New topologies in the unfolding of the Doubly Degenerate Takens-Bogdanov singularity

High-codimension bifurcations play a key role in shaping the dynamics of nonlinear models, as their unfoldings establish structured relationships between lower-codimension bifurcations and, ultimately, the attractors they generate. Owing to this unifying and predictive capacity, such bifurcations are attracting growing interest in mathematical biology, particularly in neuroscience. One notable example is the codimension-3 Degenerate Takens-Bogdanov (DTB) singularity, proposed as an organizing center for neural dynamics at both the single-neuron and population levels and playing a key role in bursting. The DTB itself arises within the unfolding of an even higher-order singularity, the Doubly Degenerate Takens-Bogdanov (DDTB), whose structure remains only partially understood, despite initial indications of its relevance in systems exhibiting a DTB point. Here, we contribute by employing numerical continuation to explore the DDTB unfolding using concentric spheres in parameter space. We confirm the conjecture that it connects DTB to another highly symmetric codimension-3 bifurcation, but through transitions that differ in part from those previously proposed in the literature, with one intermediate passage remaining unclear. We also show that using planes, rather than spheres, to explore the unfolding allows for additional bifurcation topologies. Overall, the intermediate configurations and additional bifurcation structures reveal a rich repertoire of dynamical behaviors that improve our understanding of what models with a DDTB point can do and how these behaviors are related. We draw on neuroscience for examples of these topologies in models of neural dynamics.