If two signals are phase synchronous then the respective Fourier component at each spectral band should exhibit certain properties. In a pair of artificially generated phase synchronous signals the phase difference at each frequency band changes very slowly over the subsequent frequency bands. This has been called Fourier uniformity in this paper and a measure of it has been proposed. An usefulness of this measure has been outlined in the case of cortical source localization of scalp EEG.
Deep Dive into Fourier Uniformity: An Useful Tool for Analyzing EEG Signals with An Application to Source Localization.
If two signals are phase synchronous then the respective Fourier component at each spectral band should exhibit certain properties. In a pair of artificially generated phase synchronous signals the phase difference at each frequency band changes very slowly over the subsequent frequency bands. This has been called Fourier uniformity in this paper and a measure of it has been proposed. An usefulness of this measure has been outlined in the case of cortical source localization of scalp EEG.
OW the EEG signals from different parts of the brain evolve in time with respect to each other is an important criterion to study many physiological (such as cognition) and pathological (such as seizure) brain functions. A key to study this criterion is synchronization, a loosely defined term in neuroscience, biology and even in physics [1], [2]. Since the term does not have a strict definition its measurement also varies from one application to another [3]. In this paper we will primarily be concerned with phase synchronization.
It was Huygens who first studied phase synchronization between two coupled oscillators. A signal can the thought of as a superposition of many coupled oscillators each of which is represented by a Fourier component of the signal. A dominant trend of determining (instantaneous) phase of a signal is with the help of the Hilbert transformation [4], [5]. Another trend is to determine the phase by convolution of the signal with Morlet’s wavelet [6], [7]. A Shannon entropy based measure of phase synchronization has also been proposed [8]. The notion of phase in any of these methods is not as natural and readily comprehensible as in Huygens’ work. In this paper it has been shown that the notion of phase synchronization between any two signals can be treated from Huygens’ point of view. To bridge the gap between Huygens’ phase and the phase synchronization between any two arbitrary signals a new notion has been introduced, which is named Fourier Kaushik Majumdar is with the Center for Complex Systems and Brain Sciences, Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33431, USA, (phone: ++1-561-297-2225; fax: ++1-561-297-3634; e-mail: majumdar@ccs.fau.edu).
uniformity. Since the study of synchronization in signals from different parts of the brain is an important area of research, Fourier uniformity may be a potential tool to analyze neural signals, particularly the electrophysiological signals either on the scalp or from the deep brain implants. Apart from describing Fourier uniformity theoretically in this paper we will show its application in cortical source estimation of the scalp human EEG data during median nerve stimulation.
In the next section we will describe the notion of Fourier uniformity. In section 3 we will describe the stimulation experiment, data acquisition and preprocessing. In section 4 we will be presenting the results of our studies first with simulated EEG signals generated from known sources by forward calculation. All the simulations have been done on the real head model of the subject, which has been constructed with the help of his structural MRI data. Then the method has been applied to the EEG signal of the same subject during median nerve stimulation. In the concluding section we will summarize the results with a view to future directions.
in strict sense of terms. In other words Fourier uniformity between any two signals is the condition where the phase difference between two Fourier components of the same frequency band remains exactly the same across all the bands. However ( 5) is too strict a condition. To broaden the scope of applicability we propose the following modification in (5):
…. ……..
, where 0 → δ . We will understand satisfaction of ( 6) for any two signals
as the approximate Fourier uniformity.
Following is the standard definition of phase synchronization (for motivation and detailed discussions see [2], [5]). For loosely coupled signals or systems a weaker condition is adopted for which [5].
We can rewrite this as
where δ is a small positive quantity. The condition (7) is the satisfiable condition for a pair of systems to be called loosely coupled. For them the notion of synchronization would be replaced by approximate synchronization. For convenience of calculation in this paper we shall keep 1 = = n m . The general case has been dealt in [9].
Usually the phase α of
for a given t is calculated with the help of the Hilbert transformation on ) (t x j (for detail see [4], [5]). Phase is also determined for a time window as well as for a frequency window by convolution with a suitable wavelet (for detail see [6], [7]). The notion of phase and their measurements are not same in these two most popular methods.
We have already mentioned that a signal can be viewed as superposition of many coupled oscillators. To elaborate this point consider equation (1). First the raw signal ) (t x j has been collected. To resolve the signal into its principle components integral Fourier transforms for all integers n have been performed. This has yielded the Fourier coefficients jn a and jn b with which the signal can be reconstructed as shown in (1). In reality the Fourier transform is an FFT with only a finite number of terms, which can not fully represent the signal. However the more is the number of terms the better is the representation. Therefore (1) should actually be written as
‘) However we will assume the sample frequency
…(Full text truncated)…
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