**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*.

**Figure 1**

**Figure 1**

*– 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**s (

**1**×

**1**=

**1**and

**−1**×

**−1**=

**1**; third cylinder in the first part of Movie 2). The sum of this series has the value 2

*– 1 (in our example, 2*

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

**Figure 2**

**Figure 2**

**Figure 3**

**Figure 3**

**Figure 4**

**Figure 4**

**Figure 5**

**Figure 5**

**Figure 6**

**Figure 6**

**Figure 7**

**Figure 7**

**Figure 8**

**Figure 8**

**1**s and

**−1**s (third cylinder in the second part of Movie 2). Due to the mathematical properties of m-sequences, these new series represent differently shifted versions of the original m-sequence. The sum of all

**1**s and

**−1**s within the m-sequence of each new series has the value of

**−1**(bottom right in the second part of Movie 2), implying a minimum correlation between shifted versions of an identical m-sequence (linear independence; Sutter, 1987, 2001; Sutter & Tran, 1992).

**1**represents a stimulation, whereas a

**−1**stands for a pause (step without stimulation). When cycling through the m-sequence, stimulations and pauses alternate corresponding to the m-sequence (first part of Movie 3; Figure 3; respective step highlighted in blue).

*– 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}**1**s represent the flash and

**−1**s typify pauses (Sutter, 2000, 2001; Sutter & Tran, 1992). In pattern stimulation the

**1**s and

**−1**s are usually bound to two different stimulus patterns; for example, they represent the checkerboard (

**1**) and the gray background (

**−1**) in pattern onset studies or the checkerboard (

**1**) and the reversed checkerboard (

**−1**) in pattern reversal studies (Baseler et al., 1994; Hoffmann, Straube, & Bach, 2003; Hood et al., 2003a; Unterlauft & Meigen, 2008).

**1**in the m-sequence adds the recorded data while a

**−1**subtracts the corresponding part of the recorded data. Finally, an average across all steps of the m-sequence results in a mean response to a specific shift. The cross-correlation is a concatenation of these mean responses for all possible shifts. This calculation can be accelerated by the fast m-transform presented by Sutter and Tran (1992). The waveform at the beginning of this cross-correlation trace (bottom right at the end of Movie 4) is identical to the response to a single stimulation (bottom left part of Movie 4), which can be better seen when both traces are displayed with the same temporal resolution (summary in Figure 9, first column).

**Figure 9**

**Figure 9**

*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.

**Figure 10**

**Figure 10**

**1**in this number indicates whether a specific point in time is relevant for the corresponding kernel. The prestimulus history for each order is always described by the first two digits, where the third digit is always

**1**and describes the current stimulus. In the following we superimpose these kernel responses either with a positive sign when the three digits contain an odd number of

**1**s (first order, third order) or with a negative sign when the three digits contain an even number of

**1**s (first and second slice of the second-order kernel). Whether these kernel responses are superimposed at all is determined by a bitwise logical AND operation between the m-sequence stimulus history and all kernel order binary numbers. Only when the result has an odd number of

**1**s (which coincides with an odd parity) the corresponding kernel order is superimposed (lower yellow rectangle).

**1**s represent the flash and

**−1**s typify pauses. The main response can be found in the first-order kernel using these settings. Higher order kernels usually have much smaller amplitudes than the first-order flash kernel (Figures 4 through 7). Even if higher order kernels contain additional information and induce components to lower order kernels, most of the time only the first-order kernel has been evaluated to examine the extent of impairment of retinal diseases. However, the simple evaluation has been sufficient to show that changes in mfERG correlate with morphologic changes. For example, in patients with age-related macular degeneration, central serous chorioretinopathy or retinal vascular occlusion mfERG reveals reduced amplitudes in respective affected areas (Feigl, Brown, Lovie-Kitchin, & Swann, 2004, 2006; Gerth, 2009; Gin, Luu, & Guymer, 2011; Hvarfner, Andreasson, & Larsson, 2006; Lai, Chan, & Lam, 2004; Lai et al., 2008; Vajaranant, Szlyk, Fishman, Gieser, & Seiple, 2002; Wildberger & Junghardt, 2002; Wu, Ayton, Guymer, & Luu, 2014; Wu, Ayton, Makeyeva, Guymer, & Luu, 2015; Yavas, Küsbeci, & Inan, 2014; Yip et al., 2010). Consequently, the multifocal techniques help distinguish hereditary or other rare disease. This might allow the investigator to focus adjacent genetic testing, implying a great sociopolitical advantage (Kellner et al., 2015). The sole evaluation of the first-order kernel of the mfERG already enables one to measure potential retinal impairment by toxic agents such as chloroquine or traumas (Adam, Covert, Stepien, & Han, 2012; Browning, 2013; Gjoerloff, Andreasson, & Ghosh, 2006; Kellner, Weinitz, & Kellner, 2009; Loevestam-Adrian, Andreasson, & Ponjavic, 2004; Marmor et al., 2011; Nebbioso, Grenga, & Karavitis, 2009; Park et al., 2011; Penrose et al., 2003; Stangos et al., 2007; Verdon, 2008). This potential allows perioperative monitoring in retinal surgery. In expert opinions, mfERG can be used to objectify the extent of impairment (Andreasson & Ghosh, 2014; Moschos et al., 2001; Shimada et al., 2011; Wallenten, Andreasson, & Ghosh, 2008).

**1**s and

**−1**s in pattern stimulation, the main response is found in the first slice of the second-order kernel because this kernel characterizes the inversion (see M-sequence–based stimulation). Therefore, the evaluation of pattern stimulation–based electrophysiology focuses on the respective kernel. As the Pattern ERG might be more sensitive than the Flash ERG regarding function impairment and progression, the combination has been shown to roughly appraise future progress in some diseases. Hence, monofocal Pattern ERG was already used for classifications like in Stargardt macular dystrophy (Fujinami et al., 2013; Lois, Holder, Bunce, Fitzke, & Bird, 2001).

*British Journal of Ophthalmology*, 96 (5), 723–729.

*Graefe's Archives of Clinical and Experimental Ophthalmology*, 252 (12), 1903–1909.

*Ophthalmology*, 101, 1867–1868.

*Experimental Brain Research*, 53, 285–294.

*European Journal of Ophthalmology*, 11, 41–49.

*Optometry and Vision Science*, 85 (6), 385–395.

*British Journal of Ophthalmology*, 92, 396–403.

*Cell and Tissue Research*, 253 (2), 287–296.

*Documenta Ophthalmologica*, 106, 183–188.

*Electroencephalography and Clinical Neurophysiology*, 90 (1), 65–81.

*Progress in Retinal and Eye Research*, 25 (5), 425–448.

*Current Diabetes Reports*, 14 (9), 526.

*Current Eye Research*, 22 (1), 54–63.

*Ophthalmology*, 101, 309–314.

*American Journal of Ophthalmology*, 155 (3), 418–428.

*Neurologic Clinics*, 6 (4), 657–679.

*Clinical and Experimental Optometry*, 88 (6), 410–414.

*British Journal of Ophthalmology*, 95 (11), 1560–1563.

*Clinical and Experimental Optometry*, 94 (3), 247–258.

*Archives of Ophthalmology*, 120, 870–871.

*Stem Cells International*, 2012, 263–264.

*Graefe's Archives of Clinical and Experimental Ophthalmology*, 250 (2), 261–270.

*Brain Research*, 279, 271–277.

*Documenta Ophthalmologica*, 83 (2), 163–173.

*Clinical and Experimental Optometry*, 87 (3), 163–170.

*System identification for self-adaptive control*. New York, NY: Wiley.

*Current Eye Research*, 29, 327–336.

*Current Eye Research*, 31, 635–644.

*Documenta Ophthalmologica*, 109 (1), 87–100.

*American Journal of Ophthalmology*, 155 (6), 1075–1088.

*Documenta Ophthalmologica*, 118 (1), 63–68.

*Graefe's Archives of Clinical and Experimental Ophthalmology*, 244 (1), 83–89.

*American Journal of Ophthalmology*, 133, 29–39.

*Shift register sequences*. Laguna Hills, CA: Aegean Park Press.

*Journal of Glaucoma*, 9, 10–19.

*Klinische Monatsblätter für Augenheilkunde*, 179, 173–176.

*Documenta Ophthalmologica*, 113, 21–27.

*Optometry and Vision Science*, 83 (7), 473–485.

*Graefe's Archives of Clinical and Experimental Ophthalmology*, 239 (8), 556–561.

*The Journal of Neuroscience*, 23, 8921–8930.

*Vision Research*, 45 (25–26), 3244–3252.

*Principles and practice of clinical electrophysiology of vision*(2nd ed., pp. 197–205). Cambridge, MA: MIT Press.

*Documenta Ophthalmologica*, 124, 1–13.

*Journal of Neuro-Ophthalmology*, 23 (3), 225–235.

*Journal of Neuro-Ophthalmology*, 23 (4), 279–289.

*Progress in Retinal and Eye Research*, 22 (2), 201–251.

*Documenta Ophthalmologica*, 100, 115–137.

*Documenta Ophthalmologica*, 104 (3), 303–320.

*Documenta Ophthalmologica*, 109 (2), 177–187.

*Archives of Ophthalmology*, 109, 816–824.

*Retina*, 26 (3), 292–296.

*Medical & Biological Engineering & Computing*, 40, 471–478.

*Documenta Ophthalmologica*, 121, 197–204.

*Eye*, 25 (10), 1302–1309.

*Graefe's Archives of Clinical and Experimental Ophthalmology*, 241 (6), 522.

*Principles and practice of clinical electrophysiology of vision*(2nd ed., pp. 319–340). Cambridge, MA: MIT Press.

*British Journal of Ophthalmology*, 93, 1444–1447.

*Klinische Monatsblätter für Augenheilkunde*, 232, 250–258.

*Ophthalmic Research*, 39 (5), 282–288.

*Nonlinear vision*(pp. 109–170). Boca Raton, FL: CRC Press.

*Ophthalmology*, 107, 2283–2299.

*Ophthalmology*, 114 (9), 1613–1621.

*Documenta Ophthalmologica*, 100, 155–165.

*Documenta Ophthalmologica*, 126 (1), 57–67.

*Documenta Ophthalmologica*, 100, 99–113.

*Semi-numerical algorithms*(2nd ed.). Reading, MA: Addison-Wesley.

*Documenta Ophthalmologica*, 126 (1), 45–56.

*American Journal of Ophthalmology*, 137 (5), 826–833.

*Documenta Ophthalmologica*, 116 (1), 29–40.

*Survey of Ophthalmology*, 52, 61–96.

*Biological Cybernetics*, 35 (3), 145–160.

*Multiple Sclerosis Journal*, 16, 412–426.

*Documenta Ophthalmologica*, 126 (2), 117–123.

*Documenta Ophthalmologica*, 130 (3), 197–209.

*Current Eye Research*, 34 (7), 523–535.

*Documenta Ophthalmologica*, 109 (2), 115–121.

*Archives of Ophthalmology*, 119 (3), 359–369.

*Proceedings of the Institute of Electrical and Electronic Engineers*, 35 (3), 145–160.

*Analysis of physiological systems*. New York, NY: Plenum Press.

*Documenta Ophthalmologica*, 106 (2), 105–115.

*Ophthalmology*, 118 (2), 415–422.

*Vision Research*, 47 (11), 1445–1454.

*Biological Cybernetics*, 47, 95–102.

*Journal of Neurology, Neurosurgery and Psychiatry With Practical Neurology*, 72, 523–526.

*European Journal of Ophthalmology*, 11 (3), 296–300.

*Journal of Ocular Pharmacology and Therapeutics*, 25 (3), 249–258.

*Documenta Ophthalmologica*, 108, 157–163.

*Principles and practice of clinical electrophysiology of vision*(2nd ed., pp. 479–485). Cambridge, MA: MIT Press.

*Documenta Ophthalmologica*120 (1), 111–119.

*Documenta Ophthalmologica*, 106 (3), 327–332.

*Strabismus*, 11 (4), 229–237.

*Documenta Ophthalmologica*, 112 (3), 157–168.

*Documenta Ophthalmologica*, 114 (1), 9–19.

*Korean Journal of Ophthalmology*, 25 (4), 262–267.

*Retina*, 23, 503–512.

*Graefe's Archives of Clinical and Experimental Ophthalmology*, 243 (1), 38–42.

*Documenta Ophthalmologica*, 106, 137–143.

*British Journal of Ophthalmology*, 99 (4), 500–507.

*Proceedings of the Institution of Electrical Engineers*, 114 (2), 305–308.

*Vision Research*, 26, 1461–1480.

*Clinical and Experimental Optometry*, 94 (3), 296–301.

*Archives of Ophthalmology*, 2, 296–298.

*American Journal of Ophthalmology*, 144 (1), 131–134.

*Vision Research*, 44 (1), 103–112.

*Advanced Methods of Physiological Systems Modelling*, 1, 303–315.

*Investigative Ophthalmology & Visual Science*, 29, 433.

*Documenta Ophthalmologica*, 100 (2–3), 49–75.

*Vision Research*, 41, 1241–1255.

*Vision Research*, 32 (3), 433–446.

*Klinische Monatsblätter für Augenheilkunde*, 179, 169–172.

*Der Ophthalmologe*, 105 (12), 1135–1141.

*Ophthalmology*, 109, 1243–1250.

*Optometry and Vision Science*, 85 (1), 17–26.

*Theory of functionals and of the integral and integro-differential equations*. Mineola, NY: Dover.

*Retina*, 28 (4), 558–563.

*Nonlinear problems in random theory*. Cambridge, MA: MIT Press.

*Klinische Monatsblätter für Augenheilkunde*, 219, 254–258.

*Documenta Ophthalmologica*, 121 (3), 189–196.

*Documenta Ophthalmologica*, 112 (3), 201–207.

*Documenta Ophthalmologica*, 129 (3), 167–175.

*Documenta Ophthalmologica*, 140 (2), 193–200.

*Vision Research*, 43, 1689–1698.

*Documenta Ophthalmologica*, 104 (3), 287–302.

*Journal of the Society for Industrial and Applied Mathematics*, 7, 31–48.

*Information and Control*, 13, 541–554.