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A system of dispersive representations of the Omn\`es-Khuri-Treiman-Sawyer-Wali type for the final-state interactions in the amplitudes of the $K\to\pi\pi\pi$ weak transitions is constructed, under the assumptions that CP and isospin symmetries are conserved. Both the $\Delta I = 1/2$ and $\Delta I = 3/2$ transition channels are considered. The set of single-variable functions involved in these representations is identified, and the polynomial ambiguities in their definitions are discussed. Numerical solutions for the system of coupled integral relations satisfied by these functions are provided, and the determination of the subtraction constants in terms of the experimental information on the amplitudes in the decay region is addressed., Comment: 33 pages, 3 figures, Ancillary files:i) README.pdf ii) Numerical tables for the set of fundamental solutions of the integral equations iii) Numerical values of the S and P-wave $\pi \pi$ phase shifts

Striatal cholinergic interneurons (CINs) use acetylcholine (ACh) and glutamate (Glut) to regulate the striatal network since they express vesicular transporters for ACh (VAChT) and Glut (VGLUT3). However, whether ACh and Glut are released simultaneously and/or independently from cholinergic varicosities is an open question. The answer to that question requires the multichannel detection of vesicular transporters at the level of single synaptic vesicle (SV). Here, we used super-resolution STimulated Emission Depletion microscopy (STED) to characterize and quantify the distribution of VAChT and VGLUT3 in CINs SVs. Nearest-neighbor distances analysis between VAChT and VGLUT3-immunofluorescent spots revealed that 34% of CINs SVs contain both VAChT and VGLUT3. In addition, 40% of SVs expressed only VAChT while 26% of SVs contain only VGLUT3. These results suggest that SVs from CINs have the potential to store simultaneously or independently ACh and/or Glut. Overall, these morphological findings support the notion that CINs varicosities can signal with either ACh or Glut or both with an unexpected level of complexity.