Assessment of the Amplitude-Frequency Characteristics of the Retina with Its Stimulation by Flicker and Chess Pattern-Reversed Incentives and their Use to Obtain New Formalized Signs of Retinal Pathologies

Currently, an active search for new approaches to building expert systems for diagnosing retinal pathologies based on artificial intelligence methods (fuzzy logic methods, neural network approaches, modern classification methods, simulation models) is underway. The effectiveness of such expert systems depends entirely on the completeness


Introduction
The diagnostic possibilities of modern electroretinography are determined by a change in the conditions of adaptation and stimulation and the use of various principles of stimulation of the retina with diffuse, multifocal and structured light stimulus to obtain, respectively, scotopic and photopic ganzfeld ERG [1], multifocal ERG [2] and pattern ERG (PERG) [3]. Taking  All of these approaches to the construction of expert diagnostic systems make it necessary to conduct an active search for additional formalized (digital) signs of retinal pathologies. This search should be carried out by purposeful additional mathematical processing of the recorded ERG, that is, in fact, the "data mining" of the ERG should be performed.
In all methods, the critical point is the selection and evaluation of formalized informative features and comparing them with the norm range. This approach is widely used in the study of dynamic objects and systems in engineering. However, since the retina, from a technical object, is a complex nonlinear dynamic system with strict restrictions on permissible input effects, obtaining an accurate mathematical description of such an object is extremely difficult. For example, too bright light pulses should not be applied to the human eye, or the frequency of light pulses cannot be too low.
Creation of expert diagnostic systems on any basis is associated with the need to form training samples in the form of arrays of formalized signs of specific pathologies. Most of these signs are extracted from the recorded ERG, but their number for each type of ERG is relatively small, and the values of parameters for different pathologies often have large areas of mutual intersections, which make it difficult to diagnose. One of the possible ways to obtain new informative features is an additional mathematical transformation of registered types of ERG.
The work aimed to obtain additional formalized signs of ERG responses to rhythmic flashes (flicker ERG or FERG) [1] and ERG responses to the reversal black and white cells of chess pattern (pattern-ERG or PERG) [3] by analyzing the amplitude-frequency properties of the retina. Signs that are allocated when processing ERG by building, analyzing and subsequent approximation of the amplitude-frequency characteristics (AFC) of the retina as a nonlinear dynamic object are intended both for direct visual use and for use in diagnostic systems, including those developed based on artificial intelligence.

Subject of Research
Using the mathematical processing of digitized ERGs, the spectral composition of the diffuse flicker signal and for structured signal such as a black-and-white alternating chess pattern for healthy eyes were investigated.
Since the analyzed ERGs are periodic signals, their spectrum is discrete and represents a set of harmonic oscillations, which together form the original signal. It is essential that the ERG signal subjected to decomposition in a Fourier series on a repetition period satisfies Dirichlet conditions [4]:

1.
The signal must be limited: do not have in finite values.

2.
The signal must be piecewise continuous: have a finite 3.
The number of discontinuity points of the first kind (jumps and disposable breaks).

4.
The signal must be piecewise monotonous: must have a finite number of extrema.
Thus, a periodic ERG signal can be represented by Fourier's infinite trigonometric series [4]: where are the row coefficients a 0 , a n , b n determined from ratios?
In the expression (1)

Amplitude-Frequency Characteristics (AFC) of the Retina
To describe the properties of dynamic objects in engineering, the concept of the "transfer function" of a dynamic object is widely used [5]. The transfer function of the object allows you to establish a mathematical relationship between the signal arriving at the input of the object, and the observed signal at its output.
The transfer function is a stable characteristic for a "linear object" and can be fully described by two features considered together: an amplitude-frequency characteristic (AFC) and a phase-frequency characteristic (PFC). In practice, these characteristics reflect the property of an object to transform the spectrum of the input signal into the spectrum of the observed output signal. The linear object of its frequency response and phase response are stable and do not depend on the type of signal received at its input.
A retina is a non-linear object, and for such an object, we can only talk about the frequency response and phase response for specific types of input testing effects. We studied the possibility of obtaining retinal AFC when exposed to the following diffuse and structured  The input test signal X necessarily reflects the operation of the switch that turns the flash on and off. A flash lamp with its frequency response (WL) converts the input signal X into a change in the light flux X1. Next, the retina with its frequency response (WR) converts the luminous flux into the electrical potential Y, which is recorded with the corresponding device with its frequency response (WD) as FERG (Z). Thus, the spectrum of the recorded signal FERG Z(f) represents the spectrum of the input signal X(f) converted by three dynamic links (lamp-flash, retina, and recording instrument).
This relation can reflect such a process: where f is the switching frequency, and each dynamic link in (4) is represented by a spectrum converter (or its frequency response) in the form: An analog of such a converter for linear dynamic links is its frequency response, which describes the amplifying or attenuating properties of the converter concerning to the amplitude of each specific harmonic input signal of a certain frequency or the magnitude of any particular harmonic in the spectrum of the input signal.

Spectra of input test signals X (f) or test incentives
General view of the input test stimulus X(t) for registration of the FERG and, accordingly, the test incentive for the registration of PERG is shown in (Figure 2a). The pulse duration for registering FERG is fixed τ = 0.005s. Pulse repetition period T u = 1/f. For example, for a pulse repetition rate of 10 Hz, T u = 0.1 s. The amplitude of the pulses A is a relative value and is selected, taking into account the rationing of the resulting frequency response relative to healthy subjects. This process is revealed further in work. The test stimulus is shown in Figure 2b   The spectrum is discrete. The frequency of the first harmonic is equal to the pulse repetition frequency; the second harmonic is equal to twice the frequency, etc. The sum of such an infinite series of harmonics accurately reproduces the shape of the input signal X (t). The view of the spectra of the input signal X (f) for pulse repetition frequencies of 10 Hz and 30 Hz is shown in Figure 3.
By [5], the spectra of the pulse sequences have several features [5]: a) The number of spectrum harmonics in a limited frequency range is inversely proportional to the pulse repetition rate;

Result and Discussion
The

Spectrum FERG (Z(f)) and Retinal AFC
The type of FERG as an initial signal, which is decomposed into a Fourier series on a pulse repetition period with a frequency of 10 Hz, is shown in Figure 5a. FERGs of a healthy subject with normal vision are considered. It should be noted that the use of FERG spectra for diagnosis is complicated by the fact that the magnitudes of the harmonics in them depend on the spectra of the input signals, in which, according to [5], [5] the amplitudes of the harmonics of the same name are proportional to the frequency of the supplied light pulses. Building retinal frequency response by dividing the amplitudes of the harmonics of the spectra Z(f) by the corresponding amplitudes of the harmonics of the input signals, by [5], eliminates this undesirable effect [6,7]. It should be noted that

Signs for the Diagnosis
The formalization of the signs extracted from the AFC of For research purposes, the Mathcad package was used.
Attempting to approximate the retinal frequency response by a single power polynomial over the entire frequency range up to 120 Hz gives a highly smoothed curve, regardless of the degree of smoothing polynomial being assigned. Therefore, we have proposed to approximate the frequency response separately for the two frequency ranges. In the frequency range 0 <f <50 Hz, we approximate the frequency response with a second-order polynomial, and in the frequency interval 50 <f <120 Hz, we approximate the frequency response with a first-order polynomial.
In this case, the harmonic with a frequency of 50 Hz is skipped.
Thus, the approximating curve is presented in the form: Hz), as well as the result of approximation in the form of smoothed curves, constructed from the dependencies found ( Figure 8).

Pattern Stimulus Analysis
The simulating stimulus pattern signal graph in Figure 2b was explored on a mathematical model. Modeling demonstrates short dips of the light flux to zero, which alternate through every half period. It is due to the delay due to the clock frequency of the computer when the black and white cells of the pattern are reversed (field colors). Since in this case, according to (5), the amplitudes of the harmonics of the test stimulus in the frequency range of interest are practically constant and do not depend on their number in the decomposition of such a signal in a Fourier series is determined by the expression [5]. increase by order of magnitude the amplitudes of the harmonics in the decomposition of the testing pulse. That is, it will increase its power and increase the amplitude of the retinal response. In this case, the "baseline" drift will be much less noticeable. The baseline drift itself is represented as a straight line passing through the starting and ending points of the PERG period ( Figure   9). Studies show that the linear representation of the baseline drift is in all cases reasonably accurate.    Figure 10 shows the spectrum of a PERG in the patient with glaucoma, both without lengthening and artificially lengthening the period with zero values. When constructing the AFC of the retina from the available spectrum of corrected PERG, it is advisable to normalize the ordinate axis of a relatively healthy subject, just as it was done when building the AFC of the retina using FERG (Figure 10). g) The angular size of the reversing chess pattern affects the numerical values, the type of curves, and the relationship between the calculated frequency response estimates.