Mathematics of EEG/EKG

Summary: EEGs and EKGs visually convey important information about a patient’s heart and brain.

Electrocardiography(ECG or EKG) and electroencephalography (EEG) are graphic representations of bioelectric activities of the heart and brain, respectively. EKG quantifies the rhythm of heart contraction measurements that can be used to identify damage to various myocardial muscles. EEG is used in the diagnosis of epilepsy, seizure, and encephalopathy. The production of EKG and EEG signals is grounded in mathematical analysis. Diverse mathematical and statistical techniques, including applications of calculus and chaos theory, are also used to analyze and interpret signals related to conditions such as sleep disruptions, seizures, and mental illness.

EKG

EKG is a graphic representation of the myocardial contraction (systole) and relaxation (diastole) caused by depolarization of the heart. In the myocardial muscles, depolarization is an increase of membrane potential, and repolarization is a decrease of membrane potential. A typical EKG consists of P, Q, R, S, and T waves. Atrial depolarization normally begins at the SA node and is represented as the P wave. The depolarization proceeds to ventricles, which causes the ventricular depolarization (QRS complex) and then ventricular repolarization (T wave).

EKG was first systemically studied in humans by Augustus Walker in 1887. In 1903, Willem Einthoven created a reliable EKG device based on the galvanometer. Einthoven was awarded a Nobel Prize in 1924 for his invention. EKG provides information on heart contraction and the abnormality of EKG has been used to diagnose the area of myocardial damage. Heart rate variability is a quantification of fluctuations of EKG complex; a healthier heart has higher variability.

The production of EKG signals can be explained by an idealized model in which both intracellular and extracellular currents are confined to the direction parallel to the propagation of the plane wavefront. When there are no external currents, the relationship between the potential inside the membrane V_i and the potential outside the membrane V_o can be represented as

where r_i and r_o are the intracellular axial resistance and extracellular axial resistance, respectively; and U_m is the membrane potential. During the depolarization, the transmembrane current i_m is

where the direction of positive current is defined as the direction of the positive x-axis. For the depolarization of cardiac tissue, a double layer appears at the wavefront with the dipole orientation in the direction of propagation.

A pair of electrodes can be used to produce one EKG signal; the output from the pair is called a “lead.” Usually more than two electrodes are used and combined into pairs. Clinically, a 3-lead or 12-lead EKG is used to diagnose heart diseases. For a traditional 3-lead EKG, leads I, II, and III are defined as

where LA, RA, and LL denote left arm, right arm, and left leg, respectively.

EEG

EEG is a recording of the electric potential of thousands or millions of neurons within the brain. The electrodes are placed on the scalp at certain anatomical locations. EEG was first systematically analyzed by Hans Berger in 1920, who introduced the term “electroencephalogram” to indicate fluctuations recorded from the brain. EEG waves are usually irregular and cannot be classified in the normal brain.

However, four characteristic frequencies have been identified: Alpha (8-13Hz), Beta (14-30Hz), Theta (4-7Hz), and Delta (below 3.5Hz) waves. Under pathological conditions, like epilepsy, distinct patterns can be observed and used to help predict the onset of the condition.

Using a simplified model of the brain and surrounding tissues as a sphere with several shells, it is possible to compute the EEG based on the measured intracerebral currents at the scalp. The field potential can be represented as a function of intracerebral currents or of the membrane potential. In an infinite, isotropic, and homogeneous medium, because of injected current densities j⃗_i at a point r⃗, the electrical potential at a point r⃗₀ lying at a distance, R, from r⃗ (R=|r⃗ - r⃗₀|)is the following:

where σ is the conductivity of the medium; the operator div indicates differentiation of a vector. When the injected current densities originate at the cell membrane, by assuming that the neuronal membrane is equivalent to a double layer with an intracellular membrane potential V_m, the potential at a point r⃗₀ is given approximately by

where σ_i is the intracellular conductivity, σ_e the extracellular conductivity, and

is the solid angle subtended by an infinitesimal surface on the membrane surface and seen from the extracellular point r⃗₀.

Bibliography

Malmivuo, Jaako, and Robert Plonsey. Bioelectromagnetism: Principles and Applications of Bioelectric and Biomagnetic Fields. New York: Oxford University Press, 1995.

Niedermeyer, Ernst, and Fernado Lopes da Silva. Electroencephalography: Basic Principles, Clinical Applications, and Related Fields. 5th ed. Philadelphia, PA: Lippincott Williams & Wilkins, 2004.

Sanei, Saeid, and J. A. Chambers. EEG Signal Processing. Hoboken, NJ: Wiley Interscience, 2007.