Poincar\'e inequalities for mutually singular measures

Using an inverse system of metric graphs as in: J. Cheeger and B. Kleiner, "Inverse limit spaces satisfying a Poincar\'e inequality", we provide a simple example of a metric space $X$ that admits Poincar\'e inequalities for a continuum of mutually singular measures

The Lip-lip equality is stable under blow-up

We show that at generic points blow-ups/tangents of differentiability spaces are still differentiability spaces; this implies that an analytic condition introduced by Keith as an inequality (and later proved to actually be an equality) passes to tangents. As an application, we characterize the $p$-weak gradient on iterated blow-ups of differentiability spaces.Comment: minor corrections: change of normalization of the measures; The final version will appear in Calc. Var. PD

Neuronal encoding of subjective value in dorsal and ventral anterior cingulate cortex

We examined the activity of individual cells in the primate anterior cingulate cortex during an economic choice task. In the experiments, monkeys chose between different juices offered in variables amounts and subjective values were inferred from the animals\u27 choices. We analyzed neuronal firing rates in relation to a large number of behaviorally relevant variables. We report three main results. First, there were robust differences between the dorsal bank (ACCd) and the ventral bank (ACCv) of the cingulate sulcus. Specifically, neurons in ACCd but not in ACCv were modulated by the movement direction. Furthermore, neurons in ACCd were most active before movement initiation, whereas neurons in ACCv were most active after juice delivery. Second, neurons in both areas encoded the identity and the subjective value of the juice chosen by the animal. In contrast, neither region encoded the value of individual offers. Third, the population of value-encoding neurons in both ACCd and ACCv underwent range adaptation. With respect to economic choice, it is interesting to compare these areas with the orbitofrontal cortex (OFC), previously examined. While neurons in OFC encoded both pre-decision and post-decision variables, neurons in ACCd and ACCv only encoded post-decision variables. Moreover, the encoding of the choice outcome (chosen value and chosen juice) in ACCd and ACCv trailed that found in OFC. These observations indicate that economic decisions (i.e., value comparisons) take place upstream of ACCd and ACCv. The coexistence of choice outcome and movement signals in ACCd suggests that this area constitutes a gateway through which the choice system informs motor systems