Random walks for spike-timing-dependent plasticity

Transcription

1 PHYSICAL REIEW E 70, 0296 (2004) Random waks for spke-tmng-dependent pastcty Aan Wams* euroogca Scences Insttute, Oregon Heath & Scence Unversty, 505 W 85th Avenue, Beaverton, Oregon 97006, USA odd K. Leen Department of Computer Scence and Engneerng, OGI Schoo of Scence & Engneerng, Oregon Heath & Scence Unversty, 505 W 85th Avenue, Beaverton, Oregon 97006, USA Patrck D. Roberts euroogca Scences Insttute, Oregon Heath & Scence Unversty, 505 W 85th Avenue, Beaverton, Oregon 97006, USA (Receved 23 December 2003; revsed manuscrpt receved 3 May 2004; pubshed 3 August 2004) Random wak methods are used to cacuate the moments of negatve mage equbrum dstrbutons n synaptc weght dynamcs governed by spke-tmng-dependent pastcty. he neura archtecture of the mode s based on the eectrosensory atera ne obe of mormyrd eectrc fsh, whch forms a negatve mage of the reafferent sgna from the fsh s own eectrc dscharge to optmze detecton of sensory eectrc feds. Of partcuar behavora mportance to the fsh s the varance of the equbrum postsynaptc potenta n the presence of nose, whch s determned by the varance of the equbrum weght dstrbuton. Recurrence reatons are derved for the moments of the equbrum weght dstrbuton, for arbtrary postsynaptc potenta functons and arbtrary earnng rues. For the case of homogeneous network parameters, expct cosed form soutons are deveoped for the covarances of the synaptc weght and postsynaptc potenta dstrbutons. DOI: 0.03/PhysRevE PACS number(s): 87.9.La, 87.8.Sn, 75.0.r I. IRODUCIO Spke-tmng-dependent pastcty (SDP) [] s a form of synaptc weght dynamcs found expermentay n certan neura systems [2 4]. he key feature of SDP s the dependence of synaptc weght changes on the precse reatve tmng of presynaptc and postsynaptc spkes; ths tmng dependence dstngushes SDP from earer hypotheszed forms of actvty-dependent pastcty [5 7] n whch weght changes depend ony on correatons between presynaptc and postsynaptc spke rates. Modes of SDP assume that the weght change due to each presynaptc and postsynaptc spke par s gven by some functon of the tme between them, caed the spke-tmng-dependent earnng rue [8 3]. Changes due to a pars of presynaptc and postsynaptc spke pars are then summed to gve the weght change due to presynaptc and postsynaptc spke trans. In a prevous artce [4], we nvestgated the mean weght dynamcs n a system n whch SDP has been found expermentay: the eectrosensory atera ne obe (ELL), a cerebeum-ke structure n mormyrd eectrc fsh [3]. he mormyrd fsh uses an adaptaton mechansm based on SDP to habtuate centra neura responses to the predctabe sensory nput due soey to ts own eectrc organ dscharge (EOD). In order for the adaptaton to predctabe tempora patterns to be mantanabe, the synaptc weght confguraton gvng rse to a negatve mage of predctabe patterns must be a stabe equbrum for the mean weght dynamcs *Eectronc address: waa@ohsu.edu Eectronc address: teen@cse.og.edu Eectronc address: robertpa@ohsu.edu nduced by the spke-tmng-dependent earnng rue. Condtons for the exstence and stabty of such negatve mage equbra were frst expored n [5] and extended to arbtrary spke-tmng-dependent earnng rues and arbtrary postsynaptc potenta functons n [4]. However, the equbrum weght dstrbuton n the presence of nose and n partcuar, that dstrbuton s varance s aso behavoray mportant, snce fuctuatons n the weghts due to nose ead to fuctuatons n the negatve mage, whch mpacts the detectabty of externa objects. he methods of our prevous artce [4] were suffcent to cacuate the equbrum mean, but not any hgher moments of the equbrum weght dstrbuton. hs s a serous mtaton n the boogca settng, for two reasons: frst, because n prncpe the varance coud be so arge that the fuctuatons are more physoogcay reevant than the mean and, second, because even f the varance s sma, t s mportant to be abe to cacuate t quanttatvey n order to make specfc predctons about the mpact of fuctuatons on detectabty. In the present artce, we derve mpct expressons for a moments of the equbrum weght dstrbuton and expct expressons for the varance (and the thrd and fourth moments n the snge-weght case) for SDP earnng rues wth stabe earnng dynamcs. Our approach s to express the weght dynamcs as a dscrete tme, nhomogeneous random wak. From the master equaton of ths wak we derve a dfferenta equaton for the Fourer transform of the equbrum weght dstrbuton. ayor expanson of ths equaton yeds recurrence reatons for the moments. he structure of the paper s as foows. In Sec. II we summarze the background facts about random waks, master equatons, and characterstc functons that w be used n the present paper. In Sec. III we descrbe the archtecture and /2004/70(2)/0296(6)/$ he Amercan Physca Socety

2 WILLIAMS, LEE, AD ROBERS PHYSICAL REIEW E 70, 0296 (2004) dynamca assumptons of the mode, and n Sec. I we derve the random wak for the weght dynamcs for arbtrary system parameters. In Sec. we ustrate the method for dervng recurrence reatons for the moments of the equbrum weght dstrbuton by appyng the method n the smpest possbe settng: the case of a snge synaptc weght. We then n Sec. I appy the method to the fu archtecture, wth arbtrary system parameters. In Sec. II we specaze to the case of homogeneous system parameters, dervng more expct anaytca resuts for the covarance of the equbrum weght and postsynaptc potenta dstrbutons. Fnay n Sec. III we compute the weght and postsynaptc potenta covarances for severa exampes of boogca nterest and compare our predctons wth Monte Caro smuatons. In Sec. IX we summarze our fndngs, dscuss ther boogca reevance, and suggest future experments to test the quanttatve predctons of the mode. II. RADOM WALKS, MASER EQUAIOS, AD CHARACERISIC FUCIOS he term random wak refers to any stochastc process n whch the state varabes change ony at dscrete tmes. he changes n state varabes are caed steps; from any gven poston there s a set of possbe steps, each havng a certan probabty (or probabty densty). he set of possbe steps may be dscrete or contnuous, and both the step vaues and step probabtes may depend on poston. Random waks have been used extensvey to mode other physca systems (see the bbography n [6]), and a arge body of mathematca technque has been deveoped for ther anayss [7]. But they have not prevousy been apped to SDP, where the standard approach has been to use the Fokker-Panck equaton [9,0,3]. Gven that the Fokker- Panck equaton s at best an approxmaton when apped to dscrete stochastc processes [8], whereas random wak methods are exact, we beeve t woud be prudent to expore the utty of random wak methods for the anayss of SDP. Random waks are natura modes for systems havng temporay dscrete dynamcs. Snce the synaptc weght changes n SDP are due to temporay dscrete events (spkes or spke pars), random waks are natura modes for SDP. Suppose a state varabe w undergoes a random wak. Let the possbe steps from poston w be j w x for x n some ndex set X. Let the step j w x occur wth probabty densty w x n x. Let P n w be the probabty dstrbuton for w after n steps. We wsh to derve the equaton of moton for P n w, usuay referred to as the master equaton. If the state varabe s w after n steps and w after n+ steps, then w=w+ j w x for some x. he probabty for the state varabe to be between w and w+dw after n+ steps s therefore Moreover, the condtons under whch the approxmaton s a good one, especay for the nonnear Fokker-Panck equaton, are far from cear [8]. Further dscusson of ths ssue, n the context of SDP, w be the subject of a future paper. P n+ wdw = dx w xp n wdw. Hence the master equaton s P n+ w = dx w xp n w dw dw. he quantty dw/dw compensates for any change n the densty of states from tme n to tme n+, due to poston dependence of the set of step vaues. From w=w+ jx,w we have dw dw = +, w j w x and hence the master equaton s P n+ w = dx w xp n w +. 2 w j w x Suppose the set of step vaues s ndependent of poston; then j w x/w=0, and the densty of states factor n the master equaton s dentcay. Denotng by jx the common set of step vaues, we aso have w expcty n terms of w and x: w=w jx. For such waks the master equaton takes the smper form P n+ w = dx w jx xp n w jx. A waks consdered n the present paper w turn out to be of ths type. A probabty dstrbuton Pw s an equbrum (statonary) dstrbuton for the random wak f P n = P mpes P n+ = P; n other words, the dynamcs of the wak eaves P unchanged. Hence Pw s an equbrum dstrbuton for the wak n Eq. (3), f and ony f t satsfes Pw = dx w jx xp w jx. o cacuate the moments of a probabty dstrbuton Pw, we w fnd t usefu to nvoke a property of ts Fourer transform (often referred to as the characterstc functon): Pˆ k = dw e kw Pw. akng the dervatve wth respect to k n Eq. (5) and evauatng at k=0 yeds dn Pˆ k = dk n dww k=0 n e kw Pwk= = n dww n Pw = n w n. 6 Hence the moments of Pw are, up to powers of, just the

3 RADOM WALKS FOR SPIKE-IMIG-DEPEDE PHYSICAL REIEW E 70, 0296 (2004) FIG.. Schematc of the archtecture. he postsynaptc ce receves nputs from presynaptc neurons, a repeated sensory nput x, and a nosy nput. Presynaptc ce spkes at tme x n each perod of and has synaptc weght w onto the postsynaptc ce. dervatves of the characterstc functon Pˆ k evauated at k=0. For further background on random waks, see [7]. III. FRAMEWORK he mode conssts of a snge postsynaptc ce representng a medum gangon (MG) ce, a ce type n the ELL that shows strong adaptaton to changng sensory nput [3]. he MG ce s drven by a repeated sensory nput (prmary sensory reafference), an array of presynaptc ces whose spkes are tme-ocked to the repeated sensory nput (the efference copy of the motor command), and nose (representng other unspecfed nputs) [9 2] (Fg. ). hs basc archtecture s derved from the mormyrd ELL, but s suffcenty genera to capture the dynamcs of other neura systems hypotheszed to have an array of tme-deayed, tme-ocked nputs through pastc synapses [22,23]. he framework for the neura dynamcs s the spke response (SR) mode [24,25], wthout refractorness, as descrbed n our prevous report [4]. Much of the detas of our MG ce mode have appeared prevousy [4,2], and here we sha outne our genera methods and comment on mportant dfferences between prevous treatments and the present framework. he repeated sensory nput s the postsynaptc potenta (PSP) n the postsynaptc ce due to prmary sensory afferents, over a snge EOD sweep. Each tme-ocked presynaptc ce spkes (exacty once) at a fxed tme wthn each sweep of the repeated sensory nput, causng a corrspondng PSP n the postsynaptc ce. he tota membrane potenta n the postsynaptc ce s the sum of the repeated sensory nput, the nosy nput, and the PSPs due to tme-ocked presynaptc spkes, weghted by synaptc effcaces (weghts) w. hs membrane potenta causes the postsynaptc ce to generate postsynaptc, dendrtc spkes [3] at a certan (nosy) rate. We assume that each presynaptc spke causes a constant change n the weght w (nonassocatve earnng) and each postsynaptc and presynaptc spke par causes a change n w accordng to a spketmng-dependent earnng rue.e., a functon of the tme dfference between the postsynaptc and presynaptc spkes (assocatve earnng). Let the resutng perod (puse wdth) be, and ntroduce two tme varabes: x0, for the tme wthn each perod of the sensory nput and t=n, nz, for the tme of ntaton of each such perod [20,2,26]. Genera dynamca quanttes w be functons of the par x,t. he tme-ocked presynaptc ce spkes at a fxed tme n each perod. Denote ths tme by x. Let w x,t be the synaptc weght of presynaptc ce, and et E s be the PSP evoked by a spke n ce at tme s after the spke. We w assume that E s causa: E s=0 for s0. Let be the nonassocatve weght change due to a presynaptc spke by ce and L s the assocatve weght change due to a postsynaptc spke at tme s after a presynaptc spke by ce. Let x be the perodc sensory nput and Ux,t the tota postsynaptc potenta due to the non-nosy nputs. In contrast to our prevous approach [4], we w assume that, n each perod of, ether zero or one postsynaptc spke occurs. he probabty densty (n x, for a gven t) for a postsynaptc spke to occur at x,t s assumed to be (/f(ux,t) for some postve and strcty ncreasng functon f :R 0,. he probabty of zero postsynaptc spkes n the perod begnnng at t s then / 0 dxf(ux,t). Heurstcay, the functon f s the effectve gan functon of the postsynaptc ce n the presence of the nosy nputs, wth the maxmum sope of f ndcatng the nose eve: hgh or ow nose corresponds to an f wth sma or arge maxmum sope, respectvey. We w mpement changes n weghts as dscrete steps wth no nterna tme course. We update weghts synchronousy, once per sweep of the perodc sensory nput, at tme x=0 for each t=n, nz. he vaue of w n the perod begnnng at 0,t s then ndependent of x and w be denoted w t. In the present treatment, we mpose no boundares on the weght vaues because the weght equbra and equbrum varances are such that weghts are amost aways n the regon that woud be encosed by boogca bounds. o smpfy the dervaton of the weght dynamcs, we w assume that E s,l s are zero or neggbe for s E, L, respectvey, wth E, L. We w aso mpose the sow earnng rate assumpton w, where w s the tme scae over whch weghts undergo sgnfcant reatve change. he exstence of approxmate negatve mage states requres [4] that the spacng of presynaptc spke tmes be much smaer than the wdths of E and L : E, L. hese tme-scae assumptons can be summarzed as E, L w. ypca vaues from the mormyrd ELL are ms[27], E 20 ms [3], L 40 ms [3], 80 ms [[27] (b)], and w 0 2 [3]

4 WILLIAMS, LEE, AD ROBERS PHYSICAL REIEW E 70, 0296 (2004) I. WEIGH DYAMICS We now derve the random wak for the weght dynamcs by computng the possbe weght changes w t=w t+ w t and ther correspondng probabtes. he detas here are very smar to those n [4], devatng ony n the treatment of postsynaptc spke generaton. Instead of a varabe number of spkes per EOD cyce, occurrng at a mean rate per unt tme, we now have a snge postsynaptc spke per cyce whose occurrence s gven by a probabty densty. he nonassocatve change n w t due to the snge presynaptc spke at x,t s. For the assocatve change due to presynaptc and postsynaptc spke pars, the cacuaton s dentca to that n [4]; for a par consstng of a presynaptc spke at x,t and postsynaptc spke at x,t, the change n w t s approxmatey L x x, where L s = n= L s n s the perodzaton of L wth perod. A postsynaptc spke between t and t+ occurs wth a probabty densty /f(ux,t) n x, wth the probabty of zero postsynaptc spkes beng / 0 dxf(ux,t). Hence the change n w due to postsynaptc spkes between t and t+ s L x wth densty /fux,t n x and 0 wth probabty / 0 dxf(ux,t). he tota change n w t due to both nonassocatve and assocatve earnng s therefore w t = + L x,, densty f Ux,t, probabty /0 dx f Ux,t. he cacuaton of the non-nosy component of the postsynaptc potenta, Ux,t, s the same as n [4]; we fnd that 7. OE WEIGH o ustrate the technque n the smpest possbe settng, we frst examne the case of a snge weght. If there s ony one weght w t, then wthout oss of generaty we may take x =0 by transatng f necessary. Wrtng wt,, L, and E for w t,, L, and E, the random wak, Eq. (9), for the weght dynamcs becomes where wt = + L x,, densty/f x,wt, probabty /0 f x,wt = f x + wte x. dxf x,wt, 0 From the random wak for the weght dynamcs we derve the moments of the equbrum weght dstrbuton n three steps. Frst we wrte the master equaton for the tme evouton of the probabty dstrbuton of the weght and the correspondng functona equaton for the equbrum (statonary) dstrbuton. akng the Fourer transform yeds a dfferenta equaton for the Fourer transform of the equbrum dstrbuton. ayor expanson of ths equaton yeds recurrence reatons for the moments. otce that the set of step vaues n the wak (0) s ndependent of w; hence the equbrum dstrbuton Pw must satsfy Eq. (4). From the step vaues and step probabtes n Eq. (0) we have Ux,t = x + w j te jx x j, j= 8 Pw = 0 dxf x,w Pw where E s= n= E s n s the perodzaton of E wth perod. Defnng f by f (x,wt)= f(x+ j= w j te jx), we have from Eqs. (7) and (8) the foowng expresson for the tota change n w t: w t + L x, densty/f x,w t,...,w t, =, probabty dxf x,w t,...,w t. /0 9 Equaton (9) defnes the random wak for the weght dynamcs. It s dscrete tme (steps occur ony at t=n, nz), contnuous space (steps can take a contnuum of vaues), and nhomogenous (step probabtes depend on poston). he common perodcty of the functons E, L, and s an mportant feature, aowng the systematc use of Fourer technques. + 0 dxf x,w + L x P w + L x. akng the Fourer transform dw e kw on both sdes, changng varabes, and rearrangng yeds Pˆ k e k = dxe k +L x e k 0 dwe kw f x,wpw. 2 A physoogcay pausbe spke output functon f woud take the form of a smooth, monotoncay ncreasng sgmod, but for maxma smpcty we assume f s pecewse near:

5 RADOM WALKS FOR SPIKE-IMIG-DEPEDE PHYSICAL REIEW E 70, 0296 (2004) u, fu =0, 2+ u, u,, u, 3 so that f s gven by f x,w =0, 2+ Ux, Ux, Ux,, Ux, 4 wth Ux=x +we x. We further assume that the equbrum weght dstrbuton Pw s zero or neggbe for w such that Ux or Ux. hs s a confnement condton on the equbrum postsynaptc potenta Ux and w be justfed ater. ote that the confnement condton heps justfy the pecewse near assumpton on f, snce the more confned the postsynaptc potenta Ux, the better our pecewse near f approxmates a smooth sgmod n the regon where Ux s concentrated. If the confnement condton hods, then n Eq. (2) we may repace f x,w under the ntegra by the foowng near functon of w: 2+ x + we x. Usng dw e kw wpw= Pˆ k, we then obtan Pˆ k e k dx x 0 2+ x = Pˆ k 0 dx 2 E x x, 5 where x=e k+l x e k. By Eq. (6), the moments of Pw are (up to powers of ) just the dervatves of Pˆ k at k = 0; snce those dervatves are mpcty constraned by Eq. (5), the moments of Pw are constraned by Eq. (5). Specfcay, the ayor expanson of Eq. (5) around k =0 w yed a herarchy of recurrence reatons for the dervatves of Pˆ k and hence for the moments of Pw. he ayor expansons of the exponentas are n e k = n=0 n! n k n, e k+l n x = n=0 n! + L x n k n. For the expanson of the characterstc functon Pˆ k we expand the exponenta n the defnton of Pˆ k and nvert the order of summaton and ntegraton: Pˆ k = m dw e kw Pw = m=0 m! km dw w m Pw m = m=0 m! wm k m. From ths t foows that Pˆ k = m=0 m m! wm k m m = m=0 m m! wm+ k m. By substtutng these expansons nto Eq. (5) and equatng coeffcents of k on both sdes, we obtan the foowng reatons: m m w m E m w m+ =0, m=0 = 0,,2,..., where for brevty we have defned n = n,0 n 0 n E = 0 6 dx 2+ x + L x n n, dx 2 E x + L x n n. he reatons (6) are ower tranguar 2 and hence are easy rearranged to yed expct recurrence reatons for the moments n terms of moments of ower degree ony: w = E E w m,m, where,m = m= =,2,..., m m E m m+. 7 We may now compute the centra moments M k =w w k by expressng w n n terms of the M k : n w n = w w + w n = n k w km n k. 8 k=0 Substtutng nto Eq. (7) and rearrangng yeds 2 One coud aso derve moment equatons va the more drect route of ayor expandng, n w, the equbrum condton () for Pw, but the resutng moment equatons are not tranguar. In fact they are fuy couped (each equaton nvovng a moments, n genera) and hence not ready sovabe

6 WILLIAMS, LEE, AD ROBERS PHYSICAL REIEW E 70, 0296 (2004) m E k k E M = E E E mk E m= E m=k For =2,3,4 we obtan M 2 = 2 E 2 + E 2 2 E 2,,m + M k k=2 m k,m. M 3 = 3 E 3 + E 3 3 E 2 + E E 2 E E 3, 9 x E x =E x E x x, x x L x =L L x x. he random wak for the weght vector wt takes pace n R, wth the wak for each component w t gven by Eq. (0). In vector notaton the wak for wt s then where + L x, densty/f x,wt, wt =, probabty dxf x,wt, /0 2 M 4 = E 2 3 E E E 2 4 E 4 4 E 4 + E 4 4 E 2 + E E 2 + E E 2 E E 2 3 E E E E E We can see from M 3 aone that n genera the equbrum weght dstrbuton s not Gaussan. For generc PSP E and earnng rue L there are no poynoma reatons among the coeffcents E n and n, hence M 3 s genercay nonzero. o determne the dependence of the moments on step sze, we mutpy both and L, and hence the steps of the random wak, by a scaar. he coeffcents E n and n are then both O n, and substtuton nto Eq. (20) yeds M 2 = O, M 3 = O 2, M 4 =3M O 3. Hence as 0 the skew and kurtoss approach Gaussan vaues: skew = M 3 M 2 3/2 = O/2 0, kurtoss = M 4 2 =3+O 3. M 2 I. MULIPLE WEIGHS We now appy the technque to the case of mutpe weghts w, =,2,...,. he agebra s more compcated, but the structure of the dervaton s dentca to the sngeweght case. For notatona compactness we ntroduce the vector notaton = w t wt w t, =, f x,wt = fx + wt E x and the center dot ndcates the vector dot product. Agan, the step szes are ndependent of poston, so the equbrum condton, Eq. (4), appes. We have Pw = 0 dxf x,w Pw L x P w + L x. dxf x,w 22 As before, we take the (now n-dmensona) Fourer transform on both sdes. Appyng dw e k w, changng varabes, and rearrangng yeds Pˆ k e k = dxx dwe k w f x,wpw, 0 23 where x = e k +L x e k. 24 We now assume that the postsynaptc gan functon f s pecewse near and gven by Eq. (3); hence, f s gven by Eq. (4), wth Ux=x +w E x. And as before, we assume that Pw s neggbe for w such that Ux or Ux, a confnement condton on Pw, whch w be justfed ater. hen we may repace f x,w under the ntegra by the near functon of w: Usng 2+ x + w E x. dw e k w w j Pw = Pˆ k k j, we obtan the foowng frst-order parta dfferenta equaton (PDE) for Pˆ k:

7 RADOM WALKS FOR SPIKE-IMIG-DEPEDE PHYSICAL REIEW E 70, 0296 (2004) Pˆ k e k dx x Pˆ k 0 2+ x = k j m=0 r m! m m r r w r w j + r j w r k. Pˆ k = dx E jx x j 30 x. 25 j= k j 0 2 When the expansons, Eqs. (27), (28), and (30), are substtuted nto Eq. (25), equatng the coeffcents of k on both q sdes yeds ayor expanson of both sdes of ths equaton around k=0 w yed recurrence reatons for the moments of w. he ayor expanson of a functon g on R s gven by gk,...,k = n=0 n! 0 n n s s gz,...,z s z z s z=0 k s. 26 he expansons of the compex exponentas n Eq. (25) are thus n=0 s e k = n! n n s s s k,! qw q q2 q w 2 w = where s = 0 s +, r+s=q dx 2+ x n! m! m n r s s w r w 2 r2 w r E + j r r s w w j + r j w, j= s + L x s 3 n=0 s e k +L x = n! n n s s + L x s k, 27 where n the sums on the rght, s=s s 2 s wth each s a nonnegatve nteger and = s =n. For brevty we wrte n s for the mutnoma coeffcent n Eq. (26). As before, for the expanson of the characterstc functon Pˆ k we expand the exponenta n the defnton of Pˆ k and nvert the order of summaton and ntegraton: m=0 r Pˆ k = dw e k w Pw = dw Pw w r r w r2 r w 2 w r k, m! m m r m=0 r k r = m! m m r 28 where r=r r 2 r wth each r a nonnegatve nteger and = r =m. From ths expanson of Pˆ k t foows that m=0 r Pˆ k = k j m! m m r r w r w r j r k j. Usng the combnatora dentty m m m m r j = m! r m!r r j r, we may rendex Eq. (29) to yed 29 E j s = dx E jx x j s + L x s 0 2, and q=q q 2 q, each q a non-negatve nteger, wth = q =. A sght smpfcaton foows from 0 = and E 0 =0: the quantty on the eft sde of Eq. (3) s canceed by the term on the rght sde wth s=0 and r=q. he resutng recurrence reatons are 0= r+s=q m n! m! m r n s s w r w 2 r2 w r + E r r jw w j + r j w, j= 0 q, q =, =,2, = For each choce of q we obtan a snge near equaton nvovng moments of tota order at most = q. Regardng the moments of tota order as unknowns to be soved for n terms of moments of tota order ess than, we have a near system wth the same number of equatons as unknowns. he coeffcent matrx of ths system nvoves the quanttes and E. For generc E and L there are no poynoma reatons among these quanttes; hence the determnant of the coeffcent matrx s genercay nonzero, and the system can be nverted to gve the moments of tota order n terms of, E, and the moments of tota order ess than. he compete moment herarchy can thus be obtaned: frst moments of tota order, then moments of tota order 2, and so on

8 WILLIAMS, LEE, AD ROBERS PHYSICAL REIEW E 70, 0296 (2004) A. Equbrum mean For = we must have q j = j for some. Snce n Eq. (32) ony terms wth m appear, and m= j r j, the ony possbty for r s r=0, and then s j =q j = j. he recurrence reaton Eq. (32) then becomes 0= w j 0 j= dx 2+ x L x x dx 2 E jx x j L x x. 33 Aowng to vary over a possbe vaues,2,...,, we have near equatons n the unknowns w, whch can be wrtten n vector form as Cw = d, wth the matrx C and vector d gven by C j = dx E jx x j L x x, 2 d = 0 0 dx x L x x. 36 he overa mnus sgn n the defnton of C s for ater convenence. For generc E and L the matrx C s nvertbe, and we have w=c d. he physca meanng of ths reaton can be umnated by rewrtng Eq. (33) as foows: 0= + x + dx wj j= E jx x j + L x 0 x = + dxfxl x x, 0 where fx we defne to be the vaue of fx when w=w. ow add and subtract / 0 dxfx to obtan 0= 0 dxfx + 0 dxfx + L x x = w. 37 We fnd that the equbrum mean weght vector w s that for whch the mean weght change s zero for a weghts. hs condton s obvous on ndependent grounds and coud have been used to cacuate w drecty, wthout recourse to the moment herarchy reatons. But for moments of tota order 2 or hgher, transparent condtons such as ths are not avaabe; n that case we have no choce but to sove Eq. (32). Gven the equbrum mean weghts w, we can cacuate the equbrum mean postsynaptc potenta Ux va Ux = x + E x w = x + E x C d, provded C s nvertbe. B. Equbrum varance We now take =2 and q k = k + jk n Eq. (32). After some smpfcaton, usng C, d, f, and w from above, we obtan 0= C jk w k w C k w k w j w d j w j d k= k= + dxfx + L x x j + L jx x j 0 j. hs can be rearranged to gve k= C jk w k w + C k w k w j w C jk w k k= w j C k w k = 0 k= + L jx x j + 0 w w j. In vector form ths becomes k= dxfx + L x x j dxfx j = Cww ww + w 2 ww C = w w. 38 he covarance of a vector random varabe v s cov v =vv vv. Equaton (38) then takes the compact form Ccov w + cov wc = cov w, 39 where we have used the equbrum mean condton w =0 on the rght sde. Equaton (39) s a Lyapunov equaton [28] for cov w, gvng the equbrum weght covarance n terms of C (whch depends on E and L) and covw (whch depends on f,, and L). Both C and covw can be cacuated from the parameters of the system, and then the equbrum covarance cov w, f t exsts, must satsfy Eq. (39). A theorem of Ostrowsk and Schneder [28,29] gves condtons for the exstence and unqueness of soutons to Lyapunov equatons. If S s symmetrc postve defnte and A and A have no common egenvaues, then the Lyapunov equaton AH+HA =S has a unque souton H. Furthermore, H s symmetrc and has the same nerta (number of egenvaues wth postve, zero, or negatve rea part) as A. Snce covw s necessary symmetrc postve defnte, the theorem says that a symmetrc souton cov w to Eq. (39) exsts unquey provded C and C have no common egenvaues, and cov w s postve defnte f and ony f a egenvaues of C have postve rea part. he condton that C and C have no common egenvaues s true for generc C and hence for generc E and L. he condton that cov w be postve defnte s needed n order to nterpret cov w as the covarance matrx of a probabty dstrbuton; we say cov w s physca f t s postve defnte. Denotng by C n the nth egenvaue of C, we then have the foowng physcaty condton:

9 RADOM WALKS FOR SPIKE-IMIG-DEPEDE PHYSICAL REIEW E 70, 0296 (2004) cov w physca Re C n 0 for a n. 40 A theorem of Henz [28,30] says that f a egenvaues of A have postve rea part and a egenvaues of B have negatve rea part, then the (unque) souton X to the equaton AX XB=Y s gven by X =0 ds e sa Ye sb, 4 where the matrx exponentas are defned va ayor expansons. he assumptons on the egenvaues of A and B ensure that the ntegra n Eq. (4) converges, and one can show by drect substtuton that the resutng X satsfes AX XB=Y. If the physcaty condton (40) hods, then C and C satsfy the condtons for A and B, respectvey, and we obtan cov w dse =0 sc cov we sc. 42 hs gves the equbrum covarance matrx expcty n terms of system parameters. Snce the postsynaptc potenta Ux s a determnstc functon of the synaptc weght vector w, the weght covarance cov w determnes the covarance of the postsynaptc potenta. From Ux=x+E xw, we have covux,uy = E x cov we y 43 for any par of tmes x,y n the nterva 0,. Of partcuar nterest s the dagona varance of Ux: cov Ux,Ux = E x cov we x. 44 Our dervaton of the equbrum moment herarchy equatons reed on the equbrum dstrbuton of Ux beng neggbe on the tas of the postsynaptc spke probabty functon f. We w show n the next secton, for the case of homogeneous parameters, that the confnement condton on Ux can aways be satsfed by adjustng the rates of assocatve and nonassocatve earnng. ote that for a spatay extended PSP E, Eq. (44) mpes that the dagona varance of Ux depends on the fu matrx cov w; n other words, t depends not ony on the dagona varances of the synaptc weghts w, but aso on the offdagona correatons between dfferent synaptc weghts. II. MULIPLE WEIGHS, HOMOGEEOUS PARAMEERS For maxma generaty n the foregong anayss, we have aowed the postsynaptc potenta functons and spketmng-dependent earnng rues to be dfferent for dfferent presynaptc neurons and have aowed the presynaptc spke tmes to be arbtrary. Further anaytca progress can be made n the case where the system parameters are homogeneous;.e., the postsynaptc potenta functons and spke-tmngdependent earnng rues are the same for a presynaptc neurons, and the presynaptc spke tmes are reguary spaced. For such parameters t w turn out that the matrx C, the coeffcent matrx n the Lyapunov equaton (39) for cov w, has a speca form: t s crcuant [3]. he matrx covw on the rght sde of the Lyaponov equaton for cov w s not crcuant n genera, but t s crcuant f the postsynaptc spke probabty densty fx s ndependent of x. ow t was shown n [4] that n the case of homogeneous parameters, f the spacng between presynaptc spke tmes s suffcenty sma and provded certan other constrants hod, the (mean) equbrum weght vector has the property that the mean tota postsynaptc potenta Ux s approxmatey constant. 3 In that case the mean equbrum postsynaptc spke densty fx s aso approxmatey constant, and the matrx covw s approxmatey a crcuant matrx D. he Lyapunov equaton for cov w s then approxmatey wth souton gven by Ccov w + cov wc = D, 45 cov w ds e =0 sc De sc. 46 he egenvaues and egenvectors of crcuant matrces are easy cacuated; furthermore, a crcuant matrces can be smutaneousy dagonazed. Smutaneous dagonazaton of C, C, and D n Eq. (46) w yed an expct souton for cov w n terms of the egenvectors and egenvaues of C and D, whch w themseves be wrtten as expct functons of the system parameters. Let Es, Ls, and denote the common postsynaptc potenta functon, assocatve earnng rue, and nonassocatve earnng rue, respectvey. Let the spke tme for presynaptc ce be x =, =,2,...,, =/. We then have C j = dxl x x E x x j, and for fx approxmatey the constant f we have covwd, where D j = 2 f + f 0 dx + L x x + L x x j. By perodcty of L, ths can be smpfed to D j = 2 +2f + f 0 dxl x x L x x j, 48 where =/ 0 dxl x. A matrx A s crcuant [3] f each row of A equas the row above t shfted one entry to the rght (and wrapped around at the edges); n other words, 3 he present mode dffers from the mode n [4] n havng a postsynaptc spke probabty densty nstead of a mean postsynaptc spke rate, but the argument s unaffected

10 WILLIAMS, LEE, AD ROBERS PHYSICAL REIEW E 70, 0296 (2004) A +mod,j+mod = A j for a, j. We now show that both C and D are crcuant. Frst, et gx and hx be any perodc functons of x wth perod, and et the x be reguary spaced on 0, as defned above. Let A be the matrx defned by A j =0 dx fx x gx x j. 49 akng, j to (+mod,j+mod ) n Eq. (49) shfts the argument of both functons by, and by perodcty ths does not change the vaue of the ntegra. Hence any matrx of the form (49) s crcuant. he constant matrces (a of whose entres are the same) are aso crcuant, and crcuant matrces are cosed under addton, scaar mutpcaton, and transposton. Hence by Eqs. (47) and (48), C and D are both crcuant and so s C. It s easy shown [3] that the vectors u n, n =,2,...,, wth components u n = e 2 n/, k =,2,...,, 50 are a compete set of egenvectors for any crcuant matrx A, wth correspondng egenvaue n gven by n = A j e 2j n/. 5 = he expresson on the rght n Eq. (5) s ndependent of j because A j and the compex exponenta both depend ony on j mod. It s easy checked from Eq. (5) that addng a constant matrx (a entres the same) to a nonzero crcuant matrx has no effect on ts egenvaues. Let R be the untary matrx whose nth coumn s the vector u n, and et be the dagona matrx wth entres n. hen A = RR *, where R * s the compex conjugate transpose of R. In the present context t w be convenent to defne wave numbers k n so that the argument of the compex exponenta n Eq. (50) s k n x ; ths we can arrange by takng k n =2n/, n=,2,...,. From Eq. (5), the egenvaues of C and D are then C n = e k n x j x 2 = 0 D n = f e k n x j x = 0 dxl x x j E x x, dxl x x j L x x. By perodcty of E and L and reguar spacng of the x, these can be rewrtten as C n = e k n x 2 = 0 dxl x x E x, 52 D n = f e k n x dxl x x L x. 53 = 0 Let C and D be the dagona matrces wth entres C n and D n, and et R be the untary matrx defned above wth entres R j = u j = e k x j. hen C=R C R * and D=R D R *. ransposton takes egenvaues to ther compex conjugates, so C =R C R *. From RR * =I and ayor expanson t foows that e sc =Re sc R * and e sc =Re sc R *. Substtuton nto Eq. (45) then yeds a dagonazaton of cov w: cov w = ds e R0 sc D e scr * = R w R *, where w s the dagona matrx wth entres n w =0 ds e s n C n D e s n C = n D 2Re n C, 54 provded Re C n 0. Snce D s symmetrc postve defnte (t s, by constructon, a physca covarance matrx), we have D n rea and postve for a n. Reca that n order for the souton of the Lyapunov equaton (45) to be postve defnte, a egenvaues of C must have postve rea part.e., Re C n 0 for a n. If ths physcaty condton s satsfed, then the egenvaues of cov w gven by Eq. (54) are rea and postve. hese egenvaues, wth C n and D n gven by Eqs. (52) and (53), are the varances assocated wth the ndependent components of the equbrum weght dstrbuton. he correspondng egenvectors are the u n, wth components u n j =e k nx j. Snce 0, the condton for physcaty of the covarance s Re e k n x dxl x x E x 0 for a n. = 0 hs concdes wth the condton derved n [4] for stabty of the mean weght state. Roughy speakng, t foows that f there exsts an equbrum weght dstrbuton Pw (wth fnte covarance matrx), then the mean of the dstrbuton must be stabe. We do not address the stabty of the equbrum dstrbuton (or, equvaenty, the stabty of a moments of the equbrum dstrbuton) n the present paper, but a natura conjecture woud be that f the equbrum dstrbuton Pw exsts, then t s necessary stabe. From cov w=r w R * we can now wrte down expct expressons for the equbrum covarance of any par of weghts: covw j,w = n,m= R jn w nm R * m = n= R jn R n w n = e k n x j x w n, n= 55 wth n w gven by Eq. (54) and n C, n D gven by Eqs. (52) and (53)

11 RADOM WALKS FOR SPIKE-IMIG-DEPEDE PHYSICAL REIEW E 70, 0296 (2004) ote that covw j,w depends on j and ony va the dfference x j x mod, due to perodcty and transatona nvarance of the archtecture for homogeneous parameters. Aso, the covarance of the weghts depends ony on the assocatve part L of the earnng rue, snce the nonassocatve part does not appear n Eq. (55). hs s not surprsng, snce the roe of s essentay anaagous to that of a constant externay apped force n a physca system. Such a force changes the poston of the equbrum, but does not ater the dynamcs around the equbrum. A. Confnement Our dervaton of the moment herarchy reatons, Eqs. (32), reed on the assumpton that the equbrum weght dstrbuton was neggbe on the tas of the pecewse near postsynaptc gan functon f. hs paces a constrant on the mean Ux and dagona varance cov(ux,ux) of the postsynaptc potenta: they must be such that the mean s a arge number of standard devatons away from the tas. For each x, et rx be the standard devaton of Ux dvded by the dstance from Ux to the nearest ta.e., to or. he parameter rx w be referred to as the confnement parameter for the system. he confnement condton hods provded Ux s n the nterva, and rx, for a x. We now argue that by adjustng ony the rates of nonassocatve and assocatve earnng, the confnement condton can aways be satsfed. Mutpyng the assocatve earnng rue by a postve scaar factor and both nonassocatve and assocatve components by a postve scaar factor, we have weght changes gven by + L x,densty /f x,wt, wt =, probabty dxf x,wt. /0 56 he rato of assocatve to nonassocatve earnng rate s parametrzed by, whe the overa earnng rate s parametrzed by. ow t was shown n [4] that n the case of homogeneous parameters, under certan md condtons, the equbrum mean weght vector has the property that Ux s approxmatey constant (.e., the equbrum s an approxmate negatve mage state). Hence f n Eq. (37) s approxmatey constant. If t were exacty constant, then Eq. (37) (for homogeneous parameters) woud yed, after canceng on top and bottom, f = + dxl x Provded and dxl x have opposte sgn (shown n [4] to be necessary for exstence of a negatve mage equbrum) the rght sde of ths equaton can be made to have any desred vaue by approprate choce of 0. Hence f can be made to have any desred vaue by approprate choce of. ; n partcuar, a range of exsts for whch f fas n the open nterva (f, f ). Snce f s nvertbe for arguments n, and f= fu, t foows that by approprate choce of, U can be made to have any vaue n,. Snce fx approxmatey constant mpes U approxmatey constant, t foows that the mean postsynaptc potenta Ux can aways be made to e between the tas, for a x. It remans to show that the dagona varance cov(ux,ux) can be made suffcenty sma so that the dstrbuton of Ux s neggbe on the tas. We do ths by hodng fxed and varyng. Snce the matrx C s proportona to and the matrx cov w s proportona to 2,t foows from Eq. (38) that cov w and hence cov U from Eq. (48) s proportona to. In partcuar, cov(ux,ux) can be made arbtrary sma by takng suffcenty sma. hus, by approprate choce of and, the confnement condton can aways be satsfed. he vaue of determnes the ocaton of the mean postsynaptc potenta, and the vaue of determnes the wdth of the dstrbuton around the mean. he atter fact that the wdth of the equbrum dstrbuton of the postsynaptc potenta s proportona to the overa earnng rate has drect behavora reevance to the mormyrd fsh, snce t mpes a tradeoff between speed of adaptaton and accuracy of the adapted state. 4 B. Dense spacng mt In the archtecture of the mormyrd ELL, the spacng between presynaptc spke tmes s much ess than the wdths E, L of the PSP E and earnng rue L. In the dense spacng mt the set of dscrete weghts per unt tme w / correspondng to presynaptc spkes at tmes x becomes a contnuum weght densty Wy, wth weght Wydy correspondng to presynaptc spke tmes between y and y+dy. Sums over x are repaced by ntegras over y. he matrces C and D n Eq. (45) become nfnte dmensona, wth egenvaues C n, D n gven by C n = dy e k n y dxl x ye x, D n = f dy e 0 k n y dxl x yl x, 58 0 for n=0,,... We ntroduce some usefu notaton. Let F h be the sequence of Fourer coeffcents for a functon h on 0,, gven by F h n = 0 dy e kny hy wth k n =2n/, n =0,,... Let * denote convouton on the nterva 0,, g* hx= 0 dy gx yhy. Let h denote the horzonta re- 4 he fact that the varance s proportona to the earnng rate s aso true for nhomogeneous parameters, by the same argument. But the confnement of the mean postsynaptc potenta Ux s uncear n that case, because the equbrum s not necessary an approxmate negatve mage. Further work s requred to characterze the equbrum for nhomogeneous parameters

12 WILLIAMS, LEE, AD ROBERS PHYSICAL REIEW E 70, 0296 (2004) fecton of h,h y=h y. hen Eqs. (57) and (58) can be wrtten as n C = 2 F L *E n, D n = f F L *L n. ow we nvoke the Fourer convouton theorem F g*h =F gf h and the fact that F g =F g, where z denotes the compex conjugate of z. hs gves n C = 2 F L n F E n, D n = f F L n F L n. he egenvaues of the weght covarance are therefore W n = C 2Re = f F L n F L n. 6 n ReF L n F E n n D It foows that the covarance of Wy and Wz s cov Wy,Wz = e k n y z W n = 2fF n=0 F L F L z, ReF L F E y 62 where F hx=/2 n=0 e knx h n s the nverse Fourer transform on 0,. he covarance of the postsynaptc potenta s then cov Uy,z =0 dx0 = 2f0 dxe y xcov Wx,xE z x dx0 dxe y xe z x F F L F L ReF L F E x x. 63 One speca case s worth notng: suppose the PSP and earnng rue have dentca functona form.e., are proportona to one another Lx=cEx for some (rea) constant c. hen we have F F L F L = F ReF L F E x cx = c 2 x, where x s the Drac deta functon. For such a earnng rue the covarance of the weght densty s FIG. 2. PSP and earnng rues used n the exampes. Stabty requres 3 2 2L/E Stabe exampes are drawn wth sod nes; end ponts of the stabe nterva are drawn wth dashed nes. Arbtrary unts. cov Wy,Wz = fcy z. 64 In partcuar, the covarance (and hence the correaton) of Wy and Wz s zero for yz; hence weghts correspondng to dfferent presynaptc spke tmes are statstcay ndependent. hs s surprsng, snce the coupng of weghts through the PSP E and earnng rue L has some nonzero range, gven roughy by the wdths of E and L, and wthn ths range one woud expect the weghts to necessary have some nonzero correaton. he resut just derved says that n certan exceptona cases ths correaton may vansh. he resut was derved n the dense spacng mt, but can be expected to hod approxmatey for the physca case of dscrete spacng and aso to hod approxmatey for L not qute proportona to E; ths w be verfed n the exampes cacuated beow. Gven that the best current expermenta measurement of the earnng rue n the mormyrd ELL [3] s not nconsstent wth E and L havng the same functona form, ths vanshng correaton phenomenon may have boogca reevance. III. EXAMPLES We now compute the equbrum weght covarances for a cass of PSP s and earnng rues consstent wth those measured n the mormyrd ELL, assumng homogeneous parameters. he PSP we take to be an exctatory apha functon of wdth E, and the earnng rue we take to be apha functon, depressve, and pre-before-post, of wdth L : Ex = E 2 e x/ EHx, Lx = L 2 e x/ LHx, where Hx s the Heavsde functon: Hx= f x0 and 0 otherwse (Fg. 2). In the above expressons both E and L have been normazed to unt area, but to ensure confnement of the postsynaptc potenta, the earnng rue L (and hence the sze of the earnng steps) must be made suffcenty sma so that the confnement condton s satsfed

13 RADOM WALKS FOR SPIKE-IMIG-DEPEDE PHYSICAL REIEW E 70, 0296 (2004) FIG. 3. Dagona varance of weghts, for apha functons E and L and for varous vaues of L / E. he arger of L and E was taken to be 0.2 n a cases. Dagona varance vs L / E, og-og pot. Dotted nes ndcate the boundary of the stabe nterva, L / E =3±2 2. Dmensoness unts. It was shown n [4] that n order for the mean weght dynamcs to be stabe near the (negatve mage) equbrum, the tme constants E and L must satsfy 3 2 L E For L / E n ths stabe range, we cacuated the equbrum covarance of the synaptc weghts and of the postsynaptc potenta and verfed our predctons by drect Monte Caro smuaton of the underyng random wak. he number of presynaptc ces was taken to be =50, and to ensure that the confnement condton was we satsfed, the rates of nonassocatve and assocatve earnng were adjusted so that the confnement parameter was rx=0.2 for a x (.e., the tas were fve standard devatons away from the mean postsynaptc potenta). By transatona symmetry for homogeneous parameters, the dagona varances w,w are ndependent of, and the off-dagona covarance w,w j depends ony on x x j mod. he covarance matrx s then competey descrbed by the dagona varance (a snge number) and the correaton of weght w wth the mdpont weght w /2, for =,2,...,; the correaton n ths case s just the covarance normazed by the dagona varance. he dagona varance s shown n Fg. 3, and the correaton s shown n Fg. 4, for varous vaues of L / E between and ote the approxmate vanshng of offdagona correaton for L / E near, as expected from the anaytc cacuaton n the dense-spacng mt. he manner n whch the correaton devates from an approxmate deta functon as L / E devates from aso shows an nterestng pattern: for L / E sghty greater than, the near-dagona (near-neghbor) correaton s postve, whe for L / E sghty ess than, the near-neghbor correaton s negatve. But for L / E substantay greater than or ess than, the near-neghbor correaton s postve n both cases. he magntude of off-dagona correaton tends to ncrease as L / E moves away from n ether drecton. ear the mts of the stabe range of L / E, the near-neghbor correaton s cose to and the antpoda correaton (correaton wth weghts a haf perod away) s cose to. Such strong ong-range correaton and antcorreaton was aso observed numercay n [5] n mean weght dynamcs for parameters near the boundary of the stabe regon, wth breakdown of stabty beng characterzed by the appearance of traveng waves. he correaton of the postsynaptc potenta s shown n Fg. 5. For L / E near the correaton s everywhere postve. As L / E devates from, the correaton decreases, and ong-range antcorreatons appear. As L / E devates st further, the antcorreaton decreases n range and ncreases n magntude, and a postve ong-range correaton appears. For L / E near the mts of the stabe range, the mdrange and ong-range (antpoda) correatons approach and +, respectvey, smar to the behavor of the synaptc weght correaton. he scaoped appearance of these curves for arge L / E s due to E beng not much arger than the spacng =/50 between presynaptc spke tmes, resutng n ony margna overap of adjacent PSP s. For fxed PSP wdth E, such scaopng shoud vansh as the spacng of presynaptc spke tmes goes to zero. It s beeved [27(b)] that n the mormyrd ELL the spacng of presynaptc spke tmes s suffcenty dense that ths scaopng woud be nsgnfcant. Comparson wth drect Monte Caro smuaton of the random wak reveaed exceent agreement wth predcton, provded confnement was we satsfed; resuts for L / E =5.84, near the upper end of the stabe range, are shown n Fg. 6. As above, nonassocatve and assocatve earnng rates were adjusted so that the confnement parameter rx was 0.2 for a x (.e., the tas were fve standard devatons away from the equbrum mean). Weghts were taken to be ntay uncorreated, wth mean equa to the predcted mean and varance equa to the predcted (dagona) varance; the nta correaton was then the dscrete Drac deta functon. o quantfy convergence we used the mean absoute vaue of the reatve dscrepancy between the predcted and actua (ensembe mean) correaton. ransaton nvarance of the correaton aowed us to reduce the sze of fuctuatons n the smuaton estmate by averagng not just over the ensembe but aso over the popuaton of =50 weghts n each member of the ensembe. 5 Usng ths measure, the correaton n the smuaton converged to wthn % 2% of the predcted correaton n approxmatey 0 7 tme steps (Fg. 6). IX. DISCUSSIO Snce changes n synaptc weghts n SDP are due to temporay dscrete events (spkes or spke pars), the dynam- 5 Athough the predcted correaton s transaton nvarant, the fuctuatons around the predcton are not necessary uncorreated. For our purposes ths s harmess; t smpy means that we do not obtan as arge a reducton n fuctuaton sze by popuaton averagng as we woud by usng a 50-tmes arger ensembe

14 WILLIAMS, LEE, AD ROBERS PHYSICAL REIEW E 70, 0296 (2004) FIG. 4. Correaton of weghts, for apha functons E and L and for varous vaues of L / E. he arger of L and E was taken to be 0.2 n a cases. Curves are abeed by the vaue of L / E, and for carty curves are not joned to the pont (0.5,) whch a curves have n common. (a) Correaton of w wth w /2, versus x /, for L / E sgnfcanty ess than. (b) Same for L / E sgnfcanty greater than. (c) Same for L / E near, wth expanded vertca scae. Dmensoness unts FIG. 5. Correaton of postsynaptc potenta, for apha functons E and L and for varous vaues of L / E. he arger of L and E was taken to be 0.2 n a cases. (a) Correaton of Ux wth U/2, versus x/, for L / E sgnfcanty ess than. (b) Same for L / E sgnfcanty greater than. (c) Same for L / E near, wth expanded vertca scae. Curves are abeed by the vaue of L / E. Dmensoness unts.

15 RADOM WALKS FOR SPIKE-IMIG-DEPEDE PHYSICAL REIEW E 70, 0296 (2004) FIG. 6. Convergence of weght correaton to predcted equbrum vaues n Monte Caro smuatons, for L/E=5.8, =50, confnement parameter =0.2. (a) me evouton of popuaton-averaged correaton; curves abeed by tme, t/. Dotted curve ndcates predcton. (b) Reatve dscrepancy between predcted and actua correaton, vs tme t/. Dmensoness unts. cs of such pastcty, n the presence of nose, s naturay modeed as a dscrete-tme random wak. here s a arge body of mathematca technque for the anayss of such processes [7]. From the weght dynamcs expressed as a random wak one can wrte down a master equaton for the tme evouton of the weght probabty dstrbuton. From the master equaton we obtan a functona equaton for the equbrum weght dstrbuton. akng the Fourer transform of ths equaton yeds a dfferenta equaton for the characterstc functon of the equbrum dstrbuton, and ayor expanson then yeds a herarchy of recurrence reatons for the equbrum moments. From the moments of the equbrum weght dstrbuton we aso obtan the moments of the postsynaptc membrane potenta. For the case of a snge weght, we expcty cacuate moments up to fourth order. he dstrbuton s shown to be genercay non-gaussan, but the skew and kurtoss approach Gaussan vaues as the earnng rate (sze of steps) goes to zero. For the case of mutpe weghts we expcty cacuate moments up to second order. he mean weght vector satsfes a smpe matrx-vector equaton, whch s equvaent to the condton that the mean step n the equbrum state be zero for a weghts. he weght covarance matrx satsfes a Lyapunov equaton. An expct souton to ths equaton, n the form of a matrx ntegra, s obtaned. For ths souton to be the covarance matrx of some probabty dstrbuton t must be postve defnte, whch mposes a constrant on the PSP E and the assocatve earnng rue L. For the case of mutpe weghts wth homogeneous parameters, further anaytca progress can be made. he Lyapunov equaton for the weght covarance matrx can be fuy dagonazed and the covarance of any par of weghts found n cosed form. From ths we aso obtan expct expressons for the covarance of the postsynaptc potenta between any par of tmes. he physcaty condton that the weght covarance matrx be postve defnte takes an especay smpe form n ths case, cosey reated to the condton derved n [4] for stabty of the mean-weght state. In the mt of dense spacng of presynaptc spke tmes, the expresson for the weght covarance s further smpfed. In the speca case where E and L have the same functona form, we fnd, surprsngy, that weghts correspondng to dstnct presynaptc spke tmes are statstcay ndependent. hs resut can be expected to hod approxmatey for dscrete presynaptc spke tmes and for earnng rues not qute dentca to E n functona form. umerca cacuaton of the equbrum weght covarance and postsynaptc potenta covarance was carred out for a cass of exampes reevant to the mormyrd ELL: both E and L apha functon n form, wth E exctatory and L depressve pre-before-post. For the synaptc weghts, offdagona correaton s near zero for L / E = and tends to ncrease n magntude as L / E moves away from. aues of L / E near the boundary of the stabe range show arge ong-range antcorreatons. he correaton of the postsynaptc potenta s everywhere postve for L / E =, but ongrange antcorreatons deveop as L / E moves away from. hese numerca predctons were found to be n exceent agreement wth drect Monte Caro smuatons of the underyng random wak. One of the basc resuts of ths paper s that the varance of the equbrum weght dstrbuton s proportona to earnng rate (.e., to the magntude of the weght changes nduced by ndvdua spkes or spke pars). A sow earnng rate eads to a sma varance n equbrum weght dstrbuton and hence a more accurate negatve mage; a fast earnng rate gves a arge varance n equbrum weght dstrbuton and a ess accurate negatve mage. Detectabty of sensory objects s mproved by a more accurate negatve mage; thus to optmze detectabty the earnng rate shoud be sow. However, f the fsh s own dscharge s changng (due to changes n water conductvty or body shape, for exampe), then the negatve mage must be updated to reman accurate. Such adaptabty of the negatve mage favors a fast earnng rate, to aow the negatve mage to keep up