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# Searching for Autocoherence in the Cortical Network with a Time-Frequency Analysis of the Local Field Potential

**Corresponding Author:**Samuel P. Burns Center for Neural Science New York University Room 809, 4 Washington Place New York, NY 10003 ; Email: ude.uyn.smic@snrubs

## Abstract

Gamma-band peaks in the power spectrum of local field potentials (LFP) are found in multiple brain regions. It has been theorized that gamma oscillations may serve as a ’clock’ signal for the purposes of precise temporal encoding of information and ’binding’ of stimulus features across regions of the brain. Neurons in model networks may exhibit periodic spike firing or synchronized membrane potentials that give rise to a gamma-band oscillation that could operate as a ’clock’. The phase of the oscillation in such models is conserved over the length of the stimulus. We define these types of oscillations to be *autocoherent*. We investigated the hypothesis that autocoherent oscillations are the basis of the experimentally observed gamma-band peaks: the autocoherent oscillator (ACO) hypothesis. To test the ACO hypothesis, we developed a new analysis technique to analyze the autocoherence of a time-varying signal. This analysis used the continuous Gabor transform to examine the time evolution of the phase of each frequency component in the power spectrum. Using this analysis method, we formulated a statistical test to compare the ACO hypothesis with measurements of the LFP in macaque primary visual cortex, V1. The experimental data were not consistent with the ACO hypothesis. Gamma-band activity recorded in V1 did not have the properties of a ’clock’ signal during visual stimulation. We propose instead that the source of the gamma-band spectral peak is the resonant V1 network driven by random inputs.

**Keywords:**Visual Cortex, Local Field Potential, extracellular, V1, time-frequency analysis, data analysis

## Introduction

Gamma-band (25-90Hz) oscillations occur in many parts of the brain. We are seeking to understand the underlying neural mechanisms that generate gamma-band spectral peaks in the cerebral cortex by studying gamma activity in the local field potential (LFP) in the primary visual cortex, V1. The LFP is commonly interpreted as a measure of local network activity (Kruse and Eckhorn 1996; Logothetis et al. 2001; Buzsaki 2006). Previous experimental studies of V1 have reported a peak in the gamma-band of the LFP power spectrum when the visual cortex was visually driven (Gray and Singer 1989; Gray et al. 1989; Frien et al. 2000; Logothetis et al 2001; Siegel and Konig 2003; Henrie and Shapley 2005).

The presence of gamma-band activity in many LFP measurements under stimulation led to the idea that gamma-band oscillations serve as a ’clock’ signal for the purpose of temporally encoding information (Hopfield 1995; Buzsaki and Chrobak 1995; Jefferys et al. 1996; Buzsaki and Draguhn 2004; Buzsaki 2006; Bartos et al. 2007; Fries et al. 2007; Hopfield 2004). The ’clock’ theory of gamma combined with the pervasiveness of gamma oscillations have given rise to the theory that the brain uses gamma oscillations to synchronize different regions of the brain for the purpose of ’binding’ information about a stimulus (Gray and Singer 1989; Singer and Gray 1995).

Many studies of neuronal network models have sought to explain the mechanisms underlying gamma-band activity through either quasi-periodic spike firing, synchronized oscillatory spiking, or rhythmic sub-threshold membrane potential oscillations (see Sec ). We define an oscillation that can be modeled as a sinusoid with a fixed phase that does not vary with time to be *autocoherent*. A shared feature of many network models of gamma oscillations is that they generate an autocoherent network oscillation as an equilibrium state of the network. Autocoherence is an essential feature of such models because they were designed to yield gamma oscillations that could be used as a clock signal. We refer to the hypothesis that autocoherent oscillations underlie experimentally observed gamma-band spectral peaks as the autocoherent oscillator (ACO) hypothesis.

The aim of this study was to test the ACO hypothesis. We analyzed LFP measurements recorded from V1 cortex. Visual stimulation evoked a noisy LFP response in V1, with a peak of spectral power in the gamma-band. We developed new methods of signal processing to search for either constant amplitude or amplitude-modulated autocoherent signals embedded in the LFP. Our technique uses the continuous Gabor transform (CGT) (Mallat 1999, p 69; see Methods) to investigate LFP phase portraits at each temporal frequency. Using the CGT, we formulated a statistical test to compare the ACO hypothesis with LFP data from V1 cortex. In brief, we performed a rigorous search for autocoherent oscillations in visually-driven V1 activity and did not find them; the data did not support the ACO hypothesis. This result rules out a class of models that predicts that gamma activity is generated by a deterministic mechanism that produces a constant-phase clock signal. Our interpretation is that the source of the gamma-band peak is of a stochastic nature, a hypothesis considered in the Discussion.

## Methods

### Experimental Procedure

#### Surgery and preparation

Acute experiments were performed on adult Old World monkeys (Macaca fascicularis). All surgical and experimental procedures were performed in accordance with the guidelines of the U.S. Department of Agriculture and have been approved by the University Animal Welfare Committee at New York University. Animals were tranquilized with acepromazine (50*μ*g/kg, im) and anesthetized initially with ketamine (30 mg/kg, im) and then with isofluorane (1.5-3.5% in air). After cannulation and tracheotomy, the animal was placed in a stereotaxic frame and is maintained on opioid anesthetic (sufentanil citrate, 6-12*μ*g kg^{–1} h^{–1}, iv) for craniotomy. A craniotomy (about 5mm in diameter) was made in one hemisphere posterior to the lunate sulcus ( 15mm anterior to the occipital ridge, 10-20mm lateral from the midline). A small opening in the dura ( 3×3mm^{2}) was made to provide access for multiple electrodes. After surgery, the animal was anesthetized and paralyzed with a continuous infusion of sufentanil citrate (6-18*μ*g kg^{–1} h^{–1}, iv) and pancuronium bromide (0.1mg kg^{–1} h^{–1}, iv). Vital signs, including heart rate, electroencephalogram, blood pressure, oxygen level in blood, and urine specific gravity were closely monitored throughout the experiment. Expired carbon dioxide was maintained close to 5% and body temperature kept at a constant 37°C. A broad spectrum antibiotic (Bicillin, 50,000 iu/kg, im) and anti-inflammatory steroid (dexamethasone, 0.5mg/kg, im) were given on the first day and every other day during the experiment. The eyes were treated with 1% atropine sulfate solution to dilate the pupils and with a topical antibiotic (gentamicin sulfate, 3%) before being covered with gas-permeable contact lenses. Foveae were mapped onto a tangent screen using a reversing ophthalmoscope. Proper refraction was achieved by placing corrective lenses in front of the eyes on custom-designed lens holders.

#### Electrophysiological recordings and data acquisition

The Thomas 7-electrode system (Marburg, Germany) was used to record simultaneously from multiple cortical cells in V1. The seven electrodes were arranged in a straight line with each electrode separated from its neighbor by 300 *μ*m. The electrodes had impedance values in the range 0.7-4 Megohms. Electrical signals from the seven electrodes were amplified, digitized, and filtered (0.3-10kHz) with RA16SD preamplifiers in a Tucker-Davis Technologies System 3. The Tucker-Davis system was interfaced to a Dell PC computer. Visual stimuli were generated with the custom OPEQ program (written by Dr. J.A. Henrie), running in Linux on a Dell PC with an o -the-shelf graphics card. Data collection was synchronized with the screen refresh to a precision of better than 0.01ms. Stimuli were displayed on an IIyama HM 204DTA flat Color Graphic Display (size: 40.38 × 30.22 cm^{2}; pixels: 2048 × 1536; frame rate: 100Hz; mean luminance: 53 cd/m^{2}). The screen viewing distance was 115cm.

#### Visual stimulation

Once all seven electrodes were located placed in V1 cortex, an experiment was run with drifting sinusoidal gratings (at high contrast(0.8), spatial frequency 2 cycle/deg, temporal frequency 4Hz) that covered the visual fields of all the recording sites. The stimulus was drifted in 18 different directions between 0 and 360deg, in 20deg steps. The stimulus in each condition was presented for 2 or 4 seconds, repeated between 33 and 50 times depending on the experiment.

### R-Spectrum and Spectral Shape Index (SSI)

The R-spectrum, as used here, is defined to be the visual stimulated power spectrum divided frequency by frequency by the spontaneous power spectrum,

The R-spectrum is useful for expressing the stimulated power spectrum in normalized dimensionless units that can be compared across experiments. At frequencies where *R* > 1, the stimulated spectrum has elevated power in comparison to the spontaneous activity.

The Spectral Shape Index (SSI) is a measure of how large a peak in the R-spectrum is in relation to its neighboring frequencies and is defined as (cf. Henrie and Shapley, 2005),

where *R* is the R-spectrum of equation 1. The SSI is also dimensionless. When SSI > 1 a peak seen in the gamma-band R-spectrum sits above the average LFP power and forms a power spectral ’bump’ about the maximum R-spectrum value.

### Continuous Gabor Transform (CGT)

The continuous Gabor transform (CGT) is a short time or windowed Fourier transform (also called a complex spectrogram) that retains the time dependence of the spectrum that is lost in the Fourier transform (Mallat 1999, pg. 69). The continuous transform differs from the discrete version in that the signal is oversampled in time and frequency so that neighboring points are not independent. The Gabor filter (*t*) used here is a one dimensional plane wave with frequency *ω*_{0} (in Hz) windowed with a Gaussian *g*(*t*) centered at *t*_{0},

The CGT of a signal *f*(*t*) is found by convolving the Gabor function with *f*(*t*) and results in a complex time series *R*(*t* )*e*^{iϕ(t0)} that represents the amplitude and phase of the signal at the frequency of the Gabor filter,

In time-frequency analyses the uncertainty principle limits the resolution that can be resolved in the temporal and spectral domains. This limitation is expressed by the parameter *σ* in equation (3). A balance between the time and frequency resolutions must be found that captures the characteristics of interest for the time series being studied. If the characteristic width of the Gabor filter is considered to be two e-folding lengths (the distance at which the Gaussian envelope is *e*^{–2} less than its peak value), the uncertainty condition for the CGT is

where *δ _{t}* is the characteristic time scale and

*δ*is the characteristic frequency scale. Here

_{ω}*δ*corresponds to the time scale of the LFP bursts (100ms) which gives a frequency resolution of 6.4Hz using equation (5).

_{t}### Phase Portraits and Phase Rotation

The CGT, described in equation (4), generates complex values that represent the amplitude and phase information of the signal at the center frequency of the Gabor filter at each time step. The time dependence of the data is maintained by the CGT, as opposed to the Fourier transform, and is why this type of analysis is called a time-frequency analysis. Using the amplitude and phase at each frequency as functions of time, we can plot for each frequency component a phase portrait (in polar coordinates) that tracks the time evolution of the oscillation. In computing the CGT, the phase at each time step is computed with respect to the time at the center of the Gabor filter rather than with respect to the beginning of the record. In order to determine whether a frequency component is autocoherent, one must compare the phases at different time steps; the phases must be rotated to a common time reference from the beginning of the record. This rotation is performed by finding the phase shift of each time point relative to a reference point (here the first time point in the record) by computing the number of cycles that a sine wave at frequency *ω* oscillates during the time Δ*t* that separates the time point from the reference point. The rotated phase *ϕ _{R}* of the time point is the measured local phase minus the phase shift Δ

*ϕ*

where *T* = 1/*ω* is the period of the oscillation.

As an example, consider the phase portrait of an autocoherent (constant phase) sine wave. It has a trajectory that forms a circle as the CGT tracks the propagation of the local phase of the sine wave through time. To reveal that these data points are generated by a constant-phase sine wave, a phase shift, Δ*ϕ*, that represents the propagation of the sine wave is subtract from each local phase point, *ϕ*(*t, ω*). This corresponds to rotating the local phases back to the reference point at the start of the record. Once the phases have been rotated, all points have the same rotated phase value, *ϕ _{R}*(

*t, ω*) =

*ϕ*(

*t*

_{0},

*ω*), because the oscillation is autocoherent. The phase portrait of a constant phase sine wave is thus a single point, as shown in Figure 4A.

### Test for Visual Response

For each experiment the LFP response to a blank stimulus (spontaneous) and an optimal drifting grating stimulus were recorded with 25-50 repetitions. An empirical Probability Density Function (PDF) for the power at each frequency for the spontaneous activity was estimated by bootstrapping the mean of the Fourier transform of the spontaneous recordings. The power at each frequency of the stimulated data was estimated from the mean of the Fourier transform of the data. Those frequencies with stimulated power outside of the 95th percentile with respect to the spontaneous activity were considered significant.

### Circular Variance

The degree to which an oscillation is considered autocoherent can be characterized by the localization of the rotated phase portrait (described in Methods ()). We used the circular variance (CV; Mardia 1972) of the phase portraits as a statistic to quantify this localization and hence the phase coherence of the LFP signal at each frequency. The CV has values on [0,1]. The CV statistic, as used here, takes smaller values for ACOs and larger values for more random signals.

For the constant amplitude ACO null hypothesis described in Section the CV used to quantify the phase portraits has the form,

To quantify the autocoherence of the amplitude modulated ACO null hypothesis described in Section , the projection onto the second Fourier mode is used in the following formula,

The second mode is used in the amplitude-modulated case because the time varying amplitude, *A*(*t*), of the null-hypothesis model (Equation 16) is allowed to take on negative values. This results in an amplitude modulated ACO having a phase portrait that lies along a line that passes through origin with values of the phase at *ϕ* – *π*.

### Amplitude Modulated Oscillations and Heterodyning

A sinusoid whose amplitude is modulated may not necessarily be expected to have a phase portrait that is localized in a particular sector of the polar plot. Using the Fourier expansion of the amplitude modulation, we can express any arbitrary modulating signal as a series of phase shifted sinewaves,

Each of the products in equation (10) can be expressed as a sum of two cosines using the trigonometric identity,

and are reduced to a series of constant amplitude sinusoids,

that will, under the CGT, exhibit localized phase portraits as described in Section . The expansion of an amplitude modulated sinusoid in Equation 12 into a sum of constant amplitude sinusoids is referred to as *heterodyning*. The constant amplitude sinusoid sidebands in Equation 12 are called heterodynes and have symmetric amplitudes about the carrier frequency at the sum and difference frequencies of the carrier and component frequencies of the modulation. The phases of the heterodynes are also determined by Equation 12 to be the sum and difference of the phases of the carrier with the components of the modulation.

### Modeling Amplitude Modulated Autocoherent Oscillations

If the filters used in the Gabor transform to analyze the data had su ciently narrow spectral resolution, the problem of detecting an amplitude modulated ACO would simplify to detecting a series of constant amplitude ACOs that make up the sidebands. But to have filters with temporal resolution on the order of the burst seen in the LFP data, approximately 100ms (Figure 3D), the spectral width of the filters cannot be smaller than about 6Hz, due to the uncertainty principle (see Methods ). Because of these limitations on the Gabor filters, an amplitude modulated ACO cannot be detected by the constant amplitude ACO test described in Section.

To model an amplitude modulated autocoherent oscillation a symmetric modulation power spectrum centered on the carrier frequency*ω*_{0} was fit to the significant frequencies in the gamma-band of the stimulated power spectra. The carrier frequency, *ω*_{0}, was estimated by calculating the center of mass of the significant frequencies in the gamma-band (20 - 80Hz). The symmetric spectrum was fit by averaging the power in the sidebands on either side of the carrier. The modulation signal being modeled, *A*(*t*) of equation 16, contains frequencies at the difference between the carrier, *ω*_{0}, and the sideband, *ω _{i}*. To avoid fitting noise, only frequencies that have power greater than 1/3 of the carrier were included in the envelope. To generate a real time series from the fitted symmetric modulation spectrum, the inverse Fourier transform was taken, with random phases assigned to the carrier frequency,

*ω*

_{0}, and to each component of the Fourier decomposition of

*A*(

*t*) (equation 10) which correspond to the sets of symmetric sidebands in Figure 7A according to the heterodyning relation described in Methods .

### Line Noise Filtering

In the data collected there was a strong line noise signal at 60Hz associated with the alternating current of the electrical circuitry in the laboratory. Over the length of the recordings, the amplitude of the line noise was constant but its phase drifted. To filter out the line noise signal from the LFP recording, the amplitude of the 60Hz signal line noise was estimated from the spectrum of the raw signal. This estimate was found by taking the Fourier transform of the entire record and interpolating the amplitude at 60Hz from its neighboring values as an estimate of the 60Hz component of the LFP signal. The amplitude of the line noise was assumed to be the difference between the interpolated amplitude and measured amplitude. This method removed the line noise at 60Hz effectively in most cases but it is still possible that higher harmonics (120Hz and 180Hz) of line noise were present in the data. As this study only examines the frequency band of 10-100Hz, the harmonics did not pose a problem.

## Results

### Overview of Study

In this study we have tested the idea that a deterministic oscillation, which may serve as a clock for the binding of stimuli across regions of the brain, underlies the gamma-band peak observed in the LFP recorded in V1. If it exists, this clock would supply a regular reference time for the precise temporal encoding of spikes. The gamma clock hypothesis is a prominent concept in studies of gamma activity and for this reason it is important to submit it to rigorous statistical testing using *in vivo* data from cortex.

To assist the reader in navigating the study, we provide here a step by step summary. To begin, we review the extensive literature on the experimental and theoretical work that predicts the presence of a gamma clock oscillation in Section . A demonstration of how traditional signal processing methods such as the power spectrum and spectrogram are not capable of discriminating non-autocoherent from autocoherent signals is presented in Section with an introduction to the new technique we have developed to measure the autocoherence of a signal. The cortical data are described in Sections and . In Section our first null hypothesis of a constant-amplitude autocoherent oscillator (ACO) in noise is discussed. A description of the statistical test of this hypothesis is presented in Section . The results of the statistical test, in Section , reject the constant amplitude ACO hypothesis and are plotted in Figure 6. The next model we consider, in Section , is an amplitude modulated ACO in noise. In Section we perform a statistical test of the amplitude modulated ACO hypothesis. In Section , the results of the statistical test reject the amplitude modulated hypothesis and are plotted in Figure 8. To show that autocoherent neural signals exist in the brain and that our analysis is capable of detecting them, in Section , we study an EEG recording that contains an alpha rhythm and find that it is autocoherent over several seconds. Finally, a discussion of the implications of the lack of autocoherence in gamma band activity of cortex is presented in the Discussion section.

**...**

### Oscillator Models of the LFP

An overwhelming amount of the theoretical neuroscience literature about the source of gamma band spectral peaks has been concerned with the analysis of model networks that generate autocoherent oscillations. The reasons for this focus on autocoherent models were experimental evidence and also computational goals like the ’clock’ theory of gamma oscillations.

Theorists hypothesized that oscillations in the LFP might be a ’clock’ signal used to encode spikes temporally at precise times with respect to the phase of the LFP clock (Hopfield 1995; Lisman and Idiart 1995; Jefferys et al. 1996; Hopfield 2004; Buzsaki and Draguhn 2004; Fries et al. 2007). In these theories, regular rhythmic oscillations (which we call autocoherent oscillations) in the LFP were viewed as self-organizing emergent properties of the network. If different areas of the brain shared the same autocoherent LFP ’clock’, the ’clock’ oscillation could be used for ’binding’ features of a stimulus in order to integrate information across many regions of the brain (Gray and Singer 1989; Gray et al. 1989; Singer and Gray 1995; Gray 1999; Varela et al. 2001; Buzsaki 2006).

The simplest proposed mechanism for the generation of a gamma clock signal is a population of cells each of which outputs a regular spike rate in the gamma band. For example, ’chattering’ cells have been recorded *in vivo* from cat visual cortex that exhibit rhythmic firing in the 20-70Hz range (Gray and McCormick 1996).

Early mathematical models of networks of neurons found rhythmic oscillations in certain parameter regimes that were generated by feedback between inhibitory and excitatory cells (Freeman 1975; Bressler and Freeman 1980; Leung 1982). But more recently most theoretical effort has been devoted to analyzing networks of inhibitory neurons as the source of gamma-band oscillations (Wang & Rinzel 1992, 1993) and these inhibitory models generate autocoherent oscillations. A further development of the inhibitory network model was the clustering of subpopulations of inhibitory cells into groups that each generated autocoherent oscillations but were out of phase with each other. In this model the combined output of the various clusters generated an autocoherent oscillation with a frequency higher than each of the clusters (Golomb & Rinzel 1994, van Vreeswijk 1996). Models of synchronous and asynchronous states in mean field models of networks of neurons found that autocoherent oscillations depended on inhibitory coupling (Abbott & van Vreeswijk 1993; van Vreeswijk et al. 1994; Gerstner 1995; Gerstner et al. 1996).

In addition to earlier studies of gamma-band oscillations observed in visual cortex (Gray and Singer 1989; Gray et al. 1989), gamma oscillations have also been recorded both *in vivo* (Bragin et al. 1995) and *in vitro* (Whittington et al 1995) from rat hippocampus. Models of gamma oscillations in the hippocampus also utilized networks of inhibitory neurons to generate autocoherent oscillations that are thought to be functionally important (Whittington et al 1995; Traub et al. 1996; Wang and Buzsaki 1996; White et al. 1997; Ermentrout and Kopell 1997; Chow et al. 1998; Kopell et al. 2000; Traub et al. 2000). Models in these studies of inhibitory networks generated autocoherent oscillations for either deterministic or noisy inputs whose autocoherent time-scales, the period of time over which the phase is conserved, were explicitly dependent on the time scale and properties of the external drive. A numerical simulation of a model network of inhibitory neurons is presented in the Appendix to demonstrate the autocoherent properties described in these studies.

Firing rate synchrony is another mechanism that has been proposed for the generation of autocoherent gamma-band oscillations. In models of firing rate synchrony, subthreshold membrane potentials of cells in the network exhibited rhythmic oscillations but cells fired spikes irregularly. However, individual neurons fired spikes preferentially at the peaks of the membrane potential oscillation. In this class of models, the spiking output of individual cells did not exhibit periodic firing but the network-averaged firing rate exhibited an autocoherent gamma-band oscillation (Kopell and LeMasson 1994; Brunel and Hakim 1999; Tiesinga and Jose 2000; Brunel and Wang 2003; Geisler et al. 2005; Brunel and Hansel 2006).

The ACO hypothesis, in the form of either spike or membrane potential synchrony, has been widely cited and is very influential in the experimental literature on gamma-band oscillations. The theoretical concept of emergent autocoherent oscillations has been cited in experimental studies of the hippocampus (Penttonen et al. 1998; Fisahn et al. 1998; Traub et al. 2000; LeBeau et al. 2002; Mikkonen et al. 2002; Csicvari et al. 2003; Vida et al. 2006; Mann & Paulsen 2007; Montgomery and Buzsaki 2007), visual cortex (Gray & Singer 1989, Zaksas & Pasternak 2006), prefrontal cortex (Compte et al. 2003; Durstewitz & Gabriel 2007) and somatosensory cortex (Cardin et al 2009), among others.

To demonstrate that autocoherent oscillations do exist in neural data and therefore that our statistical test of the ACO hypothesis is capable of having both positive and negative outcomes, an example of EEG data containing an alpha rhythm (11Hz oscillation) is analyzed in Section using our new statistical method (described in Sections - ). In the EEG data there was a a clear autocoherent oscillation that lasted for 3s (roughly 30 oscillations of the alpha rhythm). The EEG results established that the ACO hypothesis is a biologically plausible claim worth testing rigorously, and that our method is capable of detecting ACOs when they exist.

It is possible to conceive of models that generate gamma band peaks in which the frequency components are not autocoherent. Such stochastic models are considered in the Discussion. Testing cortical data for autocoherence is an important goal for experiments and data analysis because determining whether or not the data are autocoherent can decide between classes of models of the cerebral cortex and other brain areas.

### Need for Statistical Test of the ACO Hypothesis vs. Intuition

It is commonly thought, based on intuition, that the autocoherence of a signal can be diagnosed by visual inspection of the time series but this intuition is incorrect, as demonstrated in Figure 1. For example, plotted in Figure 1A is a simulated, amplitude-modulated ACO of the form

where the phase *ϕ*_{0} is constant in time and *ω*_{0} is set to 40Hz and *A*(*t*) is the sum of three Gaussians with peaks roughly centered at 0.2s, 1.2s and at 2.9s. Figure 1B is a plot of a simulated nonautocoherent signal, of the form

where *ω*_{0} is again equal to 40Hz, each of the three individual bursts’ amplitudes, given by the Gaussian *A _{i}* functions, are separated in time (

*A*

_{1}(

*t*) centered at 0.2s,

*A*

_{2}(

*t*) centered at 1.2s, and

*A*

_{3}(

*t*) centered at 2.9s) and the phases of the separate bursts,

*ϕ*, are unrelated. It is not possible to discriminate between the autocoherent (Figure 1A) and nonautocoherent oscillation (Figure 1B) by eye, even in this noiseless case. In Figure 1C and D, the two kinds of oscillations are added to simulated spontaneous V1 LFP activity, generated using the power spectra of data used in this study (Section 3.3.1) to recreate the noise that is commonly present in LFP recordings. In Figure 1E and 1F, the power spectra, and in Figure 1G and 1H, the spectrograms, of the two kinds of oscillations are plotted, after the two oscillatory signals were added to the simulated spontaneous LFP activity. These examples of simulated LFP signals imply that it is not possible to discriminate an autocoherent signal from a time-varying phase signal by a visual comparison of the time series, the power spectra, or the spectrograms. For this reason we developed a new method of data analysis to determine the autocoherence of a signal. The new method examines the time dependence of the phase of the signal. In Figure 1I and 1J the phase portraits (Section ) at 40Hz of the two example signals (autocoherent and drifting phase) are plotted. The phase portraits are parametric plots (time increasing along the curve) with phase represented by the angle and the amplitude by the radius. For the autocoherent oscillation, Figure 1I, each burst of activity away from the origin has the same phase (azimuthal angle) while in the case of the signal that has a phase that varies with time, Figure 1J, each burst of activity has a different phase angle. The difference between the two signals can be understood by studying their phase portraits.

_{i}### Local Field Potential Data

The local field potential (LFP) is an extracellular voltage measurement that characterizes the local network activity of the population of neurons in the neighborhood of the measuring electrode (approximately 10^{2}–10^{3} neurons) and is defined as the low frequency (≤ 250Hz) portion of the raw field potential. Specifically, the LFP signal measures the current flow due to synaptic activity while the higher frequency components of the raw data are related to action potentials (Kruse and Eckhorn 1996; Logothetis et al. 2001; Buzsaki 2006).

The LFP data analyzed here were obtained by recording with extracellular microelectrodes in macaque V1 cortex, in monkeys lightly anesthetized with the opioid sufentanil. LFP spectra and spectrograms were measured during a baseline blank (no stimulus) and also during visual stimulation with a high contrast, drifting grating pattern (using a monitor with a 100Hz refresh rate) at the ’preferred’ orientation for maximal response (further experimental details described in Methods ). The LFP response to a blank stimulus was recorded for 1 sec and the grating pattern was shown for 2 or 4 seconds, both with 25-50 repetitions depending on the trial. The data are from 90 experiments in two monkeys where LFPs were recorded with multielectrode arrays of seven electrodes. Voltage data were sampled at a rate of 25kHz. Data from the electrode with the largest response to the visual stimulus were selected for analysis as these give the largest signal to noise ratio for our statistical tests. The 60Hz line noise signal was filtered out of the LFP recordings as described in Methods . An example of a 4-second LFP recording under visual stimulation is plotted in Figure 2A and its power spectrum in Figure 2B. The power spectra were computed using a multitaper analysis (Percival and Walden 1993) with a concentration bandwidth of 1Hz and averaged over repeated identical stimuli.

**...**

In all 90 experiments, localized elevated power was observed in the gamma-band during visual stimulation. Plotted in Figure 2C is the LFP R-spectrum (see Methods ) averaged over all 90 experiments studied. The R-spectrum is the power spectrum of the visually driven response normalized by the baseline spontaneous spectrum. The averaged R-spectrum shows a clear ’bump’ in the gamma-band that begins near 20 Hz, peaks between 35-50Hz, and decreases gradually to 100Hz. After filtering, the line noise from the experiment setup (see Methods ) has a stronger relative signal in the spontaneous spectrum compared to the stimulated spectrum, resulting in a narrow notch about 60Hz in the averaged R-spectrum in Figure 2C. This notch does not effect the conclusions of the study as the line noise was well separated from the gamma-band peak power. Plotted in Figure 2D is the maximum or peak value of the R-spectrum in the gamma-band for each experiment. All visually driven LFPs had a peak gamma-band power larger than the spontaneous power (*R* > 1) meaning the LFP gamma band power was strongly visually driven by the drifting gratings (cf. Henrie and Shapley, 2005). The sharpness of the gamma band ’bump’ in the visually-driven power spectrum was quantified by comparing the peak power in the gamma-band in relation to the surrounding frequencies with a Spectral Shape Index (SSI) (see Methods ). The SSI is computed to determine whether each experiment displayed localized elevated gamma-band power (narrow ’bump’; SSI > 1) or broad-band elevated power at all frequencies (broad ’bump’; SSI < 1). In this study the characteristics of gamma activity are examined. As such, experiments that displayed a localized narrow ’bump’ of gamma-band power were most useful. Plotted in Figure 2E are the SSI's evaluated for each experiment. For all experiments analyzed in this paper the *SSI* > 1 indicating that there was a localized peaked ’bump’ of power in the gamma-band.

### Time-Frequency Analysis

We used a time-frequency analysis to examine the temporal evolution of LFP data at each frequency. Similar analyses have previously been used to study the temporal structure of brain activity recorded in EEG (Makeig 1993, Herrmann et al. 2004), LFP (Pesaran et al. 2002), and have been described as a general method for studying event-related activity in neural signals (Sinkkonen et al. 1995; Mitra and Pesaran 1999; Hurtado et al. 2004). The continuous Gabor transform (CGT) is the convolution of an enveloped complex plane wave with the time series being examined (see Methods ). The CGT is a function of *t*, the time point at the center of the convolution, and *ω*_{0}, the frequency of the underlying wave of the transform (shown schematically in Figure 3B). Scale varying wavelets, whose width in the time domain dilates with increasing scale (decreasing frequency), become too coarse in the time domain at low frequency and too broad in the frequency domain at higher frequencies for this study. To avoid the problems associated with the scale representation of wavelet transforms, we used the Gabor transform because its fixed time scale preserves the relationship to frequency.

Plotted in Figure 3C is the CGT amplitude spectrum of the LFP recording shown in Figure 3A. In comparing the amplitude spectrum with the LFP time series, the three bursts of activity in Figure 3A at times t = 1.75s, 2.8s and 3.75s correspond to high amplitude events in Figure 3C. It is clear that the CGT is capable of capturing the time evolution of the LFP recording. From inspection of Figure 3C, the LFP signal has large bursts of activity on the scale of ~100ms with power concentrated mainly near 35Hz and with a smaller peak at 70Hz. The spectral peak between 35Hz and 45Hz and the 100ms time scale were characteristics common to all LFP recordings analyzed here.

### Null Hypothesis I: Constant Amplitude Autocoherent Oscillator (ACO)

The first null hypothesis we consider is that the increased gamma-band power seen in the LFP power spectrum is the result of a constant amplitude autocoherent oscillation of the local neuronal network that becomes active under visual stimulation. Autocoherence in this case means that at a particular frequency the LFP can be modeled as a sine wave of constant phase and amplitude added to noise,

where *A* and *ϕ* are time-independent.

In the event there were multiple autocoherent oscillators in the network operating at the same frequency, the waveforms of the different oscillations would be summed when recorded with an electrode. The sum of many different constant-phase sine waves at the same frequency is a single sine wave that our method would detect as an autocoherent oscillation.

One can examine the autocoherence at a particular frequency by studying plots of the rotated phase portrait (see Methods ). When a signal has large amplitude bursts at a particular frequency, for example as seen in the LFP data in Figure 2A, the phase portrait will show a large amplitude excursion away from the origin at that frequency as shown in Figures 1I and 1J. This kind of phase portrait is illustrated in Figure 4 for idealized systems in 4A and for real LFP data in 4B. The phase of the constant amplitude sine wave appears as a single point in the phase portrait (Figure 4A-left). If noise is added to the sine wave, the rotated phases do not all fall on a single point (Figure 4A-middle) but remain localized in a common sector of the polar plane rather than exploring all quadrants as is done by the phases of a noise signal (Figure 4A-right).

Phase portraits for the LFP data shown in Figure 3 are plotted in Figure 4B at frequencies in the gamma-band from 20-90Hz. If the LFP contained constant amplitude autocoherent oscillations, as in the null hypothesis, the activity in the phase portrait should have clustered in a common sector of the phase portrait. The phase portraits of the peaks in the spectral power at 35Hz and 70Hz (Figure 3A, C) show high amplitude (large radius) events that correspond to the bursts discussed in Section . In the 35Hz and 70 Hz phase portraits the three bursts seen in the LFP time series (Figure 3A) and the CGT spectrum (Figure 3C) are visible in the three ’loops’ away from the origin. These bursts occupied different sectors of the phase portrait and therefore the signal was not autocoherent.

We used circular variance (CV) (see Methods ) as a statistic to quantify the coherence of an oscillation or, equivalently, localization of the trajectories of the phase portraits. The CV varies from zero to one and measures how tightly clustered the points are in phase angle. A CV close to zero corresponds to all points having similar phases and, for our purposes, a more autocoherent oscillation. A CV near one indicates that the different points in time have different phases and implies a nonautocoherent signal with a time-varying phase. The CV is normalized by the average amplitude of the oscillation and is dimensionless. One can use the dimensionless CV index to compare the coherence of oscillations of different frequencies and power. The CV of the phase portraits in Figure 4 are listed above the plots at each frequency. The CVs of the 35Hz and 70Hz oscillations in the LFP data from V1 were large, 0.69 and 0.63 respectively, a result which suggests there were not autocoherent oscillations at these two frequencies.

#### Null Hypothesis I: Statistical Tests

We have shown visually and using the CV statistic that the oscillations seen in the data in Figures 3 and and44 were not autocoherent. But in order to examine systematically all 90 experiments and to determine quantitatively if autocoherent oscillations were present as proposed by the null hypothesis, a statistical test had to be performed. The first step was to identify which frequencies of the LFP had significantly elevated power under visual stimulation, a step which was done as described in Methods.

We devised a Monte-Carlo-type statistical test to determine whether autocoherent oscillations were present in the visually-driven LFP signal. For all 90 experiments we performed the following simulations whose procedure is described schematically in Figure 5. A test was evaluated at all frequencies between 10-100Hz that had significantly elevated power under stimulation. At a particular frequency, the null-hypothesis model assumes that the increase in power at that frequency under visual stimulation had a constant amplitude and fixed phase and was summed with background noise. The background noise was simulated by taking the average power spectrum of the spontaneous LFP data, from 1-250Hz, assigning random phases to each frequency and taking the inverse Fourier transform. A constant amplitude ACO of the form (15) was added to the simulated background. The amplitude of the oscillator, *A*, was found by taking the difference between the stimulated and background power spectra at the frequency being tested; the phase, *ϕ*, was chosen randomly . The CGT (Mallat 1999, p 69; see Methods ) of the simulated signal was taken and the CV of the phase portrait was computed. This simulation procedure was repeated 1000 times to generate a probability density function (PDF) of the values of the CV for the null-hypothesis model.

#### Null Hypothesis I: Test Results

The null hypothesis of a constant amplitude ACO was rejected at all frequencies tested, between 10-100Hz. In particular, there was no evidence for an ACO in the gamma band where the visally-driven spectral power peaked. Plotted in Figure 6 are the differences between the CVs of the data at each frequency and the simulated 99th percentiles of the CV of the constant amplitude ACO, the null-hypothesis model. The differences between data and the null-hypothesis model are plotted in Figure 6 at all frequencies with significantly elevated power under visual stimulation for all 90 experiments. The differences were greater than zero for all experiments at all frequencies with the exception of 60Hz where a strong line noise signal was still present after filtering, and 100Hz which was the refresh rate of the computer monitor used to present the visual stimulus in the experiments. That the CVs of the data were larger than the 99th percentile of the null-hypothesis model reveals that the V1 gamma band data were significantly less autocoherent than a constant amplitude ACO.

In the range of frequencies of the gamma-band near 40Hz, where the elevated power in the data was centered, the difference between the CV of the data and of the constant amplitude ACO (denoted ’sample average’ in Figure 6) had its largest values, indicating that the oscillations in this frequency band of interest were particularly nonautocoherent. This occurred because the gamma-band was the region of the LFP spectrum where the LFP had the largest elevation of power in response to the visual stimulus. To fit the LFP data in the gamma band, the ACO in the null model had to have a larger signal-to-noise ratio than at other frequencies, hence the phase of sine wave dominated the phase trajectory and that led to a lower CV value. This caused the larger difference between the data CV and null CV in the gamma-band. The CV values of the LFP data from V1 were relatively constant across the frequencies analyzed which meant the CV difference was higher when the signal-to-noise in the null-hypothesis model rose.

From Figure 6 we conclude that the null hypothesis of a constant amplitude ACO was not supported by the V1 data analyzed here and that constant amplitude ACOs did not occur in macaque V1 even when there were strong gamma band spectral peaks.

### Null Hypothesis II: Amplitude Modulated Autocoherent Oscillator (ACO)

The constant amplitude ACO hypothesis, rejected in Section , is the simplest type of oscillatory response expected from a deterministic system. Another type of oscillation consistent with the ACO hypothesis is an amplitude modulated ACO of the form

where the modulation, *A*, is now a function of time, but *ϕ*_{0} is still time-independent and the carrier frequency, *ω*_{0}, is assumed to be in the range of the gamma band peak seen in the data (30-50Hz). Using a Fourier decomposition of *A*(*t*) and trigonometric identities we showed in Methods that an amplitude modulated ACO can be expressed as a sum of constant amplitude ACOs (for *ω*_{0} ≠ 0) through the heterodyning relation. Heterodyning is an effect where an amplitude modulated sinusoid can be expressed as a carrier friequency, *ω*_{0}, with symmetric sidebands at frequencies equal to the sum and difference of the carrier frequency and modulation frequencies and phases of the sidebands equal to the sum and difference of the phase of the carrier and modulation phases.

#### Null Hypothesis II: Statistical Tests

To provide intuition about amplitude-modulated ACOs, we simulated an example in Figure 7A-C as described in Methods . The simulation is shown in Figure 7A-C, where the fitted symmetric modulation spectrum is shown in black in Figure 7A, the random phase spectrum is plotted in Figure 7B, and the resulting real time series after taking the inverse Fourier transform is shown in Figure 7C.

A statistical test similar to that described in Section for the constant amplitude ACO hypothesis was used for the amplitude-modulated ACO hypothesis. Unlike the constant amplitude test, the amplitude-modulated case was only tested at the carrier frequency, *ω*_{0}, rather than all significant frequencies. To simulate the amplitude-modulated ACO null model, the modulation signal was fit nonparametrically to the power spectra of the data as described in Section and multiplied by a fixed phase sinewave as in Equation 16. This simulated null-hypothesis oscillation was then added to simulated spontaneous activity generated in the same manner as for the constant-amplitude null hypothesis (Section ). For example, the power spectra of the spontaneous, stimulated and simulated amplitude-modulated ACO null-hypothesis model are shown in Figure 7D, where the simulated null-hypothesis model only includes the power spectrum peak in the gamma-band and not the complete stimulated spectrum. In Figures 7E and 7F are plotted the time series and in Figures 7G and 7H the spectrograms of the data and the simulated null model respectively. The difference between the autocoherence of the data and the simulation became clear when the phase portraits at the carrier frequency (here 43Hz) (plotted in Figures 7I and 7J) were compared. The phase portrait of the data had a wandering shape corresponding to a time-varying phase signal whereas the phase portrait of the simulated signal fell along a line that signifies the presence of an amplitude-modulated ACO (see Methods ). As in the constant amplitude case, the CV (Methods ) was used to quantify this difference. A simulated PDF of the CV values for the null hypothesis was generated in the similar manner as for the constant amplitude case (Figure 5) but with the simulation of the constant amplitude ACO replaced with the amplitude-modulated simulation. The hypothesis was tested at the carrier frequency, *ω*_{0} of Equation 16, for each experiment by comparing the datas’ CVs to the 99th percentile of the PDF of the simulated null-hypothesis CV.

#### Null Hypothesis II: Test Results

The results of the amplitude-modulated ACO null hypothesis are shown in Figure 8. In Figure 8 the CVs of the data are plotted with their standard errors along with the 99th percentile of the CV for the null-hypothesis model for each experiment. The CVs of the data, including the measurement errors, were all in excess of the 99th percentile of the null hypothesis for all 90 experiments tested. These results reject the amplitude-modulated ACO hypothesis as an explanation for the observed elevated gamma band power spectra in macaque V1.

### Autocoherent Oscillations in EEG Data

Our focus in this paper has been on gamma band activity, but we include here an example of the analysis of alpha rhythms in human EEG to make the point that it is possible for some brain activity to be much more autocoherent than the gamma band activity we analyzed above, and that our analysis will find the autocoherence if it exists. An example of an autocoherent oscillation in EEG data is plotted in Figure 9. The autocoherent oscillation lasted for 3sec over the course of approximately 30 cycles. The EEG recording, bandpassed filtered from 5-50Hz, is plotted in Figure 9A. The power spectrum of the recording is plotted in Figure 9C with a peak in the alpha-band at 11Hz. The phase portrait at 11Hz of the EEG recording in Figure 9A is plotted in Figure 9B. The phase trajectory of the alpha oscillation in the phase portrait was largely confined to the region of 150-240 degrees with a CV value of 0.25 (see Methods ) indicating an autocoherent signal. The mean phase is shown by the black arrow in Figure 9B. An autocoherent 11Hz sine wave whose phase is given by the mean of the phase portrait and whose amplitude is given by the amplitude coe cients of the Gabor transform of the EEG at 11hz is overlayed on the EEG data in Figure 9A. The EEG data and the autocoherent sine wave have corresponding phases for most of the 3sec record with occasional periods where the 11hz amplitude temporarily fades (during the period of 1.4-1.75s) and then reappears with the same phase. This EEG example shows that persistent amplitude-modulated autocoherent oscillations exist in neural data and our method can identify them when they are present in data. Although this alpha rhythm has been shown to be autocoherent, cortical gamma band activity is very different because of the absence of autocoherence in gamma.

## Discussion

The results of our new time-frequency analysis of V1 local field potentials (LFPs) reject the hypothesis that gamma-band spectral peaks in the LFP of visually driven macaque V1 LFP data (Figure 3A) can be modeled as a constant-amplitude ACO (Equation 15) or as an amplitude-modulated ACO (Equation 16). In the data examined here a gamma band peak in power is induced using a drifting grating stimulus, as has been shown to occur in previous work on gamma activity in V1 (Bauer et al. 1995; Frien et al. 2000; Ray & Maunsell 2009). We conclude that elevated gamma band energy in V1 is not the result of emergent deterministic, harmonic, or relaxation, oscillations investigated in the modeling studies referenced in Section . Thus, the present results rule out a class of theoretical models for gamma activity in the V1 network.

The results in this paper call into question the idea that the gamma oscillation in V1 is operating as a ’clock’ signal for the precise temporal encoding of visual information. If there is a ’clock’ mechanism present in V1 gamma activity, the period of time over which it supplies reliable timing (time over which the phase of the oscillation is conserved) is not set by the time-scale of the stimulus but may have some intrinsic time-scale due to internal dynamics of the cortical network itself. This intrinsic time scale must be much shorter than the duration of the visual stimulus we used, on the order of 2-4 sec. Similarly, if gamma activity is involved in the ’binding’ of stimulus features across different regions of the brain by gamma, the period of time over which the brain can synchronize different areas must be very brief and is also not determined by the time scale of the stimulus being processed.

Theoretical models of gamma oscillations predict autocoherent oscillations whose time-scales are set by the properties of the external drive to the network (see Section ). Applied to the measurements in this study, these models would predict that for a constant visual stimulus (drifting grating stimulus) the time-scale of autocoherent oscillations seen in the LFP should persist for the length of the time the stimulus is presented (2-4sec). Previous measurements of gamma activity from hippocampus in rat described in Section were recorded *in vitro* in slices rather than *in vivo* as in the experiments analyzed here. It is possible that the oscillations seen in slice, driven with a tonic external drive, do generate autocoherent oscillations but the work here shows that for *in vivo* V1 networks this is not the case.

The alpha rhythm analyzed in Section exhibited an autocoherent oscillation at 11Hz. It is possible that lower frequency oscillations such as the alpha (8-12Hz) and theta (6-10Hz) rhythms may be autocoherent. The work of Wang and Rinzel (1992, 1993) was originally motivated as models of spindle rhythms with a frequency of 7-14Hz. Their model is not an accurate model of gamma activity in V1, but may be a useful model of alpha and theta rhythms.

In order for a model to generate bursts at arbitrary times of arbitrary length, as seen in the data, a fundamentally different type of nondeterministic model must be used. In the theoretical models discussed in Section , noise in the system is viewed as something that may obscure synchronous firing that allows the system to oscillate autocoherently. Rather than treating noise as a corrupting factor, we believe noise is essential to the generation of the gamma-band peak and should be treated as a leading order term in any mathematical model of gamma oscillations. To accommodate the random aspects of the timing and duration of bursts seen in the data, any biologically accurate mathematical model should include a random variable term representing noise. Doing this will necessarily cause the system to be modeled stochastically rather than the deterministic approaches of the previous theoretical studies of gamma band ’oscillations’.

While this study has shown that the ACO model does not fit the data from V1, there is structure in the data in the form of bursts of activity concentrated in particular frequencies of the gamma-band. A model that can produce the peaked, elevated gamma-band power spectra recorded in V1 and that includes a central role for noise is a resonant stochastic filter. Henrie, Kang, Shelley & Shapley 2005 and Kang et al. 2009 found that a stochastic resonant network model of V1 with recurrent connections to extrastriate cortex has a resonant response in the gamma-band. In this model the network is viewed as a resonant stochastic oscillator with a stable state for noiseless inputs corresponding to the quiescent periods (low amplitude) of the LFP data (see Figure 3B). When noise is added to the system during visual stimulation, from feed-forward and recurrent inputs, the network is randomly excited into short high energy bursts of excitation at a resonant frequency centered in the gamma-band (see also similar ideas expressed in Rennie et al. 2000). In a resonant stochastic oscillator model, the phase associated with each burst is independent, and the signal is not autocoherent for the length of the period of stimulation. The varying phases associated with the independent bursts of activity generate a broad peak in the power spectrum of the LFP, a peak centered on the resonant frequency of the network, as seen in the data presented here (Figure 2B). The gamma-band resonance in the model of Kang et al. (2009) is a consequence of the structure of the model network they considered: a recurrent excitatory-inhibitory network in which synaptic inhibition and excitation both have short time constants with the inhibitory time constant slightly slower than that of excitation. Networks with this functional connectivity have been proposed before in order to explain cortical sharpening of selectivity (for example, Kang et al 2003). The presence of gamma-band resonance in such networks may be a byproduct of their dynamics but it may also enhance their sensory function. Our future research on the elevated gamma-band response to visual stimuli will focus on testing whether the data are consistent with such a stochastic resonant oscillator model.

An alternative to the stochastic model we favor, that also generates an irregular, nonautocoherent activity, is a model of network activity with finely tuned parameters such that the network oscillates chaotically. Models of neuronal networks that exhibit chaotic oscillations have been explored by Hansel & Sompolinsky 1996 and Battaglia et. al 2007 as possible mechanisms for generating nonautocoherent network activity.

We have found that oscillations are not autocoherent on the time-scale of the visual stimulus (2-4 seconds). However on shorter time-scales, individual bursts of activity may be autocoherent. Spatially coherent bursts of gamma-band activity have been found in the rat hippocampus with a time-scale of approximately 100ms (Montgomery and Buzsaki 2007; Montgomery et al. 2008). Further work on autocoherence in LFP data should include a study to determine the intrinsic internal autocoherent time scale of individual bursts of gamma activity seen in the data. The time-frequency analysis developed here could be used for such an analysis. Also, as described in the introduction, gamma band spectral peaks have been observed in many other regions of the brain. With the development of the new time series analysis presented in this paper, data from other brain regions could be examined to determine the autocoherence of LFP activity throughout the brain.

## Acknowledgements

This work was supported by the Swartz Foundation, NIH Training Grant T32-EY007158, and NSF grant IOS-0745253. Robert M. Shapley and Dajun Xing were supported by the NIH Grant R01 EY-01472.

The authors would like to thank Dr. J. Andrew Henrie for help in the beginning of this project, as well as for his great efforts in programming the multielectrode data analysis, Dr. Lucy F. Robinson and Dr. Francesca Chiaromonte for suggestions in the development of the statistical test, Dr. Nava Rubin for helpful discussions of ’clock’ and ’binding’ models of gamma activity, Dr. Sinan Gunturk for suggestions with the time-frequency analysis and Professor James Gordon for his assistance with EEG recordings. Samuel P. Burns would like to thank Dr. Eric Shea-Brown and Dr. Alexander Casti for their time and encouragement.

## Appendix

#### Autocoherence Analysis of Inhibitory Network Model

A model of 10 inhibitory integrate and fire neurons with a constant excitatory drive was simulated numerically to demonstrate the deterministic clock mechanism referred to in Section . The details of the conductance-based model used for the simulation, and the parameters used, are described in Tao et al. 2004 (supporting text). The model was time stepped using a 4th order Runge-Kutta method (Press et al.) and the neurons were connected all-to-all with equal strength. The network was integrated for 5mins; it quickly (< 50ms) locked into a regular oscillation that persisted for the duration of the simulation. To model the influence this inhibitory network would have on the membrane potential of an excitatory neuron, the population average time course was convolved with an inhibitory synaptic conductance kernel (see Tao et al. 2004). A 500ms excerpt of the population average voltage time course, smoothed with the synapse's kernel, is plotted in Figure 10A.

**...**

This inhibitory model generates a regular clock oscillation and is a simplified version of the same network mechanism proposed by previous theorists to be the source of the gamma band peak seen in cortical LFP data as reviewed in Section . A fundamental aspect of this model is that the connections between the inhibitory cells in the network cause each of the cells to fire more slowly than they would in isolation. In this network the inclusion of inhibitory connections slows the network oscillation from 70Hz down to 42Hz which is within the gamma band frequency range (20-90Hz). The autocoherence of the model's output was calculated using the same technique as described in Section . The phase portrait at 42Hz of the time course plotted in Figure 10A is plotted in Figure 10B. The CV of the model's output is 7×10^{–10}, which indicates the model is outputting a perfect clock signal. If this inhibitory network clock mechanism were the source of the gamma activity seen in V1, strong autocoherent signals should have been present in the LFP data analyzed here, by the following argument. It is reasonable to suppose that the excitatory cells are getting added random noise and that this then would be added to the inhibitory clock signal in the LFP. The inhibitory clock signal with its 0 CV value would be equivalent to the continuous sinusoid we considered in Null Hypothesis 1 in Section . The analysis in the paper that rejected Null Hypothesis 1 therefore also rejected the hypothesis that inhibitory clock signals were the source of the gamma band peaks in the LFP.

## Contributor Information

Samuel P. Burns, Courant Institute of Mathematical Sciences & Center for Neural Science New York University, New York, NY.

Dajun Xing, Center for Neural Science New York University, New York, NY.

Michael J. Shelley, Courant Institute of Mathematical Sciences & Center for Neural Science New York University, New York, NY.

Robert M. Shapley, Center for Neural Science & Courant Institute of Mathematical Sciences New York University, New York, NY.

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