**2. The brain as a complex system**

### **2.1 Complex networks**

The human brain contains on the order of 1011 neurons and more than 1014 synaptic connections (Kandel et al., 2000). Although sparsely connected, each neuron is within a few synaptic connections of any other neuron (Buzsáki, 2006). This remarkable connectivity is achieved by a kind of hierarchical organization that is not fully understood in the brain, but is ubiquitous in nature, called scale-free or complex networks (Barabasi, 2009; Bassett and Bullmore, 2006; Ravasz and Barabasi, 2003). Complex networks are characterized by dense local connectivity and sparser long-range connectivity (Barabasi, 2009) that is fractal or selfsimilar at all scales.

Many brain disorders appear to be associated with abnormal brain connectivity that may vary between different regions and different scales (Bassett and Bullmore, 2009; Craddock et al., 2009; Noonan et al., 2009). Examples include autism (Belmonte et al., 2004; Noonan et al., 2009), schizophrenia (Raghavendra et al., 2009; Uhlhaas et al., 2008; Whittington, 2008), depression (Li et al., 2008; Sheline et al., 2009) and epilepsy (Douw et al., 2010; Percha et al., 2005). Methods for estimating neural connectivity or changes in neural connectivity might be effective diagnostic biomarkers for abnormal connectivity development that is associated with brain dysfunction. The electrical signals produced by neural networks are believed to contain information about the neural network structure on several scales in the vicinity of the EEG sensor (Raghavendra et al., 2009; Stam, 2005; Zavaglia et al., 2008).

Novel analysis methods from nonlinear systems theory are able to extract information from EEG signals that reflect the underlying network organization. System invariants will be encoded in the time series produced and measured by EEG sensors (Fuchs et al., 2007; Gao and Jin, 2009). Multiscale entropy (MSE) is one invariant measure of system dynamics that has been shown to particularly sensitive to changes in physiological systems (Costa et al., 2005b; Hu et al., 2009b; Takahashi et al., 2010), including mental disorders such as Alzheimer's Disease (Abasolo et al., 2006), schizophrenia (Takahashi et al., 2010), the effect of antipsychotic drugs (Takahashi et al., 2010) normal aging (Bruce et al., 2009) and autism spectrum disorders (Bosl et al., 2011).

A comparison of functional network properties using fMRI showed that children and young-adults' brains have similar "small-world" organization at the global level, but differ significantly in hierarchical organization and interregional connectivity (Supekar et al., 2009). The networks measured with fMRI in this study are those that are formed among correlated voxels, which may represent a different spatial scale from that measured by single EEG sensors. Transient or sustained generalized synchronization between EEG sensors is another measurement of functional connectivity in the brain. A simultaneous study of EEG and fMRI signals in patients with a degenerative type of epilepsy showed that in the non-myoclonus state, subtle abnormalities that were detected in EEG signals did not affect fMRI, suggesting that EEG measurements of connectivity may measure different connectivity or may be more sensitive to temporal synchronization that occurs on a time scale less than that of fMRI (on the order of one second).

Neuroscience has made great progress using linear methods of spectral (Fourier) analysis, it is likely that much more information is contained in the complex patterns of brain activity (see, for example, (Bruce et al., 2009). EEG may be under-utilized for clinical applications in neurology and psychiatry (Niedermeyer, 2003; Niedermeyer and Lopes da Silva, 2005) particularly now as new developments in complex systems and multivariate nonlinear time series analysis may allow previously unexplored information to be extracted from EEG signals (Kulisek et al., 2008; Mizuhara et al., 2005; Stam, 2005; Varela et al., 2001). Sensors such as EEGs measure the coordinated electrical response of many neurons to produce time series that reflect the dynamics of this complex system. In order to extract salient information from this data, methods appropriate for analyzing nonlinear time series are required. Although many useful techniques for nonlinear time series and system analysis have been developed in other disciplines (Bosl, 2000; Braha et al., 2006; Elnashaie and Grace, 2007; Holland, 1995; Stauffer, 2006), it is not immediately clear which are most appropriate for neuroscience research. Nevertheless, recent research results are quite promising.

The complexity of EEG signals was found in one study to be associated with the ability to attend to a task and adapt to new cognitive tasks; a significant difference in complexity was found between normal subjects and those with diagnosed schizophrenia (Li et al., 2008). Schizophrenic patients were found to have lower complexity than normal controls in some EEG channels and significantly higher interhemispheric and intrahemispheric cross mutual information values (another measure of signal complexity) than normal controls (Na et al., 2002).

Methods for chaotic signal and phase synchronization analysis arose from a need to rigorously describe physical phenomena that exhibited what was formerly thought to be purely stochastic behavior but was then discovered to represent complex, aperiodic selforganized dynamics (Pikovsky et al., 2001). The analysis of signal complexity and interaction between signals leading to transient synchronization may reveal information about local neural complexity and long-range communication between brain regions (Buzsáki, 2006; Stam, 2005; Varela et al., 2001).

The synchronization patterns of complex networks have been shown to be closely related to the topology of the network (Arenas et al., 2006) and are related to brain connectivity (Sakkalis et al., 2008). EEG signals are believed to derive from pyramidal cells aligned in parallel in the cerebral cortex and hippocampus (Sörnmo and Laguna, 2005), which act as

Alzheimer's Disease (Abasolo et al., 2006), schizophrenia (Takahashi et al., 2010), the effect of antipsychotic drugs (Takahashi et al., 2010) normal aging (Bruce et al., 2009) and autism

A comparison of functional network properties using fMRI showed that children and young-adults' brains have similar "small-world" organization at the global level, but differ significantly in hierarchical organization and interregional connectivity (Supekar et al., 2009). The networks measured with fMRI in this study are those that are formed among correlated voxels, which may represent a different spatial scale from that measured by single EEG sensors. Transient or sustained generalized synchronization between EEG sensors is another measurement of functional connectivity in the brain. A simultaneous study of EEG and fMRI signals in patients with a degenerative type of epilepsy showed that in the non-myoclonus state, subtle abnormalities that were detected in EEG signals did not affect fMRI, suggesting that EEG measurements of connectivity may measure different connectivity or may be more sensitive to temporal synchronization that occurs on a time

Neuroscience has made great progress using linear methods of spectral (Fourier) analysis, it is likely that much more information is contained in the complex patterns of brain activity (see, for example, (Bruce et al., 2009). EEG may be under-utilized for clinical applications in neurology and psychiatry (Niedermeyer, 2003; Niedermeyer and Lopes da Silva, 2005) particularly now as new developments in complex systems and multivariate nonlinear time series analysis may allow previously unexplored information to be extracted from EEG signals (Kulisek et al., 2008; Mizuhara et al., 2005; Stam, 2005; Varela et al., 2001). Sensors such as EEGs measure the coordinated electrical response of many neurons to produce time series that reflect the dynamics of this complex system. In order to extract salient information from this data, methods appropriate for analyzing nonlinear time series are required. Although many useful techniques for nonlinear time series and system analysis have been developed in other disciplines (Bosl, 2000; Braha et al., 2006; Elnashaie and Grace, 2007; Holland, 1995; Stauffer, 2006), it is not immediately clear which are most appropriate

for neuroscience research. Nevertheless, recent research results are quite promising.

The complexity of EEG signals was found in one study to be associated with the ability to attend to a task and adapt to new cognitive tasks; a significant difference in complexity was found between normal subjects and those with diagnosed schizophrenia (Li et al., 2008). Schizophrenic patients were found to have lower complexity than normal controls in some EEG channels and significantly higher interhemispheric and intrahemispheric cross mutual information values (another measure of signal complexity) than normal controls (Na et al.,

Methods for chaotic signal and phase synchronization analysis arose from a need to rigorously describe physical phenomena that exhibited what was formerly thought to be purely stochastic behavior but was then discovered to represent complex, aperiodic selforganized dynamics (Pikovsky et al., 2001). The analysis of signal complexity and interaction between signals leading to transient synchronization may reveal information about local neural complexity and long-range communication between brain regions

The synchronization patterns of complex networks have been shown to be closely related to the topology of the network (Arenas et al., 2006) and are related to brain connectivity (Sakkalis et al., 2008). EEG signals are believed to derive from pyramidal cells aligned in parallel in the cerebral cortex and hippocampus (Sörnmo and Laguna, 2005), which act as

spectrum disorders (Bosl et al., 2011).

2002).

scale less than that of fMRI (on the order of one second).

(Buzsáki, 2006; Stam, 2005; Varela et al., 2001).

many interacting nonlinear oscillators (Nunez and Srinivasan, 2006). As a consequence of the scale-free network organization of neurons, EEG signals exhibit complex system characteristics reflecting the underlying network topology, including transient synchronization between frequencies, short and long range correlations and crossmodulation of amplitudes and frequencies (Gans et al., 2009). A great deal of information about interrelationships in the nervous system likely remains hidden because the linear analysis techniques currently in use fail even to detect them (Drongelen, 2007).
