On December 21, 1807 a young French engineer addressed the eminent mathematicians of the French Academy, and made what seemed like an incredible claim. He stated that any arbitrary function, defined over a finite interval, could be represented as an infinite summation of cosine and sine functions. This claim was disputed by the members of the Academy, and it was some time before it was accepted for what it was - one of the major advances in Mathematics, now known as Fourier's theorem, named after it's originator: Jean Baptiste Joseph de Fourier (1768-1830).
Why use Fourier Analysis?
A ubiquitous feature of Neural Systems is the presence of Noise at all levels, from single ion channels through to fluctuations in our movements. The presence of such stochastic elements necessitates the use of a Statistical Signal Processing framework.
Part of my research work has been the development of a multivariate Fourier based framework for the analysis of neural data - both spike train and time series. This framework provides a powerful rigorous, statistical framework within which to characterize linear and non-linear interactions between stochastic signals.
In recent years there has been an increased use of spectral methods for
the analysis of neural data, both spike train and time series data. Since
the basis of this framework is the decomposition of signals into constituent
frequency components - these methods are particularly suited for the analysis
of rhythmic neural activity, and oscillatory neural systems.
Three articles provide an extensive overview of the Fourier approach to analysis of neural data. The two Progress in Biophysics and Molecular Biology (53:1-31, 1989; 64(2/3): 237-278, 1995) articles provide self contained tutorial reviews of this framework, and are illustrated throughout with examples of the application and interpretation of this approach. The 1989 article considers spike train data, the 1995 article deals with both spike train and time series data. The 1999 Modern Techniques in Neuroscience Research chapter provides a general overview of linear Fourier analysis, and includes some additional tools such as Pooled Coherence and Maximum Likelihood Analysis.
Further details and reprints are available on the publications page
Additional references including extensions of the framework describing a novel population analysis, and the use of spike train analysis to study Neural Networks, are also available for download on the publications page
A software archive is available to implement these techniques at: http://www.neurospec.org/
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