Neuropathological and Biomarker Findings in Parkinson's Disease and Alzheimer's Disease: From Protein Aggregates to Synaptic Dysfunction

There is mounting evidence that Parkinson’s disease (PD) and Alzheimer’s disease (AD) share neuropathological hallmarks, while similar types of biomarkers are being applied to both. In this review we aimed to explore similarities and differences between PD and AD at both the neuropathology and the biomarker levels, specifically focusing on protein aggregates and synapse dysfunction. Thus, amyloid- peptide (A) and tau lesions of the Alzheimer-type are common in PD and -synuclein Lewy-type aggregates are frequent findings in AD. Modern neuropathological techniques adding to routine immunohistochemistry might take further our knowledge of these diseases beyond protein aggregates and down to their presynaptic and postsynaptic terminals, with potential mechanistic and even future therapeutic implications. Translation of neuropathological discoveries to the clinic remains challenging. Cerebrospinal fluid (CSF) and positron emission tomography (PET) markers of A and tau have been shown to be reliable for AD diagnosis. Conversely, CSF markers of -synuclein have not been that consistent. In terms of PET markers, there is no PET probe available for -synuclein yet, while the AD PET markers range from consistent evidence of their specificity (amyloid imaging) to greater uncertainty of their reliability due to off-target binding (tau imaging). CSF synaptic markers are attractive, still needing more evidence, which currently suggests those might be non-specific markers of disease progression. It can be summarized that there is neuropathological evidence that protein aggregates of AD and PD are present both at the soma and the synapse. Thus, a number of CSF and PET biomarkers beyond -synuclein, tau and A might capture these different faces of protein-related neurodegeneration. It remains to be seen what the longitudinal outcomes and the potential value as surrogate markers of these biomarkers are

Structural stability of (C,A)-marked and observable subspaces

Given an observable pair of matrices (C;A) we consider the manifold of (C;A)-invariant and observable subspaces having a fixed Brunovsky- Kronecker structure. Using Arnold’s techniques we obtain the explicit form of a miniversal deformation of a marked and observable (C;A)- invariant subspace with regard to the usual equivalence relation. As an application, we obtain the dimension of the orbit and we characterize the structurally stable subspaces

Classi fication of monogenic invariant subspaces and uniparametric linear control systems

The classification of the invariant subspaces of an endomorphism has been an open problem for a long time, and it is a ”wild” problem in the general case. Here we obtain a full classification for the monogenic ones. Some applications are derived: in particular, canonical forms for uniparametric linear control systems, non necessarily controllable, with regard to linear changes of state variablesPreprin

Geometric structure of the equivalence classes of a controllable pair

Given a pair of matrices representing a controllable linear system, we study its equivalence classes by the single or combined action of feedbacks and change of state and input variables, as well as their intersections. In particular, we prove that they are differentiable manifolds and we compute their dimensions. Some remarks concerning the effect of different kinds of feedbacks are derived.Postprint (published version

Perturbations preserving conditioned invariant subspaces

Given the set of matrix pairs M ⊂ Mm,n(C) × Mn(C) keeping a subspace S ⊂ Cn invariant, we obtain a miniversal deformation of a pair belonging to an open dense subset of M. It generalizes the known results when S is a supplementary subspace of the unobservable one.Postprint (published version

Miniversal deformations of observable marked matrices

Given the set of vertical pairs of matrices M¿ Mm,n(C)×Mn(C) keeping the subspace Cd×{0} ¿ Cn invariant, we compute miniversal deformations of a given pair when it is observable and the subspace Cd × {0} is marked. Moreover, we obtain the dimension of the orbit, characterize the structurally stable vertical pairs and study the effect of each deformation parameter.Preprin

Estructura diferenciable de las clases de equivalencia de un par controlable

Dado un par de matrices que representa un sistema lineal controlable, estudiamos las clases de equivalencia por la acción individual o combinada de realimentaciones y cambio de variables de estado y de entrada, así como sus intersecciones. En particular, demostramos que son variedades diferenciables y calculamos sus dimensiones