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Visualization and Scalability
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Characterizing Brain Networks with Granger Causality
Mingzhou Ding, University of Florida
DR. DING: Thank you for coming to this last session. It takes a lot of will power to
stay on. I also belong to the neural club of this workshop, and I’m going to talk a little bit about
brain networks and how to characterize that with Granger causality. The idea of Granger
causality has been around for many years, and I will talk about our attempt at using this method
to assign directions to interactions between neural ensembles. So far in this workshop we have
heard a lot of graph theory, but the examples tend to be undirected graphs. So, hopefully, I’m
going to present some directed graphs through my combination of this method with the brain.
We’ve heard a lot of excellent neural talks so far; therefore, the groundwork has already
been laid for me. What I’m interested in is to put multiple electrodes in the brain and measure
signals from various parts simultaneously. The particular measurements that we look at are local
field potentials, as was the case for several previous talks at this meeting. This local field
potential is also recorded with an electrode directly in the brain, but it’s not looking at one
neuron’s activity, it is more like the activity of a population of neurons. People estimate that it
reflects the summed dendritic activity of maybe 10,000 or so neurons. It’s a population variable,
and particularly suited for us to look at network formations during performance of some kind of
experiment or task.
We’ve also heard a lot about rhythms in neural systems. If you look at the signal you
record as a time series and look at the spectrum, very often you see very prominent peaks in
various frequency bands; therefore in this particular study we are going to apply a spectral
analysis framework to our experimental data. Our approaches are very simple; all of them are
very standard statistical methodologies, so I have nothing new to offer in terms of statistical
methodology.
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FIGURE 1
The tools of our network analysis are listed in Figure 1. The first technique we apply is
simply the power spectrum. We collect data from each electrode and look at its power spectrum
to identify whether there are any features like peaks in a band. Next, we try to find out which if
any of those electrodes show similar features; for example, whether there are 10-hertz peaks at
multiple recording sites. In the second step, we look at their statistical correlation by computing
coherence to see whether those 10-hertz peaks measured in different parts of the brain actually
belong to the same oscillatory network.
The network I’m talking about here is defined in a statistical sense. That is, data streams
that are statistically co-varying during task performance are said to belong to a network.
Sometimes this is called a functional network. This is in contrast to another way of talking about
brain networks, which is anatomical networks, namely, how different parts of the brain are
actually wired together, linked together through physical cables (axon projections). I also want to
stress the indispensable relation between the two networks. From a graph theory perspective, our
first step will allow us to identify similar features at different nodes, and the coherence will allow
us to draw edges linking together nodes to indicate that activities at these different locations are
co-varying at certain frequencies. The result is an undirected graph. Sometimes when you look
at that graph you still don’t really know the relative contributions of different nodes in this
network, or in this case different brain areas.
The third step is more or less new in the neuroscience business, although it has been
around for many years. Through Granger causality we are trying to introduce directions among
different brain areas during a given task performance. Through those direction assessments you
can sometimes get a better view of how different areas are organized together to produce
meaningful behaviors that we experience every day.
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The final tool we use is a moving window approach. This just allows us to move through
our experimental recordings to look at various states in the brain because they change rapidly
during cognitive performance. We want to look at the temporal evolutions of that.
A very quick introduction to Granger causality is given in Figures 2 and 3. This will be a
familiar topic to the statisticians in the audience. Basically, you have two simultaneous time
series recordings, X and Y, from which you build regressive predictors. You are predicting the
current value of X based on a linear combination of a set of X measurements that has occurred in
the past where the linear coefficients are selected in some optimal sense. When the actual value
comes in you can look at the difference to see how good your predictor is compared to the real
value. The difference is captured in an error term, and the magnitude of this error term is a
signature of how good your predictor is. Obviously, the smaller it is the better your prediction.
FIGURE 2
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FIGURE 3
Then you try to see whether using the previous measurements of the X process, in
combination with Y’s previous measurements, produces a better prediction or not. In that sense
you are looking at the error term one more time to see whether introducing the other measurement
actually improved this error term. If that is true, then in some statistical sense we say Y has a
causal influence on X. You can then reverse the role of the X and the Y to examine whether there
is an influence in the other direction and assess the directional influence between these two time
series.
The idea of Granger causality came about in the early 1950s in an article written by
Wiener. He talked about the prediction of time series. That’s where he first proposed this notion,
but his article is very thick and full of measure theory; it’s unclear how you could implement this
in practice. In 1969 Clive Granger implemented this whole thing in a linear regression form as
you see in Figure 1, and that led to a lot of applications in economics. Recently we and other
people have begun to take this and try to apply it to our study of the brain. In 1982 Geweke
found a spectral version of the previously introduced Granger causality, as indicated in Figure 4.
Note that what we have seen earlier is a time-domain version. We also like to have spectral
representations, because we want to study the rhythms in the brain. In this regard we found the
spectral version by Geweke. He was able to have a very revolutionary way of looking at
decomposing the interdependence between time series. The important thing he did was to prove a
theorem that if you integrate this spectral version over frequency you get a time-domain version
back. This relation alone shows that there is something right, so we will call this Granger
causality spectrum.
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FIGURE 4
Geweke’s basic idea is this. Given two signals, X and Y, you take X signal’s power
spectrum at a given frequency f and then ask whether it can be decomposed into two parts: an
intrinsic part and a causal part. The first of these is due to some kind of intrinsic dynamics, while
the other is due to the causal dynamics from the Y signal. After performing this decomposition,
Geweke defined the causality as a logarithm of the ratio of total power to the intrinsic part of the
power. Therefore, if the causal part is 0 (i.e., all of the power is in the intrinsic part), this ratio is
1 and the log of the ratio—the causality—is 0. Therefore, there is no causal influence. On the
other hand, if the causal part is greater than 0, the ratio is greater than 1 and the log of that ratio
will be greater than 0. You do a statistical test on whether this quantity is 0 or not.
We began to work with this set of measures about six years ago. We tested various cases
trying to convince ourselves this thing is working, and it turns out we are quite convinced. We
then began to apply it to experimental data. Today I’m just going to mention one experiment
related to the beta oscillation in the sensorimotor cortex of behaving monkeys. Some of the
results below have been published in Proceedings of the National Academy of Sciences in 2004.
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FIGURE 5
Figure 5 introduces you to the experiment itself. It’s a very simple standard pattern
discrimination go/no-go experiment. The monkey comes in, sits down, and when he feels ready
to do the experiment he extends his hand and depresses a mechanical lever, depressing it and
holding it steady. The depression of the mechanical lever triggers the electronics, and then the
experiment commences. After a little while a stimulus will show up on the screen. He looks at
that screen. The stimulus is a group of 4 dots in one of 4 configurations: right slanting line, left
slanting line, right slanting diamond, of left slanting diamond, as shown in the bottom right of
Figure 5. The monkey’s job is to tell whether the dots are in a diamond configuration or in a line,
without regard to orientation. If he is able to find out which pattern he is seeing he will make a
response by indicating that he understood what is going on. The response is by releasing the
lever if he sees one pattern, by holding onto the lever if he sees the other pattern. That’s the way
we know that he really understands the situation. In this case the response itself also has two
types. One is a go response in which he lifts his hand from the lever, and the other one is a no-go
response in which he does not move. This becomes very important for us later on when we try to
understand our analysis.
The experiment was done with four monkeys, but today I’m going to report on the
analysis of two. Figure 6 shows the location of the electrodes, with each dot representing one
electrode insertion. You can see that the electrodes were distributed over many different parts of
one hemisphere. That hemisphere is chosen based on the handedness of this monkey; if the
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monkey is a right hander you record from the left brain because the hand is contralaterally
controlled. If it’s a left hander you do the right brain.
FIGURE 6
Bipolar recordings were used, meaning that two electrode tips are inserted into each
location, one tip at the surface and the other at some depth. The difference between these two tips
becomes the signal we amplify and then later analyze. It is important for us to take the difference
so that the far field effect could be subtracted out. Likewise, the volume conduction and so on
could also be subtracted out. Therefore, we are reasonably confident that the signals we are
getting from these electrodes are indeed reflecting local activities. If you want to look at
networks you don’t want to have a huge common influence (e.g., far-field effect) on the analysis
because that will make your analysis futile.
I want to show you some of the typical time series we get. Before that I want to comment
on the mathematical framework we use to model these time series. When we study stochastic
processes in the classroom we know that everything is defined first and foremost in terms of
ensembles, but in the real world, for instance in economics, you get only one realization. Here
you can have many realizations if you continue to do the same experiment over and over again.
This is precisely the case in neurobiology. Figure 7 shows the overlaying of time series from
many trials—that’s over 400 trials superimposed on the same plot. In the traditional studies
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people take all these trials and perform an ensemble average. This is the white curve you see; it’s
called the even-related potential, ERP. Traditional studies only study this average curve, and
that’s all. All these jiggly lines in Figure 7 are averaged away, branded as noise and then just
forgotten. What we are going to do is bring them back and treat them as our signal.
FIGURE 7
Time 0 in Figure 7 indicates the time when stimulus appears on the screen. The time to
the left of that is when the monkey has already depressed the lever and is looking at the screen,
getting ready to perform the experiment. Our first analysis was over this early period of the data,
from -100ms to 20ms. During this period, the monkey is already prepared to do the job but the
stimulus has not yet appeared, because the signal takes some time to transmit to the brain to
trigger a response. This time period is sometimes referred to as the ongoing period, prestimulus,
or fore period. We treat the data as coming from a stochastic process and each trial as one
realization. Then we begin to fit the parametric models and do the signal analysis.
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FIGURE 8
Among all the electrodes we simultaneously recorded, we first did a blind network
analysis to see which time series correlates with which other and in what frequency range.
Interestingly, two segregated networks popped out, as shown in red and blue in Figure 8. One is
located in the sensorimotor area, and the other is in the visual system. Today I’m going to spend
time analyzing the former, which is the most interesting one and the stronger of the two.
This network in the sensorimotor area is synchronized at about 20 hertz. That is, the
nodes tend to communicate at about 20 hertz, and therefore it’s a beta range oscillation. Among
neuroscientists there is an approximate taxonomy to describe the different frequencies: a
frequency in the range 1-3 hertz is called a delta wave, 3-7 is a theta wave, 7-14 is an alpha, 14-
25 hertz is a beta wave, and higher frequencies are called gamma waves. We have heard other
talks at this workshop dealing with gamma waves. But we are looking at a subclass of beta
oscillations in the present study. Figure 9 shows the averaged power spectrum from all these
locations; at each location we compute a power spectrum and then average them. The first plot in
the top row shows the averaged power spectrum for the two different monkey subjects. Very
clearly, they both show energy concentration in the 20 hertz range. The next plot in that row
shows coherence results averaged together from the two different monkey subjects, and again we
can see clear synchronized activities in the beta range, meaning that the signals are co-varying
also in this 20 hertz range, that is, local oscillations also appear to be the oscillations that link all
the sites together. They bind them together into one network.
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FIGURE 9
The top-right plot in Figure 9 shows our Granger causality result, which also shows quite
clearly the concentration of Granger causal influences in the 20 hertz range. It appeared to be the
same oscillation underlying the three phenomena. By the way, all the pair-wise Granger
influences 1-2, 2-1, 1-3, 2-3, 3-1 and so on are averaged together for each subject. The bottom
row shows just one pair from one subject. We are looking at the power, coherence and Granger.
The interesting thing is that power spectra showed very clear beta oscillation. Coherence was
very strong, 0.6, in the beta range, indicating that the two sites are strongly interacting but no
directional information is available from coherence. But if you look at the Granger causality, it
becomes immediately apparent that the interaction is unidirectional: one direction is very strong
and the other direction is nearly flat.
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FIGURE 10
What we did next was to draw all the pair-wise Granger plots as shown in Figure 10 on a
graph which we call the Granger causality graph which is a directed graph. A very interesting
thing happens if you look at the primary somatosensory area. It tends to send out influences
while receiving tiny ones. In contrast, the motor cortex receives all the input, and it is sending out
little. In addition, I want to say that we did a significance test. A lot of this passed the
significance test, but the strong ones are an order of magnitude bigger than the little ones.
To aid our understanding of the results, all the big and strong causal influences that are
common to the two subjects are summarized in Figure 11. At the beginning when we began to do
this study we had no idea what this network was actually doing. After we did all this directional
analysis a very clear hypothesis jumped out at us: we believe this network is actually formed to
support the monkey depressing the mechanical level and holding it steady, because holding
something steady is in itself a function. The brain needs to be involved to support that function.
The reasons for this hypothesis are several-fold. First of all, the primary somatosensory areas S1
are sitting center stage in this network, and are sending influences out to the other parts of the
network. Think about this: holding something steady is like feedback control. You are
stabilizing an equilibrium state, and when you are stabilizing an equilibrium state the key
ingredient is the sensory feedback. That is, I need to know how I’m performing right now so that
proper control signals can be exported to maintain that state.
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FIGURE 11
I want to quote a patient study that actually supports our work. In 1982 there was a group
in London that studied a patient who suffered a neuropathy and lost all sensory feedback. But his
motor system was totally intact. That is, the system that allows commands from the brain to
move muscles was completely intact, but he could not feel anything. So if you ask this person to
pick up a suitcase he would have no problem doing that. But if you ask him to hold the suitcase
steady for an extended period of time, he cannot do it unless he looks at this hand. (If he looks at
his hand, he is using visual information to supplement the lack of sensory feedback.) Therefore,
this tells us that sensory feedback is critical in maintaining the steady state.
Area 7B is a very interesting area. We later learned that it is a very important area for
non-visually guided motion. There is a network involving 7B and the cerebellum, which together
form some kind of network that helps the brain maintain goals. When we are making a
movement, the brain first has to have a goal about what to do. So we form a goal and tell the
motor system to execute and fulfill that goal. In control terms, the model is here and an idea of
what is to be achieved is here. This area receives sensory updates from S1, compares that to the
internal model, and exports the error signals to the motor cortex for output adjustment.
The third line of evidence is coming from many other studies people have done in either
humans or monkeys linking beta range neural oscillations and isometric contraction—maintaining
a steady state pressure on some kind of a gauge. People who have studied the motor cortex have
seen beta oscillation in this area very clearly. What we have done is extend the network into
post-central areas. Through the directional analysis we are able to say that the post-central areas
play an important role in organizing this network. Without the arrows we cannot really say who
is playing the organizational role in this whole system, but adding the arrows we are able to
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identify S1 as playing a more central role in organizing this network to support this behavior. At
this point one can make a prediction. Namely, when the monkey makes a movement by releasing
this pressure, this network should dissolve itself and go away. How do we test this hypothesis?
Recall that in this experiment there are two response types, a go response—the monkey sees some
stimuli he recognizes, and he lifts the lever—and there is a no-go response that consists of
holding steady until the very end. Therefore, if this oscillation network is in support of holding
something steady, then we should see this network continuing in the no-go conditions all the way
to the end, while for the go conditions, because the monkey released halfway through it, the
network should disappear halfway through the experiment.
This motivated the next study, which is the moving-window analysis, where you do a
time-frequency analysis with an 80-millisecond window and compare go versus no-go conditions.
Here the result is very clear. Figure 12 shows a time-frequency plot. This is time and this is the
frequency. The magnitude is represented by color. As you can see, for the go trials, there is 20
hertz oscillation at the beginning. The oscillation then disappeared half way through the trial.
This arrow is indicating the mean reaction time. That is the mean time at which the monkey
released the hand, so prior to lever release, the oscillation just terminated. For the no-go trials
this oscillation is maintained to the very end. This is evidence in support of our hypothesis that
the network supports the pressure maintenance on the lever.
FIGURE 12
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FIGURE 13
When we look at the time frequency plot of Granger causality it’s clear that there is
directional driving from S1 to M1 at the beginning (Figure 13). When the network disappeared
the causal influences also stopped. In the no-go trials the causal influences are maintained all the
way to the end. It even becomes stronger in the end. On the other hand, if you look at the causal
influences in the opposite direction in this 20 hertz range in Figure 14 there is nothing going on.
When we see some of this high frequency stuff we really don’t know what this means. It’s not
very well organized but the 20 hertz definitely is quite unidirectional if we compare Figures 13
and 14. Our basic conclusion here is that the time course of these beta oscillations seems to be
consistent with the lever pressure maintenance hypothesis.
FIGURE 14
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At this point I want to make a comment. That is, when people talk about networks in the
brain, there are usually two ways to talk about it. One is functional network like what I’m
discussing here. This is looking at the statistical relationships between different brain areas
during a given task. On the other hand there are also anatomical networks in the brain. One can
say that the anatomical network is there and not changing during this experiment. This
experiment is only half a second long. The anatomical network didn’t change a whole lot yet the
dynamics that is traveling on the anatomical network disappears when the behavior it supports is
no longer there.
FIGURE 15
Therefore, the dynamics are completely flexible. It’s like a highway system, the road is
always there but the traffic pattern can be very different depending on the functions it serves. It’s
the same thing here. The network is in place but the dynamics traveling on that can be changing
rapidly depending on the function it supports. This is an interesting thing.
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FIGURE 16
The next thing is my last result concerning conditional Granger causality. Here I want to
show how the anatomy can inform our functional network analysis. Specifically, this is to show
that neuroanatomy can motivate us to look at a problem that we would otherwise not have
thought of doing.
FIGURE 17
Mathematically, the problem is the following. Consider three time series. Suppose they
interact as shown in Figure 17. A pair-wise analysis, as we have been doing so far, cannot tell
these two patterns apart, because in both cases A drives C from a pairwise perspective. This issue
has relevance for the analysis result reported in Figure 11.
Consider a subset of nodes in Figure 11 which is re-plotted in Figure 18. The arrows
come from our pair-wise analysis. The question is, is the link from S1 to 7a real or is it the result
of pairwise analysis? We care about this question because anatomically there is a reason for us to
lean toward the latter. Specifically, Figure 19 shows the sensorimotor network that is published
by 1991 by Felleman and Van Essen on the macaque monkey’s brain.
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FIGURE 18
As you can see in this diagram 7A is not connected directly to S1. 7A is connected only
to 7B in this network. This suggests the hypothesis that the causal influence from S1 to 7A is
mediated by 7B.
FIGURE 19
To test this hypothesis we need additional analysis ideas. This is accomplished by
conditional Granger causality analysis. Like pairwise Granger it’s also prediction-based. Refer to
Figure 17. If, by incorporating A, we do not get any additional improvement in prediction, then
we say we have the connectivity pattern like that on the left. On the other hand, if by including A
we can still improve the predictor, we say that we have the pattern on the right. Therefore, these
two cases will be distinguished based on these two different types of computation. Spectral
versions of conditional Granger causality have been developed recently. We use the spectral
version to test our hypothesis in Figure 20. The result is shown in Figure 21.
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FIGURE 20
FIGURE 21
The blue curve shows the pairwise result from S1 to 7B which is very clearly above
threshold. If you condition 7B out, then all of a sudden, the causal influence from S1 to 7B drops
below threshold. In other words S1 to 7B appeared to be mediated by 7A.
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FIGURE 22
We then did a control study to rule out the possibility that the result in Figure 21 is a
procedural artifact. We took the same three channels and conditioned 7A out to see the effect.
As you can see in Figure 22 there is hardly any difference between the two curves. This means
that what we see in Figure 21 is not simply a procedure related artifact. It, in fact, reflects
anatomy-constrained neural communications between two areas.
FIGURE 23
Thank you very much.
[Applause.]
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QUESTIONS AND ANSWERS
DR. WIGGINS: Some of what you are doing similar to things that have happened in
genetic networks where people are trying to infer genetic networks from time series. So, just by
points of comparison, one is I’m wondering if Granger tests, which I wasn’t really familiar with,
has been validated on synthetic data? Like people generate synthetic systems.
DR. DING: We have performed extensive test on synthetic data with excellent results.
Regarding genetic networks, I have heard people talk about microarray data. We have never done
any work in this area.
DR. WIGGINS: You might be interested to look at papers from the genetic network
inference literature where they look at mutual information between different nodes, because in
that case you can, with some confidence, eliminate spurious links, like in the triangle you had if
you know for example that the mutual information between A and B is strictly less than either of
the other two. Then you can argue that that's not a direct causation.
DR. DING: Certainly. It would be interesting to look at that. On the other hand, we are
spectral people. We don’t want just pure time domain analysis. There are many reasons with
respect to the nervous system for the spectral bias.
DR. KLEINFELD: Just to make sure I understood the data was recorded with a 14
channel probe?
DR. DING: Each channel analyzed here represents a bipolar derivation. There are 14 or
15 of those.
REFERENCES
Brovelli, A., M. Ding, A. Ledberg, Y. Chen, R. Nakamura, S.L. Bressler. 2003. “Beta
Oscillations in a Large-Scale Sensorimotor Cortical Network: Directional Influences Revealed
by Granger Causality.” Proc. Natl. Acad. Sci. USA 101:9849-9854.
Felleman, D.J., and D.C. Van Essen. 1991. “Distributed Hierarchical Processing in the Primate
Cerebral Cortex.” Cerebral Cortex, 1:1-47.
Geweke, J. 1982. “Measurement of Linear-Dependence and Feedback between Multiple Time-
Series.” J. Amer. Statist. Assoc. 77:304-313.
Geweke, J. 1984. “Measures of Conditional Linear-Dependence and Feedback Between Time-
Series.” J. Amer. Statist. Assoc. 79:907-915.
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