Time–frequency analysis using damped-oscillator pseudo-wavelets: Application to electrophysiological recordings

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Abstract

The damped-oscillator pseudo-wavelet is presented as a method of time–frequency analysis along with a new spectral density measure, the data power. An instantaneous phase can be defined for this pseudo-wavelet, and it is easily inverted. The data power measure is tested on both computer generated data and in vivo intrahippocampal electrophysiological recordings from a rat. The data power spectral density is found to give better time and frequency resolution than the more conventional total energy measure, and additionally shows intricate time–frequency structure in the rat that is altered in association with the emergence of epilepsy. With epileptogenesis, the baseline theta oscillation is severely degraded and is absorbed into a broader gamma band. There are also broad 600 Hz and 2000 Hz bands which localize to hippocampal layers that are distinct from those of the theta and gamma bands. The 600 Hz band decreases in prominence with epileptogenesis while the 2000 Hz band increases in prominence. The origins of these high frequency bands await further study. In general, we find that the damped-oscillator pseudo-wavelet is easy to use and is particularly suitable for problems where a wide range of oscillator frequencies is expected.

Research highlights

▶ Time–frequency analysis using mathematical oscillators to detect data oscillators. ▶ High time–frequency resolution. ▶ Fast and convenient to use. ▶ Tested on computer generated and in vivo electrophysiological data.

Introduction

Time–frequency analysis involves monitoring the changes in the frequency spectrum of a system over time. It is of importance in nearly every field of science and engineering. The old work-horse of frequency analysis, the fast Fourier transform (FFT), can be applied to this problem by segmenting data into shorter time windows. By monitoring changes in the power spectrum in each time window, one can determine the presence or absence of activity at certain frequencies within that time window. The segmentation, however, introduces artifact and precludes finer time resolution within time windows. There have been many discussions of these issues and many refinements of the FFT approach (Thomson, 1982, Mitra and Pesaran, 1999, Groechenig, 2001). These refinements are all somewhat labor intensive, requiring judgments to be made about window sizes and windowing functions, which may be different for different frequency ranges. Mathematical generalization of time windowed FFT power spectra to a wide range of time–frequency distribution functions has been accomplished (Cohen, 1989, Oonincx, 2000). At least one of these time–frequency distribution functions has found application to the study of electroencephalograms (EEG) (Zaveri et al., 1992). Each of these time–frequency distribution functions has its own strengths and weaknesses.

Over the last 20 years, wavelet transforms have also become an important new way of doing time–frequency analysis (Daubechies, 1990, Farge, 1992, Graps, 1995, Cohen and Kovacevic, 1996). A wavelet consists of an oscillatory waveform that has a fairly well-defined frequency and which exists only for a brief period of time. By convolving time series data with a suitably chosen wavelet, one can determine whether an oscillation of a certain frequency is present at a certain interval in time, in a way that is more convenient and less susceptible to artifact than time windowed FFT. There are many kinds of wavelets, and many applications of wavelet transforms to EEG analysis (Le Van Quyen et al., 2001, Senhadji and Wendling, 2002, Le Van Quyen and Bragin, 2007, Allen and MacKinnon, 2009, Storti et al., 2009).

A defining property of wavelets is the admissibility criterion, a consequence of which is that the mean of the wavelet when averaged over all time must equal zero. This criterion insures that a stable inverse transform exists (Farge, 1992, Cohen and Kovacevic, 1996). A stable inverse transform is critical for reliable signal transmission and reconstruction. However, in many fields of science and engineering, one is not interested in signal reconstruction. Rather, one may only wish to detect whether oscillations of certain frequencies appear, at what times and for what duration of time. For example, in the brain, it is known that oscillations in the theta (4–8 Hz) and gamma ranges (30–100 Hz) are associated with cognitive activity. Oscillations from these two frequency ranges are sometimes phase-coupled, such that the faster gamma frequencies ride entirely on the crests (or troughs) of slower theta rhythms (Lisman, 2005, Canolty et al., 2006, Axmacher et al., 2010). Of increasing clinical interest are high frequency oscillations (HFOs) in the range of 200–500 Hz. These oscillations tend to occur more frequently in brain regions that are epileptogenic, and so may be useful as a marker for tissue that should be surgically resected in people with refractory epilepsy (Worrell et al., 2008, Zijlmans et al., 2009, Bragin et al., 2010). For these applications, one might consider relaxing the admissibility criterion.

Relaxing the admissibility criterion results in a wide class of waveforms called pseudo-wavelets (or quasi-wavelets), which have found application in the study of turbulence and other complex phenomena (Qiu et al., 1995, Wilson et al., 2009). Here we describe a particularly simple pseudo-wavelet that has a uniquely simple physical interpretation. Our pseudo-wavelet corresponds to using a mathematical model of a frictionally damped harmonic oscillator to detect data oscillations of the same frequency. We have previously referred to this pseudo-wavelet as the damped-oscillator oscillator detector (DOOD) (Hsu, Hsu et al., 2007). Here we develop the damped oscillator pseudo-wavelet in more detail and apply it to time series analysis of the EEG. We illustrate our approach with computer generated oscillations as well as data from in vivo rat hippocampal recordings. The rat recordings show a richness in time–frequency dynamics which invites further exploration.

Section snippets

Theory

The motivation for the DOOD pseudo-wavelet comes from considering how one might construct a mathematical oscillator detector based on analogy to mechanical oscillators. One first constructs a set of mathematical harmonic oscillators each of which has its own natural frequency of oscillation. Next, consider the time series data as a driving force that acts upon each mathematical harmonic oscillator. Start with the mathematical oscillators all at rest, and then let the data drive activity in the

Illustrative examples

Here we explore the DOOD algorithm with both computer generated data and real data from rats. We will use both X-DOOD and V-DOOD, anticipating that X-DOOD will be more sensitive to low frequency activity and V-DOOD will be more sensitive to high frequency activity.

Discussion

In this paper, our main goal is to introduce the DOOD algorithm. More detailed analysis of the time–frequency spectra of in vivo rat (and human) data will be presented in future publications, including more detailed study of the 600 Hz and 2000 Hz bands. Nonetheless, the results shown here suggest a striking richness of time–frequency structure in the normal rat, which is altered after the rat undergoes epileptogenesis. If we believe that oscillations underlie the function of the brain (Buzsaki,

Experimental methods

Adult, Sprague–Dawley rats were anesthetized with 2% isoflurane after pretreatment with atropine (2 mg/kg) and placed in a stereotaxic apparatus. The area of incision was injected with 0.5 ml of 0.5% bupivacaine for prolonged local anesthesia/analgesia and burr holes for the recording probe and ground screw were placed by conventional surgical techniques. The depth-recording electrode, a 16-channel silicon probe (NeuroNexus, Inc., Ann Arbor, MI), was implanted in the right hippocampus (3.0 mm

Acknowledgements

DH was supported by the NIH Loan Repayment Program. TPS was supported by NIH R01-25020. GAW was supported by NIH R01-NS63039-01.

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