Course Notes: Nonlinear Dynamics and Hodgkin-Huxley Equations

Transcription

1 Couse Notes: Noliea Dyamics ad Hodgki-Huxley Equatios Readig: Hille (3 d ed.), chapts 2,3; Koch ad Segev (2 d ed.), chapt 7 (by Rizel ad Emetout). Fo uthe eadig, S.H. Stogatz, Noliea Dyamics ad Chaos, Addiso Wesley, 994. I these lectues itoductoy ideas o oliea dyamics ae used to illustate the popeties o membae models based o the Hodgki Huxley (HH) equatios. The idea o educig a system to secod ode is itoduced ad applied to geeate models that ca be aalyzed i the phase plae. Hodgki-Huxley modelig iside The basic membae cicuit is daw i Fig.. This cicuit is appopiate o simple membae systems like the squid giat axo o othe axoal membaes, whee oly V C two voltage-depedet chaels ae see. I + the model thee is a capacito C, to epeset the membae capacitace, a sodium coductace G Na, potassium coductace G K, outside ad a leakage coductace G L. The membae potetial V is the potetial iside the cell mius the potetial outside ad thee ca be a cuet I ext ijected ito the cell om a electode o om othe pats o the cell. Cosistet with the usual covetio, cuets ae positive i the outwad diectio. G Na G K G L + + E Na E K E L Fig. Typical membae cicuit cotaiig active Na ad K chaels, a leakage chael ad a membae The model i Fig. epesets a patch o membae, i.e. a small aea o membae which is isopotetial, meaig that the membae potetial V is costat acoss the patch. I the last pat o the couse, we will coside moe geeal models i which the voltage vaies alog the extet o the cell. Patch models ae appopiate o systems cosistig o a lage umbe o chaels elatively evely dispesed acoss the patch. Moe speciically, two thigs ae assumed: ) the umbe o chaels is lage eough that idividual gatig evets ae aveaged out ad the Na ad K cuets ae smooth populatio cuets (see Figs. 3.6 ad 3.7 i Hille); ad 2) the chaels ae ot aaged i ay way which allows special local iteactios amog small umbes o chaels. Such iteactios most equetly occu betwee calcium chaels ad othe chael types though local calcium pools. I this model, the chaels iteact oly though V. The equatios descibig the patch i Fig. wee ist developed by Hodgki ad Huxley (HH, 952). Cosevatio o cuet at the ode o the iside o the cell gives: I ext C dv dt = Iext GNa( V ENa) GK ( V EK ) GL( V EL) ()

2 2 That is, the cuet though the capacitace is the exteally applied cuet mius the cuets though the io chaels. C ad G L ae costats. G Na ad G K ae uctios o membae potetial ad time, give the ollowig HH equatios: 3 dm m ( V) m GNa = GNam h = dt τ m( V) dh h ( V) h = dt τ h( V) 4 d ( V) GK = GK = dt τ ( V) (2) whee G Na ad G K ae costats. Eqs. 2 wee developed by HH to model the gatig behavio o the sodium ad potassium chaels. They deped o the uctios x (V) ad τ x (V) (o x = m, h, o ) which ae uctios o membae potetial oly. The x (V) uctios ae the steady state values o m, h, ad at a paticula membae potetial ad the τ x (V) uctios ae the time costats with which m, h, ad chage i esposes to chages i V. Figue 2 shows plots o these uctios o the oigial HH model. The ull equatios o the HH model ae give i the appedix. Fig. 2 Paametes o the Hodgki-Huxley model o squid giat axo membae. m 3 h ad 4 ca be itepeted as the pobability that a chael s gate is ope. The Na chael has two sets o gates, activatio gates, epeseted by m, ad iactivatio gates, epeseted by h. The activatio gates ope ad the iactivatio gates close whe the membae depolaizes. I Fig. 2, this is epeseted by the act that m iceases ad h deceases with V. I the model, the Na chael is assumed to have 3 m gates ad h gate; because all ou gates must be ope o the chael to be ope, the pobability that a chael is ope is m 3 h. The K chael has oly a sigle activatio vaiable ad thee ae ou o them, leadig to the 4 expessio. Figue 3 shows the evolutio o the paametes o the HH model duig a actio potetial. Explaatios o the actio potetial based o this model ae give i the eeeces (Hille, chapte 2 ad Johsto ad Wu, chapte 6) ad will ot be epeated hee. Istead, these otes ocus o developig a udestadig o the popeties o oliea systems like the HH equatios usig the phase-plae appoach om oliea dyamics. Questio. Assume that the membae potetial has bee ixed at V o a log time. What is the steady state value o m? At time, the membae potetial is stepped (voltage-clamped) to V.

3 3 Solve Eq. 2 o m(t) o t>; expess the solutio i tems o m ( V) ad τ m( V). Fom this solutio, you should see that the behavio o the dx/dt equatios is to cause the HH vaiables m,, ad h to tack the x uctios as they vay with membae potetial. Would you solutio method be valid i V wee vayig i time, as i the actio potetial? Membae pot., mv HH vaiables m Millisecods µa/cm 2 cuet stim. h Coductace, ms/cm 2 Cuet, ma/cm I Cap G Na I Leak I K G K I Na Millisecods Fig. 3 Time couse o the impotat vaiables i the HH model o eve actio potetials duig a actio potetial. Show ae the membae potetial (uppe let), the HH vaiables (lowe let), the coductaces (uppe ight), ad the membae cuets (lowe ight). Questio 2. Thee is a otch i the sodium cuet (I Na ) i the lowe ight plot o Fig. 3. It occus about the time the sodium coductace ( G Na ) eaches its peak. Explai the oigi o the otch. Similaly, the shape o the potassium cuet waveom (I K ) is dieet om the potassium coductace waveom (G K ). Explai why. Questio 3. HH cosideed m, h, ad to speciy the actio o gatig elemets that wee ope i the two chael types. Each gatig elemet could udego tasitios om closed to ope accodig to the ollowig ist-ode kietic scheme C α x (V) β x (V) O (3) whee C ad O ae closed ad ope states ad α ad β ae voltage-depedet ate costats o the opeig ad closig tasitios, espectively. Deive expessios o x (V) ad τ x (V) i tems o α x (V) ad β x (V). The HH model is speciied i tems o the ate costats α x (V) ad β x (V) i the appedix. Questio 4. Takig the ou-gate model o the Na chael seiously, gives 2 4 o 6 possible states o the Na chael (e.g. CCC-C, CCC-O, CCO-C, etc. whee the ist thee symbols, to the let o the dash, epeset the states o the m gates ad the outh symbol, to the ight o the dash,

4 4 epesets the h gate). Wite out all sixtee states ad daw aows betwee them to epeset the elemetay tasitios like Eq. 3. Assume that oly oe o the ou gates chages i each tasitio, i.e. the likelihood o two gates switchig simultaeously is. May o these states ae equivalet (e.g. OCC-C, COC-C, ad CCO-C). Collapse you state diagam ad show that it is equivalet to (4) below. Also show that this state model is equivalet to the HH model o the Na chael i Eq. 2. I3 3α m (V) I2 2α m (V) I α m (V) I β m (V) 2β m (V) 3β m (V) β h (V) α h (V) β h (V) α h (V) β h (V) α h (V) β h (V) α h (V) (4) 3α m (V) 2α m (V) α m (V) C3 C2 C O β m (V) 2β m (V) 3β m (V) HH models have ow bee deived o a lage umbe o chaels (see Yamada et al., 998 o examples). Geeally the om o the equatios is simila to those o Eq. 2. Howeve, as the detailed study o vaious chael systems has evolved, some o the eatues o the HH model have bee chaged, o both the squid axo (e.g. Clay, 998) ad o othe systems (Hille, chaptes 8 ad 9).. Fo a iceasig umbe o chaels, the kietic scheme o Eq. 4 has bee oud to be iaccuate. Moe complex chael state models ae eeded ad these ca ow be deived om sigle chael ecodigs. This meas that explicit dieetial equatios o the state model have to be witte ad these usually caot be collapsed ito the compact om o the HH equatios (Eq. 2). This chage i the model geeally elects subtle chages i the gatig behavio o chaels. 2. The HH model assumes that the istataeous cuet-voltage elatioship o a chael is liea, i.e. that I x =G x (V,t). (V-E x ). As discussed i the otes o chael pemeatio models, the cuet-voltage elatioship does ot ecessaily take this liea om. Usually a moe accuate istataeous cuet voltage model is the Goldma-Hodgki-Katz costat ield equatio: zfv/ RT [ outside] I m h V C e i p q iside C i = ( cost.) zfv RT e i / whee m ad h ae HH vaiables as i Eq. 2 ad Eq. 5 is substituted o the liea cuet voltage elatioships i Eq.. Equatio 5 is most commoly used i modelig Ca ++ chaels, because the lage cocetatio atio C out /C i leads to stog ectiicatio. I these otes, the liea om o the equatio, i.e. Eq., will be used o simplicity o aalysis. State vaiable models ad educed-state models: the Mois-Leca Equatios The state vaiables o the HH model ae V, m, h, ad. These om the state vecto = [ V, m, h, ] T ad the dieetial equatios (Eqs. ad 2) ca be witte as ollows: (5)

5 5 V V ( V, m, h, ) V ( ) m m( V, m, h, ) m( ) = = F ( ) = = h h( V, m, h, ) h( ) ( V, m, h, ) ( ) [ ] 3 4 whee, o example, ( V, m, h, ) = I G m h( V E ) G ( V E ) G ( V E ) C ad [ ] m( V, m, h, ) = m ( V) m τ m( V). V ext Na Na K K L L Equatio 6 is a ou-dimesioal oliea system. The methods o aalysis that ae applied below wok best whe the system is two-dimesioal, i.e. with oly two state vaiables. This meas elimiatig two dieetial equatios om the HH system. Geeally, this is doe by takig advatage o the act that the time costats i the ou dieetial equatios that make up Eq. 6 ae quite dieet. Fo example, the time costats o the HH gatig vaiables ea the estig potetial ae τ m.4 ms, τ h 8 ms, ad τ 5 ms (Fig. 2). The time costat o the membae potetial equatio vaies ove a age slightly lage tha τ m. Thus V ad m ae expected to luctuate aste tha the othe two state vaiables. The dieeces i time scale o the vaiables ca be see i the let had plots i Fig. 3, whee V ad m chage apidly i espose to the depolaizatio ad ad h chage slowly. A appoximate system ca be deied which has state vaiables V ad m with = (V ) ad h=h (V ), i.e. ad h ae costat at thei iitial values. Such a system might be valid o the ist ms o two ate a sudde depolaizig cuet is applied to a cell sittig at est. O couse, as soo as ad h begi to chage, this system becomes ivalid. Results usig this method to aalyze the theshold o the HH model ae descibed late. Aothe way i which the HH equatios might be educed is to assume that τ m is so ast that m.8 m(t) chages essetially istataeously to ollow m (V), i.e. that m(t)=m (V(t)). This elimiates the dieetial equatio o dm/dt. The eithe o h ca be.4 m assumed to be slow ad set equal to x (V ) as above (V(t)) to get dow to two equatios. These assumptios poduce a appoximate system which is valid o Millisecods loge times tha the m-v system ad ca eve Fig. 4 Compaiso o m(t) (solid lie) duig a poduce actio potetials. Two examples o this appoach ae discussed below. The extet to which the assumptio about m is tue o the HH equatios ca be judged om Fig. 4 which compaes the actual tajectoy o m duig a actio potetial (solid lie) with the tajectoy o m (dashed lie). (6)

6 6 As expected om Eq. 2, m leads m. The appoximatio o holdig eithe h o costat ca be evaluated om the lowe let plot i Fig. 3. Fo the iitial discussio o phase-plae aalysis, a two-dimesioal set o equatios simila to the educed HH equatios will be used. These ae the Mois-Leca equatios (MLE) i the om used by Rizel ad Emetout (998). The membae model o Fig. is still appopiate, except that the model cotais a calcium chael istead o the sodium chael. The calcium activatio vaiable, m, is assumed to have a ast time costat, so that m=m (V) ad the calcium coductace is GCa = GCa m ( V). Thee is o calcium iactivatio vaiable, which is equivalet to assumig that h is costat. The potassium chael has a sigle activatio vaiable w, aalogous to, so the state vaiable equatios ae as ollows: C dv dt dw dt = I G m ( V)( V E ) G w ( V E ) G ( V E ) ext Ca Ca K K L L w ( V) w = φ τ ( V) w (7) whee m ( V) = 5. + tah V V V w ( V) = 5. + tah V V V [ (( ) 2) ] [ (( 3) 4) ] τ w( V ) = cosh( ( V V3) V4) (8) The uctioal oms chose o m, w, ad τ w have the same geeal shape as the aalogous paametes o the HH model ad have the advatage that they ae easy to compute. The paamete φ i Eq. 7 is a tempeatue acto, which ca be used to chage the elative time costats o V ad w. Expeimetally, the time costats o chael gatig (dw/dt) ae moe sesitive to chages i tempeatue tha the time costat o dv/dt. The model will be used with two sets o paametes: Paamete set # G Ca =4.4 G K =8. G L =2. C=2 E Ca =2 E K =-84 E L =-6 φ=.4 (9) V =-.2 V 2 =8 V 3 =2 V 4 =3 Paamete set #2 G Ca =4. G K =8. G L =2. C=2 E Ca =2 E K =-84 E L =-6 φ=.667 () V =-.2 V 2 =8 V 3 =2 V 4 =7.4 Uits ae mv o potetials, ms/cm 2 o coductace, µa/cm 2 o cuet, ad µfd/cm 2 o capacitace. I ext will be vaied to simulate cuet iput to the cell.

7 7.6 dw/dt= w, potassium activatio.4.2 dv/dt= dv/dt= x x x V, membae potetial, mv) Fig. 5. Phase plae o the MLE with paamete set #. The gee dashed lies ae the ullclies dw/dt= ad dv/dt=. The blue aows show the diectios ad low velocities o tajectoies at the poits coespodig to the tails o the aows. The mageta solid lies ae example tajectoies o V =-2 mv, -4 mv, -3.9 mv, ad mv, espectively om let to ight. The coespodig V vesus time plots ae show i Fig. 6. The phase plae Because Eqs ad 2 ad Eqs 7 ad 8 ae oliea, it is ot possible to obtai solutios except by umeical simulatio. Howeve, cosideable isight ito the atue o the solutios ca be gotte usig simulatio ad phase plae aalysis. A phase plae, o a ode 2 system like Eq. 7, is a Catesia plot with the state vaiables o the axes. Figue 5 shows a example o the MLE with paamete set #. Thee ae seveal eatues i this plot, which will be descibed oe by oe. At each poit (V, w) i the plae, the velocity vecto (dv/dt, dw/dt) ca be computed om Eq. 7. The blue aows show these velocity vectos o a evely spaced gid i the plae. The vectos poit i the diectio that the solutios to Eq. 7 will move i the system s state vecto is located at the tail o the aow; the legth is the speed o movemet. A geat deal about the atue o the system s behavio ca be ieed om the aows. Ideed, the pocess o simulatio to obtai solutios is basically computig these aows o a suicietly dese gid ad the ollowig the aows.

8 8 The solid mageta lies show actual tajectoies o the system om ou iitial poits (the x s). That is, Eqs. 7 ad 8 wee solved umeically with the iitial coditios V()=-2, -4, -3.9, ad mv ad w()=.495 (its value at the estig poit). Plots o the esultig time uctios V(t) ad w(t) ae show i Fig. 6 o two o the iitial coditios (-4 ad -3.9 mv). The mageta tajectoies i Fig. 5 ae plots o w(t) vesus V(t) with time t as a paamete. Time is ot explicitly maked o the tajectoies ad thee is o ecessay elatioship betwee distace alog the tajectoies ad the passage o time. Notice i Fig. 2 that the tajectoies ollow the diectios pedicted by the blue aows. All ou tajectoies ed up at the gee dot (V=-6.2 mv, w=.49), which is the estig state o the model. V, mv 4-4 V() = -4 mv V w V() = -3.9 mv V w.4.2 w, potassium activatio Time, ms Time, ms. Fig. 6. Plots o membae potetial V (blue, let odiates) ad potassium activatio w (geee, ight odiates) vesus time o two o the tajectoies i Fig. 5 (the two begiig at the middle x ea 4 mv). The iitial voltage value is give i the leged. The dashed lies show the values o V (blue) ad w (gee) at the estig state Fo iitial values o V o 2 ad 4 mv, the membae potetial decays diectly back to the estig value. This is illustated by the solutio i the let plot o Fig. 6, o 4 mv iitial coditio. I the phase plae, the tajectoies move to the let, back to the estig state (gee dot). Fo the lage iitial values (-3.9 ad mv), the system poduces a actio potetial, see i the ight plot i Fig. 6. I that case, the tajectoy i Fig. 5 moves to the ight, evetually etuig to the est state ate ciclig the phase plae. Make sue you udestad the elatioship betwee the time plots i Fig. 6 ad the tajectoies i Fig. 5. Questio 5. The tajectoies i Figs. 5 ad 6 wee computed om a iitial value i which the membae potetial was displaced om the est potetial hoizotally, so that w did ot chage om its value at est. Show that this maipulatio is equivalet to begiig the system at its estig potetial ad ijectig a impulse o cuet. That is, show that the ollowig ae equivalet iitial coditios o the MLE (whee (V, w ) is the estig poit): V() = V, w() = w ad Iext() t = qδ() t vesus V() = V + q/ C, w() = w ad I () t = It is wothwhile thikig about how the diectios o low i the phase plae, idicated by the blue aows i Fig. 5, come about. Coside ist the aows ea the V-axis, whee w. Movig ext

9 9 om let to ight alog the V-axis poduces a stog icease i dv/dt, idicated by the iceasig legth o the ight-poitig hoizotal compoet o the aows. This icease esults om the icease i m (V) with V i the secod tem o the.h.s. o the dv/dt potio o Eq. 7 (emembe that (V-E Ca ) is egative), i.e. by the iwad calcium cuet which depolaizes the cell. Movig vetically aywhee i the phase plae, coespodig to iceasig w, yields let-poitig aows, a eect caused by the thid tem o the.h.s. o the dv/dt potio o Eq. 7, coespodig to the outwad potassium cuet which hypepolaizes the cell. The vetical compoets o the aows ca be udestood i tems o the dw/dt potio o Eq. 7 ad the w ullclie, descibed late. A popety o liea systems is that the amplitude o the solutios is liealy elated to the amplitude o the iitial values. I a liea system is stated om two iitial coditios vey ea oe aothe, the solutios will also be vey close togethe. Noliea systems do ot ecessaily behave this way, as illustated by the examples i Figs. 5 ad 6. The MLE s behavio vaies quite stogly with iitial value i the age betwee 3.9 mv ad-4 mv. This stog depedece o iitial coditios is oe o seveal popeties o oliea systems which dieetiate them om liea systems. The behavio o the phase plae is cotolled by the isoclies o ullclies. These ae the locus o poits whee the time deivatives o the state vaiables ae zeo, dv/dt= ad dw/dt= o the MLE. The MLE s ullclies ca be obtaied om Eq. 7 with some algeba: dv dt dw dt Iext GCa m ( V)( V ECa) GL ( V EL) = wv ( V) = G ( V E ) = w ( V) = w ( V) w K K () The ullclies w V (V) ad w w (V) ae plotted i Fig. 5 with dashed gee lies. Note that the ullclie o dv/dt= goes egative betwee 6 ad 2 mv ad is ot plotted. The cuve is cotiuous acoss this age, howeve. The ullclies povide two kids o iomatio about the system:. Because a deivative is zeo o a ullclie, the system is costaied to coss the ullclies movig eithe hoizotally (acoss the dw/dt= ullclie) o vetically (acoss the dv/dt= ullclie). This ca be see i Fig. 5 by the diectios o the aows ad the tajectoies ea the ullclies. 2. The system s velocity chages diectio acoss the ullclies. Thus, below the dv/dt ullclie, the diectio o low alog the V axis is positive, the blue aows i Fig. 5 poit to the ight. Above the dv/dt ullclie, the low is to the let ad the blue aows poit letwad. Similaly, below the dw/dt ullclie, the low is upwad ad above this ullclie, the low is dowwad. Note that the diectios o low ea the ullclies ca be misleadig i oliea systems. Fo example, i Fig. 5, oe might imagie that the dividig lie betwee tajectoies that etu diectly to the est state ad tajectoies that poduce actio potetials would be the dv/dt= ullclie. That is, iitial values to the let (ad above) the ullclie should move to the let towad the est poit ad iitial values to the ight (ad below) the ullclie should move to the ight towad a actio potetial. As the tajectoies i Fig. 5 show, howeve, the actual tasitio om

10 subtheshold behavio to actio potetials occus slightly to the ight o the ullclie at about 4 mv. This occus because the behavio o the system is detemied by the elative size o the vetical ad hoizotal motios; the vetical motio is lage tha the hoizotal motio ea the ullclie, whee hoizotal velocity is. The gee dot at about 6 mv is at the poit whee the two ullclies itesect. At this poit, both time deivatives ae zeo ad so the system should ot move i placed thee; this poit is a est state o the system. A poit whee all time deivatives ae zeo is called a equilibium poit. I the case o eve membae systems, the estig potetial o the euo is a equilibium poit. Fo paametes set #, the equilbium poit o the MLE is globally attactig, meaig all tajectoies i the phase plae lead to it. This ca be see by examiig the lows pedicted by the blue aows. Thee is a existece ad uiqueess theoem o iitial value poblems ivolvig dieetial equatios like Eq. 6 o 7 (Stogatz, 994, p. 49). The theoem says that i the uctios deiig the time deivatives o the state vaiables (the ight had side o Eqs. 6 o 7) ae cotiuous ad have cotiuous deivatives, the a uique solutio exists ove some iite time iteval. The act that the solutio is uique is impotat, because it implies that tajectoies caot coss i the phase plae. I two tajectoies wee to coss at a poit, the that poit would have to be the iitial value o two dieet solutios, i violatio o the uiqueess theoem. Thus we ae guaateed that tajectoies caot coss i the phase plae. Questio 6. Coside the ollowig system: dx dt y dy = x+ e ad = y (2) dt Daw a phase plae o this system, icludig ullclies ad equilibium poits. Daw aows to idicate the geeal diectios o lows (Stogatz, Fig ad 6..4). Limit cycle Aothe kid o behavio emeges o the MLE i exteal cuet is applied. Fig. 7 shows the phase plae ad, i the iset at top ight, the membae potetial vesus time, whe a exteal cuet (I ext ) o 95 µa/cm 2 is applied. The equilibium poit (gee dot) still exists ad is still a est poit o the system, but it is o loge attactig. Oe o the tajectoies show i mageta i Fig. 7 begis vey close to the equilibium poit. As ca be see i the time waveom at uppe ight, the membae potetial oscillates with a iceasig amplitude ad the beaks ito a lage oscillatio which is epeated exactly om cycle to cycle. This oscillatio is called a limit cycle. The coespodig tajectoy i the phase plae spials aoud the equilibium poit with a iceasig amplitude ad the meets the limit cycle, which is the lage closed cuve i the plae. I this case, the limit cycle is the oly stable elemet i the phase plae ad the system will always ed up i exactly the same limit cycle, egadless o the iitial value. As a example, a secod tajectoy is show i Fig. 7 begiig outside the limit cycle; this tajectoy moves towad the limit cycle ad jois it just ate the peak membae potetial.

11 A limit cycle is a secod eatue o oliea systems which is ot obseved i liea systems. It is possible to have a oscillatio like that show i Fig. 7 i a liea system, but the amplitude o the oscillatio is liealy popotioal to the iitial value. I a oliea system the limit cycles have a ixed amplitude, which is a chaacteistic o the system itsel, ad is ot detemied by the iitial values. I a liea system, the oscillatio ca have ay size, depedig o the iitial values. O couse a oliea system ca have moe tha oe limit cycle, but each limit cycle will be a isolated cuve like the oe show i Fig. 7. By a isolated cuve, we mea a closed cuve which is ot immediately adjacet to aothe limit cycle. Istead, tajectoies slightly lage o slightly smalle tha a stable limit cycle will spial ito the limit cycle, as i Fig mv ms dv/dt = w, potassium activatio.4.2 dw/dt = dv/dt = V, membae potetial, mv) Fig. 7 Phase plae o the MLE with a exteal cuet I ext =95 µa/cm 2. Two tajectoies ae show, oe begiig ea the equilibium poit (gee dot at the itesectio o the ullclies) ad the secod begiig at (,.4). The iset at top shows the time waveom o the tajectoy begiig ea the equilibium poit. Lieaizatio aoud equilibium poits A impotat dieece i the behavio o the MLEs betwee Figs. 5 ad 7 is that the equilibium poit i Fig. 5 (with zeo cuet) is attactig, i.e. tajectoies low ito it, wheeas the

12 2 oe i Fig. 7 is ot attactig, tajectoies low away om it. I act, a geat deal ca be leaed about the popeties o oliea systems om thei popeties i the viciity o equilibium poits. The appoach is to compute a liea appoximatio to the oliea system i the viciity o a equilibium poit. I the uctios deiig the time deivatives o the oliea system ae suicietly smooth, the i some small eighbohood o the equilibium poit, the liea appoximatio should behave like the oliea system. Coside the oliea system at ight. is the system s -dimesioal state vecto. Eq. 3 is exactly aalogous to Eqs. 6 o 7, o =4 o =2, espectively. The uctios i ( ) ca be expaded i a Taylo seies aoud ay poit ; the ist tems o such a expasio ae show i Eq. 4. The otatio i meas the i th compoet o. I is a equilibium poit, the i ( ) = o all i, om the deiitio o a equilibium poit, so that Eq 4 simpliies to the liea equatio i Eq. 5. The matix J o patial deivatives is called the Jacobia matix o the system. Note that the Jacobia is a matix o scala costats. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (4) = ( ) ( ) ( ) x 2 2 J (5) ( ) ( ) ( ) = = 2 2 (3)

13 3 [ ] I the ightmost pat o Eq. 5 the vecto x = ( ), ( 2 2),, ( ) is a vecto o small deviatios o the state vecto om the equilibium poit. This is, the lowe case x deotes the value o usig the equilibium poit as oigi. Fially, ote that = x, because the time deivative o the costat tems i x ae zeo. Thus the lieaized system ea the equilibium poit is ẋ = Jx o x = ad Jdeied above Questio 7. Coside a secod ode system ẋ = J x with iitial value x ( ) = x. Show (by substitutio) that the solutios o this equatio take the om λ t t xt ()= Ae λ e + Be 2 e2 (6) whee λ ad λ 2 ae the eigevalues o J ad e e ad 2 ae the eigevectos o J. Explai how to id the scala costats A ad B (hit: o λ λ 2, the eigevectos ae liealy idepedet ad om a basis set o the phase plae R 2 ). Questio 8. Wite the solutios o the system ẋ = J x, x ( ) = x o T J= 2 4 ad 2 4 = x J = ad o the same with x (7) 2 2 Questio 9. Coside the system x = y x + x y = x y 3 (8) Daw a phase plae o this system, icludig ullclies ad equilibium poits. Compute the lieaized system aoud the equilibium poits, icludig the eigevalues. Popeties o systems lieaized aoud a equilibium poit The popeties o the lieaized system ẋ = J x ea the equilibium poit ca be detemied om the behavio o tajectoies i the viciity o the equilibium poit; it is suiciet to coside what happes i the state vecto is moved slightly away om the equilibium poit, i.e. to coside a iitial value poblem o the system with iitial value ea the equilibium poit. The popeties o such a system ae detemied by the eigevalues. Thee ae ive impotat cases, listed below. The sketches at ight show the atue o tajectoies om iitial values ea the equilibium poit. I these sketches, the equilibium poit is at the oigi o the plot ad the axes ae i the diectios o the eigevectos.

14 . Stable ode: λ ad λ 2 both eal ad egative. I this case the expoetials i Eq. 6 both decay with time, so the tajectoies move smoothly ad expoetially towad the equilibium poit. Such a equilibium poit is a attacto o tajectoies i its viciity ad may be a attacto o a lage pat o the phase plae. 2. Ustable ode: λ ad λ 2 both eal ad positive. I this case the expoetials i Eq. 6 both icease without boud, so the tajectoies move away om the equilibium poit. The system is stable i placed exactly o such a equilibium poit, but ay eo will lead to a tajectoy that moves away om the equilibium poit. 4 e 2 e 2 e e 3. Stable ad ustable spial: λ ad λ 2 both complex; complex eigevalues must occu i complex cojugate pais. Thee ae two cases, a stable spial occus whe the eigevalues have a e 2 e 2 egative eal pat ad a ustable spial occus whe the eigevalues have a positive eal pat. The solutios i this e Re λ t case take the om Ae cos(im λ t θ) stable ustable gives a oscillatio at equecy Im[λ]/2π whose amplitude gows o shiks accodig to the expoetial multiplie exp(re[λ]t). The sig o Re[λ] detemies which. I the phase plae, such solutios will spial aoud the equilibium poit, sice thee is a phase shit betwee the time waveoms alog e e ad 2. The equilibium poit i Fig. 5 is a stable spial (the expoetial decay is aste tha the peiod o the oscillatio i this case, so the spial is ot see; this case is vey simila to a stable ode) ad Fig. 7 is a ustable spial. 4. Saddle ode: oe eal positive eigevalue ad oe eal egative eigevalue. The two expoetial tems i Eq. 6 ow behave oppositely. Oe decays expoetially, the othe gows expoetially. I the igue at ight, λ is e egative, so the tajectoies decay towad the equilibium poit alog the diectio o e ; λ 2 is positive, so the tajectoies move away om the equilibium poit alog e 2. Most tajectoies ollow a hypebolic path, as sketched o the ou tajectoies show with a sigle aowhead. Howeve, thee ae ou tajectoies which ollow the diectios o the eigevectos e e ad 2 i the viciity o the equilibium poit. These ae idicated by the double aowheads i the sketch. These tajectoies ae poduced by iitial coditios exactly o oe o the eigevectos, so that thee is oly oe tem i Eq. 6. The tajectoies daw above cotiue outside the age o the lieaized system. The tajectoies alog the eigevectos o a saddle ode lead eithe to aothe equilibium poit o to a limit cycle (they caot lead to iiity i a HH-type system; to see why, coside the behavio o the dieetial equatios at,, E K ad E Na see Questio 2 below). I the lage oliea system, they ae called the ustable ad stable maiolds, espectively, o the oe leadig away om ad the oe leadig towad the saddle ode. Fo the lage oliea system, they sepaate the tajectoies o the system ad ca ote seve as a tue theshold, as descibed below. e 2 e

15 Figue 8 shows a phase plae o the MLE with the paametes o set #2. The compoets o the phase plae ae maked as i Fig. 5. Note that thee ae ow thee w (potassium activatio) equilibium poits, maked by the gee dots. The oe ea 42 mv is a stable ode, the oe ea 2 mv is a saddle ode, ad the oe ea 4 mv is a ustable spial. The stable ad ustable maiolds ae show as the ed tajectoies with double aowheads. These wee computed as ollows: o the ustable maiolds, the system was stated with iitial coditios ea but ot quite o the dv/dt = dw/dt= dv/dt = V (membae potetial, mv) Fig. 8 Phase plae o the MLE with the paametes o set 2. All compoets ae as o Fig. 5 except that the ed lies ae the stable ad ustable maiolds. Note that thee ae ow thee equilibium poits (gee dots). saddle ode. These tajectoies wee the computed diectly i the usual way. Oe poduced a subtheshold etu to the stable ode ad the secod a actio potetial which also etued to the stable ode. The stable maiolds caot be computed diectly i this way, because thee is o way to id a iitial value o a stable maiold at a distace om the equilibium poit. Istead, the system was u i evese time, i.e. the equatios ( ) ( ) x = F x ad x = x wee solved. The oly place that time appeas i the dieetial equatio is i the time deivative o the let had side, so eplacig t with t chages the system oly by evesig the sig o the time deivatives. With this chage, all tajectoies evese diectio ad ow the stable maiolds become ustable (i.e. lead away om the equilibium poit) ad ca be computed like ay othe tajectoy. Figue 9 shows ou tajectoies i the viciity o the saddle ode. This plot shows the same phase plae as Fig. 8, except that the axis scale is magiied cosideably. Oly the stable ode ad the saddle ode ae show (gee dots), alog with the ullclies (gee) ad maiolds (ed). The mageta lies show ou tajectoies. Note how the tajectoies ollow the diectios o the maiolds. This is typical o a saddle ode. A impotat aspect o a saddle ode is that tajectoies caot coss the maiolds, sice the maiolds ae themselves tajectoies (the uiqueess theoem). Thee ae two impotat implicatios o this act:. The stable maiold uig vetically dowwad om the saddle ode acts as a tue theshold o this system. This act is appaet om Fig. 9. Coside tajectoies om iitial values (V, w ) cosistig o a depolaized membae potetial V ad a estig value o the potassium chael state w. Such iitial coditios lie alog a lie to the ight o the estig potetial. The two iitial coditios maked with x s i Fig. 9 ae o opposite sides o the stable maiold.

16 6 Oe gives a subtheshold etu o membae potetial to est ad oe gives a actio potetial, by ollowig the ightwad diected ustable maiold. I this case, iitial values to the let o the maiold, by howeve small a amout, give subtheshold esposes ad iitial values to its ight give actio potetials. Thus cossig the maiold is the coditio o theshold i this system. Futhemoe, ulike the actio potetials show i Fig. 5 o paamete set #, the peak membae potetial o the actio potetial does ot deped o the iitial coditio, because the actio potetial tajectoies all vey close to the ustable maiold. 2 You should be able to covice yousel, om Fig. 8 ad Fig. 9, that thee is o limit cycle i this system. Because the membae potetial caot coss ay o the maiolds, thee is o way to costuct a loop i this phase space that is cosistet with the blue aows. Fig. 9 Same phase plae as Fig. 8 show magiied i the viciity o the stable ode ad the saddle ode. Gee ad ed lies ae ullclies ad maiolds as i Fig. 8. Mageta lies ae tajectoies om ou iitial values ea the maiolds I the discussio above, it was assumed that the system has two o-idetical eigevalues. Whe the eigevalues ae eal ad equal o whe the eigevalues ae puely imagiay, the the lieaized system tus out equetly ot to be a good appoximatio to the oliea system, eve w, potassium activatio dv/dt = dw/dt= dv/dt = xx Membae potetial, mv vey close to the equilibium poit. This situatio does ot aise i pactice, but cae should be take i cases whee the eigevalues ae close to oe o these degeeate cases. A moe detailed discussio o this poit is give i Stogatz, pp Questio. Repeat Questio 8 o the ollowig system. 5 J = 4 ad 2 x = (9) 2 I case the eigevalues o this system ae complex; show that the solutios take the om Ae [ ] [ ] + (2) Re λ t cos( Im λ t θ)

17 7 Questio. Classiy the equilibium poits o the system o Questio ad sketch tajectoies i the phase plae o this system, based o the behavio o the equilibium poits. The chapte by Rizel ad Emetout (Koch ad Segev, 998, chapt. 7) cotais seveal othe iteestig applicatios o phase-plae aalysis to the MLE, icludig a vey clea demostatio o how bustig ca be added to this system usig a physiologically ealistic calcium-depedet potassium chael. I the ollowig, phase plae aalysis is applied to educed HH systems i ode to illustate othe applicatios o the method. Necessay ad suiciet coditios o a limit cycle I ode to have a limit cycle, it is clea that the diectio o the aows i the phase plae must poit i a cicula low. Howeve, this qualitative statemet is ot pecise ad it is ot immediately clea why the cicula low i Fig. 7 allows a limit cycle, but those i Figs. 5 ad 8 do ot. The idea o a cicula low ca be made slightly moe igoous o systems o ode 2 with idex theoy. The idex o a closed cuve C i the phase plae is the cumulative chage i the agle φ = ta - ( y / x) though oe couteclockwise obit o the cuve, see Fig. A. φ is the agle, elative to the positive abscissa, o the velocity vectos whee they itesect the cuve (blue aows). The agle is measued i uits o 2π, so the idex takes o oly itege values. Fo the case i Fig. A, the idex is zeo, because φ stats at a agle ea 45, luctuates i the positive ad egative diectio though the obit, but etus to its oigial agle without makig a complete tu i eithe diectio. Fo a cuve suoudig a stable o ustable equilibium poit (gee dots i Fig. B), the idex is + because i eithe case, the aow makes oe ull couteclockwise evolutio as the cuve is obited i the couteclockwise diectio. Fially, o a saddle ode, the idex is, as i Fig. C. Β Α C C Fig.. Closed cuves (black cicles) i a phase plae. Blue aows ae velocity vectos. A. No equilibium poit. B. Stable (let) ad ustable equilibium poits (gee dots). C Saddle ode. φ ẋ ẏ The idex o a cuve i the phase plae has two impotat popeties:. I C is deomed ito aothe closed cuve C without passig though a equilibium poit, as i Fig., the idex does t chage. This popety ca be udestood as a cosequece o the act that we assume a cotiuously dieetiable vecto ield o the phase-plae velocity. As a cuve is deomed theeoe, the idex must also chage i a cotiuous ashio, except at a equilibium poit. Howeve, the idex ca oly take o itege values, so it caot chage as the cuve is deomed, except at equilibium poit.

18 8 2. The idex o a cuve is equal to the sum o the idices aoud all the equilibium poits cotaied withi the cuve. This act ollows diectly om popety ad is illustated i Fig., which shows a cuve C cotaiig thee equilibium poits (gee dots). C is deomed ito C without cossig a equilibium poit, ad theeoe without chagig its idex. The idex o C is equal to the sum o the idices o the small cicles aoud the equilibium poits plus the idices o the staight lie segmets. Each pai o staight lie segmets ca be bought as close togethe as possible, so the idex accumulated duig tasit i oe diectio alog a lie is exactly compesated by the tasit i the othe diectio. This leaves oly the idices o the thee equilibium poits. C Fig.. Deomig a closed cuve C ito a secod cuve C without cossig a equilibium poit. C A limit cycle is a closed cuve i the phase plae. Its idex must be +, because the velocity vectos must be taget to the limit cycle at evey poit (ote that it does t matte which diectio the limit cycle us, its idex still must be +). As a esult, popety 2 implies that a limit cycle must cotai a umbe o equilibium poits whose idices sum to +. That is, a limit cycle must cotai a odd umbe o equilibium poits, with oe moe stable/ustable poit tha saddle ode; o example two stable o ustable equilibium poits plus a saddle ode satisy the coditio. Note that this is a ecessay coditio; a limit cycle does ot ecessaily exist i a phase plae which meets this equiemet (as i Figs. 5 ad 8). Fo the phase plae o Fig. 8, the maiolds o the saddle ode ae positioed i such a way that a limit cycle caot exist, because thee is o way to complete a loop aoud oe o thee equilibium poits without cossig oe o the maiolds. Limit cycles ca exist i systems with a saddle ode, as i the example show i Fig. 7.8 o Rizel ad Emetout (998). A suiciet coditio o a limit cycle o ode 2 systems (oly) is povided by the Poicaé-Bedixso Theoem. R Theoem: Suppose that. R is a closed, bouded subset o the plae (shaded at ight); 2. dx / dt = ( x) is a cotiuously dieetiable vecto ield o a ope set cotaiig R; 3. R does ot cotai ay equilibium poits; C

19 9 4. Thee exists a tajectoy C that is coied i R, i the sese that it stats i R ad stays i R o all time. The eithe C is a closed obit (limit cycle) o it spials towad a closed obit as t. I eithe case R cotais a closed obit. This theoem imagies a situatio i which the tajectoy o the system is coied withi a space R that is bouded o the outside, but also excludes a egio i the iside, so that thee ca be a equilibium poit iside C to satisy the idex citeio. I coditios ca be set up such that the velocity vectos poit ito the iteio o R eveywhee o both boudaies, the tajectoies that ae withi R must stay thee. Because thee ae o equilibium poits i R, the tajectoy caot appoach a ixed positio, which leaves oly a limit cycle. The poo o this theoem is advaced; eeeces to the poo ae give by Stogatz (994, p. 24). Questio 2. Ague that HH type systems have a atual oute bouday o R set by [,] alog the HH vaiable axis (e.g. w i the MLE) ad by the equilibium potetials o the ios alog the V axis. Ude what coditios ca you show that a acceptable ie bouday ca be set up (Hit, coside the situatio i Fig. 7)? The Poicaé-Bedixso theoem ca be applied to show that a limit cycle must exist i the case o Fig. 7, o example. Theshold behavio om the V-m educed HH system The ollowig discussio is based o Fitzhugh s (96) aalysis o the HH system i educed om. Coside ist the coditios o theshold whe a cell is depolaized by a bie cuet ijectio. As i Figs. 5 ad 9, this situatio is modeled by usig a iitial coditio i which membae potetial is displaced away om the estig equilibium poit without chagig the othe state vaiables. Figue 3 shows that the iitial evet i espose to such a stimulus is a apid icease i m ad V, with chages i h ad that occu moe slowly. Thus it seems easoable to eplace the ull HH system with a educed system cosistig oly o two state vaiables, V ad m ad hold h ad costat at thei estig values. The the system becomes C dv dt 3 4 = I G m h ( V )( V E ) G ( V )( V E ) G ( V E ) ext Na R Na K R K L L dm dt m ( V) m = τ ( V ) m (2) whee V R meas the membae potetial beoe the applicatio o cuet, o example the estig potetial. Note that h ad ae ow held costat at thei steady-state values at V R. Coside ist the behavio o the educed system statig om the estig potetial, -6 mv. Figue 2 shows a phase plae o the (V, m) system with h=h (-6)=.596 ad = (-6)=.38. The covetios ad colo code ae the same as o phase plaes i pevious igues. Thee ae thee equilibium poits (gee dots) i this system, a stable poit at the estig potetial (-6 mv, labeled ), a saddle ode just above the estig potetial (-57.3 mv, labeled s) ad aothe stable poit at depolaized potetials (+53.9 mv, ot labeled). The maiolds associated with the saddle

20 2 ode ae daw i ed. The expaded view at ight shows the stuctue o the phase plae ea the estig potetial ad the saddle ode m.4 dm/dt=.2 dv/dt= s Membae potetial, mv dm/dt= dv/dt= Membae potetial, mv Fig. 2. Phase plae o the educed system with V ad m as state vaiables. Expaded view show at ight. The stable maiold that exteds dowwad om the saddle ode acts as a theshold sepaatix, just as it did o the MLE system i Figs. 8 ad 9. Tajectoies begiig at membae potetials just to the let o ad just to the ight o this theshold (ot show) ollow the ustable maiolds, edig at oe o the othe stable equilibium potetial. The ull system behaves similaly, i that it has a shap theshold i the same egio. Thus, it seems easoable to study theshold behavio i the ull model by assumig that theshold coespods to cossig the stable maiold i the educed system. The similaity is qualitative, howeve, i that the theshold o membae potetial displacemet pedicted by the educed model ( mv) is somewhat dieet om the theshold o the ull model ( mv). s The behavio o the educed system is compaed with the ull system i Fig. 3, which shows two tajectoies i the (V, m) phase plae. Both tajectoies ae om iitial values just above the elevat theshold; the oe om the educed system (V-m sys, black) ollows the ustable maiold to the lagedepolaizatio equilibium poit, wheeas the othe (HH, mageta) ollows a slightly dieet path to depolaized potetials ad the etus to the equilibium poit. Note that the mageta tajectoy is ot the actual tajectoy o the system, which is a cuve i 4- dimesioal space. Istead it is a pojectio o that tajectoy o the V-m plae. O couse, it is ot expected that the educed system should poduce a ull actio potetial, because to do so equies eithe h o o both. The pupose o the educed system is oly to aalyze the.8 HH.6 m.4 V-m sys..2 s Membae potetial, mv Fig. 3. Compaiso o tajectoies o the educed system (black) ad the ull system (mageta) o the V-m phase plae. Both tajectoies begi just above voltagedisplacemet theshold.

21 2 coditios o theshold, ude the cotol o V ad m. why. Questio 3. The mageta tajectoy i Fig. 3 is icosistet with the blue aows. Explai The behavio o the educed system depeds o the values chose o the state vaiables that ae held costat. Figue 4 shows the eect o the value o h o the theshold. Show is the m ullclie (log dashes) ad the V ullclie (shot dashes) o two values o h, as labeled. The eect o h o the V ullclie ca be see om the equatio o the V ullclie, dv dt 4 Iext GK VR V EK GL V E ( )( ) ( L ) = mv ( V) = GNa h VR V E ( )( Na) 3 / (22) which is obtaied by solvig Eq. 2 o m with dv/dt=. As h deceases the V ullclie moves vetically upwad. The esult is to move the saddle ode to the ight alog the m ullclie. As the saddle moves, the dowwad-poitig stable maiold is dagged alog. Because that maiold is the theshold sepaatix, the eect is to icease the theshold. The hoizotal black lie shows whee the thesholds ae, at its itesectio with the stable maiolds, ad shows clealy the icease i theshold as h deceases. This behavio i the phase.8.6 m.4.2 dm/dt= h=. h= Membae potetial, mv Fig. 4. Eect o vayig h o the educed system i the V-m phase plae..596 is the value o h at the HH estig potetial Note the chage i the stable plae coespods physiologically to the act that as h deceases, sodium chaels ae iactivated ad theeoe uavailable to paticipate i actio potetials. The ull model behaves qualitatively similaly, with the theshold goig om mv o h()=.596 to 38.7 mv o h()=.. Questio 4. It is appaet om Fig. 2 that the ea-theshold itesectio egio o the V ad m ullclies is vey small. Ague om Eq. 22 that by ijectig positive cuet it is possible to elimiate the itesectio o the ullclies ea the estig potetial, so that the oly equilibium poit i the V-m plae is the oe ea +5 mv. (Hit: what is the sig o V-E Na?) How will the educed system behave i this case? What does the educed system pedict about the ull system i this situatio?

22 22 The V-m educed system povides isight ito the pheomeo o aode-beak excitatio, see i both the squid giat axo ad the HH model. Figue 5 shows the pheomeo, i the ull model, at uppe let. The membae is hypepolaized by a 2.8 µa/cm 2 cuet ijectio ove the time peiod to 4 ms (the heavy ba o the abscissa). Whe the cuet is tued o, the membae potetial etus towad est, but the oveshoots the est potetial ad ies a actio potetial. Qualitatively, the aode-beak eect ca be explaied by the act that h iceases ad deceases duig the maitaied cuet ijectio, so that the membae is moe excitable at the ed o the cuet (time t ed ); the icease i h meas moe sodium chaels ae available o activatio ad the decease i meas that ewe potassium chaels ae ope to stabilize the membae. Membae pot., mv HH vaiables t ed.8 h m Millisecods.8.6 m dm/dt= dv/dt= Fig. 5. Aode-beak excitatio. Top let shows time waveoms o membae potetial ad the HH vaiables o the ull HH model i espose to a 4 ms log 2.8 µa/cm 2 cuet ijectio (heavy black ba). At ight ae V-m phase plaes o the istat just ate the cuet is m. dm/dt= dv/dt= Membae potetial, mv The phase plaes at ight i Fig. 5 show the educed V-m system with ad h set at thei values just at the ed o the hypepolaizatio, i.e. at time t ed i Fig. 5 ( is.272 istead o the estig value o.38 ad h is.695 istead o its estig value o.596). The eect o educig

23 23 ad iceasig h is to move the dv/dt ullclie dowwad, elimiatig the itesectio o the ullclies ea the est potetial. As a esult, thee is ow o stable equilibium poit i the educed system, except ea +5 mv. The lows i the viciity o est potetial ae show at lowe ight i Fig. 5. The mageta lie shows the tajectoy o the educed system though this egio om a iitial coditio at the values o V ad m at the ed o the hypepolaizatio. The explaatio o the aode-beak spike, i tems o the phase plae aalysis, is that the stable equilibium poit at the estig potetial is abolished by the chages i ad h so the system udegoes a apid depolaizatio which leads to a actio potetial i the ull system. Biucatio theoy illustated with the V- educed HH system A educed system simila to the MLE ca be costucted om the HH equatios by elimiatig m ad h as state vaiables, leavig V ad. This appoach was take by Cao ad collaboatos (993) i a model o cetai muscle disodes liked to loss o iactivatio i the sodium chaels i the muscle. The ist assumptio is that the time costat o m is ast, so that m ca be appoximated by m (V). The secod assumptio is to set h to zeo, which is a cude appoximatio o the situatio duig the epolaizatio phase o the actio potetial, ate about 2.5 ms i Fig. 3, whe most sodium chaels ae iactivated. The muscle disodes studied by Cao et al. show a loss o epolaizatio, so this phase o the actio potetial was o pimay iteest i thei study. The model assumes that some sodium chaels, a actio, have lost thei iactivatio gates, so that thei coductace is gated oly by the activatio gate m. The esultig model cotais oly two state vaiables ad esembles the MLE: dv dt C I G m 3 V V E G 4 = V E G V E ext Na ( )( Na) K ( K ) L ( L ) d ( V) = dt τ ( V ) [ ] (23) Note that the sodium tem assumes that o the sodium chaels do ot iactivate ad the emaide ae pemaetly iactivated. The uctioal om o m (V) ad (V) used by Cao et al. ae the same as i the oigial HH model, but the paametes ae dieet. The paametes ae listed i the appedix. (Note that with these paametes, the behavio o the model is qualitatively the same as published, but the quatitative behavio, i.e. values o at the biucatio poits, is slightly dieet; pesumably this dieece elects some dieece i paamete values actually used by Cao et al. vesus the oes gleaed om thei pape.) The educed -V model poduces actio potetials that ae qualitatively simila to the ull HH model. Figue 6 shows a compaiso o the actio potetial ad (t) waveoms betwee the ull model (solid lies) ad the educed model (dashed lies) with =.25. Fig. 6. Compaiso o the membae potetial (top) ad (t) waveoms betwee the ull (dashed) ad educed (solid) Cao models (Fig. 9A o Cao et al, 998)

24 The phase-plae o the educed model is show i Fig. 7A o a age o values o. The V ullclie is show by the solid lies. As i the MLE, the V ullclie has two legs, a egative slope potio below 8 mv which does ot vay as vaies, ad a covex upwad potio betwee about 6 ad +4 mv which is quite sesitive to. The ullclie is show by the dashed lie. As beoe, this ullclie is a plot o (V) ad does ot deped o. Thee ae thee equilibium poits i the system, idicated by the symbols. Oe, the est potetial (.p.), is stable, ad emais ea 85 mv o all values o. A secod equilibium poit is ea 6 mv (ope tiagles); this poit moves with the V ullclie as chages. It is a saddle ode with a stable maiold that seves as a theshold sepaatix, as i Fig. 8. The maiolds ae ot show, but the stable theshold maiold ollows vey closely the dv/dt= ullclie i the dowwad diectio. A thid equilibium poit is ea ad 24 Fig. 7. A. Phase plae o the educed Cao et al. model with vayig om. to.. Show ae ullclies; oly the dv/dt= ullclie vaies with. Symbols makig equilibium poits vay i shape accodig to eq. poit type..p. is equilibium poit at estig potetial. B. Biucatio diagam o this system. (modiied om Fig. o Cao et al., 998.) above 4 mv. It ca be a stable (illed squaes) o ustable (uilled squaes) spial, depedig o. As deceases, the uppe two equilibium poits move towad oe aothe alog the d/dt= ullclie util they coalesce ito a sigle poit o =.3 ad disappea o smalle values. Note that thee is oly oe equilibium poit, the.p., o the smallest value o show i Fig. 7A. The biucatio diagam i Fig. 7B summaizes the behavio o the system o a plot o the membae potetial at the equilibium poits (odiate) vesus the value o, which is the biucatio paamete i this case. I oliea dyamics, a biucatio meas a sudde chage i the qualitative behavio o a system with a small chage i some paamete. The lie maked stable ode shows the membae potetial at the.p., which does ot chage with ; this poit emais