Fourier Analysis for Neuroscientists

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 chapters 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 presented in three steps: In the first step, we will demonstrate that sinusoidal inputs to linear, shift-invariant (LSI) 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 “eigenfunctions” of LSI systems. In the second step, we show that most functions can be represented as linear combinations of sine and cosine functions of various frequencies; these may be conveniently written as complex exponentials. Both ideas combine in the third step: that is, the convolution theorem, which states that the convolution of two functions can also be expressed as the simple multiplication of the respective Fourier transforms. This is also the reason why linear shift-invariant systems are often described as “filters” removing some frequency components from a signal and passing others.