Nature's forms are frilly, flexible, and functional.

A ubiquitous motif in nature is the self-similar hierarchical buckling of a thin lamina near its margins. This is seen in leaves, flowers, fungi, corals, and marine invertebrates. We investigate this morphology from the perspective of non-Euclidean plate theory. We identify a novel type of defect, a branch-point of the normal map, that allows for the generation of such complex wrinkling patterns in thin elastic hyperbolic surfaces, even in the absence of stretching. We argue that branch points are the natural defects in hyperbolic sheets, they carry a topological charge which gives them a degree of robustness, and they can influence the overall morphology of a hyperbolic surface without concentrating elastic energy. We develop a theory for branch points and investigate their role in determining the mechanical response of hyperbolic sheets to weak external forces.

Noise-induced tipping under periodic forcing: Preferred tipping phase in a non-adiabatic forcing regime.

We consider a periodically forced 1D Langevin equation that possesses two stable periodic solutions in the absence of noise. We ask the question: is there a most likely noise-induced transition path between these periodic solutions that allows us to identify a preferred phase of the forcing when tipping occurs? The quasistatic regime, where the forcing period is long compared to the adiabatic relaxation time, has been well studied; our work instead explores the case when these time scales are comparable. We compute optimal paths using the path integral method incorporating the Onsager-Machlup functional and validate results with Monte Carlo simulations. Results for the preferred tipping phase are compared with the deterministic aspects of the problem. We identify parameter regimes where nullclines, associated with the deterministic problem in a 2D extended phase space, form passageways through which the optimal paths transit. As the nullclines are independent of the relaxation time and the noise strength, this leads to a robust deterministic predictor of the preferred tipping phase in a regime where forcing is neither too fast nor too slow.

A retinal code for motion along the gravitational and body axes.

Self-motion triggers complementary visual and vestibular reflexes supporting image-stabilization and balance. Translation through space produces one global pattern of retinal image motion (optic flow), rotation another. We examined the direction preferences of direction-sensitive ganglion cells (DSGCs) in flattened mouse retinas in vitro. Here we show that for each subtype of DSGC, direction preference varies topographically so as to align with specific translatory optic flow fields, creating a neural ensemble tuned for a specific direction of motion through space. Four cardinal translatory directions are represented, aligned with two axes of high adaptive relevance: the body and gravitational axes. One subtype maximizes its output when the mouse advances, others when it retreats, rises or falls. Two classes of DSGCs, namely, ON-DSGCs and ON-OFF-DSGCs, share the same spatial geometry but weight the four channels differently. Each subtype ensemble is also tuned for rotation. The relative activation of DSGC channels uniquely encodes every translation and rotation. Although retinal and vestibular systems both encode translatory and rotatory self-motion, their coordinate systems differ.