Electrophysiological Mapping and Source Imaging

Abstract

In this chapter, we present the physical and physiological basics behind EEG and MEG signal generation and propagation. We first start by presenting the biophysical principles that explain how the coordinated movements of ions inside and outside neuronal cells result in macroscale phenomena at the scalp, such as electric potentials recorded by EEG and magnetic fields sensed by MEG. These physical principles enforce EEG and MEG signals to have specific spatial and temporal features, which can be used to study brain’s response to internal and external stimuli. We continue our exploration by developing a mathematical framework within which EEG and MEG signals can be computed if the distribution of underlying brain sources is known, a process called forward problem. We further continue to discuss methods that attempt the reverse, i.e., solving for underlying brain sources given scalp measurements such as EEG and MEG, a process called source imaging. We will provide various examples of how electrophysiological source imaging techniques can help study the brain during its normal and pathological states. We will also briefly discuss how combining electrophysiological signals from EEG with hemodynamic signals from functional magnetic resonance imaging (fMRI) helps improve the spatiotemporal resolution of estimates of the underlying brain sources, which is critical for studying spatiotemporal processes within the brain. The goal of this chapter is to provide proper physical and physiological intuition and biophysical principles that explain EEG/MEG signal generation, its propagation from sources in the brain to EEG/MEG sensors, and how this process can be inverted using signal processing and machine learning techniques and algorithms.

Homework

• 1. What is the Nyquist frequency? How is it related to the sampling frequency of a band-limited signal?

• 2. If we believe that our signals of interest in the EEG/MEG recordings are within the 1–50 Hz frequency bands, what would be the minimum sampling rate you propose that will allow the recovery of the full information content within this particular frequency band?

• 3. Could you think of a way to define the minimum number of EEG/MEG sensors necessary to avoid aliasing the spatial frequency content of surface recordings?

• 4. We only record from two EEG electrodes, say C₃ and C₄, for two conditions A and B. These two conditions are elicited when stimuli A and B are presented to our experiment subject. Each stimuli is presented 100 times, and the voltage recorded from C₃ and C₄ at 100 ms poststimulus is recorded in a vector, ${\mathcal{V}}_{\phi }={\left[{\phi}_{{\mathrm{C}}_3}(100),{\phi}_{{\mathrm{C}}_4}(100)\right]}^{\mathrm{T}}$ where ${\phi}_{{\mathrm{C}}_3}(100)$ and ${\phi}_{{\mathrm{C}}_4}(100)$ are the recorded signals from C₃ and C₄ electrodes at 100 ms poststimulus, and is plotted below:

  

  • (i) How could you distinguish between condition A and B if you were only given ${\mathcal{V}}_{\phi }$?

  • (ii) Let us assume that ${\mathcal{V}}_{\phi }$ under conditions A and B has the same exact distribution except that for condition A the distribution is centered around the point (1,1)^T and for condition B around the point (−1,−1)^T. Let us denote this probability distribution with p(x, y) and also let us assume symmetry with respect to origin, that is, p(x, y) = p(−x, −y), and indifference to input variables’ order, i.e., p(x, y) = p(y, x). Now if we want to fit the line $y-\alpha x-\delta =0$, such that any point lying on one side of this line is designated as condition A and the other side as condition B, how should we find (α, δ)?

  • (iii) Based on your answer in (ii), find the optimal set of (α, δ), if any.

• 5. If we assume that a dipole is placed at coordinates (x, y, z), the distance between the dipole source and field space, $({x}^{\prime},{y}^{\prime},{z}^{\prime})$, is defined as $r=\sqrt{{\left(x-{x}^{\prime}\right)}^2+{\left(y-{y}^{\prime}\right)}^2+{\left(z-{z}^{\prime}\right)}^2}$.

  • (i) Calculate $\nabla \left(\frac{1}{r}\right)$, where ∇ is the gradient operator defined as $\nabla f={\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right)}^{\mathrm{T}}$.

  • (ii) Calculate ${\nabla}^{\prime}\left(\frac{1}{r}\right)$, where ∇^′ is the gradient operator with respect to $({x}^{\prime},{y}^{\prime},{z}^{\prime})$, i.e., ${\nabla}^{\prime }f={\left(\frac{\partial f}{\partial {x}^{\prime }},\frac{\partial f}{\partial {y}^{\prime }},\frac{\partial f}{\partial {z}^{\prime }}\right)}^{\mathrm{T}}$.

  • (iii) Show that $\nabla \left(\frac{1}{r}\right)=-{\nabla}^{\prime}\left(\frac{1}{r}\right)$.

• 6. Assuming that a current dipole is placed at the origin of an infinitely homogeneous space pointing toward the z-direction, i.e., (x, y, z)^T = (0, 0, 0)^T and $\overline{{\mathcal{J}}^{i}}$ = (0, 0, 1)^T, using Eq. 13.3:

  • (a) Can you calculate the potential field generated by this dipole in any point $({x}^{\prime},{y}^{\prime},{z}^{\prime})$?

    Hint. $\underset{v}{\int}\nabla \left(\frac{1}{r}\right).\overline{{\mathcal{J}}^{{i}}}\left(x,y,z\right)\ dv=\nabla \left(\frac{1}{r}\right).\overline{{\mathcal{J}}^{{i}}}$, where $\overline{{\mathcal{J}}^{{i}}}$ is the current dipole moment at the origin and $\nabla \left(\frac{1}{r}\right).\overline{{\mathcal{J}}^{{i}}}$ is the inner product of the dipole moment and the gradient of the reciprocal of field point distance to dipole. This equality is due to the fact that we assumed the dipole source to be a point source at the origin. This basically is the impulse response of the Poisson’s equations, more generally referred to as the Green’s function. The inner product between vectors A = (A _x, A _y, A _z)^T and B = (B _x, B _y, B _z)^T is defined as follows: A. B = A _xB _x + A _yB _y + A _zB _z.

  • (b) Assuming that the EEG sensor is located at (0, 0, 1)^T, what number would it read as the potential (ideal conditions, noise is nonexistent)?

  • (c) What if the sensor is located at (1, 0, 0)^T?

  • (d) What if the sensor is located at (0, − 1, 0)^T?

• 7. Repeat problem 6 to calculate the magnetic field an MEG magnetometer would sense at the same locations. Use ($B=\frac{\mu }{4\pi}\int \overline{{\mathcal{J}}^{i}}\times \nabla \left(\frac{1}{r}\right) dv$) and the Green’s function hint given before. The cross product between vectors A = (A _x, A _y, A _z)^T and B = (B _x, B _y, B _z)^T is defined as follows: A × B = (A _yB _z − A _zB _y, A _zB _x − A _xB _z, A _xB _y − A _yB _x).

• 8. Based on problems 6 and 7, can you explain [and prove mathematically] why EEG signals are less sensitive to tangential sources and MEG signals to radial sources?

• 9. Let us simply assume that the lead field matrix A, of an MEG recording system with two recording channels and 3 possible sources, is given as follows:

  • (i) Assuming that the given lead field matrix models the relationship between source current density and the magnetic field in z-direction, what is the relationship between the recorded magnetic field (in z-direction) B at these sensors and the current density S = (S ₁,S ₂,S ₃)^T? Assume ideal conditions where no noise exists.

    

  • (ii) What would the MEG sensors record if S = (1,1,1)^T?

  • (iii) What if S = (2,−1,2)^T?

  • (iv) What if S = (3,0,3)^T?

  • (v) What if S = (1,1,1)^T + $\mathcal{t}$(2,−1,2)^T, $\left(\mathcal{t}\in \mathcal{R}\right)$?

  • (vi) Can you calculate the null space of matrix A, that is, all vectors x such that Ax = 0?

  • (vii) Can you briefly explain why the inverse problem is not unique? You can mathematically prove this using the concept of null space of a matrix.

• 10. Combining Eqs. 13.5 and 13.8:

  • (i) Can you formulate the relationship between estimated, X, and true source, S?

  • (ii) Based on the relation derived in (i), what should be the relationship between A and B, for the estimated source to be exactly the same as the true source?

  • (iii) Can linear methods, as studied in this problem, ever truly estimate the true source without any further priors or assumptions?

• 11. Can you derive Eq. 13.16 from Eq. 13.15 by differentiating Eq. 13.15 and setting it to zero?

• 12. Let us study the Bayesian approaches in more detail (Eqs. 13.20, 13.21, and 13.22). Let us assume that $\phi = Ax+{n}$ and that ${n}$ is a white Gaussian noise, ${n}\sim N\left(0,{\sigma_n}^2\right):$

  • (a) What is the probability distribution function (pdf) of ${n}$?

  • (b) What does p(ϕ| x) mean? Convince yourself that $p\left(\phi |x\right)=p(n)\propto {e}^{-\frac{1}{2{\sigma_n}^2}{\left\Vert \phi - Ax\right\Vert}^2}$.

  • (c) If we assume x ∼ N(0, σ _s ²), what is p(x)?

  • (d) Using Bayes’ rule (Eq. 13.20), formulate the posterior distribution p(x| ϕ).

  • (e) Define the likelihood of a distribution as $\mathcal{L}(x)=\ln p(x)$. Derive the posterior likelihood calculated in (iv).

  • (f) Formulate $\hat{x}=\underset{x}{\mathrm{argmax}}\mathcal{L}\left(x|\phi \right)$ and derive a similar formula to Eq. 13.15, showing that weighted minimum-norm (WMN) solutions are a form of maximum a posteriori (MAP) estimators.

• 13. The L₂-norm of a 2D vector, (x, y), is defined as $\sqrt{x^2+{y}^2},$ and the L₁-norm is defined as ∣x ∣  +  ∣ y∣. The level sets of norm functions are closed curves partitioning the space to inside and outside. On the other hand, some functions, such as lines or hyperplanes, partition the space to above and below. We will explore the level sets of these functions in simple cases and in a two-dimensional space. We will examine how these simple functions can be combined to form optimization problems, in later questions.

  • (a) Can you plot $\sqrt{x^2+{y}^2}=1$ and |x| + |y| = 1?

  • (b) Can you plot and describe the set of lines described by y + 2x = K ₀ for ${K}_0\in \mathcal{R}$? If K ₀ decreases, which direction will the line move toward? What happens when K ₀ increases?

• 14. Assuming x, y ≥ 0, how would you describe the following optimization problem?

  • (a) $\underset{x,y}{\mathrm{argmin}\ }\left(y+2x\right)$

    Subject to |x| + |y| = 1 x, y ≥ 0

    Hint. Basically, you want to minimize K ₀ (where y + 2x = K ₀) for nonnegative x, y with L₁-norm of 1.

  • (b) Can you graphically depict this optimization problem, by varying the values of K ₀?

  • (c) Based on (b), can you propose a systematic way to solve this type of an optimization problem? What are the optimal values of x ^∗, y ^∗, and K ₀ ^∗ in this problem?

• 15. Repeat problem 6 for the following optimization problem:

  $$ \underset{x,y}{\mathrm{argmin}\ }\left(y+2x\right) $$

  Subject to $\sqrt{x^2+{y}^2}=1\ x,y\ge 0$

• 16. From problems 14 and 15, can you explain why you would expect L₁-norm regularizations to induce sparsity in the solution? Sparsity in case of a 2D signal means only 1 nonzero element!