Fourier Analysis for Neuroscientists

Abstract

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.