-task, cross-age, and cross-cohort functional change.

We study the dynamics of a low-dimensional system of coupled model neurons as a step towards understanding the vastly complex network of neurons in the brain. We analyze the bifurcation structure of a system of two model neurons with unidirectional coupling as a function of two physiologically relevant parameters: the external current input only to the first neuron and the strength of the coupling from the first to the second neuron. Leveraging a timescale separation, we prove necessary conditions for multiple timescale phenomena observed in the coupled system, including canard solutions and mixed mode oscillations. For a larger network of model neurons, we present a sufficient condition for phase locking when external inputs are heterogeneous. Finally, we generalize our results to directed trees of model neurons with heterogeneous inputs. [ABSTRACT FROM AUTHOR]