Functional Connectivity and the Neurophysics of EEG Ramesh Srinivasan Department of Cognitive Sciences University of California, Irvine
Outline Introduce the use of EEG coherence to assess functional connectivity in cognitive and clinical neuroscience Introduce methods to model the relationship between activity in the brain and measurement of EEG on the scalp Discuss the uses, limitations, and alternatives to source localization (solving the inverse problem) Introduce the use of spatial filters (surface Laplacians) to improve functional connectivity and source localization estimates with EEG
EEG recording Every EEG recording is the difference in potential between two points on the surface of the head. Thus, it is a measure of current in the scalp due to sources in the brain. The potential fluctuates on a millisecond time scale, reflecting extracellular currents in (mainly) pyramidal cells in the cortex.
Spectral Analysis of EEG UNIVARIATE V ( t) A sin 2 f t F exp( jw t) n n n n n n n EEG studies usually report findings in different frequency bands, typically Delta 0.5-3 Hz Mu 10-14 Hz Theta 3-7 Hz Beta 14-30 Hz Alpha 8-12 Hz Gamma 30-50 Hz Given an ensemble {V k (t)} of k = 1,..,K observations K 1 ( t) Vk ( t) K k 1 Stationarity means t Then the power spectrum is P f K K 2 F f F f 2 K K F f 2 n k n k n k n k1 k1
Spectral Analysis of EEG BIVARIATE Given an ensemble of multichannel observations {V mk (t)} consisting of k = 1, K observations in m = 1, M data channels I can define the cross spectrum between any pair of channels as coherence = 0.99 K 2 C f A e F f F f juv ( ) uv n uv uk n vk n K k1 And the coherence between channels as 2 C f 2 uv n uv fn Pu fn Pv fn coherence = 0.01
Coherence as a measure of functional connectivity Coherence is a squared correlation coefficient that measures at one frequency the proportion of variance in one channel that can be accounted for by the other channel with a linear transformation. A linear transformation means constant relative amplitude (amplitude ratio) and constant relative phase. Coherence is not strictly synchronization, which in the neurophysiological literature usually implies zero phase difference between signals At the macroscopic scale of scalp EEG measured over the whole brain, finite transmission delays along axons connecting distant regions of the brain range will impose delays ranging from 5-50 ms We may expect there are differences in the phase of oscillations in functionally connected areas of the brain, which can be detected using coherence.
Coherence and Functional Connectivity in Neuropsychiatric Disorders Functiona connectivity has been implicated in neuropsychiatric disorders (schizophrenia, ADHD, Autism, etc.). A number of studies have found significant differences in connectivity in different frequency bands. Very typically, these changes involve both increases and decreases in coherence in different frequency band, suggesting a reorganization of connections and/or changes in transmission delays. Murias, et al. (2007) Biol Psychiatry. ;62(3):270-273.
Can functional connectivity at rest predict behavior? EEG coherence with motor cortex In the beta band (20-30 Hz) PLS Resting-state EEG L R behavioral score New Subject Predict behavioral score
Motor Cortex coherence at rest predicts learning in motor tasks (rotor pursuit task) 40 35 Actual % Improvement 30 25 20 15 10 5 0 0 10 20 30 40 Predicted % Improvement Cross-validated R 2 = 0.81 Wu, et al., Resting-state cortical connectivity predicts motor skill acquisition Neuroimage 91:84-90, 2014
Motor Cortex coherence at rest predicts learning in motor tasks (sequenced wrist movement task)
Stroke One of the main motivations of this research is to Develop tools to rapidly diagnose stroke develop tools to assess the state of the motor system in stroke patients and optimize strategies for rehabilitation. to understand individual differences in rehabilitation after stroke. Stroke is heterogeneous in that the location and size of the lesion varies considerably. Yet, 80% of patients have motor deficits acutely, and 50% of patients have persistent motor deficits.
EEG coherence measures of connectivity as a biomarker of motor function in chronic stroke
Stroke EEG coherence is a biomarker for motor impairment Injured M1 tract Uninjured M1 By conventional we have used the left side to represent the ipsilesional hemisphere and the right side the contralesional hemisphere. Actual Fugl-Meyer score Fitted R 2 = 0.96 Validated R 2 = 0.78 60 55 50 45 40 35 30 25 20 20 30 40 50 60 Predicted Fugl-Meyer score There was a significant correlation between % M1 Cerebrospinal Tract injury and motor status. PM-M1 coherence and %M1 CST showed independent (partial) correlation with motor status and did not correlate with each other. Infarct volume was not correlated with motor status. Motor status (FM score) 60 50 40 30 20 0 20 40 60 80 100 % M1 CST injury
Changes in coherence predicts motor status rehabilitation in chronic stroke patients EEG coherence prior to rehabilitation predicts improvement due to rehabilitation therapy. Patients who exhibit connectivity with the contralesional motor areas improve with rehabilitation. Fitted R 2 = 0.97 Validated R 2 = 0.79 Ipselisional M1-PM coherence increases over time reflect improvement in motor status during rehabilitation therapy
Brain activity and EEG EEG coherence is a robust metric of brain function in healthy and diseased brains. In order to make stronger inferences about brain activity from EEG coherence, we must develop models of the relationship between current sources in the brain and the potentials measured on the scalp. These models can facilitate qualitative inferences about scalp measurements or potentially form the bases of methods to estimate brain activity from the scalp potentials.
Current Sources in the Brain W s( r, w, t) dw ( w) 0 1 P( r, t) ws( r, w, t) dw ( w) W W Within this cylinder there are perhaps 10 5-10 6 neurons and 10 8-10 10 synapses The far-field approximation allows us to approximate the complex source sink configuration of the cylinder by a dipole moment per unit volume, P. The strength of the source P depends on microsources s(r,w,t) The distribution of positive (inhibitory) and negative (excitatory) sources Synchrony (zero-phase lag) of microsources
The Brain is a (folded) sheet of dipole current sources in a volume conductor ( r, t) G ( r, r, h ) P( r, t) dr V GP B E The scalp potential at each electrodes is a weighted average of the dipole current sources. The weighting function G E contains all the electrical information about volume conduction through the tissues of the head. The most important tissue compartment is the skull which has conductivity 20-80 times lower than the scalp or brain. E
Volume Conduction Models of the Head In order to mathematically model the relationship between current sources in the brain and scalp EEG a volume conduction model is needed. The simplest such model is a concentric spheres model which captures the essential feature of a poorly conducting skull layers The most common model in the literature today is a Boundary Element Model, typically with 3 layers brain, skull, scalp derived from an MRI image. Many easy to use MRI packages (e.g., FSL) will automatically generate these meshes which can be used with libraries such as Open M/EEG. The most accurate models are Finite Element Models that can incorporate far more detailed tissue properties.
EEG recordings favor superficial sources in the gyral crowns
Spatial Filtering Implies Temporal Filtering: EEG emphasizes different spatial scales of brain activity than ECoG or fmri ECoG ECoG
Spatial Filtering and EEG Coherence r1, r2, r1, r1 r1, r r2, r C f G C G dsds E E B 2 E 2 1 2 S S B 1 2 Lets assume a spherical source distribution in a concentric spheres model. Then we can define a spatial white noise as,,,, 2 cos cos C f p f 1 1 2 2 1 2 1 2 Then in a concentric spheres model we can predict coherence that is due to volume conduction. Hn( rz) Pn cos12 2 n1 2n 1 V ( 12) Hn( rz) n1 2n 1 2
Volume conduction effects are independent of frequency
Sensitivity of EEG electrodes in a realistic BEM S E ( r, r) GE ( r, r) max G ( r, r) E
EEG coherence measures functional connectivity is robust for widely spaced electrodes Volume conduction poses a serious challenge to making estimates of functional connections using coherence or any correlation measure between EEG or MEG channels. In practice only widely separated channels, at least 5 cms for EEG and 7-8 cms for MEG (in surface coordinates), can be simply interpreted as functional connectivity. This effect is not a simple inflation of correlation; it reflects the fact that closely spaced electrodes or sensors are picking up from the same sources; thus coherence is correlated with power and is elevated at all frequencies
The 3-layer structure of skull Measurements of the resistance of skull plugs (or living skull flaps Akhtari et al., 2002) have shown that resistance across the skull layers is either uncorrelated or negatively correlated to thickness. Thus, head models that use a single skull layer of variable thickness are erroneously introducing greater resistance at locations where the skull is thicker. These realistic models are actually less accurate than models that assume uniform layers
Skull thickness mostly depends on spongy skull (diploe)
Finite Element Model
Can we measure connectivity between brain areas rather than electrodes? The problem of the Inverse problem of EEG. Solution requires additional information you are trying to reconstruct information in 3-D from measurements on a surface. Sometimes that additional information could come from fmri measurements that indicate which brain areas are active during a particular task. These could act as prior information in a Bayesian model. L2-norm minimum norm estimates L1-norm sparse source estimates Hauk O (2004) Keep it simple: a case for using classical minimum norm estimation in the analysis of EEG and MEG data. Neuroimage. 2004 Apr;21(4):1612-21.
A single-source Localization experiment with simultaneous EEG and MEG
A single-source Localization experiment with simultaneous EEG and MEG
Spatial Filtering - Theoretical Basis of Surface Laplacian I S I 4 0 j S J S ds S SS ds S C C j1 d C S dd S 4 SdS [4 0 j] 4 j1 S 4 C V d J K J S [4 ] 0 j IS j1 K 2 S S 2 S S S J d d L ( d / 2) d 4
Surface Laplacians focus each electrode on localized superficial sources
Surface Laplacians and EEG coherence Model High Frequency EEG (> 50 Hz)
Software for computing surface Laplacians on realistic heads One limitation to the adoption of surface Laplacians has been the estimation of derivatives We have developed a software tool in MATLAB ssltool which implements a technique we developed to use a triangulated mesh scalp to estimate the derivatives taking into account the realistic curvature of the head. http://hnl.ss.uci.edu/software
If you really must estimate brain sources use a surface Laplacian EEG LAPLACIAN EEG LAPLACIAN
Conclusions Measures of connectivity obtained from EEG are robust predictors of behavior and disease. EEG signals depend strongly on synchronization of synaptic current sources Reconstructing these sources from EEG (or for that matter MEG) is a difficult problem, possibly unsolvable, unless you have strong prior information. Spatial filtering methods, like the surface Laplacian (or beamforming), improve the resolution of the EEG. Even with the use of spatial filters, EEG connectivity estimates are really macroscopic estimates of connectivity between regions of the brain
Future Directions: Global Fields NEURAL FIELD THEORY (NUNEZ) t1 t2 t4 t3 Lamme, VAF and Roelfsema PR (2000) The distinct modes of vision offered by feedforward and recurrent processing Trends in Neurosciences, 23:571-579. Local circuits and are immersed in a globally connected environment by the corticocortical fiber systems of the brain.
Directions: Neural Field Theory Localizing EEG data and measuring functional connectivity is a difficult, if not impossible problem. What would be of greater interest, especially in applications to whitematter disease, is to fit neural field models to EEG data, to estimate not only the connectivity, but also the delays between brain areas. r r1 H E( r, t) RE( r, r1, v) G( r, r, t ) dvds( r1) v S