Supporting Tet Evoution of the Averge Synptic Updte Rue In this ppendi e evute the derivtive of Eq. 9 in the min tet i.e. e need to ccute Py ( ) Py ( Y ) og γ og. [] P( y Y ) P% ( y Y ) Before e strt et us rec some nottion. The verge of n rbitrry function f ith rguments nd y is by definition f ( y ) = p( y ) f( y ) [2] y y here p( y ) denotes the oint probbiity of the pir ( y) to occur nd the sum runs over configurtions of nd y. The subscript indictes tht both the probbiity distribution p nd the function f my depend on prmeter. By definition e hve p( y ) = p( y p ) ( ) here p( ) is given input distribution nd p( y ) the (prmeter-dependent) condition probbiity of generting n output y given. Hence Eq. 2 cn be trnsformed into f ( y ) = p ( ) p( y ) f( y ) = p( y ) f( y ) y y y If e no te the derivtive ith respect to the prmeter the product rue yieds to terms f( y ) = p( y ) f( y) [4] y y + p( y ) og p( y ) f( y) y The first term contins the derivtive of the function f heres the second term contins the derivtive of the condition probbiity p. We note tht Eq. 4 cn so be ritten in the form f( y ) = f( y ) + og p( y ) f( y ) [5] y y y i.e. s n verge over the oint distribution of nd y. This formution i be usefu for the probem t hnd. The grdient in Eq. contins sever terms nd for the moment e pic ony one of these. The others i then be treted nogousy. Let us focus on the term og Py ( Y X ) nd ppy steps competey nogous to those [3]
eding from Eqs. 2-5. og ( Py ) Y X = og Py ( Y X ) + og PY ( X ) og P( y Y X ) [6] We no evute the verges using the identity = y of Eq. 6 vnishes since og Py ( Y X ). We find tht the first term on the right-hnd side y = og Py ( ) Py ( ) y {0 } = Py ( ) = 0 [7] y {0 } becuse of the normiztion of probbiities. The sme rgument cn be repeted to sho tht 0= og Py ( Y ). The reference y distribution Py % ( Y ) is by definition independent of. Hence the ony term tht gives nontrivi contribution on the right-hnd side of Eq. 6 is the second term. With n nogous rgument for the other fctors in Eq. e hve Py ( ) Py ( Y ) og γ og P( y Y ) P% ( y Y ) og P( Y X ) P( y ) P( y Y ) = og γ og [8] P( y Y ) P% ( y Y ) Y X An identifiction of the fctors F nd G in the min tet is strightforrd. From Eq. 4 in the min tet e hve og ( Py ) = y og( ρ ) + ( y )og( ρ ) [9] Hence e cn evute the fctors P y Y X ρ F = og = y og + ( y ) og ( ) ρ P( y Y ) ρ ρ P( y Y ) ρ ρ = og og ( ) og P% = ( y Y ) % ρ + % ρ G y y Furthermore e cn ccute the derivtive needed in Eq. 8 using the chin rue from Eq. 6 of the min tet i.e.
( ) = ( ) PY X P y Y X [0] = hich yieds og PY ( X ) = og Py ( ) [] = y y = ρ ε ( t t ) = ρ ρ n n n [2] We note tht in Eq. 8 the fctor og PY ( X ) hs to be mutipied ith F or ith G before ting the verge. Mutipiction genertes terms of the form y y = y y Y X y y ith Y X X For ny given input X the utocorretion < of the postsynptic neuron i hve trivi vue y y = y y for > Y X Y X Y X [3] here t is the idth of the utocorretion. As consequence y y F γ G 0 for ρ ρ = > Y X [4] Hence for > e cn truncte the sum over in Eq. 2 i.e. = hich yieds ecty the coincidence mesure = introduced in the min tet; cf. Eq. in the min tet nd hich e repet here for convenience y y n n = ρ ε ( t t ) = ρ ρ [5] n From Averges to n Onine Rue The coincidence mesure counts coincidences in rectngur time indo. If e repce the rectngur time indo by n eponenti one ith time constnt nd go to continuous time the summtion in = t Eq. 5 turns into n integr dt ep[ ( t t )/ ] hich cn be trnsformed into differenti eqution d () t ( t δ ) ( f ) = + ε( tt ) () ( ˆ St δ tt δ) gut ( ()) Rt () ; dt [6] f cf. Eq. 5 in the min tet. Bsed on the considertions in the previous prgrph the time constnt shoud best be chosen in the rnge
t 0 t. Simiry the verge firing rte ρ () t = g() t R() t cn be estimted using running verge dg() t g = gt () + gut ( ()) [7] dt ith time constnt g. In Fig. 6 e compre the performnce of three different updte schemes in numeric simutions. In prticur e sho tht (i) the ect vue of the trunction of the sum in Eq. 5 is not reevnt s ong s t is rger thn the idth of the utocorretion; nd (ii) tht the onine rue is good pproimtion to the ect soution. To do so e te the scenrio from Fig. 3 of the min tet. For ech segment of s e simute 00 pirs of input nd output spie trins. We evute numericy Eq. 8 by verging over the 00 smpes. After ech segment of second (=000 time steps) e updte the eights using rue ithout trunction in the sum of Eq. 5. We c this the fu btch updte; compre Fig. 6 (Top). Second e use the definition of ith the truncted sum nd repet the bove steps; Fig. 6 (Midde). The trunction is set to t = 200 ms hich is e bove the epected idth of the utocorretion function of the postsynptic neuron. We c this the truncted btch rue. Third e use the onine rue discussed in the min body of the pper ith = s; Fig. 6 (Bottom). omprison of top nd center grphs of Fig. 6 shos tht there is no difference in the evoution of men synptic efficcies i.e. the trunction of the sum is oed s epected from the theoretic rguments. A further comprison ith Fig. 6 Bottom shos tht updtes bsed on the onine rue dd some fuctutions to the resuts but its trend cptures nicey the evoution of the btch rues. Suppement to the Pttern Detection Prdigm In Fig. 3 e presented pttern detection prdigm here ptterns defined by input rtes ere chosen rndomy nd ppied for one second. After erning the spie count over one second is sensitive to the inde of the pttern. Fig. 7A shos the histogrm of spie counts for ech pttern. Optim cssifiction is chieved by choosing for ech spie count the pttern hich is most iey. With this criterion 8 percent of the ptterns i be cssified correcty. The updte of synptic efficcies depends on the choice of the prmeter γ in the erning rue. According the the optimity criterion in Eq. 8 of the min tet high eve of γ impies strong homeosttic contro of the firing rte of the postsynptic neuron heres o eve of γ induces ony e homeosttic contro. In order to study the roe of γ e repeted the numeric eperiments for the bove pttern detection prdigm ith vue of γ = 00 insted of our stndrd vue of γ =. Fig. 7B shos tht the output firing rte is sti moduted by the pttern inde the modution t
γ = 00 is hoever eer thn tht t γ =. As resut pttern detection is ess reiby ith 45 percent correct cssifiction ony. We note tht this is sti significnty higher thn the chnce eve of 25 percent.