# On why a few points suffice to describe spatiotemporal large-scale brain dynamics

###### Abstract

An heuristic signal processing scheme recently introduced shows how brain signals can be efficiently represented by a sparse spatiotemporal point process. The approach has been validated already for different relevant conditions demonstrating that preserves and compress a surprisingly large fraction of the signal information. In this paper the conditions for such compression to succeed are investigated as well as the underlying reasons for such good performance. The results show that the key lies in the correlation properties of the time series under consideration. It is found that signals with long range correlations are particularly suitable for this type of compression, where inflection points contain most of the information. Since this type of correlation is ubiquitous in signals trough out nature including music, weather patterns, biological signals, etc., we expect that this type of approach to be an useful tool for their analysis.

###### pacs:

In the analysis of complex spatiotemporal patterns, such as large scale brain dynamics, an important challenge is the adequate coarse graining of the data. In the case of brain imaging the dataset is composed of several thousand time series, of the so called BOLD (“blood oxygenated level dependent”) signal, covering the entire brain. The usual question in this analysis revolves around the detection of burst of correlated activity across certain regions, which requires extensive computations, in part due to the usually humongous size of the data sets.

Recently it was uncoveredT1 (); T2 (); T3 (); caballero (); allan () that these type of problems can be efficiently analyzed using only the timings of the peak amplitude signal events, i.e., a point process (PP). Subsequent work using similar approachesli (); liu1 (); liu2 (); chen (); jiang (); Amico (); Wu () further confirmed that the method entails a large compression of the original signals. Overall these findings not only suggest a way to speed up computations, but most importantly highlight the need to clarify which aspects or features of the brain imaging signals contain the most relevant information.

The present work is dedicated to clarify the reasons underlying the effectiveness of this approach. The results show that the key lies in the correlation properties of the time series under consideration. In synthesis, it is found that signals with long range correlations are particularly suitable for this type of compression, where inflection points contains most of the information. The results applied as well to other signals from any origin as long as their correlation features are similar.

Figure 1 summarizes the basic process that has been used inT1 (); T2 (); T3 () to define the point process in brain signals. The data consists in time series representing the activity of one of many thousands small brain regions, recorded from the brain using functional magnetic resonance imaging (fMRI). This imaging technique measures in each small region a “blood oxygenated level dependent” signal (i.e., “BOLD”), that is an estimation of the blood’ saturation of oxygen, which itself is proportional to the local neuronal activity. As shown in the figure, time points are selected at the upward threshold (here at unity) crossings of the signal (filled circles). The point process can be constructed also by selecting the local peaks of the BOLD time series. The temporal co-occurrence of the points defines the co-activation matrix (bottom graphs) which can be further averaged to estimate the correlation matrix of the system under study.

It has been established already, in different circumstancesT1 (); T2 (); T3 (), that the co-activation matrix obtained with the PP methods is very similar to the correlation matrix computed from the full (i.e., continuous) BOLD signal. Since this implies a large compression, the question is why a few points are enough to compute results similar to those obtained with the full signal. Figure 2 shows an example constructed from BOLD time series from an experiment in which the subject is resting T3 (). The results demonstrate that as few as 4 points are already sufficient to define clusters of co-activation, as demonstrated previously in T1 (); T2 (); T3 (). In addition, the results here show how de-activations (i.e., blueish colors) are also evidenced by the PP approach.

A simple visual inspection of the BOLD traces reveals that the type of signals we are dealing with are temporally correlated. This is very well known, the neuronal activity is temporally and spatially correlated, and furthermore the activity is convoluted by the hemodynamic transfer function which in itself introduce additional temporal correlations. Therefore, for any time series with that properties, it seems natural to think that the most informative points are those in which its derivative changes sign. The rest of the points are redundant, since they can be predicted, up to a degree, by a linear estimator.

This is illustrated in Figure 3 using as an example two minutes of BOLD recording (normalized by its standard deviation (S.D)). After setting a threshold the inflection points larger than a given value are identified. These points constitutes the marked point process in question. Now we ask how much of the raw signal is left out if these points are used to extrapolate a piece-wise linear time series. To answer that we analyze BOLD time series from the brain of a subject during an experiment in which fMRI data is collected at restT2 (). We proceed to compute the linear correlation between the two time series, the raw and the piece-wise linear one. In panels B and C are shown the results for different values of threshold (in units of S.D.) as well as for the correlation of the time series, estimated by the value of the first autocorrelation coefficient . Panel D shows that as the BOLD signal’ autocorrelation increases the similarity between the piece-wise linear and the raw signals increases, evaluated in two ways: by the error and the linear correlation between both time series. As expected, the raising of the threshold from zero (i.e, less information from the signal is considered) it is followed by a monotonic increase of the and a decrease of the values (see Panel E).

According with the present hypothesis, the functional dependences shown by the BOLD signals in Panel D and E shall be replicated by using synthetic signals with similar autocorrelation properties. For that we generate artificial time series with autocorrelation values identical to those of the BOLD signals using the routine `_alpha_gaussian.m }`

rom MATLAB. Panels B and C show that the dependence with the threshold and exhibit very similar behavior.
The results show that the key lies in the correlation properties of the time series under consideration. In synthesis, it is found that signals with long range correlations are particularly suitable for this type of compression, where inflection points contains most of the information. The results shall apply as well to other signals from any origin as long as their autocorrelation features are similar.

In summary, the success and the merits of the PP approach to represent spatiotemporal dynamics are related to a very trivial fact: in the case of autocorrelated signals the only informative points are those with zero derivative (inflection points); remaining ones are more or less straight lines which can be in principle, and for certain applications, ignored. Applications of these ideas to a diversity of fields should be expected.

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