Currently, there is no disease-modifying treatment available for Alzheimer’s and Parkinson’s disease (AD and PD) and that includes the highly controversial approval of the Aβ-targeting antibody aducanumab for the treatment of AD. Hence, there is still an unmet need for a neuroprotective drug treatment in both AD and PD. Type 2 diabetes is a risk factor for both AD and PD. Glucagon-like peptide 1 (GLP-1) is a peptide hormone and growth factor that has shown neuroprotective effects in preclinical studies, and the success of GLP-1 mimetics in phase II clinical trials in AD and PD has raised new hope. GLP-1 mimetics are currently on the market as treatments for type 2 diabetes. GLP-1 analogs are safe, well tolerated, resistant to desensitization and well characterized in the clinic. Herein, we review the existing evidence and illustrate the neuroprotective pathways that are induced following GLP-1R activation in neurons, microglia and astrocytes. The latter include synaptic protection, improvements in cognition, learning and motor function, amyloid pathology-ameliorating properties (Aβ, Tau, and α-synuclein), the suppression of Ca2+ deregulation and ER stress, potent anti-inflammatory effects, the blockage of oxidative stress, mitochondrial dysfunction and apoptosis pathways, enhancements in the neuronal insulin sensitivity and energy metabolism, functional improvements in autophagy and mitophagy, elevated BDNF and glial cell line-derived neurotrophic factor (GDNF) synthesis as well as neurogenesis. The many beneficial features of GLP-1R and GLP-1/GIPR dual agonists encourage the development of novel drug treatments for AD and PD.

Much thought has been given to the impact of Amyloid Beta, Tau and Alpha-Synuclein in the development of Alzheimer's disease (AD) and Parkinson's disease (PD), yet the clinical failures of the recent decades indicate that there are further pathological mechanisms at work. Indeed, besides amyloids, AD and PD are characterized by the culminative interplay of oxidative stress, mitochondrial dysfunction and hyperfission, defective autophagy and mitophagy, systemic inflammation, BBB and vascular damage, demyelination, cerebral insulin resistance, the loss of dopamine production in PD, impaired neurogenesis and, of course, widespread axonal, synaptic and neuronal degeneration that leads to cognitive and motor impediments. Interestingly, the acylated form of the hormone ghrelin has shown the potential to ameliorate the latter pathologic changes, although some studies indicate a few complications that need to be considered in the long-term administration of the hormone. As such, this review will illustrate the wide-ranging neuroprotective properties of acylated ghrelin and critically evaluate the hormone's therapeutic benefits for the treatment of AD and PD.

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We investigate an ultraweak variational formulation for (parameterized) linear differential-algebraic equations with respect to the time variable which yields an optimally stable system. This is used within a Petrov-Galerkin method to derive a certified detailed discretization which provides an approximate solution in an ultraweak setting as well as for model reduction with respect to time in the spirit of the Reduced Basis Method. A computable sharp error bound is derived. Numerical experiments are presented that show that this method yields a significant reduction and can be combined with well-known system theoretic methods such as Balanced Truncation to reduce the size of the DAE.