The one-dimensional ray equation, the differential description of Fermat’s principle, is deduced directly from Snell’s law using two methods. In the first method, we obtain the ray equation from a differential equation relating the spatial coordinates derivative with the index of refraction field. In the second method, the ray equation is deduced from the proper generalization of Snell’s law for a refractive field, that is, a differential equation relating the index of refraction field and the refraction angle. Additionally, we used an intermediate expression of the first method to find a straightforward analytical solution of the ray path to an inferior mirage.

Introduction: This study investigates how traumatic injuries alter joint movements in the ankle and foot. We used a brain injury model in rats, focusing on the hippocampus between the CA1 and dentate gyrus. Materials and Methods: We assessed the dissimilarity factor (DF) and vertical displacement (VD) of the ankle and metatarsus joints before and after the hippocampal lesion. We analyzed joint movements in rats after the injury or in rats treated with resveratrol, exercise, or a combination of both. Results: Resveratrol facilitated the recovery of DF in both legs, showing improvements in the ankle and metatarsus joints on the third and seventh days post-injury. The hippocampal lesion affected VD in both legs, observed on the third or seventh day after the injury. Both exercise and resveratrol partially recovered VD in the ankle and metatarsus joints on these days. These effects may be linked to increased hippocampal neurogenesis and reduced neuroinflammation. Conclusions: The study highlights the benefits of resveratrol and exercise in motor recovery following brain injury, suggesting their potential to enhance the quality of life for patients with neurological disorders affecting motor function and locomotion. These findings also suggest that resveratrol could offer a promising or complementary alternative in managing chronic pain and inflammation associated with orthopedic conditions, thus improving overall patient management.

Laura Paulina Osuna-Carrasco, Sergio Horacio Dueñas-Jiménez, Carmen Toro-Castillo, Braniff De la Torre, Irene Aguilar-García, Jonatan Alpirez, Luis Castillo and Judith Marcela Dueñas-Jiménez. Exp Neurobiol 2022;31:89-96. https://doi.org/10.5607/en21019

This study investigates how traumatic injuries alter joint movements in the ankle and foot. We used a brain injury model in rats, focusing on the hippocampus between the CA1 and dentate gyrus. We assessed the dissimilarity factor (DF) and vertical displacement (VD) of the ankle and metatarsus joints before and after the hippocampal lesion. We analyzed joint movements in rats after the injury or in rats treated with resveratrol, exercise, or a combination of both. Resveratrol facilitated the recovery of DF in both legs, showing improvements in the ankle and metatarsus joints on the third- and seventh days post-injury. The hippocampal lesion affected VD in both legs, observed on the third or seventh day after the injury. Both exercise and resveratrol partially recovered VD in the ankle and metatarsus joints on these days. These effects may be linked to increased hippocampal neurogenesis and reduced neuroinflammation. The study highlights the benefits of resveratrol and exercise in motor recovery following brain injury, suggesting their potential to enhance the quality of life for patients with neurological disorders affecting motor function and locomotion. These findings also suggest that resveratrol could offer a promising or complementary alternative in managing chronic pain and inflammation associated with orthopedic conditions, thus improving overall patient management.