Proceedings of a Workshop on Statistics on Networks (CD-ROM) (2007) / Chapter Skim
Currently Skimming:
Visualization and Scalability--Characterizing Brain Networks with Granger Causality--Mingzhou Ding, University of Florida
Visualization and Scalability--Characterizing Brain Networks with Granger Causality--Mingzhou Ding, University of Florida
Pages 375-395
The Chapter Skim interface presents what we've algorithmically identified as the most significant single chunk of text within every page in the chapter.
Select key terms on the right to highlight them within pages of the chapter.
From page 375... ...
Visualization and Scalability 375
|
From page 376... ...
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.
|
From page 377... ...
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.
|
From page 378... ...
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.
|
From page 379... ...
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.
|
From page 380... ...
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.
|
From page 381... ...
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.
|
From page 382... ...
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.
|
From page 383... ...
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.
|
From page 384... ...
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.
|
From page 385... ...
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.
|
From page 386... ...
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.
|
From page 387... ...
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.
|
From page 388... ...
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.
|
From page 389... ...
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.
|
From page 390... ...
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.
|
From page 391... ...
Specifically, Figure 19 shows the sensorimotor network that is published by 1991 by Felleman and Van Essen on the macaque monkey's brain.
|
From page 392... ...
Spectral versions of conditional Granger causality have been developed recently. We use the spectral version to test our hypothesis in Figure 20.
|
From page 393... ...
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.
|
From page 394... ...
It, in fact, reflects anatomy-constrained neural communications between two areas. FIGURE 23 Thank you very much.
|
From page 395... ...
2003. "Beta Oscillations in a Large-Scale Sensorimotor Cortical Network: Directional Influences Revealed by Granger Causality." Proc.
|
Key Terms
This material may be derived from roughly machine-read images, and so is provided only to facilitate research.
More
information on Chapter Skim is available.