We obtain Central Limit Theorems in Functional form for a class of time-inhomogeneous interacting random walks on the simplex of probability measures over a finite set. Due to a reinforcement mechanism, the increments of the walks are correlated, forcing their convergence to the same, possibly random, limit. Random walks of this form have been introduced in the context of urn models and in stochastic algorithms. We also propose an application to opinion dynamics in a random network evolving via preferential attachment. We study, in particular, random walks interacting through a mean-field rule and compare the rate they converge to their limit with the rate of synchronization, i.e. the rate at which their mutual distances converge to zero. Under certain conditions, synchronization is faster than convergence.

We consider a system of urns of Pólya type, containing balls of two colors; the reinforcement of each urn depends on both the content of the urn and the average content of all urns. We show that the urns synchronize almost surely, in the sense that the fraction of balls of a given color converges almost surely as time tends to ∞ to the same limit for all urns. A normal approximation for a large number of urns is also obtained.

The multidimensionality of chronic pain forces us to look beyond isolated assessment such as pain intensity, which does not consider multiple key parameters, particularly in post-operative Persistent Spinal Pain Syndrome (PSPS-T2) patients. Our ambition was to produce a novel Multi-dimensional Clinical Response Index (MCRI), including not only pain intensity but also functional capacity, anxiety-depression, quality of life and quantitative pain mapping, the objective being to achieve instantaneous assessment using machine learning techniques. Two hundred PSPS-T2 patients were enrolled in the real-life observational prospective PREDIBACK study with 12-month follow-up and received various treatments. From a multitude of questionnaires/scores, specific items were combined, as exploratory factor analyses helped to create a single composite MCRI; using pairwise correlations between measurements, it appeared to more accurately represent all pain dimensions than any previous classical score. It represented the best compromise among all existing indexes, showing the highest sensitivity/specificity related to Patient Global Impression of Change (PGIC). Novel composite indexes could help to refine pain assessment by informing the physician’s perception of patient condition on the basis of objective and holistic metrics, and also by providing new insights regarding therapy efficacy/patient outcome assessments, before ultimately being adapted to other pathologies.