NeurostimulationNeurostimulation is the purposeful modulation of the nervous system's activity using invasive (e.g. microelectrodes) or non-invasive means (e.g. transcranial magnetic stimulation or transcranial electric stimulation, tES, such as tDCS or transcranial alternating current stimulation, tACS). Neurostimulation usually refers to the electromagnetic approaches to neuromodulation.
Functional (mathematics)In mathematics, a functional (as a noun) is a certain type of function. The exact definition of the term varies depending on the subfield (and sometimes even the author). In linear algebra, it is synonymous with linear forms, which are linear mappings from a vector space into its field of scalars (that is, they are elements of the dual space ) In functional analysis and related fields, it refers more generally to a mapping from a space into the field of real or complex numbers.
Continuous linear operatorIn functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator. Continuous function (topology) and Discontinuous linear map Bounded operator Suppose that is a linear operator between two topological vector spaces (TVSs). The following are equivalent: is continuous.
Deep brain stimulationDeep brain stimulation (DBS) is a neurosurgical procedure involving the placement of a medical device called a neurostimulator, which sends electrical impulses, through implanted electrodes, to specific targets in the brain (the brain nucleus) for the treatment of movement disorders, including Parkinson's disease, essential tremor, dystonia, and other conditions such as obsessive-compulsive disorder (OCD) and epilepsy. While its underlying principles and mechanisms are not fully understood, DBS directly changes brain activity in a controlled manner.
Antilinear mapIn mathematics, a function between two complex vector spaces is said to be antilinear or conjugate-linear if hold for all vectors and every complex number where denotes the complex conjugate of Antilinear maps stand in contrast to linear maps, which are additive maps that are homogeneous rather than conjugate homogeneous. If the vector spaces are real then antilinearity is the same as linearity.
Linear formIn mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If V is a vector space over a field k, the set of all linear functionals from V to k is itself a vector space over k with addition and scalar multiplication defined pointwise. This space is called the dual space of V, or sometimes the algebraic dual space, when a topological dual space is also considered.
Discontinuous linear mapIn mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions (see linear approximation). If the spaces involved are also topological spaces (that is, topological vector spaces), then it makes sense to ask whether all linear maps are continuous. It turns out that for maps defined on infinite-dimensional topological vector spaces (e.g.
Functional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous or unitary operators between function spaces.
Transcranial direct-current stimulationTranscranial direct current stimulation (tDCS) is a form of neuromodulation that uses constant, low direct current delivered via electrodes on the head. It was originally developed to help patients with brain injuries or neuropsychiatric conditions such as major depressive disorder. It can be contrasted with cranial electrotherapy stimulation, which generally uses alternating current the same way, as well as transcranial magnetic stimulation. Research shows increasing evidence for tDCS as a treatment for depression.
Open mapping theorem (functional analysis)In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result which states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map. This proof uses the , and completeness of both and is essential to the theorem. The statement of the theorem is no longer true if either space is just assumed to be a normed space, but is true if and are taken to be Fréchet spaces.
