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Bruno Schein,1,2 Gerardo Beltran,1– 3 Bárbara Regina França,1,2 Paulo RS Sanches,4 Danton P Silva Jr,4 Iraci Lucena Torres,5,6 Felipe Fegni,7 Wolnei Caumo1,2,5,8 1Post-Graduate Program in Medical Sciences, School of Medicine, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Rio Grande do Sul (RS), Brazil; 2Laboratory of Pain and Neuromodulation, Hospital de Clínicas de Porto Alegre (HCPA), Porto Alegre, RS, Brazil; 3Institute of Neurosciences, Universidad Catolica de Cuenca, Cuenca, Ecuador; 4Laboratory of Biomedical Engineer, HCPA, Porto Alegre, RS, Brazil; 5Pain and Palliative Care Service, HCPA, Porto Alegre, RS, Brazil; 6Laboratory of Pharmacology of Pain and Neuromodulation, Experimental Research Center, HCPA, Porto Alegre, RS, Brazil; 7Laboratory of Neuromodulation and Center for Clinical Research Learning, Physics and Rehabilitation Department, Spaulding Rehabilitation Hospital, Boston, MA, USA; 8Department of Surgery, School of Medicine, UFRGS, Porto Alegre, RS, BrazilCorrespondence: Wolnei Caumo, Laboratory of Pain and Neuromodulation, Hospital de Clínicas de Porto Alegre at UFRGS, Ramiro Barcelos, 2350 - CEP 90035-003 Bairro Rio Branco, Porto Alegre, RS, Brazil, Tel/Fax +55 51- 33598083, Email wcaumo@hcpa.edu.brObjective: We compare the effect of HAS, a-tDCS on the left dorsolateral prefrontal cortex (l-DLPFC), and rest-testing on pain measures [(cold pressor test (CPT) (primary outcome) and heat pain threshold]. We also compare their effects on the motor evoked potential (MEP) (primary outcome), short intracortical inhibition (SICI), intracortical facilitation (ICF), and cortical silent period (CSP).Methods: This randomized, blind, crossover trial included 18 women with fibromyalgia, aged from 18 to 65 years old. They received at random and in a crossover order a-tDCS over the l-DLPFC (2mA), HAS, or a rest-testing.Results: HAS compared to a-tDCS increased the pain tolerance with a moderate effect size (ES) [Cohen’s f=− 0.78; (CI 95%; − 1.48 to − 0.12)]. While compared to rest-testing, HAS increased the CPT with a large ES [Cohen’s f=− 0.87; (CI 95%; − 1.84 to − 0.09)]. The a-tDCS compared to HAS increased the MEP amplitude with large ES [Cohen’s f=− 1.73 (CI 95%; − 2.17 to − 0.17)]. Likewise, its ES compared to rest-testing in the MEP size was large [Cohen’s f=− 1.03; (CI 95%; − 2.06 to − 0.08)].Conclusion: These findings revealed that HAS affects contra-regulating mechanisms involved in perception and pain tolerance, while the a-tDCS increased the excitability of the corticospinal pathways. They give a subsidy to investigate their effect as approaches to counter regulate the maladaptive neuroplasticity involved in fibromyalgia.Clinical Trial Registration: www.ClinicalTrials.gov, identifier – NCT05066568.Keywords: tDCS, hypnosis, hypnotic analgesia, chronic pain, fibromyalgia, pain threshold, cold pressor test, CPM-test

Gerardo Beltran Serrano,1– 3 Laura Pooch Rodrigues,2 Bruno Schein,1,2 Maxciel Zortea,1,2 Iraci Lucenada Silva Torres,1,4,5 Felipe Fregni,6 Wolnei Caumo1,2,5 1Post-Graduate Program in Medical Sciences, School of Medicine, Universidade Federal Do Rio Grande Do Sul (UFRGS), Porto Alegre, Brazil; 2Laboratory of Pain and Neuromodulation at Hospital De Clínicas De Porto Alegre (HCPA), Porto Alegre, Brazil; 3Psychology Department, Universidad Catolica De Cuenca, UCACUE, Cuenca, Ecuador; 4Department of Pharmacology, Institute of Health Sciences (ICBS), Universidade Federal Do Rio Grande Do Sul (UFRGS), Porto Alegre, Brazil; 5Pharmacology of Pain and Neuromodulation: Pre-Clinical Investigations Research Group, Universidade Federal Do Rio Grande Do Sul (UFRGS), Porto Alegre, Brazil; 6Department of Neurology, Beth Israel Deaconess Medical Center, Harvard Medical School, Boston, USACorrespondence: Wolnei Caumo Fax + (55) 51- 3359.8083Email wcaumo@hcpa.edu.brObjective: We evaluated whether active(a)-tDCS combined with hypnotic analgesia suggestion (HS) would be more effective than a single active(a)-tDCS, and/or sham-(s)-tDCS and s-tDCS/HS on the following outcomes: function of descending pain modulatory system (DPMS) during the conditioned pain modulation test (CPM-test) (primary outcome), heat pain threshold (HPT), heat pain tolerance (HPTo) and cold pressor test (CPT) (secondary outcomes). We also examined whether their effects are related to neuroplasticity state evaluated by serum brain-derived-neurotropic factor (BDNF).Materials and Methods: Forty-eight females received one session of one of the four interventions (a-tDCS/HS, s-tDCS/HS, a-tDCS, and s-tDCS) in an incomplete randomized crossover sequence. The a-tDCS or s-tDCS was applied over the left dorsolateral prefrontal cortex (DLPFC) for 30 minutes at 2mA.Results: A generalized linear model revealed a significant main effect for the intervention group (P < 0.032). The delta-(Δ) pain score on the Numerical Pain Scale (NPS0-10) during CPM-test in the a-tDCS/HS group was − 0.25 (0.43). The (Δ) pain score on NPS (0– 10) during CPM-test in the other three groups was a-tDCS=− 0.54 (0.41), HS − 0.01 (0.41) and s-tDCS/HS=− 0.19 (0.43). A-tDCS/HS intervention increased the CPT substantially compared to all other interventions. Also, higher baseline levels of BDNF were associated with a larger change in CPT and HPTo.Conclusion: These findings indicate that the HS combined with a-tDCS mitigated the effect of the a-tDCS on the DPMS. The a-tDCS up-regulates the inhibition on DPMS, and the HS improved pain tolerance. And, together they enhanced the reaction time substantially upon the CPT.Clinical Trial Registration: www.ClinicalTrials.gov, identifier NCT03744897.Keywords: tDCS, hypnotic analgesia, conditioned pain modulation, pain perception

The following problem was posed by the second author in 'Semigroup Forum', Vol. 1, No. 1, 1970, p. 91: Describe the structure of semigroups S such that for all a, b, c ε S, abc=ab, ac or bc. [ABSTRACT FROM AUTHOR]