Department of Neurology and Neuroscience, Joan and Sanford I. Weill Medical College of Cornell University, New York, New York 10021
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ABSTRACT |
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Schiff, N. D.,
K. P. Purpura, and
J. D. Victor.
Gating of Local Network Signals Appears as Stimulus-Dependent
Activity Envelopes in Striate Cortex.
J. Neurophysiol. 82: 2182-2196, 1999.
Neuronal activity often is treated
as a composition of a stimulus-driven component and a second component
that corrupts the signal, adding or deleting spikes at random. Standard
quantitative methods such as peristimulus histograms and Fourier
analysis use stimulus-locked averaging to enhance detection of the
driven component of neuronal responses and de-emphasize the
"noise." However, neural activity also includes bursts,
oscillations, and other episodic events that standard averaging methods
overlook. If this activity is stimulus independent, it can be
characterized by standard power spectral analysis (or autocorrelation).
But activity that is excited by (but not temporally locked to) the
visual stimulus cannot be characterized by averaging or standard
spectral analysis. Phase-locked spectral analysis (PLSA) is a new
method that examines this "residual" activitythe difference
between the individual responses to each cycle of a periodic stimulus
and their average. With PLSA, residual activity is characterized in
terms of temporal envelopes and their carriers. Previously, PLSA
demonstrated broadband interactions between periodic visual stimuli and
fluctuations in the local field potential of macaque V1. In the present
study, single-unit responses (SUA) from parafoveal V1 in anesthetized
macaque monkey are examined with this technique. Recordings were made
from 21 neurons, 6 of which were recorded in pairs along with multiunit activity (MUA) from separate electrodes and 8 of which were recorded along with MUA from the same electrode. PLSA was applied to responses to preferred (orientation, direction, and spatial frequency) and nonpreferred drifting gratings. For preferred stimuli, all cells demonstrated broadband (1-10 Hz and higher) residual activity that
waxed and waned with the stimulus cycle, suggesting that changes in the
residual activity are introduced routinely by visual stimulation.
Moreover, some reconstructed envelopes indicate that the residual
activity was sharply gated by the stimulus cycle. Oscillations
occasionally were seen in the power spectrum of single units.
Phase-locked cross-spectra were determined for 3 SUA/SUA pairs and 11 SUA/MUA pairs. Residual activity in the cross-spectra was generally
much less than the residual activity determined separately from each
neuron. The reduction in the residual activity in the cross-spectra
suggests that nearby neurons may gate inputs from distinct and
relatively independent neuronal subpopulations that together generate
the background rhythms of striate cortex.
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INTRODUCTION |
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The issue of what constitutes "signal"
and what constitutes "noise" is present at all levels of
investigation of neuronal activity, from single (SUA)- and multiunit
(MUA) recordings to recordings of local (LFP) and far field (ECoG, EEG)
potentials. The usual approach in neurophysiological investigations is
to equate signal with the "average response"whatever is
reinforced with peristimulus averaging of Fourier analysis. Variations
of individual responses from this average response necessarily are eliminated. This residual activity may include oscillations, bursts, and changes in the variability of firing rate (mean-variance
relationship) (Gershon et al. 1998
).
At present, considerable attention has been given to oscillations in
both LFP and SUA/MUA recordings as an example of the type of temporal
structure in the noise of neuronal responses (Freeman and Barrie
1994; Singer and Gray 1995
). However, as
Bullock (1996)
recently noted, spectral analysis of
brain activity typically does not reveal narrow peaks suggestive of
independent oscillators. Other types of episodic dynamics, such as
bursts (Lisman 1997
), transient responses
(Friston 1995
; Victor and Purpura
1996
), stereotypical patterns of intervals
(Abeles 1988
), and maintained elevated discharges (Moran and Desimone 1995
; Treue and Maunsell
1996
), may be more representative of ongoing brain activity. It
is crucial not only to demonstrate the existence of such phenomena in
residual fluctuations but also to define their relationship to external stimuli.
Examination and quantification of such episodic events in the context
of their possible roles in information coding and processing requires
appropriate tools. We present a method, phase-locked spectral analysis
(PLSA), that investigates the relationship of average and residual
components of neuronal responses and apply it to recordings from
macaque primary visual cortex (V1). This method is a natural
generalization of the standard Fourier methods used in the calculation
of power spectra and can be applied to both field potential and spike
train data. Previous studies of the LFP in macaque V1 using PLSA showed
evidence of interaction of average activity with the ongoing LFP in the
form of such residual components (Victor et al. 1993).
Here we examine SUA and MUA recordings from macaque V1 for evidence of
such changes in residual activity. The analysis demonstrates an
interaction of periodic stimulation and residual activity that is
typically broadband. We also examine interaction of neurons in paired
SUA/MUA and SUA/SUA recordings. Our results demonstrate that
stimulus-induced changes in the residual activity of one cell are
correlated only weakly with such changes in nearby cells. This implies
that residual cortical activity reflects multiple, relatively
independent, cell assemblies.
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METHODS |
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Phase-locked spectral analysis
PLSA is a natural extension of spectral analysis to
situations in which a periodic stimulus is present, and its possible
influence on residual activity is of interest. In analogy with the
standard power spectrum (see Eq. A16), we define the
phase-locked spectrum of order n at the frequency by
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(1) |
Interpretation of PLSA in terms of envelope and carrier
A nonzero value for Pn() implies
that the joint appearance of components at these frequencies is more
likely than expected from their individual representation in the power
spectrum. When n/S is small compared with
,
the joint appearance of these frequencies can be interpreted as a
"beat" with carrier frequency
and envelope period
S/n. That is,
Pn(
), for each n, describes
the power at the frequency
that varies with the nth
harmonic of the external stimulus. Note that
Pn(
) is a complex-valued quantity. Its
amplitude and phase are the amplitude and phase of the beat's
envelope. We emphasize that the phase of the
carrier has no preferred value with respect to the stimulus
cycle because such a preferred phase would have contributed to the
average response, which was subtracted out (see Eq. A9).
Figure 1, A-D, illustrates
several time series the statistics of which are revealed by PLSA. In
all cases, the average over the stimulus cycle is a sinusoid the period
of which is that of the periodic stimulus, S. The top
line of Fig. 1A illustrates a signal containing
variability that is independent of the stimulus. Subtraction of the
period-averaged signal (2nd line) reveals a broadband noise
(3rd line) the statistics of which are independent of the
stimulus. For this situation, P0()
is the spectrum of this broadband noise, and
Pn(
) (n > 0) is zero,
indicating the lack of modulation of the residual activity at the
frequency of the stimulus or any of its harmonics. The top
line of Fig. 1B illustrates a signal containing
variability that waxes and wanes at the first harmonic of the stimulus
frequency. This kind of variability is consistent with the
mean-variance relationship seen in visual cortical neurons
(Gershon et al. 1998
; Tolhurst et al.
1983
) (see DISCUSSION). The period-averaged signal
(2nd line of Fig. 1B) is sinusoidal as in Fig.
1A. However, subtraction of this average yields (3rd
line of Fig. 1B) residual activity that peaks at the
start of each stimulus cycle. P1(
)
is thus nonzero, indicating that the residual activity (noise) is
modulated at the first harmonic of the stimulus frequency.
P0(
) is also nonzero and represents
the average spectrum of this noise. However, Pn(
) (n > 1) is zero,
indicating the lack of modulation of the noise at higher harmonics of
the stimulus frequency. Figure 1C is similar to Fig.
1B, except that the residual activity is broadband rather
than narrowband. As in Fig. 1B,
P0(
) and
P1(
) are nonzero, but their
dependence on
is different, reflecting the different spectral
distribution of the noise. Figure 1D, like Fig.
1B, is an example of a signal containing modulated
narrowband residual activity. But in contrast to Fig. 1B,
there are two noise bursts per stimulus cycle. Thus
P0(
) and
P2(
) are nonzero and the remaining Pn(
) are zero.
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Significance testing
We use two approaches to determine whether experimentally
measured values of the phase-locked spectra
Pn() (for n > 0) are significantly different from zero. One approach is to compare empirically estimated values of Pn(
)
when a periodic stimulus is present with estimates obtained when the
periodic stimulus is removed (i.e., blank screen) and the "true"
value of Pn(
) is guaranteed to be zero.
The second approach makes use of the interpretation of
Pn(
) in terms of an envelope frequency
2
n/S and a carrier frequency
. If
Pn(
) reflects power in some frequency range around
that systematically varies at the envelope frequency 2
n/S, then the phases of
Pn(
) should be coherent across a range of carrier frequencies
and should reflect the phase of the common envelope. To assess phase coherence, we used the Rayleigh phase criterion (RPC) across bands of frequencies
(Mardia
1972
).
General physiological methods
We recorded single-unit activity in the parafoveal
representation in cortical area V1 of six anesthetized, paralyzed
macaque monkeys. Twenty-one single units were isolated and stable
recordings maintained for sufficient time (2-3 h) for the studies
reported here. Other studies were performed on additional cells in
these animals (Victor and Purpura 1996, 1998
). All
procedures involving the animals were performed in accordance with
National Institutes of Health guidelines for the care and use of
laboratory animals. General physiological preparations, methods, and
recording techniques are described elsewhere (Victor and Purpura
1998
; Victor et al. 1994
).
In 15 recordings, units were isolated from one electrode using criteria
implemented by a Tucker-Davis hoop discriminator. In three recordings
(6 cells), recordings were made from separate microelectrodes,
independently manipulated and laterally displaced by 2 mm. These six
single units were recorded as part of pair recordings, along with
discrete MUA that could be separated from the single unit but not
resolved into individual unit activity. We classified cells as simple
or complex by standard Fourier analysis of their responses to drifting
gratings and the criteria of Skottun et al.(1991).
For measurement of phase-locked spectra, drifting gratings at a
temporal frequency of 1.06 Hz, 100% contrast
[(Lmax Lmin)/(Lmax + Lmin)] unless otherwise noted,
optimal orientation, direction, and spatial frequency were presented to
each cell (mean luminance of the computer-driven CRT display 150 cd/m2 with a green phosphor, subtending 4 × 4° at a viewing distance of 114 cm). For most cells (see following
text), a second grating stimulus and/or a blank was presented in
interleaved runs.
Orientation experiment
For six cells, stimuli were interleaved with drifting gratings at the orthogonal orientation but optimal spatial frequency as well as a blank condition. Eight additional cells were studied with two (interleaved) orientations but without the blank condition. Here and in the following text, optimal indicates a maximal fundamental response for simple cells and a maximum elevation of the mean rate for complex cells.
Spatial frequency experiment
For three cells, stimuli were interleaved with drifting gratings at a nonoptimal spatial frequency and a blank condition. The nonoptimal spatial frequency differed from the optimal spatial frequency by at least an octave, and both gratings were presented at the optimal orientation and direction. Ten additional cells were studied with two (interleaved) spatial frequencies but without the blank condition.
Data collection and analysis
Data were collected in continuous streams of 1 min per
condition. Thirty-two examples of each condition were interleaved, with
~20 s of uniform illumination at the mean stimulus luminance between
runs. Spike times were recorded at the resolution of the frame interval
of the display (3.7-ms bins). Responses from each stimulus presentation
were segmented into analysis periods of length L = 3.79 s (1,024 bins); the stimulus period S was 0.947 s
(256 bins). Spikes were considered to be delta-functions, and Fourier
estimates (at frequencies <122) = 2n/L)
were calculated following detrending and windowing with a raised cosine
function. By choosing an analysis period that is four times the length
of the sampling period (L = 4S) and using
the raised cosine bell window, spurious correlations due to the leakage
of the spectral estimates away from their nominal frequencies are
eliminated. Consecutive analysis segments overlapped by half their
length to compensate for the effects of windowing (Blackman and
Tukey 1959
). Rayleigh phase criterion (RPC) calculations
(Mardia 1972
) were based on nine adjacent frequencies
separated by 1/L=0.26 Hz, thus spanning a 2-Hz range.
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RESULTS |
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Analysis of single-unit data
Figure 2 displays the analysis of
residual activity obtained from a simple cell in macaque V1. The
stimuli consisted of drifting gratings at preferred and orthogonal
orientations. The peristimulus time histogram (PSTH) of responses to
drifting gratings of preferred, orthogonal, and blank runs (Fig.
2A) demonstrates a maintained discharge in the "blank"
condition, and both modulation and elevation for the preferred
orientation. Figure 2B shows the zeroth-order phase-locked
spectrum, P0() in the three
conditions. If the stimulus merely influenced neuronal activity by
adding a periodic signal to an autonomous noise,
P0(
) would be the same in all
conditions. However, as shown in Fig. 2B,
P0(
) is threefold larger for the preferred grating (top) than for the nonpreferred grating or
the blank stimulus. This difference in power is present over the entire frequency range examined (with a greater difference at frequencies <50
Hz), thus indicating broadband residual activity.
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P1() (Fig. 2C) describes
the components of the residual activity that are modulated at the first
harmonic of the stimulus cycle (1.06 Hz). For the blank condition (Fig.
2C, bottom), the "true" value of
Pn(
) must be zero (for n
>0), and thus the measured P1(
) is
an indicator of the uncertainty of the statistical estimate. In this
cell, for the nonpreferred stimulus, the estimated
P1(
) (middle) is no
different from the estimate obtained in the blank condition. However,
for the preferred stimulus (top),
P1(
) is again approximately
threefold higher than in the blank condition, indicating that much of
the residual power waxes and wanes with the stimulus cycle, as in Fig.
1, B and C. Analogously,
P2(
) describes the components of
the residual activity that are modulated at the second harmonic of the
stimulus frequency (Fig. 1D). As shown in Fig.
2D, the behavior of P2(
)
is similar to that of P1(
).
The phase-locked spectral components reveal an aspect of neuronal
activity that is distinct from the average response. That is, the
average response (the PSTH) is very nearly sinusoidal, but the residual
activity is modulated by an envelope that is not sinusoidal [i.e., not
confined just to P0() and
P1(
)]. The spectral distribution of the
residual activity (the carrier) is indicated by the dependence of
P1(
) on
and is broadband. If oscillations were present, they would be manifest in a narrowband dependence on
.
The phases of Pn() for this simple
cell (Fig. 3, A and
B) form relatively straight-line clusters of values,
more so for n = 1 than for n = 2. This supports the interpretation of
Pn(
) as modulation of the residual
activity by a common envelope because the phases of
Pn(
) (which indicate the phase of the
envelope at each frequency) are coherent over the range of frequencies
present in the modulated residual activity.
|
Phase coherence for Pn() up to the
n = 2 to n = 4 was typical of
the data set as a whole (21 cells). However, five cells showed
modulation of the residual activity up to at least the n = 6 harmonic (the highest harmonic routinely
analyzed). The presence of phase coherence of the carriers up to high
harmonics of the stimulus cycle indicates that the envelope of the
residual activity has rapidly varying components. That is, the residual activity is sharply gated by the periodic stimulus.
Estimation of the envelope
Figure 1, B-D, provides examples of signals in
which the phase-locked spectra are nonzero only for n = 0 and one other low value; in these simple cases, the envelope of the
residual activity is a sinusoid. In our data,
Pn() is nonzero for many values of n; this indicates that the envelope of the residual activity
is not sinusoidal. Because the phases of
Pn(
) are approximately coherent over a
broad range of carrier frequencies for a given n (Fig. 3),
we make the approximation that the residual activity within each
frequency range rises and falls at the same time over the stimulus
cycle. This allows us to combine the individual values of
Pn(
) across a range of carriers
(
lo to
hi), to obtain
a more reliable quantity
Pnaverage, that represents the
nth harmonic of the (assumed) common envelope
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(2) |
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(3) |
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Nonpreferred responses in single-unit recordings
In most datasets, residual activity was most prominent for the
preferred responses, as illustrated in Fig. 2. Figure
5A
displays data from a complex cell that shows an atypical response to a nonpreferred grating. The PSTH (Fig. 5A) shows a maintained
discharge during the presentation of the orthogonal grating and the
blank, and modest overall elevation of the mean rate for the preferred condition. P0() (Fig.
5B) is decreased in the orthogonal condition relative to the
preferred and blank conditions. However, the higher-order spectra
(n = 1, 2) for the orthogonal condition demonstrate the appearance of a distinct "hump" in the low-frequency range (1-10 Hz) that is not evident in the other conditions (Fig. 5, C
and D). The low-frequency hump also is seen in higher-order
phase-locked spectra (Fig. 5E), and phase coherence in the
10- to 50-Hz range is observed up to n = 10 (data not
shown).
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The reconstructed envelope for the preferred response (Fig. 6A) is similar to the PSTH of the preferred response (Fig. 5A). This contrasts with the reconstructed envelope for the orthogonal response (Fig. 6B), which has a greater amount of rapid variations than the preferred-response envelope. This indicates sharp gating of the residual activity during the presentation of the orthogonal stimulus. This gating is not evident in the orthogonal-response PSTH, which is minimally different from the PSTH recorded in the blank condition (Fig. 5A).
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The strong modulation of residual activity in this case shows that the cell is receiving inputs from other orientations. However, this cross-orientation interaction does not change the average response, but rather consists of peaks of variability at specific times (Fig. 6B) of the stimulus cycle. Another (simple) cell recorded without a blank condition demonstrated a similar phenomenon (sharper gating of the residual activity of the response to a nonpreferred grating compared with the response to a preferred grating). That is, these two units showed evidence of a novel aspect of cross-orientation interaction: a modulation in the residual activity with little or no modulation in the average response.
Simultaneous MUAs and SUAs
In the datasets with simultaneous recordings of SUA pairs and MUA,
we examined not only the phase-locked spectra within channels but also
the phase-locked cross-spectra. As detailed in the
APPENDIX, the phase-locked cross-spectra describe how
cross-covariances between channels vary throughout the stimulus cycle.
We use Pn;j,k() to denote the
phase-locked cross-spectrum of order n between channels j and k. Pn,j(
) denotes the
phase-locked spectrum of order n on channel j
considered in isolation.
Figure 7, A-C, shows
phase-locked spectra and cross-spectra for two cells recorded
simultaneously from electrodes 2 mm displaced (laterally) in parafoveal
V1. Channel j = 1 is a simple cell, and channel
j = 2 is a complex cell. For a drifting grating at a
preferred orientation, the simple cell's PSTH showed a strongly modulated response and the complex cell's PSTH showed an elevation of
maintained discharge. In both single units, phase coherence was strong
for P1;j() and obvious but
weaker for P2;j(
) (Fig. 7,
A and B). The RPC statistic showed statistically
significant coherence for n = 1, 2, 3, and 5 in both
units. Figure 7C shows the phase-locked cross-spectra,
Pn;1,2(
) for n = 0, 1, and 2. For n = 0, there is phase-coherence,
indicating correlated activity between the two recorded units. [The
Fourier transform of this spectrum, essentially the shuffle-corrected
cross-correlogram, shows that this cross-correlation is maximal at 0 time lag, as would be expected from the clustering of the phases of
P0;1,2(
) around 0]. The
phase-locked cross-spectrum, Pn;1,2(
) for n > 0, measures the component of this correlation
that varies with the nth harmonic of the stimulus cycle. As
can be seen in Fig. 7C, the amplitudes are lower than that
of P0;1,2(
), and there is little,
if any, phase coherence. The finding of a reduction or loss of phase
coherence in the phase-locked cross-spectra (compared with individual
phase-locked spectra) was typical of SUA/SUA recordings from
independent electrodes. It indicates that although the residual activity on the two channels is correlated (n = 0),
this correlation is not detectably gated by the stimulus. The
reconstructed envelopes for each single unit and the cross-spectra are
displayed in Fig. 8, A-C. The
envelope of the cross-spectra (Fig. 8C) is lower in amplitude than that of either single unit (A or
B).
|
|
For the three pairs (6 cells) recorded in SUA pairs, common gating of
residual activity was weaker than gating of residual activity in
individual units. As in this case, some common gating was observed, but
the cross-spectral residual components were never as prominent as those
seen in either individual unit. Correlations of individual SUA and MUA
were also typically independent of the stimulus, even though individual
SUA and MUA residual activities were strongly modulated by the
stimulus, up to the n = 5 or n = 6 harmonic of the stimulus cycle. In five SUA/MUA recordings from the
same electrode, the cross-spectra
Pn;1,2() (n > 0) were
insignificant even though analysis of individual channels
[Pn;1(
) and
Pn;2(
)] revealed modulation of the
residual activity. In a similar analysis for another pair recording of
two complex cells (j = 1, 2), recorded from two
electrodes during stimulation with a preferred drifting grating at a
low spatial frequency of 0.25 cycles/°, but not others at higher
spatial frequencies, both cells exhibited very strong gating by the stimulus.
Some recordings also showed a narrow band of activity consistent with
oscillations superimposed on the broadband activity that was
universally observed. Figure 9 shows the
phase-locked spectrum Pn;1()
(n = 0, 1, 2) of the simple cell shown in Fig.
7A plotted on a linear scale to emphasize this narrowband activity at ~10 Hz (thin arrows). This peak is present for
n > 0, but less distinct, indicating that the
oscillatory activity was only partially gated by the stimulus cycle. In
addition, a broader hump, spanning ~15 Hz (thick arrows) between
carrier frequencies 20-35 Hz, is present up to at least
n = 4 (only n = 1, 2 shown), indicating
nonoscillatory activity in the same frequency range is gated by the
stimulus. The second single unit in this recording did not show the
sharp peak at 10 Hz, but did show a slight "hump" for
n = 1 in the 20- to 35-Hz range of frequencies.
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DISCUSSION |
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Summary of results
Stimulus-dependent spectral components in the residual activity
were identified using PLSA in single-unit responses of 21 macaque V1
cells. In most of the cells, we found that residual activity was gated by the stimulus during presentation of a
periodic drifting grating. This gating was not merely a "side
effect" of modulation of the average firing rate: in many
cases, modulation was present up to the n = 6 harmonic of the stimulus cycle, even though the stimulus
induced either a nearly sinusoidal average response (PSTH) or primarily
an elevation of the maintained discharge. Furthermore, two cells showed
gating of the residual activity during the presentation of a
nonpreferred stimulus, thus indicating a novel form of
cross-orientation interactions. These SUA recordings from macaque V1
are consistent with previously identified PLSA components in local
field potentials from this region (Victor et al. 1993).
Oscillations were only occasionally seen in the average power spectra
P0(
) of individual single units. In
this regard, it may be important that all recordings were done with sufentanil anesthesia in light of recent evidence that opiate anesthetics may block high-frequency oscillations in the gamma range
(Whittington et al. 1998
).
In six cells, cross-spectra derived from SUA/SUA pair recordings
revealed that gating of the residual activity of individual units by
the stimulus typically was reduced (or in some cases eliminated) in the
phase-locked cross-spectra. Similar results were seen in 11 SUA/MUA
pairs. While some single-unit pairs did demonstrate strong gating of
their covariances, this was true only for selective stimulus
parameters. The decrease in phase-locked residual activity in the
cross-spectra of both SUA/SUA and SUA/MUA recordings suggests that
envelopes of residual activity seen in different individual single
units gate substantially independent sources of ongoing activity.
Furthermore, the appearance of joint modulation for certain stimulus
parameters but not others suggests that the common gating reflects
dynamic changes in functional connectivity rather than hard wiring
(Vaadia et al. 1995).
Mean-variance relationship
Previous workers also have noted that the variability of cortical
neurons is stimulus dependent and depends on the mean firing rate
(Tolhurst et al. 1983). This mean-variance relationship
predicts that the residual activity is nonzero but does not by itself
predict the form of the residual activity because it does not model the dynamics of the dependence. One way to compare the mean-variance relationship to our analysis is to hypothesize that the variance depends on the instantaneous (expected) mean firing rate and that the
variance at nearby times is independent. Under this hypothesis, the
phase-locked spectra Pn(
) are predicted
to be independent of the carrier frequency
, and the synthesized
envelope will be a scaled replica of the average response histogram
(adjusted by the windowing function). Thus the mean-variance
relationship accounts for the shape of the synthesized envelope
observed in most units. It also could account for our observations that
phase-locked cross-spectra between units tend to be small, if the
variability in the two units' responses are independently related to
their means. However, the mean-variance relationship, along with these simple dynamics, does not account for the dependence of the
phase-locked spectra Pn(
) on
. This
dependence requires that variability at distinct times are
correlated
the phase-locked spectra represent the Fourier transform of
this autocorrelation structure. This mean-variance relationship also
does not account for the behavior of the few units in which the
phase-locked spectra are prominent for stimuli that do not change the
mean firing rate or for the sharp gating of the envelopes that also are
seen occasionally (e.g., Fig. 6B). These observations appear
to require a separate source of variability not directly related to the
average response.
Comparison of PLSA with related methods
Several methods of signal analysis share a common goal of a search for an interaction between autonomous and stimulus-driven activity. These techniques all examine second-order statistics of signals and have as their null hypothesis that trial-to-trial variability can be explained by stimulus-independent additive noise. In each analysis, the residual activity may be considered to be a carrier, and its possible interaction with the stimulus is considered to be described by a window of modulation (envelope). The distinction between these methods is whether carrier and envelope are viewed in the time domain or in the frequency domain and whether or not a periodic or transient event may interact with the signal.
To make these similarities more explicit, we introduce a new quantity
B(t, ), closely related to the
shuffle-corrected cross-correlogram, a well-known technique for
analyzing spike trains (Perkel et al. 1967
). We call
this function as the "time-resolved autocorrelogram" and define it
formally as
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(4) |
Figure 10 displays the connection
between different analytic techniques in the family that includes PLSA.
B(t, ), when averaged over the stimulus cycle
to remove the dependence on t, is the shuffle-corrected
autocorrelogram, SAC(
):
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(5) |
![]() |
(6) |
|
A second Fourier transformation, this time in the carrier variable ,
transforms An(
) into
Pn(
). This transformation makes sense
if the stimulus S is periodic. That is, for a periodic
stimulus, the time-resolved autocorrelogram B(t,
) is related to Pn(
) by a double
Fourier transform
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(7) |
![]() |
(8) |
The above paragraphs provide for a common framework to discuss several,
apparently quite diverse, investigations. In each case, properties of
the residual activity are analyzed by one of the above techniques,
depending on the nature of the stimulus signal (periodic or transient)
and on whether the phenomena of interest are best described in the time
or frequency domain. For example, Vaadia and colleagues
(1995) examined multiple single neuron responses during the
performance of a GO/NO-GO paradigm with the
joint peristimulus time histogram, JPSTH (see Eq. 4 and surrounding discussion). These studies demonstrated that dynamic patterns of correlation (without modulation of the mean firing rate)
could identify different behavioral responses based on the appearance
of the patterned activity in time referenced to the behavioral event.
Friston (1995)
carefully examined this result and
further suggested that changes in correlation may reflect neuronal
excitation due to transient responses. This conclusion was reached by
principal components analysis of a matrix derived from the JPSTH. Other
approaches to the analysis of contributions to the JPSTH based on
second-order statistics recently have been developed by Brody
(1998)
and allow for a separation of contributions of circuit
properties and intrinsic membrane properties under certain circumstances.
Heinrich and Bach (1997) calculated
Q(t,
) from steady-state visual evoked
potentials (VEPs) recorded from the human electroencephalogram (EEG) at
selected values of
(30-40 Hz). They labeled these components "residual frequency components." Their temporal envelopes showed that they were stimulus dependent and differed from the standard VEP
waveform. Tallon-Baudry and colleagues (1996)
analyzed
human VEPs obtained during presentations of illusory contour figures. They augmented time-frequency analysis with a wavelet transform to
improve frequency and temporal resolution. Their study demonstrated significant non-phased-locked activity in several frequency bands and
at different times. Some of the activity in these bands strongly correlated with the perception of an illusory contour.
These studies, which identified interactions of stimulus and residual
activity, complement those of Arieli and colleagues (1996). In their study, ongoing brain activity was examined
with optical imaging and extracellular recording techniques. Results were interpreted as consistent with linear summation of the
deterministic response and the residual activity. However, in this
study the high-frequency response was superimposed on slow-frequency
components of ongoing activity, or noise. It did not address whether
the correlation structure of single-trial responses could in fact be
accounted for by the correlation structure measured in the absence of
stimulation. In terms of the present study, this approach discards
information about Pn(
) for
n > 0 and does not compare P0(
) in stimulus and blank
conditions. Thus interactions of the residual activity and the average
response were likely to have been overlooked.
Implications for models of brain functions
Several investigators have emphasized the importance of
oscillations (Singer and Gray 1995), bursts
(Lisman 1997
), or other transient, organized dynamic
events in neuronal responses to stimuli (Aertsen et al.
1989
; Bullock 1996
; Friston 1995
,
1997
; Victor and Purpura 1996
). In
striate cortex, oscillations have been described in response to
coherent stimuli (Gray 1994
). Such activity also has
been proposed to reflect formation of neuronal assemblies and to a play
a role in linking multiple brain regions and local patches of neuronal
circuits to facilitate binding (Llinas et al. 1994
;
Singer and Gray 1995
). These stimulus-dependent events have been postulated to reflect important aspects of brain function other than rate codes. The results here, of residual activity that is
strongly modulated by an external stimulus, demonstrate rich
spatiotemporal dynamics in neuronal responses. Stimulus-dependent changes in brain activity are not restricted to narrowband, single population phenomena such as oscillations. The envelopes and spectra calculated from PLSA allow us to visualize the dynamics of complex local network inputs that shape individual neuronal responses. Moreover, the diminished stimulus-gated residual activity in
cross-spectra, compared with its prominence in individual
single-channel phase-locked spectra, suggests that the envelopes of
residual activity seen in individual units gate signals from multiple
neuronal assemblies with only partial overlap.
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APPENDIX: THEORY |
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Phase-locked spectral analysis
PLSA extends spectral analysis to situations in which a periodic stimulus is present. PLSA is a two-step procedure: estimation of the average response to a periodic stimulus and modified spectral analysis of the residual activity, i.e., the difference between individual responses and the estimated average response. To develop and motivate our method, we begin by revisiting the concepts behind the power spectrum when there is no external stimulus and then extend these concepts to the situation where a periodic stimulus is present. The discussion of the power spectrum is somewhat elaborate, to facilitate an extension to the situation in which a periodic stimulus is present.
No external stimulus
We assume that s(t), the time series to be analyzed, is autonomous and stationary. That is, its statistics are independent of external stimuli and also of our choice of a time t0 at which to begin the analysis.
Estimation of the power spectrum depends on the statistics of Fourier
components obtained from finite segments of the time series
s(t). We use Z(, L, T,
t0) to denote the Fourier component at the
frequency
determined from a finite segment of length L
that begins after an elapsed time T after the arbitrary
start time t0. Formally
![]() |
(A1) |
![]() |
(A2) |
![]() |
(A3) |
![]() |
(A4) |
![]() |
(A5) |
![]() |
(A6) |
![]() |
(A7) |
![]() |
(A8) |
Because (as we have just seen) the average values of Fourier components
are necessarily 0, we examine the next simplest candidate to
characterize the statistics of the Fourier components: their variances
and covariances. We denote an averaged covariance of Fourier components
at two frequencies 1 and
2 by
![]() |
(A9) |
![]() |
(A10) |
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(A11) |
![]() |
(A12) |
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(A13) |
![]() |
(A14) |
When 1 +
2 = 0 in
Eq. A14, we may choose
1 =
and
2 =
. Equation A11 then
becomes
![]() |
(A15) |
![]() |
(A16) |
Periodic external stimulus
When a periodic external stimulus is present, shifting the start time t0 of the analysis by arbitrary amounts may change the observed statistics, because the stimulus cycle provides a periodic marker (see Fig. 1, B-D). However, for a stationary system, shifts of the start time by a multiple of the stimulus period will leave the observed statistics unchanged. We assume that the analysis period L is a multiple of the stimulus period S.
The definition (Eq. A11) of the covariance
Xav(1,
2,L,
t0) still makes sense, but the average over
elapsed time T must be restricted to values of T
that are multiples of S to avoid averaging together estimates that begin at different phases of the external stimulus. Furthermore, because the external stimulus provides a time marker with
period S, equality in Eq. A13 only can be
guaranteed for shifts
T that are integer multiples of
S. Thus the relationship Eq. A14 holds only when
T is a multiple of S. It follows that
Xav(
1,
2,L, t0) can have nonzero values whenever
1 +
2 = (2
n)/S, for any integer n (and not
just n = 0, as in the "no external stimulus" case).
We let = (
1 +
2)/2. The condition for a nonzero covariance,
1 +
2 = 2
n/S, becomes
1 = (
n/S)
and
2 = (
n/S) +
. In analogy with the standard
power spectrum (Eq. A16), we define the phase-locked
spectrum (with start time t0=0) as
![]() |
(A17) |
Note that the condition that time-shift by a stimulus period
S leave the observed statistics unchanged explicitly
excludes the possibility of period-doubling (Feigenbaum
1983) and other related phenomena. However, our analysis could
be generalized to this situation by replacing the stimulus period
S by a suitable multiple MS in the formalism in
the preceding text.
Phase-locked spectra for multiple-channel recordings
The above procedures may be extended to multichannel data. We
use Zj(,L,
T,t0) to denote the Fourier component at the
frequency
determined from a segment of length L on
channel j, beginning at elapsed time T after the
arbitrary start time t0 (Eq. A19). In analogy to Eq. A5, we use
Zj,av(
,L,
t0) to represent their average over a set of
intervals parameterized by T. The set of covariances of
these components includes not only cross-products within a channel
j but also between pairs of channels j and
k. Thus for multichannel data, the definition of the
cross-covariances, analogous to Eq. A9, is
![]() |
(A18) |
![]() |
![]() |
(A19) |
Note (from Eqs. A18 and 19) that the phase of the
cross-spectrum is the combined result of two influences. One
contribution, analogous to the phase of the single-channel phase-locked
spectrum, is the phase of the modulation envelope. This contribution is independent of . The second contribution is specific to the
multichannel situation and arises if there is a phase difference of the
carriers on the two correlated channels. If, for example, the residual activity on two channels is correlated with a delay
between channel
1 and channel 2 (channel 1 earlier than channel 2), then this delay
contributes a phase of e
i
to
P0;1,2(
). Phase coherence remains a
valid test for the significance of
Pn;j,k(
). A band of consistent phases
indicates a band of frequencies in which the residual activity on two
channels is correlated at particular times in the stimulus cycle (the
1st contribution). A nonzero slope of the phases within this band indicates a temporal shift of the carriers between the channels (the
second contribution).
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ACKNOWLEDGMENTS |
---|
We thank P. Mitra for helpful comments and suggestions and for pointing out some limitations of the Wigner transform. We acknowledge the programming assistance of A. Canel. We thank D. Reich, F. Mechler, and S. Kalik for helpful comments and review.
This work was supported by National Institutes of Health Grants NS-02014 to N. Schiff, EY-9314 to J. Victor, and NS-01677 and NS-36699 to K. Purpura.
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FOOTNOTES |
---|
Address for reprint requests: N. Schiff, Dept. of Neurology and Neuroscience, New York Presbyterian Hospital-Cornell, 1300 York Ave., New York, NY 10021.
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 20 August 1998; accepted in final form 28 June 1999.
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REFERENCES |
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