16th Mediterranean Conference on Control and Automation Congress Centre, Ajaccio, France June 25-27, 2008
From Synchronization to Network Theory: A Strategy for MEG Data Analysis Maide Bucolo Member IEEE, Federica Di Grazia Student Member IEEE, Mattia Frasca Member IEEE, Francesca Sapuppo Member IEEE, David Shannahoff-Khalsa. Abstract—The functional connectivity of the brain has been studied here using knowledge from two different scientific fields. The methods of Synchronization Likelihood and network theory are applied to magnetoencephalography (MEG) data in an effort to analyze the brain as a complex network. These studies show an interesting small-world phenomenon in functional connectivity. Network and head-map images of the results are presented.
I. INTRODUCTION The dynamics of neurophysiologic systems and the temporal and spatial variability of neural signals generated by the cortex make the analysis of brain activity a challenge that is similar to other complex systems. Numerous scientific discoveries, especially in physics, have led to the development of new techniques for the measurement and characterization of cerebral physiological parameters [1,2]. However, despite this rapid increase in knowledge, the study of brain processes and their relationship to higher brain function [3] remain obscure. This gap has inspired a search for novel methods for studying the brain in fields that are beyond the neurosciences. Work in mathematics and physics have contributed to the study of complex systems [4-6]. A method used here is based on the modern theory of networks derived from graph theory [7]. It is combined here with the concept of interactions between dynamical systems that have been commonly quantified using linear techniques, eg. Coherency [22,23], and other methods [24,25] like “synchronization likelihood” (SL) [9]. Here we employ the SL method, which is based on the statistical interdependence of brain activity signals, as an index of "functional connectivity" [8] for studying MEG data. The MEG signals used here were obtained using a 148channel whole-head MEG instrument. Signals were recorded from a subject during a yogic breathing exercise that has been used to successfully treat obsessive compulsive disorder (OCD)[27]. The matrices of pair wise SL values, representing the Manuscriptum received 31.01.08 M. Bucolo, F. Di Grazia, M. Frasca, F. Sapuppo are with the Dipartimento di Ingegneria Elettrica, Elettronica e dei Sistemi, Università degli Studi di Catania, V.le A, Doria 6, 95125 Catania, Italy. (e-mail:
[email protected]). D. Shannahoff-Khalsa is with the Institute for Nonlinear Science, University of California, San Diego, La Jolla, California, USA. (email:
[email protected])
978-1-4244-2505-1/08/$20.00 ©2008 IEEE
connectivity between couplets of MEG channels, have been converted to un-weighted graphs and opportunely represented by networks. These networks were then characterized by parameters like the cluster coefficient, path length L and degree, with an effort to investigate small world like structures. These parameters reflect an optimal situation associated with rapid synchronization and information transfer, minimal wiring costs, the balance between local processing and global integration, all of which are important for synchronizing the neural activity between different brain regions [12]. Anatomical studies suggest that neural networks, ranging from the central nervous system of C. elegans [6] to cortical networks in the cat and macaque, may be organized as small-world networks [29]. Lago-Fernandez et al. showed [11] that neural network models with small-world structures facilitate a fast system response and the emergence of coherent oscillations. Sporns and Tononi used a genetic algorithm [13] to select artificial neural networks with maximal entropy, integration or complexity. The aim of this work is to investigate the possibility of finding the small-world architectures in functional connectivity. Therefore the starting point of this study is not the neural network, but the MEG data. The architectures emerge from the data analyses. Here we present the work in four sections: a description of the case study; an introduction of the SL method and the modern network theory underlying the small-world phenomenon; the results; and the conclusion. II. CASE STUDY Recordings were made using a whole-head 148-channel MEG instrument (4-D Neuroimaging, San Diego, California) located at The Scripps Research Institute (La Jolla, CA). Each of the 148 pick-up coils in this instrument is a 2-cm diameter magnetometer, with a distance of 2.2 cm between coils, center to center. Each coil is connected to a SQUID that produces a voltage proportional to the magnetic field radial to the head, resulting in preferential sensitivity to neural electrical sources tangential to the surface of the scalp emanating from cortical sulci. This MEG system is contained in a magnetically shielded room that helps reduce the contribution of magnetic fields from more distant sources, and this significantly increases the signal-to-noise ratio and improves the ability to detect deeper signal sources in the brain. Trained MEG technicians
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positioned the subject, applied electro-oculogram leads, and performed head shape digitization. An individual was employed who is highly trained with yogic breathing techniques and in being a subject for MEG studies. Headshape was digitized, based on known locations on the subject's head (tragus of left and right ears and nasion). Head shape data is for later co-registration between measurement coil locations, electrode locations, and scalp landmarks (Figure 1 (a),(b)). Eye movements were recorded with electrodes placed above and below the right eye. Electrode impedances were set below 5 kohms. MEG data was recorded with a sampling rate of 251 Hz, with an analog filter band pass of 1 to 100 Hz.
Dataset Reduction. The computational burden of the entire data set (the dynamics of a single channel consists of 15,240 samples per minute) has led to data analysis reduction in both the space and time domain. Assuming that neighbouring channels are highly correlated, we have selected 25 channels as representatives of their neighbourhood with radius one, see Figure 3. With respect to time reduction, we have selected two-minute time slots from each phase of the protocol, as shown in Table 1. The time series have also been re-sampled using 16 samples per second to be comparable with the SL method application in other studies [2].
Figure 3. The red circles indicate the 25 selected channels.
a Table. 1. Selected minutes of the three protocol phases.
Rest Phase I 5-6
Exercise Phase 25-26
Rest Phase II 45-46
III. METHODS
b Figure 1. Spatial distribution whole head MEG channels on the scalp. (a) Numeric code and position of the channels. (b) Digital head reconstruction via software (Matlab).
Yogic protocol. The subject was recorded while reclining and supported at 45 degrees. He followed a well-practiced protocol that involves a 10 minute resting baseline recording (rest phase I), followed by a 31 minute exercise recording phase, followed by a second 10 minute resting recording (rest phase II), see Figure 2. The three phases are separated by a one-minute recording pause. The exercise phase consists of selectively breathing through only the left nostril (using a plug for the other side, with both arms resting in the lap) at a respiratory rate of one breath per minute for 31 min that uses a repeating pattern of 15s slow inspiration, 15s breath retention, 15s slow expiration, and 15s breath hold out) [14,26,27].
Figure 2. Yogic Protocol Timeline.
The developments in the theory of complex networks have motivated new applications in the field of neuroscience [15]. Recent studies have led to important results in understanding the relationship between the structural properties of networks and the nature of dynamics taking place on these networks [16]. The study of models of neural networks, anatomical connectivity, and functional connectivity, based on fMRI, EEG and MEG, have been determined by using graph spectral analysis [12]. This suggests that the human brain can be modelled as a complex network, and may have different structures both at the level of anatomical as well as functional connectivity. In this paper the “synchronization likelihood” method has been used to characterize the functional interactions as ‘‘functional connectivity,’’ by performing the measure of statistical interdependencies between signals of brain activity. The underlying assumption is that such correlations reflect, at least in part, functional interactions between different brain areas. Synchronization Likelihood methods. Let us consider M simultaneously recorded time series xk with length of N, where k denotes channel number (k = 1, . . .,M). The first step in the computation of the SL is to convert the series related to each channel k in a matrix Xk in which the rows are the time series of state space vectors obtained using the method of time delay embedding [17], where L is the time lag, m the embedding dimension and xk,i represents the
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starting point of the series where i = 1,2,..., N-(m-1)*L as reported in equation (1).
X k,i = (x k,i ,x k,i + L ,x k,i + 2 L ,...,x k,i +(m-1 )xL )
(1)
Assuming two generic different channels and their related matrices X and Y, SL is defined as the conditional likelihood that the distance between Yi and Yj is smaller than a cut-off distance ry, given that the distance between Xi and Xj is smaller than a cut-off distance rx. In the case of maximal synchronization, this likelihood is equal to 1; in the case of independent systems, it is a small number Pref, different from zero. This number is the likelihood that two vectors, randomly chosen, Yi (or Xi) are closer than the cut-off distance r. In practice, the cut-off distance is chosen such that the likelihood of random vectors being close is fixed at Pref, which is assumed to be invariable for Yi and Xi. To understand how Pref is used to fix rx and ry, it is necessary to consider, at first, the correlation integral of a specific channel k, identified by X:
Cr =
N N −ω 2 ∑ ∑ θ (r − | X i − X j |) N ( N − ω ) i =1 i + w
(2)
Here the correlation integral Cr is the likelihood that two randomly chosen vectors are closer than r. The |.| operator represents the Euclidean distance between the vectors, N is the number of vectors, w is the Theiler correction for autocorrelation [20], and θ is the Heaviside function: θ(X)=0 if X>0 and θ(X)=1 if X1 and λ ≈1, thus σ ≥1. Many types of real networks have been shown to have small-world features [18]. Moreover, different patterns of anatomical connectivity in neuronal networks show a high clustering and a small path length [19]. IV. RESULTS
SL is a symmetric measure of the strength of synchronization between Xi and Yi. In the equation (3), the average is calculated over all i and j; by doing the averaging only over j, SL can be computed as a function of time i. From equation (3), it can be seen that in the case of complete synchronization SL=1; in the case of complete independence SL=Pref. In the case of intermediate levels of synchronization Pref
