Decision-making with multiple alternatives

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1 Deision-mking with multiple lterntives Anne K Churhln, Roozeh Kini & Mihel N Shlen 28 Nture Pulishing Group Simple pereptul tsks hve li the grounwork for unerstning the neuroiology of eision-mking. Here, we exmine this fountion to explin how eision-mking iruitry justs in the fe of more iffiult tsk. We mesure ehviorl n physiologil responses of monkeys on two- n four-hoie iretion-isrimintion eision tsk. For oth tsks, firing rtes in the lterl intrprietl re ppere to reflet the umultion of eviene for or ginst eh hoie. Eviene umultion egn t lower firing rte for the four-hoie tsk, ut rehe ommon level y the en of the eision proess. The lrger exursion suggests tht the sujets require more eviene efore mking hoie. Furthermore, on oth tsks, we oserve time-epenent rise in firing rtes tht my impose eline for eiing. These physiologil oservtions onstitute n effetive strtegy for hnling inrese tsk iffiulty. The ifferenes pper to explin sujets ury n retion times. Orgnisms fe eisions of vrying omplexity. In simple eisions, pereptul oservtions llow n niml to hoose etween tion n intion, or etween two lterntive tions. These re simple instnes of omplex ognitive proesses, whih my require itionl informtion from the environment or from memory. The ility to ely response to onsier inoming informtion is hllmrk of higher rin funtion. Deisions etween two hoies in pereptul-motion tsk 1 3 emonstrte mehnisms of eision-mking. In the tsk, humns or monkeys reporte the net iretion of motion in pth of moving rnom ots. In the retion-time version 2, sujets ommunite their eision with se when they were rey. This version ientifies the perio when sujets re umulting eviene for eision, ut hve not yet ommitte to n lterntive. How rpily eviene umultes epens on the strength of motion. The proess ens when the eviene rehes threshol or oun orresponing to one lterntive. These oune umultion of eviene moels enompss multiple mehnisms tht hve een propose to explin hoie n eision time 1,4,5. Severl oservtions re onsistent with the ie tht eviene umultes to oun. First, forml moel of oune umultion ounts quntittively for sujets spee n ury 3,6.Seon, neuronl firing rtes in the lterl intrprietl re (LIP) re onsistent with eviene umultion 2,7 9.Speifilly,whenstimulusmotion fvors the trget in the response fiel of n LIP neuron, firing rtes progressively uil up s monkeys form their eisions; the rte of uilup is proportionl to motion strength. Finlly, in the retiontime version, firing rtes re similr t eision en when the monkey selets trget in the neuron s response fiel, for ll motion strengths n retion times. The stereotype firing rte my reflet oun tht is ommon to oth esy n iffiult motion strengths. Beuse two lterntive hoie tsks re simple, they my offer limite insight into eision-mking in generl; orgnisms regulrly fe eisions with multiple lterntives. Here, we ompre responses on four-hoie eision tsk with two-hoie tsk. Our results rgue tht the oune umultion frmework n e extene to explin more omplex eisions, n they egin to revel how eision-mking iruitry justs to inresingly iffiult eisions. RESULTS Behvior Two monkeys were trine on two-hoie n four-hoie motionisrimintion tsk (Fig. 1). We mesure oth the ury n spee of their hoies. In oth tsks, ury ws nerly perfet t high motion strengths, ut fell towr hne levels with lower motion strengths (Fig. 1). The performne with strong motion inites tht the monkeys unerstoo the reltionship etween stimulus iretion n hoie trgets. Therefore, the more frequent errors t low motion strengths on the four-hoie tsk my e the result of filure to isriminte the iretion of motion, rther thn onfusion out tion seletion. Like ury, eision spee lso epene on oth motion strength n the numer of hoies. Retion times were longer on the four-hoie tsk 1. These ifferenes were lrgest t lower motion strengths, ut were signifint t ll motion strengths (P o.1; Fig. 1f). We mesure responses on n itionl onition with two trgets, spe 91 prt (Fig. 1). This onfigurtion (91 ontrol) uses suset of the trgets n motion iretions in the four-hoie tsk. Aury on this tsk emulte the stnr two-hoie tsk (Fig. 1e), leit with longer retion times, whih fell etween those on the stnr two- n four-hoie onitions (Fig. 1g). Howr Hughes Meil Institute, Deprtment of Physiology n Biophysis, Ntionl Primte Reserh Center, University of Wshington Meil Shool, Settle, Wshington 98195, USA. Corresponene shoul e resse to A.K.C. (nne99@u.wshington.eu). Reeive 29 Otoer 27; epte 14 April 28; pulishe online 18 My 28; orrete online 17 June 28; oi:1.138/nn.2123 NATURE NEUROSCIENCE VOLUME 11 [ NUMBER 6 [ JUNE

2 28 Nture Pulishing Group Proility orret f Retion time (ms) Two hoie Two hoie Four hoie Four hoie Two hoie: 15,937 trils Four hoie: 33,268 trils Premotion Motion strength (% oh) Motion Physiologil responses on the four-hoie tsk The 7 neurons reore in our experiment h sptilly seletive persistent tivity on elye n memory-guie eye-movement tsks (Supplementry Fig. 1 online). Multi- n single-unit responses with these hrteristis re ommon in the ventrl portion of LIP trgete in these experiments 11. In the motion-isrimintion tsk, one of the hoie trgets (T in ), ws entere in the neuron s response fiel. The other hoie trgets (T out n two orthogonl trgets, T 9 ) were evenly spe roun entrl fixtion point (see Methos). On ny one tril, the iretion of motion ws towr one of the trgets. Figure 2 introues the pttern of tivity seen over the ourse of the tril on the four-hoie tsk. Trget pperne use lrge, trnsient inrese in the firing rte (Fig. 2,), followe y the estlishment of seline firing rte s the monkey wite the onset of the rnom ot motion. Shortly fter the onset, the responses unerwent rief ip, followe y grul rise in the firing rte. The ip, oserve in other stuies 2,12 14, seems to mrk the eginning of eviene umultion. After the ip, the rte of inrese in the firing rte epene on motion strength. Ner the se, however, this neurl orrelte of motion strength 1,2,15 vnishe (Fig. 2,). The stereotype firing rte t the en of the eision ws lerer when responses were groupe y retion time (four hoie, Fig. 2,; two hoie, Supplementry Fig. 2 online), inste of y motion strength. Anlysis of vriility ross ifferent retion-time groups provie quntittive support for ommon firing rte lose to the en of the tril (Supplementry Fig. 2). This pttern of LIP tivity in the four-hoie tsk resemles the proess unerlying two-hoie eisions 1,2,16. It is roly onsistent with oune umultion of eviene mong ompeting response mehnisms. Next, we ompre responses on two- n four-hoie efore the motion, uring motion viewing n just efore the se 9 ontrol e g ontrol 1 1 Two hoie Four hoie Two hoie: 7,22 trils 9 ontrol: 5,961 trils Four hoie: 15,247 trils 1 1 Motion strength (% oh) Figure 1 Tsk n performne. ( ) Sequene of events on two- n fourhoie iretion-isrimintion tsks. The monkey fixtes entrl point until the rnom ot motion ppers n then inites its eision y mking si eye movement to hoie trget. The motion is in one of two or four iretions (trils rnomly interleve in 1:2 rtio). A liqui rewr is given for hoosing the trget long the xis of rnom ot motion, or it is given with proility 1/2 or 1/4 when the motion strength is zero. Rnom intervls (trunte exponentil istriutions) seprte fixtion, pperne of hoie trgets n motion onset. The rnom-ot motion is extinguishe when the monkey initites se to one of the hoie trgets. One of the hoie trgets is in the response fiel of n LIP neuron reore uring the tsk (shing). The iretions were 91 prt in the four-hoie tsk (). The iretions were 181 prt in the two-hoie tsk (). One iretion is towr the trget in the neuron s response fiel (T in ). The 91 ontrol tsk is shown in. ( g) Spee n ury of eisions. Smooth urves in ll pnels re fits to the oune iffusion moel. The fits were performe seprtely for n f n for e n g. Psyhometri funtions re shown in. The proility of orret hoie is plotte s funtion of motion strength. All experiments ontriute to these grphs. At % motion strength, hoies were rewre rnomly (open symols). Psyhometri funtions for the 29 experiments tht inlue the 91 ontrol re shown in e. Chronometri funtions re shown in f. Men retion time for orret trils is plotte s funtion of motion strength. Eh point reflets orret responses from ll experiments. Error rs for s.e.m. re smller thn the symols. Chronometri funtions for the 29 experiments tht inlue the 91 ontrol re shown in g. to etermine whether physiologil ifferenes on the two tsks n ount for the ehviorl ifferenes tht we oserve (Fig. 1). Comprison of responses efore motion Responses efore the stimulus reflet the stte of eision-mking iruitry efore the rin egins umulting eviene. Trget pperne ws the first ue initing whether the isrimintion woul involve two or four iretions. Between trget pperne n motion onset, there ws ler ifferene in firing rtes etween the two- n four-trget onitions (Fig. 3). In single neuron (Fig. 3), responses were 11.9 ± 4. spikes s 1 lower on the four-hoie tsk (P o.3). The reue firing rte on the four-hoie tsk ws evient ross the popultion of neurons teste (Fig. 3,). It ws sutly pprent in the trnsient trget onset response n then ws prominent until motion onset (men ifferene ¼ 16.1 ± 1.6 spikes s 1, P o 1 5 ; Fig. 3). This effet ws sttistilly signifint in oth monkeys (Monkey I: men ifferene ¼ 15.5 ± 1.8 spikes s 1, P o 1 5 ; Monkey S: men ifferene ¼ 18.8 ± 2.6 spikes s 1, P o 1 4 ). Firing rte ws lso reue in the 91 ontrol tsk, lthough muh less thn in the four-hoie tsk (responses were 3.7 ± 2.1 spikes s 1 lower on the 91 ontrol tsk thn on the two-hoie tsk, P o.5; Fig. 3). We onlue tht the smller ngulr seprtion etween the trgets ontriutes to the reution in tivity seen in the four-hoie tsk in this epoh, ut tht the reution is lrgely expline y the numer of hoies. Responses erly in the motion epoh The reue firing rte on the four-hoie tsk persiste fter motion onset. Rell tht the motion ws presente outsie the neuron s response fiel, n its erliest effet on LIP ws ip in the firing rte (Fig. 2). During this ip, firing rtes were 9.17 ± 1.5 spikes s 1 lower for the four-hoie thn for the two-hoie tsk (P o 1 5 ; Supplementry Fig. 3 online). Thus this epoh retins muh of the ifferene in firing rte seen efore motion onset. Following the ip, there ws time-epenent rise in the firing rte ( uilup ). The rte of this uilup offers insight into the onversion of sensory informtion into eision vrile: tht is, the form of the 694 VOLUME 11 [ NUMBER 6 [ JUNE 28 NATURE NEUROSCIENCE

3 28 Nture Pulishing Group Trget onset n = 1 Trget onset n = 7 Neuron I % 9.% eviene tht unerlies the hoie n eision time. Four ftors ffet uilup: eision outome, motion strength, the numer of hoies present n the pssge of time (Figs. 4 n 5). The effet of hoie is most onspiuous lte in the eision proess (for exmple, Fig. 5 ). Our smple ws sreene for sptilly seletive responses on elye eye-movement tsk, so ll neurons re expete to inite the eision outome. We were intereste in the hnge in firing rte ompnying eision formtion. Erly on, eision outome hs only wek effet on the neurl responses. We therefore nlyze groups of trils with the sme motion strength n iretion, regrless of the monkey s eventul hoie (our onlusions lso hol if trils re groupe y motion strength n hoie; Supplementry Fig. 4 online). To quntify how sensory informtion is onverte into eision eviene, we estimte firing-rte uilup t eh motion strength (uilup rte ¼ slope of line fit to the she portion of the response; RF 25.6% 1 ms 1 ms % % 9.% Figure 3 Neurl responses in the pre-motion epoh re lrger on the twohoie tsk. () Averge firing rte from single neuron uring the pre-motion epoh when two or four hoie trgets were isplye. Vertil lk line inites the onset of the hoie trgets. Insets re shemti of the trget onfigurtions use in this experiment. One trget is in the neuron s response fiel (shing). () Popultion verge response. The sme onventions re use s in, exept tht the tres re verge firing rtes from 7 neurons. All trils ontriute to these verges. Insets illustrte tht one trget is in the response fiel of the neuron; the lotion of this response fiel vries from neuron to neuron. () Comprison of firing rtes from iniviul neurons on the two- n four-hoie tsks. Responses were mesure from 2 to 3 ms fter hoie trget onset. The green irle mrks the neuron shown in. Points for three neurons with high kgroun firing rtes re omitte from the plot to filitte n pproprite sle for the remining points ((227, 174), (132, 99) n (144, 127)). Error rs re s.e.m. Histogrm shows the firing-rte ifferenes for ll 7 neurons. Shing inites signifine (P o.5). () Comprison of firing rtes from iniviul neurons on the two-hoie n the 91 ontrol tsks. The sme onventions re use s in. Two neurons were omitte from the stter plot ((131, 138) n (227, 226)) , ms 6 ms 8 ms 85 ms 65 ms 75 ms 55 ms 4 ms 1 ms 1 ms 45 ms Figure 2 Responses of LIP neurons on the fourhoie tsk re onsistent with oune umultion. Firing rtes re ligne to key events in the ourse of tril, whih re mrke y vertil lines. Motion ws either rnom (% oherene) or in the T in iretion. () Averge firing rtes from one neuron. Left, responses ligne to the onset of the hoie trgets. Mile, responses ligne to the onset of stimulus motion. Right, responses ligne to se initition. For se-ligne responses, only T in hoies re shown. Tres were smoothe with 3-ms exponentil filter. () Responses reflet termintion of the eision. Responses re groupe y retion time t the vlues inite (±25 ms). The verges re ligne to se initition n exlue neurl tivity in the first 2 ms of motion onset. Only orret T in hoies were inlue n, for lrity, every other retiontime group is not isplye. (,) Popultion verge responses (n ¼ 7 neurons). Sme onventions s in n, exept tht no smoothing ws performe; firing rtes were ompute in 2-ms nonoverlpping ins. Arrow in inites the ip in firing rte seen shortly fter motion onset. Arrow in inites the time when responses ppere to olese, pproximtely 6 ms efore the se. Fig. 4 ). Builup rtes sle pproximtely linerly s funtion of motion strength for motion towr n wy from the neuron s response fiel. This reltionship ws similr on the two- n four-hoie tsks. For motion towr T in, these slopes iffere y only.17 ±.45 spikes s 2 per unit hnge in motion strength (tht is, per 1% rnom ot oherene; %oh 1 from here on) (P ¼.71; Fig. 4f). For motion towr T out, inresing motion strength suppresse uilup rte slightly more in the two-hoie onition, ut the ifferene ws not relile (ifferene ¼.5 ±.31 spikes s 2 %oh 1 ; P ¼.11, see Methos, eqution (3)). These trens were lso pprent in single Firing rte: four hoie (spikes per s) Trget onset n = 1 2 hoie 4 hoie 1 ms Neuron I Firing rte: 9 ontrol (spikes per s) Trget onset n = 7 1 ms 2 hoie 4 hoie Firing rte: two hoie (spikes per s) Firing rte: two hoie (spikes per s) NATURE NEUROSCIENCE VOLUME 11 [ NUMBER 6 [ JUNE

4 28 Nture Pulishing Group Figure 4 Neurl responses uring motion viewing epen on iffiulty. ( ) Popultion verge firing rtes (n ¼ 7 neurons) for three motion strengths. Trils re groupe on the sis of the iretion of motion n the numer of hoies (insets). Corret n inorret trils re inlue in these verges. The tres for % oherent motion re ientil in n n in n. Only three motion strengths re shown for lrity. She retngle inites the epoh use to estimte the uilup rtes (19 32 ms fter the onset of stimulus motion). Two-hoie trils where the motion iretion ws towr T in re shown in n trils where the motion iretion ws towr T out re shown in. Arrow inites the stereotype firing rte ip tht ours fter motion onset. Four-hoie trils where the motion iretion ws towr T in () ort out (, she line) or towr n orthogonlly positione (T 9 ) trget (ot-sh line) re shown. The five tres in re lrgely superimpose. The single yn tre is for % oherent motion verge ross ll hoies (T in,t out n T 9 ; sme s yn tre in ). (e g) Effet of motion strength on uilup rtes. A single neuron exmple is given in e. Builup rtes were estimte for eh motion strength; these uilup rtes (± s.e.m.) re plotte s funtion of motion strength. The slope of the fitte line estimtes the effet of unit hnge in motion strength on the uilup rte (lk, two hoie; re, four hoie). Five motion strengths were teste for this neuron (Methos). Builup rtes were lulte in iniviul neurons n then verge ross the popultion (n ¼ 7 neurons) efore fitting the line (f). Error rs re s.e.m. of uilup rtes ross neurons. Popultion nlyses for the 29 neurons teste with the 91 ontrol onition in ition to two- n fourhoie trils re shown in g. Blue lines orrespon to the 91 ontrol onition; error rs re s.e.m. of uilup rtes ross neurons. Points orresponing to 51.2% motion strength re not inlue on this plot euse this motion strength ws teste in only three neurons. neurons (Supplementry Fig. 5 online). We onlue tht LIP registers eviene long the T in -T out xis similrly for two- n four-hoie tsks. Thus, ifferenes in ehvior re not expline y ifferenes in the mpping of motion informtion onto hnge in LIP firing rte. This nlysis lso suggests tht LIP is only wekly ffete y motion in iretions orthogonl to the T in -T out xis on the four-hoie tsk. Stronger motion reue the firing rte slightly (slope of T 9 tre ¼.33 ±.9 spikes s 2 %oh 1, P o 1 3 ; Fig. 4f). Although either of the T 9 iretions might provie wek positive eviene for T in hoie in ny one experiment (for exmple, Fig. 4e), the net effet of orthogonl motion ws, on verge, wek negtive eviene. There is lso prominent uilup in firing rte tht oes not epen on motion strength or iretion n is lso not expline y the monkey s hoie, s seen in % motion strength trils, whih fvor ll iretions eqully (Fig. 4 ). Although the monkeys istriute their hoies with nerly equl frequeny to ll (two or four) hoie trgets, firing rtes inrese s funtion of time. The rte of this uilup ws onsierly lrger for two- thn for four-hoie (the y-interept for two-hoie ws 5.7 ± 8.2 spikes s 2 greter thn the y-interept for fourhoie, P o 1 5 ; Fig. 4e,f; see Disussion n Supplementry Fig. 6 online). This time-epenent rise is not expline y rnom flututions in motion energy, s the inrese ws evient when trils ening in T in n T out hoies were verge together (s in Fig. 4 ). We interpret this rise s refletion of the ost ssoite with the pssge e Builup rte (spikes per s 2 ) n = 1 Two hoie, T in Neuron I ms 25.6% 9.% % Two hoie, T out 1 ms Four hoie, T in Four hoie, T out n T 9 1 ms Motion strength (% oh) f 1 ms T T in in 4 3 n = 7 n = 29 T in T T in in T in 3 T in T 9 T out 2 1 T 9 1 T 9 T 9 T T out out T T out out Tout Motion strength (% oh) Motion strength (% oh) of time, orresponing to the psyhologil sense of urgeny, u(t) (refs. 17,18). The positive uilup rtes t ll motion strengths n for ll iretions effetively impose eline on the eision proess, eline tht is impose erlier when there re just two hoies. The 91 ontrol tsk (n ¼ 29) llowe us to sertin whether the ifferenes in LIP responses on the two- n four-hoie tsks re expline y ifferenes in the numer of hoies per se or the ifferene in ngle etween iretions of motion. Rell tht efore motion onset, tivity on the 91 ontrol trils ws slightly reue in omprison with the stnr two-hoie tsk. This reution eme more moest throughout the ip uring the eginning of motion (two-hoie responses were 2.66 ± 2. spikes s 1 higher thn 91 responses, P ¼.1), followe y uilup in firing rtes leing to the eision. The effet of motion strength on these uilup rtes ws similr to two-hoie responses for motion towr T in (for the popultion, slopes iffere y.23 ±.32 spikes s 2 %oh 1, P ¼.47; Fig. 4g). For motion towr T 9, however, 1% inrese in motion strength erese uilup rte. This suggests tht motion towr T 9 onstitutes wek eviene ginst the T in hoie. It lso rises the possiility tht the ner sene of n effet of orthogonl motion on the four-hoie tsk is n rtift tht results from verging the two orthogonl iretions. Alterntively, the T 9 iretion my e more likely to ontriute negtively to T in hoies without T out lterntive. g VOLUME 11 [ NUMBER 6 [ JUNE 28 NATURE NEUROSCIENCE

5 28 Nture Pulishing Group Two hoie, T in.% 25.6% 9.% 1 ms Four hoie, T in 1 ms As efore, some of the uilup in firing rte on the 91 ontrol tsk is not the result of n umultion of motion eviene, s it is present even on the trils with % motion strength. The mgnitue of this urgeny signl, u(t), fell etween those mesure on the two- n fourhoie tsks (Tle 1 n Supplementry Fig. 6). Responses lte in the motion epoh On vriety of two-hoie eision tsks, threshol rossing is ssoite with termintion of the eision 1,2,19.Wefouneviene for threshol on the four-hoie tsk s well (Fig. 2). We ske whether this threshol iffers for two- n four-hoie tsks. Firing rtes just preeing T in hoies were similr for two- n fourhoie trils (Fig. 5,; responses in she region were on verge only 3.1 ± 1.3 spikes s 1 higher for four hoie thn for two hoie, P o.3). Although signifint, this ifferene mounts to only 3.7% of the firing rte in this epoh. Furthermore, unlike the uilup rtes t the eginning of the eision, the firing rtes t the en of eisions for T in i not vry s funtion of motion strength (P 4.57 for two n four hoie) n were the sme for orret n error trils (P 4.78 for two n four hoie). We onlue tht the termintion of eisions is ssoite with fixe firing rte when the monkey hooses the trget in the response fiel. Firing rtes just preeing T out n T 9 hoies were hrer to interpret. Firing rtes tene to e lower on the four-hoie tsk (ifferene ¼ 9.81 ± 4.8 spikes s 1, P o.2). Beuse firing rtes were lrey lower t the eginning of these trils, the persistene of this ifferene is Tle 1 Moel prmeters only notle in ontrst with the T in hoies. Wht remins unler is whether the firing rtes hieve stereotype level efore the se or whether they reflet the motion Boun strength n retion time. We oserve (spikes s 1 ) wek epenene of firing rte on motion n ¼ 7 strength, ut the effets were not sttistilly relile (P 4.7). In the ontext of oune umultion, wek reltionship etween n ¼ 29 motion strength n firing rte rises the possiility tht the losing eision proess (or proesses) onveys informtion out the egree of iffiulty 2. Two hoie, T out 1 ms Four hoie, T out n T 9 1 ms Figure 5 Neurl responses just preeing the eye-movement responses. Popultion verge firing rtes (n ¼ 7 neurons) re shown ligne to the initition of ses (vertil line). Trils re groupe on the sis of the iretion of the se with respet to the response fiel of the neuron (insets). Only three motion strengths re shown for lrity. () T in hoies in the two-hoie tsk. () T out hoies in the two-hoie tsk. () T in hoies in the four-hoie tsk. () T out (she) n T 9 (ot-sh) hoies in the fourhoie tsk. Firing-rte exursion Firing rtes t the eginning n en of the motion-viewing perio suggest tht LIP neurons unergo lrger hnge in firing rte uring eisions mong four possile lterntives. When the monkey hose the trget in the neuron s response fiel, firing rtes for the two- n four-hoie tsk ultimtely rehe similr vlue ner the time of the se (Fig. 2 n Supplementry Fig. 2). However, the initil response when the motion egn ws lower for the four-hoie tsk thn for the two-hoie tsk (Fig. 3). We onfirme this y estimting the firing-rte exursion from eh neuron in the popultion (Fig. 6). The omine ifferene in firing rtes t the eginning n en of motion viewing ws 6.11 ±.21 spikes s 1 lrger for the four-hoie tsk thn for the two-hoie tsk (see Methos, eqution (2)). This ifferene, present in iniviul neurons (for exmple, Fig. 6), ws highly signifint ross the popultion (P o.5; Fig. 6,) n in oth monkeys iniviully (Monkey I: exursion ifferene ¼ 4.7 ±.27 spikes s 1, P o.2; Monkey S: ifferene ¼ 11.9 ± 1. spikes s 1, P o.5). The mgnitue of this ifferene epene on whih intervl we use to estimte firing rtes t the en of the tril, ut the effet ws sttistilly signifint for wie rnge (Supplementry Fig. 2). These estimtes were otine using only orret trils, ut we otine similr estimte when we inlue ll of the trils me to the T in trget. The ifferene in exursion is ttriute to the numer of hoies. The firing-rte exursion on the 91 ontrol tsk ws nerly ientil to tht seen on the stnr two-hoie tsk (men ifferene ¼.13 ±.37 spikes s 1, P ¼.53; Fig. 6). The hnge in exursion in the fourhoie tsk ws lso onfirme in this suset of neurons (men ifferene ¼ 4.2 ± 2.3 spikes s 1 lrger for four hoie thn for two hoie, P o.4). Thus, the ngulr seprtion etween hoie trgets is, y itself, insuffiient to use n inrese in firing-rte exursion of the mgnitue seen on the four-hoie tsk. Reltionship etween LIP tivity n ehvior The preeing nlyses expose the similrities n ifferenes in how LIP tivity reflets oune umultion of eviene uring two- n four-hoie eisions. We next ttempte to relte these Estimte from neurl reorings Urgeny (t 1/2, u N ) (ms, spikes s 1 ) Noneision time (T )(ms) Fitte prmeters K (spikes s 2 %oh 1 ) s (spikes s 1.5 ) 2 hoie , ± ± ±.15 4 hoie , ± ± ±.1 2 hoie , ± ± ±.28 4 hoie , ± ± ±.1 91 ontrol , ± ± ±.5 NATURE NEUROSCIENCE VOLUME 11 [ NUMBER 6 [ JUNE

6 28 Nture Pulishing Group Exursion: four hoie (spikes per s) n = 1 1 ms Neuron I4215 oservtions to the ehviorl mesurements of hoie n retion time. We hve lrey note tht firing rtes uil up fster in ssoition with strong motion n fst retion time. Moreover, this uilup egins t fixe time fter the onset of motion n signls the en of eision (for T in ) y hieving ritil level. Here, we provie two itionl nlyses in support of this mehnism. First, we exmine the reltionship etween uilup rte n retion time for single trils tht terminte with T in hoie (Fig. 7). For eh neuron, the single-tril estimtes of uilup rte n the mesure retion time were etrene to remove the effet of motion strength (see Methos) n omine in stter plot (Fig. 7,). Tril-to-tril vriility in uilup rte ws inversely orrelte with tril-to-tril vriility in retion time, espite the noisy estimte of uilup rte n lrge vrition in retion time. This oservtion ws typil of the t (two hoie: men orreltion oeffiient ¼.22 ±.2, P o 1 5, n ¼ 7 neurons; four hoie: men orreltion oeffiient ¼.26 ±.2, P o 1 5, n ¼ 69; Fig. 7,). Next, we ttempte to reonile LIP tivity with the spee n ury of the Exursion: 9 ontrol (spikes per s) n = 7 1 ms Builup rte (stnrize) Numer of neurons Exursion: two hoie (spikes per s) Exursion: two hoie (spikes per s) Figure 7 Neurl responses n retion times re inversely orrelte on single trils. () Tril-ytril orreltion etween retion time n uilup rte for one representtive neuron (Kenll, t ¼.3, P o 1 4 ). Vlues re expresse in units of s.. from men. This etrening ws performe for eh motion strength using ll orret T in hoies n ll T in hoies for % oherene. () As in, ut for four-hoie trils (t ¼.42, P o 1 4 ). (,) Distriution of orreltion oeffiients (Kenll, t) for eh neuron in the tset for the two-hoie () n four-hoie tsks (). Gry shing inites iniviul neurons with signifint orreltion (P o.5). Arrow inites the men. 5 Two hoie Figure 6 Firing-rte exursion is lrger on the four-hoie tsk. () Firing rtes on two- n four-hoie trils t the eginning n en of the motionviewing perio for single exmple neuron. Responses re ligne to motion onset (left) n se initition (right). Blk n re tres inite responses on the two- n four-hoie tsks, respetively. Tres re smoothe with 3-ms exponentil filter. Exursion is the ifferene etween firing rtes in the she regions. All responses leing to orret T in hoies n with retion time 445 ms were use in this nlysis. Firingrte exursion ws 29.7 ± 8.4 spikes s 1 lrger for four hoie thn for two hoie. () As in, exept tht tres reflet the verge firing rte from 7 neurons n no smoothing ws performe; firing rtes were ompute in 2-ms nonoverlpping ins. () Comprison of firing rte exursion on twon four-hoie tsks. Points re estimtes of the firing-rte exursion from single neurons. One point (17, 159) ws omitte from the stter plot to filitte sling of the remining points. Error rs show stnr error of the exursion (eqution (2)) n re osionlly osure y the points. Green irle mrks the exmple neuron in. Histogrm epits the ifferenes in exursion on two- n four-hoie tsks. Arrow inites the men. Only orret responses to T in trgets were use for this nlysis. Gry shing inites iniviul neurons with signifint ifferenes (P o.5). () Comprison of firing rte exursion on two-hoie n 91 ontrol tsks. The sme onventions were use s in. The sme neuron (17, 159) ws omitte from the stter plot. monkeys eisions (Fig. 1 g). The exursion of LIP firing rtes throughout the eision ws lrger on the four-hoie tsk (Fig. 6). The hnge in LIP uilup rtes proue y hnge in motion strength towr T in ws similr for the two- n four-hoie tsks (Fig. 4). The rise in firing rte tht epens on time, ut not on motion strength or iretion (u(t), urgeny), ws lrger in the two-hoie tsk (Supplementry Fig. 6). All of these oservtions re onsistent with longer retion times on the four-hoie tsk; it tkes longer for LIP firing rtes to unergo the lrger exursion. These oservtions hve less intuitive effets on ury. For exmple, the lrger exursion n less pronoune time-vrying inrese woul le to etter ury on the four-hoie tsk were it not for the lrger numer of hoies. Our moel implements re etween two n four hoie mehnisms to etermine whih trget will e selete (Supplementry Fig. 6). Eh mehnism umultes momentry eviene in 3 Two hoie Kenll τ =.3 Four hoie Kenll τ = Neuron I Retion time (stnrize) Correltion (τ) Four hoie 2 4 Retion time (stnrize).5 1. Correltion (τ) 698 VOLUME 11 [ NUMBER 6 [ JUNE 28 NATURE NEUROSCIENCE

7 28 Nture Pulishing Group fvor of the hoie trget in its response fiel. A eision is rehe when one of the umultors rehes oun. We llowe the monkeys physiology to onstrin oun height n u(t). Three itionl prmeters were juste to fit the hoie n retiontime t (see Methos n Tle 1). We re utious out interpreting the vlues of the fitte prmeters, s we hve fixe the most importnt moel prmeters using the LIP mesurements. The min point is tht they re resonly similr for the two- n four-hoie tsks. The moel provies n equte ount of oth the retion time n ury seen on the two- n four-hoie tsks (Fig. 1 g). This fit supports the hypothesis tht the frmework of oune umultion n e extene to explin eisions with more thn two lterntives. Moreover, the physiologil ifferenes on the two- n four-hoie tsks re of the right size to explin the ehviorl ifferenes tht we oserve. The moel lso ounts for the longer retion times tht we oserve on error trils for oth the two- n four-hoie tsk (Supplementry Fig. 7 online). The moel, espeilly the hyri of fixe n free terms, is not unique, ut unersores the onsisteny of physiologil n ehviorl mesurements in the frmework of oune umultion. DISCUSSION Simple experimentl moels tht uil on knowlege out sensory n motor funtions (reviewe in ref. 1) hve the ownsie tht they rely on mehnisms tht my e inequte for more omplex eisions. Reltive to two-hoie eisions 2,8,9,21, our four-hoie tsk oules the numer of lterntives n hlves the ngulr seprtion etween iretions. Although this ngle remins lrge ompre with the fine isrimintions tht humns n monkeys n mke 22 24,it i inrese iffiulty slightly, s shown y 91 seprtion in twohoie tsk. Longer viewing urtions were require for the sme ury (Fig. 1f,g). To hieve n eptle rewr rte, we use mixture of motion strengths tht fvore esier onitions, ompre with previous hoie retion time experiments (for exmple, see ref. 2). The set ws ientil in ll onitions, so the two-hoie tsk ws esy, proly explining the fster retion times tht we mesure. The monkeys oul ffor to terminte hoies using more lx riterion, wht we interpret s shorter exursion to termintion oun, without inresing the verge proility of error. Mny moels hve een propose to explin the oorintion of retion time n hoie ury in simple two-hoie tsks 4,18, These moels shre ommon feture: termintion oun is pplie to some form of eviene representtion, terme the eision vrile. A puttive neurl orrelte of this opertion is evient in re LIP; for ll retion times (n motion strengths), the firing rtes of LIP neurons rehe stereotype level shortly efore the monkey initites se to inite its hoie for T in (Fig. 2, n Supplementry Fig. 2). Our finings suggest tht this sme oune umultion frmework pplies to eisions mong four hoies. This hypothesis hs een suggeste y theorists 5,28,29, ut hs not een teste until now. A mjor ifferene etween two- n four-hoie responses ws seen in the firing rte t the eginning of the eision proess (Fig. 3). This effet resemles the inverse reltionship etween the numer of possile eye movement iretions n firing rtes in the superior olliulus preeing se instrution 3. We think tht the reution in firing rte on four-hoie response is unlikely to e the result of low-level ftor, suh s the presene of trget in the neuron s inhiitory surroun. This possiility ws refute in the superior olliulus y issoiting unertinty from the numer of trgets. We trie to ple the monkey in two-hoie tsk while proviing four trgets (two were irrelevnt), ut this mnipultion introue uneptle ises in susequent testing. Nonetheless, we out tht surroun suppression explins the firing-rte reution. First, no inhiitory surroun is seen in LIP response fiels 31,32. Seon, responses to the T 9 trget on our sreening se tsk were not lower thn responses to the T out trget, s woul e expete if they were in n inhiitory surroun (Supplementry Fig. 1). Thir, in two experiments, we oserve reution in firing rte on the four-hoie tsk even when the response fiels were lrge enough to enompss the orthogonl trgets in their exittory regions (t not shown). In our four-hoie tsk, this lower firing rte t the eision s eginning onferre higher threshol for terminting the eision. Three experimentl finings support this onlusion. First, the firing rte t the eision s en ws similr in the two- n four-hoie onfigurtions. Thus, the lower strting point implies lrger exursion in firing rte from the strt to the en of the eision. Seon, the rte of uilup in LIP tht n e ttriute to motion strength ws similr on the two- n four-hoie onfigurtions for motion towr T in (Fig. 4) n for T in hoies (Supplementry Fig. 4). This rules out the possiility tht the ifferene in exursion ws ompenste for y hnge in the sling of umulte eviene into units of LIP firing rte. Thir, the urgeny ws stronger in two- thn in four-hoie onitions. This mplifies the effetive ifferene in exursion etween the two- n four-hoie onitions euse it effetively shrinks the exursion; less of the rnge of LIP firing rtes etween strt n en is use to represent the umulte eviene. In the frmework of oune umultion, lrger exursion improves signl to noise t eision termintion, or equivlently, reution in unertinty, t the expense of eision time. When humn sujets rry out the two-hoie motion tsk, they pper to just the oun (or exursion) when they tre off spee n ury 3. When the numer of lterntives is inrese, ll else eing equl, the level of unertinty is initilly inrese. Thus, it is not surprising tht the rin woul rue more eviene efore terminting the eision. The ost is eision time, ut this is tempere y the implementtion of eline tht is impose y the urgeny signl. We ientifie this urgeny signl s the time-epenent rise in firing rte tht is seen on ll trils regrless of motion strength n iretion. For exmple, when the eviene is neutrl (% motion strength), ssuming no hoie is, the oune umultion moel pproximtes n unise iffusion of eviene: rising vrine, ut no hnge in the men eviene. The urgeny signl is the ifferene etween the oserve firing rtes n this expettion from unise iffusion. It is n essentil feture of the moel use to explin hoie n retion time in these experiments. A time-epenent rise of firing rte in LIP n elsewhere hs een interprete s representtion of elpse time in the form of hzr rte or ntiiption funtion 12, In the frmework of oune umultion 17,ituses eisions to terminte s time elpses, regrless of the eviene. In the four-hoie tsk, time is ostly. Long eision times woul improve performne ove hne (25% orret), ut not for the % motion-strength trils n not y muh for the next wekest motion strength. It is therefore sensile to ple time limits on the eision proess. Similr resoning pplies to the two-hoie onfigurtion. Beuse hne represents 5% orret, there is less impetus to omplete the tril t wek motion strengths, whih proly explins the weker urgeny signl seen previously in two-hoie experiments 2,36. Here, however, our esy set of motion strengths renere fst eisions less ostly in the two-hoie tsk. NATURE NEUROSCIENCE VOLUME 11 [ NUMBER 6 [ JUNE

8 28 Nture Pulishing Group One question tht remins is how signls from iretion-seletive neurons in the mile temporl re re omine into momentry eviene. Oservtions from the stnr two-hoie tsk suggest tht LIP neurons reflet the ifferene in firing rtes etween neurons tune to the two opposing iretions of motion 21. This oppositionl reltionship ws emonstrte iretly y stimulting neurons in the mile temporl re n mesuring the effet on hoie n retion time. Builup rtes in LIP reflet, t lest in prt, this sutrtion; hnge in motion strength in the iretion fvoring T in les to proportionl rise in uilup rte in LIP, n n opposite hnge in motion strength les to omprle eline. Our results suggest tht the omputtion tht LIP performs on the motion signls tht it reeives from the mile temporl re (n elsewhere) is similr in the two- n fourhoie tsks. In the ltter, the ontriution of the two nonopposing iretions (T 9 ) ws smll (Fig. 4f), ut we out tht this will generlize to more iretions n hoies. Our experiment is not efinitive, in prt euse the ngle etween lterntives is lrge ompre with the rnge of ngles tht primtes n isriminte Our stuy suggests tht some of the priniples glene from simple inry eisions proly exten to eisions with more thn two hoies. Speifilly, we oserve neurl orrelte of eviene umultion in strutures tht represent the hoie. In ition, the oserve oun on the umultion ties the ontent of eision with the time require to mke it together in single mehnism. This ie lso reeives support from reent stuy in humn psyhophysis 37.It remins to e seen whether the frmework will exten to eisions mong more thn four hoies 28,38,39 or to eisions mong ontinuous rry of hoies, in whih prmeter n tke on ny vlue, like the heing of ompss 4. It my seem tht, oring to the oune umultion frmework, eisions mong ontinuous hoies woul require n infinite numer of umultors. This frmework my simply not e le to ount for eisions mong ontinuous hoies, n my nee to e reple with more suitle moel 27,41. Alterntively, eisions mong ontinuous hoies might e explinle using finite numer of umultors. The resolution of ognitive n motor systems my not require tht the rin represent n infinite rry of possiilities with preision; the egree of unertinty reution nee not exee tht of the systems tht use the estimte. If so, oune umultion frmework with finite numer of umultors my e suffiient, even for eisions mong ontinuous hoies. METHODS Behviorl tsks. Stimuli were shown on CRT monitor with refresh rte of 99 Hz positione 59 m from the monkeys. Stimuli were generte using Mtl 5.2 (Mthworks) n the Psyhophysis toolox 42. For the rnom ot motion tsk, trils were similr to those reporte elsewhere 2,8,36 (Fig. 1 ). Following suessful fixtion (±1.51) ofentrl white trget, two or four highly visile, re peripherl hoie trgets ppere (imeter ¼.51). Choie trgets were 181 prt (two-hoie tsk) or 91 prt (four-hoie tsk n 91 ontrol tsk) n ll hoie trgets were eqully eentri (rnge ¼ 4 151, men¼ 1.51). Two- n four-hoie trils were rnomly interleve. The ynmi rnom ots were isplye in 51 irulr perture entere on the fixtion point. We use vrile oherene rnom ot isply 2,8,43 (ot ensity ¼ 16.7 ots per eg 2 per s). Coherently moving ots were isple to proue 61 s 1 motion. For the first 2 neurons exmine, six motion strengths were use (%, 3.2%, 6.4%, 25.6%, 51.2% n 76.8%). For the remining 5 neurons, only five motion strengths (%, 3.2%, 9.%, 25.6% n 72.4%) were use, so s to mximize the numer of trils per onition ollete. For grphs, omine t from the highest motion strengths re shown t 72.4% oherene. The motion stimulus ws extinguishe when the monkey s gze exite the fixtion winow. s ening in winow (rnge, ± 1.51 to ± 3.51 epening on eentriity) roun the hoie trget orresponing to the orret iretion of stimulus motion were rewre with juie or wter. Otherwise, no rewr ws given n the orret hoie trget riefly oule in size to provie feek. For Monkey S, ely of 1, ms ws impose fter inorret trils efore the next tril egn to isourge fst hoies. We lso use overlp n memory se tsks. In the overlp se tsk, peripherl trget ws ple in the visul fiel, while the monkeys mintine fixtion of entrl spot uring ely perio. The ely perio ws rnom, smple from trunte exponentil istriution. When the entrl spot ws extinguishe, the monkeys me se to the hoie trget. In the memory se tsk, the peripherl trget ws flshe only riefly n the monkeys me se to its rememere lotion. Memory n overlp se tsks were rrie out efore the isrimintion tsk to verify the sptil seletivity of the neuron; trils using the overlp se tsk were usully rnomly interleve with motion isrimintion trils to test for stility of the response fiel n the neurl responses (Supplementry Fig. 1). Eletrophysiology. Two monkeys (I, 51 neurons; S, 19 neurons) were prepre for hroni single-neuron reoring using stnr surgil proeures, s esrie previously 2,8. All surgil n experimentl proeures were in orne with the US Ntionl Institutes of Helth Guie for the Cre n Use of Lortory Animls n were pprove y the University of Wshington Animl Cre Committee. Eletroes (Alph Omeg) were introue into re LIP t eh ily reoring session. Neurl responses were mplifie onventionlly, n wveforms orresponing to ifferent neurons were sorte online using the Plexon Sort Client (Plexon). The Plexon Offline Sorter ws use fter eh experiment to onfirm tht ll reore spikes me from single well-isolte neurons, etermine y wveform shpe n the presene of refrtory perio. Neurons were selete oring to ntomil n physiologil riteri. Antomil lnmrks were provie y mgneti resonne imges of the monkeys rins tht were ompre to ortil prtitioning shemes using Cret softwre 44. These oservtions were onfirme y physiologil oservtions of white mtter, gry mtter n lumen rossings. Neurons inlue in the stuy ll h sptilly seletive responses uring the ely on the overlp n memory se tsks 45 (Supplementry Fig. 1). For 7 neurons, we reore responses on n interleve lok of two- n four-hoie trils. For 29 of these neurons, responses were ollete on seon lok of trils tht ontine 91 ontrol trils interleve with either two- or four-hoie trils. Dt nlysis. We mesure firing rtes in severl time winows efine with respet to the tsk epohs: efore motion, erly motion n efore ses. For the pre-motion epoh, firing rtes were otine 2 3 ms fter the onset of the hoie trgets. An erly winow (17 to 21 ms fter motion onset) n lte winow (4 8 ms efore se initition) provie estimtes of the firing rtes t the eginning n t the en of the eision proess (Figs. 3 n 6). These points were estlishe from n nlysis of firing-rte vrine ssoite with the ifferent motion stimuli n retion times (Supplementry Fig. 2). For oth the two- n four-hoie tsk, firing rte vrine remine t stle, low vlue until 19 ms fter the onset of stimulus motion, fter whih it inrese mrkely, refleting the ivergene of responses to ifferent stimuli. We use 4-ms winow entere t this point, ut the estimte of the firing rte ws nerly ientil for vriety of intervls. The en of the eviene umultion ws tken to e the point t whih the vrine ws lowest just efore the se (Supplementry Fig. 2), s in previous stuies 2. We use 4-ms time winow entere on the point orresponing to miniml vrine. For nlyses of firing rtes roun the se, the first 2 ms of the response to the ot motion were remove to exlue the stereotype ip tht ws time-loke to the onset of the rnom ot isply (Fig. 2). For the nlysis of firing-rte exursion n firing rte t the en of the eision, trils with retion times shorter thn 45 ms were exlue. A wie vriety of intervl efinitions yiele nerly ientil results, with one exeption. Estimtes of firing rte t the en of eviene umultion were sensitive to the intervl use euse the firing rtes hnge mrkely ner the 7 VOLUME 11 [ NUMBER 6 [ JUNE 28 NATURE NEUROSCIENCE

9 28 Nture Pulishing Group time of the se. However, the effet of hoie numer on firing-rte exursion tht we report here ws present for rnge of time intervls, entere up to 13 ms efore the se (Supplementry Fig. 2). Two- n four-hoie firing-rte exursions were more similr for intervls entere t longer time points efore the se, ut these erlier time points re unlikely to orrespon to the en of eviene umultion 2,46. To estimte the uilup in firing rte uring eision formtion, we first onstrute peri-stimulus time histogrm (1-ms ins) for eh neuron from responses ligne to the onset of rnom ot motion, exluing the epoh eginning 6 ms efore se initition. The uilup rte is the slope of the line fit to peri-stimulus time histogrm in the epoh from ms fter motion onset using regression. We use n erly epoh to inlue s mny trils s possile in the verges, minimize ttrition of portions of trils from the verges (preeing the se) n minimize the potentilly onfouning effet of hoie on the stimulus-epenent uilup rte. Anlyses employing ifferent time intervls ffete the reporte vlues of uilup rte only slightly; the lrgest effet ws on uilup rtes orresponing to the highest motion strength, owing minly to the ttrition of trils with short retion times. Dt from the 3.2% n 6.4% oherenes were omine, s were t from the 72.4% n 76.8% oherenes so tht popultion verges woul not e skewe y vlues ssoite with motion strengths tht were only use on minority of ys. On the four-hoie tsk, the uilup rtes for orthogonl motion were generte from omine responses to the two orthogonl motion iretions use. We otine similr results whether uilup rtes were from iniviul neurons (s esrie ove) or were ompute from the verge popultion response (t not shown). Popultion nlyses tht group trils with ommon motion iretion (reltive to T in ), s well s the estimte of u(t), oul e ffete y hoie is. We therefore nlyze the istriution of hoies me on the % oherene motion trils. No relile is ws foun. For two-hoie trils, the numers of T in n T out hoies were 1,452 versus 1,387 (P ¼.22, inomil istriution, H : p ¼.5). For four-hoie trils, the numers of T in, T out,t 9 n T 9 hoies were 1,461, 1,421, 1,387 n 1,426 (P ¼.65, w 2 test, H : equl proportions). To estimte the reltionship etween uilup rte uring eision formtion n retion time, we nlyze single trils (Fig. 7), using wek motion strengths ( 25.6% oherene) leing to orret T in hoies (ll T in hoies for % oh). We estimte the firing rte on eh tril y onvolving the spike trin with n lph-like funtion ð1 e t=g Þe t=,where ¼ 25 ms n g ¼ 1 ms. The uilup rte is the slope of the est fitting line to the smoothe firing rte in the epoh from 19 ms fter motion onset to 12 ms efore se initition. The uilup rtes n the ssoite retion times were stnrize for eh motion strength (ifferene from the men in units of smple s..). The stnrize uilup rtes n retion times were omine for ll oherenes to estimte orreltion oeffiient (Kenll, t) for eh neuron. We require tht t lest three spikes e present in tril to estimte uilup rte n tht t lest five trils e present in given neuron to estimte the orreltion oeffiient. One neuron ws exlue on the four-hoie tsk for filing to meet these riteri. After estimting the orreltion oeffiient for eh neuron, we teste whether the men of the popultion ws signifintly less thn (t-test). We use regression nlyses to estimte the effet of the numer of hoies on the LIP firing rtes 7. Unless otherwise stte, ll fitting uses mximum likelihoo uner Gussin ssumptions for noise (tht is, weighte lestsqures regression with known vrine); stnr errors of prmeter estimtes were otine y inverting the Hessin mtrix of erivtives of the log likelihoo with respet to the prmeters, evlute t the mximum likelihoo solution. For these regression nlyses, tests of signifine were etermine y t-sttistis ompute from the estimte oeffiients n their stnr errors. For mny nlyses, single-neuron mesurements were gthere in frequeny histogrm (Figs. 3, n 6,; see lso Supplementry Figs. 1, 4, n 5), on whih we use t-test (effetively pire t-test on the ssoite stter plots). For ll nlyses, we lso use mximum likelihoo to generte popultion mens n ssoite t-sttistis. These two methos sometimes yiele slightly ifferent estimtes, ut the iretion n signifine of ll of the results reporte here were lwys in greement. For simple omprisons of firing rtes in speifie epoh, the regression is effetively t-test, y ¼ 1 I n ; where I n ¼ or 1 for two n four hoie, respetively, n y is the spike rte on iniviul trils. i re fitte oeffiients. The null hypothesis is tht hoie numer oes not ffet the firing rte (H : 1 ¼ ). To ompre firing-rte exursion for two verses four hoie, we fit the moel: y ¼ 1 I n 2 I T 3 I n I T ð2þ where the i re fitte oeffiients, y is the spike rte mesure in the eginning n lte epohs (efine ove) n the I x re initor vriles, I n ¼ or1 for two n four hoie, respetively, n I T ¼ or 1 for erly n lte epohs, respetively. The oeffiients estimte the following prmeters: is the firing rte for the two-hoie onition in the erly epoh, 1 is the firing rte in the four-hoie onition in the erly epoh, 2 is the firing rte in the two-hoie onition in the lte epoh n is the firing rte in the four-hoie onition in the lte epoh. 2 furnishes n estimte of the firing-rte exursion in the two hoie n 3 estimtes the ifferene in exursion on the four-hoie tsk. The null hypothesis tht the numer of hoies oes not ffet the exursion is H : 3 ¼. To ompre responses etween the two- n four-hoie tsks uring the motion epoh, we ompute the reltionship etween motion strength n uilup rte using weighte regression, y ¼ 1 I n 2 C 3 I n C where y is the uilup rte either verge ross ll the neurons in the popultion (Fig. 4f,g) or for single neuron (Fig. 4e n Supplementry Fig. 5) for eh motion strength, I n is n initor for the numer of trgets (s ove) n C is motion strength (% oherene). The uilup rtes ssoite with % oherene motion re furnishe y n 1 for two- n fourhoie tsks; 2 n 2 3 provie estimtes for the effet of hnge in motion strength on the uilup rtes. The null hypothesis is tht the numer of hoies oes not ffet this reltionship (H : 3 ¼ ). Eye movements. Eye position ws smple t 1 khz using the slerl serh-oil metho 47, s esrie previously 2. To ientify ifferenes in the oulomotor responses on two-hoie versus four-hoie trils, we ompre pek se veloity, se mplitue n se error on eh onition. Some of these mesures were ffete y the numer of hoies 48,49 (Supplementry Fig. 8 online). To test whether these ftors expline (s potentil onfouners) the effet of hoie numer on LIP responses, we inorporte the three signifint prmeters into the regression esrie ove (eqution (1) n (2)): y ¼ 1 I n 1 S 2 Ṡ 3 E ð4þ y ¼ 1 I n 2 I T 3 I n I T 1 S 2 Ṡ 3 E where y is the spike rte mesure on eh tril either t the eginning or t the en of the eision for ll trils in ll ells (tht is, eh tril ontriutes two. mesures), i re itionl fitte oeffiients, n S, S n E re mplitue, pek veloity n enpoint error, respetively. The null hypotheses were reevlute with these extr terms in the regression. The null hypothesis, tht the numer of hoies i not ffet the exursion, ws gin rejete oth t the popultion level (P o.1) n in ll ut one of the iniviul ells where the effet of firing-rte exursion ws signifint. In this ell, firing-rte exursion ws foun to e longer for two hoie thn for four hoie, n exeption to the tren tht we orinrily oserve. Diffusion moel. We fit ehviorl t using n umultor moel (Tle 1 n Supplementry Fig. 6). An umultor orresponing to eh trget integrtes the momentry eviene towr eision oun. We ssume the eviene is liner omintion of mile temporl re tivity for motion towr the trgets. The liner weights were tken from the osine of the ngle ifferene etween the iretions 22. The oun height n the motion-inepenent signl u(t) were otine from the neurl responses to the two- n four-hoie tsks, s expline elow. Three free prmeters ð1þ ð3þ NATURE NEUROSCIENCE VOLUME 11 [ NUMBER 6 [ JUNE 28 71