Searching for Temporal Phase Coherence in the Cortical Network

with a Time-Frequency Analysis of the Local Field Potential

Samuel P. Burns 1,2, Robert M. Shapley 1,2, Michael J. Shelley 1,2, Dajun Xing 2

1Courant Institute of Mathematical Sciences & 2Center for Neural Science

New York University, New York, NY

email : sburns@cims.nyu.edu

May 23, 2008

1 Introduction

Gamma-band (25-90Hz) oscillations occur in manyparts of the brain. We are seeking to understand theunderlying neural mechanisms that generate gammaoscillations in the cerebral cortex by studying gammaactivity in the Local Field Potential (LFP) in V1cortex. The LFP is a measure of the average mem-brane potential of neurons in the neighborhood of therecording electrode and is commonly interpreted as ameasure of local network activity (Kruse and Eckhorn1996; Logothetis et al. 2001; Buzsaki 2006).

Previous experimental studies of V1 have reporteda peak in the gamma-band of he LFP power spectrumwhen the visual cortex was visually driven. (Grayand Singer 1989; Frien et al. 2000; Logothetis etal 2001; Siegel and Konig 2003; Henrie and Shapley2005). In modeling studies of neuronal networks ofinhibitory neurons, and with both inhibitory and ex-citatory neurons, theorists have found that neuronsin these networks may exhibit synchronized spikingthat gives rise to deterministic, harmonic network os-cillations (Abbott and van Vreeswijk 1993; Somersand Kopell 1993; Wang and Rinzel 1993; Somers andKopell 1995; Wang and Buzsaki 1996; Traub et al.1996a; Traub et al. 1996b; Ermentrout and Kopell1998; White el al. 1998; Kopell et al. 2000; Maex andDe Schutter 2003; Borgers and Kopell 2005; Borg-ers et al. 2005; Tuckwell 2006, Kumar 2008). Sev-eral other modeling studies (Brunel and Hakim 1999;Brunel 2000; Tiesinga and Jose 2000; Brunel andWang 2003; Geisler et al. 2005; Brunel and Hansel2006) have found that there are parameter regimes inwhich individual cells do not exhibit periodic firingbut the network-averaged firing rate does, and more-over exhibits a temporally coherent, periodic oscilla-tion in the gamma-band. These studies find that thenetworks in these studies have a harmonic or relax-ation oscillator response in their subthreshold mem-

brane potential, and noise in the network causes in-dividual neurons to fire preferentially at the peaks ofthe membrane oscillation. A common feature of thesemodels is that the coherent network oscillation is anemergent behavior of the system and is an equilib-rium or steady dynamical state of the network as longas the statistics of the network inputs remain con-stant. That such temporally coherent oscillations un-derlie the experimentally observed gamma-band peakwe refer to as the coherent oscillator hypothesis.

The coherent oscillator hypothesis, in the form ofeither spike or membrane potential synchrony, hasbeen widely cited in the experimental literature ongamma-band oscillations but the temporal coherenceof these oscillations has not been examined statisti-cally. The concept of emergent coherent oscillationsappears in studies of the hippocampus in vivo andin vitro (Penttonen et al. 1998; Fisahn et al. 1998;Traub et al. 2000; LeBeau et al. 2002; Mikkonen etal. 2002; Csicvari et al. 2003; Mann et al. 2005a;Mann et al. 2005b; Vida et al. 2006; Mann andPaulsen 2007; Montgomery and Buzsaki 2007; Senioret al. 2008), visual cortex (Henrie and Shapley 2005,Zaksas and Pasternak 2006; Disney and Aoki 2008),neocortex (Giugliano et al. 2004; Morita et al. 2008),prefrontal cortex (Compte et al. 2003; Durstewtizand Gabriel 2006), olfactory systems of insects andmammals (Macleod and Laurent 1996; Macleod etal. 1998; Lagier et al. 2004; Bathellier et al. 2006;Schoppa 2006) and other regions of the brain (Buhlet al. 1998; Edwards et al. 2005; Sohal and Hugue-nard 2005; Joelving et al. 2007). In the EEG litera-ture there have been studies of the phase-locking be-tween visual stimulus and the gamma-band responsein EEG signals recorded in humans (Tallon-Baudryet al. 1996; Frund et al. 2007; Senkowski et al. 2007,Schadow et al. 2007, Ohla et al. 2007)

Here we analyze LFP measurements recorded frommacaque V1. Visual stimulation evokes a noisy re-

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sponse in the LFP with peak spectral power in thegamma-band near 40Hz. We have developed a spe-cialized time series analysis technique to analyze thecoherence of the temporal phase content of a signal.This technique uses the continuous Gabor transform(CGT) (Mallat 1999, p 69; see Methods) to inves-tigate a signal at each temporal frequency. Usingour CGT technique, we formulate a statistical testto compare the coherent oscillator hypothesis againstthe data: measurements of LFP in vivo in macaqueV1 cortex. In short, we have performed a rigor-ous search for emergent coherent oscillations and findthat the data do not support the coherent oscillatorhypothesis. Our interpretation is that the source ofthe gamma-band spectral peak is a signal of a morestochastic nature, for instance, arising from a reso-nant network driven by random inputs.

2 Results

2.1 Local Field Potential Data

The local field potential (LFP) is an extracellu-lar voltage measurement that characterizes the lo-cal network activity of the population of neurons inthe neighborhood of a measurement (approximately102−103 neurons) and is defined as the low frequency(≤ 250Hz) portion of the raw field potential. Specifi-cally, the LFP signal measures the current flow due tosynaptic activity while the higher frequency compo-nents of the raw data are related to action potentials(Kruse and Eckhorn 1996; Logothetis et al. 2001;Buzsaki 2006).

The data analyzed here are single electrode record-ings from macaque V1 under sufentanil-anesthesia.The recordings are made with a baseline blank (nostimulus) and with a high contrast, drifting gratingpattern (using a monitor with a 100Hz refresh rate)at the ’preferred’ orientation for maximal response(further experimental details described in Methods4.1). The grating pattern is shown for 2 or 4 seconds,with 1 to 33 repetitions depending on the trial. TheLFP response to a blank stimulus is recorded for 1sec with 12 repetitions for each trial. The single elec-trode data are taken from six experiments in whichdata were recorded with multielectrode arrays ofsevenelectrodes. Voltage data were sampled at a rate of20kHz. Data from the electrode with the largest re-sponse to the visual stimulus were selected for anal-ysis. The 60Hz line noise signal was filtered out ofthe LFP recordings as described in Methods 4.6. Anexample of a 4 second LFP recording under visualstimulation is plotted in Figure 1B.

In order to test the coherent oscillator hypothesis,

recordings with spectra that exhibit elevated powerin the gamma-band with a distinct peak are chosen infavor of recordings that have a broad band response(see Figure 1A) as the peaked spectra are more con-sistent with the hypothesis of a coherent oscillator.The power spectra are computed by windowing thedata with gain corrected Welch window (Harris 1978)and averaged over repeated identical stimuli.

2.2 Time-Frequency Analysis

A time-frequency analysis is used here to examinethe temporal evolution of LFP data at each fre-quency. This method is chosen over a spectral anal-ysis in order to study the properties of the signal asa function of time. Similar analyses were previouslyused to study the temporal structure of brain activ-ity recorded in EEG (Makeig 1993, Herrmann et al.2004, Busch et al. 2005; Morup et al. 2006; Ohla etal. 2007), LFP (Pesaran et al. 2002), and describedas a general method for studying event-related activ-ity in neural signals (Sinkkonen et al. 1995; Mitra andPesaran 1999; Hurtado et al. 2004). The continuousGabor transform (CGT) is the convolution of an en-veloped complex plane wave ψ against the time seriesbeing examined(see Methods 4.2) and is a function oft, the center of the convolution, and ω0, the frequencyof the underlying wave (shown schematically in Fig-ure 1C). Scale varying wavelets, whose width in thetime domain dilates with increasing scale (decreasingfrequency), become too coarse in the time domain atlow frequency and too broad in the frequency domainat higher frequencies for this study. To avoid theseproblems associated with the scale representation ofwavelet transforms, the Gabor transform is used be-cause its fixed time scale preserves the relationship tofrequency.

Plotted in Figure 1D is the CGT amplitude spec-trum of the LFP recording shown in Figure 1B. Incomparing the amplitude spectrum with the LFPtime series, the three bursts of activity in Figure 1Bat times t = 1.75s, 2.8s and 3.75s can be seen in Fig-ure 1D and it is clear that the CGT is capable ofcapturing the time evolution of the variability of theLFP recording. From inspection of Figure 1D, theLFP signal has large bursts of activity on the scaleof 100ms with power concentrated mainly near 40Hzand with a smaller peak at 70Hz. These spectralpeaks and time scale are characteristics common toall LFP recordings analyzed here. The phase spec-trum of CGT is plotted in Figure 1E and shows thetime evolution of the local phase of each frequency ofthe transform. The phase of the peaks at 35Hz and70Hz have less variability than at other frequency and

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Figure 1: Local field potential data recorded from macaque V1. A: Average power spectra of the LFP with a blank stimulus (black)and with a drifting grating stimulus (magenta), B: The time course of a 4 sec stimulated LFP recording, C: Schematic diagram ofcontinuous Gabor transform, D: The amplitude spectrum from the continuous Gabor transform of the time course in (B.) with a 0.5Hzspectral resolution and 2ms time resolution, E: The phase spectrum from the continuous Gabor transform with the phase rotated tothe beginning of the sample and unwrapped to remove 2π jumps.

in this study we quantitatively examine the coherenceof these frequency components.

2.3 Statistical Tests

This study is designed to investigate the temporalcoherence of this signal at each frequency. The nullhypothesis tested is whether the measured LFP sig-nal can be modeled as coherent oscillator in noise atany frequency in the gamma-band. Coherence, as

discussed here, means at a particular frequency thesignal can modeled as an amplitude modulated sinewave of constant phase plus noise over the course ofthe record,

A(t) sin(ω0t+ φ0) + noise, (1)

where the case of a constant amplitude coherent os-cillator is included in this more general model. Asdescribed in the introduction, the coherent oscillator

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20Hz, CV = 0.859 25Hz, CV = 0.898 30Hz, CV = 0.960

35Hz, CV = 0.690 40Hz, CV = 0.870 45Hz, CV = 0.857

50Hz, CV = 0.930 55Hz, CV = 0.837 60Hz, CV = 0.672

65Hz, CV = 0.829 70Hz, CV = 0.626 75Hz, CV = 0.885

80Hz, CV = 0.915 85Hz, CV = 0.926 90Hz, CV = 0.822

Sine WaveCV = 1.34e−006

Sine Wave + NoiseCV = 0.0389

Noise CV = 0.840

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Figure 2: A: Phase portraits for simulated data of a pure sine wave, a sine wave in noise and noise only, B: Phase portraits of thedata shown in Figure 1 for several frequencies in the gamma-band. Each portrait is for a 4 second period.

response is a steady-state behavior that persists for aslong as the statistics of the driving inputs remain con-stant. In the data examined here the drifting gratingvisual stimulus is presented for a constant 2-4 sec-onds depending on the experiment. According to thenull hypothesis, the V1 network from which the dataare recorded will undergo a temporally coherent os-cillation of the type described in (1) for the lengthof the stimulus. The statistical test performed hereexamines the coherence of the data over a period de-termined by the length of time the visual stimulus

is presented. From a visual inspection of the foursecond LFP voltage recording in Figure 1B it is am-biguous whether the prominent oscillation at 35Hz istemporally coherent or not. It is for this reason thatwe conduct a rigourous quantitative analysis of thenature of the phase of this signal.

The coherence at each frequency is examined byplotting the polar phase portrait (see Methods 4.3)of the CGT’s amplitude and phase parameterized bytime. When the LFP has large amplitude bursts asseen in Figure 1 the phase portrait will have a cor-

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responding large amplitude excursion away from theorigin at that frequency. The phase portraits of apure sine wave, a sine wave in noise, and noise areplotted in Figure 2A. The pure sine wave appearsas a single point in the phase portrait representationbecause it has a constant amplitude and fixed phase.When noise is added to the sine wave the rotatedphases do not all fall on a single point but remain lo-calized in a common sector of the polar plane ratherthan exploring all quadrants as is done by the noisesignal.

In Figure 2B, the phase portraits for the LFP datashown in Figure 1 are plotted at frequencies in thegamma-band from 20-90Hz. If the LFP has phasecoherent oscillations, as in the null hypothesis, thelarge amplitude events will cluster in a common sec-tor of the phase portrait. The phase portraits of thepeaks in the spectral power at 35Hz and 70Hz (Fig-ure 1A) show high amplitude (large radius) eventsthat correspond to the bursts discussed in Section2.2. In comparision to the CGT spectrum, additionalinformation about the phase is included in the phaseportraits. In the 35Hz phase portrait the three burstsseen in the LFP time series (Figure 1B) and the CGTspectrum (Figure 1D) are visible in the three ’loops’away from the origin. From visual inspection thesebursts occupy different sectors of the phase portraitand therefore are not phase coherent. Similarly, inthe 70Hz phase portrait the three bursts are presentand again are not phase coherent.

The circular variance (CV) (see Methods 4.5) isused as a statistic to quantify the coherence of anoscillation or localization of the trajectories of thephase portraits. The CV is normalized by the averageamplitude of the oscillation and is dimensionless, thisis convenient as it allows the coherence of oscillationsat different frequencies to to be compared. The CVof each phase portrait is listed above the plots at eachfrequency and as seen here more coherent oscillationhave CV values near zero and noisier signals closer toone.

At frequencies near the peaks in the power spec-trum, 35Hz and 70Hz, the phase portraits have higheramplitudes but their phases wander in a circular man-ner. This is the result of being slightly off the actuallypeak and is an artifact of the CGT decomposition.These wandering off-peak phase portraits have a CVvalues closer to one and so are not be mistaken forcoherent oscillations.

A Monte Carlo type statistical test is devised todetermine whether coherent oscillations are presentin the recorded LFP signal. The test, which is per-formed at all frequencies between 10Hz and 100Hz foreach experiment, simulates the recorded LFP signal

under the blank and drifting grating visual stimuliusing the coherent oscillator null hypothesis model.A schematic diagram of the simulation procedure isshow in Figure 3.

In the left hand column of Figure 3 the simulationof the blank stimulus data is described. The blankcase is assumed to have no structure and is modeledas noise. The blank stimulus time series is generatedby taking the inverse Fourier of the average amplitudespectrum of the unstimulated LFP data with randomphases drawn from a uniform distribution on the in-terval (0,2π] assigned to each frequency in the rangeof 1-250Hz. The CGT of the simulated time series iscomputed over the frequency band between 10-100Hzand, at each frequency, the CV of the phase portraitis calculated. An ensemble 10,000 simulations is gen-erated at each frequency for each experiment to cre-ate an estimate of the simulated probability densityfunctions (PDF) for the blank stimulus CV.

The LFP under drifting grating stimulus has anelevated power spectrum with a strong peak in thegamma-band centered on 30-40Hz. The null hypoth-esis tested here assumes that this increased power isthe result of a coherent oscillation of the local neu-ronal network of type described in (1) that becomesactive in under visual stimulation in addition to thenoise of the blank stimulus. The simulation of thedata under the drifting grating stimulus is describedin the right hand column of Figure 3. The simulatedtime series for the drifting grating stimulus is gener-ated by simulating the background noise of the blankstimulus and adding to it a constant amplitude sinewave at the frequency being tested. The amplitude ofthis coherent sine wave is given by the square root ofthe difference between the average power spectra ofthe blank and drifting grating data and whose phaseis random. As in the blank stimulus case, the CGTof the simulated time series is computed and the CVis computed at each frequency. An ensemble 10,000simulations is generated at each frequency for eachexperiment to create an estimate of the simulatedPDFs for the drifting grating stimulus CV. Ampli-tude modulated sine waves can be expressed as a sumof constant amplitude sine waves, described in Meth-ods 4.4, and as a result the test described here willalso detect amplitude modulated sinusoidal signals.

Before the CV values of the data are compared tothe simulated null hypothesis CV PDFs, it must beshown that the CV statistic is able to identify thata coherent oscillation is present in a signal relativeto the noise of the blank stimulus. The ability ofthe CV to identify the presence of a coherent oscil-lation in the simulated LFP signal is quantified bythe distance between the simulated blank and drift-

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ing grating CV PDFs. If the two PDFs are similarand have significant overlap it will not be possible todetermine if a given CV value belongs to one PDFand not the other. The inability to discern to whichPDF the sample belongs results in the CV being a lowpower statistic and the test will not be meaningful(it is important to note that as used here statisticalpower and the power spectrum are not the same). Ingeneral, the simulated drifting grating signal on av-erage has a lower mean CV than the blank signal dueto the addition of the coherent oscillator. Given thischaracteristic of the simulated CV PDFs, the statis-tical power of the CV is measured at each frequencyby comparing the 95th percentile of the simulateddrifting grating CV PDF to the 5th percentile of thesimulated blank PDF. If the 95th percentile of thedrifting grating CV is less than the 5th percentile ofthe blank CV than there is sufficient statistical powerto test the LFP at that frequency otherwise no testis performed. Intuitively the statistical power is re-lated to the difference between the blank and drifting

grating power spectra and the statistical power calcu-lation is a way to quantify how elevated the driftinggrating power spectrum must be over the blank spec-trum to detect statistically that a signal is present.

2.4 Test Results

The results of the statistical test of the LFP data forthe temporally coherent null hypothesis described in(1) are shown in Figure 4. Tests are performed bycomputing the CV of the data at the frequencies thathave sufficient statistical power, and percentiles withrespect to the simulated CV PDFs are determined.In Figure 4, the lower portions of the plots containthe blank and drifting grating power spectra and theupper portions contain the percentiles of the CV ofthe data with respect to the null hypothesis. Alsoplotted in each pane of Figure 4 are the 95% confi-dence limits.

For all six experiments the test rejects the coher-ent oscillator null hypothesis in the neighborhood ofthe gamma-band peak between 20Hz and 60Hz as the

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Figure 4: Statisical results from six experiments. In each pane the bottom two curves are the average power spectra of the blankstimulus (black) and the drifting grating stimulus (magenta). In the top portion of each pane is plotted the percentile of the CV of thedata at that frequency with respect to the coherent oscillator null hypothesis (red squares). The 95th percentile is plotted for reference.

percentile of the CV of the data with respect to thenull hypothesis is greater than 95% at these frequen-cies. These results reject the coherent oscillator hy-pothesis as an explanation for the observed elevatedpower spectra in macaque V1 under stimulation.

In the experiment shown in Figure 4D there areseveral frequency whose CV fall in the 85-95% rangeabove 70Hz which not seen in the other five experi-ments. In this experiment the drifting grating stim-ulus was presented only once in comparison to the

higher number of repeated presentations in the otherexperiments. In addition the stimulus was shown fora period of two second in comparsion to the foursecond periods of the other experiments. For thesereasons we will consider the borderline results in theexperiment shown in Figure 4D to be a result of un-dersampling.

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3 Discussion

The results shown here reject the hypothesis that os-cillations seen in the gamma-band of visually drivenmacaque V1 LFP data (Figure 1A) can be modeledas a temporally coherent oscillation of the type de-scribed by (1). We conclude that the elevated gammaoscillations are not the result of spontaneous deter-ministic, harmonic or relaxation network oscillationsas seen in modeling studies. The results rule out aclass of theoretical models for V1 that claim steady-state temporally coherent oscillations can be sus-tained in a local neuronal network. These results alsohave implications for the existence of coherent oscil-lations originating from cellular mechanisms in V1.We do not see evidence of the influence of pacemakercells such as ’chattering cells’ (Gray and McCormick1996) or gap junctions (Traub et al. 1999; Traub andBibbig 2000; Traub et al. 2003).

This study has found that oscillations are not tem-porally coherent on the time-scale of the visual stim-ulus (2-4 seconds). However on shorter time-scales,individual bursts of activity may be coherent. Spa-tially coherent bursts of gamma-band activity havebeen found in the rat hippocampus with a time-scaleof approximately 100ms (Buzsaki ref.). The tempo-ral coherence and time-scale of the individual burstsseen in the data can be quantified using the methoddescribed here and their determination we plan topursue in future research.

While this study has shown that the temporallycoherent oscillator model does not fit the data fromV1, there is structure in data in the form of burstsof activity concentrated in particular frequencies ofthe gamma-band. Another model that could pro-duce the peaked, elevated gamma-band power spectrarecorded in V1 is a resonant stochastic filter. Henrie,Kang, Shelley & Shapley 2005 and Kang et al. 2008found that a stochastic resonant network model ofV1 with recurrent connections to extrastriate cortexhas a resonant response in the gamma-band. In thismodel the network is viewed as a resonant stochas-tic oscillator with a stable state for noiseless inputscorresponding to the quiescent periods (low ampli-tude) of the LFP data (see Figure 1B). When noiseis added to the system, from feed-forward and re-current inputs, the network is randomly excited intoshort high energy bursts of excitation at a resonantfrequency centered in the gamma-band (see also Ren-nie et al. 2000). In the resonant stochastic oscillatormodel, the phase associated with each burst is inde-pendent and the signal is not temporally coherent forthe length of the record in the sense described here.The varying phases associated with the independent

bursts of activity generate a broad peak about theresonant frequency of the network as seen in the datapresented here (Figure 1A). Our further research onthe elevated gamma-band response to visual stimu-lus will focus on testing whether the data support astochastic resonant oscillator model.

The finding that coherent oscillations are notpresent in V1 activity also may have consequencesfor theories of synchrony in brain activity (Gray 1999;Shadlen and Movshon 1999; Engel et al. 2001; Varelaet al. 2001, Buzsaki and Draguhn 2004). Gammaoscillations in EEG and LFP recorded in differentparts of the brain are theorized, via temporal binding(or ’binding-by-synchrony’), to be a mechanism bywhich different regions of brain synchronize in orderto transmit information, an example of which may beperception (Gray and Singer 1989; Gray et al. 1989;Singer and Gray 1995; Buzsaki 2006). Possible theo-ries include a bottom-up model for coherent networksynchronization in which synchrony originates as aprolonged coherent oscillation at lower levels of thebrain and propagates to higher regions. This studyhas shown that V1 does not contain coherent oscil-lations under anesthesia when higher functioning re-gions of the brain are inactive. As a result, this studydoes not support the idea of bottom-up synchroniza-tion originating as a coherent oscillation in V1 asthere is no evidence of a coherent signal emergingout the stimulated response.

As described in the introduction, gamma oscilla-tions have been observed in many other regions ofthe brain. With the development of this new timeseries technique we plan to examine data from re-gions outside of V1 to determine the nature of LFPoscillations measured in other regions.

4 Methods

4.1 Experimental Procedure

4.1.1 Surgery and preparation

Acute experiments were performed on adult OldWorld monkeys (Macaca fascicularis). All surgicaland experimental procedures were performed in ac-cordance with the guidelines of the U.S. Depart-ment of Agriculture and have been approved by theUniversity Animal Welfare Committee at New YorkUniversity. Animals are tranquilized with acepro-mazine (50μg/kg, im) and anesthetized initially withketamine (30 mg/kg, im) and then with isofluorane(1.5-3.5% in air). After cannulation and tracheotomy,the animal is placed in a stereotaxic frame and ismaintained on opioid anesthetic (sufentanil citrate,

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6-12μg kg−1 h−1, iv) for craniotomy. A craniotomy(about 5mm in diameter) is made in one hemisphereposterior to the lunate sulcus ( 15mm anterior to theoccipital ridge, 10-20mm lateral from the midline). Asmall opening in the dura ( 3x3mm2) is made to pro-vide access for multiple electrodes. After surgery, theanimal is anesthetized and paralyzed with a contin-uous infusion of sufentanil citrate (6-18μg kg−1 h−1,iv) and pancuronium bromide (0.1mg kg−1 h−1, iv).Vital signs, including heart rate, electroencephalo-gram, blood pressure, oxygen level in blood, andurine specific gravity are closely monitored through-out the experiment. Expired carbon dioxide is main-tained close to 5% and rectal temperature is kept at aconstant 37oC. A broad spectrum antibiotic (Bicillin,50,000 iu/kg, im) and anti-inflammatory steroid (dex-amethasone, 0.5mg/kg, im) are given on the first dayand every other day during the experiment. The eyesare treated with 1% atropine sulfate solution to dilatethe pupils and with a topical antibiotic (gentamicinsulfate, 3%) before being covered with gas-permeablecontact lenses. Foveae are mapped onto a tangentscreen using a reversing ophthalmoscope. The visualreceptive fields of isolated neurons are then mappedon the same tangent screen, keeping reference to thefoveae. Proper refraction is achieved by placing cor-rective lenses in front of the eyes on custom-designedlens holders. The eyes are stabilized with ophthalmicrings to prevent drift of visual field position.

4.1.2 Electrophysiological recordings anddata acquisition

The Thomas 7-electrode system (Marburg, Germany)was used to record simultaneously from multiple cor-tical cells in V1. The seven electrodes are arrangedin a straight line with each electrode separated fromits neighbor by 300 μm. Each electrode consists ofa platinum/tungsten core ( 25 μm in diameter, and1 μm at the tip) covered with an outer quartz-glassshank ( 80 μm in diameter), and has an impedancevalue of 0.7-4 Megohms. The multi-electrode matrixis precisely positioned before recordings - the tip ofthe matrix is placed approximately 3mm above thecortical surface. Electrical signals from the sevenelectrodes were amplified, digitized, and filtered (0.3-10kHz) with RA16SD preamplifiers in a Tucker-DavisTechnologies System 3 configured for multi-channelrecording. The Tucker-Davis system was interfacedto a Dell PC computer that ran a multi-channel ver-sion of the OPEQ program (written by Dr. J.A. Hen-rie) to acquire both spike and local field potentialdata. Visual stimuli were generated with the cus-tom OPEQ program, running in Linux on a Dell PC

with an off-the-shelf graphics card. Data collectionwas synchronized with the screen refresh to a preci-sion of better than 0.01ms. Stimuli were displayed onan IIyama HM 204DTA flat Color Graphic Display(size: 40.38 x 30.22 cm2; pixels: 2048 x 1536; framerate: 100Hz; mean luminance: 53 cd/m2). The screenviewing distance was 115cm. The basic attributesof each cell were estimated using small drifting sinu-soidal gratings surrounded by gray background (boththe gratings and the gray background had a mean lu-minance of 53 cd/m2).

4.1.3 Visual stimulation

Once all seven electrodes were located in the samelayer, an experiment was run with drifting sinusoidalgratings (at high contrast, spatial frequency at 2cycle/deg, temporal frequency at 4Hz) covering allrecording sites. The stimulus drifted at different di-rection between 0 and 360deg with 20deg step. Thestimulus at each condition was presented for 2 or 4seconds, repeated between 1 and 33 times dependingon the experiment.

4.2 Continuous Gabor Transform(CGT)

The continuous Gabor transform (CGT) is a shorttime or windowed Fourier transform (also called aspectrogram) that retains the time dependence of thespectrum that is lost in the Fourier transform (Mallat1999, pg. 69). The continuous transform differs fromthe discrete version in that the signal is oversampledin time and frequency so that neighboring points arenot independent. The Gabor filter ψ(t) used here isa one dimensional plane wave with frequency ω0 (inHz) windowed with a Gaussian g(t) centered at t0,

ψ(t, ω0) = g(t)e2πiω0t =1

σ√

2πe−

12 σ−2(t−t0)

2e2πiω0t.

(2)The CGT of a signal f(t) is found by convolving theGabor function with f(t) and results in a complextime series R(t0)eiφ(t0) that represents the amplitudeand phase of the signal at the frequency of the Gaborfilter,

G[f ](t, ω0) =∫f(s)ψ(s− t, ω0)ds

= R(t;ω0) eiφ(t;ω0). (3)

In time-frequency analyses the uncertainty principlelimits the resolution that can be resolved in the tem-poral and spectral domains. This limitation is ex-pressed by the parameter σ in equation (2). A bal-ance between the time and frequency resolution must

9

be found that captures the characteristics of interestfor the time series being studied. If the characteris-tic width of the Gabor filter is considered to be twoe-folding lengths (the distance at which the Gaussianenvelope is e−2 less than its peak value), the uncer-tainty condition for the CGT is

δtδω =2π, (4)

where δt is the time scale and δω is the frequencyscale. Here δt corresponds to the time scale of theLFP bursts (100ms) which gives a frequency resolu-tion of 6.4Hz using equation (4).

4.3 Phase Portraits and Phase Rota-tion

The CGT, described in equation (3), generates com-plex values that represent the time local amplitudeand phase information of the signal at the center fre-quency of the Gabor filter (with some error due tothe finite spectral width of the filter) at each timestep. Using the amplitude and phase from the CGT,we can plot a polar phase portrait at each frequencythat parametrically tracks the time evolution of theoscillation. In computing the CGT, the phase at eachtime step is computed with respect to the center ofthe Gabor filter and not the beginning of the record.To compare the phases at different time steps theymust be rotated to a common time. This rotationis performed by finding the phase shift of one pointrelative to the other by computing the non-integernumber of cycles that the sine wave oscillates dur-ing the time Δt that separates the two points (if thetwo points are chosen an integer number of periodsof the sine wave apart they will have the same localphase). The rotated phase φR of the later point isthe measured local phase minus the phase shift Δφ

φR(t, ω) = φ(t, ω)−Δφ = φ(t, ω)− mod(T,Δt)Δt

, (5)

where T = 1/ω is the period of the oscillation. Forthe case of a coherent sine wave, the local phase atone point and the rotated local phase of a secondpoint with respect to the first will always be equal. Ifall the local phases are rotated to a common referencetime, such as the beginning of the time record, theywill all have the same value as the phase of the sinewave at the beginning of the record as shown in phaseportrait of the sine wave in Figure 2B.

4.4 Amplitude Modulated Oscilla-tions

A sinusoid whose amplitude is modulated would notnecessarily be expected to have a phase portrait thatis localized in a particular sector. Using the Fourierexpansion of the modulated amplitude, we may ex-press any arbitrary modulating signal as a series ofsines and cosines,

A(t) sin(ω0t+ φ0) ={ N∑i=1

aisin(ωit+ φi)}

sin(ω0t+ φ0). (6)

Each of the products in equation (6) can be expressedas a sum of two sinusoids using the trigonometricidentity,

sinu sinv =12[cos(u− v) − cos(u + v)

], (7)

and are reduced to a series constant amplitude si-nusoids that will, under the CGT, exhibit localizedphase portraits as described in section (4.3).

A(t) sin(ω0t+ φ0) =N∑

i=1

aicos[(ωi − ω0)t+ (φi − φ0)

]

− aicos[(ωi + ω0)t+ (φi + φ0)

](8)

4.5 Circular Variance

The degree to which an oscillation is considered co-herent can be characterized by the localization of therotated phase portrait (described in Methods (4.3)).The circular variance (CV) of the phase portraits isused as a statistic to quantify the this localizationand hence the phase coherence of the LFP signal ateach frequency. The CV has values on [0,1] and is thesecond moment of the von Mises distribution, whichis the circular equivalent of the normal distributionon a line (Mardia 1972),

CV = 1 − |B|, with B =∑

tR(t;ω0) eiφ(t;ω0)∑tR(t;ω0)

.

The CV may also be viewed as the projection of theCGT coefficients on to the first nonconstant Fouriermode. The CV statistic, as used here, takes smallervalues for coherent oscillations and larger values formore random signals.

0 ≤ CV ≤ 1 (9)

CV ∼ 0.8 − 1 → random (10)

CV ∼ 0 − 0.3 → coherent (11)

10

4.6 Line Noise Filtering

In the data collected there is a strong line noise signalat 60Hz associated with the alternating current of theelectrical circuitry in the laboratory. It is assumedthat over the length of the recordings made here (2-4sec) the amplitude of the line noise is constant butits phase may drift, as is seen in the data. In orderto filter out this signal from the LFP recording, theamplitude of the 60Hz signal line noise is estimatedfrom the spectrum of the raw signal. This estimateis found by taking the Fourier transform of the entirerecord and interpolating the amplitude at 60Hz by itsneighboring values to estimate the 60Hz componentof the LFP signal. The amplitude of the line noiseis assumed to be the difference between the interpo-lated amplitude and measured amplitude. In orderto determine the phase of the line noise a sine wavewith the computed line noise amplitude is regressedon to the subsamples of the record used in the analy-sis. This method appears to remove the line noise at60Hz effectively in most cases but it is still the possi-ble that higher harmonics (120Hz and 180Hz) may bepresent in the data. As this study only examines thefrequency band of 10-100Hz these harmonics shouldnot pose a problem.

5 Acknowledgements

This work was supported by the Swartz Foundationand NIH Training Grant T32-EY007158. Robert M.Shapley and Dajun Xing were supported by the NIHGrant R01 EY-01472. Thank you to Dr. J. AndrewHenrie for help in the beginning of this project, aswell as for his great efforts in programming the multi-electrode data analysis, to Lucy F. Robinson and Dr.Francesca Chiaromonte for suggestions in the devel-opment of the statistical test and to Dr. Sinan Gun-turk for suggestions with the time-frequency analy-sis. Samuel P. Burns would like to thank Dr. EricShea-Brown and Dr. Alex Casti for their time andencouragement.

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