The Fermat-Torricelli problem in normed planes and spaces

We investigate the Fermat-Torricelli problem in d-dimensional real normed spaces or Minkowski spaces, mainly for d=2. Our approach is to study the Fermat-Torricelli locus in a geometric way. We present many new results, as well as give an exposition of known results that are scattered in various sources, with proofs for some of them. Together, these results can be considered to be a minitheory of the Fermat-Torricelli problem in Minkowski spaces and especially in Minkowski planes. This demonstrates that substantial results about locational problems valid for all norms can be found using a geometric approach

Dorsal Periaqueductal gray ensembles represent approach and avoidance states

Animals must balance needs to approach threats for risk assessment and to avoid danger. The dorsal periaqueductal gray (dPAG) controls defensive behaviors, but it is unknown how it represents states associated with threat approach and avoidance. We identified a dPAG threatavoidance ensemble in mice that showed higher activity farther from threats such as the open arms of the elevated plus maze and a predator. These cells were also more active during threat avoidance behaviors such as escape and freezing, even though these behaviors have antagonistic motor output. Conversely, the threat approach ensemble was more active during risk assessment behaviors and near threats. Furthermore, unsupervised methods showed that avoidance/approach states were encoded with shared activity patterns across threats. Lastly, the relative number of cells in each ensemble predicted threat avoidance across mice. Thus, dPAG ensembles dynamically encode threat approach and avoidance states, providing a flexible mechanism to balance risk assessment and danger avoidance

Dorsal premammillary projection to periaqueductal gray controls escape vigor from innate and conditioned threats

Escape from threats has paramount importance for survival. However, it is unknown if a single circuit controls escape vigor from innate and conditioned threats. Cholecystokinin (cck)-expressing cells in the hypothalamic dorsal premammillary nucleus (PMd) are necessary for initiating escape from innate threats via a projection to the dorsolateral periaqueductal gray (dlPAG). We now show that in mice PMd-cck cells are activated during escape, but not other defensive behaviors. PMd-cck ensemble activity can also predict future escape. Furthermore, PMd inhibition decreases escape speed from both innate and conditioned threats. Inhibition of the PMd-cck projection to the dlPAG also decreased escape speed. Intriguingly, PMd-cck and dlPAG activity in mice showed higher mutual information during exposure to innate and conditioned threats. In parallel, human functional magnetic resonance imaging data show that a posterior hypothalamic-to-dlPAG pathway increased activity during exposure to aversive images, indicating that a similar pathway may possibly have a related role in humans. Our data identify the PMd-dlPAG circuit as a central node, controlling escape vigor elicited by both innate and conditioned threats

Shared Dorsal Periaqueductal Gray Activation Patterns during Exposure to Innate and Conditioned Threats.

The brainstem dorsal periaqueductal gray (dPAG) has been widely recognized as being a vital node orchestrating the responses to innate threats. Intriguingly, recent evidence also shows that the dPAG mediates defensive responses to fear conditioned contexts. However, it is unknown whether the dPAG displays independent or shared patterns of activation during exposure to innate and conditioned threats. It is also unclear how dPAG ensembles encode and predict diverse defensive behaviors. To address this question, we used miniaturized microscopes to obtain recordings of the same dPAG ensembles during exposure to a live predator and a fear conditioned context in male mice. dPAG ensembles encoded not only distance to threat, but also relevant features, such as predator speed and angular offset between mouse and threat. Furthermore, dPAG cells accurately encoded numerous defensive behaviors, including freezing, stretch-attend postures, and escape. Encoding of behaviors and of distance to threat occurred independently in dPAG cells. dPAG cells also displayed a shared representation to encode these behaviors and distance to threat across innate and conditioned threats. Last, we also show that escape could be predicted by dPAG activity several seconds in advance. Thus, dPAG activity dynamically tracks key kinematic and behavioral variables during exposure to threats, and exhibits similar patterns of activation during defensive behaviors elicited by innate or conditioned threats. These data indicate that a common pathway may be recruited by the dPAG during exposure to a wide variety of threat modalities.SIGNIFICANCE STATEMENT The dorsal periaqueductal gray (dPAG) is critical to generate defensive behaviors during encounters with threats of multiple modalities. Here we use longitudinal calcium transient recordings of dPAG ensembles in freely moving mice to show that this region uses shared patterns of activity to represent distance to an innate threat (a live predator) and a conditioned threat (a shock grid). We also show that dPAG neural activity can predict diverse defensive behaviors. These data indicate the dPAG uses conserved population-level activity patterns to encode and coordinate defensive behaviors during exposure to both innate and conditioned threats