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Journal of Neuroscience Methods 180 (2009) 171–184

Contents lists available at ScienceDirect

Journal of Neuroscience Methods

journa l homepage: www.e lsev ier .com/ locate / jneumeth

method for mapping response fields and determining intrinsic referencerames of single-unit activity: Applied to 3D head-unrestrained gaze shifts

erald P. Keitha,b, Joseph F.X. DeSouzaa,b,c,e, Xiaogang Yana,ongying Wanga, J. Douglas Crawforda,b,c,d,e,∗

Canadian Action and Perception Network, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3Department of Psychology, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3Department of Biology, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3Department of Kinesiology & Health Sciences, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3Neuroscience Graduate Diploma Program, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3

r t i c l e i n f o

rticle history:eceived 14 January 2009eceived in revised form 8 March 2009ccepted 9 March 2009

eywords:uperior colliculuseural activityimulationon-parametricegressionitting

a b s t r a c t

Natural movements towards a target show metric variations between trials. When movements combinecontributions from multiple body-parts, such as head-unrestrained gaze shifts involving both eye andhead rotation, the individual body-part movements may vary even more than the overall movement. Thegoal of this investigation was to develop a general method for both mapping sensory or motor responsefields of neurons and determining their intrinsic reference frames, where these movement variationsare actually utilized rather than avoided. We used head-unrestrained gaze shifts, three-dimensional (3D)geometry, and naturalistic distributions of eye and head orientation to explore the theoretical relationshipbetween the intrinsic reference frame of a sensorimotor neuron’s response field and the coherence ofthe activity when this response field is fitted non-parametrically using different kernel bandwidths indifferent reference frames. We measure how well the regression surface predicts unfitted data using thePREdictive Sum-of-Squares (PRESS) statistic. The reference frame with the smallest PRESS statistic was

ernel bandwidthain field

categorized as the intrinsic reference frame if the PRESS statistic was significantly larger in other referenceframes. We show that the method works best when targets are at regularly spaced positions within theresponse field’s active region, and that the method identifies the best kernel bandwidth for responsefield estimation. We describe how gain-field effects may be dealt with, and how to test neurons withina population that fall on a continuum between specific reference frames. This method may be applied to

gle-u

any spatially coherent sinbehaviors.

. Introduction

In any goal-directed action the location of the goal in sensoryoordinates must be transformed into a motor command in anffector-specific frame (Andersen, 1995; Crawford et al., 2004). Theeference frame of the sensory input is that of the part of the bodyo which the sensor is embedded (i.e., the eye for vision). The move-

ent reference frame is generally the more stable insertion point

f a muscle—i.e., the end that moves less when the muscle con-racts. So, for example, to use the eyes and head to look toward aisual stimulus, an eye-centered visual signal must be transformednto the appropriate commands for the eye muscles fixed in the

∗ Corresponding author at: York Centre for Vision Research, Computer Scienceuilding, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3.el.: +1 416 736 2100x88621; fax: +1 416 736 5857.

E-mail address: jdc@yorku.ca (J.D. Crawford).

165-0270/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.jneumeth.2009.03.004

nit activity related to sensation and/or movement during naturally varying

© 2009 Elsevier B.V. All rights reserved.

head, and the neck muscles fixed to the torso (Klier et al., 2003;Martinez-Trujillo et al., 2004).

These problems are well-defined at the level of ‘black box’input–output, but it is much less clear which, if any, referenceframes are used by individual neurons involved in the interme-diate transformations. Some have questioned the validity of thenotion that individual neurons encode information in a particu-lar reference frame (Pouget et al., 2002; Scott, 2008). However,the characterization of neural reference frames remains one of ourbest methods for understanding how the brain encodes space andproduces spatial transformations at the level of individual neu-rons (Stricanne et al., 1996; Klier et al., 2001; Avillac et al., 2005;Mullette-Gillman et al., 2005; Pesaran et al., 2006; Porter et al.,

2006; Blohm et al., 2008).

The current paper develops a novel method for determining theintrinsic reference frame of single-unit sensory or motor responsefields. In principle this method can be applied to any neural activ-ity that is involved in encoding spatial information, but we have

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72 G.P. Keith et al. / Journal of Neuro

eveloped the method specifically for the example of the rota-ion of eyes and head associated with gaze shifts. Most previoustudies of intrinsic neural reference frames have used either 1Dr 2D approaches to characterizing the response field and vary-ng initial eye position, generally in head-fixed preparations (e.g.,tricanne et al., 1996; Avillac et al., 2005; Porter et al., 2006). Thesepproaches have limitations that motivated our development of theurrent analytic technique. In particular they (1) cannot distinguishetween head- and body- or space-fixed reference frames; (2) doot allow for the coordinated eye and head movements that com-rise normal gaze shifts made in head-unrestrained conditions; (3)sually limit eye position between a few discrete positions; (4) doot account for torsional variations in eye and head position thatecome significant in head-unrestrained conditions; and (5) nei-her account for, nor allow, the exploration of models related tohe non-linear elements of rotation that predominate in the largerange of head-unrestrained gaze shifts.

Another approach to the question of intrinsic reference framess to electrically stimulate a group of neurons in a given brain area,dentifying the reference frame in which the consequent move-

ent response is most coherent across a variety of initial positions.or microstimulation, experimental and analytic techniques haveeen developed which address some of the above problems—i.e.,ead-unrestrained 3D recordings have been combined with non-

inear 3D analysis and comparison with model simulations (Kliert al., 2001; Martinez-Trujillo et al., 2004; Constantin et al., 2007).hese studies have demonstrated the importance of the non-linearD components in head-unrestrained conditions. However, stim-lation and unit recording provide different, complementary setsf information about the system, especially when 3D geometryequires that the neurons involved in reference frame transforma-ions should show different frames when tested using these twoifferent techniques—i.e., unit recording showing the input frameo the neuron, and stimulation showing the output frame (Smithnd Crawford, 2005; Blohm et al., 2008). Thus there remains theroblem of developing a complementary technique for single-unitecordings.

Determining the intrinsic reference frame for a neuron is closelyinked to the concept of its response field (i.e., its sensory receptiveeld or movement field). A neuron’s receptive field is the region ofpace in which the presence of a stimulus will change the firingate activity of the neuron (Hubel and Wiesel, 1959). Similarly, aeuron’s movement field is the region of space that contains theovements associated with the neuron’s non-background activ-

ty (Wurtz, 1969). These concepts are only meaningful when theyre defined in some specific reference frame, which compels uso ask what that intrinsic reference frame for a given neuron’sesponse field is. Moreover, in structures like the superior colliculus,here some neurons show sensory properties, some show motorroperties, and many show both (Munoz and Wurtz, 1995a,b), theapping of the neuron’s receptive and movement fields and the

etermination of their intrinsic reference frames provide separateieces of information that are clues to the function of both individ-al neurons and the overall neural population (Schlag et al., 1980;ouget et al., 2002; Smith and Crawford, 2005; Blohm et al., 2008).herefore, we developed a method that is applicable to determin-ng the intrinsic reference frames of both sensory receptive fieldsnd movement fields.

When this method is applied to head-unrestrained conditions,hile advantages (such as separation of the head frame from the

ody/space frame, and more natural behavior) arise, complications

o as well. The method common to all approaches for determininghe intrinsic reference frame of a neuron’s activity has been to mea-ure this activity over widely separated positions in the receptiver movement field while dissociating the possible reference frames.or example, a neuron’s activity in response to visual targets at reg-

ce Methods 180 (2009) 171–184

ularly spaced positions can be measured using different initial eyepositions, so that the target positions for different trials form a dis-crete set in both eye and head coordinates (e.g., Stricanne et al.,1996). The underlying assumption (which we support here) is thatthe response field will be more coherent – i.e., show less variabilityacross trials within spatial positions averaged across all positions –if plotted in the correct (intrinsic) reference frame. Most previousstudies have approached this problem by measuring activity repet-itively from a small number of initial and final positions, and thenusing an analytic method that treats position as a discrete quan-tity. However, this approach is not possible in behaviors such asnaturally coordinated head-unrestrained gaze shifts, because anygaze direction in space is naturally produced in successive trialsby combining different head and eye-in-head positions (Glenn andVilis, 1992). Thus, a different analytic method must be used in suchconditions—one that treats position as a continuous quantity.

The use of head-unrestrained conditions also results inincreased analytic opportunities and problems related to 3D rota-tion. The most obvious of these relates to torsional rotation of theeye and head around the line of sight. In head-fixed conditions,head torsion does not change and eye torsion is strictly limitedby Listing’s law (Ferman et al., 1987; Tweed et al., 1990; Tweedand Vilis, 1990). In head-unrestrained conditions, however, headand eye torsion are much more variable, resulting in eye-in-spacetorsion values as large as 20◦ (Tweed et al., 1998). As a result, thedirection of any space-fixed target will vary in eye coordinates (andnearly as much in head coordinates) for successive trials using asingle gaze direction. Moreover, if one does not account for thenon-linear aspects of rotation in 3D when performing coordinatetransformations on gaze, eye, and head orientations in the head-unrestrained range (for example, if the computation of target ineye coordinates is approximated as a vector subtraction of gazeposition from target position in space) then mathematical errorsin target direction as large as 90◦ are possible. Failure to account forthese various factors would result in large systematic and variableerrors, which would obscure the signal and mislead the experi-menter. Fortunately, the necessary experimental techniques andanalysis required to account for these rotational aspects in head-unrestrained monkeys are essentially already available (Crawford etal., 1999; Martinez-Trujillo et al., 2004), and allow testing betweenmodels that are indistinguishable in linear approximations (Klieret al., 2001).

The goal of the current study was to develop an analytic methodby which the response fields of neurons could be mapped and theirassociated intrinsic reference frames identified, using single-unitrecording methods in head-unrestrained monkeys. The methodtreated gaze, head, and eye positions as continuous variables, andused the correct geometry of 3D rotations in its transformations.Our method was based on the same assumption as that used inprevious 1D and 2D head-fixed studies (e.g., Stricanne et al., 1996;Avillac et al., 2005; Porter et al., 2006), namely that a neuron’sintrinsic reference frame is the reference frame in which the activ-ity was most coherent at each position when averaged across allpositions.

We developed this method for the purpose of analyzing supe-rior colliculus unit activity during head-free gaze shifts (DeSouzaet al., 2008), but soon found that we were dealing with a verycomplex and general set of issues. Hence, the current paper. Andalthough we are already applying this method for use with realdata, we develop the method here strictly with the use of simulateddata so that we can fully control the test set and compare results

with known underlying properties of the model (which cannot bedone for experimental situations). We have, however, taken greatcare to incorporate our years of experience using this experimen-tal paradigm and data into the simulations in order to maintainmaximum realism.

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We begin by developing a method for determining the spatialoherence of response field activity in different reference framessing non-parametric regression on the neural activity data, andeasuring how well the regression surface predicted the unfit-

ed data using the PREdictive Sum-of-Squares (PRESS) statistic. Theeference frame with the smallest PRESS statistic was identifieds the intrinsic reference frame if the PRESS statistic was signifi-antly larger (determined using the Brown–Forsythe test) in otherandidate reference frames. We applied this method to the head-nrestrained gaze-shift paradigm, using the correct geometry ofotations in 3D on all spatial quantities, and considered the questionf gain fields, intermediate reference frames, and the combining ofata from neuron populations. Allowing for differences in exper-

mental measurements and geometry, the same analytic methodould be applied to any neural receptive or movement field dataelated to the motion of two or more egocentric reference framesuring natural behaviors.

. Methods and results

.1. The basic approach using non-parametric regression

To develop and test the basic properties of a general methodor estimating a neuron’s response field and identifying its intrinsiceference frame, we used 1D simulated data before applying theethod to realistic simulations using 3D geometry. Fig. 1A shows a

imulated neuron’s 1D visual response field (black dashed curve)efined in space (i.e., lab) frame. Fifty trials were generated. Inach trial, while the subject fixated a visual home-target locatedt the zero (straight-ahead) direction, a visual destination-targetso-called because in the paradigm we are considering the subject

akes a gaze shift to that target) was flashed at a random posi-ion defined in space frame. The activity of the neuron in each trialas generated from the response field value at the position of theestination-target using a Poisson probability distribution, wherehe expected value of the distribution was equal to the responseeld value. This probability distribution simulated the stochasticariation (‘noise’) observed in the firing rate of neurons.

Because the response field of a neuron cannot be assumed ariori to have a specific shape, we used a non-parametric fittingechnique with the Nadaraya–Watson estimator (Nadaraya, 1964;

atson, 1964) to estimate the field:

ˆ (x; h) =∑n

i=1Fiw(x − xi; h)∑n

i=1w(x − xi; h)(1)

ere n was the number of trials, and m̂ the estimated responseeld at each position, x. This estimate was a weighted average ofhe firing rates of all trials, the firing rate of the neuron for each trial,i, being multiplied by a weight, w(x − xi; h), that was a function ofhe distance between the destination-target position in the trial, xi,nd the position where the field was being estimated, x. While theernel, w, may have any shape, we used a Gaussian kernel in ordero ensure good smoothing:

(x − xi; h) = e−|x−xi |2/h2(2)

he kernel in Eq. (2) gives preferential weighting to trials whoseestination-target positions are closest to the position at whichhe receptive field was being estimated. The kernel bandwidth, h,etermines the scale at which the target-to-field-position distanceroduces a significant decrease in weighting (i.e., |x − xi|/h ≈ 1).

hus, twice this distance may be approximated as the range overhich trial firing rates were averaged to obtain the regression fit atgiven position. Regression fits were performed using kernel band-idths in 1◦ steps from 1◦ to 15◦. Fits of the 50 simulated trials are

hown in Fig. 1A for a narrow kernel bandwidth (1◦; dotted red line)

e Methods 180 (2009) 171–184 173

which over-fits the data (i.e., it follows the noise in individual trialfiring rates because the region over which it averages is very small);a very broad kernel bandwidth (15◦; dashed red line) which fails tocapture the full response field (having a large positive bias at thetails and a negative bias at the peak because it averages over toobroad a region); and an intermediate bandwidth (4◦; solid red line)which reproduces the underlying response field almost exactly.

The goodness-of-fit may be estimated as the root-mean-square(RMS) of the differences between the regression curve and theactual response field (RF) at each trial position. This RMS fit-to-RFresidual is shown for kernel bandwidths 1◦ to 15◦ in Fig. 2A (black;mean ± SD from 100 data sets). As can be seen, the best fit (smallestRMS residual) occurred at an intermediate kernel bandwidth (3◦;filled black circle).

The fitting thus far was performed for the simulated data plot-ted in the frame in which the neuron’s response field was defined(i.e., its intrinsic reference frame: space frame for this neuron). Howwould the fitting be affected if the data were plotted in anotherframe, such as eye frame? We simulated destination-target posi-tions in eye frame by having initial eye position vary randomlybetween trials within a range of ±R, centered about the zero home-target position. The destination-target’s position in eye frame wasits position in space frame minus initial eye position. Fig. 1B showsregression fits of the data plotted in eye frame where the rangeof initial eye position was ±5◦. The fit obtained using a kernelbandwidth of 4◦ (solid red curve) reproduced the response field(dashed black curve) less well than for the data plotted in spaceframe (Fig. 1A). When the range of initial eye position was increased(±10◦ in Fig. 1C and ±20◦ in Fig. 1D), as range increased the 4◦ ker-nel bandwidth regression curve reproduced the response field moreand more poorly. The goodness-of-fit, represented by the RMS fit-to-RF residual, reflected this, increasing with the range of initial eyeposition (Fig. 2A; blue, green and red curves). The reason for this isthat, since the neuron’s response field was defined in space frame,the shifts in destination-target position in eye frame arising frominitial eye position injected a random spatial ‘noise’ that reducedthe spatial coherence of the activity.

It should be noted that the reduction of spatial coherencedepended on the ratio of the range of initial eye position to the widthof the response field. If both quantities were scaled identically, thenno change in coherence would result (although the kernel band-width of best fit would). When the range of shift was greater thanthe response field width, as in Fig. 1D (the full shift range, between±20◦, was 40◦; the response field width was 20◦) the spatial ‘noise’obliterated the response field more or less completely.

While the RMS fit-to-RF residual of the non-parametric fit iden-tified both the best-fit kernel bandwidth and the intrinsic referenceframe (at which the residual was a minimum), since the underlyingresponse field is not known for actual neurons, this residual is notavailable to the experimenter. A solution to this problem presentsitself in the PRESS (PREdictive Sum-of-Squares) statistic, which is aform of cross-validation. One trial is left out and the remaining trialsfitted. The residual activity of the left-out trial relative to the fit atthat position, squared and summed over repeating this procedureleaving out each trial in turn, yields the PRESS statistic. Thus, thePRESS statistic is a measure of how well the fit predicts new data.Since only the response field portion of a neuron’s activity that maybe predicted (as opposed to the stochastic component which, beingnoise, is not predictable), the PRESS statistic is a measure of howwell the fit estimates the underlying response field. This fact is con-firmed by the RMS PRESS residuals (defined as the square root of the

PRESS statistic divided by the number of trials) obtained for the datain Fig. 1 and plotted in Fig. 2B, which show a variation with kernelbandwidth and range of initial eye position variation very similarto that observed for the RMS fit-to-RF residuals in Fig. 2A. We usedthe kernel bandwidth that produces the smallest PRESS statistic (or

174 G.P. Keith et al. / Journal of Neuroscience Methods 180 (2009) 171–184

Fig. 1. Non-parametric fitting of 1D simulated data in intrinsic and non-intrinsic reference frames. (A) Activity was generated for 50 random destination-target positions(open blue circles) between 5◦ and 45◦ for a simulated neuron having a 1D Gaussian visual response field (dashed black curve) defined in space frame, centered at 25◦ with1/e width 10◦ , and a maximum (minimum) mean firing rate of 20 (2). The firing rate for each destination-target position trial was generated using a Poisson probabilityf . Nonb ye poz

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test comparing the fits of the data in the different reference framesis related to the ratio of the variation of the relating spatial variableover the characteristic width of the response field.

unction where the expected value was equal to the response field for that positionandwidths. (B) The same 50 trials and fits are plotted in eye frame, where initial eero) within the range ±5◦ . (C) The range ±10◦; and (D) the range ±20◦ .

MS PRESS residual) as the best-fit estimate of the response field forata plotted in a given reference frame.

.1.1. Identifying the intrinsic reference frameWe compared the best-fit PRESS statistics obtained from the

ata plotted in the different reference frames, identifying the best-andidate for the neuron’s intrinsic reference frame as the framen which the PRESS statistic was smallest. To exclude the othereference frames, we tested whether the difference in PRESS statis-ic between these other frames and the best-candidate frame wasignificant (i.e., whether the fit is significantly better in the best-andidate reference frame). We used the Brown–Forsythe test toompare the spread of PRESS residuals (where a PRESS residualas defined as the residual of a single excluded trial relative to

he fit obtained from the remaining trials) obtained in the best-andidate with each other reference frame in turn. (We used therown–Forsythe test because the distribution of these residuals wasot in general normal.)

In the Brown–Forsythe test, a t-test was performed between thebsolute value of each PRESS residual minus the median PRESSesidual for the data plotted in two reference frames (the best-andidate and one other). We used two-tailed t-test since we could

ot use our knowledge that the best-candidate frame had the small-st residuals in the test. If the test produced a p-value less than.05 we inferred that the fit in best-candidate reference frame wasignificantly better than that obtained in the other frame, and theatter could be excluded as the intrinsic reference frame of the neu-

-parametric regression fits of data (red curves) are shown for 1◦ , 4◦ , and 15◦ kernelsition varied randomly between trials about the home-target position (which was

ron. Since we sought to exclude each non-best-candidate referenceframe separately, we did not have to account for family-wise type Ierrors.

The results of this test on our simulated data of Fig. 1, wherethe best-fit kernel bandwidth of the data in eye versus space framewere compared for each of the different ranges of initial eye positionvariation, are shown in Fig. 2C, where the median and the central67% of the range of p values from the 100 data sets are plotted. Ascan be seen, for ranges of initial eye position variation ±10◦ or ±20◦

the fit of the data in space frame was reliably significantly betterthan in eye frame. This allowed space frame to be inferred (cor-rectly) as the intrinsic reference frame of the data and eye frame asa non-intrinsic reference frame.1 When initial eye position range ofvariation was ±5◦, however, this inference could not be made formost data sets. Thus we see that the ability to distinguish the intrin-sic reference frame of data is dependent on the size of variation ofthe spatial quantity that relates one reference frame to another.More generally, the effect size that determines the power of the

1 This identification is actually limited to the candidate frames compared. It ispossible that the intrinsic reference frame is none of those considered. This issueis addressed when we consider the possibility of reference frames intermediatebetween the canonical eye, head, and space reference frames.

G.P. Keith et al. / Journal of Neuroscienc

Fig. 2. Fit residuals of the 1D data in intrinsic and non-intrinsic reference frames.(A) The root-mean-square (RMS) residuals between the non-parametric regressioncurve and the underlying response field (shown in the inset as the red and blackcurves, respectively) plotted as the mean ± SD from 100 data sets for the simulatedneuron of Fig. 1 and kernel bandwidths from 1◦ to 15◦ . The residuals are shownfor data plotted in space frame (black; equivalent to data in eye frame where therewas zero variation in initial eye position) and in eye frame when random initial eyeposition variation was ±5◦ (blue), ±10◦ (green), ±20◦ (red). (B) RMS PRESS residualsfor the same conditions as in (A). (C) Results of the two-tailed Brown–Forsythe testcomparing the variance of PRESS residuals of data plotted in eye frame versus spaceframe for fits using the best-fit kernel bandwidth. Plotted are median with error barsshowing the central 67% of the range of the p values from the 100 data sets.

e Methods 180 (2009) 171–184 175

2.1.2. Statistical considerationsSince the neural data being used for fitting in the different ref-

erence frames are the same, it may be argued that the two samplesare not independent. To address this we performed a comparisonof PRESS residuals using a non-parametric method, the methodand results of which are given in the Supplementary Materials. Wefound no improvement in our findings using the non-parametrictest, however, and, since the Brown–Forsythe test took much lesstime to run on a current high-end PC, we used it in all our subse-quent analyses.

We also noted that using the Poisson distribution to generatethe noise component of the firing rate for each trial meant that thestandard deviation of this distribution was equal to the square rootof the mean firing rate at any position (the response field value).Thus, for different response field values an uneven variation inactivity will occur, which could affect the fitting and distributionof PRESS residuals. We tested this (method and results are givenin the Supplementary Materials) but found no distorting effect. Weconcluded that our analysis is not sensitive to the shape of noise inthe firing rate of neurons.

2.2. Optimal placement of home- and destination-target positions

We next considered a question of practical importance to theexperimental physiologist: How should home- and destination-targets be distributed across trials for the best results?

2.2.1. Effect of the number of active-region trialsAs we saw, determining the intrinsic reference frame of a neu-

ron depended on the variability of the spatial quantity relating thedifferent reference frames. There is a second requirement embed-ded in this, however: that the neuron’s activity varies with targetposition. (By ‘target position’ here, we mean the destination-targetposition. That is, the position of the visual target when the sub-ject fixates the home target, the destination-target being the goalor destination of the subsequent gaze shift.) Thus, to discriminatebetween intrinsic and non-intrinsic reference frames, the data mustbe collected from targets positioned within the active region of theneuron response field—i.e., that region in which the response fieldvaries with position.

Using the same simulated neuron shown in Fig. 1, we first con-sidered trials in which target positions were located within theactive region of the neuron’s response field: from 10◦ to 40◦ (Fig. 3A,red box). We then considered trials in which target positions werelocated outside this region, in the non-active region of the responsefield: 40◦ to 70◦ (blue box). We also chose target positions that wereregularly spaced within these regions, since we found that this gavemore consistent results (this point is discussed in the next section).

The p values of the Brown–Forsythe test comparing the PRESSresiduals of the fits of the data plotted in eye frame and space frame(the latter being the intrinsic reference frame of the simulated neu-ron) are shown in Fig. 3B for data sets of 50, 100, 200, and 300trials. The first 50 trials of each data set were always placed withinthe active region of the response field. The additional 50, 150, or250 trials were placed either within the active region as well, or inthe non-active region. As can be seen, when the additional trialswere located in the active region (red), the significance of the testimproved, while when they were located in the non-active region(blue), it did not. (Effects are shown for data sets of two differ-ent initial eye position ranges: 6◦ and 10◦, dashed and solid linesrespectively.)

The mechanism underlying why trials in the response field’sactive region contribute to the significance of the test distinguish-ing intrinsic from non-intrinsic reference frames while trials in thenon-active region do not is described in detail in the SupplementaryMaterials.

176 G.P. Keith et al. / Journal of Neuroscience Methods 180 (2009) 171–184

Fig. 3. Optimal destination-target positions in 1D data. (A) Simulated data created for 100 regularly spaced destination-target positions and using the same response field asFig. 1 are plotted in space frame (gray circles) and eye frame (black circles), where eye-frame positions are space-frame positions minus initial eye position. Initial eye positionvaried randomly between trials within ±3◦ of the home-target position (at zero). Destination-target positions between 10◦ and 40◦ were defined as lying in the responsefield’s active region (the red box), while those lying between 40◦ and 70◦ were defined as lying in the non-active region (the blue box). (B) The p values of Brown–Forsythetests between fit PRESS residuals for data in eye and space frame are shown for data sets having 50, 100, 200, or 300 trials, where the range of variation of initial eye positionwas either within ±3◦ (dashed line) or ±5◦ (solid line). The destination-target positions of the first 50 trials in each data set were within the active region. The additional 50,150 or 250 trial destination-target positions were either within the active region (red) or within the non-active region (blue). Plotted p values are the median values for 100data sets in each condition, the error bars showing the central 67% of the range of p values. (C) An example data set generated from the same response field, where the 50trials had four destination-target positions: 10◦ , 20◦ , 30◦ , and 40◦ in space frame (gray circles). The range of variation of initial eye position was ±3◦ , so that the trials plotted ineye frame (black circles) are still clustered around these four positions. The fitted curves of these trials in both frames (gray and black curves) made with a kernel bandwidthof 4◦ reflect this clustering and do not reproduce the underlying receptive field very well (dashed black curve). (D) The p values (median of 100 data sets where the rangeof variation of initial eye position was ±3◦ , with error bars indicating the central 67%) of the test comparing the residuals of the fits in eye and space frame, show a steadyi trialso rge un(

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mprovement across increasing range of variation of initial eye position when all 50nly four destination-target positions are used (solid curve), the p values remain la10◦).

.2.2. The effect of clustering trialsThus far we have used data sets in which only one trial was

erformed for each destination-target position. The traditionalpproach to determining the intrinsic reference frame of neurons,owever, used a small number of widely spaced destination-targetositions with multiple trials being made to each target position.o test which approach produced better results, we created dataets using the same Gaussian 1D response field with 50 trialsnd N destination-target positions, these positions being evenlypaced in the active region of the response field. N < 50 meant

hat target positions were used for more than one trial. Fig. 3Chows a data set where N = 4 and initial eye position had a vari-tion of 6◦. The non-parametric regression fits of this data using4◦ kernel bandwidth are shown for the data plotted in space

nd eye frame (gray and black curves). As can be seen, these

have distinct, regularly spaced destination-target positions (dashed curve). Whentil this range of variation becomes as large as the interval between these positions

fits do not reproduce the underlying response field (dashed blackcurve), one good reason not to use such spatial clustering oftrials.

The results of the Brown–Forsythe test are shown in Fig. 3D fordata using 50 and 4 destination-target positions for different rangesof initial eye position. As can be seen, while the p-value of this testimproves with any increase in this range for the 50-destination-target position data (dashed), for the 4-destination-target positiondata (solid) the p-value only improves when the range of initialeye position variation approaches the interval between target posi-

tions (10◦). We performed the same analysis on data sets usingN target positions between 4 and 50, and found that using adifferent destination-target position for each trial (N = 50) and reg-ular spacing between these positions consistently produced betterresults.

G.P. Keith et al. / Journal of Neuroscienc

Fig. 4. Optimal home-target positions in 1D data. The same response field as Fig. 1was used to generate 50 trials made to regularly spaced destination-target positionswithin the active region of the response field. Data sets were collected for varioushome-target position ranges and different numbers of regularly spaced home-targetpositions within these ranges. The range of initial eye position was ±3◦ , centered onefb5

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meridians of the retina (gray axes), and the secondary-target thatappears horizontally to the right in space frame, is shifted so that ineye frame (Fig. 5D) it is not horizontally rightward. Thus, in order to

ach home-target position. Home-target positions were chosen pseudo-randomlyor each trial. The p-value of the Brown–Forsythe test comparing the residuals of theest fits for data plotted in eye and space frame are shown for 3 (gray), 7 (black), or0 (black, bold) home-target positions.

.2.3. Optimal placement of home-target positionsAs we have seen, a greater variation in initial eye position

mproves the effect size that distinguishes intrinsic and non-ntrinsic reference frames (such as space and eye). This variationas simulated to reflect the variation that arises as a natural prod-ct of the animal’s behavior when fixating a single home-targetosition. It may be further increased, however, by varying the home-arget position between trials. The question posed here is: Whereo place such targets?

We began with three regularly spaced home-target positions,arying this position between trials pseudo-randomly,2 and usinghe same response field as that shown in Fig. 1. Initial eye positionaried randomly in each trial, centered on the home-target position,ithin a range of 6◦. Fig. 4 shows results of the Brown–Forsythe

est as a function of the total range of the three home-target posi-ions, which was varied from 0 to 80◦ (gray). For smaller ranges ofhe three home-target positions (up to 20◦) the p-value of the testomparing the fits in eye and space frame became more significants home-target position range increased. Beyond 20◦, however, the-value began to increase again. This was due to the separationf the target positions in eye frame that were associated with thehree home-target positions (the width of the response field itselfeing only 20◦). We found that this effect could be eliminated, how-ver, by increasing the number of home-target positions withinhe range (and thus narrowing the step-size between home-targetositions). The p values for the same overall ranges of home-targetosition are plotted for seven regularly spaced home-target posi-ions (black curve), and no increase in the p-value occurred at

arger-than-20◦ ranges. Increasing the number of regularly spacedome-target positions to 50 (bold black curve) did not improvehe p values further. Thus, the best differentiation between ref-rence frames can be achieved when the number of home-target

2 Pseudo-random variation of position requires that all positions are used theame number of times, but that the order in which they are used is random.

e Methods 180 (2009) 171–184 177

positions is increased to the point where the distance between adja-cent home-target positions is less than the width of the responsefield.

It should be noted that the increase in variation of initial eyepositions due to multiple home-target positions increases the pos-sibility of significant gain-field effects. Fortunately, these effects canbe dealt with in our analytical method. This is described below forthe more realistic 3D simulations.

2.3. Using 3D geometry

Thus far we have described our method for mapping responsefields and determining intrinsic reference frames in 1D. We nowturn to applying the method to actual 3D situations. The task wesimulated here was that of head-unrestrained gaze shifts madeto visible targets. In each trial the subject fixated a home-targetlocated in the straight-ahead direction. A destination-target is thenflashed elsewhere and the subject immediately makes a gaze shift(combined saccade and head rotation) to fixate that target. Sinceboth gaze (eye-in-space) and head orientations rotate during thegaze shift, both must be measured (in 3D). The full algebra ofrepresenting such orientations, as well how these quantities aretransformed between reference frames, is given in detail in theSupplementary Materials.

The important property of gaze and head orientation in thehead-unrestrained gaze-shift task is that both orientations varyrandomly between repeated fixations of a single home-target posi-tion. Moreover, as noted in the Introduction, when the head isunrestrained the eye-in-space orientation that directs gaze in space(lab) frame generally has a significant torsional component. Inmonkeys trained to perform such head-unrestrained gaze shiftsto visible targets (DeSouza et al., 2005), the initial gaze fixatingthe home target typically showed variations in the horizontal andvertical components between ±4◦, and a variation in the torsionalcomponent between ±6◦.

Eye and head motion include translational components thataffect the relative orientation of earth-fixed targets as an inversefunction of their distance (Vibert et al., 2001; Medendorp et al.,1999). In the current paper we simulated targets as if they wereat an infinite distance in order to simplify this issue. In the experi-mental lab these effects would be minor for the small variationsin initial head orientation and distant targets,3 but would needto be accounted for with larger distributions of initial head ori-entation, especially for near targets. For an account of the mathrequired to deal with the translational geometry of this sys-tem and near targets, see (Viirre et al., 1986; Medendorp et al.,2003).

The effect of the rotational variations alone on the destination-target position (direction) in eye frame is shown in Fig. 5. If the initialgaze is rotated from the straight-ahead orientation by horizontaland vertical components only (represented by the cross-hairs inFig. 5A), then the secondary-target in eye frame (Fig. 5B) is shiftedfrom its position in space frame by the negative of the deviationof this initial gaze from the straight-ahead direction. If initial gazeis rotated torsionally (Fig. 5C), this rotates horizontal and vertical

correctly transform target positions from space to eye frame, gaze

3 If this is not done, then these effects appear as noise in the response field maps.In the case of a single home target position, however, the range of initial head ori-entation components lying within approximately ±5◦ (DeSouza et al., 2005), thisnoise will be too small to significantly affect the distinction between intrinsic andnon-intrinsic reference frames.

178 G.P. Keith et al. / Journal of Neuroscien

Fig. 5. The effect of 3D initial gaze orientation on destination-target position ineye frame. The 3D gaze orientation is represented as a cross centered at the 2D(horizontal and vertical) gaze direction relative to the straight-ahead directionindicated by intersection of the two axes, black. The third, torsional componentof gaze orientation is represented by the tilt of the cross (gaze orientation anddirection are described in detail in the Supplementary Materials). (A) When ini-tial gaze orientation is rotated in its horizontal and vertical components relative tothe straight-forward direction, gaze direction is shifted relative to the origin accord-ingly. A visual target (�) located at a given direction in space frame, in eye frame (B)is shifted oppositely to the shift of initial gaze direction relative to the origin in spaceframe (A). (C) When initial gaze orientation is rotated in terms of its torsional com-ponent, the horizontal and vertical meridians of the eye (indicated by the gray tiltedaxes) are rotated. The effect of this tilt on the destination-target located directly totatf

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he right of the origin in space frame, considered in eye frame (D) is that it is rotatedbout the origin opposite to the initial gaze orientation torsional rotation. Thus allhree components of initial gaze orientation affect the location of a target in eyerame.

ust be recorded in 3D. The same is true for head frame, since headrientation during initial home-target fixation also varies between5◦ in horizontal, vertical and torsional directions (DeSouza et al.,005).

All of this means that a destination-target will appear at differentocations in space (lab), eye and head frames. These differences maye used to determine the intrinsic reference frame of the neuron,ince in non-intrinsic reference frames this position variation is aource of spatial noise that reduces the coherence of the neuron’sctivity in those frames.

.3.1. Different spatial representationsWe simulated a head-unrestrained gaze-shift task using realistic

D eye and head orientations, and the response fields of simulated

eurons of the type found in the deeper layers of the superior col-

iculus, which have been shown to have visual responses, motoresponses, or both (Mays and Sparks, 1980; Munoz and Wurtz,995a).4 In a visual gaze-shift task these two responses are not

4 By this we mean that our simulated neurons had gaze-shift-related activity,ither visual or motor (since we envisaged visual saccades no distinction was madeetween these activities). All of response fields were Gaussian, whose values variedith 2D direction vector, x, and had a center response field direction, x0, and 1/e

idth, �: RF(x; x0, �) = e−|x−x0|2/�2.

ce Methods 180 (2009) 171–184

temporally well separated, so that the saccade-related activityassociated with the gaze shift may be either sensory or motorrelated. While the sensory activity would relate to the position ofthe visual destination-target described above, the motor activitywould relate to the movement made towards that target. Mon-keys trained to perform this head-unrestrained gaze-shift taskshowed final gaze directions that not only varied between trials(by about ±3◦ in the horizontal and vertical components), buton average undershot the destination-target position by severaldegrees as well (DeSouza et al., 2005). Motor-related responsefields, therefore, encode position values in a given reference framethat are distinct from the sensory destination-target position—i.e.,the final gaze direction.5 This means that a total of six possiblespatial representations of the data are possible, corresponding tothe two spatial quantities (visual target or final gaze direction)plotted in three reference frames (space, head, or eye). Any ofthese could be the intrinsic spatial representation of the neuron’sactivity.

2.3.2. A simulated neuron in 3DFig. 6A shows the simulated 2D response field of a neuron (col-

ored field) that was defined in terms of destination-target positionin eye frame (the response field in this example was defined asGaussian with a maximum value of 20, located 20◦ to the rightof the fovea in eye frame, and with a 1/e width of 10◦). Thedestination-target positions (black circles) were regularly spacedin a square array, and each target position was used for one trialonly. The ranges and distributions of 3D initial head and eye orien-tation, and final eye orientation, were generated to mimic thosedescribed above and actually observed in monkeys performingthis task (DeSouza et al., 2005). The firing rate of the neuron ineach trial was generated from the response field values at the tar-get positions in eye frame using a Poisson probability function asdescribed earlier for the 1D data. The estimates of the responsefield from the data using the non-parametric fitting approach areshown in Fig. 6B for the destination-target position in space, head,and eye frame representations, using kernel bandwidths 1◦, 4◦,and 10◦. The fitted surfaces are shown as colored fields and thefiring rates of all trials used in generating the fits are shown asblack circles, centered at the destination-target position in eachframe, with the diameter being proportional to the firing rate.What is immediately evident from the fitted surfaces producedusing 1◦ kernel bandwidths is that the data is over-fitted: it showshighly variable values that bear little resemblance to the under-lying response field of Fig. 6A. The fitted surfaces produced usinga 4◦ kernel bandwidth, on the other hand, do roughly resemblethis response field; and the fitted surfaces produced using a 10◦

kernel bandwidth are poor, showing a negative bias in the cen-ter thus failing to reproduce the activity seen in the responsefield.

The RMS PRESS residuals of the fits performed on the simulateddata using 1◦ to 15◦ kernel bandwidths in the six representa-tions (final gaze and destination-target position in space, head,and eye frame) are shown in Fig. 6C. As was the case with the1D data, the best fit in each representation (smallest residual)occurred at intermediate bandwidth values. Also, the target ineye-frame representation (red filled circles) showed the best fitamong all the representations (and thus was the best-candidate

for the intrinsic representation). This was consistent with the factthat the response field was defined in terms of the destination-target position in eye frame representation. Fig. 6D shows thep values of the statistical Brown–Forsythe tests comparing the

5 Gaze direction is distinct from gaze orientation. This is described in theSupplementary Materials.

G.P. Keith et al. / Journal of Neuroscience Methods 180 (2009) 171–184 179

Fig. 6. Analysis of a simulated neuron’s response field in 3D. (A) Data was generated using a simulated Gaussian receptive field defined in the eye reference frame, shown bycolored field, having a 1/e width of 10◦ . Destination-target positions (open black circles) were set in rectangular array with 5◦ horizontally and vertically between positions.One trial was made to each destination-target position. The simulated data used initial 3D gaze and head orientation that varied randomly within ranges specified in Section2 text. Destination-target positions were transformed into head and eye frame using the transformations described in the Supplementary Materials. (B) Regression fits ofthe data plotted in terms of target position in space, head and eye frame using kernel bandwidths of 1◦ , 4◦ , and 10◦ are shown as colored fields, with the activity for all trialsp propo ◦ ◦

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lotted as black circles centered at the destination-target position and having radiiernel bandwidths with the data plotted in the different representation—i.e., in termgreen) and eye (red) frames. (D) Brown–Forsythe test results comparing variance otarget in eye frame) with that of the best-fit in each other representation.

est-fit PRESS residuals in the best-candidate representation withhose of the other five representations. Notably, the best fits of thenal gaze in eye-frame and target in space-frame representationsere not significantly worse than that obtained in the destination-

arget in eye frame representation (having p values greater than.05).

Since the variations in initial and final eye orientation and initialead orientation are characteristic of the animal’s behavior, therere only three ways in which the above results could be improved tohe point that all five non-best-candidate representations could beliminated as the intrinsic representation. First, the number of tri-ls could be increased so as to discriminate smaller effect sizes. Thisay not always be possible—for example, if the neuron is lost dur-

ng the recording after this number of trials. Alternatively, the datarom multiple neurons could be combined (so that the total num-er of trials is increased). Finally, the effect size could be increasedy varying the home-target position between trials in the mannerescribed earlier for 1D data.

rtional to the firing rates. (C) RMS PRESS residuals are shown for fits using 1 to 15rget position (solid circles) or final gaze direction (open circles) in space (blue), headSS residuals for the best-fit kernel bandwidth in the best-candidate representation

2.3.3. Population analysis: combining data across neuronsAs already noted, the number of trials may be increased by com-

bining data from multiple neurons so as to discriminate smallereffect sizes. This approach, however, means that information codedin individual neurons is lost. What is produced is an average ref-erence frame over the population of neurons. When the neuronsmay be posited as having similar intrinsic reference frames andgreater power is needed in the statistical test, it would be worth-while doing this. Since different neurons in general have differentsized and shaped response fields, fits must be performed for eachneuron separately. What is combined, then, are the PRESS residu-als of the fits. These residuals must be transformed (as describedfor the Brown–Forsythe test) and normalized across the neurons.

Normalization is done by scaling the residuals so that the meantransformed PRESS residual in one representation is the samefor all neurons. In this way the relative goodness-of-fit betweenrepresentations is preserved across the neuron population. The tri-als from all neurons can then combined and the Brown–Forsythe

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80 G.P. Keith et al. / Journal of Neuro

est carried out in the same manner as described for a singleeuron.

.4. Dealing with gain fields in 3D geometry

As noted in our analysis of 1D simulated data, varying the home-arget position across trials to increase the effect size distinguishinghe intrinsic from non-intrinsic reference frames, introduces suf-cient variation in initial gaze and head orientation to produceossible significant gain-field effects—i.e., overall modulations ofhe response field values that depend on initial gaze and/or headrientation.

To test our method using gain fields, we introduced gain-fieldffects into our simulated data by modifying the activity of theeuron described earlier using a gain field that was linear in bothorizontal and vertical dimensions of the 2D initial gaze direc-ion (rather than the 3D initial gaze, eye-in-space, orientation)or simplicity. (More complex gain fields, such as those involvingigher-order, non-linear terms, can be dealt with in the same man-er.) The horizontal and vertical linear gain-field coefficients, gfc/H

nd gfc/V, we used were −0.04 and 0.03, respectively. The gain-fieldactor, gf, for each trial was calculated from the initial gaze direction,i = (Gi/H, Gi/V), and these gain-field coefficients:

f (Gi) = 1 + gfc/H × Gi/H + gfc/V × Gi/V (3)

his meant that when Gi = (0,0), the straight-ahead direction, gf = 1.he response field of the neuron for each trial was generatedy multiplying the basic, without-gain-field response field valuewhich depended only on the destination-target position) by theain-field factor for that trial. We used home-target positions thatomprised a 3 × 3 array, at 0 and ±5◦ in both horizontal and ver-ical directions about the straight-ahead direction in space frame.he gain-field factors for initial gaze direction at each of these targetositions are shown in the inset of Fig. 7.

In generating the simulated data for this neuron-with-gain-field,he initial gaze orientation was varied from trial to trial within theange defined for the previously described neuron, centered on theome-target position of each trial. This home-target position forach trial was selected pseudo-randomly from the array of possibleositions.

The gain-field modulation of the simulated neuron’s activityhus introduced depended on the initial gaze direction rather thanhe destination-target position. Since our fitting of the responseeld of the neuron in different representations depends on theestination-target position, the gain-field effect would appear asoise in any fit of the data where the gain field was not explicitlyaken into account. We hypothesized that the effect of this noiseould be to obscure the distinction between the intrinsic and non-

ntrinsic reference frame fits. To both identify the gain-field effectnd to remove it so that the intrinsic reference frame can be bet-er identified, we parsed through a range of gain-field values. Forhis example analysis we restricted ourselves to linear gain-fieldffects in horizontal and vertical initial gaze direction components,he coefficients gfc/H and gfc/V in Eq. (3). The same method can besed, however, with higher-order gain-field effects.

The range of gain-field coefficients was chosen such that, forach pair of gain-field coefficients chosen, gfc/H and gfc/V, the gain-eld factor, gf in Eq. (3), was never less than zero for any trialwhich would produce negative modulated firing rates). For theange of initial gaze directions in the data of the current examplehis range was defined by: (gfc/H)2 + (gfc/V)2 < (0.09)2. We stepped

ach gain-field coefficient (gfc/H and gfc/V) by 0.01. For each combi-ation of gain-field coefficient values the gain-field factor for eachrial was computed as per Eq. (3), and the actual trial activity fir-ng rate was divided by this factor in order to remove the effectf the hypothesized gain field from the activity being fitted. For

ce Methods 180 (2009) 171–184

each gain-field coefficient pair of values, fits were performed forthe data plotted in all six representations for kernel bandwidths offrom 1◦ to 15◦, the best-fit bandwidth selected for each representa-tion, and the PRESS residuals of the best-candidate representationand each other representation performed. (Note that the best-candidate representation was set to be the same for all gain-fieldcoefficient values, and selected as the representation for which, atzero gain-field coefficients, the best-fit RMS PRESS residual was thesmallest—i.e., the fit was the best.)

The p values of these tests across all gain-field coefficient valuesare shown in Fig. 7A comparing each non-best-candidate repre-sentation with the best-candidate representation. The gain-fieldcoefficients actually used in generating the data (©) are seen tobe close to the coefficient values for which the smallest p valuesoccur. Thus, our hypothesis was proven correct: failing to take intoaccount gain-field effects (by using zero or incorrect gain-field coef-ficients in the fitting) does obscure the distinction between intrinsicand non-intrinsic reference frames, and the most significant p-valuemay be used to estimate the coefficients of the gain field in the data.

Fig. 7B shows the p values at the gain-field coefficient valuesthat were actually embedded in the data (©), which it may be seenare consistently better than the values obtained when no gain fieldwas taken into account (�). Also importantly, these p values arebetter than those obtained using the same simulated neuron whenno variation of home-target position was used (Fig. 6D). Thus, vary-ing home-target positions does increase the discrimination of theintrinsic representation of a neuron, and that gain-field effects maybe taken into account as well.

In actual experimental data, it is important to determinewhether there is a significant gain-field effect. This may be doneusing the standard bootstrapping method, where a number of sam-ple sets (say, 100) are extracted from the actual data (trials ineach sample being selected at random until the sample set has thesame number of trials as the original data set, using selection withreplacement). The gain-field coefficients corresponding to the bestp-value for each sample set are extracted, and confidence inter-vals of the horizontal and vertical coefficients from all sample setsdefined. If the confidence interval does not intersect zero, then thatgain-field coefficient is significant and should be removed from thedata before testing for the intrinsic reference frame.

2.5. Intermediate reference frames

Up to this point we have assumed that our neurons’ activity wasdefined in one of three cardinal reference frames: space, head andeye. However, accumulating neurophysiological evidence (Kahlonand Lisberger, 1996; Duhamel et al., 1997; Metzger et al., 2004;Avillac et al., 2005) suggests that in many brain areas the activity ofneurons is defined in intermediate reference frames. Fortunatelyour analytic method works just as well for intermediate framesas it does for cardinal reference frames. One need only transformthe spatial quantity (target position for a receptive field, or finalgaze direction for a movement field, or indeed some other spa-tial quantity—such as the relative position of target and hand asused in Pesaran et al. (2006)), into an intermediate reference frame,and then calculate the spatial coherence of activity by means ofthe PRESS residuals for the data plotted in this frame. The algebrafor this transformation is described in detail in the SupplementaryMaterials.

Any reference frame that is intermediate between two cardinalreference frames may be characterized by a parameter. For example,

for a reference frame intermediate between head and eye frame,˛HE may be defined as being 0 for space frame and 1 for eye frame. Aparameter value between 0 and 1 thus lies between these frames, avalue outside this range lies beyond them, but in the same rotationalcontinuum. In order to distinguish the best-candidate intermediate

G.P. Keith et al. / Journal of Neuroscience Methods 180 (2009) 171–184 181

Fig. 7. Analyzing gain-field effects in 3D data. To the simulated response field of Fig. 6 we added a gain field that depended linearly on the horizontal and vertical componentsof initial gaze direction (the linear gain-field coefficients being −0.04 and 0.03, respectively). Home-target positions were at 0 and ±5◦ horizontally and vertically. The gain-fi ned frb f eachh locatd ro) by

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eld factors at these target positions are shown in the inset. (A) The p values obtaiandwidth of the best-candidate representation (target in eye frame) with those oorizontal and vertical gain-field coefficient values (−0.09 to +0.09 in 0.01 steps). Theata is indicated by ©; that corresponding to no gain field (the coefficients being ze

eference frame from along this continuum, a fit is performed and aet of PRESS residuals obtained for the data plotted in intermediateeference frames at stepped values of the parameter ˛HE. The best-andidate intermediate reference frame is that reference frame forhich the RMS PRESS residual is a minimum. The PRESS residuals in

his frame are compared with those obtained for all other parameteralues using the Brown–Forsythe test, each test producing a p value.

Since our simulated head-unrestrained gaze-shift paradigmnvolved three cardinal reference frames: space, head and eye, con-inua exist between each pair of these frames (which in ‘framepace’ are not collinear, since head and eye position are only some-hat correlated, DeSouza et al., 2005). The p values comparing the

est-candidate intermediate reference frames with other referencerames are plotted for all three continua for a simulated neuron withhe response field characteristics of the neuron shown in Fig. 6, andhose response field was defined as target position in eye frame

Fig. 8A), averaged over 20 data sets. The p values for a similar sim-lated neuron whose response field was defined in the referencerame midway between head and eye frame (˛HE = 0.5) are shownn Fig. 8B. The uncertainty in the inferred intrinsic reference frameits confidence interval) was taken to be the range of parameteralues for which the p-value was greater than 0.05 (this range ofntermediate reference frames could not be excluded).

We simulated data sets for six neurons, whose intrinsic refer-nce frames were space, head, and eye frame, and midway betweenpace-and-head, space-and-eye, and head-and-eye frames, and

lotted the best-candidate and confidence intervals along each con-inuum in Fig. 8C. We then developed a heuristic for inferring thentrinsic reference frame from these values (described in detail inhe Supplementary Materials). Using this heuristic, the inferredntrinsic reference frame for each neuron (+) was in all cases quite

om the Brown–Forsythe test comparing the PRESS residuals for the best-fit kernelother frame. Fits were performed after removing gain-field effects for a range of

ion that corresponded to the gain-field coefficients used in generating the simulated�. The p values at these two gain-field coefficient values are plotted in (B).

close to the actual intrinsic reference frame (©), indicating that thismethod could accurately estimate intermediate intrinsic referenceframes.

3. Discussion

3.1. Overview

Previous studies determining the intrinsic reference frame ofneuron activity did so by rigidly controlling the spatial positionsat which this activity was measured. Head position was generallyfixed, and the remaining spatial position variables, such as eye ortarget position, limited to a small set of widely spaced values. A sta-tistical comparison of the coherence of the neuron’s activity at thesediscrete positions defined in different reference frames allowed theintrinsic reference frame to be inferred: it was the frame in whichmaximum coherence occurred. The extension of such intrinsic ref-erence frame determination to head-unrestrained experimentalconditions, however, precludes the use of such discrete positionvalues since any given gaze direction is generally produced by arange of combinations of head and eye position. Coherence mustbe measured using some sort of regression, or curve fitting, of theactivity measured at values sampled from a continuous set of spatialpositions.

Because a response field may not be assumed to have a spe-cific shape a priori, we used non-parametric regression (with the

Nadaraya–Watson estimator and a Gaussian kernel) to estimatethe response field from the position and neuron activity data. Weestimated the spatial coherence of the activity using the PRESSstatistic, a measure of the predictability of the data relative to theregression fit of the data plotted in each reference frame. A sta-

182 G.P. Keith et al. / Journal of Neuroscience Methods 180 (2009) 171–184

Fig. 8. Analyzing intermediate reference frames in 3D. Fits may be performed for the data plotted in intermediate references frames on the continua between space-and-head, space-and-eye, or head-and-eye frames, defined by parameters ˛SH, ˛SE, and ˛HE, respectively (described in the Supplementary Materials). (A) The p values of theBrown–Forsythe test comparing the PRESS residuals obtained at the best-candidate intermediate reference frame (indicated by the vertical arrow) with those frames atother parameter values in each continuum are plotted for a neuron whose response field was identical to that of Fig. 7 and was defined in terms of target position in eyeframe. Confidence intervals (indicated by the horizontal arrow) were defined for the intervals of the parameters over which the p values remained above 0.05. (B) The pvalues determined in the same manner as in (A) are plotted for a neuron whose response field was defined in the intermediate reference frame midway between head ande etweet oloredr ation u(

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ye frames (˛HE = 0.5). (C) The three continua are plotted in an agnostic format briangle in ‘reference-frame space’. The best-candidate and confidence intervals (ceference frames are indicated in each corresponding panel by ©. A heuristic calcul+; described in Supplementary Materials).

istical comparison was performed between the spread of PRESSesiduals obtained in the reference frame producing the best fitminimum PRESS statistic) with those obtained in the other ref-rence frames. If the best-fit reference frame showed statistically

n space, head and eye frames (S, H and E) placed at the corners of an equilateraldots and lines) are plotted along each continuum for six neurons whose intrinsicsed this information to estimate the intrinsic reference frame from the test results

smaller spread of PRESS residuals (greater coherence) than those ofother reference frames, then the former was be identified as the bestcandidate for the intrinsic reference frame of the neuron’s activity,the other frames being excluded. Even if the results for individual

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eurons are not statistically significant, they show trends, and weave described how to combine the data at a neuron population

evel.Since it is the coherence of activity in a response field map-

ing that distinguishes the intrinsic from non-intrinsic referencerames, it is an important feature of this method that no require-

ent is made that the full response field be mapped. Rather whats required that the data be collected from within the active regionf the response field. (Details of the mechanism are given in theupplementary Materials.)

.2. Applications of this method

It has been shown that the individual units in neural networkodels of the transformations associated with generating saccades

Smith and Crawford, 2006) and reaching (Blohm et al., 2008) mayhow different reference frames when characterized using standardensory receptive field methods versus microstimulation. Theseheoretical studies predicted that neurons/regions involved in aeference frame transformation should show a sensory receptiveeld that correlates the reference frame of its input (such as theye) when tested with unit recording, but movements that corre-ate best to the output effector (head, shoulder, etc.) when testedsing stimulation.

Since our lab has already characterized gaze shifts evokeduring stimulation of the SC (Klier et al., 2001), supplementaryye field (Martinez-Trujillo et al., 2004), laterial intraparietal areaConstantin et al., 2007) and frontal eye field (Ascencio-Monteont al., 2006), the use of the method developed here would allow airect comparison between unit data (receptive field and move-ent field) and stimulation data using a complete 3D analysis

hould provide the best possibility to test these fundamental mech-nisms of reference frame transformation for gaze, arm movement,r any other sensorimotor system (Blohm et al., 2008). We havepplied this method to response fields of neurons in the interme-iate SC (DeSouza et al., 2008) and can extend this to other brainegions, such as area LIP, SEF and FEF, including situations in whichultiple sensory modalities are combined.This method is well-suited for identifying which movement a

euron’s activity is best associated with. In addition to SC neu-al activity encoding gaze movement (Cowie and Robinson, 1994;reedman and Sparks, 1997; Bergeron et al., 2003), there is evidencehat some SC neurons have activity associated with head-only

ovements (Walton et al., 2007). It is possible, therefore, thatifferent neural populations encode gaze, head-, and eye-only com-onents. A head-unrestrained study (Freedman and Sparks, 1997)xamined the variation of SC motor neuron activity with movementizes during gaze shifts under conditions where the eye, head, andaze movements were clearly dissociated, in order to determinehich movement best correlated with neuron activity. This distinc-

ion could be made equally well using our analytical method, sincehe non-spatial variation of a neuron’s activity would be smallesthen plotted in terms of the movement best associated with that

ctivity.The manner in which target position is encoded in various brain

reas in such reaching tasks as well generally gives clues to howuch actions are programmed, and the reference frame of the activ-ty of neurons in brain areas associated with this programming is anssential element of this (Blohm et al., 2008). Pesaran et al. (2006)iscovered neurons whose activity is encoded in a reference frame

dentified with the difference between hand and target position

sing head-fixed conditions. Neural activity encoding such positionifferences could be explored further in head-unrestrained condi-ions using an analysis nearly identical to that shown here, in whichatural gaze positions and movements would be available, and theffect of gaze and head position on the activity of such relative-

e Methods 180 (2009) 171–184 183

position encoding neurons be examined. This opens the door toexploring the neural mechanisms for reach during naturalistic con-ditions (Flanagan et al., 2008; Mennie et al., 2007).

As previously stated, any spatial quantity could be analyzed inthis way. For example, neurons in the brain encode not only gazeshifts and reaching movements, but also whole body movements,such as the place cells in the hippocampus and surrounding cortex.These structures have also been identified as containing head-direction cells in primates (Robertson et al., 1999) and in freelymoving rats (Taube et al., 1990a,b); also, in rat cells whose activityshowed modulation both by position and head direction (Cacucciet al., 2004). All of these individual and combined spatial propertiesare amenable to analysis using our method, which is ideally suitedfor behavioral conditions where it is not possible or desirable tolimit positions to a few points.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.jneumeth.2009.03.004.

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