Skip to main content

2013 | OriginalPaper | Buchkapitel

Fourier Analysis for Neuroscientists

verfasst von : Hanspeter A Mallot

Erschienen in: Computational Neuroscience

Verlag: Springer International Publishing

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

In this Chapter, we introduce a piece of mathematical theory that is of importance in many different fields of theoretical neurobiology, and, indeed, for scientific computing in general. It is included here not so much because it is a genuine part of computational neuroscience, but because computational and systems neuroscience make extensive use of it. It is closely related to systems theory as introduced in the previous chapter but is also useful in the analysis of local field potentials, EEGs or other brain scanning data, in the generation of psychophysical stimuli in computational vision and of course in analyzing the auditory system. After some instructive examples, the major results of Fourier theory will be addressed in two steps:

Sinusoidal inputs to linear, translation-invariant systems yield sinusoidal outputs, differing from the input only in amplitude and phase but not in frequency or overall shape. Sinusoidals are therefore said to be the “eigen-functions” of linear shift invariant systems. Responses to sinusoidal inputs or combinations thereof are thus reduced to simple multiplications and phase shifts. This is the mathematical reason for the prominent role of sinusoidals in scientific computing.

The second idea of this chapter is that any continuous function (and also some non-continuous functions) can be represented as linear combinations of sine and cosine functions of various frequencies. Alternatively to the use of sine- and cosine functions, one may also use sinusoidals with a phase value for each frequency, or complex exponentials from the theory of complex numbers.

Both ideas combine in the convolution theorem, stating that the convolution of two functions can also be expressed as the simple product of the respective Fourier transforms. This is also the reason why linear shift-invariant systems are often considered “filters” removing some frequency components from a signal and passing others.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Metadaten
Titel
Fourier Analysis for Neuroscientists
verfasst von
Hanspeter A Mallot
Copyright-Jahr
2013
Verlag
Springer International Publishing
DOI
https://doi.org/10.1007/978-3-319-00861-5_3