Functional connectivity estimation from

trical activity, practically free of artifacts, directly from the brain structures. In the past, this research was mostly concentrated on the analysis of spike trains; the role of slow cortical activity was mostly neglected. Nowadays, the role of oscillatory activ- ity of neural networks has become more and more recognized. There is a consensus that the higher cortical functions depend on dynamic interplay between spatially dis- tributed multiple cortical regions coupled by the transmission of oscillatory activity, e.g., [Singer, 1993, Buzsaki and Draguhn, 2004].

The most popular method to study the interaction between cortical areas is coher- ence (in spite of the limitations of bivariate coherence). To investigate the dynamics of interactions between brain structures, temporal fluctuations in coherence were studied, e.g., by [Bullock et al., 1995], and oscillatory synchrony was considered by [Tallon-Baudry, 2003]. Recently attention has focused on the directionality of interactions between brain regions. Among the methods used for the study of direc- tionality, those based on Granger causality seem to be the most appropriate.

For the analysis of signals recorded from specific brain structures it is important to determine the direct interaction. Direct directed transfer function (Sect. 3.3.2.3.1) was introduced by [Korzeniewska et al., 2003] for the purpose of investigating the propagation of LFP recorded from electrodes chronically implanted in the brain of a behaving animal. The patterns of propagation were recorded from four brain struc- tures involved in processing of emotions. The patterns of transmission were studied for an animal walking on a runway and for walking on a runway accompanied by a stressful stimulus (bell ringing). More flows between the brain structures appeared in the case of the stressful stimulus and most of them were reciprocal.

The investigation of the intracranial human EEG is limited to cases of patients who are being prepared for surgical intervention. The limitations in the iEEG re- search are caused by the fact that the electrode placement is dictated solely by clin- ical concerns. However, the area covered by electrodes is usually not confined to the location of the diseased tissue because this location is not precisely determined before implantation and also the neighboring areas have to be checked to be sure

surgery will not seriously disturb sensory, motor, or cognitive functions. iEEG pro- vides unprecedented opportunity for studying indices of cortical activation, since it is characterized by high temporal resolution and mesoscopic spatial resolution that is intermediate between the macroscopic scale of EEG/MEG and multiunit recording of neuronal activity.

iEEG allows for investigation of high-gamma activity, which is hard to observe us- ing scalp electrodes. Causal interaction between signals in high-gamma range (above 60 Hz) for the word repeating task were studied by [Korzeniewska et al., 2008]. The authors introduced a new measure—short-time direct directed transfer func- tion (SdDTF), which combined the benefits of directionality, directedness, and short- time windowing. This function appeared to be an effective tool for analyzing non- stationary signals such as EEG accompanying cognitive processes. The performance of the function was tested by means of simulations, which demonstrated that the SdDTF properly estimates directions, spectral content, intensity, direct causal inter- actions between signals, and their time evolution. To evaluate event-related changes in SdDTF, that is, event-related causality (ERC), a new statistical methodology was developed to compare prestimulus and poststimulus SdDTF values. This procedure is described inSect. 4.1.7.3.7.

In order to quantitatively describe the transmissions, ERC for high gamma activ- ity was integrated in the 82–100 Hz range, which was empirically derived based on the mean ERC over all time points and all pairs of analyzed channels.Figure 4.24 shows the magnitudes of the interaction in the form of arrows of different widths. The most prominent identified connections involved: in the first phase (listening) flows from the auditory associative cortex to mouth/tongue motor cortex, and in the second phase (repeating of the word) propagation from the Brocas area (responsible for speech) to mouth/tongue motor cortex.

One of the problems important for neuroscience is the relation between the spike trains and local field potentials. For the spike train evaluation the methods developed in the field of point processes analysis are customarily applied [Brown et al., 2004]. Nevertheless there is a possibility of also using the broad repertoire of stochastic continuous signal analysis methods, described in this book, to spike trains.

In the approach proposed by [Kocsis and Kaminski, 2006] the spike trains were processed in the following way: the spikes were low-pass filtered by an order 1 But- terworth filter with cutoff frequency at 10% of Nyquist frequency (the filtering pro- cedure was applied as zero phase filter; Sect. 2.1); then 10% of stochastic noise uncorrelated with the signal was added in order to make the spike train better match the stochastic character of the AR model. The procedure is illustrated inFigure 4.25. The described approach was used in the experiment where LFP was recorded from hippocampus and spike trains from the supramammilliary nucleus (SUM) of a rat with the aim of finding the dynamic coupling between the structures in a situation when the sensory stimulus was applied. The MVAR model was fitted to the spike signal from SUM (approximated in the above described way) and the hippocam- pal LFP; then the SDTF functions were estimated. The temporal dynamics of the direction of influence revealed sharp reverses in the direction of the theta drive in association with sensory-elicited theta rhythm. It was found that in this situation the

FIGURE 4.24: (SEE COLOR INSERT) Integrals of ERC for frequency range 82–100 Hz calculated for three stages of an auditory word repetition task. (a) Audi- tory perception stage, (b) response preparation, (c) verbal response. Arrows indicate directionality of ERC. Width and color of each arrow represent the value of the ERC integral. Color scale at the left. For clarity, only integrals for event-related flow in- creases are shown. From [Korzeniewska et al., 2008].

FIGURE 4.25: Transforming spike train into continuous signal. A, LFP recorded from hippocampus, B, standardized spike train, C, low-pass filtered spike train, D, low-pass filtered spike train with 10% noise added. From [Kocsis and Kaminski, 2006].

subpopulation of SUM neurons contains information predicting future variations in the LFP rhythm in hippocampus. In contrast, during slow spontaneous theta rhythm it was the SUM spike train that can be predicted from the hippocampal activity.

The described procedure of turning the spike trains to the continuous signals opens new perspectives of application of the methods of signal processing to point pro-

cesses. In particular this approach may be useful for spike train analysis and for the investigation of the relations between point processes and continuous signals. 4.1.7.3.7 Statistical assessment of time-varying connectivity Functional con- nectivity is expected to undergo rapid changes in the living brain. This fact should be taken into account when constructing the statistical tool to test the significance of the changes in the connectivity related to an event. Especially, it is difficult to keep the assumption that during the longer baseline epoch the connectivity is stationary. ERC involves statistical methods for comparing estimates of causal interactions during pre-stimulus baseline epochs and during post-stimulus activated epochs that do not require the assumption of stationarity of the signal in either of the epochs. Formally the method relies on the bivariate smoothing in time and frequency defined as:

Yf,t= g( f ,t) + εf,t (4.34)

where g( f ,t) is modeled as penalized thin-plate spline [Ruppert et al., 2003] rep- resenting the actual SdDTF function and εf,t are independent normal random

(N(0,σ2

ε)) variables. The thin-plate spline function can be viewed as spatial plates joined at a number of knots. The number of knots minus the number of spline pa- rameters gives the number of the degrees of freedom.

To introduce the testing framework proposed in [Korzeniewska et al., 2008], we first introduce some notations. Denote by f1,..., fmthe frequencies where fmis the

number of analyzed frequencies. Also, denote by t= t1,...,tnthe time index cor-

responding to one window in the baseline, where tnis the total number of baseline

windows. Similarly, denote by T = T1,...,Tnthe time index corresponding to one

window in the post-stimulus period, where Tn is the total number of post-stimulus

windows.

The goal is to test for every frequency f , and for every baseline/stimulus pair of time windows(t,T ), whether g( f ,t) = g( f ,T ). More precisely, the implicit null hypothesis for a given post-stimulus time window T at frequency f is that:

H0, f ,T: g( f ,t1) = g( f ,T ) or g( f ,t2) = g( f ,T ) or ,...,g( f ,tn) = g( f ,T ) (4.35)

with the corresponding alternative

H1, f ,T: g( f ,t1) = g( f ,T ) and g( f ,t2) = g( f ,T ) and ,...,g( f ,tn) = g( f ,T ) (4.36)

To test these hypotheses a joint 95% confidence interval for the differences

g( f ,t) − g( f ,T ) for t = t1,...,tn is constructed. Let ˆg( f ,t), ˆσ2g( f ,t) be the penal-

ized spline estimator of g( f ,t) and its associated estimated standard error in each baseline time window. Similarly, let ˆg( f ,T ), ˆσ2

g( f ,T ) be the penalized spline esti-

mator of g( f ,T ) and its associated estimated standard error in each post-stimulus time window. Since residuals are independent at points well separated in time, the central limit theorem applies and we can assume that for every baseline/stimulus pair of time windows(t,T )

( ˆg( f ,t) − ˆg( f ,T )) − (g( f ,t) − g( f ,T ))_ ˆσ2

g( f ,t) + ˆσ2g( f ,T )

approximates a standard normal distribution. A joint confidence interval with at least 95% coverage probability for g( f ,t) − g( f ,T) is:

ˆg( f ,t) − ˆg( f ,T ) ± m95 

ˆσ2

g( f ,t) + ˆσ2g( f ,T ) (4.38)

where m95is the 97.5% quantile of the distribution

MAX(tn,Tn) = maxt∈{t1,...,tn},T∈{T1,...,Tn}, f ∈{ f1,..., fm}|Nt,T, f| (4.39)

where Nt,T, f are independent N(0,1) random variables. This is equivalent to apply-

ing the Bonferonni correction for tnTnfm tests to control the family-wise error rate

(FWER).

The utility of the ERC approach was demonstrated through its application to hu- man electrocorticographic recordings (ECoG) of a simple language task [Korze- niewska et al., 2008]. ERC analyses of these ECoG recordings revealed frequency- dependent interactions, particularly in high gamma (>60 Hz) frequencies, between brain regions known to participate in the recorded language task, and the temporal evolution of these interactions was consistent with the putative processing stages of this task.