aguirre 2011 de bruijn cycles for neural decoding

Technical Note

de Bruijn cycles for neural decoding

Geoffrey Karl Aguirre !, Marcelo Gomes Mattar, Lucía Magis-WeinbergDepartment of Neurology, University of Pennsylvania, Philadelphia, PA 19104, USA

a b s t r a c ta r t i c l e i n f o

Article history:Received 6 January 2011Revised 30 January 2011Accepted 1 February 2011Available online xxxx

Keywords:BOLD fMRICarry-over designsMVPAAdaptationDe BruijnM-sequences

Stimulus counterbalance is critical for studies of neural habituation, bias, anticipation, and (more generally) theeffect of stimulus history and context.We introduce de Bruijn cycles, a class of combinatorial objects, as the idealsource of pseudo-random stimulus sequences with arbitrary levels of counterbalance. Neuro-vascular imagingstudies (such as BOLD fMRI) have an additional requirement imposed by the !ltering and noise properties of themethod: only some temporal frequencies of neural modulation are detectable. Extant methods of generatingcounterbalanced stimulus sequences yield neural modulations that are weakly (or not at all) detected by BOLDfMRI. We solve this limitation using a novel “path-guided” approach for the generation of de Bruijn cycles. Thealgorithm encodes a hypothesized neural modulation of speci!c temporal frequency within the seeminglyrandomorder of events. By positioning themodulation between the signal and noise bands of the neuro-vascularimaging method, the resulting sequence markedly improves detection power. These sequences may be used tostudy stimulus context and history effects in a manner not previously possible.

© 2011 Elsevier Inc. All rights reserved.

Introduction

Neuroscience experiments routinely examine the relationshipbetween a set of stimuli and the neural responses they evoke. Whenpresented sequentially and repeatedly, the ordering of the stimuli isfundamental to the inferences drawn. For the neuroimaging modalityof BOLD fMRI, methodological advancement has been tied to novelsequence design, with increasingly sophisticated inferences followingthe introduction of “blocked” (Bandettini et al., 1993), sparse andrapid “event-related” (Dale and Buckner, 1997), and “adaptation”(Grill-Spector and Malach, 2001; Henson and Rugg, 2003) studies.While multi-voxel pattern analysis (MVPA; Haxby et al., 2001) canuse data derived from any of these sequencing approaches, stimulusordering still has important consequences for inference and repro-ducibility (Aguirre, 2007).

Stimulus ordering plays an especially critical role in studies of“carry-over effects” (Aguirre, 2007): the impact of stimulus historyand context upon neural response. All studies of neural adaptation(Desimone, 1996) are measures of carry-over effects, as are studies ofanticipation, priming, bias (Kahn et al., 2010), contrast (Gescheider,1988), and temporal non-linearity. These effects are measuredef!ciently and without bias in the setting of counterbalanced stimulussequences (Aguirre, 2007).

A sequence in which every stimulus appears immediately beforeevery other stimulus is counterbalanced at the n=2 level, meaningthat every possible sequential pairing of stimuli is included an equal

number of times. Higher-level counterbalancing is possible, in whichevery triplet (or n-tuple) of possible sequential stimulus orderings ispresent. In fact, n=3 counterbalancing or greater is preferred in BOLDfMRI studies that seek tomeasure carry-over effects. This requirementderives from the ability to model only the history of stimuli, asopposed to the history of neural states of the system (see Appendix Afor a complete examination of this issue).

Experiments using BOLD fMRI (and other neuroimaging methodsbased on vascular response; e.g. near-infrared optical tomography)face an additional challenge. The neural signal must be detected in thepresence of both a low-pass hemodynamic response function andpink (1/f) autocorrelation in the noise (Zarahn et al., 1997). As aconsequence, only neural modulations at certain temporal frequen-cies are detectable.

Extant methods of generating counterbalanced stimulussequences yield neural modulations that are weakly (or not at all)detected by BOLD fMRI, and this limitation grows with ever higherlevels of counterbalance. There is no way currently to design neuro-vascular imaging experiments that provide both statistical power andthe inferential power of higher levels of stimulus counterbalance.

Here we describe a new foundation for the design of imagingstudies, based upon the guided construction of de Bruijn cycles. DeBruijn cycles are a class of combinatorial objects that provide arbitrarylevels of stimulus counterbalance in a maximally compact sequence.The ordering of the stimuli can be pseudo-random and therefore notpredictable by the subject. They are generally useful for anyneuroscience study that examines or controls the in"uence ofstimulus order upon neural response.

To fully realize the potential of de Bruijn cycles for BOLD fMRI,however, an additional development is needed. We have created a

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! Corresponding author at: Department of Neurology, University of Pennsylvania, 3West Gates, 3400 Spruce Street, Philadelphia, PA 19104, USA. Fax: +1 215 349 5579.

E-mail address: [email protected] (G.K. Aguirre).

YNIMG-08058; No. of pages: 8; 4C:

1053-8119/$ – see front matter © 2011 Elsevier Inc. All rights reserved.doi:10.1016/j.neuroimage.2011.02.005

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Please cite this article as: Aguirre, G.K., et al., de Bruijn cycles for neural decoding, NeuroImage (2011), doi:10.1016/j.neuro-image.2011.02.005

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novel, “path-guided” algorithm for the creation of de Bruijn cycleswhich solves the problem of poor statistical power. The resultingsequence encodes in the seemingly random order of stimuli a neuralmodulation at one or more speci!c frequencies. These second-ordertemporal "uctuations may be positioned to be optimal for the signaland noise properties of the imaging technique. In application, theencoded modulation will only be detected if the hypothesized neuralrepresentation is present. This solution provides a many-foldimprovement in statistical power over extant approaches.

Using path-guided sequences, an entire domain of study of therepresentation and effect of stimulus context and history is renderedtractable. We begin with the properties of de Bruijn cycles and ourpath-guided algorithm before considering an example experiment.

Theory

de Bruijn cycles

Consider an experiment with k different stimuli (including any“blank” or null stimuli, which themselves constitute an experimentalcondition). We consider the order in which stimuli are presented byassigning each a label; for convenience we will use the naturalnumbers {0,1,2,3,…,k-1}. De Bruijn sequences are a cyclic ordering of klabels of order-n such that travel through the sequence yields everypossible sub-sequence of n consecutive labels. Equivalent statementsof this property are that de Bruijn sequences contain every “word” oflength n in the alphabet of k labels, or that de Bruijn sequences arecounterbalanced at the level of n. The sequence has a length of kn.Cyclic sequences of this kind are a subject of study in discretemathematics and have been discovered independently many times.Nicolaas Govert de Bruijn (pronounced roughly “duh-BROUN” or"duh-BROIN"), with Tania van Ardenne-Ehrenfest, formalized thesequences for k>2, and determined that the number of distinctsequences for a given k and n is:

k!kn!1! "

kn

which is clearly vast for even small values of k and n.De Bruijn sequences may be represented, and thus constructed, as

a closed path that visits every vertex of a de Bruijn graph: a network ofkn connected nodes in which each node is one of the length n wordsthat may be assembled from the k labels (Fig. 1). One vertex has adirected edge to another if the word obtained by deleting the symbolof the former word is the same as the word obtained by deleting thelast symbol of the latter word. The de Bruijn cycle is given by aHamiltonian cycle through the directed graph, which is a path thatvisits each vertex once and returns to the starting vertex.

Finding Hamiltonian circuits is an NP-complete problem (Karp,1972), and thus generally solved by brute-force search algorithms.One approach begins at a random node in the graph and then selectsrandomly amongst the available neighboring nodes to trace a path. If adead-end is reached from which there is no available next node, a“backtracking” process is used to re-route. This search approach is

guaranteed to produce a valid Hamiltonian path (and thus a de Bruijn)cycle within a de Bruijn graph.

The de Bruijn cycle for a given k and n de!nes the sequentialpresentation of k different stimuli with the guarantee of nth-ordercounterbalance. Cycles of this form are of utility to a broad range ofneuroscience experiments, although there has been only limited use ofgeneral de Bruijn cycles for this purpose (Brimijoin and O'Neill, 2010;Appendix B describes a taxonomy of de Bruijn sequences, encompass-ing counterbalanced sequences that have been used previously inneuroimaging studies). de Bruijn cycles hold the same potentialbene!ts for neuro-vascular imaging studies, providing both rigor and"exibility in experimental design to support more robust and subtleinferences. To realize this potential, however, the idiosyncratic signaland noise properties of the measure must be addressed.

Detection power

The properties of BOLD fMRI (and other neuro-vascular methods)render some experimental designs more statistically powerful thanothers. Speci!cally, variations in neural activity which approximatesinusoids of ~0.01–0.1 Hz are well detected (Zarahn et al., 1997).

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Fig. 1. A k=3, n=2 de Bruijn graph and cycle. For an alphabet of three letters {0,1,2},there are 9, two-letter words. Each node (light blue) in the de Bruijn graph is a possibletwo-letter word. The directed edges between the nodes indicate available sequentialtransitions between one word and the next, each of which shifts the word by oneregister and adds a new letter. The red lines indicate one of 24 possible Hamiltoniancycles through the graph. Below the graph is the necklace of nodes indicated by the redpath. In the bottom-right is the de Bruijn cycle, derived from the last register of eachbead on the necklace. Travel about the cycle yields every possible two-letter word in thethree-letter alphabet. (For interpretation of the references to colour in this !gurelegend, the reader is referred to the web version of this article.)

Fig. 2. An example experiment with a path-guided de Bruijn sequence. (a) The experiment calls for 17 stimuli (16 and a null-trial) and n=2 counterbalance. The corresponding deBruijn graph is shown, with an enlargement to the right of a small portion. In the enlargement, a segment of a Hamiltonian circuit is shown in red (the entire path visits every node).The grey background is theweb of the connections between nodes. (b) The example experiment uses 16 stimuli drawn from a di-octagonal sampling of a two-dimensional space. Thelabels for each of the 16 stimuli in the space are shown, along with the matrix that is the distance between each stimulus in the space (black near, white far). An example stimulusset—gratings varying in spatial frequency and orientation—is shown to the right. (c) The Hamiltonian cycle through the de Bruijn graph was guided by a sinusoidal function with aperiod of 32 elements (48 s). The sequence generated is pseudo-random. The sequence of labels for the portion of the plot shaded in green is shown to the right. The sequenceelements marked in red correspond to the nodes and path indicated in red in panel a. The !nal value of the sequence is enclosed in parentheses as it is also the !rst value of thesequence, completing the cycle. (d) Given the distance matrix in panel b, the path-guided de Bruijn cycle encodes a series of sequential stimulus transitions that vary systematicallyin their distance. The plot demonstrates the sinusoidal modulation encoded in the seemingly random sequence. The green portion of the plot is expanded to the right, andcorresponds to the sequence elements shown in panel c. The di-octagonal schematics show the location and sizes of the transitions through the stimulus space that produce overallsinusoidal modulation. Note that some transitions (e.g., between “blank” stimuli, or from a “blank” to a stimulus) have an unde!ned distance, and so are omitted from the plot. (Forinterpretation of the references to colour in this !gure legend, the reader is referred to the web version of this article.)

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Higher frequencies are attenuated by the dispersed hemodynamicresponse, while lower frequencies are lost within the pink (1/f) noiseof the system.

Several techniques have been used to search for, or iterativelycraft, sequences with improved detection power while retaining apseudo-random order (Aguirre, 2007; Wager and Nichols, 2003; Liu,2004). These approaches provide at most n=2 counterbalance (or anapproximation thereof; Wager and Nichols, 2003; optseq2; http://www.nitrc.org/projects/optseq/).

Ideally, we would use a stimulus ordering de!ned by a de Bruijncycle with the desired level of counterbalance, and pick a particularcycle that provides for optimal detection power. The vast space ofpossible de Bruijn sequences renders a random search inef!cient. Wedescribe here a novel approach to the generation of de Bruijn cycles, inwhich the Hamiltonian cycle through the de Bruijn graph may beguided to craft a sequence with the desired properties.

Path-guided sequences to increase decoding detection power

Given variation in the amplitude of expected responses fordifferent stimuli or transitions, a sequence of stimuli may be createdthat will produce a modulation of neural activity within the band oftemporal frequencies that are optimal for the imaging method. This isdone by de!ning a guide function, which concentrates power withinthe desired band of temporal frequencies, and then biasing the paththrough the de Bruijn graph to follow that function. The guidefunction may be simply a sinusoid of the desired frequency, or thesum of sinusoids of random phases within a frequency range.

A path-guided sequence is generated by ranking the nodes that areavailable as the next point in the growing Hamiltonian path. Theranking favors nodes (and thus transitions) predicted to producelarger or smaller neural responses and this bias varies over the courseof the sequence according to the guide function. (See Appendix C fordetails of the algorithm; code available for download at http://cfn.upenn.edu/aguirre.)

The resulting sequences are ideal for a great many neuroimagingexperiments: pseudo-random, with arbitrary levels of counterbal-ance, but with a tailored statistical power many times greater thanthat obtained by even extensive computational searches of possiblestimulus orders.

Example experiment

We consider an example experiment designed to detect neuralhabituation (adaptation) that is proportional to stimulus dissimilarity(or distance within a stimulus space) (Jiang et al., 2006; Drucker andAguirre, 2009;Drucker et al., 2009). As is discussed inAppendix A, this isbest examined with at least n=3 counterbalance. We will show thatthis is not possiblewith extantmethods, but canbe achievedusingpath-guided de Bruijn sequences. We begin with an example of the n=2case—as this is more readily illustrated and compared to existingmethods—before considering a third-level counterbalanced design.

Second-level counterbalance

Consider a set of 16 stimuli drawn from a two-dimensionalstimulus space. We wish to test if the response to a given stimulus ismodulated in proportion to its similarity to the preceding stimulus, ashas been examined in many studies (e.g., Jiang et al., 2006; Druckerand Aguirre, 2009; Drucker et al., 2009); such an experimentexamines population codes for stimulus properties. Including ablank or “null” trial yields 17 different experimental states, requiringa k=17 sequence for presentation order. If each stimulus is to bepresented for 1.5 s, a !rst-order counterbalanced (n=2) sequenceyields ~8 min of data collection.

A k=17, n=2 de Bruijn cycle may be de!ned as the Hamiltoniancycle through the corresponding de Bruijn graph (Fig. 2a). There areover 8 ! 10244 possible cycles, rendering a search for sequences withgood detection power inef!cient. Instead, wewill guide a Hamiltoniancycle predicted to induce a sinusoidal modulation of the neuralresponse with a period of 32 s (0.02 Hz).

The stimuli are sampled from the two-dimensional space on a di-octagonal grid (Drucker et al., 2009) (Fig. 2b; the demonstration doesnot depend in any special way upon this sampling). The particularstimuli studied could be faces that differ in features, or shapes thatvary in aspect ratio or curvature, or, as illustrated, gratings that vary inorientation and spatial frequency. The perceptual distance betweenany pairing of the stimuli is de!ned by a dissimilarity matrix. We biasthe search along the growing Hamiltonian circuit to rank neighboringnodes by perceptual dissimilarity following the sinusoidal guidefunction.

A resulting path-guided sequence (Fig. 2c) is n=2 counter-balanced across the 17 stimuli. The sequence is pseudo-random, withessentially no sequential predictability (conditional entropy,2H2=4.08 bits; Liu, 2004). Encoded in this ordering of stimuli is theguide function, which is revealed by plotting the perceptual distancebetween sequential stimuli (Fig. 2d). A sinusoidal modulation ofsequential distance is readily apparent. Relative detection power(DPrel; Liu, 2004; Friston et al., 1999) is calculated with respect to ahemodynamic response function (Aguirre et al., 1998) and a model oflow-frequency noise (here a 0.01 Hz high-pass !lter). If neuraladaptation is proportional to perceptual distance in this experiment,this example provides DPrel=0.64 (a perfect sinusoidal modulation atthis frequency is an eigenmode of the system and has a DPrel of 1.0).Varying the frequency of encoded sinusoid and searching across path-guided sequences, we have found sequences for this experimentaldesign with a DPrel as high as 0.74.

Experimental measurement of the sinusoidal signal modulationproportional to stimulus dissimilarity would be the target of a study ofneural adaptation and population coding. Perhaps surprisingly, thissequence also has important utility in an MVPA study of thedistributed pattern of voxel responses to the stimuli themselves.Using this sequence, the nuisance carry-over effects may be ef!cientlymodeled and removed from the data, revealing the neural coding ofthe individual stimuli free of preceding context.

Table 1 presents the best performance of sequences obtained usingextant methods for generating counterbalanced sequences. The bestsequence had less than half of the relative detection power of thepath-guided de Bruijn sequence. Additional measures of sequencerandomness and balance are presented, and it can be seen that thepath-guided de Bruijn sequence does not differ in any substantial wayfrom the randomly generated de Bruijn sequences or an m-sequencein these measures.

Third-level counterbalance

The advantage of the path-guided de Bruijn approach becomesmore apparent with higher levels of stimulus counterbalance. Assequence length increases geometrically with n, shorter stimuluspresentation times are needed if the experiment is to be of practicalduration. Continuing with our example, n=3 counterbalance for the17 stimuli would require a ~20 min study if each stimulus werepresented for 250 ms. Stimulus presentations of this brevity readilyproduce neural adaptation along the visual pathway (Kourtzi &Huberle, 2005), and may enhance early adaptive neural responses,which have greater stimulus speci!city (Verhoef et al., 2008).

Previously, this experiment could only be conducted by de!ning anm-sequence of the appropriate length. The "aw of such a design is thatthe rapid stimulus presentation and requirement for counterbalanceplaces the bulk of signal modulation at high temporal frequenciesrelative to the hemodynamic response function (Fig, 3a). Searching



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across 1,000,000 k=17, n=3, m-sequence variations (permuting theassignment of sequence label to stimulus; Aguirre 2007, Appendix A.3)we !nd the best DPrel is 0.048. This is suf!ciently low to render theexperiment infeasible.

With the path-guided approach, however, we may encode alower-frequency modulation within the pattern of rapid presenta-tions of stimuli. We guided a de Bruijn sequence with a sinusoidalperiod of 289 elements (0.014 Hz). The resulting sequence hasDPrel=0.32, a nearly 7-fold improvement over the best extanttechnique for stimulus ordering. The power spectrum of the path-guided sequence shows the encoded sinusoidal modulation posi-tioned between the !ltering and noise properties of the BOLD fMRIsystem (Fig, 3b).

Discussion

Context is a key determinant of neural representation. Stimuluscarry-over is therefore either quarry or confound in essentially everyneuroscience experiment. De Bruijn cycles, providing arbitrarycontrol of counterbalance for any number of stimuli, are offundamental utility in decoding neural representation.

The characterization of a neural response function measures the“direct effect” of each stimulus upon neural activity, theoreticallyindependent of stimulus context. In such studies, carry-over is anuisance which can bias the results if not properly modeled (Kahn etal., 2010). Stimulus counterbalance allows the direct and carry-overeffects to be modeled independently and without bias (Kempton etal., 2001), providing the opportunity to resolve the confound.

Multi-voxel pattern analysis makes use of the distribution of directeffects across brain locations. Un-modeled carry-over effects decreasethe reproducibility and statistical power of the technique if differentstimulus sequences are used during “training” and “test” phases.Incorrect inferences result if sequence patterns are repeated acrossphases. More generally, the attribution of neural coding to the effect ofthe stimuli themselves, as opposed to their status within the contextof the other stimuli, requires attention to counterbalance.

Finally, carry-over effects are themselves the target of study inmanyneuroscience experiments. Our example experiment was tailored todetect a hypothesized neural adaptation effect. Carry-over effects cantakemany other forms (bias, contrast, etc.), and the techniqueswe haveintroduced may be used to create sequences optimized for the study ofany of these phenomena.

The implementation of path-guided sequence constructionmay beexpanded greatly beyond that presented here. Using a guide function

composed of a range of sinusoids, the encoded (second order)modulationmay be rendered more stochastic without substantial lossof detection power.Moreover, differentmodulationsmay be guided atdifferent carrier frequencies, creating a sequence simultaneouslyoptimized for the detection of two orthogonal neural responsepatterns.

The construction of sequences that encode multiple possibleresponse patterns can provide greater generality and "exibility thanour example design. The example study is designed to test for thepresence of neural adaptation that is proportional to stimulussimilarity. In this case, the design is precise: it is optimized to test aparticular hypothesis, and has less power to detect other forms ofneural response. This precision should not be mistaken for “bias,” inthat the experiment will only detect the encoded pattern to a degreethat re"ects its match to the neural response.

It should further be understood that this speci!city is a feature ofthe example experiment, and not of carry-over designs or path-guided de Bruijn sequences in general. The entire space of possiblein"uences of stimulus history upon response is captured in a carry-over matrix (Aguirre 2007), and may be decomposed into a basis set

Table 1Properties of k=17, n=2 sequences.

Sequence DPrel-adapt DPrel-main Autocorrelation Balance Uniformity

Path-guidedde Bruijn

0.74 0.31 1.09 1918 95

Randomde Bruijn

0.29 0.34 0.99 1309 96

Type 1, index 1 0.28 0.34 1.03 293 26m-sequence 0.32 0.32 1.22 1135 94

Sequence properties calculated for a single, path-guided de Bruijn sequence, aswell as thebest performing of each of 1,000,00 randomly sequences of each type. The random deBruijn sequences were generated from randomHamiltonian cycle; the type 1 index 1 andm-sequenceswere produced by searching across permutations of the assignment of labelsto stimuli. DPrel-adapt calculated by taking the variance of the hypothesized neuralmodulation proportional to sequential stimulus distance as the denominator, and thenumerator as the variance of that modulation following application of a 0.01 Hz high-passnotch !lter and convolutionwith a hemodynamic response function (Aguirre et al., 1998).DPrel-main was calculated for the main (direct) effect of presentation of any of the 16stimuli versus thenull-trials.Autocorrelation is the square rootof the SSQof the correlationof a sequence with itself at every lag except zero. Balance is a measure of the evendistribution of labels across the sequence; small numbers indicate more balance (Hsieh etal., 2004). Uniformity is ameasure of spacing of labels across the sequence; small numbersindicate a more uniform distribution of labels (Hsieh et al., 2004).

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Fig. 3. Power spectra of k=17, n=3 sequences. (a) 1,000,000 label permutations of anm-sequence were searched for the sequence with the greatest DPrel-adapt for thestimulus space shown in Fig. 2b, given a stimulus presentation time of 250 ms. Thepower spectrum of the distance transitions for this best sequence is shown in blue. Inblack is the transfer function of the hemodynamic response, which also incorporates a0.01 Hz high-pass notch !lter to represent the loss of sensitivity for the elevated 1/fnoise range. Effectively, only neural variance under the curve of the black line will berepresented in the BOLD fMRI signal. The shaded green region (0–0.25 Hz) is expandedin the inset. (b) A path-guided de Bruijn sequence was created, encoding a 0.014 Hzmodulation in the distance transitions. The power spectrum is shown in red.Experimental variance is concentrated at the ideal location with respect to the !lteringproperties of the system. (For interpretation of the references to colour in this !gurelegend, the reader is referred to the web version of this article.)





of matrices that represent particular hypothesized in"uences ofstimulus properties and timing upon neural response. A path-guidedde Bruijn sequence may be generated that is sensitive to a set of thesematrices, thus providing the ability to "exibly estimate the space ofpossible responses. Unavoidably, the more "exible and general anexperimental design, the less power it has to test for any particulartype of neural response.

Despite their "exibility and power, path-guided de Bruijnsequences have limits. In general, the !delity of the encodedmodulation to the guide function will be related to the range anddistribution of steps possible in the distance matrix. Anotherlimitation is that provision of ever higher levels of counterbalancewill reduce the !delity of the modulation for shorter (higherfrequency) cycles, as the available neighbors in the Hamiltoniancircuit become further constrained by the path history. Fortunately,this limitation is ameliorated by using a lower-frequency guidefunction, which is required regardless by the use of shorter stimuluspresentations in long sequences.

A deeper limitation relates to the distribution of higher, uncon-trolled levels of counterbalance. Our method establishes counterbal-ance at a speci!ed level and encodes a particular modulation ofstimuli in that sequence. To encode the modulation, however, thesequence will necessarily induce imperfect counterbalance at stillhigher levels. This is not a limitation of our method in particular, but isinstead an unavoidable property of any sequence which is notin!nitely long and utterly uncorrelated.

The application of de Bruijn cycles to decode neural states followsfrom their use in cryptography. For example, a de Bruijn cycle providesan ef!cient brute-force attack upon certain kinds of digital locks (thosethat do not use an “enter” key and accept the last n digits entered). Theef!ciency that a de Bruijn cycle provides in traversing the space ofpossible input states to a system is the basis for its usefulness in aneuroscience context. Beyond the general application of thesesequences for neuroscience experiments, the path-guided approachdramatically improves the statistical power of neuro-vascular studies bytailoring experimental design to the signal and noise properties of theimaging method. The resulting sequences permit previously unachie-vable experiments, in which higher-order counterbalance reveals theeffect of stimulus history and context on neural response.

Acknowledgments

Thanks to Shivakumar (Shiva) Vishwanathan for prompting thediscussion included in Appendix A, and to Joshua Cooper of theUniversity of South Carolina for general assistance with graph theoryand terminology. This work was supported a Burroughs-WellcomeCareer Development Award and the Integrated InterdisciplinaryTraining in Computational Neuroscience program, 5-R90-DA023424.

Appendix A. The level of counterbalance in neuro-vascular studies

Generally, neuro-vascular imaging studies require stimulus coun-terbalance at a level one greater than the level at which inference ismade. That is, measures of direct-effects should be made in thecontext of a n=2 counterbalanced sequence, while the carry-overeffects of immediately preceding stimuli (e.g., in a study of neuraladaptation) are ideally obtained in a sequence that is at least n=3counterbalanced.

This additional level of counterbalance is mandated by a modelinglimitation in neuro-vascular studies. While one can model the carry-over effects of the prior stimulus (t-1) upon the current stimulus (t), itis actually the prior neural state of the system that modulates theresponse to the current stimulus. In BOLD fMRI studies, anindependent measure of the prior neural state of the system isgenerally not available. A possible concern is that the same priorstimulus (t-1) might leave the neural system in two different

amplitude states (perhaps, dependent upon the effect of the stimulusthat preceded the prior stimulus; e.g., the stimulus two back, t-2). Forexample, the t-1 stimulus may have been the same (along somerelevant modeled dimension) as the t-2 stimulus, and thus the neuralstate is relatively adapted. Presentation then of the current (t)stimulus may result in a different signal response depending upon theprior neural state. If this effect were unbalanced across the possiblestimulus pairings, it could introduce bias (the effect being dependentupon the particular sequence used and how second-order transitionswere distributed across !rst-order transitions).

This issue is best addressed by employing a sequence with anadditional level of counterbalance. This will ensure that the neuralamplitude state of the system prior to the current stimulus is alwaysbalanced across the possible transitions from the prior to currentstimulus (presuming that the systematic neural state of the systempriorto the current stimulus is entirely determined by the t-2 stimulus).

Appendix B. Speci!c forms of de Bruijn cycles

Speci!c forms of cyclic, counterbalanced sequences are containedwithin the larger class of de Bruijn sequences.

A modi!ed (or “punctured”) de Bruijn sequence is created byremoving one label (by convention, the label “zero”) from the singleoccurrence of the length n word composed of all zeros, yielding asequence of length kn!1. The set of maximum length sequences (orm-sequences) are contained within this set of punctured de Bruijnsequences. In addition to the counterbalance properties of all deBruijn sequences, m-sequences have the additional property of anearly "at power spectrum (alternatively stated, the autocorrelationfunction of the sequence approaches a delta function). Introduced foruse in BOLD fMRI studies by Buracas and Boynton (2002), m-sequences provide ideal ef!ciency for estimating the shape of thehemodynamic transfer function, although relatively weak detectionpower. M-sequences are generated with maximal linear feedbackshift registers, and exist only for alphabets in which k is the power of aprime number; there are no constraints upon n. An example m-sequence for the k=17, n=2 case is shown in Fig. B1a.

Type 1, Index 1 (T1I1) sequences are n=2 counterbalancedsequences of k labels. T1I1 sequences have the property of presentingpermuted blocks of the labels over the course of the sequence, withrepetition of the labels at the boundaries of each block. The sequenceswere described by Finney and Outhwaite (1956), and in Nonyane andTheobald (2008) presented an algorithm for the production of T1I1sequences with 6 or more labels. T1I1 sequences have some desirableproperties for BOLD fMRI experiments (Aguirre 2007). T1I1 sequenceshave excellent balance and uniformity (Hsieh et al., 2004), meaningthat labels are evenly distributed across the sequence. Consequently,neural effects are relatively insensitive to order effects across theentire experiment. In other words, the T1I1 sequence has relativelylittle power at very low frequencies, thus avoiding the elevated (1/f)noise range of BOLD fMRI. Limitations include the non-stochasticstructure of the sequence (each stimulus is guaranteed to appeartwice within 2k!2 trials, and there is a stimulus repetition every ktrials), the restriction to n=2 (i.e., no higher-order counterbalancingis possible as, for example, the word [010] is prohibited), and theabsence of T1I1 sequences for k=3, 4, or 5. An example T1I1 for thek=17, n=2 case is shown in Fig. B1b.

In contrast to T1I1 sequences which have a nearly evendistribution of labels across the sequence, the lexicographically leastde Bruijn sequence (also known as a “Ford” sequence) is highlydiscrepant (Cooper and Heitsch, 2010). This sequence is generated asthe concatenation of the primitive parts of the Lyndon words of the klabel alphabet in lexicographic order (Arndt 2010). There exists a “lexleast” de Bruijn sequence for every k and n. An example lex leastsequence for the k=17, n=2 case is shown in Fig. B1b.





A Kautz sequence (Kautz, 1968) is a non-de Bruijn sequence whichis essentially a de Bruijn cycle without self-adjacency of labels. It iscreated by !nding a Hamiltonian circuit through a Kautz Graph, whichis a modi!ed de Bruijn graph that eliminates those vertices that wouldlead to repetition of the !nal register of the nodes.

Appendix C. An algorithm for guided-path sequences

The pair-wise distances between k labels is speci!ed in a k!kmatrixD. We wish to create a de Bruijn sequence of k labels containing everyword of length n in which the distances corresponding to sequentialtransitions in the sequence approximate a guide function G of length kn.Thismay be done by guiding aHamiltonian circuit through the deBruijngraph composed of kn directionally connected nodes.

We !rst discretize the guide function G into b values. Next, thevalues in D are ranked and then divided into b bins, such that each bin

has an equal (or as close as possible) number of values. If the currentnode in the growing path is i, then:

1) Start at a random node (i=1)2) Identify them available neighbors (where 0"m" k). Ifm=0 then

initiate a back-track algorithm.3) If m=1, select the only available node. If mN1, select the bin

corresponding Gi. If there are no available nodes in that bin, selectthe next closest bin (randomly choose in the event of a tie).

4) Travel to a random neighbor within the selected bin, increment i.5) If i=kn and the current node is connected to the i=1 node, then

the sequence is !nished. Otherwise, continue with step 2.

Someminor stochasticity in node selection is desirable. This avoidsthe generation of a sequencewhich closely conforms to guide functionearly in the sequence, with a progressive departure from desired formas the path progresses and the set of available nodes decreases. For

0

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16

1 35 69 103 137 171 205 239 273

b Type 1, Index 1 sequence

c Lex-least sequence

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1 35 69 103 137 171 205 239 273

Labe

l

Sequence position

a M-sequence

0

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1 35 69 103 137 171 205 239 273

Fig. B1. Examples of particular k=17, n=2 de Bruijn sequences. Note that the apparent “ordering” at the start some of the sequences is a consequence of the use of the naturalnumbers as sequence labels. In application, there is no requirement that the initial stimuli presented to a subject will have a perceptual ranking, as the assignment of labels to stimulimay be arbitrary. (a) M-sequence. (b) Type 1, index 1 sequence. Gray bars indicate the permuted “blocks” of the labels, and the repeated labels at the borders of the blocks areindicated by blue points. (c) Lexicographically least sequence.





transitions between some stimuli (e.g., the stimuli that follow “null”trials or target stimuli), the distance matrix value may be unde!ned;these transitions can be randomly distributed throughout thesequence without disrupting the primary sinusoidal variation that isthe target of optimization.

The approach may also be described as assigning lengths to theconnections in the de Bruijn graph equal to the predicted distances inD. The path-guiding algorithm then favors “trips” of certain lengthsfollowing the guide function G.

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