In multiple sclerosis (MS) brains, inflammation induces specific abnormalities of synaptic transmission, collectively called inflammatory synaptopathy. Such synaptopathy consists in unbalanced glutamatergic and GABAergic transmission and in remarkable changes in synaptic plasticity, causing excitotoxic neurodegeneration and impairing the clinical compensation of the ongoing brain damage, thereby exacerbating the clinical manifestation of the disease. In progressive MS (PMS), synaptopathy is characterized by pathological potentatiation of glutamate-mediated synaptic up-scaling (Centonze et al., 2008; Rossi et al., 2013) and loss of long-term synaptic potentiation [LTP (Weiss et al., 2014)], both caused by proinflammatory molecules (released by microglia, astroglia, and infiltrating T and B lymphocytes) (Malenka et al., 2004; Di Filippo et al., 2017; Stampanoni Bassi et al., 2019). The combination of increased up-scaling and decreased LTP has a significant impact on the clinical manifestations of PMS, often presenting with signs and symptoms indicating length-dependent degeneration of neurons of the corticospinal tract. Altered LTP expression impairs brain ability to compensate ongoing neuronal loss (Stampanoni Bassi et al., 2020), and pathological TNF-mediated up-scaling may directly promote excitotoxic damage and neurodegeneration (Rossi et al., 2014). In addition, up-scaling and LTP are mutually exclusive at a given synapse through a mechanism of synaptic occlusion (i.e., pre-existing up-scaling saturates and prevents subsequent LTP expression), further promoting neurodegeneration by preventing the pro-survival effect of LTP, the induction of which activates intracellular anti-apoptotic pathways (Bartlett & Wang, 2013). It follows that a neuromodulation approach that can chronically (over several months) dampen up-scaling expression in the primary motor cortex (M1) of PMS patients could be beneficial by preventing excitotoxic neurodegenerative damage triggered by up-scaling itself (Centonze et al. 2008, Rossi et al. 2014), and also by promoting LTP induction and LTP-dependent functional compensation of deficits, thereby reducing the speed of the neurodegeneration process through increased LTP-dependent neuronal survival and preservation of dendritic spines (Ksiazek-Winiarek et al., 2015). Our study aims to test whether transcranial static magnetic field stimulation (tSMS) could represent such a therapeutic approach, as recently proposed in patients with amyotrophic lateral sclerosis (ALS) (Di Lazzaro et al, 2021). Forty (40) ambulatory patients with PMS, presenting with the ascending myelopathy phenotype of the disease, will be recruited at the MS Center of the Unit of Neurology of the IRCCS Neuromed in Pozzilli (IS). In this randomized, sham-controlled, double-blind, within-subjects, cross-over study (allocation ratio 1:1), we will test the ability of repeated sessions of tSMS applied bilaterally over the M1 to safely reduce disability progression in patients with PMS. Patients will be randomly assigned to either real or sham tSMS. Each patient will participate in two experimental phases (real or sham stimulation). Each patient will self-administer tSMS over right and left M1, two session per day, 60 minutes each. The order will be randomly established and counterbalanced across participants. Both investigators and participants will be blinded to stimulation parameters. In the "real stimulation" phase, tSMS will be applied for 120 minutes each day, at home, for 12 consecutive months. In the "sham stimulation" phase, sham tSMS will be delivered with non-magnetic metal cylinders, with the same size, weight and appearance of the magnets. Clinical evaluations, including the Multiple Sclerosis Functional Composite measure (MSFC) will be performed before, during and after each experimental phase ("real" and "sham"). In addition, blood levels of neurofilaments, excitability and plasticity of M1, and MRI measures of cortical thickness will be measured before, during and after each stimulation phase.

Inclusion Criteria:

• Ability to give written informed consent to the study
• Age range 18-65 years
• Diagnosis of primary of secondary progressive MS according to 2017 revised Macdonald's criteria (Thompson et al., 2017), presenting with signs of symptoms of progressive dysfunction of the corticospinal tract
• EDSS ≤ 6,5
• Ability to participate to the study protocol
• No or stable (at least six months) DMT or rehabilitative treatments before study entry, and willingness not to change these therapies (including cannabinoids, SSRI, baclofen) during the study.

Exclusion Criteria:

• Relapsing-remitting MS or progressive MS presenting with signs of symptoms other than those typical of the ascending myelopathy phenotype (i.e. progressive cerebellar or cognitive involvement)
• Female with positive pregnancy test at baseline or having active pregnancy plans
• Comorbidities for which synaptic plasticity may be altered (i.e., Parkinson's disease, Alzheimer's disease, stroke)
• Contraindications to TMS
• History or presence of any unstable medical condition such as malignancy or infection
• Use of medications with increased risk of seizures (i.e. Fampridine, 4-Aminopyridine)
• Concomitant use of drugs that may alter synaptic transmission and plasticity (L-dopa, antiepileptics)

All patients, caregivers, and investigators assessing outcomes will be blind to the intervention assignment.