**The aim of this review is to use the multimedia aspects of a purely digital online publication to explain and illustrate the highly capable technique of m-sequences in multifocal ophthalmic electrophysiology. M-sequences have been successfully applied in clinical routines during the past 20 years. However, the underlying mathematical rationale is often daunting. These mathematical properties of m-sequences allow one not only to separate the responses from different fields but also to analyze adaptational effects and impacts of former events. By explaining the history, the formation, and the different aspects of application, a better comprehension of the technique is intended. With this review we aim to clarify the opportunities of m-sequences in order to motivate scientists to use m-sequences in their future research.**

*maximal length shift-register sequences*or

*m-sequences*, first applied to this scope by Sutter and colleagues.

*1*and

*0*). M-sequences are pseudorandom in the sense that they seem random but follow a strict generation rule using digital shift registers with a linear feedback (Golomb, 1982; Keating & Parks, 2006; Marmarelis & Marmarelis, 1978; Sutter & Tran, 1992). The following section describes this specific generation of pseudorandom binary m-sequences by means of Movie 1 (Figure 1). The generation of m-sequences starts with a shift register consisting of a definite number of digits (

*n*= 7 in Movie 1) holding the states

*1*or

*0*(initially often called

*seed*). The modulo 2 sum (parity) of a specific subset of this shift register (tap register) is fed back into the serial input. In the present example (Movie 1), the first and the last three digits were chosen for this subset (bold rectangles). The modulo 2 sum represents the rest after dividing the sum of the chosen digits by 2. If the sum is even, the rest after division by 2 is

*0*(first step of Movie 1). If it is odd, the formula results in

*1*. The resulting digit is positioned at the end of the binary number representing a new step (linear feedback). After shifting the sequence by one digit each time, the following steps are created in the same way (digital shift registers). Depending on the length of the shift register and the chosen subset to be fed back, the content of the shift register will recur and the sequence repeats at a specific point. The longest possible sequence has the maximal length of 2

*– 1 steps before it repeats, meaning that each of the possible 2*

^{n}*– 1 arrangements of*

^{n}*1*and

*0*can be found somewhere in the sequence. Consequently, generated sequences with a length of 2

*– 1 steps (in our example, 2*

^{n}^{7}– 1 = 127) are referred to as

*maximum length feedback shift register sequences*or simply

*binary m-sequences*(Sutter & Tran, 1992). The only missing arrangement from all possible 2

*configurations is the zero state (i.e., all digits in the register are*

^{n}*0*). The zero state is a cycle in itself as the respective modulo 2 sum always results in

*0*.

*– 1 steps is reached. In this case the resulting stimulation sequence would depend on the initial values of*

^{n}*0*s and

*1*s within the shift register (often called

*seed*). Moreover, the repetition of parts of the sequence is a correlation that can lead to a cross-contamination (also known as

*cross-talk*) of responses from different parts of the visual field when different fields are stimulated with shifted versions of the same sequence (Ireland, Keating, Hoggar, & Parks, 2002; Keating & Parks, 2006). Thus, it is mandatory to ensure that a subset is chosen that generates sequences with a length of 2

*– 1 steps regardless of the initial seed.*

^{n}*1*as −1 and the opposite state

*0*as 1, the mathematical properties are preserved because the multiplication of the elements 1 and −1 is equivalent to a modulo 2 sum of the elements

*0*and

*1*using this substitution.

*– 1 (in our example, 2*

^{n}^{7}−1 = 127), implying maximum correlation.

*– 1 independent fields can be stimulated with the same m-sequence because 2*

^{n}*– 1 different starting points are possible. But the duration of an evoked response has to be considered. In the second part of Movie 3 we chose a response waveform with a length of four steps in time. If the temporal delay between different stimulation fields was less than four steps, then the corresponding waveforms of these fields would always be superimposed in the same way and could not be separated by any data analysis procedure. This would result in a cross-contamination between different stimulus locations (Keating & Parks, 2006). As a consequence, the number of independent fields for an m-sequence with 2*

^{n}*– 1 steps depends on the length of the evoked response. A longer response leads to a smaller number of independent fields.*

^{n}*higher order kernels*without any additional data analysis (Nemoto, Momose, Kiyosawa, Mori, & Mochizuki, 2004; Sutter, 2000, 2001). Further mathematical details of kernel analysis can be found in respective publications (Klein, 1992; Voltera, 1959; Wiener, 1958).

*first slice of the second-order kernel*. The top cylinder shows the m-sequence, and the two blue rectangles indicate two subsequent stimulation steps. The bottom cylinder shows the product of the multiplication between the two stimulation steps (e.g., 1 × −1 = −1) for the current steps. As the movie progresses in time, a corresponding correlation of two neighboring steps is calculated and the results fill the bottom cylinder in the first half of the movie. While this operation may seem trivial, the important aspect is shown in the second half of the movie. When we calculate the cross-correlation between the sequences of the two cylinders similar to Movie 2, a maximum correlation (127) is found for a shift of 87 steps where all other shifts show the same minimal correlation (−1). The simple explanation of this result is that the time course of the first slice of the second-order kernel is exactly identical to the original m-sequence but shifted by a specific number of steps—in this case, 87 steps.

*second slice of the second-order kernel*, where the interval between the two observed points in time (blue rectangles) is two steps (Movie 6a in Figure 6). The cross-correlation shows that this interaction coincides with a shift of the m-sequence by 47 steps. In Movie 6b (Figure 6) a stimulus two steps before the current stimulus (white, black, and white rectangles in the bottom left part of Movie 6b) leads to a smaller modulation of the response than a stimulus that directly precedes the current stimulus (black, white, and white rectangles in the bottom left part of Movie 6b). When both preceding stimulations contain a stimulus (three white rectangles in the bottom left part of Movie 6b), the amplitude reduction is simply a summation of the amplitude reductions of both situations with only one preceding stimulus. The cross-correlation result shows two significant second-order kernel responses. Compared with Movie 5b, an additional second slice of the second order kernel response appears exactly 47 steps after the first-order response, as expected from Movie 6a.

*third-order kernel*indicates a processing that depends on three subsequent points in time (blue rectangles in Movie 7a of Figure 7). Again, this interaction turns out to occur with the same m-sequence as the original stimulation, in this case shifted by 97 steps. An additional third-order kernel response (bottom right part of Movie 7b of Figure 7) is seen with a delay of 97 steps after the first-order kernel in the cross-correlation result. Compared with Movie 6b, this third-order kernel is present because the response to three subsequent stimulations (here a flat waveform) cannot be derived by adding the effect of both individual preceding stimulations. Thus, all three steps are required to explain the response pattern in the bottom left part of Movie 7b.

- Start with a full-field stimulation and a long m-sequence that allows one to study the higher order kernels up to the fifth or sixth order. This allows one to (a) identify the number of relevant higher order kernels and (b) estimate the maximum duration of the evoked responses of these relevant kernels.
- Then an appropriate m-sequence can be chosen that has a minimum delay that is longer than the maximum response duration for all relevant higher order kernels and for all stimulation fields. In this way a cross-contamination between different stimulation fields and between different kernel orders is avoided.

*induced components*and can be demonstrated in Movie 5b, 6b, and 7b. For better visibility of these higher order kernels we combined the individual responses and the different kernel responses for Movies 4, 5b, 6b, and 7b in Figure 9.

*yes*. If we have available all relevant higher order kernels, we can predict the response of the visual system to any arbitrary sequence of stimulation (Sutter, 2000, 2001). This is illustrated in Movie 10a (Figure 10). Here we use the response pattern as shown in Movie 7b with significant responses up to the third-order kernel. In the top part of Movie 10a we synthesize a response to some arbitrary single, double, or triple stimulations in the same way as before. Responses to specific prestimulus conditions are superimposed and lead to the recorded waveform in the top yellow rectangle.

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*Information and Control*