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Critical phase transitions have proven to be a powerful concept to capture the phenomenology of many systems, including deeply non-equilibrium ones like living systems. The study of these phase transitions has overwhelmingly relied on…
Topologically ordered phase has emerged as one of most exciting concepts that not only broadens our understanding of phases of matter, but also has been found to have potential application in fault-tolerant quantum computation. The direct…
Torus canards are solutions of slow/fast systems that alternate between attracting and repelling manifolds of limit cycles of the fast subsystem. A relatively new dynamic phenomenon, torus canards have been found in neural applications to…
The wiring diagram of the mouse brain has recently been mapped at a mesoscopic scale in the Allen Mouse Brain Connectivity Atlas. Axonal projections from brain regions were traced using green fluoresent proteins. The resulting data were…
Brain imaging data mapping onto human connectome networks enables the investigation of global brain dynamics, where the brain hubs play an essential role in transferring activity between different brain parts. At this scale, the…
Topology identification and inference of processes evolving over graphs arise in timely applications involving brain, transportation, financial, power, as well as social and information networks. This chapter provides an overview of graph…
Living systems exhibit complex yet organized behavior on multiple spatiotemporal scales. To investigate the nature of multiscale coordination in living systems, one needs a meaningful and systematic way to quantify the complex dynamics, a…
Neural oscillations have long been considered important markers of interaction across brain regions, yet identifying coordinated oscillatory activity from high-dimensional multiple-electrode recordings remains challenging. We sought to…
This work addresses a modification of the random geometric graph (RGG) model by considering a set of points uniformly and independently distributed on the surface of a $(d-1)$-sphere with radius $r$ in a $d-$dimensional Euclidean space,…
A rectangular dual of a plane graph $G$ is a contact representation of $G$ by interior-disjoint rectangles such that (i) no four rectangles share a point, and (ii) the union of all rectangles is a rectangle. In this paper, we study…
Many real-world complex systems including human interactions can be represented by temporal (or evolving) networks, where links activate or deactivate over time. Characterizing temporal networks is crucial to compare such systems and to…
Advances in imaging techniques enable high resolution 3D visualisation of vascular networks over time and reveal abnormal structural features such as twists and loops, and their quantification is an active area of research. Here we showcase…
Weakly coupled oscillators are used throughout the physical sciences, particularly in mathematical neuroscience to describe the interaction of neurons in the brain. Systems of weakly coupled oscillators have a well-known decomposition to a…
Neural network design has utilized flexible nonlinear processes which can mimic biological systems, but has suffered from a lack of traceability in the resulting network. Graphical probabilistic models ground network design in probabilistic…
Dynamic functional connectivity (DFC) analysis involves measuring correlated neural activity over time across multiple brain regions. Significant regional correlations among neural signals, such as those obtained from resting-state…
We consider random graphs on the set of $N^2$ vertices placed on the discrete $2$-dimensional torus. The edges between pairs of vertices are independent, and their probabilities decay with the distance $\rho$ between these vertices as…
Functional connectivity of cognitive tasks allows researchers to analyse the interaction mapping occurring between different regions of the brain using electroencephalography (EEG) signals. Standard practice in functional connectivity…
Multivariate phase relationships are important to characterize and understand numerous physical, biological, and chemical systems, from electromagnetic waves to neural oscillations. These systems exhibit complex spatiotemporal dynamics and…
Time-varying graph signals are alternative representation of multivariate (or multichannel) signals in which a single time-series is associated with each of the nodes or vertex of a graph. Aided by the graph-theoretic tools, time-varying…
The analysis of electrical impulse phenomena in cardiac muscle tissue is important for the diagnosis of heart rhythm disorders and other cardiac pathophysiology. Cardiac mapping techniques acquire local temporal measurements and combine…