Role of NMDA conductance in average firing rate shifts caused by external periodic forcing. Nikita Novikov 1 and Boris Gutkin 1,2 1 ABSTRACT

Transcription

1 Rol of conductnc n vg fng t shfts cusd by xtnl podc focng Nkt Novkov 1 nd Bos Gutkn 1,2 1 Cnt fo Cognton nd Dcson Mkng, Ntonl Rsch Unvsty Hgh School of Economcs, Moscow, Ru, Goup fo Nul Thoy, LNC INSERM U960, Dptmnt of Cogntv Studs, Ecol Noml Supu PSL Rsch Unvsty, Ps, Fnc, BSTRCT wdly ccptd vw of computtons n th bn ls on populton codng, wh th nul nsmbl fng t s modultd n stbl mnn to tnsmt nfomton nd pfom vous cogntv tsks. t th sm tm, osclltoy nul ctvty s spcfclly modultd n fquncy, cohnc nd pow dung cogntv pfomnc. How th fng t nd osclltons ntct mns slnt quston. In ths pp, w dvlop thoy fo th ntctons btwn osclltoy sgnls nd th fng t of nul popultons bsd on ctvty of non-ln voltg-dpndnt synp. Notbly, w show und whch condtons osclltoy nputs cn contol th mn fng t wthout lo of stblty. Usng mthmtcl nly nd smultons of mn-fld modls, w dmonstt tht psnc of synp on both th xcttoy nd th nhbtoy nuons s ctcl fo snusodl osclltons to sgnfcntly nd stbly ncs th fng t. W chctz th oscllton-nducd mn fng t shft s functon of th fst nd slow synptc wghts nd dmonstt th pmt gon, n whch th ffct und nvstgton s mostly ponouncd. Rsults of ou wok my hlp dntfy th popts of nul ccuts tht llow fo constuctv contol of th fng t cods by lg-scl nul osclltons. I. INTRODUCTION Osclltoy ctvty s bundnt co nul ntwoks of th bn, nd ts pofl s snstv to vous xpmntl nd ntul condtons (Wng, 2010). Howv, th lton btwn nul osclltons nd th most bsc functons of th bn ccuts such s ncodng nd dstbutng of nfomton s pooly undstood. On of th most pomnnt pncpls of nfomton ncodng n th bn s populton t codng (Rolls, 2011). ccodng to ths pncpl, nfomton s stod s dstbuton of fng ts co populton. Fng t n ths cs s dfnd s th vg numb of spks mttd n ltvly polongd tm ntvl. In ths contxt, quston of how mght th fng t modultons, ncy fo nfomton tnsmon nd tsk xcuton, my ntct wth th osclltoy pofl of th nuonl ctvty, coms to th fofont. In mny c, osclltoy pofl s closly ltd to populton fng t (zouz, Gy, 2003). lso, th s vdnc tht osclltoy ctvty s not mly bypoduct of t codng, s t could b modultd ndpndntly of fng ts (Fs t l., 2001). In pncpl, osclltoy ctvty could b ltd to t codng n svl wys. Fst, t could sv s n ndpndnt dmnson of nul cod. Scond, t could sv s mtdt tht contols tnsmon of t-codd nfomton n th bn (km, Kullmnn, 2014; Fs, 2005). Thd, t could povd modultoy scffold fo t codng whch stblzs th cod o mks th codng mpobl by dstblzng th ncy fng t dynmcs (Dpopp, Gutkn, 2013; Schmdt t l., 2018). In od to lgothmclly mplmnt th fomntond functons, th should b mchnsms of tnsfomton btwn populton t cod nd osclltoy pofl (km, Kullmnn, 2010; km, Kullmnn, 2014). Th convson fom fng t to osclltoy popts s qut wll undstood t th lvl of smpl spkng ntwoks (Bunl, Wng, 2003); lso ctn bophyscl mchnsms w dscbd fo mcoccuts (Roopun t l., 2008), s wll s fo sngl nuons. Howv, th oppost convson,.. th ffct of osclltoy popts on th mn fng t, s much l studd. To chng th mn fng t of systm by osclltoy ntnmnt, th systm should b nonln. On cnddt popty s non-lnty of th f-i cuvs (gn functons) of nuons n th subthshold stt. Th postv hlf-wv of nput osclltons bngs nuon clos to th thshold, thus ncsng th fng pobblty, whl th ngtv hlf-wv dos not stongly ffct ths pobblty.

2 Thus, n vg, osclltoy nput modulton could ncs th mn fng t (Rolls, 2011; Slns, Sjnowsk, 2000); ths ffct hs bn xplod mthmtclly n th lmt of wk osclltons (Voonnko, Lndn, 2017). Howv, gvn tht n vy nosy stt (typcl fo cotcl ntwoks) th gn functons clos to ln, th fomntond mchnsm should poduc qut subtl ffct. noth poblty to lnk osclltons nd fng t modulton s to consd slow non-ln mchnsm, such s conductnc o voltg-dpndnt on chnnls; ths d ws poposd n th study (km, Kullmnn, 2010), but hs not bn mplmntd n tht study. In th psnt pp, w thotclly nvstgt th ol tht slow voltg-dpndnt cunts could ply n shftng th tm-vgd stt of n xcttoy-nhbtoy systm n th psnc of xtnl zo-mn snusodl focng. In od to spt th ffct of non-lnty fom th fomntond ffct of gn functon non-lnty, w consd low-dmnsonl populton modl wth th gn functons lnzd bout th unfocd qulbum. Slops of th gn functons, s wll s th tm constnts tht dfn th dynmcs of populton fng ts nd voltgs, numclly dvd fom smultons of th uncoupld ntwoks of lky ntgt-nd-f nuons. In ou study, w mnly concntt on xcttoy ffct poducd by th xtnl focng. W dmonstt tht n th psnc of -cptos on th xcttoy nuons only, th postv shft of th fng t could only b vy slght (1-2 Hz) wthout lo of stblty, whl th postv nhbtoy fng t shft could b mo ponouncd. Howv, whn -cpto lso psnt on th nhbtoy nuons, th systm could b md mo xctbl wthout lo of stblty, nd th lg postv shfts of both xcttoy nd nhbtoy fng ts could b chvd. Th pp s ognzd s follows. (1) W dscb ou modl nd ts lnzd vson. (2) W povd nlytcl xpons fo th lton btwn th pmts of th xtnl focng nd th shft of th tm-vgd stt of th systm tht th focng poducs. (3) W dscb th pocdu of pmt slcton tht povds lstc stt of th systm. (4) W pfom gomtcl nly of th phs pln fo th systm wth -cptos on th xcttoy populton only. W fnd th tmvgd stdy-stts fo th unfocd nd fo th focd systm. W lso dscu th stblty-ltd lmttons fo th stdy-stt shft tht follows fom ou gomtcl consdton. (5) W confm pdctons of ou gomtcl nly by xplctly smultng th cospondng populton modl. (6) W pfom th gomtcl nly nd th confmng smulton fo th systm wth -cptos on both popultons (xcttoy nd nhbtoy). W dscu how ddng -cptos to th nhbtoy populton hlps to ovcom th lmttons on th stdy-stt shft. (7) Fnlly, w comp th two fomntond c fo ng of vlus of systm pmt tht contols xctblty of th fst subsystm. II. THEORY. Th low-dmnsonl nul ccut systm modl In od to dscb th nul populton dynmcs of n xcttoy-nhbtoy nul ccut w us low dmnsonl modl. Th two dffnt nul popultons cuntly ntcoupld wth nstntnous xcttoy nd nhbtoy GB, s wll s slow non-ln xcttoy connctons. Th ntwok lso cvs xtnl xcttoy nputs psntd by sum of wht-nos sgnl wth non-zo mn nd snusodl sgnl wth zo mn. W dscb ou systm by th followng sx vbls: (1) th xcttoy nd nhbtoy populton fng ts:, (2) th mn mmbn voltgs vgd ov th xcttoy nd th nhbtoy popultons: V, V, nd (3) th mn cunts cvd by th xcttoy nd th nhbtoy nuons: I, I ; ths cunts dpnd on th psynptc fng t, s wll s on th postsynptc populton voltg. Dynmcs of ths vbls gvn by systm of fst-od dffntl qutons wth th vbls volvng ound th stt-dpndnt nstntnous qulbum vlus,0,0,0,0,0,0,, V, V, I, I on tm scls dfnd by th tm constnts,,,, :, V V

3 d dt 1 d 1 dt ( u (,, I )),0 ( u (,, I )),0 dv 1,0 V ( u(,, I )) V dt V dv 1,0 V ( u (,, I )) V dt V di 1 I (, V ) I dt di 1,0 I (, V ) I dt H, u, u,0, (1) dnot th mn nstntnous nputs to th xcttoy nd th nhbtoy popultons spctvly (xpd n tms of voltg). Th functons ( u ), ( u ), V ( u ), V ( u ) tk th,0,0,0,0 mn nputs s ts gumnts nd dpnd both on th popts of th nuons nd on th lvl of nos n th systm (w dv ths functons fom smultons of ndvdul nuons, s blow). Pls not tht w do not modl th nos xplctly; nstd, ts ffct s ncopotd n th functons ( u ), ( u ), V ( u ), V ( u ). Th functons gvng th nonln -dpndnt xctton,0,0,0,0,0,0 ( I (, V ), I (, V ) ) tk th xcttoy fng t nd th mn voltg of th nput populton s ts gumnts nd xpd s follows:,0 I (, V ) g ( V ), (2) wh th ndx could b plcd by fo th xcttoy populton o by fo th nhbtoy populton (ths notton wll b usd long th pp). H s th stngth of th -ltd couplng btwn th xcttoy populton nd th populton ; th functon g dscbs th dpndnc of -cunt on th mmbn voltg: 1 g ( V ) 1 xp( V ) 3.57, (3) wh V s xpd n mllvolts. Th nputs dpnd on th fng ts u, u, nd th cunts I, I s follows:,, th xtnl tonc nputs h, h u (,, I ) h I g, (4) m wh, th stngths of th fst synptc couplngs (th fst ndx dfns th nput populton, nd th scond ndx th output populton), g m s th mmbn conductnc of nuons fom th populton. Th synptc couplng stngths xpd s follows: j K m j K, (5) jk m wh j s th mpltud of th nstntnous nhbtoy postsynptc potntl on th nuons fom th populton ; j s th mpltud of th nstntnous xcttoy -postsynptc potntl onto th nuons n th populton ; j - th mpltud of th -postsynptc cunt stp onto th

4 nuons fom th populton ; K b b - numb of nputs tht nuon fom populton m cvs fom nuons tht blong to populton (,b =,), - mmbn tm constnt th nuons fom th populton. Th tm constnt of -synptc nput s much lg thn th tm constnts tht govn th dynmcs of th populton fng ts nd th mmbn voltgs,..,, V, V. Thus, w cn consd ou systm s slow-fst systm, nd spt th tm scls. t th fst tm scl, th stt of th slow subsystm ( I, I ) cn b consdd s constnt. Thn th fst subsystm (,, V, V ) convgs to n qulbum (,, V, V ) tht dpnds on th stt of ( I, I ). t th slow tm scl, th slow subsystm ( I, I ) volvs, nd ts dynmcs ums tht th fst systm s convgd to n nstntnous qulbum t ch tm momnt,.. (,, V, V ) (,, V, V ). s sn blow, w us th tm scl spton to nlyz th stblty of th fst nd th slow subsystms ndpndntly, s wll s to nlyz th ffct of th focd osclltons of th fst subsystm on th dynmcs of th slow subsystm. B. Modllng umptons In th modl dscbd bov, w md th followng umptons: (1) fst nd GB synp nstntnous; (2) cunts usd s dynmc vbls, nstd of conductncs; (3) -cunts non-stutng; (4) voltg nd fng t dynmcs dscbd by ln dffntl qutons, wthout tkng th ffct of spk-to-spk synchonzton nto ccount; (5) tm constnts,, V, V do not dpnd on th stt of th systm; (6) th mn souc of nos s,0,0,0,0 xtnl, nd t s ccountd fo n th functons ( u), ( u ), V ( u), V ( u ). Th umpton (4) s vld whn th systm opts n subthshold gm, n whch th ctvty of th cospondng spkng ntwok s nos-dvn nd gul (n ths wok, w chck tht th pmts w us stsfy ths condtons). Not tht ths gm s dffnt fom sup-thshold on, wh th cospondng spkng ntwok would show gul osclltoy spkng. W n gu tht th umpton (6) s vld f th fst synptc wghts suffcntly smll, whch s lso tu n ou cs. Whl umptons (1) nd (5) my b poblmtc n gnl, thy only ffct th ltonshp btwn th xtnl focng nd th focd osclltons of th fst systm. In ou nly, w stt fom slctng th pmts of th focd osclltons (s opposd to dvng thm fom th pmts of th xtnl focng), so th fomntond ltonshps not mpotnt n ou cs. Th umptons (2) nd (3) cucl, s thy ffct th non-ln dpndnc of th -cunts on th fng ts nd th mmbn voltgs. W us ths umpton to smplfy ou nly. W should lso not tht w th xpon (3) fo g ( V ) s vld fo sngl-nuons. In th ntwok, th s dstbuton of voltgs co nuons V k (, ), wh k s nuon numb; th mn voltg of populton s dnotd by V k k V. Thfo, ch nuon s chctzd by k ts own vlu of th voltg-dpndnt conductnc g ( V ). In th xpon fo th mn k -cunt co nuons, on should us th tm g ( V ), w mk smplfyng umpton nd us g k V g V k k k g V g( V ) k fnd th ppopt functon. Of cous, fom th non-lnty of g, t follows tht k k. Howv, und qus-sttc ppoxmton, t should b pobl to k g such tht g ( V ) g ( V ) nly unwldy whl hvng psumbly smll ffct on ou mn conclusons. C. Th lnzd systm k.yt ths would mk ou Th gol of ou study s to nvstgt th mpct of zo-mn podc focng on shft of th,0,0 qulbum. In th systm (1), ths shft cn b ccountd fo by non-lnty of th functons, V

5 (govnng th dynmcs of th fst subsystm), s wll s by non-lnty of th functons pp, w concntt on th ffct of th lnz,0,0, V,0,0,0 I,0 I. In ths non-lnty. Thus, to nlyz ths ffct sptly, w bout th qulbum. W should not, howv, tht th umpton of lnty of, V s non-lstc, spclly n th cs whn th nducd osclltons stong; n th lstc stuton th nflunc of both shft should b consdd., V,0,0 nd,0 I non-lnts on th focng-nducd qulbum W now dscb th lnzton pocdu. Lt us um tht th full systm hs stbl fxd pont dnotd s ( 0, 0, V 0, V 0, I 0, I 0) nd th cospondng vlus of nputs s ( u0, u 0).,0,0,0,0 Now lt us lnz th functons ( u), ( u ), V ( u), V ( u ) bout ths fxd pont, nd substtut ths functons n (1) by th lnzd vsons, whch w dnot s ( u ), ( u ), V ( u ), V ( u ), spctvly (lt us dfn,0,0 (, ) (, ), (, ) I V I V I I V fo nottonl convnnc): d 1 ( u (,, I )) dt dv 1 V ( u (,, I )) V. (6) dt V di 1 I (, V ) I dt Blow, w wll f to (6) s th unfocd systm. Th lnzd functons xpd s follows: ( u ) 0 c u u0, (7) V ( u ) V 0 cv u u0 wh th ppopt dvtvs :,0 d ( u) c du u 0. (8),0 dv ( u) cv du u 0 D. Fxd ponts of th unfocd systm Lt us now duc th systm (6) to two lgbc qutons wth th xcttoy t nd nd th cunt n th xcttoy populton I s th vbls. In th nxt sctons, w dmonstt tht plottng th cuvs dfnd by ths qutons on th ( - I ) phs pln povds usful gomtcl ntuton bout xstnc of solutons (whch dfnd by ntsctons of ths cuvs), s wll s bout stblty of ths solutons. Lt us dfn th functons V( ) nd V( ) n such wy tht, fo n nput (, ), th followng dntty s tu: V ( ( u )) V ( u ), (9) u.., f th nput movs th fng t towds, thn th sm nput movs th mmbn voltg towds V( ). Fom (7), t follows tht: cv V ( ) V 0 0. (10) c u

6 Nxt, lt us fnd th stt of th fst subsystm (,, V, V ) towds whch t convgs t th fst tm scl fo gvn p of fxd vlus of th slow vbls I, I (w dnot ths stt s (,,, ) V V. By sttng th ght-hnd pt of th fst quton of (6) to zo, w gt: wh ( I, I ) P Q I Q I V ( I, I ) V ( I, I ) c(1 c ) Q gmq cc Q gmq cc Q gmq c (1 c ) Q gmq Q (1 c )(1 c ) c c P Q I Q I P Q I Q I 0,0,0 0,0,0, (11). (12) Now w cn conclud tht th stdy-stt fng ts nd -cunts should stsfy: I I, V. (13) ( I, I ) Usng (11), lt us xp nd I fom (13) s functons of nd I : Q (, I ) P QI P QI Q, (14) I (, I ) I, V (, I ) Fnlly, n od obtn slf-contnt qutons fo nd I, lt us put (14) bck nto (13): P QI QI (, I ). (15) I I, V s w mntond pvously, th fst nd th scond qutons of (15) dfn two cuvs on th (, I ) -pln, ntsctons of whch cospond to th fxd ponts of th systm (6). Futh n ths txt, w wll f to ths cuvs s th -cuv nd th I -cuv, spctvly. W should not tht ths cuvs not nullclns, lthough thy ntsct t th fxd ponts of th systm. E. nly of th lnzd systm wth xtnl podc nput: focd osclltons Ou gol s to undstnd und wht condton xtnl podc focng my chng th mn ctvty of th nuonl popultons n ou modl. W stt by nlyzng th lnzd systm (6) wth podc xtnl focng. H w dv th mpltud nd phs ltons btwn th xtnl podc sgnl nd th focd osclltons of th xcttoy nd nhbtoy popultons.

7 Lt us pply n xtnl podc focng to th systm (6), wth th ccul fquncy nd th complx-vlud mpltuds of th osclltoy nputs to th E- nd I-popultons qul to nd, spctvly: d 1 osc ( u (,, I, t)) dt dv 1 osc V ( u (,, I, t)) V dt V di 1 I R( ),R( V ) I dt osc t u (,, I, t) I gm h h Usng th tm scl spton, w cn um tht: (1) th slow vbls I nd I do not gt ntnd by th xtnl focng (whch s sonbl whn th focng fquncy s not xtmly low); nd (2) fo ctn combnton ( I, I ), th dynmcs of th fst vbls could b psntd s hmonc osclltons bout th ttctng fxd pont of th unfocd fst subsystm (,, V, V ) (s (11)). Thus, f w dnot th complx-vlud mpltuds of th focd fng t osclltons s V, V, h h. (16),, V, V, nd th complx-vlud mpltuds of th focd mmbn voltg osclltons s, thn th dynmcs of th fst subsystm could b xpd s follows: t ( t) ( I, I ). (17) t V ( t) V ( I, I ) V Th dynmcs of (, ), dfnd by th systm (16), could b xpd n th mtx fom: wh c I g h h u t d m 0 0 M dt t c I gm h h u0 0, (18) c 1 c M. c ( c 1) (19) Now w put th xpon fo fom (17) nto (18) nd tk nto ccount th fct tht () t ( I, I ) s th soluton of (18) whn h h 0. ft ll cnclltons, w gt th followng quton fo th mpltuds (, ) : h c M. (20) h c Usng (20), w cn fnd th xtnl mpltuds ( h, h ) tht would poduc th focd osclltons of th fng t wth th mpltuds (, ) : 1 h 1. (21) c s th nxt stp, w wnt to xp th osclltoy pt of th totl nput (, ). Lt us put th xpon h fom (21) nto th xpon fo t ( t) ( I, I ) osc u s functon of fom (17) nd th xpon fo osc u fom (16):

8 osc t t 1 t,,,,, 1 c u I t u I, (22) u I s th totl nput cvd und ctn stt of th slow subsystm, gvn tht wh,, th fst subsystm convgd to ts stdy-stt (n th bsnc of th focng). H, w omttd th dpndnc of on I, I fo nottonl smplcty. Now lt us fnd th how th mpltud of th focd voltg nd th focd fng t ltd t nd. Lt us put (22) nd th xpon V ( t) V ( I, I ) V fom (17) nto th V quton fo, V V V u(,, I ), w gt: fom (16). Usng th xpon (7) fo V, nd tkng nto ccount tht 1 c V V 1 V. (23) V c Solvng (23) wth spct to ylds: cv 1 V c 1 V V. (24) In ou nly, w found t convnnt to pmtz ou systm by,, nstd of h, h Th ltt p of pmts could b dvd usng (21). Th lton (24) wll b qud n th nxt scton, wh w xp th focng-nducd shft of th tm-vgd qulbum. F. Systm wth xtnl podc nput: focd shft of th tm-vgd qulbum In ths scton, w nlyz th nflunc of th focd osclltons of th fst (,, V, V ) - subsystm on th dynmcs of th slow ( I, I ) -subsystm du to ts non-lnty. Fst, lt us ntoduc shot nottons fo th functons tht govn th dynmcs of th cunts (s (6) nd (2)): 1 G,, V I I, V I. (25) Now lt us pply th tm scl spton nd wt th qutons fo th dynmcs wthout podc focng, substtutng th fst vbls, V n (25) by th vlus, V, towds whch ths vbls convg t th fst tm scl: di G ( I, I ) dt, (26) G ( I, I ) G ( I, I ), V ( I, I ), I wh, V gvn by (11). To ccount fo th ffct of th xtnl focng, w wt th qutons fo th slow dynmcs wth, V not qul to, V (s n (26)), but osclltng bout, V (s (17): di osc G ( I, I, t) dt, (27) osc G ( I, I, t) G cos( t ), V V cos( t V ), I wh g, g V V. In od to stmt th ffct of th focd osclltons on th slow dynmcs, w pply th vgng mthod (Stogtz, 2015). W ntoduc th nw slow vbls I nd I tht dscb th vlus of I nd I, spctvly, vgd ov th focng pod T 2. Fom th tm.

9 scl spton, t follows tht T s much smll thn th chctstc tm of th slow dynmcs. Consquntly, th dynmcs of th tm-vgd vbls I, I could b xpd s follows: di dt tt 1 osc G ( I, I, t) dt T. (28) Lt us wt n xplct xpon fo t osc G, usng (25) nd (27): osc G ( I, I, t) cos( ) cos( ) 1 g V V t t I V In od to nlytclly stmt th ntgl n (28), lt us xpnd 1 g V g V V k ( ) k k 0 k! dg( V ) gk dv V Fom th xpnson (30), w gt: cos( ) cos( ) g V V t t V ( )cos( V )cos( ) g bout V :. (29). (30) g( V ) g( V )cos( t ) V g( V )cos( t V ), (31) V g V t t V g ( V )cos ( t V ) o 2 wh w um tht,, V, V hv th sm od of mgntud. Now w put (31) nto (29), omttng th hgh-od tms, nd clcult th ntgl n (28). s w ntgt ov th pod of osclltons, ll tm-dpndnt cosn tms cncl out, nd w gt n utonomous quton: di G I, I dt D, V G I, I G I, I, (32) 1 2 D, V 2 V g V cos V g V 4 wh V s th phs lg btwn th voltg nd th fng t osclltons, nd, V nwly ntoducd vbls such tht: I, I. (33) V V I, I W cn consd oscllts und th xtnl focng. Fom now on, w wll f to th s slowly movng cnt, ound whch th fst subsystm -systm dscbd by (32), togth wth th vbls tht functonlly dpnd on, s th tmvgd focd systm.

10 Th fst tm n th squ bckts n (32) (contnng V ) flcts th fct tht cunt dpnds both on th psynptc fng t nd th postsynptc voltg, whl th scond tm (contnng V 2 ) flcts th fct tht -cunt dpnds non-lnly on th postsynptc voltg. Ou numcl sults (s th followng sctons) show tht, fo th slctd pmts, th fst tm s much lg thn th scond on. Thus, th focng-nducd shft of th tm-vgd qulbum s mnly ltd to th jont ffct of th psynptc fng t osclltons nd th postsynptc voltg osclltons, occung wth smll phs lg btwn thm. W pmtz ou systm by nd In od to do ths, w dv fom (24) th mpltuds V c c V V, so w wnt to xp V, V : D, D n th tms of, (34) nd th cosns of th phs lgs: 2 V 1 V cos cos g V 1, (35) 2 V 1 V cos V sn cos cos g V 1 wh s th ngl btwn nd. Tkng nto ccount (25) nd (26), th stdy-stts of (32) should stsfy: I I, V ( ) D, V ( ), (36) I, I whch s th nlog of (13) n th cs of th tm-vgd focd systm. Now lt us xclud I fom (36), smlly to (14), nd gt th slf-contnt xpon fo th stdy-stt vlus of nd I, whch s nlogous to (15):,,,,, ( ) P Q I Q I I QD V I I I V D V, (37) wh nd I th sm functon s t ws ntoducd n (14). Sml to th unfocd cs, th stdy-stts of th tm-vgd focd systm could b found s ntsctons of th -cuv nd th -cuv on th -pln, whch dfnd by th fst nd th scond qutons of (37), spctvly. Both g nd g postv n th physologcl ng of voltgs. lso, n th sults scton w dmonstt tht, fo th slctd pmts, cos,cos lso postv. Consquntly, both D nd D postv. lso, fo bng smll nough, Q 0. Thus, fom (37) nd (15), t follows tht th -cuv s shftd upwds ltvly to th I -cuv, nd th -cuv s shftd to th lft ltvly to th -cuv. In th followng sctons, w wll dmonstt tht n cs of 0, such chct of th - nd th -cuv shfts mpls tht th xtnl focng lwys nc th mn fng ts. Futh n ths pp, w dnot th fxd pont of th tm-vgd focd systm tht cosponds to th fxd pont ( 0, 0, V 0, V 0, I 0, I 0) of th ntl systm s (,, V, V, I, I ) G. Stblty nly of th unfocd nd th podclly focd systms,. nd

11 Lt us now nvstgt th stblty condtons fo th systms wth nd wthout th podc focng. W wnt to dntfy condtons und whch th podc focng chngs th tm-vgd ctvty wthout dstblzng t. Fo th unfocd systm, w cn pply tm scl spton nd nlyz stblty of th fst nd th slow subsystms ndpndntly. Th fst (,, V, V ) subsystm s stbl whn ll th gnvlus of th mtx (gvn by (19)) hv ngtv l pts. Stblty of th slow ( I, I ) subsystm M of th unfocd systm could b dtmnd by nlyzng (26). In th cs of subsystm s stbl, f: dg ( I ) di In th cs of M I 0 0 dg di dg di 0, th slow 0. (38), on should consd th mtx: dg di dg di I 0, I 0 Th slow subsystm s stbl f ll th gnvlus of. (39) M hv ngtv l pts. pplyng th tm scl spton to th focd systm, ts symptotc dynmcs could b consdd s focd osclltons of th fst (,, V, V ) subsystm ound th pont ( 0, 0, V 0, V 0) tht functonlly dpnds on th qulbum ( I 0, I 0) of th tm-vgd slow subsystm ( I, I ). s th mtx s constnt, stblty of th fst subsystm dos not dpnd on th stt of th slow subsystm. Consquntly, f th fst subsystm of th unfocd systm s stbl, t s guntd tht th focd osclltons wll hv fnt mpltud (t lst n th cs whn ths mpltud s smll nough, so w do not nd to tk fxd ponts oth thn ( I 0, I 0) nto consdton). Th stblty condtons fo th tm-vgd focd slow subsystm sml to (38) nd (39). Fo 0 w gt: dg ( I ) di I 0 M 0, (40) nd fo 0 w qu th l pts of th gnvlus of th followng mtx to b ngtv: M dg di dg di dg di dg di I 0, I 0. (41) W should not tht th tm-vgd focd systm my hv no qulb (.. th -cuv nd th I -cuv do not ntsct). In ths cs, th condtons (40) nd (41), obvously, not pplcbl, nd th systm dvgs. Lt us tnsfom th stblty condtons (38) nd (40) to fom tht s sutbl fo futh gomtcl nly. W cn us th xpons (26) nd (25) fo G nd G to xplctly wt down th dvtv dg di, thn pply th chn ul: d di d d d di, nd tk nto ccount tht, ccodng to (11), d di Q. Th sm pocdu could b ppld to dg di

12 , wth th xcpton tht G dffs fom G by th functon D. s th sult, th dvtvs n (38) nd (40) could b xpd s: ( ) 1 di, V dg I 1 Q di d I 0 0. (42) dg ( I ) 1 d 1 I, V D, V Q di I d 0 0 Fo bng smll nough,. Consquntly, tkng nto ccount (42), th stblty condtons Q (38) nd (40) (tht cospond to th cs 0 0 ) could b wttn s follows: di, V 1 d Q 0. (43) d 1 I, V D, V d Q 0 W futh dscu (43) n th sults scton, wh w consd th systm wth ; w lso povd gomtcl ntptton fo ths condtons n tms of postons of th,, I, I -cuvs on th phs pln. In th subsqunt sults sctons, wh w consd th cs of w chck th stblty condtons (39) nd (41) numclly. III. PRMETER SELECTION ND VLIDTION. Pmt slcton 0 In ths scton, w bfly dscb th slcton of th pmy pmts of ou modl, s wll s clculton of th pmts dvd fom th pmy ons. In ths wok, w mnly ntstd n th ffct of th cunt synptc wghts on th systm s blty to chng ts tm-vgd stdy stt und xtnl focng. To spt ths ffct fom pobl nflunc of th unfocd stdy stt chng cusd by chngs n th synptc wghts, w po slct th stdy stt of th unfocd systm nd thn vy th synptc wghts, utomtclly -tunng th xtnl nput stngths fo ch combnton of th wghts n such wy tht th p-slctd stdy-stt s kpt constnt. Mo pcsly, w fxd (.. st po) th totl mn nputs u0, u 0 nd th totl nput vncs 0, 0, nd clcultd th xtnl mn nputs nd th xtnl nput vncs x, x s functons of th totl nput pmts nd of th synptc wghts. Th sm ws don fo th osclltoy pt of th xtnl nput: w fxd th mpltuds, h, h of th focd osclltons of th xcttoy nd nhbtoy popultons, s wll s th phs lg btwn thm, nd thn w dvd th mpltuds h, h of xtnl osclltons dlvd to th popultons (nd th cospondng ph) s functons of,, nd of th synptc wghts.,0,0 Th lnzton coffcnts c, c V tht dfn th slops of, V t u 0 gvn 0 (s (8) ) w dvd fom smultons of ndvdul lky ntgt-nd-f (LIF) nuons (on xcttoy nd on nhbtoy). Ths LIF nuons w dscbd by th followng pmts: th mmbn tm th constnt m, th stng potntl E L, th spkng thshold V, th st voltg V. W pfomd,0,0 smultons fo vlus of u clos to u 0, obtnd numcl stmtons of ( u), V ( u ) fo ch of thm, nd thn clcultd c, c usng ths stmtons. lso, w pfomd sml smultons fo u V u nd clcultd CV vlus CV 0 fom th obtnd spk tns. 0 0,

13 In od to fnd populton tm constnts nd V, w consdd populton of uncoupld LIF nuons wth th cospondng pmts. W dlvd smll nput stp tht shftd th totl nput fom, nd msud th dynmcs nd tht th ntwok dmonsttd n spons to ths u 0 nput stp. Thn w fttd constnts nd V. () t nd () t V () t V () t wth xponntl functons, fom whch w obtnd th Th fomntond pocdus of pmt dvton dscbd n mo dtls n th ppndx. To pusu ou nly nd mk cl th ol of th non-ln synptc cunts, w consdd two modls. Th fst modl (Modl 1) contnd cunts on th xcttoy nuons only ( j 0 th nhbtoy nuons ( ), nd th scond modl (Modl 2) contnd cunts both on th xcttoy nd j 0 ). Th pmts of both modls w slctd n such wy tht povdd lstc mod of opton, s wll s xstnc nd stblty of th tm-vgd focd qulbum. Th synptc wght ws dcsd n th Modl 2 (compd to th Modl 1) n od to dmonstt tht th fst subsystm could b md mo xctbl n th psnc of th slow nhbton wthout lo of stblty (whch lds to mo ponouncd focng-nducd shft of th mn xcttoy fng t). ll th pmy pmts of th Modls 1 nd 2 modls, xcpt of nd, th sm. j j Dung th slcton of pmts, w usd th followng convnnt popty of ou systm: on cn not tht th -cuv dos not dpnd on th fst synptc wghts,,,. Thus, w w bl to contol th fxd pont of th tm-vgd focd systm by chngng th fst synptc wghts (t th sm tm utomtclly djustng th xtnl nputs, s dscbd bov). Chng of th fst wghts cusd otton of th -cuv ound th unfocd qulbum ( 0, I 0), not ffctng th -cuv, so th poston of th tm-vgd focd qulbum ws sy to pdct. Th pmy pmts lstd n th Tbl 1; th dvd pmts, s wll s chctstcs of qulb lstd n th Tbl 2. Tbl 1. Pmy pmts K 800 m 20 ms u mv K 200 m 10 ms u mv 200 ms gm gm 100 µs/cm mv j 0 mv V V -70 mv 13 mv j 0.15 µ/cm 2 V th th -1 mv L L E 0 V -55 mv 2 20 Hz E -70 mv 10 Hz Modl 1 0 mv j mv Modl2 j 0 µ/cm 2 5 Hz j 0.06 mv j µ/cm 2 0 Tbl 2. Dvd pmts nd chctstcs of qulb Hz V mv CV Hz V mv CV c c ms 2.8 ms V 2.37 mv V 0.60 mv c V V 7.9 ms cos 0.993

14 c V Modl 1 h x h x h mv mv mv V x x g h 4.4 ms mv 9.12 mv 0.16 cos I 0 I µ/cm 2 0 µ/cm 2 h 7.96 mv g h Modl 2 h x h x h 27 Hz V mv I 0 23 µ/cm 2 39 Hz mv mv mv V 0 x x g h mv mv 9.17 mv 0.16 I 0 I 0 I 0 0 µ/cm µ/cm µ/cm 2 h 3.42 mv g h Hz Hz V mv I µ/cm 2 V mv I µ/cm 2 B. Pmt vldton In ths scton, w dmonstt tht th pmts w slctd physologclly plusbl nd ld to lstc bhvo of th systm. Post-synptc potntls poducd by ctvton of th fst cptos gvn by th synptc wghts,,,. Post-synptc potntls poducd by th cptos ctvton qul to 1.16 mv fo th E-E connctons nd to mv fo th E-I conncton. ll ths vlu l n th physologcl ng oughly blow 1.5 mv. In ou modl, w dfn th xtnl nputs by th mn vlus nd stndd dvtons. Howv, th nputs n l ntwoks psntd not by contnuous sgnl but by spk tns. Lt us um tht nuons fom populton cv xcttoy nd nhbtoy xtnl nputs whch x K x x x x hv th synptc wghts qul to, nd th psynptc fng ts qul to,, spctvly. W wnt to chck whth t s pobl to slct sonbl vlus fo th fomntond pmts of th xtnl nputs tht povd th vlus of nd tht w usd n th modl. On of th ppopt h x x combntons of ths pmts s th followng: x x x x x x x x x K K 4000, K K 1000, 2.5Hz, 5Hz, 0.44mV, x x x 1.57 mv, 0.62mV, 1.6mV. W suggst tht ths pmts fll nto physologclly sonbl ng of vlus. s w us lnzd vsons of th gn functons n ou modl, th fng ts could, n pncpl, bcom ngtv, whch s non-sns. Howv, w fx th stdy-stt nd th mpltud of th nducd osclltons n such wy tht t dos not hppn. Spcfclly, th mnml fng t fo th xcttoy / nhbtoy populton dung th osclltons ( 0 ) quls to 26.8 Hz / 65.9 Hz n th Modl, nd 17 Hz / 34 Hz n th Modl B. Th vlus of CV (numclly obtnd fom th smultons of ndvdul LIF nuons) t th unfocd stdy-stt ( 0, 0, V 0, V 0, I 0, I 0) qul to 1.1 fo th xcttoy nuons nd 0.9 fo th nhbtoy nuons, whch cosponds to Poon-lk spkng tht s xpmntlly obsvd fo most of th cotcl nuons. Vlus of CV und xtnl focng not fully tctbl, s th spk tns ptlly modultd by osclltons. Howv, w cn consd systm tht hs th qulbum stt (,, V, V, I, I ) n th psnc of puly synchonous nput wth ppoptly x K h x x

15 ncsd mn. Fo th pmts of th Modl, th CV n such systm quls to 1.1 fo th xcttoy nuons nd 0.9 fo th nhbtoy nuons; fo th pmts of th Modl B, th CV vlus of th xcttoy nd nhbtoy nuons qul to 1.0 nd 0.8, spctvly. Fom ths, w cn conclud tht th qulbum shft poducd by th xtnl focng dos not mov th systm out of th physologclly plusbl gon of synchonous fng. Th dffnc btwn th spkng thshold nd th tm-vgd focd qulbum voltg fo th xcttoy / nhbtoy nuons ( ) quls to 19.3 mv / 13.7 mv n th Modl nd to 16.7 mv V th V 0 / 9.6 mv n th Modl B. Ths vlus lg compd to th mpltuds V of th focd voltg osclltons (2.37 mv fo th xcttoy nuons nd 0.6 mv fo th nhbtoy nuons). Consquntly, vn dung pks of osclltons, th nuons wll blow th spkng thshold, so spkng s dvn by ndom fluctutons of th nput nd not by osclltons thmslvs. Futhmo, th to btwn th fng t mpltud nd ts tm-vgd qulbum fo th xcttoy / nhbtoy nuons ( 0 ) quls to 0.37 / 0.13 n th Modl nd 0.27 / 0.07 n th Modl B. Th vlus fo th xcttoy populton (0.37 nd 0.27) lg thn usully obsvd xpmntlly, howv, thy stll suggst tht th systm s f fom unlstc ovsynchonzton nd ll-o-non spkng pttn dvn by osclltons. In summy, th systm dmonstts hllmks of th spsly synchonous gm chctzd by modt podc fng t modulton nd gul fluctuton-dvn spkng of ndvdul nuons whch s blvd to b typcl fo osclltoy cotcl ntwoks. IV. NLYSIS OF SYSTEM WITH SYNPSES ON THE EXCITTORY POPULTION ONLY (MODEL 1) In ths chpt, w nlyz th systm wth -cptos loctd on th xcttoy nuons ). Fst, w plot th - nd -cuvs nd dscu th lton to xstnc nd only ( 0 N I stblty of th qulbum n th tm-vgd focd systm. Nxt, w confm th pdctons of th phs-pln nly by dct smulton of th systm. Fnlly, w dscu ltons btwn th focngnducd shft of th qulbum nd th pmts of th modl.. Phs-pln nly Gvn th condton 0, th qutons (15) fo th - nd I -cuvs of th unfocd systm xpd s follows: P Q I, I I, V (44) nd th sml qutons (37) fo th - nd I -cuvs of th tm-vgd focd systm could b xpd s follows: P Q I. I I, V D, V ( ) (45) Fom (44) nd (45), t s sn tht th - nd -cuvs dntcl nd both psntd by stght ln. In th Fg. 1(), th phs pln s shown: th sold blck ln psnts th / -cuv, th sold blu ln psnts th I -cuv, nd th sold d ln psnts th -cuv. Th pont S 0 s th qulbum ( 0, I 0) of th unfocd systm, nd S 0 s th qulbum ( 0, I 0) of th tmvgd focd systm. Th unstbl fxd pont of th tm-vgd focd systm s dnotd by S 1. zoomd n gon of th phs pln contnng th qulbum ponts s psntd n th Fgu 1B (ths gon s dnotd by th gy ctngl n th Fgu 1). Th dshd d ln s th tngnt ln to th I -cuv tht ps though S 0 (th tngncy pont t th I -cuv s dnotd by ). Th mpotnc of ths ln s dscud blow.

16 Now w povd smpl gomtcl ntptton fo stblty of th slow subsystm of th unfocd systm. Tkng look t th stblty condton gvn by th fst nqulty n (43), on cn s tht ts ght-hnd pt dfns th slop of th -cuv, whl th lft-hnd pt dfns th slop of th ln tngnt to th I -cuv t S 0 Consquntly, th slow unfocd subsystm s stbl t 1B) gos stp thn th I (s th qutons (44) fo th S 0, f th -cuv (sold blu ln n th Fgu 1B) t - nd th I -cuvs). -cuv (sold blck ln n th Fgu S 0. lso, fom ths condton t follows tht, f S 0 s stbl, thn 0 0 (s th I -cuv gos bov th I -cuv),.. th focng hs n xcttoy ffct on th systm. Smlly, fom th scond nqulty n (43) (whch povds th stblty condton fo th tmvgd focd slow subsystm), on cn s tht ts ght-hnd pt, gn, dfns th slop of th -cuv, nd th lft-hnd pt dfns th slop of th ln tngnt to th qutons (45) fo th stbl t S 0, f th d ln n th Fgu 1B) t - nd th I I -cuv t S 0 (s th -cuvs). Thus, th tm-vgd focd slow subsystm s -cuv (sold blck ln n th Fgu 1B) gos stp thn th S 0. I -cuv (sold FIG 1. () Pt of th phs pln wth,, I, I -cuvs fo Modl 1. (b) Zoomd n gon of ntst (dnotd n th pnl by th gy ctngl). Sold blck ln - -cuv; sold blu lns -

17 I -cuv; sold d lns - focd systms, spctvly; ln - tngnt ln to th S 1 I -cuv; S, S qulb of th unfocd nd th tm-vgd - unstbl fxd pont of th tm-vgd focd systm. Rd dshd I -cuv contnng Th condton fo xstnc of th tm-vgd focd qulbum S 0 ; - th cospondng tngncy pont. fomultd s follows. Th dynmcs of th focd systm s boundd only f th wth th to th I I -cuv,.. th S 0 could b gomtclly -cuv should go stp thn th ln png though -cuv (th ltt s psntd by th dshd d ln th fst subsystm mo xctbl (by ncsng, -cuv dc, nd vntully ths cuv concds wth th ( S 0 ) o dcsng ( S 0 ) -cuv ntscts S 0 nd tngnt n th Fgu 1B). s w mk sddl-nod bfucton n th tm-vgd focd systm (th stbl fxd pont unstbl fxd pont S 1 S 1, ), th ngl of th ln, whch cosponds to th S 0 mgs wth th t th pont ). Wth th futh ncs of xctblty, th fxd ponts dspp, nd th dynmcs of th focd systm dvgs. S 0 nd B. Numcl smultons of th low-dmnsonl systm To confm th pdctons of th gomtcl nly, w pfomd numcl smulton of th systm dscbd by (16). Smulton ws pfomd dung wth th tm stp t 0.1 ms. Podc xtnl focng ws tund on t tosc 200 ms. Rsults of th smulton psntd n th Fgu 2; top pnl psnts th fng t tcs, nd th bottom pnl th mmbn voltg tcs (n both c, th l pt s shown). Rd tcs cospond to th xcttoy populton nd th blu tcs to th nhbtoy populton. Thn sold lns psnt th smultd dynmcs nd dmonstt focd osclltons fo t tosc. Thck sold lns psnt th smultd tcs smoothd wth 250-ms tm wndow. Thck dshd lns psnt pdctons obtnd fom th gomtcl nly. W cn s tht th vgd smultd tcs ppoch th vlus tht vy clos to th pdctd ons. T 10s

18 FIG 2. Rsults of smulton of Modl 1. Rd: xcttoy populton; blu: nhbtoy populton. Thn sold lns - smultd tcs; thck sold lns smultd tcs vgd ov 250 ms; thck dshd lns pdctons of th gphcl nly. C. Dpndnc of th stdy-stt shft on pmts In ths scton, w nlytclly dscb how ctn pmts of th systm ffct th focdnducd shft of th tm-vgd qulbum. Lt us wt th systm (45) n th followng fom: I P Q p(, ), (46) I I, V D, V ( ) q(, ) wh s vcto of pmts tht dfn th functons p nd q. Th stdy-stt 0 of th tm-vgd focd systm s gvn by th quton: p(, ) q(, ). (47) 0 0 If pmt s such tht q dpnds on t, but p dos not, thn th dpndnc of 0 on could b obtnd fom th followng xpon: 0 q p q. (48) 0 0 s w mntond n th pvous scton, f 0 s stbl (whch s th cs of ntst), thn th tm n th dnomnto of th ght-hnd pt of (48) s postv. Consquntly, s gows, sgn of chng of 0 nd q s th sm. Usng (2), (32), nd (34), w cn xp q s follows:

19 q g V ( ) g V ( ) cos g V ( ) 2 4, (49) cv 1 c 1 V. wh On cn s tht q gows wth cn conclud tht 0 lso gows s On th oth hnd, f p could b obtnd fom th xpon: Consquntly, nd nd dpnds on pmt. s p dos not dpnd on ths pmts, w ncs., but q dos not, th th dpndnc of p p q. (50) 0 gows wth s p dc wth Usng (12), w cn clcult th dvtvs of p, nd vc vs. by th fst synptc wghts: p 0 0 gm p 0 0 gmc (1 c ). p 0 0 gmc (1 c ) 2 p 0 0 gmc 2 (1 c ) (51) Usng th fct tht 0 0 whn both nd stbl, b cn conclud fom (51) tht nc wth, (.. whn slf-xctton n th systm gts stong), nd dc wth 0, (.. whn slf-nhbton gts stong). s w hv sn, th qumnt of stblty of th tm-vgd focd systm stongly lmts th pobl shft of th xcttoy fng t poducd by th xtnl focng. Howv, th shft of th nhbtoy fng t could b md suffcntly lg. Usng (11) fo I I 0 nd I I, w cn xp th fng t shfts s follows: Q I 0 I 0. (52) Q I I Consquntly, th nhbtoy fng t shft could b dvd fom th xcttoy fng t shft: Q (53) Q If w ncs nd dcs s such wy tht th poduct dos not chng 2 2 (nd lso djust th xtnl nputs n od to kp u, u,,,, constnt, s t ws dscbd bov), thn Q wll ncs, whl Q nd wll sty th sm. Consquntly, by ncsng 0 on 0 (wth th cospondng djustmnt of oth pmts), w cn mk th dvton 0 0 s lg s w wnt. V. NLYSIS OF SYSTEM WITH SYNPSES ON THE EXCITTORY ND INHIBITORY POPULTIONS (MODEL 2)

20 In ths scton, w nlyz th Modl 2 tht contns cptos both on th xcttoy nd nhbtoy nuons. Fst, w not tht th xcttoy-to-nhbtoy (E-I) couplng wk n th modl thn th xcttoy-to-xcttoy (E-E) couplng s much, so th mn ffct of th xtnl focng on th systm s stll xcttoy. Th ol of th E-I couplng h s to mk th systm mo obust, whch llows th fst subsystm to b mo xctbl wthout hvng dvgnt dynmcs n th psnc of th xtnl focng. In th Modl 2, th fst subsystm ws md mo xctbl compd to th Modl 1 by dcsng th E-I couplng. FIG 3. Pt of th phs pln wth,, I, I -cuvs fo th Modl 2. Sold blck ln - -cuv; dshd blck ln - -cuv; sold blu ln - I -cuv; sold d ln - I -cuv; 0 0 S, S - qulb of th unfocd nd th tm-vgd focd systms, spctvly.. Phs-pln nly Th phs pln fo th Modl 2 s psntd n th Fgu 3. Th lgnd s th sm s n th Fgu 1 (whch psnts th sm pctu fo th Modl 1), wth th dffnc tht th -cuv now dffs fom th -cuv; th -cuv s psntd n th Fgu 3 by th dshd blck ln, nd th -cuv s psntd by th sold blck ln. Compng th Fgus 1 nd 3, on cn s two mjo dffncs btwn th Modl 1 nd th Modl 2. Fst, s t ws dmonsttd bov, th -cuv fo th Modl 1 s stght ln, whl ths cuv fo th Modl 2 s concv. Consquntly, th poston of th tm-vgd focd qulbum S 0 n th Modl 1 s lmtd by th pont, dnotd n th Fgu 1B s (whch s th pont on th I -cuv such tht th ln ( S 0 ) s tngnt to th I -cuv); t th sm tm, th s no such lmtton n th Modl 2. Th scond dffnc s tht n th Modl 1 th unstbl fxd pont systm xsts n th qulbum pont S 0, whl n th Modl 2 th S 1 of th tm-vgd focd -cuv quckly gos wy fom th I -cuv n th gon bov S 0, so th s no unstbl pont S 1 n S 0. Thus, w xpct tht no sddl-fold bfucton (nd, consquntly, no dvgnt dynmcs) wll occu f w mk th fst subsystm mo xctbl (s w obsvd n th Modl 1). In th followng scton, w dmonstt tht t s ndd th cs, nd tht th systm lo ts stblty v Hopf bfucton, nd stts to gnt vy slow osclltons.

21 W should not tht stblty of S 0 nd S 0 fo th cs of 0 dos not follow fom smpl gomtcl consdtons, so t should b chckd xplctly by numclly clcultng th gnvlus of th mtcs gvn by th xpons (39) nd (41). W dd t fo th Modl 2, nd confmd tht both S 0 nd S 0 stbl. FIG 4. Rsults of smulton of Modl 2. Rd: xcttoy populton; blu: nhbtoy populton. Thn sold lns - smultd tcs; thck sold lns smultd tcs vgd ov 250 ms; thck dshd lns pdctons of th gphcl nly. B. Numcl smultons of th low-dmnsonl systm W confmd th pdctons of th gomtcl nly fo th Modl 2 by numcl smulton of th systm (16), usng th sm smulton pmts s w dd fo th smulton of th Modl 1. Th sults of smulton psntd n th Fgu 4; th lgnd s th sm s n th Fgu 2. On cn s tht th smulton sults n th good gmnt wth th pdctons; howv th pdcton o s bt lg thn fo th Modl 1. Ths o s cusd by th fct tht th pdcton s obtnd usng tm scl spton whch ums th nfnt to btwn th slow nd th fst tm scls, whl n ou systm ths to s, obvously, fnt. To pov tht th o s cusd by th fnt tm scl to, w ncsd (mkng t ght tms lg) nd popotonlly dcsd, (mkng thm ght tms smll), so, dd not chng (s (5)), nd th pdctd shft ws th sm s fo th Modl 2. Smulton of th modfd modl (s ppndx) dmonsttd lmost pfct mtch wth th pdcton. C. Dpndnc of th stdy-stt shft on pmts In ths scton w numclly xplo dpndnc of th focd-nducd shft of th tm-vgd qulbum on ctn pmts of th systm hvng cptos both on th xcttoy nd th nhbtoy populton.

22 Th dpndnc of 0 on th xtnl focng mpltuds on th slow synptc wghts, s psntd n th Fg. 5(),, n th Fg. 5 (b), on th fst xcttoy synptc wghts, n th Fg. 6(), on th fst nhbtoy synptc wghts n th Fg. 6(b). Th bods of nstblty of th unfocd slow subsystm dnotd by th blck lns; of th unfocd fst subsystm by th gn lns; of th tm-vgd focd systm by th d lns. Fg. 5,6 suggst tht nc wth,, (.. whn th s mo xctton n 0 th systm) nd dc wth,, (.. whn th s mo nhbton n th systm). lso, 0 nc wth dc wth (.. whn th xcttoy populton s mo ctvtd by th xtnl focng) nd (.. whn th nhbtoy populton s mo ctvtd by th xtnl focng). It s sn tht th mxml 0 s lwys obsvd n th bods of nstblty of th slow subsystm. Th tm-vgd focd systm bcoms unstbl s,,, ncs nd,,, dcs. Th unfocd slow subsystm bcoms unstbl s,, ncs nd,, dcs. Th unfocd fst subsystm bcoms unstbl fo vy lg vlus of nd fo combntons of vy modtly lg vlus of nd vy smll vlus of. Intstngly, th s n optml combnton of,, fo whch th mxml chd [s Fg. 5 ()]. 0 s FIG 5. Dpndnc of th focng-nducd shft of th tm-vgd qulbum 0 on th followng pmts: () th slow synptc wghts,, (b) th xtnl focng mpltuds,. Blck ln: th bod of stblty of th unfocd slow subsystm; d ln: th bod of stblty of th tmvgd focd systm. Th nstblty gon s flld by th dk-blu colo.

23 FIG 6. Dpndnc of th focng-nducd shft of th tm-vgd qulbum 0 on th followng pmts: () th fst xcttoy synptc wghts,, (b) th fst nhbtoy synptc wghts,. Blck ln: th bod of stblty of th unfocd slow subsystm; d ln: th bod of stblty of th tm-vgd focd systm; gn ln: th bod of stblty of th unfocd fst subsystm. Th nstblty gon s flld by th dk-blu colo.

24 FIG. 7. (,b) Bfucton dgm of th tm-vgd focd systm. Hozontl xs:, vtcl xs:. Blck ln BCD: sddl-nod bfuctons; B nd C th cusp ponts. Rd ln CG: Hopf bfuctons. Gn ln: Focus-to-nod tnstons. Th pont E s th ntscton btwn CG nd B. In th gon to th lft of th blu ln, th unfocd systm s unstbl. Hozontl lns: th pobd vlus of ; thy dnotd by th numbs 1 9 t th ght pts of th pnls. (c,d) Blu / gy lns: on-dmnsonl (1-D) bfucton dgms (dpndnc of th tm-vgd focd ) fo vous vlus of qulb on. Dgms mkd by th numbs tht cospond to hozontl lns n th Fg. 7(). Sold lns: n qulbum s stbl; dshd lns: unstbl. Blu colo: th unfocd qulbum s stbl; gy colo: unstbl. Supmposd thck lns: n qulbum s focus. Rd ln CG conncts ll Hopf bfucton ponts; blck ln BCD conncts ll sddl-nod ponts; th mnng of,b,c,d,g s th sm s n th Fg. 7(). E 1, E 2 ponts on th b nd CG lns, spctvly, t vlu tht cosponds to th pont E n th Fg. 7(). Th numbs 1 9 n th lft nd th ght pts of th pnls cospond to th numbs n th Fg. 7(,b) nd lnk th cuvs wth th slctd lvls. Now w dscb th dpndnc of th systm s bhvo on th slow E-I couplng wght nd on th xctblty of th fst subsystm (vd v th fst E-I couplng wght ). W nlyzd th bfuctons tht occu n th tm-vgd focd systm whn nd vd nd xplod th consquncs of ths bfuctons fo th blty to xct th systm by xtnl

25 podc focng wthout dstblzng t. Th mn pupos of ths nly s to dfn how stongly th qulbum could b shftd by th xtnl focng wthout lo of stblty und vous pmt combntons. In th Fg. 7(), th two-dmnsonl (2-D) bfucton dgm fo th tm-vgd focd systm s psntd, wth on th hozontl xs nd on th vtcl xs. In th Fg. 7(b), gon 0 of th dgm s zoomd n. Th gon to th lft of th blu ln n th Fg. 7(,b) cosponds to nstblty of th unfocd qulbum, nd t s lvnt fo ou nly. Th blck ln cosponds to sddlnod bfuctons: two fxd ponts mg nd dspp s th ln B o BC s cod fom th ght to th lft, nd s th ln BC s cod fom th lft to th ght. In th gon bov th gn ln n th Fg. 7(,b), on of th fxd ponts s focus, so th systm dmonstts th dmpd o sustnd slow osclltons. Ths focus s stbl to th ght of th d ln CG nd unstbl to th lft of t; th d ln CG tslf cosponds to th supctcl Hopf bfuctons. Th Intscton of th B nd CG lns s dnotd by E. To povd dp undstndng of th systm s bhvo, w fxd t vous lvls nd nlyzd how th fxd ponts of th systm dpnd on. Th vlus of w usd shown n th Fg. 7(,b) by hozontl lns nd mkd by numbs. Th cospondng on-dmnsonl (1-D) bfucton dgms shown n th Fg. 7(c) by th blu / gy lns [nd mkd by th sm numbs s n th Fg. 7(,b)], wth on th hozontl xs, nd th stdy-stt vlus of on th vtcl xs. gon of ntst of th Fgu 5(c) s shown n mo dtl n th Fg. 7(d). Stbl pts of th dgms n th Fg. 7(c,d) psntd by sold lns, nd th unstbl pts by dshd lns. Supmposd tnspnt thck lns dnot pts of th dgms tht cospond th to stbl o unstbl foc. dgm pts hv gy colo f thy loctd n th ng of vlus fo whch th unfocd systm s unstbl. ll ponts of sddl-nod bfuctons connctd by th blck ln, nd ll ponts of Hopf bfuctons by th d ln; th ponts,b,c,d,g hv th sm mnng s n th Fg. 7(,b). Pls not tht pont E n th Fg. 7(,b) s not codmnson 2 bfucton, bcus th lns B nd CG cospond to bfuctons tht occu on th dffnt bnchs of 1-D dgms. In th Fg. 7(d), th bfucton ponts on th two stbl bnchs fo th dnotd by E 1 nd E 2. vlu tht cosponds to th pont E, Fo ll vlus blow th cusp pont B [s lns 1 6 n th Fg. 7()], th cospondng 1- D dgms contn stbl bnch tht tmnts by sddl-nod bfucton [s th lowst pts of th dgms 1 6 n th Fg. 7(c,d) tht loctd blow th blck ln B]. Fo vlus blow th cusp pont C [lns 1 nd 2 n th Fg. 7()], ths s th only stbl bnch of th 1-D dgm. Fom th Fg. 7(c,d), on cn s tht th low bnchs go up s dc. Consquntly, th mxml vlus of fo th dgms 1 nd 2 chvd t th vlus tht cospond to sddl-nod bfucton; ths mxml vlus l on th blck ln B. Fo vlus btwn th cusp ponts C nd B [lns 4 6 n th Fg. 7()], th cospondng 1-D dgms stll contn th fomntond low stbl bnch, but, n ddton to tht, pt of noth bnch wth hgh s lso stbl; n th Fg. 7(c,d), ths stbl bnch pts loctd btwn th blck ln BC nd th d ln CG. Bstblty occus fo combntons of nd tht l n th gon CEB n th Fg. 7(,b). In pncpl, focd osclltons my cus th systm sttl onto th upp stbl bnch, but ou numcl smultons dmonsttd tht n most c th low bnch s slctd (howv, count-xmpl s psntd n th ppndx). Futhmo, th bstblty gon CEB s vy now, so, vn f slcton of th upp bnch s pobl, t qus xtmly fn tunng of pmts. Fo vlus blow th pont E [lns 4 nd 5 n th Fg. 7()], th Hopf bfucton ln CG (d) n th Fg. 7(,b) ls to th ght of th sddl-nod bfucton ln B (blck). ccodngly, s dc, th upp stbl bnch of th 1-D dgm [s dgms 4 nd 5 n th Fg. 7(c,d)] lo ts stblty (cong th d ln CG) bfo th low stbl bnch dspps (cong th blck ln

26 B). Consquntly, f w um tht th upp bnch s nv slctd und bstbl condtons, thn th mxml vlus of tht could b chd by vyng, gn, loctd on th B ln. Fo 0 vlus btwn th ponts E nd B [ln 6 n th Fgu 5()], th Hopf ln CG gos to th lft of th sddl-nod ln B. In ths cs, s dc, th low stbl bnch of th 1-D dgm [s dgm 6 n th Fg. 7(c,d)] dspps (cong th blck ln B), but fo vn smll vlus, th upp stbl bnch stll xsts, untl t lo stblty v Hopf bfucton (cong th d ln CG). Consquntly, th mxml vlus of on th CG ln. Fo 0 tht could b chvd by vyng, loctd vlus bov th cusp pont C [lns 8 nd 9 n th Fg. 7()], only on stbl bnch of th 1-D dgm xsts [s dgms 8 nd 9 n th Fg. 7(c,d)]; ths bnch lo ts stblty v Hopf bfucton s dc (cong th d ln CG). Th mxml vlus of tht could b chvd by vyng,, gn, loctd on th CG ln. In summy, w cn dstngush two gons of 0 vlus, bsd on th locton of th mxml tht could b chvd by vyng. Fst, fo vlu blow th pont E n th Fg. 7(,b), th mxml s chvd t th vlu of tht cosponds to th sddl-nod bfucton tht tmnts th low stbl bnch of th 1-D dgm t ts lft sd [s dgms 1 5 n th Fg. 7(c,d)]. ll such vlus loctd on th cuv E 1 n th Fg. 7(d). Scond, fo vlu bov th pont E n th Fg. 7(,b), th mxml s chvd t th vlu of tht cosponds to th Hopf bfucton [s dgms 6,7 n th Fg. 7(c,d)]. ll such Fg. 7(d). Thus, th dshd ln E 1E 2G n th Fg. 7(d) dfns mxml vlus loctd on th d ln E 2G n th vlus tht could b, n pncpl, chvd by th xtnl focng wthout dstblzng th systm. s on cn s, th lgst of ths vlus s chd t th pont E 2; ths vlu cosponds to th fng t shft (ltvly to th unfocd qulbum) tht s qul to ppoxmtly 20 Hz. VI. CONCLUSION Numous thotcl concpts of nul pocng (.g., slctvty to ctn nfomton) fomultd n tms of vg fng ts. t th sm tm, bn ctvty dmonstts collctv osclltoy pttns tht colt wth functonl stts, tsk qumnts nd bhvol ftus. In od to buld thoy tht concls t-bsd nul codng wth functonl ol of osclltons n computtons nd nfomton outng, on should consd non-ln mchnsm tht convts osclltoy pow nto tonc fng t shfts. In ths pp, w xplod potntl ol tht -synp wth non-ln bhvo could ply n shftng of th vg lvl of populton ctvty n th psnc of xtnl osclltoy nput. W consdd n xcttoy-nhbtoy populton modl wth th populton fng ts, mn mmbn voltgs, nd -cunts s th dynmcl vbls. -cunt dpndd both on psynptc fng t nd postsynptc voltg. In od to dlnt th -ltd ffcts fom th ffcts of non-lnty of spk gnton mchnsm, w lnzd th dpndncs of th fng t nd th mn voltg on th synptc nput bout th unptubd qulbum of th systm. W ppld th tm scl spton nd th tm-vgng mthod, whch llowd us to nlytclly dv xpons fo th mn fng t shft, nducd by th osclltoy nput to th systm. Und lstc modl pmts, w found tht such shft s mostly poducd by jont ffct of th psynptc fng t nd th postsynptc voltg osclltons occung lmost n phs, nd not by th non-ln dpndnc of -cunt on th postsynptc voltg. Ou nlytcl pdctons w confmd by dct numcl smultons. W consdd two modls. In th Modl 1, cptos w loctd xclusvly on th xcttoy nuons, whl n th Modl 2, thy w loctd both on th xcttoy nd th nhbtoy nuons. Usng phs pln nly bsd on ou nlytcl sults, w gomtclly povd tht th oscllton-nducd fng t shft s stongly lmtd by stblty qumnts n th Modl 1. ddng vn vy wk nput to th nhbtoy populton llows to ovcom ths lmtton nd chv ponouncd fng t shft (up to 20 Hz) wthout dstblzng th systm. Fnlly, w xplod th

27 pmt spc nd found optml gons of pmts ( nput wghts to both popultons, nput wght to th nhbtoy populton), n whch th stongst fng t could b chvd und th sm mpltud of th ntnd osclltons. It s cl tht mny nul pocs could potntlly lnk osclltoy ctvty to mn fng t modultons. s th xmpls, on could popos non-lnty of spk gnton, non-ln bhvo of vous slow voltg-dpndnt on chnnls, tnston btwn spkng nd bustng, o synptc plstcty. Futhmo, th chct of such lnk should ctclly dpnd on mcoconnctvty pttns. Nvthl, th -bsd mchnsm tht w poposd h s th gnl nd should ply ctn ol n osclltoy-nducd fng t shfts longsd wth oth potntl mchnsms n most nul confgutons. Howv th xtnt to whch th -ltd non-lnty s nvolvd n ths shfts s subjct of futu xpmntl sch. PPENDIX : PRMETER DERIVTION 1. Lnzton coffcnts nd th CV In ou numcl computtons, w umd tht th souc of th nput vnc s mnly xtnl, so ths vnc dos not dpnd on th stt of th systm. Followng ths umpton, w smultd sngl xcttoy (nd sngl nhbtoy) lky ntgt-nd-f nuon tht cvd Gun wht nos wth th mpltud nd th tonc nput. W pobd svl vlus of ound. Th 2 0 u smultons w govnd by th followng qutons: dv m v EL u 0 () t dt, (1) th v V : v V wh - th mmbn voltg, - th mmbn tm constnt, - th stng potntl, () - v m wht nos wth zo mn nd unt stndd dvton, - th spkng thshold, - th st voltg.,0 Fo ch vlu of, w clcultd th sultng mn mmbn voltg ˆ V ( u ) nd th fng u,0 t ˆ ( u ). Gvn ths mpcl dpndncs, w numclly stmtd th dvtvs t dtmnng th constnts c nd c V. th V E L u V u 0 u 0 t, thus lso, fom th smultd spk tn fo u u0 w clcultd th coffcnt of nt-spk ntvl vton whch chctzs gulty of th spk tn. CV 0 In od to fnd populton tm constnts 2. Populton tm constnts nd V, w consdd N 5000 uncoupld nuons whos dynmcs govnd by (1) wth u u0. W ptd ths smulton 20 tms. Fo ch tm bn, w vgd th mmbn voltgs ov nuons nd tls, thus computng th mpcl tmpol dynmcs of th populton mmbn voltg Vˆ () t. W lso vgd th numbs of spks poducd by th ntwok ov tls, thus computng th mpcl tmpol dynmcs of th populton fng t ˆ () t. ft ˆ () t nd Vˆ () t stblzd (gvn u u0 ), w ncsd u by u 2 mv, contnud th smultons, nd clcultd ˆ () t nd Vˆ () t fo th subsqunt tm momnts, untl thy stblzd gn. Thn w fttd th tnstons of ˆ () t nd Vˆ () t poducd by th ncs of u by xponntl functons, fom whch w got nd V, spctvly. Th sultd ˆ () t nd Vˆ () t fo xcttoy nd nhbtoy ntwok (nd th cospondng xponntl fts) psntd n th Fgus 8() nd 8(b), spctvly.

28 FIG. 8. Rsults of smulton of n uncoupld populton of xcttoy () nd nhbtoy (b) nuons. Top pnls: dynmcs of th populton fng t, bottom pnls: dynmcs of th mn mmbn voltg. Blu lns: smulton sults; d lns: xponntl fts. 3. Pmts of th xtnl nputs Usng th sngl-nuon smulton (s n th pvous pgphs), w numclly dfnd th stdy-stt populton fng t nd th populton mmbn voltg gvn th nput u u : 0 0,0 ˆ 0 ( u0). ˆ,0 V 0 V ( u0) (2) Fom th found vlus of nd, usng (2), w clcultd th stdy-stt -cunt 0 V 0 I 0 : I I, V. (3) Thn w dtmnd mn totl nputs 2 2 vncs, : x x h, h h u K K I g h u0 Km Km hx h EL hx h EL x 0 K m K m x 0 K m K m 2 0 m m,0 m V 0, mn xtnl nputs h, h, nd xtnl nput x. (4) x

29 2 2 lthough hx, hx, x, x do not ptcpt dctly n ou nly, w us thm to chck whth ths vlus could b chvd und lstc condtons. In od to clcult th xtnl osclltoy nputs, w usd th l-vlud mpltuds of th focd osclltons dlvd to th xcttoy nd nhbtoy popultons, spctvly, s wll s th phs lg btwn thm, nd constuctd th cospondng complx-vlud mpltuds s follows:. (5) xp( ) Thn w clcultd th complx mpltuds h, h of th osclltoy xtnl nputs usng th xpon (21). PPENDIX B: MODEL WITH VERY SLOW DYNMICS Fo th Modl 2, w obsvd dvton btwn th pdcton nd th smulton sults (s Fgu 4). In od to confm tht ths dvton s du to fnt to btwn th slow nd th fst tm scls, w consdd modfd vson of Modl 2, n whch th tm constnt of dynmcs ws md vy lg. Th slow synptc wghts w ppoptly scld, so th pdctd ffct of th xtnl focng ws th sm s fo th Modl 2. In summy, w followng pmts dffd fom th Modl 2: 1600 ms μ cm μ cm Du to vy slow dynmcs, w smultd th modfd modl fo long pod of tm T 120s. Th sult s psntd n th Fgu 9. On cn s tht th pdcton o s much smll thn n th Fgu 4 (fo th Modl 2), so th pdcton s lmost pfct., FIG. 9. Rsults of smulton of modl nlogous to Modl 2, but wth vy slow tm constnt.