Seizure Detection and Advanced Monitoring Techniques

Nicholas Fisher

Sachin S. Talathi

Alex Cadotte

Stephen Myers

William Ditto

James D. Geyer

Emery E. Geyer

Paul R. Carney

EPILEPSY: A DYNAMIC PROCESS

The EEG is a complex set of signals with statistical properties variable in terms of both time and space.¹ The individual characteristics of the EEG, such as bursting events (during stage II sleep), limit cycles (alpha activity, mu activity, ictal activity), amplitude-dependent frequency behavior (the smaller the amplitude, the higher the EEG frequency), and frequency harmonics (particularly in association with photicdriving conditions), are but a few of the vast array of concerns related to the properties typical of nonlinear systems. The EEG of the epileptic brain is a nonlinear signal with numerous confusing deterministic and possibly even chaotic properties.²^,³^,⁴

The voltage of the EEG can be represented by a series of numeric values over time and space (multielectrodes), known as a multivariate time series. The standard methods for time series analysis (eg, power analysis, linear orthogonal transforms, and parametric linear modeling) fail to detect the critical features of a time series generated by a nonlinear system. In some cases, this can even result in the false indication that most of the series is random noise.⁵ While we are unable to measure all of the relevant variables in the case of a multidimensional, nonlinear system such as the generators of EEG signals, this problem can be addressed mathematically. Since all dynamic systems have variables related over time, one may obtain information about the important dynamic features of the whole system by analyzing a single variable (eg, voltage) over time. By analyzing more than one variable over time, we can follow the dynamics of the interactions of different parts of the system. Neuronal networks can generate a variety of activities, some of which are characterized by rhythmic or semirhythmic signals that are reflected in the corresponding local EEG field potential. An essential feature of these networks is that variables of the network have both a strong nonlinear range and complex interactions.

Characteristics of the dynamics can depend strongly on small changes in the control parameters and/or the initial conditions. Real neuronal networks behave with complex nonlinear characteristics and can display changes between states such as small-amplitude, quasirandom fluctuations and large-amplitude, rhythmic oscillations. These dynamic state transitions are observed during the transition between interictal state and epileptic seizure onset. A functional system must stay within a given range in order for the system to maintain a stable operation. The most essential difference between a normal and an epileptic network can be conceptualized as a decrease in the distance between operating and bifurcation points.

In considering epilepsy as a dynamic disorder of neuronal networks, Lopes da Silva and colleagues proposed two scenarios of how a seizure could evolve.¹ The first is that a seizure could be caused by a sudden and abrupt state transition, in which case it would not be preceded by detectable dynamic changes in the EEG. Such a scenario would be conceivable for the initiation of seizures in primary generalized epilepsy. Alternatively, this transition could occur as a gradual change or a cascade of changes in dynamics, which could in theory be detected and possibly even anticipated.

SEIZURE DETECTION

Most of the current techniques used to detect or “predict” an epileptic seizure involve linear or nonlinear transformation of the signal using one of several mathematical
models and subsequently trying to predict or detect the seizure based on the results. These models include some purely mathematical transformations, such as the Fourier transform, and machine learning techniques, like artificial neural networks, or some combination of the two. In this section, we review some of these modeling techniques for detection and prediction of seizures.

Seizure prediction models have used a variety of techniques in an attempt to detect the EEG signature of epileptic seizures and predict their occurrence. The goal of a model is to perform mathematically the kind of analysis performed visually by skilled EEGers using the concepts presented in preceding chapters. This could improve our understanding of seizure mechanisms and our ability to detect them in complex settings. The majority of these techniques use some kind of time series analysis to detect seizures offline. Time series analysis of an EEG signal falls into one of the following two groups:

1. Univariate time series analysis refers to time series that consist of a single observation recorded sequentially over equal time increments. Time is an implicit variable in the time series. Information on the start time and the sampling rate of the data collection can allow one to visualize the univariate time series graphically as a function of time over the entire duration of data recording. The information contained in the amplitude value of the recorded EEG signal sampled in the form of a discrete time series x(t) = x(t_i) = x(iΔt) (i = 1, 2, …, N and Δt is the sampling interval) can also be encoded through the amplitude and the phase of the subset of harmonic oscillations over a range of different frequencies.
2. Multivariate time series analysis refers to time series that consist of more than one observation recorded sequentially in time. Multivariate time series analysis is used when one wants to understand the interaction between the different components of the system under consideration. As in univariate time series, time is also an implicit variable in the multivariate time series.

REVIEW

22.2: Univariate time series analysis consists of:

a. Multiple observations recorded sequentially over equal time increments

b. A single observation recorded sequentially over variable time increments

c. Multiple observations recorded sequentially over different time increments

d. A single observation recorded sequentially over equal time increments

View Answer

22.2: d. Univariate time series sample a single parameter (eg, voltage at one electrode) sequentially at a defined increment of time (the sampling frequency).

UNIVARIATE TIME SERIES ANALYSIS

Short-Term Fourier Transform

Power spectral analysis of the EEG is one of the more widely used techniques for detecting or predicting an epileptic seizure. The basic hypothesis is that the EEG signal when partitioned into its component periodic (sine/cosine waves) elements has a signature that varies between the ictal and the interictal states. In order to detect this signature, the Fourier transform of the signal is calculated, and then the frequencies that are most prominent (in amplitude) are identified. There is a relationship between the power spectrum of the EEG signal and ictal activity.⁶ Although there is some correlation between the power spectrum and ictal activity, the power spectrum is not effective as a stand-alone detector of a seizure. Most models couple power spectrum with some other time series prediction technique or machine learning modality to detect a seizure.

The Fourier transform breaks up any time-varying signal into its frequency components of varying magnitude, defined in Eq. (22.1).

Euler’s formula allows this to be written as shown in Eq. (22.2) for any complex function f(t), where k is the kth harmonic frequency.

Utilizing this system, any time-varying signal can be represented as a summation of sine and cosine waves of varying magnitudes and frequencies.⁷ The Fourier transform is represented by the power spectrum. The power spectrum has a value for each harmonic frequency, which indicates how strong that frequency is in the given signal. The magnitude of this value is calculated by taking the modulus of the complex number that is calculated from the Fourier transform for a given frequency (|F(k)|).

Stationarity must be considered when using the Fourier transform. A stationary signal is one that is constant in its statistical parameters over time, and the Fourier transform assumes that stationarity is present. A signal that is made up of different frequencies at different times will yield the same transform as a signal, which is made up of those same frequencies for the entire time period considered. As an example, consider two functions f₁ and f₂ over the domain 0 ≤ t ≤ T, for any two frequencies ω₁ and ω₂ shown in Eqs. (22.3) and (22.4).

and

When using the short-term Fourier transform, the assumption is made that the signal is stationary for some small period of time, T_s. The Fourier transform is then calculated for segments of the signal of length T_s. The short-term Fourier transform at time t gives the Fourier transform calculated over the segment of the signal lasting from (t – T_s) to t. The length of T_s determines the resolution of the analysis. There is a trade-off between time and frequency resolution. A short T_s yields better time resolution; however, it limits the frequency resolution. The opposite of this is also true; a long T_s increases frequency resolution while decreasing the time resolution of the output. Other modalities, such as wavelet analysis, can alleviate this limitation. Wavelet analysis provides a model that maintains both time and frequency resolution.⁷

Discrete Wavelet Transforms

Wavelet transforms follow the principle of superposition, just like Fourier transforms, and assume EEG signals are composed of various elements from a set of parameterized basis functions. Wavelets must meet certain mathematical criteria, which allow the basis functions to be far more general than simple sine/cosine waves as in the Fourier transform. Wavelets make it substantially easier to approximate sharply contoured waveforms such as spikes, as compared to the Fourier transform. Fourier transforms have a limited ability to approximate a spike because of the sine (and cosine) waves’ infinite support (ie, stretch out to infinity in time). In the case of wavelets, there is the possibility of finite support, allowing estimation of the spike by changing the magnitude of the component basis functions.

The discrete wavelet transform is similar to the Fourier transform in that it will break up any time-varying signal into smaller uniform functions, known as the basis functions. The basis functions are created by scaling and translating a single function of a certain form. This function is known as the mother wavelet. In the case of the Fourier transform, the basis functions used are sine and cosine waves of varying frequency and magnitude. The only requirements for a family of functions to be a basis is that the functions are both complete and orthonormal under the inner product. Consider the family of functions, Ψ = {ψ_ij | -∞ < i, j < ∞}, where each i value specifies a different scale and each j value specifies a different translation based on some mother wavelet function. Ψ is considered to be complete if any continuous function f, defined over the real line, x, can be defined by some combination of the functions in Ψ as shown in Eq. (22.5).⁷

A family of functions must meet two criteria to be orthonormal under the inner product. It must be the case for any i, j, l, and m where i ≠ l and j ≠ m that <ψ_ij, ψ_lm> ≥ 0 and <ψ_ij, ψ_ij> ≥ 1, where <f, g> is the inner product and is defined as in Eq. (22.6) and f(x)^* is the complex conjugate of f(x).

The wavelet basis is very similar to the Fourier basis, with the exception that the wavelet basis does not have to be infinite. In a wavelet transform, the basis functions can be defined over a certain window and then be zero everywhere else. As long as the family of functions defined by scaling and translating the mother wavelet is orthonormally complete, that family of functions can serve as the basis. With the Fourier transform, the basis is made up of sine and cosine waves that are defined over all values of x where –∞ < x < ∞.

One of the simplest wavelets is the Haar wavelet (Daubechies 2 wavelet). In a manner similar to the Fourier series, any continuous function f(x) defined on [0, 1] can be represented using the expansion shown in Eq. (22.7). h_j,k(x) is known as the Haar wavelet function and is defined as shown in Eq. (22.8) and p_j,k(x) is known as the Haar scaling function and is defined in Eq. (22.9).⁷

The combination of the Haar scaling function at the largest scale, along with the Haar wavelet functions, creates a set of functions that is an orthonormal basis for functions in R².

Spectral entropy calculates some feature based on the power spectrum. Entropy was first used in physics as a thermodynamic quantity describing the amount of disorder in a system, but can also be used to calculate the entropy for a given probability distribution.⁸ The entropy measure that Shannon developed with can be expressed as in Eq. (22.10).

Entropy is a measure of how much information there is to learn from a random event occurring. Events that are unlikely to occur yield more information than events that are very probable. For spectral entropy, the power spectrum is considered to be a probability distribution. The spectral entropy is an indicator of the number of frequencies that make up a signal. A signal made up of many different frequencies (eg, white noise), would have a relatively uniform distribution and therefore yield high spectral entropy. Conversely, a signal made up of a single frequency would yield low spectral entropy.

A wavelet filter can be used to partition the EEG between seizure and nonseizure states. It flagged any increase in power or shift in frequency regardless of cause, whether this change in the signal was caused by an artifact, normal EEG activity, interictal epileptiform discharges, or ictal activity. The signals were then passed through a second filter that tried to isolate the seizures from the other activity. By decomposing the signal into components and passing it through the second step of isolating the seizures, the authors were able to detect all seizures with an average of 2.8 false positives per hour.⁹ This system was unable to predict seizure onset reliably.

Statistical Moments

It is possible to describe an approximation to the distribution of a random variable using moments and functions of moments, even when a cumulative distribution function for such a variable cannot be determined.¹⁰ Statistical moments relate information about the distribution of the amplitude of a given signal. In probability theory, the kth moment is defined as in Eq. (22.11), where E[x] is the expected value of x.

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