The impact of complex network structure on seizure activity induced by depolarization block

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1 Th mpact of complx ntwork structur on szur actvty nducd by dpolarzaton block Duan Nykamp School of Mathmatcs Unvrsty of Mnnsota Chrstophr Km Laboratory of Bologcal Modlng NDDK/Natonal nsttuts of Halth

2 Msoscopc nuronal ntworks 10 5 nurons pr mm 3 n prmat cortx. Communcat wth spks; connctd va synapss. What dtrmns th nsmbl actvty of nural ntworks? du dt = F (u ) + j xc W j φ(u j ) k nh Sngl nuron dynamcs: F ( ) Connctvty structur: W j W k φ(u k ), xctatory and nhbtory nurons: +/ = 1, 2,..., N Dvlop rducd qn that capturs populaton actvty whn nhbtory nurons ar pathologcal (F ) scond ordr connctvty structur (W j ).

3 Spkng nuron modl Morrs-Lcar modl: (1) smpl, ralstc on channl dynamcs, (2) ntrs dpolarzaton block whn strong stmulus appld. C dv dt = xt g Na m(v )(V Na ) g K w(t)(v K ) }{{}}{{} upstrok downstrok dw = φ w nf (V ) w dt τ w m(v ) = 1 ( ( )) V V1 1 + tanh 2 V 2 w nf (V ) = 1 ( ( )) V V3 1 + tanh 2 V 4 g Cl (V Cl ) }{{} lak

4 Dpolarzaton block Morrs-Lcar plptc bran slc Goal: nvstgat pathologcal xctatorynhbtory ntwork dynamcs whn nh. nurons brak down du to dpolarzaton block. Zburkus t al. 06

5 Outln 1. ntwork dynamcs nducd by dpolarzaton block Smulat rcurrntly connctd ntworks Man fld modl Comparson of smulatons and man fld analyss 2. Dpolzaton block n complx ntwork structur Scond ordr ntwork motfs ffcts of th convrgnt and chan motfs Gnralzd man fld modl

6 Smulat rcurrntly connctd ntworks Randomly connct N xctatory and N/4 nhbtory ML nurons (N = 3000) wth conncton probablty p = For ach nuron, smulat dv dt dg xc τ xc dt dg nh τ nh dt = F (V, w ) + g xc (t)( xc V ) + g nh (t)( nh V ) = g xc + J j W j δ(t tj k ) = g nh j xc + J j W j j nh k k δ(t t k j ) Rcord th avrag frng rat of nurons n xc and nh populatons. Compar wth man fld analyss.

7 Man fld:. Populaton rspons Add hgh suscptblty to dpolarzaton block to nhbtory nurons Stmulat unconnctd ML nurons (N = 1000) wth random spks xctatory nurons: monotonc rspons to stmulus nhbtory nurons: non-monotonc rspons to stmulus Us ths faturs to dvlop a man fld modl that capturs larg-scal smulaton rsults.

8 Man fld:. Modfy Wlson-Cowan q Can w com up wth a rducd modl that capturs larg-scal smulatons? Modfy Wlson-Cowan qn: dr dt τ dr dt = r + φ (J r J r + ) = r + φ (J r J r + ), r / : actvty of xc/nh populatons. φ / : transfr functon convrts nput to output Mak nhbtory transfr functon, φ, non-monotonc to captur DB. φ (x) = x, φ (x) = x k(x θ).

9 Compar man-fld qn and smulatons Can ad hoc man-fld modl xplan ntwork smulatons? Man-fld prdcts that ntwork stat can b bstabl. Normal stat (b 1 ) and szur stat (b 2 ) coxst. Aftr brf xtrnal stmulus, nh nurons ntr dpolarzaton block and xc nurons fr at max rat. Ths corrsponds to szur stat (b 2 ).

10 Non-oscllatory transton Phas plan ncras xt Bfurcaton dagram Dcras xt Ntwork smulatons

11 Oscllatory transton - couplng ncrasd. ncras xt Dcras xt

12 Dynamcs around Bogdanov-Takns HB HC homoclnc bfurcaton J osc s z u r BT saddl nod bfurcaton normal SN xtrnal nput

13 Conclusons, frst half 1 Man fld modl capturs dffrnt typs of transtons to szur-lk actvty that ar sn n ntwork modl. oscllatory, non-oscllatory, and tonc-clonc transtons

14 Outln 1. ntwork dynamcs nducd by dpolarzaton block Smulat rcurrntly connctd ntworks Man fld modl Comparson of smulatons and man fld analyss 2. Dpolzaton block n complx ntwork structur Scond ordr ntwork motfs ffcts of th convrgnt and chan motfs Gnralzd man fld modl

15 Scond ordr ntworks (SONTs) W j = 1 dnots a conncton from nod j to. ls W j = 0. P(W j = 1) = p. For an rdős-rény graph, all dgs ar ndpndnt: P(W j = 1, W kl = 1) = p 2 Scond ordr statstcs of W j ar trval. Go byond -R: (1) fx conncton probablty p (2) gnrat W j ; scond ord stat quals to prscrbd α s. [Zhao t al 11] α rcp = cov(w j,w j ) α p 2 conv = cov(w j,w k ) p 2 α dv = cov(w j,w kj ) α p 2 chan = cov(w j,w jk ) p 2

16 Proprts of scond ordr ntworks: rdős-rény Th rdős-rény random ntwork N = 20, p = 0.3, α rcp = 0.1, α conv = 0, α dv = 0, α chan = 0 probablty xpctd ncomng dgr dstrbuton ncomng dgr probablty xpctd outgong dgr dstrbuton outgong dgr N = 3000, p = 0.01, α rcp = 0, α conv = 0, α dv = 0, α chan = 0

17 Proprts of scond ordr ntworks: rcprocal Add rcprocal connctons: N = 20, p = 0.3, α rcp = 2.0, α conv = 0, α dv = 0, α chan = 0 probablty xpctd ncomng dgr dstrbuton ncomng dgr probablty xpctd outgong dgr dstrbuton outgong dgr N = 3000, p = 0.01, α rcp = 3, α conv = 0, α dv = 0, α chan = 0

18 Proprts of scond ordr ntworks: convrgnt Add convrgnt connctons: N = 20, p = 0.3, α rcp = 0.1, α conv = 0.5, α dv = 0, α chan = 0 probablty xpctd ncomng dgr dstrbuton ncomng dgr probablty xpctd outgong dgr dstrbuton outgong dgr N = 3000, p = 0.01, α rcp = 0, α conv = 3, α dv = 0, α chan = 0

19 Proprts of scond ordr ntworks: dvrgnt Add dvrgnt connctons: N = 20, p = 0.3, α rcp = 0.1, α conv = 0, α dv = 0.5, α chan = 0 probablty xpctd ncomng dgr dstrbuton ncomng dgr probablty xpctd outgong dgr dstrbuton outgong dgr N = 3000, p = 0.01, α rcp = 0, α conv = 0, α dv = 3, α chan = 0

20 Proprts of scond ordr ntworks: no chans Add convrgnt and dvrgnt connctons, rduc chans: N = 20, p = 0.4, α rcp = 0.9, α conv = 0.3, α dv = 0.4, α chan = 0.3 probablty xpctd ncomng dgr dstrbuton ncomng dgr probablty xpctd outgong dgr dstrbuton outgong dgr N = 3000, p = 0.01, α rcp = 0, α conv = 3, α dv = 3, α chan = 0.9

21 Proprts of scond ordr ntworks: chans Add convrgnt and dvrgnt connctons wth chans: N = 20, p = 0.3, α rcp = 1.0, α conv = 0.3, α dv = 0.3, α chan = 0.3 probablty xpctd ncomng dgr dstrbuton ncomng dgr probablty xpctd outgong dgr dstrbuton outgong dgr N = 3000, p = 0.01, α rcp = 0, α conv = 3, α dv = 3, α chan = 3

22 Prvous rsults Chan motfs Lnar thory works! Whn on lnar rgm of transfr functon, chans hav strongst nflunc on synchrony [Zhao t al 11, Nykamp t al 16]. ncras spk-tm corrlatons (lnar rspons thory) [Hu t al 13]. Convrgnt motfs Rduc ntwork synchrony du to htrognty. [Zhao t al 11] Promot synchrony n low actvty rgm. [Roxn 11] Nonlnarty of transfr functon mattrs!

23 SONTs and dgrs n-dgr d n, out-dgr d out normalzd n- and out-dgr: x = d n (d n ), y = d out (d out ) d out =5 d n =4 SONT statstcs ar varancs of th dgr dstrbuton: α conv var(x) α dv var(y) α chan cov(x, y) α conv > 0 α dv > 0 α chan < 0 α chan > 0

24 A rat quaton Gvn a postsynaptc nuron wth n-dgr x a prsynaptc nuron wth out-dgr ỹ conncton probablty s proportonal to xỹ. d n =4 d out =5? Lt r(x, y, t) b frng rat of nuron wth dgr (x, y). Gt rat quaton nonlnarty couplng strngth { { prsynaptc frng rat dgr dstrbuton nput Couplng strngth from prsynaptc dgr ( x, ỹ) onto postsynaptc dgr (x, y) s Jxỹ.

25 Gnralzd quaton - Drvaton Th dynamcs of sngl nuron frng rat s gvn by ( ) dr (x, y, t) + r(x, y, t) = φ Jxỹρ( x, ỹ)r( x, ỹ, t)d xdỹ + dt Dfn populaton synaptc actvty h Thn, S(t) = yr(x, y, t)ρ(x, y)dxdy dr dt + r = φ (JxS(t) + ) Multply by yρ(x, y) and ntgrat to obtan gnralzd qn ds(t) dt + S(t) = yρ(x, y)φ(jxs(t) + )dxdy.

26 Lnar trm dpnds on chan motf ds(t) dt + S(t) = yρ(x, y)φ(jxs(t) + )dxdy Lnar trm: covaranc btwn n and out dgr (α chan ) JSφ xyρ(x, y)dxdy = J(1 + α chan )Sφ Rsult: robust dpndnc on α chan ffct of convrgnc α conv n hghr-ordr trms 0.2 hghr-ordr drvatvs of φ nonlnarty shap mattrs Synchrony ˆα chan ˆαconv

27 Gnralzd quaton - ntwork f n-dgr dpnds on out-dgr (α chan 0), th gnralzd qn has four synaptc varabls, S, S, S, S, rprsntng ach dg. Consdr rducd qns: assum n-dgr s ndpndnt of out-dgr (α chan = 0). Th populaton rat actvty S k (t) = ρ(x k )r(x k, t)dx k satsfs ds + S = ρ (x )φ (J x S J x S + )dx dt ds + S = ρ (x dt )φ (J x S J x S + )dx. whr ρ( ) s n-dgr dstrbuton. [Nykamp t al 16; Roxn 11]

28 Convrgnt motfs n ntwork Consdr convrgnt motfs across th ntwork. R α conv, > 0 Non-monotonc rspons of nh nurons lads to strong dpndnc on α conv,. R α conv, > 0 Htrognty of nh synapss dsrupts synchrony n hgh actvty rgm.

29 Smulatons: facltats szur onst Nar szur, nh nurons wth hgh xc n-dgr ntr DB, low xc n-dgr hav wak nh actvty Ovrall nh actvty rducd. α conv, α conv, > 0 facltats transton to szur stat.

30 Smulatons: favors non-osc transton Htrognous nh synaptc nputs dsrupt oscllatons at hgh actvty rgm. R α conv, > 0 α conv, supprsss oscllatons nar th transton. non-osc oscllatory

31 Global bfurcaton, Saddl-nod and homoclnc bfurcatons occur at rducd

32 Global bfurcaton, Oscllatory rgm pushd away nar th transton.

33 Analytcal rsults Assum (1) th - couplng s strong (J J > J J ) and (2) Φ, Φ, Φ, Φ < 0 at hgh actvty rgm. α, conv facltats a saddl-nod and a Hopf bfurcaton. (Rd < 0, Blu > 0) [ 1 dtl α conv = dtl 0 + α conv,, τ J Φ J 2 S Φ J2 J SΦ ] 2 J2 S2 Φ Φ ( J J + J J ) α conv, trl α conv, = trl 0 + α conv, 1 2τ J2 J S( Φ 2 ). favors non-oscllatory transton to szur. trl α conv, = trl 0 + α conv, 1 2 J J 2 S2 Φ.

34 Conclusons 1 Man fld modl capturs dffrnt typs of transtons to szur-lk actvty that ar sn n ntwork modl. oscllatory, non-oscllatory, and tonc-clonc transtons 2 nflunc of ntwork structur on ths transtons Nonlnarty of transfr functon lads to strong dpndnc on convrgnt motfs Htrognous synaptc nputs can supprss synchrony. Gnralzd quaton can captur such ffcts.

35 Acknowldgmnts Natonal nsttuts of Halth Chrstophr Km Carson Chow Brnstn Cntr Frburg Arvnd Kumar Ulrch grt Fundng Sourc

36 Tonc-clonc transton Transton occurs whn k rcovrd toward physo valu. Smulatons Jnsn, Yaar 97 TONC (patho K ) CLONC (physo K ) r r

37 Chans can modulat th ffct of Corrlat nh nuron s outdgr wth ts xc n-dgr. α chan > 0 α chan < 0 Corrlatd out-dgr can amplfy/supprss ffcts of α conv,.

38 Chans can modulat th ffct of Corrlat xc nuron s outdgr wth ts nh n-dgr. α chan > 0 α chan < 0 Corrlatd out-dgr acclrats/dlays szur onst.