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Neurons of the cerebral cortex exhibit precise inter spike timing
in correspondence to behavior
Tomer Shmiel [1] , Rotem Drori [2] , Oren Shmiel [1] , Yoram Ben-Shaul [2] , Zoltan
Nadasdy [3] , Mina Teicher [1] & Moshe Abeles [1,2]
[1] Bar-Ilan University, Ramat-Gan 52900, Israel
[2] Hebrew University, Jerusalem 91120, Israel
[3] California Institute of Technology, Pasadena, 91125, USA
Abstract
We show that times of spikes can be very precise! In the cerebral cortex, where each
nerve cell is affected by thousands of others (1, 2), it is the common belief that the
exact time of a spike is random up to an averaged firing rate over tens of milliseconds
(3). In a brain slice, precise time relations of several neurons have been observed (4).
It remained unclear whether this phenomenon can also be observed in brains of
behaving animals. Here we show, in behaving monkeys, that time intervals between
spikes, measured in correspondence to a specific behavior, may be controlled to within
3 milliseconds.
1) Introduction
As known, most nerve cells in the brain communicate with each other by standard pulses
called action potentials (or spikes). The common belief, so far, is that each neuron
represents one aspect of the mental processes not by precise firing time, but by elevating its
firing rate (3). However, if time relations among different neurons could be precisely
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controlled and read out, complex representations could be built from simpler ones
efficiently and very fast (5-7). In a brain slice, precise time relations among several
neurons have been observed (4). Could this phenomenon be also observed in brains of
behaving animals? We use here data-mining techniques and rigorous statistic testing to test
how precise are time intervals between spikes of different neurons. We show, in behaving
monkeys, that when time intervals between spikes of different neurons are measured in
correspondence to a specific behavior, timing may be controlled to within 3 milliseconds.
2) Experiments Description and Drawings Analysis
In our experiments single unit activity was recorded from 8 microelectrodes inserted into
the motor and pre-motor cortices of a monkey while it was freely scribbling. Repeated
scribbling paths were extracted by data-mining algorithms (8, 9). These paths are called
drawing components. In a typical day there are 12 to 25 such drawing components. Figure
1 illustrates the monkey’s drawings and two simple drawing components.
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Figure 1. Examples for two drawing components. 30 seconds of drawing is shown in
each box. The hand position was sampled 100 times per second (dots in the drawing). The
monkey mostly drew in a counter clockwise direction. Scale bar is 50 mm. a, Occurrences
of the drawing component: "transitions of drawing direction from a range of 180o-210o to a
range of 210o-240o" are marked by thicker dots. b, Occurrences of the drawing component:
"transitions of drawing velocity from a range of 20-30 cm/sec to a range of 10-20 cm/sec"
are marked by thicker dots.
3) The Main Idea
To determine whether there are any precise timing relations between the spikes of two
neurons and the drawing, we selected a 300ms time slice before the start of each drawing
component. For a given pair of neurons, we counted how many times a spike in the first
neuron was followed by a spike in the second neuron within 0-1 ms, 2-3 ms… 98-99 ms.
The interval that repeated the largest number of times was hypothesized to show precise
firing times in relation to this particular drawing component. For example, the interval 9091 ms between neuron #1 from electrode 8 (noted by 8.1) and neuron #2 from electrode 1
(noted by 1.2) repeated 372 times within the time window 400ms to 100ms prior to the
drawing component that was illustrated in Fig. 1.b. 62 of these repetitions are depicted in
Figure 2 (uniformly distributed such that 1 of each 6 is shown). This figure shows one of
the least likely relations between a certain drawing component and a certain pair of neurons
in this recording day.
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Figure 2. Dot display showing occurrences of a frequent inter-spike interval around
occurrences of the drawing component that was shown in Fig. 1.b. Top panel shows
the firing times of unit 8.1. The bottom panel shows the firing times of unit 1.2. Each
linelet represents a single spike. Each of the 62 lines in both panels shows spikes occurred
around the appropriate 62 (chosen out of 372) occurrences of the drawing component, in
which it was preceded at least once by the interval 90-91 ms in the window of 0.4 to 0.1
seconds before the start of the drawing component. Linelets representing spikes which took
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part in the selected interval are colored red. The rasters were aligned on the first spike of
the selected interval. If the same interval appeared twice around the same movement, the
first occurrence was used for alignment, and the other occurrences appear as red linelets
somewhere else along the raster. The time of onset of the drawing is colored blue. Trials
are sorted by increasing delays between the neural intervals and the drawing components.
The gray line in each panel represents the average firing rate considering all 372 common
occurrences.
Are these 372 occurrences random? To assess the probability of chance events we
generated 1000 surrogate spike trains by randomly teetering the time of each spike within
10ms around its real time. For each of these surrogates we used the same idea for counting
all possible intervals between neurons 8.1 and 1.2, which were repeated around the same
drawing component during the same time slice. Similarly, the maximal frequency of these
intervals was taken as a representative of that surrogate. In the example shown in Fig. 2
these 1000 maximal frequencies tended to be significantly smaller than 372. Their mean
and their variance were used to estimate the probability of these 372 repetitions, assuming
normal distribution of counts. For this drawing component and pair of neurons the mean
was 350.678 and the standard deviation was 5.01, yielding a probability of 0.00001.
Is this truly so unlikely? Probably not! The counting process is most probably not
distributed normally, so that assessing probability by the mean and variance may be
misleading. A much more complicated issue involves finding a rare event for all 12
drawing components recorded that day. We analyzed all the 50 possible pairs of neurons
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on that day, all 50 possible time intervals for each pair, and for 7 different time slices
around the start of each of the drawing components. Picking the rarest event out of all
these possibilities should yield a highly unlikely event. Hence, we need to assess the
likelihood of finding such low probabilities when multiple trials are conducted.
4) Statistical Analysis
A solution which can be used to test the null hypothesis that spike times are random within
a window of width W was offered (10, 11). If this null hypothesis is true, then replacing the
time of each spike by a randomly selected time within W around its true time should not
affect any of the statistics extracted from the spike times. To use this idea we need to
describe the entire set of relations between firing intervals and drawing by one statistic. To
do so we defined a statistic based on the ten least likely relations between pairs of spikes
and any of the drawing components. We termed this the relations-score. For each
recording day we computed a relations-score for the actual data. Then we randomly
teetered all spike times within some time window W and recomputed it for the teetered
data. Teetering was done 5000 times independently and a histogram of the relation-scores
for teetered data was constructed. Figure 3 illustrates this histogram for W=10ms. Note
that all computations of relations-score for each teetered data were done de novo by the
same process as was done for the actual data (including multiple trials of all possible
drawing components, pairs of neurons, time intervals and time slices).
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Figure 3. Distribution of relations-scores for surrogate spike trains and the actual
data. Five thousand surrogate spike trains were independently generated by teetering spike
times within 10ms. For each of these a relation-score was extracted. The distribution of
these relations-score values was estimated by a histogram. The actual data had a value of
106.37 (arrow). None of the 5000 surrogate trains had a value above it. Hence the p-value
for the actual data was estimated as less than 1/5000.
This method was used to estimate the probability p of the relations-score value of the actual
data. From this p we derived the surprise-value which was defined as the value of –
log2(p). Figure 4 shows the surprise values for one recording session, obtained when spike
times were teetered within different windows between 2ms and 8ms. Clearly, teetering
within 3ms already had a significant effect on the surprise value. Thus the spike times of
the cortical neurons are accurate within 3ms.
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Figure 4. Surprise values for different teetering windows. Abscissa is the teetering
window, ordinate is the surprise value. The horizontal line shows the surprise value for
significance of 0.05. Thus, teetering within 3ms already had a significant effect.
Significant relations-score values were observed for all 3 recording days in this study. The
smallest windows W producing significant results were 3ms (shown here), 6ms and 12ms
respectively. Note that these represent an upper bound for the resolution. One may find a
different statistic that can indicate even higher time precision.
When the same procedure was repeated step by step for the neural data around randomly
selected points in time (instead of time occurrences of drawing components), no significant
surprise values were found (for a window W of 10ms). Thus we only obtained significant
time relations by relating the neural intervals to specific features in the behavior.
Furthermore, no significant surprise values were found when the same procedure was
repeated taking a teetered neural data instead of the original data.
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5) Discussion
Numerous studies have reported precise time relations among spikes in the cortex (11-17).
In some cases it was claimed that the results were attributed to insufficient statistics (1819). In some cases the analysis was limited to precise synchrony, and in some cases the
relations to behavior were not clear.
Our null hypothesis that spike times are not determined within W milliseconds can be
easily tested by teetering. We were able to show that the null hypothesis could be rejected
for W=3ms. However, there are cellular mechanisms and experimental artefacts that may
generate precise spike timing. These include: a. neurons can recover rapidly from the
refractory period; b. spike intervals within a burst may repeat with high precision; c.
periodic activity driven by internal pace-maker processes (not due to network oscillations);
d. dead time for spike detection when recording is made through the same electrode; e.
sharp on or off responses to an external stimulus with abrupt onset (or offset). By
considering only intervals between spikes of units recorded with different electrodes, we
avoided precise intervals that could be ascribed to a., b., c., or d. above. The continuous
drawing motion does not generate abrupt time markers that might be responsible for e.
Examinations of dot displays as given in figure 2 show that the involved spikes (red
linelets) were isolated, not periodic, and not located at the onset of sharp changes in activity
rates.
Figure 2 provides further indications that the relations between the neural interval and the
drawing component were not random or due to trivial artefacts. First, in both panels the
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firing rate is stationary. In this condition, had the red marks been random, the spike density
around these dots marks should have approximated the autocorrelation function which must
be symmetric. However the little troughs on both sides of the peak (relative refractoriness)
are not symmetric. The difference is significant at 0.0002. Second, in dot displays
showing another unlikely relations, the delay between the neuronal component and the
drawing component (blue marks) is not evenly distributed between -0.4 and -0.1 seconds,
as might be expected for chance relations.
One would hope to see 1 to 1 relations between precise firing intervals and drawing
components. This is not found here, probably because the sparse sampling. We record
from a score of neurons from a region containing hundreds of thousands. However, the
relation of our findings to behavior is underscored by the fact that when random time points
(rather than start of drawing components) were chosen, no significant relations scores were
found.
These findings, highlighting the precision of spike times in the cortex, raises three types of
questions concerning their generation, the way they are read out and their use. Previous
research has shown that synfire chains may produce and readout such precise timing (20)
and this property may be used for the dynamic binding of components into a whole (6).
Whether the nervous system takes advantage of these features remains open to debate.
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6) References
1. M. Abeles, Corticonics: Neural Circuits of the Cerebral Cortex (Cambridge Univ.
Press, New-York, 1991).
2. V. Braitenberg, A. Shuez, Cortex: Statistics and Geometry of Neuronal Connectivity
(Springer, Berlin, ed. 2, 1998).
3. H. B. Barlow, Single units and sensation: a neuron doctrine for perceptual psychology.
Perception 1, 371-394 (1972).
4. Y. Ikegaya et al. Synfire chains and cortical songs: temporal modules of cortical
activity. Science 23, 559-564 (2004).
5. C. Malsburg, The correlation theory of brain function. MPI Biophysical Chemistry,
Internal report 81-2 (1981). Reprinted in Models of Neural Networks II (E. Domany,
J.L. vanHemmen and K. Shulten eds.) 95-119 (Springer, Berlin 1994).
6. E. Bienenstock, BrainTtheory: Biological Basis and Computational Properties (A.
Aertsen, V. Braitenberg, Eds.) 269-300 (B. V. Elsvier Sci., Amsterdam, 1996).
7. L. Shastri, V. Ajjanagadde, From simple associations to systematic reasoning. Behav.
Brain Sci. 16, 417-494 (1993).
8. H. Mannila, H. Toivonen, A. I. Verkamo, Discovering frequent episodes in sequences.
KDD 210-215 (1995).
9. H. Mannila, H. Toivonen, Discovering generalized episodes using minimal occurrences.
KDD 146-151 (1996).
10. A. Date, E. Bienenstock, S. Geman, A statistical tool for testing hypothesis about the
temporal resolution of neural activity. Soc. Neurosci. Abstr. 26, 828.6 (2000).
11. N. Hastopoulos, S. Geman, A. Amarasingham, E. Bienenstock, At what time scale does
the nervous system operate? Neurocomputing 52-54, 25-29 (2003).
12. C. M. Gray, W. Singer, Stimulus specific neuronal oscillations in orientation columns
of cat visual cortex. Proc. Natl. Acad. Sci. 86, 1698-1702 (1989).
13. W. Singer, C. M. Gray, Visual feature integration and the temporal correlation
hypothesis. Annu. Rev. Neurosci. 18, 555-586 (1995).
14. A. Riehle, S. Grun, M. Diesmann, A. Aertsen, Spike synchronization and rate
modulation differentially involved in motor cortical function. Science 278, 1950-1953
(1997).
15. M. Abeles, I. Gat, Detecting precise firing sequences in experimental data. J. Neurosci.
Methods 107, 141-154 (2001).
16. Z. Nadasdy, H. Hirase, A. Czurko, J. Csicsvari, G. Buzsaki, Replay and time
compression of recurring spike sequences in the hippocampus. J. Neurosci. 19, 9497507 (1999).
17. A. E. Villa, I. V. Tetko, B. Hyland, A. Najem, Spatiotemporal activity patterns of rat
cortical neurons predict responses in a conditioned task. Proc. Natl. Acad. Sci. 96,
1106-1111 (1999).
18. M. W. Oram, M. C. Wiener, R. Lestienne, B. J. Richmond, Stochastic Nature of
Precisely Timed Spike Patterns in Visual System Neuronal Responses. J. Neurophysiol.
81, 3021-3033 (1999).
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19. S. Baker, R. N. Lemon, Precise spatiotemporal repeating patterns in monkey primary
and supplementary motor areas occur at chance level. J. Neurophysiol. 84, 1770-1780
(2000).
20. M. Abeles et al. Dynamics of neuronal interactions in the frontal cortex of behaving
monkeys. Concepts in Neurosci. 4, 131-158 (1993).
Acknowledgements:
This project was funded in part by grants from ISF, GIF, DIP, Horowitz Fund, and the RICH center. MA is a
Shamoon professor at Bar-Ilan University. TS was responsible for the analysis and was aided by OS. RD
was responsible for the experimental part and was aided by YBS, ZN, and MA. MT overviewed the analysis.
MA overviewed the entire project.
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Attachment A: Methods Overview
A Maccaca fascicularis monkey was trained to hold a two-joint manipulandum and scribble
in the horizontal plane. Continuous movement was reinforced by juice rewards. After
training the monkey, a recording chamber and a head holder were attached to the skull
under full surgical anesthesia in aseptic conditions. The chamber position was verified by
MRI, cortical microstimulation, and histology. After recovery, daily recording of cortical
activity with 8, individually movable, micro electrodes commenced. Monkey handling and
treatment conformed the law and was approved by the institutional ethics committee.
Single unit activity was extracted and isolated on line by MSD (Alpha Omega, Nazareth).
Only shapes which were regarded as well-isolated or at most as a mixture of two spike
shapes were used. Spike times were recorded at a time-resolution of 1ms. The three
recording days with the largest number of isolated units and stationary activity were
analyzed.
Directions of drawing were quantized into 12 sections of 300 each. Similarly, velocity of
drawing was quantized into 5 sections of 10 cm/sec each (0-10, 10-20, … , 40-50).
Transitions of the direction or of the velocity from one section to another were considered
as elementary events. Sequences of such events were revealed by data-mining techniques
(8, 9). One or more elementary events which repeated at least 750 times were considered
as drawing-components. The time interval between the first and the last events
constructing a drawing component was limited to 0.5sec at most.
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The relations-score evaluating the relations between time-coordination of spikes and
drawing components was computed in the following way. For each drawing component D
we marked all times at which D occurred. Around these occurrences and for each pair of
neurons, we looked for the most frequent time interval between the spikes generated by
these neurons. We estimated a p-value for the frequency of this interval (defined by the
two neurons) assuming a normal distribution whose mean and variance were estimated by
another 1000 independently teetered spike trains. The mean and the variance were
computed for every drawing component D and for every pair of neurons using the teetered
spikes. Finally, a relations-score was defined as the sum of the highest ten "–log2(pvalue)" values that evaluated relations consisting of distinct intervals alone.
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Attachment B: Supporting Detailed Material
Our main idea was to look for unknown relations between hand motion and patterns of
spikes. In our analysis we focused on finding relations that involve exact time delays
between spikes. The relations were found using various statistical methods including datamining techniques, and their probability of occurring by chance was evaluated using 5000
surrogate spike trains having the same firing-rate undulations as the actual data.
1. The experiment
Maccaca fascicularis monkeys were trained to hold a two-joint manipulandum and scribble
in the horizontal plane. One out of 19 hexagonal patches which tiled the working space
was randomly selected as a target. When the monkey hit the target, a short beep was
sounded, the monkey obtained a juice reward and the target jumped at random to another
location. Each piece of drawing till a reward was considered a single trial. In this way the
monkey was encouraged to produce continuous movements. After training, a recording
chamber and a head holder were attached to the skull under full surgical anesthesia in
aseptic conditions. The chamber position was verified by MRI. Proximal arm regions were
identified by cortical microstimulation and the border between M1 and PreM by histology.
After recovery, daily recordings of cortical activity with 8 individually movable microelectrodes commenced. After recording for several weeks, the monkey was trained to
alternate between trials of scribbling and trials of a standard center-out motion. Data were
initially stored in the computer memory and after 10 rewards, or one minute with no
reward, were transferred into disk files. Each file only contained trials with similar
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behavior. Single unit activity was extracted on-line by MSD (Alpha Omega, Nazareth).
Only shapes which were regarded as well-isolated or as a mixture of two spike shapes at
most were used. Times of spikes were recorded at a time-resolution of 1ms.
Simultaneously, the manipulandum position was sampled 100 times per second. Monkeys’
handling and treatment conformed the law, were approved by the institutional ethics
committee, and were supervised by a veterinarian.
Here, only data obtained for blocks of scribbling motion were analyzed. We felt that the
statistics would benefit from having long stretches of stationary recording with multiple
neurons. Therefore, we looked for recording days with penetrations in which many cells
showed arm related activity based on intra-cortical microstimulations (train duration 100200 ms, 300 pulses per second each lasting, 0.2ms, amplitude <= 50A) and/or passive
manipulations of arm joints. Only days with at least 400 correct trials were selected. All
isolated units were tested for stability of total spike count in a fixed window ranging from
1000ms before the reward event until the reward. For each unit we used only a continuous
range of files in which its activity was judged to be stable. The results in this article are
based on the 3 recording days which best fit the above criteria. These days contained 8 to
12 isolated units recorded through 4-8 electrodes that showed stationary activity in a range
of 10 to 44 files.
2. Drawing components and neural components
The first goal was finding repeating patterns for each kind of data independently. These
patterns are termed drawing components and neural components respectively.
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2.1. Drawing events and drawing components
In order to find repeated patterns of drawing we used data-mining techniques (1, 2). For
this purpose the continuous drawing must be converted into a sequence of events (drawing
events). In one experimental day there are hundreds of such events. Searching for
repeating sequences of events is greatly facilitated by algorithms of data-mining. A
drawing event was marked as occurring at the time at which a certain drawing-property
changed from one range of values to another. The property itself may be arbitrarily chosen.
For example, it could be defined as a change in the drawing direction from a range of 0o30o to a range of 30o-60o. Other definitions can be based on changes in the curvature or in
the velocity of the drawing. Once a set of criteria for identifying drawing events is defined,
the drawing data is translated into a sequence of these events along the time axis. Then,
data-mining algorithms are activated to detect repeating subsequences in the translated data.
The repeating subsequences are called drawing components. Naturally, different
definitions of the set of criteria lead to different drawing components. Figure 5 illustrates
three drawing components found in one recording day, where two types of drawing events
were used: changes in the direction and changes in the curvature.
In this article, only drawing events which were defined using direction changes and
velocity changes were used. All possible drawing directions were quantized into 12
sections of 300 each. Transitions from one section to another were considered as drawing
events. Similarly, drawing velocity was quantized to 5 sections of 10 cm/sec each.
Sequences of such events which repeat at least 750 times during the recording day and last
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for at most 0.5sec were considered as drawing components. There were 12 to 25 different
drawing components for a given day.
2.2. Neural components
The basic entity of the neural data is a single spike generated at a specific time by a specific
neuron. A neural component was defined as a triple (n1, n2, delta) where n1 and n2 are two
neurons and delta is a time-interval (between spikes generated by these neurons). In this
way, each pair of spikes in the neural data could be interpreted as an occurrence of some
neural component. For the results given in this article, time-intervals between spikes were
limited to 100ms and were quantized to 2ms (i.e. 38ms and 39ms were both interpreted as
the same interval). Hence each pair of neurons yielded 50 distinct neural components.
Spike sorting by shapes that were recorded through a single electrode can result in
confusions. Intracellular properties which may generate precise time intervals can be
confused with precise timing which is generated by the organization of activity in the
network. Therefore, we considered only neural components consisting of two neurons that
were recorded through different electrodes. For example, if we have 3 electrodes
recording spikes from 2 neurons each, there are 30-6=24 valid pairs of neurons (note that
the pairs <n1,n2> and <n2,n1> are different). Altogether, we have 24*50=1200 potential
neural components, some of which may never occur. In the days analyzed there were 1938,
2142 and 6195 such neural components.
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Figure 5. Examples of three drawing components found by data-mining techniques.
Each row represents several occurrences of the same component which are marked by
thicker lines. The drawing components were based on direction changes and on curvature
changes of the drawing.
3. Relations between drawing components and pairs of neurons
Once we had the time occurrences of all the drawing components and the neural
components, we were interested in finding relations between drawing components and pairs
of neurons. For each drawing component A and for each possible pair of different neurons
<n1, n2> we counted the occurrences of each neural component around A that consisted of a
spike from n1 and a spike from n2. In other words, we were interested in the total
occurrences of each relevant time-interval between a spike of n1 and a spike of n2. The
time regions in which we counted these intervals were determined relative to the
occurrences of A by two external parameters Tfrom and Tto. Formally, suppose that during a
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recording day A occurred at {T1,T2,T3,…,Tn}, then the time regions are [Ti+Tfrom, Ti+Tto]
where 1≤i≤n.
Eventually, we defined the support of a relation to be the total number of occurrences of the
most frequent interval. Figure 6 illustrates the count of a specific time-interval around 4
occurrences of a drawing component where [Tfrom,Tto] was [-0.4sec,-0.1sec]. Note that in
practice, the supports of the relations were computed for each [Tfrom,Tto] in { [-1.4,-1.1], [1.2,-0.9], [-1.0,-0.7], …, [-0.2, 0.1] }. At a later stage, the range with the strongest result
was selected for each recording day.
Figure 6. An example for counting neural components around drawing components.
Bottom trace-times show the occurrences of a drawing component A. Vertical linelets
show the times at which neurons 1.2 and 8.0 generated spikes. For each occurrence of A
we look at a window that started 400ms before it and lasted for 300ms (marked in red), and
count the occurrences of each possible neural component (that consists of these two
neurons) whose first spike is inside this window. Thicker blue and green linelets show the
occurrences of a particular neural component in which neuron 8.0 emitted a spike and then,
within 38 to 39 milliseconds, neuron 1.2 emitted a spike. In this illustration, the illustrated
neural component repeats 4 times around A.
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Figure 7 shows appearances of the drawing component that was shown in figure 1.a (in the
article) as well as the spike activity around these appearances. This figure provides further
indications that the relations between the neural interval and the drawing component were
not random or due to trivial artefacts. The delay between the neuronal component and the
drawing component (blue marks) is not evenly distributed between -0.4 and -0.1 seconds as
might be expected for chance relations. Second, in the lower panel the firing rate is
stationary. In this condition, had the red marks been random, the spike density around
these dots marks should have approximated the autocorrelation function which must be
symmetric. However the little troughs on both sides of the peak (relative refractoriness) are
not symmetric. The difference is significant at 0.013.
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Figure 7. Dot display showing occurrences of a frequent inter-spike interval around
occurrences of the drawing component that was shown in Fig. 1.a in the article. The
top panel shows the firing times of unit 8.0. The bottom panel shows the firing times of
unit 1.2. Each linelet represents a single spike. Each of the 63 lines in both panels shows
spikes occurred around the appropriate 63 occurrences of the drawing component, in which
it was preceded at least once by the interval 38-39 ms in the window of 0.4 to 0.1 seconds
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before the start of the drawing component. For all other details about the dot display, see
the legend of figure 2 in the article.
4. Evaluations using teetered neural data
Using the computed support for each relation in a recording day, a statistic called the
relations-score was extracted for this day (details are given in the next section). Intuitively,
relations-score gets larger as the likelihood of the support for the relations decreases. Once
a relations-score was computed for the actual data (denoted by S0), we evaluated its
probability of occurring by chance.
As relations between hand motion and firing rates of neurons have been studied
extensively (3, 4), we wanted to test whether S0 was significantly higher than what we
would expect from random data that have similar firing rates. In order to simulate
random data that preserves firing rates of neurons, we randomly teetered the time of each
original spike within a small window of W milliseconds (5). For example if W=10ms and
the original time of some spike was 125ms, its new time after teetering may be any time
within [120ms,130ms]. Using this technique, we generated 5000 such surrogate spike
trains. Each surrogate train was given a relations-score (S1,S2,S3,…,S5000 respectively)
following exactly the same procedure as for computing S0 (including re-selection of the
best range [Tfrom, Tto] that leads to the largest relations-score). These 5000 values were used
for estimating the probability (denoted by p(S0)) of getting the value S0 by chance. For
example, if only 50 surrogate trains exceeded the relations-score of the actual data, then
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p(S0) was estimated by 50/5000=1%. Table 1 shows the minimal width of the teetering
window for each day that led to a significant value of p(S0).
Day
16jul00
28jun00
27jun00
Minimal width of teetering window
3ms
6ms
12ms
Table 1. The minimal widths of the teetering windows used in each of the three recording
days to obtain a value of p(S0) which was less than 5%. Note that no significant results
were achieved when smaller windows were used, and even higher significant results were
achieved when larger windows were used.
A significant value for this probability indicates that the relations between drawing
components and pairs of neurons (in that day) are damaged as a result of teetering within W
milliseconds. By this we can conclude that around the occurrences of similar behavior,
pairs of spikes tend to prefer some specific time delays.
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5. The computation of relations-score
The computation of the relations-score statistic for a recording day involves the following
steps:
A) Generate a set of 1000 independent teetered neural data (denoted by J1000)
B) Recognize all potential relations between a drawing component and a pair of neurons.
C) For each relation R do the following:
1) Compute the support of R (denoted by Rsupport).
2) Skip R if its support is less than a predefined noise threshold. This step is carried
out for both actual and teetered data in order to prune noisy relations. Note that
the values used for this threshold were 60, 40 and 20 for the 3 different recording
days (depending on the firing rates of the neurons in that day)
3) Compute the support of R for each neural data in J1000 (using the same drawing
component).
4) Compute the mean  and the variance  2 of these 1000 values.
5) Estimate the probability of Rsupport assuming the normal distribution N(  ,  2 )
6) Set Rsurprise to –log2(Prob{Rsupport})
D) Sort all relations by descending order of their computed Rsurprise.
E) If several relations involve the same two neurons and the same interval between them,
delete them all except from the first (the one whose Rsurprise is the highest).
F) Set relations-score by the sum of the Rsurprise values of the first 10 relations.
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Note that step E is done to reduce dependencies between the most unlikely relations that are
used in step F. For example, if the 10 best relations contain a relation that connects a neural
component N to some drawing component C1 as well as a relation that connects the same N
to another drawing component C2, we do not consider both relations for computing
relations-score. The logic beyond this idea is that C1 and C2 may be dependent. Suppose
that C1 is based on direction changes and C2 is based on velocity changes, this dependency
is supported by the two-thirds-power-law.
6. Verifications
Several tests and verifications were made in order to ensure the reliability of the given
results.

We repeated the entire process described in this article in order to evaluate p(S0) on
sham data. These sham data were built by teetering the actual neural data within 10ms
and then using them as the input for the entire algorithm (including re-teetering the
sham data for 5000 times etc…). Note that no significant results were found for any of
the recording days in any of the 7 tested [Tfrom,Tto] windows.

No significant results were found when we invented the occurrence times of each
drawing component. For this operation, if a drawing component i repeated Ni times,
we selected Ni random times, and repeated the entire procedure.

No significant results were found when the time region around the drawing component
was later than the occurrence of the drawing component (i.e. Tfrom>0).
27

We observed that the same significant results can be achieved using much smaller
number of teetered data. This fact may indicate a small variance in the estimation of
p(S0).
7. References
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2. H. Mannila, H. Toivonen, Discovering generalized episodes using minimal occurrences.
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3. Evarts, E.V. Pyramidal tract activity associated with a conditioned hand movement in
the monkey. J. Neurophysiol. 29, 1011-1027 (1966).
4. A. P. Georgopoulos, J. F. Kalsaska, R. Caminity, J. T. Massey, On the relations
between the direction of two dimensional arm movement and cell discharge in primate
motor cortex. J. Neurosci. 2, 1527-1537 (1982).
5. A. Date, E. Bienenstock, S. Geman, A statistical tool for testing hypothesis about the
temporal resolution of neural activity. Soc. Neurosci. Abstr. 26, 828.6 (2000).