View source for Talk:Hilbert transform for brain waves

== Reply to comments by Reviewer on 8 Jan 07 ==

The Reviewer writes: " Such backward rotation ideally should not occur in a true analytic phase signal -- because the analytic phase should always increase, not going backwards."  Indeed that restriction holds within a given state. However, we provide evidence that the analytic phase is subject to discontinuiites that manifest state transitions. In fact, these state transitions offer a key to understanding the sequential changes in the spatial patterns of amplitude modulation of the beta and gamma carrier waves. Each such AM pattern is accompanied by a PM (phase modulation) pattern in the form of a cone. Each successive cone has an apex, for which the location and sign are random variables. Therefore at any one electrode location, the replacement of a cone by its successor gives an abrupt change in phase that can go either forward or backward. Yet the changes are not actually simultaneous, because the phase cones have a slope (spatial phase gradient) in rad/mm that for a given beta or gamma carrier frequency in rad/sec gives phase velocity in m/sec, which is in the range of the conduction velocities of cortical axons that mediate the spread of phase transitions. AS I conclude in my Section 3: "The main use of Δpj(t) phase differences is to locate the onsets of state transitions by SDX(t). " 

The Reviewer is quite correct that the mixing of frequencies inadequatelye excluded by temporal filtering is a major cause of 'phase slip' unrelated to state transitions. As I take pains to illustrate in my Fig. 4 D, the 'side-band' frequencies typically have differing spatial distributions of AM (amplitude modulation), which results in the ladder-like divergence of unwrapped phase values, for the reason that I explain in the text. 

As the Reviewer surmises, ['…Because the filtered signal include at least two possible oscillations (beta and gamma) having independent phases…"] independent wave packets with carrier frequencies in the beta and gamma ranges can and do overlap, and they can be separated by narrower band pass filtering, as I have cited in my references (Freeman, 2005, 2006), where I describe the classification of their AM patterns with respect to behavior. 

Incidentally, as I write in my article, the great advantage of the analytic phase is that it allows for variation in the frequency, which is not the case with the Fourier components. The "always positive" phase change the Reviewer has in mind is compatible with slowing (or hastening) the frequency even close to zero. The Hilbert Transform of course, as Hwang among many others has stated, removes negative frequencies, which is another of its advantages over the FFT. However, physiologically a state transition can precipitate a sudden jump in phase that exceeds the mean step size, thus giving a negative value to the instantaneous frequency, but only at the transition and not sustained, so that Hwang's constraint is not violated. As I try to make clear, the advantage of the clinical mode decomposition over the EMD is the relevance to the detection of the temporal and spatial locations of the state transitions, which the H-HT cannot do. 

I ask that the Reviewer evaluate my reasoning on physiological grounds and agrees or still disagrees. Meanwhile, I will modify my article to state that 'negative frrequencies' are not in violation of the HT in the case of state transitions, and I will introduce briefly the concept of the conical phase gradient, which I did not do before owing to space limitations.  I hope the Editor will accept a modest increase in the length of my already long article.