Journal of Computatonal Neuroscence 5, 71 90 (1998) c 1998 Kluwer Academc Publshers. Manufactured n The Netherlands. A Mathematcal Model of the Cerebellar-Olvary System II: Motor Adaptaton Through Systematc Dsrupton of Clmbng Fber Equlbrum GARRETT T. KENYON, JAVIER F. MEDINA AND MICHAEL D. MAUK Department of Neurobology and Anatomy, Unversty of Texas Medcal School at Houston, Houston, Texas 77030 mmauk@nba19.med.uth.tmc.edu Receved March 5, 1997; Revsed June 3, 1997; Accepted May 16, 1997 Acton Edtor: John Rnzel Abstract. The mplcatons for motor learnng of the model developed n the prevous artcle are analyzed usng dealzed Pavlovan eyeld condtonng trals, a smple example of cerebellar motor learnng. Results suggest that changes n gr Pkj synapses produced by a tranng tral dsrupt equlbrum and lead to subsequent changes n the opposte drecton that restore equlbrum. We show that these opposng phases would make the net plastcty at each gr Pkj synapse proportonal to the change n ts actvty durng the tranng tral, as nfluenced by a factor that precludes plastcty when changes n actvty are nconsstent. Ths yelds an expresson for the component of granule cell actvty that supports learnng, the across-trals consstency vector, the square of whch determnes the expected rate of learnng. These results suggest that the equlbrum mantaned by the cerebellar-olvary system must be dsrupted n a spec and systematc manner to promote cerebellar-medated motor learnng. Keywords: eyeld condtonng, Purknje, ncttatng, LTP, LTD In the precedng artcle we used a mathematcal model to study how the synaptc organzaton of the cerebellarolvary system (cerebellum and clmbng fber nputs from the nferor olve) nfluences plastcty at granule cell to Purknje cell (gr Pkj) synapses n the cerebellar cortex. The results suggest that the plastcty observed emprcally at these synapses they decrease n strength when actve durng clmbng fber nput (long-term depresson, LTD) and ncrease when actve wthout a clmbng fber nput (LTP) (Ekerot and Kano, 1985; Hrano, 1990; Ito and Kano, 1982; Kano and Kato, 1988; Lnden et al., 1991; Sakura, 1987, 1989; Saln et al., 1996; Schreurs and Alkon, 1993; Shbuk and Okada, 1992) mantans an equlbrum level of clmbng fber actvty at whch LTD and LTP balance and the expected net change n the weghts of gr Pkj synapses s zero (Kenyon et al., 1997). Here, we consder how motor learnng may be affected by a self-regulatng equlbrum of clmbng fber actvty. The cerebellum s clearly mportant for the approprate executon of movements as revealed by the severe motor mparments assocated wth cerebellar pathology (Dow and Moruzz, 1958; Glman et al., 1981) and by the numerous examples of motor defcts produced by expermental manpulatons of the cerebellum (McCormck and Thompson, 1984a, 1984b; Nagao, 1983; Optcan and Robnson, 1980; Robnson, 1976; Thach et al., 1992; Westhemer and Blar, 1973; Zee et al., 1981). Recently, doubts regardng the role of the cerebellum n motor functon have stemmed from studes mplcatng cerebellar nvolvement n nonmotor tasks (Bracke-Tolkmtt et al., 1989; Fez et al., 1992; Ivry et al., 1988; Km et al., 1994; Lener et al., 1991; Mddleton and Strck, 1994). Although the queston of motor and nonmotor roles for the cerebellum s
72 Kenyon, Medna and Mauk sometmes portrayed as an ether/or ssue (Gao et al., 1996), there s no compellng reason to beleve that they are mutually exclusve. Snce the regularty of the cerebellum s synaptc organzaton suggests that all regons perform a common computaton, the applcaton of ths computaton to both motor and nonmotor behavors seems nether mplausble nor nconsstent wth current data. Indeed, ths potental versatlty represents an especally exctng aspect of current cerebellar research. The spec mechansms of cerebellar nformaton processng whch may be most easly dentfed by studyng movements may also provde nroads nto a deeper understandng of the mechansms of cogntve processes. Many theores have attempted to relate the anatomy and physology of the cerebellum to dfferent aspects of sensory-motor functon (Bloedel, 1992; Bower and Kassel, 1990; Bullock et al., 1994; Fujta, 1982; Gao et al., 1996; Glbert, 1974; Ito, 1982; Kawato and Gom, 1992; Llnas and Welsh, 1993; Moore et al., 1989; Pellonsz and Llnas, 1980; Sejnowsk, 1977; Thach et al., 1992; Thompson, 1986). One nfluental class of theores suggests that the cerebellar-olvary system contrbutes to motor learnng. These theores share basc features that were orgnally proposed by Marr (1969) and Albus (1971): (1) stmul that precede movements are encoded by mossy fber drven patterns of actvty n spec subsets of cerebellar granule cells, (2) clmbng fber nputs to Purknje cells are actvated by movement errors and as such sgnal the need for adaptaton of the recently executed movement, (3) gr Pkj synapses that are coactve wth the clmbng fber nput are modfed n strength such that subsequent movements under the same crcumstances are mproved. In support of these features, results from several preparatons ndcate that gr Pkj synapses undergo clmbng fber-dependent LTD (Ito, 1989; Lnden et al., 1991; Sakura, 1987) and that several forms of motor learnng, ncludng Pavlovan eyeld condtonng (McCormck and Thompson, 1984a, 1984b; Perrett et al., 1993; Perrett and Mauk, 1995; Thompson, 1986), adaptaton of the vestbular-ocular reflex (VOR) (du Lac et al., 1995; Ito, 1982; Lsberger, 1988; Nagao, 1983; Raymond et al., 1996; Robnson, 1976), and learnng of wrst movements (Glbert and Thach, 1977; Thach, 1980; Thach et al., 1992), depend on the cerebellum n a manner consstent wth these theores. There s not, however, complete agreement regardng role of the cerebellum n motor learnng (Bower and Kassel, 1990; De Schutter, 1995; Llnas and Welsh, 1993; Pellonsz and Llnas, 1980; Houk and Wse, 1995). There even reman debates about the prmary leson, stmulaton, and recordng data that support a role for the cerebellum n motor learnng (Kelly et al., 1990; Welsh and Harvey, 1989). Although these debates are beyond the scope of ths artcle, the present analyss s motvated n part by a number of the emprcal observatons and conceptual arguments that dentfy apparent nconsstences wth basc aspects of cerebellar motor learnng theores. These potental problems have been addressed to varyng degrees by certan cerebellar models, but no sngle model addresses all crtcsms. We do not attempt an exhaustve revew of these models but only touch on representatve examples to hghlght the potental problems of exstng motor learnng theores of cerebellar functon. Also, analyss of both eyeld condtonng and adaptaton of the vestbulo-ocular reflex ndcate that plastcty can occur n both the cerebellar cortex and nucle (see Raymond et al., 1996; Mauk, 1997). Here, we focus only on the contrbutons of plastcty n the cerebellar cortex. One argument relates to an array of dffcultes posed by the spontaneous actvty of clmbng fbers (Keatng and Thach, 1995). Although the orgnal Marr/Albus theores dd not address spontaneous clmbng fber actvty, wth the addton of such actvty these theores appear to predct that all gr Pkj synapses wll saturate at maxmum (Marr) or mnmum (Albus) values, whch would preclude the possblty for storng motor memores. It s also unclear how the occasonal error-evoked clmbng fber nputs can convey nformaton aganst a background of the more abundant spontaneous nputs. Glbert (1975) addressed ths ssue by assumng that LTD only perssts when ts nducton occurs durng a noradrenergc nput. However, ths dea s drectly contradcted by numerous n vtro LTD studes (see Ito, 1989). Another crtcsm relates to the nablty of most cerebellar theores to explan bdrectonal adaptaton of movements, whch s requred to explan the ablty of the VOR to both ncrease or decrease n gan, and by the ablty of Pavlovan eyeld responses to be acqured and extngushed. Although gnored by the Marr and Albus theores, ths ssue was addressed by more recent models that employ bdrectonal plastcty controlled by the level of clmbng fber actvty. For example, both Fujta (1982) and Sejnowsk (1977) assume that gr Pkj synapses decrease n strength when actve durng clmbng fber actvty above a crtcal
Cerebellar Model II: Motor Adaptaton 73 level and ncrease n strength when actve durng lower levels of clmbng fber actvty. It s clear how movement errors could drve clmbng fber actvty above the crtcal value, whch would lead to acquston or ncreased ampltude/gan of the response. However, nether model spees how spontaneous clmbng fber actvty could fall below the crtcal value to permt extncton or decreased ampltude of the movement. Moreover, nether model spees how the clmbng fber actvty remans at the crtcal value, whch s essental for mantanng the present pattern of synaptc weghts when movements do not requre adaptaton. In general, an nablty to explan how synaptc weghts reman constant when movements are approprate s a weakness of all cerebellar motor learnng models. As noted above, some models touch on ths ssue by assumng plastcty does not occur when clmbng fber actvty s at a crtcal level, although how ths level s mantaned s not speed. Alternatvely, some models employ the bologcally unrealstc smplfcaton that actvty s lmted to dscrete learnng and retenton or readout perods (for example, Bullock et al., 1994). Plastcty controlled by error sgnals s only operable durng the learnng phase, and the effects of ths plastcty are subsequently probed n the readout phase wthout affectng synaptc weghts. Ths ensures that changes n synaptc weghts wll relably survve untl the readout phase and thus ensures the stablty n the pattern of synaptc weghts encodng memores. However, the cerebellum represents a system n whch the assumptons that both actvty and plastcty are lmted to dscrete learnng perods are untenable. Evdence ndcates that LTD and LTP at gr Pkj synapses s actvty dependent; synapses are elgble to change only when they are actve (for example, Lnden and Connor, 1993). Gven the varety of stmul that can actvate mossy fbers (Bloedel and Courvlle, 1981), t seems lkely that there s always a nonzero amount of mossy fber nput to the cerebellum and that there s always a subset of granule cells that s actve at any gven tme. Thus, actvty-dependent plastcty suggests that there are ongong opportuntes for plastcty at these synapses and that stablty n the pattern of synaptc weghts encodng memores cannot arse from a lack of such opportuntes. We suggest that realstc attempts to understand or model the cerebellum must apply the same rules for plastcty at all tmes and must consder the ongong actvty of cerebellar neurons. Another apparent problem that has not been addressed by theores of the cerebellum relates to the dscrepancy between the rate of nducton of LTD and the rate of acquston of cerebellar-medated responses. Although LTD can be nduced n vtro n sx pared presentatons of transmtter and postsynaptc depolarzaton ntended to mmc a clmbng fber response (Lnden and Connor, 1993), the acquston of Pavlovan eyeld responses generally requres more than 100 tranng trals. If LTD s a putatve mechansm of motor learnng, how can ths dscrepancy be explaned? Fnally, n attemptng to address certan lmtatons, many models have departed from the drect, neuronbased language that contrbuted to the elegance of the Marr/Albus theores. The rules for plastcty at gr Pkj synapses were stated n terms of the local and mmedate effects that synaptc nputs produce, wthout assumng crtcal ntermedate steps. For example, Albus assumed that gr Pkj synapses decrease n strength when coactve wth a clmbng fber nput. In contrast, many of the more recent models employ less drect language n whch crtcal ntermedate steps are assumed but are not speed n bologcal terms. For example, both Fujta (1982) and Sejnowsk (1977) assume gr Pkj synaptc changes are related to clmbng fber actvty that s dfferent from a crtcal level, but t s not clear how gr Pkj synapses could detect such changes. Here we extend the analyss presented n the precedng artcle to address the crtcsms outlned above wth a contnuous actvty model that acknowledges spontaneous clmbng fber actvty and unformly apples the same plastcty rules at all tmes. The model bulds on the precedng results n whch (1) plastcty rules are stated n terms of local and drect effects of synaptc nputs and (2) clmbng fber actvty s regulated by the cerebellar-olvary system to an equlbrum at whch gr Pkj synaptc weghts undergo no net changes (Kenyon et al., 1997). Usng a smple representaton of Pavlovan eyeld condtonng, we show how error-evoked clmbng fber nputs can be detected aganst a background of spontaneous nputs, how ths spontaneous clmbng fber actvty can be ncreased or decreased to produce acquston or extncton, respectvely, and how the rate of motor learnng should be sgnfcantly slower than the rate of LTD nducton. Thus, our results suggest that cerebellar-medated motor learnng can be understood n terms of the condtons under whch cerebellar-olvary equlbrum must be dsrupted: (1) condtonng trals dsrupt clmbng fber equlbrum and nduce plastcty at gr Pkj synapses, (2) ths ntates a second phase of plastcty
74 Kenyon, Medna and Mauk after the tral that restores equlbrum, and (3) the net effect of these two phases allows learnng to occur only when the clmbng fber equlbrum s repeatedly dsrupted durng a pattern of granule cell actvty that s relatvely consstent from one tral to the next. These processes cast new lght on many of the apparent problems of motor learnng theores of cerebellar functon. Methods Results from Prevous Model In the precedng artcle (Kenyon et al., 1997), we developed a mathematcal model of the cerebellar-olvary system usng dscrete tme steps of duraton t, correspondng to the tme over whch a clmbng fber nput can nduce LTD n coactve grpkj synapses. For completeness, the basc mathematcal formalsm and man results of ths analyss are summarzed here (Fg. 1B). The actvty of a Purknje cell, denoted by P pc and defned as the number of spkes per tme step, s assumed to be a functon of the lnear sum of ts actve nputs weghted by the strength of each synapse. Thus, P pc = m P w, (1) =1 where P represents the background lkelhood of actvty and w the synaptc weght of the th gr Pkj synapse, m gves the total number of such synapses, and the weghts are scaled such that 0 P pc 1. Smlarly, the actvty of the assocated clmbng fber, denoted by P, s gven by the sum of ts nput from cerebellar output, P pc, and ts nput related to the uncondtoned stmulus n eyeld condtonng, E US, such that Plastcty at gr Pkj Synapses P = P pc + E US. (2) Based on emprcal evdence, gr Pkj synapses are assumed to dsplay LTD n whch the synapses decrease n strength when actve wth a clmbng fber nput and LTP n whch they ncrease n strength when actve n the absence of a clmbng fber nput. As n the precedng artcle, ths plastcty can be expressed by an equaton of the form w = P (δ + + δ ) [ P ( ) P ], (3) n whch δ and δ + are constants that represent the step decreases and ncreases n w resultng from LTD and LTP events, respectvely, and P ( ) denotes the equlbrum level of clmbng fber actvty, P ( ) = δ + /(δ + + δ ), (4) at whch the effects of LTD and LTP balance. For convenence, the tme step t s taken to reflect the duraton of nfluence of a sngle clmbng fber nput to a Purknje cell. Clmbng Fber Equlbrum The analyss descrbed n the precedng artcle suggests that these forms of LTD/P combne wth the synaptc organzaton of the cerebellar-olvary system to regulate spontaneous clmbng fber actvty to the equlbrum level, P ( ). Snce, the effects of LTD and LTP would balance at ths level of clmbng fber actvty, both spontaneous Purknje cell and clmbng fber actvty would also reman at ther respectve equlbrum levels. The precedng results also suggest that any perturbaton of ths equlbrum results n net changes n gr Pkj synapses that restore clmbng fber and Purknje cell equlbrum. Mathematcal Representaton of Pavlovan Eyeld Condtonng Although we expect the present results to apply to other forms of cerebellar-medated motor learnng (Raymond et al., 1996), we make use of the relatvely drect assocaton between the stmul used n Pavlovan eyeld condtonng and the nput/output pathways of the cerebellum (Fg. 1). As shown n Fg. 1A, eyeld condtonng typcally nvolves the pared presentaton of a condtoned stmulus (for example, a tone) wth an uncondtoned stmulus (puff of ar to the eye). Under approprate condtons, repeated presentaton of these condtoned stmulus + uncondtoned stmulus trals promotes the acquston of condtoned eyeld responses that are elcted by the condtoned stmulus (Gormezano et al., 1962), as also shown n Fg. 1A. A number of laboratores have provded evdence that the condtoned stmulus s conveyed to the cerebellum va mossy fbers (Lews et al., 1987; Solomon et al., 1986; Stenmetz et al., 1985, 1986, 1987, 1988), the uncondtoned stmulus by clmbng
Cerebellar Model II: Motor Adaptaton 75 fbers (Mauk et al., 1986; McCormck et al., 1985), and that cerebellar output drves the expresson of condtoned responses (McCormck and Thompson, 1984a). Ths s most drectly demonstrated by studes n whch normal condtonng occurs when the condtoned stmulus and uncondtoned stmulus are replaced by drect stmulaton of mossy fber and clmbng fber pathways, respectvely (Stenmetz et al., 1989). Gven these relatonshps between the mossy fber nputs and the condtoned stmulus, clmbng fber nputs and the uncondtoned stmulus, and the output of the cerebellum and the expresson of condtoned responses, eyeld condtonng can be represented relatvely smply n the present model (Fg. 1B). As llustrated schematcally n Fg. 1C, the presentaton of a condtoned stmulus and the correspondng actvaton of certan mossy fbers can be represented by actvty n granule cells that may be dfferent from the background actvty. Thus, through the actvaton of mossy fbers, the condtoned stmulus s assumed to change the actvty of the th granule cell from P to P (CS), and thus the granule cell actvty vector durng the condtoned stmulus, P (CS), may be dstnct from the background actvty vector P. Uncondtoned stmulus presentatons can be represented by transently assgnng a postve value to E US, whch brefly ncreases the actvty of the clmbng fber (Fg. 1C). Fnally, snce Purknje cells nhbt the output cells of the cerebellar nucle, a decrease n Purknje actvty durng the condtoned stmulus can be taken as a measure of the magntude of the condtoned response. Evdence suggests that ths pattern and tmng of stmul s common to other forms Fgure 1. The relatonshps between eyeld condtonng, the synaptc organzaton of the cerebellum, and ther mathematcal representaton n the present model. A. Eyeld condtonng nvolves the pared presentaton of a condtoned stmulus, such as a tone, wth a uncondtoned stmulus, such as a puff of ar n the eye. Under approprate crcumstances, the presentaton of many such trals promotes the acquston of a condtoned eyeld response. B. Smplfcatons of the synaptc organzaton of the cerebellum employed n the the analyss. Snce ncreased cerebellar output appears the to drve expresson of condtoned eyeld responses and snce Purknje cells nhbt the output cells (not shown), we assume that decreases n Purknje cell actvty produce condtoned responses. Moreover, snce cerebellar output also nhbts clmbng fbers, we have smplfed the pathway between the Purknje cells and clmbng fbers, whch contans two nhbtory synapses n seres, to a smple exctatory connecton. The stmul used n eyeld condtonng are conveyed to the cerebellum va two cerebellar afferents the uncondtoned stmulus va clmbng fbers, whch project to the Purknje cells, and the condtoned stmulus va mossy fbers, whch make exctatory connectons onto the granule cells. C. The present analyss gnores the mossy fber synapses onto granule cells and smply assumes that the condtoned stmulus can alter granule cell actvty. Each of the m granule cells s assumed to have a background level of actvty P and a potentally dfferent actvty durng the condtoned stmulus P (CS). Each granule cell contacts the Purknje cell wth a strength or weght w that can change when the synapse s actve dependng on the presence or absence of the clmbng fber nput (see text).
76 Kenyon, Medna and Mauk of cerebellar motor learnng such as VOR adaptaton (Raymond et al., 1996). Results Eyeld Condtonng Trals Intate Two Phases of Plastcty To analyze the expected consequences of eyeld condtonng trals on the cerebellar-olvary system, we frst consder the effects of a condtoned stmulus + uncondtoned stmulus tral on a sngle gr Pkj synapse. Wth the smplfyng assumpton that the uncondtoned stmulus relably elcts a clmbng fber nput (P = 1 durng the uncondtoned stmulus), Eq. (3) descrbng the expected change n synaptc weghts becomes w (CS) = P (CS) (δ + + δ ) [ 1 P ( ) ], (5) where w (CS) s the change n synaptc weght of the th synapse durng the condtoned stmulus + uncondtoned stmulus tral. Usng Eq. (4) to substtute for, Eq. (5) smplfes to P ( ) w (CS) = δ P (CS), (6) showng that each synaptc weght s expected to decrease to the extent that t s actve durng the condtoned stmulus, whch s smply a stochastc expresson of the plastcty rule proposed by Albus (1971). Snce gr Pkj synaptc weghts can only decrease n strength f they are actve durng the condtoned stmulus, and reman unchanged f they are not, Eq. (6) shows that the condtoned stmulus + uncondtoned stmulus tral wll necessarly dsturb the Purknje cell and clmbng fber equlbra exstng before the tral. Both Purknje cell and clmbng fber actvty wll be lower than ther equlbrum levels snce the reducton n some gr Pkj synaptc weghts wll decrease the sum of the synaptc nputs ( P w ) below equlbrum. Ths mples that (1) a second phase of plastcty occurs after each condtoned stmulus + uncondtoned stmulus tral durng whch synaptc weghts ncrease accordng to ther background actvty untl equlbrum s restored, and (2) the effects of a condtoned stmulus + uncondtoned stmulus tral can be understood n terms of the net dfference between these two phases, as s shown schematcally n Fg. 2. Snce ths renormalzaton of synaptc weghts wll occur over tme, we Fgure 2. Pared condtoned stmulus + uncondtoned stmulus trals dsrupt the clmbng fber equlbrum exstng before the tral and promote two phases of plastcty. The gr Pkj synapses actve durng the tral undergo net LTD due to the ncrease n clmbng fber actvty produced by the uncondtoned stmulus. Ths decreases clmbng fber actvty below equlbrum after the tral. Durng ths phase gr Pkj synapses undergo net LTP accordng to ther background actvty untl clmbng fber equlbrum s restored. let w (R) denote the total change n synaptc weght durng the return to clmbng fber equlbrum, such that w (R) = t n w (t n ), (7) where the sum s over all tme steps followng the condtoned stmulus + uncondtoned stmulus tral. From our general expresson for LTD/P (Eq. (3)), ths sum s gven by w (R) = P (δ + + δ ) [ P (t n ) P ( ) ], (8) t n where P (t n ) denotes clmbng fber actvty as a functon of tme after the condtoned stmulus + uncondtoned stmulus tral. Equaton (8) shows that durng return to equlbrum the total change n the weght of each synapse s a functon of ts background actvty and a sum that s the same for all synapses. Thus, lettng ths sum be denoted as x, Eq. (8) can be wrtten more smply as w (R) = xp. (9) Durng return to equlbrum the change n background synaptc nput to the Purknje cell produced by the condtoned stmulus + uncondtoned stmulus tral s reversed (that s, P w CS = P w (R) ). Ths equalty permts a soluton for x n terms of P and, such that Eq. (9) becomes P (CS) w (R) = δ sp, (10)
Cerebellar Model II: Motor Adaptaton 77 where s s gven by s = Pj P (CS) j P 2 j. (11) The expected net change n synaptc weghts produced by both phases of plastcty ( w (net) ) s then gven by combnng Eqs. (6) and (10): w (net) = δ [ sp P (CS) ]. (12) Equaton (12), whch s the man result of ths secton, shows that grpkj synaptc weghts generally decrease when they are more actve than normal durng the condtoned stmulus and ncrease when less actve durng the condtoned stmulus. However, ths relatonshp s modfed by the factor s, the mplcatons of whch we consder next. Net Plastcty s Selectvely Senstve to the Across-Trals Consstency of Granule Actvty Any form of Pavlovan condtonng requres a degree of across-trals consstency n the neural actvty elcted by the condtoned stmulus. Condtonng would clearly be retarded or prevented should the condtoned stmulus actvate a dfferent subset of synapses each tral. Wth no background actvty the expected change n the strength of the th synapse produced by a condtoned stmulus + uncondtoned stmulus tral would be related to ts condtoned stmulus actvty,, whch s also a reasonable defnton of the across- P CS trals consstency for that synapse. Wth the arguments that follow we suggest that the expected change n synaptc weghts durng condtonng ndcated by Eq. (12) represents a more general expresson for across-trals consstency necesstated by the presence of background actvty. In partcular, the term s prevents learnng when there s no consstency but when the condtoned stmulus produces an overall change from background n the number of actve granule cells. To llustrate ths we use the hypothetcal scatter plot shown n Fg. 3A n whch the condtoned stmulus actvty for each synapse, P (CS), s plotted as a functon of ts background actvty P. Ths scatter plot llustrates three ponts. Frst, the term s from Eq. (12) defnes the slope of the least-squares lnear regresson lne that s constraned to pass through the orgn. Second, for any gven value of background actvty P, the correspondng y-axs value of the pont on the regresson lne (the term sp n Eq. (12)) s proportonal to the amount of LTP that a synapse wth that background actvty wll contrbute to the plastcty that restores clmbng fber equlbrum (Fg. 3A, dotted arrow). The slope term would be relatvely hgh for large amounts of gr Pkj actvty durng the condtoned stmulus, ndcatng that a relatvely large amount of LTD was nduced durng the tral, there was a relatvely large dsplacement of clmbng fber actvty away from equlbrum, and therefore a relatvely large amount of LTP wll be requred to return to equlbrum. Whatever the total amount of LTP requred to restore equlbrum, t wll be dstrbuted across synapses accordng to ther background actvtes, as reflected by the lne descrbed by s. Thrd, the dfference between each pont and the correspondng pont on the regresson lne (Fg. 3A, sold arrow) represents the amount of net plastcty for that synapse, and thus the condtonablty of a condtoned stmulus s related to the sum of these vertcal dstances over all synapses. Therefore, synapses whose condtoned stmulus and background actvtes place them above the lne wth slope s wll undergo more LTD durng the tral than LTP durng return to equlbrum (lghter regon n Fg. 3B). Conversely, those below the lne wll undergo less LTD than subsequent LTP (darker regon n Fg. 3B). Fgure 3 llustrates that the expected net changes n synaptc weghts produced by a condtoned stmulus + uncondtoned stmulus tral are related to the acrosstrals consstency of the granule cells actvated by the condtoned stmulus. Wth perfect across-trals consstency the subset of granule cells actvated by the condtoned stmulus s dentcal each tral, whch means that each P (CS) s ether zero or one. In contrast, t s more dffcult to defne precsely the complete lack of across-trals consstency. Although wthout background actvty across-trals consstency s related to the condtoned stmulus actvty (P CS ), wth background actvty, across-trals consstency s presumably related n some way to the dfferences between background and condtoned stmulus actvtes for each synapse. For example, a synapse that s always actve durng the condtoned stmulus (perfect acrosstrals consstency wthout background actvty) could not make a sgnfcant contrbuton to a condtoned response f ts background actvty s equally hgh. Ths suggests the possblty that dfference between background and condtoned stmulus actvty (P P (CS) )
78 Kenyon, Medna and Mauk may approxmate the degree of across-trals consstency. However, the apparent mportance of the term s n Eq. (12) s that a smple actvty dfference rule can napproprately predct across-trals consstency when the condtoned stmulus changes the overall number of granule cells that are actve. Consder an example n whch the condtoned stmulus actvates a random subset of granule cells each tral (that s, no consstency), yet makes twce as many cells actve as durng background. Ths s the stuaton shown n Fg. 3C (larger dots). The overall ncrease n granule cell actvty would make each P (CS) greater than ts assocated P, despte the absence of across-trals consstency. In ths case the actvty dfference rule would ndcate a hgh degree of condtonablty. By addng the slope term s to the actvty dfference rule, Eq. (12) compensates for such changes n the overall granule cell actvty. In summary, condtoned stmulus actvty s comprsed of two components that correspond respectvely to the two phases of plastcty ntated by a condtoned stmulus + uncondtoned stmulus tral. The term P (CS) n Eq. (12) represents the granule cell actvty that leads to LTD durng the tral and sp represents the actvty Fgure 3. A graphcal vew of Eq. (12) and ts relatonshp to acrosstrals consstency of the granule cells actvated by the condtoned stmulus. Each graph shows a hypothetcal scatter plot n whch the probablty of condtoned stmulus-evoked actvty for each synapse (P (CS) ) s plotted as a functon of ts probablty of background actvty (P ). A. An example showng a nonzero amount of across-trals consstency. The lne represents the least squares regresson lne descrbed by the ponts and constraned to pass through the orgn. The slope of ths lne s the term s from Eq. (12). As also mpled by Eq. (12), the condtoned stmulus actvty for each synapse can be decomposed nto two components. Frst, the expected actvty durng the condtoned stmulus, gven the background actvty of the synapse and the behavor of all other synapses, s the heght of the regresson lne for that level of background actvty (sp, dotted arrow). Ths also corresponds to the amount of net LTP that a synapse wth that background actvty wll undergo durng return to equlbrum. Second, the dstance from the regresson lne (P (CS) sp = P (atc), sold arrow) represents both the dfference between the actual and expected condtoned stmulus-related actvtes and the amount of net plastcty expected for that synapse. B. Snce the lne passng through the orgn and wth slope s represents the actvtes for whch LTD durng a condtoned stmulus + uncondtoned stmulus tral s exactly reversed by LTP durng return to equlbrum, (1) all synapses whose actvty falls above the lne (lghter shadng) wll decrease n strength snce LTD s expected to be greater than LTP, and (2) all synapses whose actvty falls below the lne (darker shadng) wll ncrease n strength snce LTP exceeds LTD. C. Two examples of zero across-trals consstency. The small dots llustrate the stuaton n whch the condtoned stmulus related actvty P (CS) s the same as the background actvty P for all synapses (s = 1). The large dots llustrate a stuaton n whch the condtoned stmulus ncreases the actvty of all synapses by the same factor (that s, doubles), such that the predcted condtoned stmulus related actvty sp equals the actual actvty P (CS) (s = 2). In both cases the amount of LTD expected durng the condtoned stmulus + uncondtoned stmulus trals s equal to the expected amount of LTP durng the subsequent return to equlbrum.
Cerebellar Model II: Motor Adaptaton 79 assocated wth the net LTP durng the return to equlbrum. The dfference, whch we call the across-trals consstency, P (atc), represents the component of the condtoned stmulus granule cell actvty that s assocated wth the net changes n synaptc weghts produced by a condtoned stmulus + uncondtoned stmulus tral and the subsequent return to equlbrum. Thus, we can wrte the condtoned stmulus related actvty, summed over all granule cells, as the sum of two components, P (CS) = P (atc) + s P (13) When the condtoned stmulus-evoked granule cell actvty P (CS) s the same as that component that s ncreased n strength durng return to equlbrum (s P), then the across-trals consstency s zero, and no learnng s expected (as llustrated n Fg. 3C). In the next secton we llustrate further that t s the across-trals consstency component P (atc) that contrbutes to the acquston of condtoned responses. Acquston and Extncton Here we llustrate how acquston and extncton of condtoned responses are nfluenced by the two phases of plastcty and by the across-trals consstency of the gr Pkj synapses actvated by the condtoned stmulus. Snce evdence ndcates that condtoned responses are elcted by ncreases n the approprate cerebellar nucleus cells (McCormck and Thompson, 1984a) and snce Purknje cells nhbt nucleus cells, the decrease n Purknje cell actvty durng the condtoned stmulus can be taken as a convenent measure of condtoned responses. Thus, we defne the term R = P pc ( ) P pc (CS), where P pc ( ) s the equlbrum actvty of the Purknje cell and P pc (CS) s the Purknje actvty durng the condtoned stmulus. For smplcty we make four assumptons: (1) the ntal values of the synaptc weghts are such that there s no change n Purknje cell actvty durng the condtoned stmulus and thus no condtoned response (that s, ntally R = 0), (2) the strength of the uncondtoned stmulus s ntally such that a clmbng fber nput s relably actvated (ntally P (CS+US) = 1), (3) condtoned stmulus-related granule cell actvty s dfferent from background actvty n a manner that makes acrosstrals consstency nonzero ( P (atc) 0) and (4) the tme between condtoned stmulus + uncondtoned stmulus trals s suffcent to restore equlbrum. Fgure 4 llustrates the processes predcted to occur durng condtoned stmulus + uncondtoned stmulus tranng under these condtons (see the appendx for mathematcal detals). The flled crcles n Fg. 4A are a conventonal representaton of the acquston and extncton curves (the expected magntude of R) durng smulated condtoned stmulus + uncondtoned stmulus acquston (left half of fgure), and condtoned stmulus-alone extncton tranng (rght half). The acquston curve shows a negatvely acceleratng ncrease n R that s qualtatvely smlar to acquston curves observed n eyeld condtonng (Gormezano et al., 1962). The extncton curve s also qualtatvely smlar to emprcally observed extncton curves. For comparson, the expected clmbng fber actvty elcted durng the tral s shown by the flled crcles n Fg. 4B. Each condtoned stmulus + uncondtoned stmulus tral produces a transent ncrease n clmbng fber actvty due to the presentaton of the uncondtoned stmulus. One useful feature of ths mathematcal analyss s the ablty to plot a contnuous representaton of the condtoned response expected had the condtoned stmulus been presented at any gven tme, as s shown n the sold trace n Fg. 4A. Ths contnuous dsplay of R llustrates that through the nducton of LTD, each condtoned stmulus + uncondtoned stmulus tral elcts a step ncrease n the magntude of the expected condtoned response (see nset, Fg. 4A). Ths LTD also produces a step decrease below equlbrum n the probablty of a clmbng fber nput (Fg. 4B). As seen n Fg. 4B, and shown n more detal n the nsets, there follows a second phase of plastcty durng whch net LTP restores clmbng fber equlbrum. Durng return to equlbrum the expected magntude of the condtoned response (R) declnes. These two phases of the expected ampltude of R llustrate the relatve contrbutons to condtoned responses made by the two components of condtoned stmulus-actvated granule cell actvty as descrbed n the prevous secton (that s, P (CS) = s P + P (atc) ). The LTD nduced at gr Pkj synapses actve durng the condtoned stmulus ( P (CS) ) produces the step ncrease n R and the step decrease n clmbng fber actvty. The subsequent reducton of R durng return to clmbng fber equlbrum s produced by net LTP at the gr Pkj synapses proportonal to the background actvty (s P). Thus, the net change n R s due to net LTD produced by the dfference of these two components ( P (atc) ), whch s related to the across-trals consstency of the granule
80 Kenyon, Medna and Mauk Fgure 4. The role of across-trals consstency n the acquston and extncton of Pavlovan condtoned responses. A. A measure of the condtoned ampltude R s plotted as a functon of repeated condtoned stmulus + uncondtoned stmulus tranng trals. The dots ndcate the condtoned response expected durng the presentaton of each condtoned stmulus + uncondtoned stmulus tranng tral and as such represent a conventonal representaton of acquston and extncton curves. The curves exhbt the negatvely acceleratng changes n respondng that are observed emprcally. The curves between the dots show a contnuous representaton of the condtoned response ampltude had the condtoned stmulus been presented at that tme. Ths trace llustrates that durng acquston (left half of fgure) each tral s assocated wth two phases of plastcty: (1) an LTD-domnated phase durng the condtoned stmulus + uncondtoned stmulus presentaton that produces a step ncrease n the expected ampltude of the condtoned response and (2) a subsequent LTP-domnated phase n whch clmbng fber equlbrum s restored and n whch the condtoned response can be partally or fully extngushed. The nset on the left shows ths behavor more clearly for one condtoned stmulus + uncondtoned stmulus tral. Condtoned stmulus-alone extncton trals (rght half of fgure) exhbt a complementary seres of events. B. A plot of clmbng fber actvty, P, durng the same acquston and extncton tranng. Each condtoned stmulus + uncondtoned stmulus tranng tral elcts (1) a bref ncrease n clmbng fber actvty due to the uncondtoned stmulus, (2) a subsequent decrease n actvty below the equlbrum level due to the net LTD nduced at gr Pkj synapses durng the tral, and (3) a subsequent return to equlbrum due to net LTP at certan gr Pkj synapses. The nset shows these three phases more clearly for a sngle tral. Followng the tral, P returns to P ( ) wth an exponental tme course. As acquston proceeds the step ncrease n clmbng fber actvty due to the uncondtoned stmulus decreases because of the response-related nhbton from the cerebellum (modeled here as a decrease n exctaton). The rght sde of the fgure agan shows the complementary processes that occur durng extncton. There s a transent decrease below equlbrum durng the tral due to the response-related nhbton of clmbng fbers that s unopposed by the exctatory drve from the uncondtoned stmulus. C. The contrbutons from s P and P (atc) are plotted separately as a functon of tranng trals. The partal extncton of condtoned responses followng each pared tral s due entrely to net LTP produced n synapses accordng to the contrbuton from s P, whereas the contrbuton from P (atc) s unaffected by the return to cells actvated by the condtoned stmulus. Ths s shown n Fg. 4C where the contrbutons to R made by s P and P (atc) have been plotted separately (agan, a convenence permtted by the mathematcal model). Durng the return to equlbrum the contrbuton of s P s completely reversed, whle the contrbuton due to P (atc) s unchanged. In the absence of across-trals consstency, the entre step ncrease n R produced by the condtoned stmulus + uncondtoned stmulus tral would be completely reversed durng the return to equlbrum (snce P (CS) = s P), and there would be no ncrease n condtoned respondng (Fg. 4D). These results agan llustrate that the across-trals consstency vector of gr Pkj actvty, P (atc) determnes the net amount of learnng produced by condtoned stmulus + uncondtoned stmulus trals. Although these two phases of plastcty are repeated each condtoned stmulus + uncondtoned stmulus tral throughout acquston, the uncondtoned stmulus-evoked ncrease n clmbng fber actvty decreases, as does the step ncrease n condtoned response magntude (Fgs. 4B and 4A, respectvely). These decreases are due to the response-related nhbton n clmbng fber actvty durng the uncondtoned stmulus. As the condtoned response-related ncrease n cerebellar output (here a decrease n Purknje actvty) ncreases durng acquston, the ablty of the uncondtoned stmulus to actvate clmbng fbers decreases and the rate of further acquston slows. Ths llustrates that the exctatory drve from the uncondtoned stmulus s effectvely canceled by the decreased Purknje cell actvty durng the condtoned stmulus. Ths s consstent wth evdence that the uncondtoned stmulus-evoked clmbng fber responses decrease as condtoned responses are acqured (Sears and Stenmetz, 1991). Thus, nhbtory feedback from the cerebellum to the clmbng fbers (see Fg. 1B) can explan (1) the negatvely acceleratng acquston curves observed n eyeld condtonng, as has been prevously equlbrum. The nset shows the separate contrbutons from P (atc) and s P at an expanded tme scale for a sngle tral. D. The ncrease n the condtoned response for a sngle tral s shown for a case wth zero (dotted) and nonzero (sold) across-trals consstency. Wth no across-trals consstency the ncrease n the expected magntude of the condtoned response produced by the condtoned stmulus + uncondtoned stmulus tral s completely reversed by the subsequent LTP that returns clmbng fber actvty to equlbrum. Wth nonzero across-trals consstency the return to equlbrum only partally reverses the ncrease n the expected condtoned response.
Cerebellar Model II: Motor Adaptaton 81 suggested (Donegan et al., 1989; Sears and Stenmetz, 1991), and (2) the attenuaton of clmbng fber actvty as movement performance mproves (Glbert and Thach, 1977; Thach, 1980). These processes also llustrate that once acquston s complete, clmbng fber actvty s n equlbrum durng background and durng the condtoned stmulus + uncondtoned stmulus tral, despte the exctatory nfluence of the uncondtoned stmulus on the clmbng fbers. The rght halves of Fgs. 4A and 4B llustrate that the processes predcted to occur durng condtoned stmulus-alone extncton tranng are dentcal to those of acquston, but opposte n polarty. Early n extncton tranng, clmbng fber actvty s less than equlbrum durng the condtoned stmulus-alone tral (Fg. 4B, rght sde). Ths s the result of the condtoned response-related ncrease n nhbton of clmbng fbers durng the condtoned stmulus (modeled here as a decrease n exctaton) that s no longer opposed by the exctatory drve from the uncondtoned stmulus. Snce clmbng fber actvty s no longer n equlbrum durng the condtoned stmulus, gr Pkj synapses must undergo net LTP to restore equlbrum. Thus, each condtoned stmulus-alone tral produces a step decrease n R that s partally reversed after the tral to restore equlbrum durng background (see nset, rght half of Fg. 4A). These processes contnue untl clmbng fber actvty s agan n equlbrum both durng the condtoned stmulus and durng background. These results ndcate the mportance of response-related nhbton of clmbng fbers n extncton. Wthout the response-related nhbton, clmbng fber actvty could not fall below equlbrum n the absence of the uncondtoned stmulus, net LTP would not occur, and extncton would not be possble. be shown by defnng a condtoned response as the change n condtoned stmulus-related Purknje cell actvty (P CR = Ppc CS ) and combnng the expresson for synaptc weght changes ( w CS = δ P CS ) wth the expresson for Purknje actvty durng the condtoned stmulus (Ppc CS = P CS w ). Ths yelds the expresson for the amount of learnng produced each tral: P CS pc = δ ( ) P CS 2. (14) Here we show that wth background actvty the same general result apples; the rate of learnng s proportonal to the square of the across trals consstency vector (( P atc ) 2 = (P atc ) 2 ). Fgure 5 shows that the predcted rate of acquston s related to across-trals consstency as descrbed by the term β = P (atc), (15) P where β represents a scalar measure of the magntude of P (atc) relatve to the magntude of the background actvty vector P. Usng R as a measure of the ampltude of condtoned responses, acquston curves for several values of β are shown n Fg. 5. Results demonstrate that the rate of acquston ncreases as a functon of β 2. Indeed, t s possble to show (see appendx) that n the lnear model the number of trals to approach asymptotc response levels, denoted by N CR,sgven Across-Trals Consstency Determnes the Magntude of Synaptc Changes and the Rate of Learnng The above results demonstrate that the across-trals consstency vector P (atc) represents the only component of the granule cell representaton of the condtoned stmulus that contrbutes to the acquston of condtoned responses. It follows that the magntude of P (atc) should determne the rates of acquston and extncton. In the absence of background actvty, across-trals consstency for each synapse s ts condtoned stmulus-related actvty (P CS ) and the rate of learnng s proportonal to the sum of the squares of the condtoned stmulus actvtes. Ths can Fgure 5. The role of across-trals consstency n the number of trals to acqure condtoned responses. Acquston, as measured by R, s plotted as a functon of the number of tranng trals for three values of β (.3,.6,.9). The nset shows that the step ncrease n respondng s the same for hgh and for low values of β. Rather, β nfluences the extent to whch the plastcty/responses nduced durng the tral are reversed durng return to equlbrum.
82 Kenyon, Medna and Mauk Fgure 6. A summary dagram showng predcted changes n neural actvty durng acquston and extncton tranng. Idealzed representatons of neural actvty are shown to the rght of the correspondng neuron n the smplfed crcut dagram. In the crcut dagrams, the nucleus cells are shown n gray to hghlght that ther nfluence was modeled as a vrtual exctatory connecton between Purknje cells and clmbng fbers. For the clmbng fber actvty shown at rght, lght gray shadng depcts equlbrum actvty, wherease darker gray denotes actvty above equlbrum, and no shadng denotes subequlbrum actvty. A. The events predcted to occur durng acquston tranng are shown n panels A1 through A3. (1) Before tranng, presentaton of a tone condtoned stmulus (stmulaton protocol llustrated at top of fgure) produces no effect on ether Purknje cell or clmbng fber acton potental actvty. (2) Pared presentaton of tone and a renforcng uncondtoned stmulus produces an ncrease n clmbng fber actvty durng the uncondtoned stmulus. Owng to the LTD nduced by ths tral, there follows a perod durng whch both clmbng fber actvty and Purknje cell actvty may be at tmes reduced below equlbrum levels. The magntude of ths reducton, whch depends both on the number of gr Pkj synapses actve durng the tone as well on the overlap between synapses actve durng the tone and the perod followng the tone, has been exaggerated for clarty. Eventually, gr Pkj synaptc weghts ncrease n proporton to ther background actvty untl equlbrum s restored. (3) When condtoned responses have been acqured, clmbng fber actvty remans n
Cerebellar Model II: Motor Adaptaton 83 by N CR = N β 2, (16) nablty of synaptc weghts to obtan the extreme values requred to produce condtoned responses gven the low degree of across trals consstency. where N denotes the number of tme steps to restore equlbrum (see precedng artcle). As n Fg. 4, Fg. 5 shows a contnuous representaton of R, depctng the condtoned response expected should the condtoned stmulus be presented at any gven tme. Ths trace llustrates that the step ncrease n condtoned responses elcted by pared condtoned stmulus + uncondtoned stmulus trals at the begnnng of tranng s ndependent of β. Instead, Eq. (6) suggests that the ampltude of ths step s determned by the total condtoned stmulus-related actvty ( P CS ) and by the ampltude of LTD events (δ ). Thus, the larger number of trals to acqure wth smaller values of β s not due to a decrease n the magntude of plastcty nduced by condtoned stmulus + uncondtoned stmulus trals; rather, t reflects that more of the plastcty s erased durng the return to equlbrum. Ths analyss also demonstrates that the amount of across-trals consstency nfluences the average change n gr Pkj synaptc weghts requred to produce a crteron or asymptotc level of condtoned responses. Wth hgh across-trals consstency, where gr Pkj synapses relably sgnal the condtoned stmulus, the changes n gr Pkj synaptc weghts requred to produce a gven ncrease n R are relatvely small compared to lower levels of across-tral consstency. As across-trals consstency decreases ether because each synapse sgnals the condtoned stmulus less relably or because there are fewer synapses whose actvty s consstently altered by the condtoned stmulus or both each synaptc weght must change by a greater amount to produce the same condtoned response. In the bologcally realstc case where synaptc weghts are bounded (that s, there are maxmum and mnmum values) decreasng amounts of across-trals consstency also affects the maxmum level that R can reach (Medna and Mauk, 1995). Ths lmt would be mposed by the Dscusson Results of the precedng artcle paper suggest that spontaneous clmbng fber actvty may be regulated to an equlbrum level at whch LTD and LTP balance at gr Pkj synapses, and the net plastcty at each synapse s zero ndependent of ts actvty. Here we have examned the consequences of ths equlbrum on cerebellar-medated motor learnng usng an dealzed representaton of Pavlovan eyeld condtonng. The man results are that eyeld condtonng trals may dsrupt cerebellar-olvary equlbrum and n dong so elct two competng phases of plastcty. The net effects of these two phases would make the condtonng-nduced changes n synaptc weghts selectvely senstve to the consstency from one tral to the next n the gr Pkj synapses actvated by the condtoned stmulus. Thus, the systematc dsrupton of cerebellar-olvary equlbrum requred for condtonng s comprsed of two components: (1) nonequlbrum clmbng fber actvty and (2) granule cell actvty that s suffcently consstent from one tral to the next. For eyeld condtonng these condtons occur durng acquston tranng when a condtoned stmulus precedes an unexpected uncondtoned stmulus and durng condtoned stmulus-alone extncton tranng when the condtoned stmulus predcts an expected uncondtoned stmulus that s omtted. Studes of both eyeld condtonng and of VOR adaptaton have provded evdence that plastcty can occur n both the cerebellar cortex and cerebellar nucle durng motor learnng. For example, prevously learned changes n the gan of the VOR can be partally retaned followng lesons of the vestbular cerebellum. Smlarly, lesons of the cerebellar cortex have on partal effects on the expresson of prevously learned eyeld responses. Specally, cerebellar cortex lesons equlbrum durng the tone (due to ncreased clmbng fber nhbton assocated wth expresson of the response) as well as before and after the tone. B. The events predcted to occur durng extncton tranng are shown n panels B4 to 5. (4) In a well traned anmal, presentaton of a tone-alone extncton tral brngs clmbng fber actvty below equlbrum durng the expresson of the response. The reducton, whch reflects the response-related nhbton of clmbng fbers n the absence of the uncondtoned stmulus, nduces LTP at gr Pkj synapses. In a manner smlar to acquston tranng, ths produces a tendency for ncreased clmbng fber actvty (above equlbrum) for a perod followng the tral. (5) As extncton tranng contnues and the LTP reduces response ampltude, clmbng fber actvty agan returns to beng n equlbrum, before, durng, and after the tral.