A-waves, being a new evoked-response phenomenon, raise a number of issues, none of which can be definitively settled in an introductory paper such as this. Instead, we hope to indicate in this discussion what questions the findings generate, and in what ways these new phenomena might be useful.
Trivial coincidence, or tantalizing clue?
Mindful that "coincidence implies causality but does not prove it", the consistency of waveform differences in the visual, auditory, and somatosensory systems on either side of the STZ provides powerful motivation for producing some wide-ranging speculation. We demonstrate our primary speculation by means of Fig. , which shows a group of grey dots of different sizes. The presentation is steady (at the refresh rate of the screen you are watching). You are now going to see the same screen flashed, where one of the dots will move back and forth a distance of about its radius.
(Disclaimers: 1) Because of the characteristics of computer monitors,we cannot duplicate the experimental conditions that were used in our visual experiments. For example, we are limited to just the frame-rate for changes, and for the duration of/the stimulus. In our experiments the stimuli were brief, allowing manipulation of the SIs. 2) There are many factors that can influence this effect. For the purposes of this discussion, only the rate effect will be at issue.)
This demonstration is intended to provide you with the answer to the following question: Does the rate of presentation affect your ability to determine which dot moves?
1) First go to Fig. and then try MovieB (see Fig. legend). The repetition- rate is 2.4 S/s with a 20% duty cycle.
2) Next try MovieC (see Fig. legend). The repetition-rate for MovieC is 12 S/s with a 100% duty cycle. Hopefully you now answer the question in the affirmative, and that it is easier to see the dot move under the conditions of Button C. Note that seeing the dot move did not require any conscious effort, or any prior use of "attention". The detection of the moving dot is automatic (and presumably a relatively low-level of extraction of a changing stimulus embedded within a background that appears unchanging because of fusion).
The presence of this psychophysical phenomenon raises two questions:
1) What are the neurophysiological mechanisms that underly this psychophysical experience?
2) What functional role might such mechanisms play? Since we cannot immediately answer the first question, let's start with the second.
With regard to the "functional role" that this neurophysiological mechanism plays, a "scene-presentation" with most elements having a repetition rate above STZ provides a means to rapidly identify a change in the visual field. Whereas, when one is presented with the same stimuli at a subSTZ repetition-rate, it is very difficult to identify the dot that moves, even when one knows which dot to look at. For ease of speculation about neurophysiological mechanisms, let us assume that a "single" stimulation is followed by a single firing of the cells in the early part of the response (as is possible if the stimulus magnitude is adjusted to be moderate – neither near threshold nor near saturation – and the stimuli are brief). Certainly this kind of firing can be found at sensory cells in the PNS (Peripheral Nervous System). We assume, in this case, that the firing of subsequent post-synaptic cells in the CNS (Central Nervous System) is not at a slower rate than the rate at which the PNS cell is being driven.
If the assumption of a one-to-one correspondence between stimulation and firing be granted, then the stimulus repetition-rate is also the firing-rate of these early cells (PNS and CNS) in the response. In this way, a change in stimulus repetition-rate is equivalent to changing the intensity of a steady, continuous stimulus to the PNS. From this we can conceptualize the following hypothetical "rule": Any part of a sensory field that is firing at a uniform rate above the STZ is "Ground", whereas the parts of the sensory field that are the "Figure" have one or more of the following characteristics:
1) A firing rate below the STZ,
2) A firing rate, though above the STZ, that is changing. (Our attempts to define "Figure" and "Ground" have always lead to either obvious or subtle circular definitions; we therefore will purposely avoid rigor.)
The effect of Fig. relies on the Ground being presented
above STZ, while the Figure is presented below STZ. If we imagine that the sensation of "fusion" involves the detection of unchanging "sameness", then there must be a memory of the immediate past, and an estimate for the duration of that memory can be made from our experiments. Based on our data, we would
roughly estimate the
longest duration of this "fusion-memory" is about 80 ms for peripheral vision, and about 60 ms for the auditory system. (Based upon the work of Lalanne [
2] and Brecher [
3] the value would be 56 ms [the period of 18 Hz].) That is, we predict that any sensory inputs that are repeated at
shorter unchanging intervals than some short interval, will give the "supraSTZ response", where the word "unchanging" implies "less than the just-noticeable-difference for that stimulus repetition-rate" (a criterion apparently met by a low-jitter QSD because there is a fusion effect despite the jitter).
"Fusion-memory" needs be compared with the memory that occurs when the stimuli are separated by 150 ms or more (the number 150 is arbitrarily chosen to be larger than 100 ms, to avoid contentious arguments about alpha waves that may distract from this exposition). We will call this second memory "flash-memory" because the presentation that initiates it "comes and goes in a flash". Thus, brief auditory stimuli can also generate "flash-memory". A stimulus that is a step-function change probably generates a combination of flash-memory (transient) and fusion-memory (new steady-level), such as that shown in the firing rate for Cell "A" in Fig. , relative to the step-increase in light. Thus, we hypothesize that the CNS response to the PNS activity indicated by Cell "A"'s firing rates will be different for the peak of differing spike-intervals at the onset of the step, as compared with the CNS response to the more uniform firing after adaptation to the new intensity. The CNS difference we imagine is that the changing firing rate corresponds to "Figure" whereas the more uniform rate corresponds to "Ground". Note that "Ground" takes some time to stabilize (after adaptation of the sensory ending's response to the step), which could correspond to the time for the subSTZ waveform to morph into the supraSTZ waveform.
When comparing the quality of "fusion-memory" with "flash-memory", fusion-memory is much more accurate for some aspects of the stimulus. For example, even though there is memory of a single flash of Fig. , it is not possible to remember enough detail to determine that one of the dots is changing position. The accuracy of fusion-memory is shown when something changes in an otherwise "stationary" scene. Look out a room-window at a scene in which nothing seems to be changing. A small movement of something in any part of the scene is rapidly noticed, even when the details of the scene are complex, unfamiliar, or even random. Somehow fusion-memory retains the "current-state" of the pattern of sensory-input, so that change is readily detected.
On the other hand, if change in a scene is sufficiently slow, it will go unnoticed – a neural phenomenon which is utilized by many predators who use slow approaches to prey, at a speed below that which triggers an "alerting" response in the sensory system of the prey.
To allow you to compare the accuracy of fusion-memory, with flash-memory, we offer a demonstration in the auditory system. For this demonstration, the sounds must be played through loud speakers, notheadphones. If you have a stereo computer system, space the computer's speakers about 1.3 meters, or more, apart. Put one speaker at least 0.5 meter closer to you than the other, so that the "Dau-chirps" will appear to originate between the speakers, but closer to the near speaker (even though the "Dau-chirps" in both speakers will actually occur simultaneously). If you have difficulty observing the effects described, then try either moving the speakers a bit farther apart, placing yourself more asymmetrically relative to the speakers, and/or trying it in a smaller space, such as a closet. (The senior author has added the mention of the closet here, so that if someone finds you listening to buzzing sounds in a closet, you can produce written evidence to them that you are not totally crazy.)
To start stimulation at 2 S/s (uniform) use the following link [see
Additional file 2] Adjust the intensity to be comfortably loud. Also set the audio-player to "loop" so that the sound plays continuously if it is not doing that.
Point your finger to the spatial location from which the sound seems to originate. Now rotate your head, left and right, over about a 60° range. Note that despite movements of the head relative to the speakers, the "location" of the sound is unchanged, and easily indicated by your pointing finger. Further, note that the subjective quality of the sound does not change with this head movement. Confirm the same observations by moving your head closer and farther from the speakers by about 15 cm. You have experienced what we call "flash-memory".
Now start stimulation at 100 S/s (uniform) using the following link [see
Additional file 4]. Although the sound is raspy, a low-pitched tone is perceptible, in addition to higher-frequency timbre. Repeat the observations you made after pushing the 2 S/s button. Does the quality or loudness of the stimulus sensation change
with even small changes in head position or rotation? (If so, return to "2 S/s (uniform).mov" to verify that you
cannot hear these differences at the slower rate. [see
Additional file 2]).
Is the accuracy of your locating the "source" the same as with the low rate, or has the "location" broadened? You have experienced what we call "fusion-memory".
(You might remember this the next time you encounter the sound of a solitary cricket's "chirp" and find it difficult to physically locate the cricket solely by its sound. The senior author presumes that the frequency of the cricket chirp is above your STZ, but somehow starts and stops without energizing flash-memory in predators (while having a different effects in other crickets). Another example is the lack of "location effect" for a sub-woofer in a multi-speaker sound system, where the sounds are cyclic repetitions that are supraSTZ.)
Did you notice that when you were listening to "100 S/s (uniform)" (fusion-memory) that you could hear the "glitch" when the sound-player on the computer reaches the end of the track and takes a moment to loop to the re-start? If not, try again: [see
Additional file 4]. This is the
very feature of the sensory input that fusion-memory is very good at detecting. Can you hear the glitch listening to "2 S/s (uniform)" (flash-memory)? The same timing "glitch" is there, too, but not detectable by flash-memory. You can verify this: [see
Additional file 2].
These effects are important in that the subjective differences observed can be hypothesized to be due to differences in memory functionality between the shorter fusion-memory (at repetition-rates above STZ) and the longer flash-memory (at repetition-rates well below STZ). We hypothesize that these psychophysical differences are due to differences in neural processing which are reflected in A-wave differences. Another important aspect of these differences is that up to now time-domain waveforms from evoked-response research have been limited to those observable at subSTZ stimulus repetition-rates – so the conclusions from such research only apply to flash-memory. The issue of SS studies of supraSTZ rates is discussed later, in a separate section.
To demonstrate that the psychophysical effects are still present even though there is a small amount of jitter in the stimulus-intervals, as is required by QSD, we offer the same stimuli here, but with stimulus repetition-rates which are
jittered 12% as compared with the uniform rates. "2 S/s (jitter)." [see
Additional file 1] "100 S/s (jitter).mov" [see
Additional file 3]
Flicker-fusion and visA-waves
We have not done any formal testing to establish the relationship of A-waves to well-defined psychophysical phenomena. However, the region of stimulus repetition-rates above and below which the A-waves show clear changes in waveform is the STZ, which in vision can be described without much specificity as "where the flicker changes to fusion". When we tried, in a dark room, manipulating the flash-rate of a simple tachometer-flash system (no jitter), it was clear that there are many possible end-points that can be called "fusion". The central region of the visual field seemed to "go smooth" at lower frequencies than the peripheral vision which still could detect a flicker. There were moving "strings", "tendrils", or "webs" which ultimately "blended away", but at rates higher than that needed for fusion of central vision. For these reasons, we consider that there is no single "fusion" rate in our visual experience, and suspect that stimulus parameters, plus subject variables (such as accommodation and possible hysteresis) are likely to lead to different endpoints. Although we cannot provide this experience via computer monitors, we offer audio demonstrations in Fig. for listening to sounds at different repetition-rates, with either "clicks" or Dau-chirps. These files are accessed via the Figure Legend of Fig. .
Also in Fig. , we provide some sequences with increased jitter, not used in our experiments, to show the psychophysical effects of increased jitter. Note that the sounds of the non-jittered (uniform) 40 S/s [see
Additional file 48] tend to form a low-frequency tone. That tone is less in the 12% jitter that we used [see
Additional file 58]. At higher jitters the tone is gone – 24% [see
Additional file 59] and 36% [see
Additional file 60]. It is also missing with the MLS sequence [see
Additional file 57]. This observation suggests to us that we were lucky to have not started with a larger jitter, and that future A-wave research needs to verify whether the waveforms differ at percentage jitters less than 12%.
We did not try to find an A-waveform "at
the fusion-point" in our studies because such an endpoint might be highly variable, could differ with intensity, and, like threshold measurements, could involve many long runs. Our choices of stimulus repetition-rates were based upon guesses in the hopes of staying on either side of the STZ. It may be that the unusual audA-waveform at 15 S/s in Fig. is within the STZ since it is unlike the waveforms at rates either
above or below it.
There may well be interesting changes occurring within the range of about 12–20 S/s, as indicated by Fig. . In 1936, v. Bekesy reversed the usual procedure and kept the stimulus magnitude constant while varying frequency while searching for a fusion threshold, using a closed ear-canal stimulator [
13]. He had this to say about fusion threshold:
"Careful examination revealed that the auditory threshold for low tones reflects the quantal character of neural processes. Thus if the frequency of the alternating pressures was changed slowly, and without any variation of magnitude [emphasis added], from 2 to about 50 cps, it was possible to observe that the loudness and the pitch did not vary continuously, but were altered in a stepwise manner.
"This discontinuity was most clearly perceptible in the region of 18 cps. As the higher frequencies were approached, there appeared a sudden increase in loudness, corresponding approximately to a doubling of the sound pressure. At the same time there was a doubling of pitch; the number of pulses, which were separately perceptible below 18 cps, suddenly became doubled, and the whole sensation became fused and acquired a tonal character (Brecher) [
3]. This tone was still extremely rough, and the roughness gradually declined as the frequency was raised further. This frequency therefore can properly be designated as the threshold of fusion (Brecher) [
3]. It is practically the same for all the sensory modalities..."
For further work related to the quantal nature of this and other data, see Geissler [
24]. For a review, see Kompass [
1].
We might expect that A-waves might be affected by these factors, although the A-waves we have shown are all recorded "above threshold" in terms of intensity and repetition-rate.
The relevance of QSD to psychophysical research on the phenomenon of "fusion" is that QSD provides a means of correlating observable brain activity with psychophysical endpoints (if the endpoints can be adequately defined and determined). Whether the correlation will be exact remains to be seen. But it is clear that in those research or clinical areas where flicker-fusion shows interesting and/or useful results, QSD may make a contribution. For example, visA-waves might be helpful in those patients in whom the subjective CFF measure is unreliable, such as Parkinson's [
4-
6]. Even where the patient's CFF is reliable, the objective measure of visA-waves by QSD might augment or replace the psychophysical measurement of CFF in a variety of clinical conditions, such as migraine, Alzheimer's, reading disabilities, hypertension, drug side-effects, and visual deficits [
7-
11].
Latency shift?
It is of interest that there seems to be a latency "shift" in the peaks of the larger A-waves when comparing waveforms at rates above and below the STZ (Figs. , , , , , ). Such a shift can only be known for certain by research which shows which peaks in the subSTZ and supraSTZ waveforms are functionally comparable. But, for the purpose of this section, let's assume that such latency differences are present. Given that we have associated subSTZ and supraSTZ waveforms with different memories (fusion-memory and flash-memory), it is but a small additional leap to consider whether the differences which are a function of repetition-rate are somehow connected with some mechanism that we will imagine as being similar to the spike-timing dependent plasticity of LTP (Long-Term Potentiation) and LTD (Long-Term Depression).
At excitatory cortical synapses, induction of synaptic plasticity is dependent both on the rate and the timing of input activities. While experimental protocols for study of these phenomena tend to emphasize the timing of activation rather than the rate, it is clear that both are jointly responsible for the induction of synaptic plasticity [
25,
26]. While this plasticity is generally studied and conceptualized with respect to changes lasting minutes to hours, in
our model we assume that the
mechanisms that trigger these longer effects
may also trigger shorter memory mechanisms, as well. So, we note that, with respect to rate, LTP occurs with higher stimulation rates (e.g., 60–100 S/s), while LTD occurs with low rate stimulation (e.g., 13–20 S/s). The sign of plasticity (LTP or LTD) is dependent on the temporal order of synaptic activity relative to the back-propagation of the action potential. This temporal order might be affected by the latency-shifts we are assuming. The magnitude of the shift that we "eyeball" between the subSTZ and supraSTZ waveforms is 70–80 ms. This is of an appropriate size to move from the LTD window (75–50 ms before the action potential) to the LTP window (10–15 ms after the action potential) [
25,
26]. Hence, it is conceivable that cellular mechanisms could be triggered by the latency shift that distinguishes subSTZ and supraSTZ responses, and by implication might distinguish fusion and flash memories.
Since we are far out on a speculative limb, the incremental risk of further speculations seems small:
1) The effects triggered by the LTP/LTD mechanism with respect to fusionmemories would be predicted to be very short (if not enhanced by attention, emotions, etc.), such as less than 75 ms.
2) We wonder whether the time needed from the start of a rapid stimulus train, to develop the supraSTZ waveform (Fig. ) should have some equivalent time at the cellular level. Such equivalent time might be the time necessary for activity in the dendritic tree, at higher input frequencies, to induce a prolonged depolarization in the cell that, together with continued synaptic activity, induces LTP [
26].
3) Since the senior author hypothesizes that evoked-responses obtained at stimulus repetition-rates above about 6 S/s are almost entirely due to action potentials, he cannot resist commenting here that an increased peak latency of A-waves at supraSTZ firing rates (as controlled by stimulus repetition-rate) could be a measure of timings of the action potentials causing the back-propagation required for the "LTP-triggered" model suggested here. If so, then some aspects of the timing of these cellular processes could be detected and measured on the human scalp for research and clinical purposes. Note, however, the large number of "if's" needed to reach this notion.
4) The senior author also conjectures that the EEG may be the "ground" brain activity which registers the "current status" of unchanging sensory inputs via A-wave oscillations time-locked to the steady firing-rate of an given sensory input (as a function of the steady stimulus intensity at each sensory input, independently). When a changing sensory input results in a markedly-uneven firing rate, then the "figure" thus identified is rapidly analyzed with the brief A-wave responses (below STZ). The analysis involves associative memory.
Unclear to the senior author at this stage of the investigation are the following:
a) Does the large amount of "ground" activity affect the affect the associative memory search?
b) Is "ground" the "context" of the resulting association?
The "oscillatory nature" of A-waves
A-waves indicate a new source of data about brain activity, obtained by a technique that can directly stimulate and record sustained oscillations of more than 1000 ms after each stimulus in a rapid stimulus train. This data may contribute to global theories of brain function that utilize "oscillations" as a generalization underlying many aspects of brain functioning, as described in several books [
27-
31]. Some articles report research on brain oscillatory behavior based upon EEG or ERP data,
e.g., [
32-
34] while other articles describe theoretical approaches,
e.g. [
35-
37]. Our data suggest that QSD methods may be applied across a considerable range of studies directed toward understanding neural oscillations, with the hope that this new approach may complement and deepen the interpretation of previous results and hopefully uncover new phenomena.
A-waves as probability functions
At first thought, an "oscillation" might seem to indicate repetitive firing from neurons driven by the stimuli. Indeed, we described such neurons with reference to Fig. -Right. There is much evidence to indicate that PNS cells can be driven in timing with the repetitive stimuli, as can CNS cells that are innervated by such cells. But as one ascends the neurons of a sensory system, towards the cortex, it becomes more and more difficult to achieve a simple one-to-one correspondence between the timing of a simple stimulus and the timing of the cellular response. Such observations are relevant to considerations of what cellular activity underlies scalp-recorded A-waves.
If we assume that a given cortical cell fires at the same phase of each cycle in a sustained A-wave oscillation, there might be some stimulation rate at which the cell is about to fire due to the most recent stimulus, but has just fired as a later "cycle" to an earlier stimulus. In such a case, the refractory period of the cell may prevent a response to the most recent stimulus. A simulation of such a possibility is shown in Fig. , where it can be seen that there are multiple opportunities for this "conflict" to occur. However, we might not be able to detect a loss of such a response because our data is formed from averages of hundreds of stimuli, and responses from many thousands of cells. (We have proven [
14], that variation in
signal cannot be detected in the poor signal-to-noise conditions under which we record A-waves.) From such considerations, it may be a better "mental model" to imagine A-waves as representing the probability of synchronous firing in populations of cortical cells.
If such a model is accepted, what is the significance of a negative potential, in contrast to a positive potential, as a measure of probability? The senior author has previously shown that an AP (Action Potential) will produce one polarity at far-field electrodes when the AP is initiated, and the opposite polarity at the termination of the axon [
38-
40]. Since these potentials are dipoles rather than quadrupoles, these events are more easily detected at distant electrodes than is conduction along the axon (which can be quadrupolar). The consequence would be that the half-cycle time of the A-waves would be the conduction time from the initial segment to the axonal termination, as measured in a population of neurons. If the AP from neuron "A" activates neuron "B" and neuron B's AP travels subsequently in the opposite direction to the AP from neuron A, the initiation of the AP in B will have the same polarity as the termination of the AP of B. In such a case, the repetitive oscillations seen in A-waves are consistent with cyclic activity between two brain areas, such as could arise from thalamo-cortical or cortical-cortial reciprocal connectivity.
"Can the brain really do THAT?"
We have received this type of comment from reviewers, and we feel it important to describe the limitations that affect
any waveshapes that are obtained by
averaging. It should be clear that an average may not represent any particular individual datum. Consider that although the mean number of children per family may be 2.3, there is no family with that number of children. This fact does not negate the usefulness of the mean value, but does limit its interpretation to the
population of families rather than to any one family. So, while it is easy to imagine that the mean evoked-response occurs with
every stimulus, this may not be the case. As mentioned in the previous paragraph, in the case of an initially poor signal-to-noise ratio, it is not possible to detect signal variation from run-to-run variation (see Appendix of QSD paper [
14]). So the interpretation of the "meaning" of a waveform in terms of the neuronal generators which created it during a run of repeated stimuli may be different for different evoked responses. Note that these statements refer to
averaging, which is the first step in QSD. Deconvolution of the
average is the next step, but does not change the basic problem that has already been generated by the average. Said in another way:
QSD shares with averaging of evoked-responses the same ambiguities with regard to whether the average-waveform occurs with each stimulus or not.
"How can a nonlinear brain response be detected by a purely linear mathematical scheme?"
This is another reasonable question that we have received from reviewers. It is clear that A-waves are non-linear responses with respect to stimulus repetition-rate. It is also true that all computations in QSD are linear. However, as shown in Fig. , a nonlinear response can be detected by repeated runs in which the shape of the nonlinear response is estimated at a number of points, each using a linear approximation over a small excursion-range. This is a standard technique in physics and engineering. In our experiments, all stimulus parameters are kept constant during a run, except for the small excursion of the repetition-rate (12% jitter). The smaller the excursion, the more accurate is the estimate. The jitter excursions are somewhat smaller than the changes in repetition-rate necessary to show changes in A-waveforms.
"Steady-State" Potentials compared with QSD waveforms
Starting in the early 90's, phenomena and theoretical excitement about the functional role of cortical oscillations (alluded to in the previous section), there was an expansion of the range of application of the SSVEP (
Steady-State Visual Evoked Potential). The SSVEP was combined with cortical localization and topographic analysis, where the SSVEP was used as a "probe stimulus" that revealed activity in various areas of the brain under conditions of sensory and cognitive processing [
41-
44]. In the probe-SSVEP studies, the SS (
Steady State) response amplitude is considered to vary inversely with intensity of processing in any area, according to the "processing capacity model" put forward by Papanicolaou [
45]. The "spare" resources available to process the probe response go down as task processing load increases. This may have the same physiological mechanism as the well studied inverse variation of alpha amplitude with increased processing activity found in "Evoked Response Desynchronization" studies [
46-
52]. Several established researchers have developed the SS technique with their own technical variations and created new experimental designs [
48,
53,
54] to apply the SSVEP to diverse fields of study [
43,
50-
52,
55,
56], with clinical applications to areas such as migraine [
57,
58], schizophrenia [
55,
59], and Attention Deficit Hyperactivity Disorder [
60,
61].
Recently researchers have begun to compare the localization derived from electrical measures to localization using fMRI [
48,
62]. Techniques are now in use that permit simultaneous measurements of both SSVEP and fMRI [
62]. Using these combinations of techniques [
42-
44,
50,
52,
61,
63-
67], it is now possible to study:
1) Oscillatory neural processing over all parts of the cortex,
2) Cognitive processing from early sensory discrimination, recognition, and attentional processing, to complex cognitive tasks,
3) Working and long term memory as related to decision processes, and
4) Motor output sequencing and coordination. At the root of all this capability and these techniques is the use of the SS stimulation.
There are differences between the data presented in this paper and that obtained by SS stimulation:
1) The stimulus intervals in QSD are jittered, whereas in the SS response they are uniform.
2) The stimuli used in this paper, are brief, whereas "probe-SSVEP" stimulation uses sinusoidal stimulation.
Although these differences make direct comparisons between published results and ours problematic, the overlapped waveform average (i.e., the "raw data" before deconvolution) approximates to the SS average which would be obtained using our brief stimuli (with a uniform repetition-rate). For this reason, we call it the qSS (quasi-Steady-State) average. The peak in the frequency-domain at the stimulus repetition-rate in the qSS average has a peak that is equivalent to the SSVEP magnitude. So, if experimental conditions are similar, the results of the two methods can be reasonably compared in the frequency-domain.
Another method for comparing QSD visA-waves with SSVEP results is to simulate the SSVEP result-magnitude using frequency-domain analysis of the visA-waves, as we will now do. In Fig. we show the frequency-domain power of the deconvolved time-domain visA-wave shown in Fig. at 30 S/s. Note that the time-domain data used to compute Fig. is circular, so that there is no distortion due to windowing; the frequencies are those of the signal, within the passband 8–50 Hz. In this frequency-analysis the prominent peak is just passed 10 Hz, with lesser peaks in the range of 13–17 Hz,even though the stimulus repetition-rate was 30 S/s.
We will now visualize the SSVEP results that would be obtained recording
this brain response. As we have already proven [
14], averaging overlapping waveforms is temporal convolution. In the frequency-domain, temporal convolution becomes just complex multiplication of the magnitude of the Fourier coefficients at each frequency in the frequency-spectra of the two circular vectors. So, if we want to know the
frequency-domain result
if the temporal waveform of Fig.
(30 S/s) were uniformly convolved,
we need only multiply the frequency-spectrum of the signal (Fig. )
by the frequency-spectrum of the uniform stimulus repetition-rate, which is a
comb filter. The "comb filter" is so-named because the identical-height amplitudes in the frequency-spectrum of the uniform stimulus pattern look like the teeth of a gap-toothed comb. Comb filters for five uniform repetition-rate stimulation sequences, are shown at the bottom of Fig. .
Because the frequencies between the "teeth" of the comb filter are zero, the magnitude of the product resulting from the multiplication of zero times the visA-wave amplitude, no matter what it is, will be zero. Hence, there are no "results" from these frequencies, only from those frequencies that have "teeth". So we need only look at the products that will result at these frequencies. Starting with the comb for a uniform stimulus repetition-rate of 5 S/s, the first product will be very small, the second very large, the third about 50% of the second, the fourth but a quarter of the third, and the rest being as small or smaller than the first. These products are the totality of the frequency-domain information available from the time-domain average from the uniform repetition-rate. This limited information is too sparse to recover the time-domain waveform from the frequency-domain data.
Changing the repetition-rate merely changes the "tooth frequencies" whose limited number cannot reveal the details of the response. Nor can the magnitudes of different "tooth frequencies" observed by changing the repetition-rate be reasonably compared, because they are probing different parts of the signal. To better understand this, it is suggested that the reader repeat the process of identifying the parts of the signal-frequencies that are probed, for each of the stimulus repetition-rate comb filters shown at the bottom of Fig. . The reader can then confirm the following statements:
1) At 10 S/s, only two frequencies (10 and 20 Hz) contribute significantly to the products.
2) At 15 S/s and stimulus repetition-rates above 15 S/s, only the product at the stimulation frequency has much magnitude.
3) At a repetition-rate of 30 S/s, no frequencies of the response from 8–29 Hz (that were actually occurring when the brain was stimulated at 30 S/s) would contribute to the result!
4) If the usual practice in SS analysis were done, namely that only the product at the frequency of stimulationis used, then the data obtained from the 6 runs at the bottom of Fig. would show marked variation in amplitude even though the actual brain response is the same in every run!
Thus, if changes in amplitude of the "probe frequency"occur as repetition-rate is changed, one can conclude either that:
1) The response changed, or
2) The response didn't change (i.e., a different part of the response is being probed).
In consequence, the inherent information limits in SS data as a function of repetition-rate must be recognized. This error occurs when the experimental variable is repetition-rate. If the repetition-rate is held constant while some other variable is changed, then changes in the magnitude of the product at the stimulus repetition-rate may indicate changes in brain activity if the change in the experimental variable causes no changes in the general waveshape (time-domain), but only changes the magnitude of the entire brain response. But the waveshape must be determined using QSD, in order to validate such SS data.
(Technical note: The critique centered on Fig. has not included the 1/N factor in Fourier Transformations, nor whether the magnitude of the comb filter varies with repetition-rate because of repeated use of the same sweep length in the average. The general conclusion would be the same, should these have been included.)
Because the limitations imposed by data collection at a uniform rate are important when considering SS data, we have animated the differences between SS analysis and QSD, as shown in demonstrations accessed from the Legend of Fig. . In each of these demonstrations, in the lower left is shown a red waveform which is the brain's response to the stimulus (time-domain). On the lower right (in the box) in red are the magnitudes of the brain's response in the frequency-domain. The vertical lines indicate the frequencies of the comb filter. Across the top is the time-domain data that will occur from repeated stimulation, as computed from the convolution of the comb filter and the brain's frequency-domain magnitudes. Recall that in an SS recording, this waveform cannot be deconvolved. On the other hand, in a QSD recording this waveform approximates to the SS recording, so we call it "qSS" (quasi-Steady State), and it can be deconvolved, as shown in the middle left. This waveform (middle left) is the time-domain waveform that occurs either as deconvolved brain response in QSD, or as a 500 ms window for SS.
As you watch the SS animations, you can see that as the repetition-rate changes, different frequencies of the brain response (lower right) make up the convolved waveform (across the top). As the stimulus repetition-rate gets faster and faster, the frequencies "probed" by the comb-filter become less and less, and the waveform of "the response" (middle left) becomes more and more simple, until it is just a sine wave (when only a single tooth of the comb-filter is within the frequency of the brain response).
In contrast, as you watch the QSD animations, you will see that the frequencies that are "probed" are always numerous because of the jittered sequence of the stimuli. Note that the deconvolved waveform recovered by QSD (middle left) is the same as the brain's response. In the absence of noise the two waveforms would be identical. Since you might not believe that we were actually computing the deconvolved waveform, we added some noise within the passband, so that the waveform changes slightly.
On repeated viewings, the reader can verify that whereas in the SS animations the convolved waveform becomes simpler and simpler as the repetition-rate increases, in the QSD animations there is continued complexity in the convolved waveform (across top). It is this complexity that the QSD method utilizes to recover the brain's response. The use of a uniform repetition-rate destroys such information.
What we did not find
Like the dog that did not bark in one of the Sherlock Holmes' mysteries, what we did not observe may also be of some importance. Although oscillations with periods in the alpha-band were often observed, no prolonged or sustained oscillations in the gamma band were seen. Note that in Fig. , G-waves show only 1.5 "cycles" in the gamma frequency range (between the peaks G0 and G2, and between the valleys G1.3 and G3). The G-waves are not prolonged oscillations, as seen in the A-waves. If the data is recorded with a passband of 30–120 Hz (as we have done for G-waves) then there can be summation of the 25 ms periods of the G-waves (peaks adding to peaks) since the larger A-waves are filtered out. If so, the decreased amplitude above and below 40 S/s with this passband can easily be due to peaks adding to valleys. Since we have recorded with an "open passband" in Fig. , one can see that the "G-wave portion" of the audA-wave recording is very small. So, if the "alpha-rate oscillations" seen in the open passband are removed by a high-pass filter, then the remaining waves may sum in the time-domain, as just described. If the observations of this explanation are replicated, then the lack of gamma activity in our recordings will be viewed in retrospect as not surprising. In which case we would have to conclude that 40 Hz may not be a critical stimulus repetition-rate to whatever part of the CNS that is responding in synchrony to our jittered stimuli . Note however, that this critique applies only to "40 Hz evoked responses" recorded from the scalp, not to data from single cells or cell groups. Thus, we hypothesize that it is possible for scalp-recorded evoked-responses to seem to support single unit data, when the "support" is actually artifactual, based upon a fortuitous period between peaks in the ABR-AMLR, not upon cortical firings. Note further, that these comments do not apply to any induced oscillations which the stimulation may have caused and which we did not measure. ( What is notable is that some of our supraSTZ stimulus repetition-rates are in the gamma range. Thus, our results can be interpreted as showing long, synchronized "alpha waves" due to prolonged stimulation at gamma rates. However, the waveforms obtained at these rates are not unique to "gamma-rate"stimulation since similar waveforms were recorded to "below-gamma" rates. Our only sure conclusion is that QSD methodology offers a new way to study stimulus repetition-rate effects in sensory systems.
1,4 Toryalai Hart,1 Linda J Larson-Prior,2 Bill Baird,3 Marram Olson,1 Michael Trumpis,1 Katherine Makayed,1 and Payam Bavafa1
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(t), then b(t) can be "estimated" by deconvolution. The brain response b(t) cannot be fully recovered because there is always some noise contamination, so the estimated brain response is expressed as 
(t).
(f) = [B(f)·Q(f)] + N(f) [Eq2]
(f)] by dividing by the frequency-domain equivalent of q(t) (which is Q(f)), as shown by the Fundamental Equation of QSD:
