Mixed Mode Oscillations as a Mechanism for Pseudo-Plateau Bursting

Transcription

1 Mixd Mod Oscillatios as a Mchaism for Psudo-Platau Burstig Richard Brtram Dpartmt of Mathmatics Florida Stat Uivrsity Tallahass, FL

2 Collaborators ad Support Thodor Vo Marti Wchslbrgr Joël Tabak Uivrsity of Sydy Florida Stat Uivrsity Full articl: Vo t al., J. Comput. Nurosci., 2010 Supportd by NIH grat DA ad NSF grat DMS

3 Ctral Aim Udrstad th dyamics of a ovl form of burstig i a modl for xcitabl docri clls

4 Psudo-Platau Burstig Oft Occurs i Edocri Clls Prforatd patch rcordig from a GH4 pituitary cll li Simulatio from a mathmatical modl (Toporikova t al., 2008)

5 Thr ar Svral Modls That Produc Psudo-Platau Burstig Lactotrophs: Tabak t al., J. Comput. Nurosci., 2007 Toporikova t al., Nural Computatio, 2008 Somatotrophs: Tsava-Ataasova t al., J. Nurophysiol., 2007 Corticotrophs: LBau t al., J. Thortical Biology, 1998 Short t al., J. Thortical Biology, 2000 Sigl β-clls: Zhag t al., Biophysical J., 2003

6 Typical Psudo-Platau Fast- Subsystm Bifurcatio Structur V (mv) [Ca] (μm) Ulik squar-wav burstig (platau burstig), th top brach of th z-curv is stabilizd; thr is o stabl spikig brach.

7 O Form of Psudo-Platau Burstig has a Diffrt Structur V (mv) [Ca] (μm) What is th mchaism of this burstig oscillatio?

8 Th Mathmatical Modl Calcium coctratio has b rmovd; ot cssary for th burstig. C dv dt = g Ca m ( V )( V V ) + g ( V V ) Ca K K + g A a ( V ) ( V K V ) + g L ( V K V ) τ d dt = ( V ) d τ = dt ( V ) Th ad variabls chag o a slowr tim scal tha V. Thr ar 2 slow variabls ad 1 fast variabl.

9 Burstig Occurs Ovr a Rag of g K Valus burstig spikig V (mv) PD 1 HP 1 PD 2 SN 1 HM SN g K (S) Black = statioary Rd = priodic (spikig)

10 Burstig Occurs i a Rgio of th g K - g A Paramtr Spac g A (S) dp/hyp dp SN 1 hyp HP 1 /PD 1 PD 2 burstig spikig g k (S) Paramtrs ar th maximum coductacs corrspodig to th two slow variabls.

11 Mchaism of Burstig: Go to th Sigular Limit ( C 0) Paramtrs st to produc spikig. Th spikig solutio bcoms a rlaxatio oscillatio o th critical maifold; this is th quasi-quilibrium surfac for th V variabl.

12 Th Rducd ad Dsigularizd Systms ) ( ),, ( L A K Ca I I I I V f RHS of V-ODE: { } 0 ),, ( : ),, ( 3 = R V f V S Critical maifold: 0 ),, ( dt d V f dt d = Dyamics o S: Dsigularizd systm: ),, ( V F f f d dv + = τ τ τ V f d d = τ τ t V f 1 τ with Rducd systm: f f dt dv V f + = τ τ = V dt d τ ) ( ), ( V = with =0 o folds Goal: Driv quatios for th flow o th critical maifold.

13 Foldd Sigularity A foldd sigularity of th rducd systm is a quilibrium of th dsigularizd systm that occurs o a fold curv, ad satisfis f ( V,, ) = 0 o th critical maifold F( V,, ) = 0 V tim drivativ is 0 i dsigularizd systm f V = 0 o a fold curv of S

14 Foldd Nod Sigularity A foldd od sigularity (FN) is a foldd sigularity with gativ ral igvalus. For small valus of C (larg, but ot ifiit, tim scal sparatio) th slow maifold is twistd i th ighborhood of th FN. From Dsrochs t al., Chaos, v. 18, 2008

15 Th Sigular Ful Th sigular ful of a foldd od is dlimitd by th fold curv (L + ) ad th strog sigular caard (SC). This is th trajctory that is tagt to th igdirctio of th strog igvalu of th FN. SC FN ful L + I th sigular limit, trajctoris trig th ful mov through th FN i fiit tim ad follow th middl sht of S for som tim. For small, but o-zro C, corrspodig trajctoris xhibit small oscillatios du to th twistd slow maifold.

16 Rlaxatio Oscillatios (Spikig) do ot Etr th Sigular Ful g A =0.2 S ful

17 Mixd-Mod Oscillatios (Burstig) Etr th Sigular Ful ful g A =4 S For small C, th trajctory oscillats oc it jumps up. Th small oscillatios i combiatio with th larg jumps is a mixd-mod oscillatio. Th small oscillatios ar th spiks of th psudo-platau burst.

18 μ = 0 Caard Thory Provids th Burstig Bordrs μ = 0 FN bcoms a foldd saddl, oly o caard For C>0 this is a Hopf bifurcatio μ is igvalu ratio λ μ = 1 λ 2 ( 0,1) =1 μ Dgrat FN. To th right it bcoms a foldd focus, with o caards. δ is th distac of th sigular orbit from th SC, withi th sigular ful. δ>0 i th MMO rgio.

19 Spiks Emrg as C is Icrasd Du to jump up Du to th FN

20 Maximum Numbr of Small Oscillatios For C sufficitly small, is Giv by a Formula s max = μ + 1 2μ whr [ ] is th gratst itgr fuctio. For drivatio s Wchslbrgr, SIAM J. Dy. Syst., This is issitiv to chags i g A. Blu: s max =1 Gr: s max =2 Yllow: s max =5 Rd: s max =12

21 Actual Numbr of Small Oscillatios Dtrmid by Whr th Sigular Trajctory Etrs th Ful δ For C small ad S max =3: SC S=1 S=2 scodary caards FN L + S=3 Numbr of small oscillatios icrass with δ, th distac from th SC.

22 Numrical Simulatio Agrs with th Caard Thory Numbr of spiks pr burst, with C=2 pf. s max δ Small, dark blu = spikig Small, light blu = burst with 2 spiks Larg, rd = burst with 53 spiks

23 Thak You! (Postdoc positio availabl)