Jurnal f Bilgical Physics (preprint versin, see www.springerlink.cm fr final versin) Rbert Clewley Inferring and quantifying the rle f an intrinsic current in a mechanism fr a half-center bursting scillatin A dminant scale and hybrid dynamical systems analysis Accepted: February 2011 Abstract This paper illustrates an infrmatic technique fr inferring and quantifying the dynamic rle f a single intrinsic current in a mechanism f neural bursting activity. We analyze the patterns f the mst dminant currents in a mdel f half-center scillatin in the leech heartbeat central pattern generatr. We find that the patterns f dminance change substantially ver a cycle, allwing different lcal reductins t be applied t the mdel. The result is a hybrid dynamical systems mdel, which is a piecewise representatin f the mechanism cmbining multiple vectr fields and discrete state changes. The simulatin f such a mdel tests explicit hyptheses abut the mechanism, and is a nvel way t retain bth mathematical clarity and scientific detail in answering mechanistic questins abut a cmplex mdel. Several insights int the central mechanism f escape-release in the mdel are elucidated by this analysis and cmpared t previus studies. The brader applicatin and extensin f this technique is als discussed. Keywrds Dminant scale analysis Hybrid dynamical systems reductin Bursting dynamics Mdel inference Neurinfrmatics Central pattern generatin 1 Intrductin Biphysical mdels f neural bursting activity in small circuits knwn as central pattern generatrs (CPGs) are cmmnly used t hypthesize rhythmgenic mechanisms behind lcmtr r autnmic cntrl [7,12, 16]. We will study ne aspect f an existing mdel f the leech heartbeat elemental half-center scillatr (HCO), which is part f the heartbeat CPG circuit cnsisting f tw reciprcally-cupled neurns that burst alternately [8,14]. Mdels f small neural circuits are already sufficiently high dimensinal and nnlinear that they present great challenges t existing mathematical analysis tls, and typically the functinal understanding f the circuits invlves reductins t simpler mdels f sme kind (e.g., see [10]). As mdels becme mre sphisticated there is an increasing demand fr tls that aid in the develpment and clear understanding f a mdel s functin. Simulatin capabilities fr large, detailed mdels are currently far in advance f ur analytical methds fr deeply understanding their functin [15,17]. Our primary aim here is t prmte a frmalized, hypthesis-driven mdeling technique that may be able t redress the balance. In essence, this wrks by explring parsimnius, abstracted descriptins f a mechanism as a frm f reverse engineering f the full-dimensinal dynamics. Supprted by NSF CISE/CCF-0829742 R. Clewley Neurscience Institute and Dept. f Mathematics and Statistics Gergia State University Atlanta, GA 30303, USA E-mail: rclewley@gsu.edu
2 Rbert Clewley We apply a dminant scale analysis f a central mechanism behind the heartbeat CPG scillatins [4]. This analysis decmpses a single, high-dimensinal vectr field t a piecewise lw-dimensinal reductin knwn as a hybrid dynamical system [26]. In such a system, smth vectr fields gverning the dynamics f the system in different regimes are switched at discrete times indicated by zer-crssing transitin events, at which pint a discrete mapping may be applied t the system s state befre initializing the next vectr field. In cntrast t making glbally valid assumptins abut the dynamics in rder t reduce a cmplex mdel t a single reduced mdel, a dminant scale analysis facilitates multiple, lcal reductins. These are typically f lwer rder than a single, glbal reductin, and yet can als capture mre details f the dynamics [5]. This apprach dissects functinal relatinships between parameters, variables, and mdel behavirs at a finer granularity than is pssible with traditinal mdel manipulatins. As with any frm f reverse engineering, the user must prvide sme initial assumptins and hyptheses abut the mechanisms based n initial bservatins f the system [1]. This wrk des nt fcus n hw a user can generate such infrmatin in a purely bjective r ptimal fashin, and mstly cncentrates n the technique t synthesize mechanistic hyptheses int hybrid dynamical mdels in rder t evaluate them. Indeed, iteratin is a central aspect f a reverse engineering methdlgy, which shuld validate r falsify the initial hyptheses and lead t a mre refined and rbust descriptin that can be tested further. The prpsed methdlgy is nt intended as a standalne replacement f all subjective analytical steps with bjective nes. Instead, it attempts t assist a human user in perfrming a lgical, and smetimes qualitative, analysis f cmplex dynamics, with greater cnfidence and reslutin. We will restrict urselves t a simple case study invlving a single intrinsic current f the leech heartbeat CPG mdel, namely the hyperplarizatin-activated current I h. We d this fr the purpses f expsitin, despite the presence f several functinally relevant currents t explre in this CPG. Nnetheless, we will uncver sme nvel insights int the rle f this current in this CPG as well as verify previusly established insights by ur methd. Mtivated by sme brad numerical bservatins, initial mechanistic hyptheses are presented in Sectin 4, but their precise chices are nt critical t this wrk. First f all, the iterative nature f the methdlgy is such that bad chices shuld rapidly becme apparent during testing. There is naturally mre ambiguity in the ther chices. Hwever, a key aspect f the methdlgy is use f cncrete means t evaluate a reduced mdel ( metrics ) that are suited t the reslutin f investigatin. On further iteratin f the reverse engineering prcess, mre subtle questins may be psed, fr which bth the metrics and the reduced mdel decmpsitin can be refined accrdingly. We will suggest future directins f this kind based n the initial findings fr this case study demnstratin withut pursuing them here. 1.1 Leech heartbeat case study Previus studies f this HCO mdel and electrphysilgical studies f the leech heartbeat have addressed fundamental questins abut the mechanism f bursting [8,14,18 20,24,25]. Regarding I h specifically, these questins can be summarized in terms f rles: R1: What rle des I h have in the existence f a stable HCO rhythm? R2: What rle des I h have in the mdulatin f the rhythm? T these we add tw questins, which investigate I h mre thrughly: R3: What rle des I h have in the rbustness f the rhythm? R4: What rle d the details f activatin / deactivatin dynamics f I h have in the perfrmance f the rhythm? The primary manipulatin f I h in previus mdeling studies was t adjust its maximum cnductance ḡ h, including setting it t zer t remve the effect f the current altgether. In cnjunctin with manipulatins f ther currents, this apprach has facilitated natural language descriptins f the mechanistic rle f I h. The maximal cnductance is a physilgically relevant parameter t vary, but this wrk dissects the dynamics in a different way. We keep ḡ h at its cannical value frm Hill et al. [14], and infer frm the behavir f the mdel that there are different regimes during ne scillatin f the HCO in which different currents dminate. Frm this we hypthesize different rles fr I h in each regime, and explre the replacement f I h in ne cell f the HCO with a simple but functinally equivalent behavir fr each regime. We will reduce nly ne neurn f the cupled pair, in a similar fashin t the neural-silicn circuit f Srensen et al. [24]. The purpse f this is t use the unreduced neurn as a reference, s that the quality f a reductin in the ther neurn can be measured by the degree f symmetry preservatin in the cupled system.
Inferring and quantifying a mechanism fr half-center bursting 3 I h V I h V 2 2 I h V reduce m h in neurn 1 1 V1 I h Fig. 1 The half-center scillatr mdel fr the leech heartbeat central pattern generatr, shwing the tw frms f reciprcal inhibitry synaptic cupling present in the circuit ( graded and spike-mediated ), and the intrinsic current f fcus fr the analysis: the hyperplarizatin activated current I h. On the left is the full 19-dimensinal mdel M F. On the right, the reduced hybrid dynamical systems mdel M R utilizes a piecewise cnstant reduced frm fr the activatin m h in I h, applied nly t neurn 1. Functinal criteria n the behavir f the HCO mdel are required t characterize the rle f I h in the CPG rhythm. We will use five functinal metrics: cycle perid, relative cycle phase between the tw cells ( phase symmetry ), duty cycle, inter-spike interval (ISI), and phase respnse prperties f the circuit t weak perturbatins. Previus mdeling studies f this mdel have nt cnsidered the phase respnse prperties in detail. The metrics will be investigated ver a range f maximal synaptic cnductances ḡ SynS fr the spikemediated cupling, which is a physilgically imprtant parameter used in previus cmputatinal studies f this circuit [20]. We will then make a cmparisn f these metrics between the riginal HCO mdel and the reduced frm. We briefly review the HCO mdel in Sectin 2. In Sectin 3 we make brad and infrmal bservatins in relatin t previus wrk, based n measurements f time scales and the cmparisn f the current terms in the mdel that cntrl the mtin f the membrane ptentials. Frm these bservatins, explicit hyptheses are develped int a hybrid dynamical mdel in Sectin 4, which are tested in Sectin 5 and discussed in Sectin 6. 2 The HCO mdel Each interneurn in the leech heartbeat HCO mdel cnsists f a single smatic cmpartment with membrane ptential V, whse rate f change is determined by a current balance equatin made up f several intrinsic in channel currents and tw frms f inhibitry synaptic input current riginating frm the ther neurn. It is illustrated in Figure 1. The existing, full versin f the mdel will be dented M F and has been develped ver several studies. The versin f the equatins and parameters used here is described fully in [8], based n the mdel f Hill et al. [14]. The frm f the rdinary differential equatin (ODE) fr V is C V = I inic I applied I syn, (1) fr a membrane capacitance C = 0.5nF, a sum f inic currents and an applied current (which is zer unless therwise stated). Thrughut this paper, the units f V are vlts, thse f current are pa, and thse f cnductance are ns. In using the Hdgkin-Huxley frmalism, each intrinsic inic current takes the frm I = ḡm p h q (V E) fr channel reversal ptential E, maximum cnductance ḡ, activatin gating variable m and, if present, inactivatin gating variable h, fr sme nn-negative integer pwers p and q. The kinetics f each gating variable are gverned by a first-rder differential equatin. The mdel cntains a passive leak current I L = ḡ L (V E L ) and 8 active inic currents: a fast Na + current (with activatin m Na and inactivatin h Na ), three types f K + current (m K1, h K1 ; m K2, h K2 ; and m KA ), a persistent Na + current (m P ), a hyperplarizatin-activated current (m h ), and a rapidly and slwly inactivating lw-threshld Ca 2+ current (m CaF, h CaF ; m CaS, h CaS ). Because we fcus n I h, we detail its definitin further: I h = ḡ h m 2 h (V E h), where ḡ h = 4.0nS and E h = 0.021V. The activatin is gverned by m h = (m h, m h )/τ h. The time scale and steady state activatin
4 Rbert Clewley functins are given by τ h (V ) = 0.7 + 1.7 1 1 + e 100(V +0.073) and m h, (V ) = 1 + 2e 180(V +0.047). + e500(v +0.047) The cells are reciprcally inhibited by bth slw-acting graded (G) and fast-acting spike-mediated (S) frms f synaptic inhibitin [18]. The synaptic current mdels use a ttal f 5 variables but I SynG and I SynS appear in Eq. (1) accrding t I SynG = ḡ SynG P 3 C + P 3 ( V ESyn ), ISynS = ḡ SynS MY ( V E Syn ), where P, M, and Y are three f the synaptic variables, E Syn = 0.0625V, ḡ SynG = 30nS, ḡ SynS = 150nS, and C = 1 10 32 is cnstant. With the 5 synaptic variables there are a ttal f 19 dynamic variables per neurn in mdel M F. The remaining details f the currents are nt imprtant t the analysis undertaken here. 3 Preparatry bservatins Busting activity in this CPG is mst fundamentally split int the active r bursting state (heren dented B) in which the cell generates actin ptential spikes, and the recvery r inhibited state (heren dented I). Previus wrk has identified that I h is imprtant fr an escape -based mechanism underlying the creatin f a rhythmic cycle in the HCO, and fr determining its perid [14]. Briefly, the mechanism wrks as fllws. A neurn inhibited by its partner in the HCO is in the inhibited state I. Eventually, it pr-actively vercmes the inhibitin, spikes again itself, and thereby stps its partner frm cntinuing its burst f spikes. I h acts t deplarize the neurn frm I int B due t E h being mre deplarized than the threshld fr actin ptentials. I h appears t play a majr rle in balancing the hyperplarizing effect f I L (fr which E L = 0.0635) and ultimately vercmes the effect f I SynS caused by actin ptential spikes arriving frm the neurn in B. As nted in previus studies, the situatin is mre cmplex, and there are aspects present f a release -based mechanism, primarily invlving the decay f I CaS. Thrugh analysis f M F we nw fcus n which currents dminate the activity f the membrane ptential V 1 f neurn 1 during I. The dynamics f the membrane ptentials lie within a high cnductance regime because τ V = C/ i ḡ i m p i i h q i i, the effective time cnstant fr the membrane, stays small thrughut each cycle [22]. We will nt fcus n the rle f I h at the reslutin f dynamics within single spikes, and at slwer time scales in the cycle τ V remains within a range f 0.02 0.03s n average, ver bth B and I. It is therefre apprpriate t this study t use the type f dminant scale analysis perfrmed n bursting cells in a crustacean stmatgastric system by cmparing the changing magnitudes f the currents ver a cycle [5]. (In a purely high-cnductance regime this prvides equivalent infrmatin t studying the influence sensitivity quantities als cnsidered in [4,5].) Over ne cycle, m h (t) rises and falls with kinetics at cmparable time scales, in that τ h remains between 2.0 and 2.4s. The range f m h values ver ne cycle perid varies between apprximately (0.14,0.53) and (0.13,0.59) as ḡ SynS increases ver the interval [100,300]. The steady state activatin m h, (V ) reduces t zer almst entirely by V = 0.04 frm its maximum near 1, ccurring at the bttm f the actualized vltage range f a burst (V 0.06). The cnsequence f this is that m h will relax expnentially twards zer when V > 0.04, i.e., during spiking. Thus, B and I exhibit qualitatively distinct frms f I h dynamics. Nrmalized current data fr ne entire I state is shwn in Figure 2 in a lg-scale plt. These data are cmputed by nrmalizing all currents at time t t whichever is the greatest in magnitude, and separating the inward and utward currents fr clarity. The largest current magnitude is therefre always represented by the value 1 (if inward) r -1 (if utward). Data fr B is nt shwn because the nrmalized lg 2 ( I h ) at the beginning f B is smaller than 1/8 and diverges belw 1/16 after nly three spikes, s that we will cnsider I h t play a minr rle in this state, at mst. A subtle aspect f the mechanism f escape during I invlves an interplay between the hyperplarizing spike-mediated synaptic input (I SynS ) that activates m h and the deplarizing effect f I h as it is increasingly activated. As is visible in Figure 3a, the inhibited neurn is mst hyperplarized near the beginning f I when the ISIs frm the active neurn are shrtest. Due t its lng activatin time cnstant, the hyperplarizatinactivated I h des nt reach its maximum until the latter part f I. At this time, the ISIs between the synaptic input pulses have increased, and the membrane ptential is deplarizing due t the resulting reductin in
Inferring and quantifying a mechanism fr half-center bursting 5 Inward Signed, nrmalized current magnitudes K1 CaF CaS P h CaF K2 S (*) G L KA K1 K2 G h L KA S P CaF CaS K2 Outward time (s) * Fig. 2 Signed and nrmalized current magnitudes during the entire inhibited state f a single burst cycle at ḡ SynS = 150, shwn in lg scale as a fractin f the largest current (f either sign) at every time step. The psitive scale shws inward currents, while the negative scale shws utward currents. Fr clarity, nly ne representative spike in the spike-mediated synaptic current I SynS is shwn at the instant marked with the asterisk. Nearly identical spikes are present at every ther crrespnding psitin in the plt, fitting in the sharp ntches in the lines fr the ther currents, including a sle spike that des nt becme mst dminant at t 0.1s. inhibitin and the existing activatin f I h. Althugh the deplarizatin causes m h, (V ) t return twards 0, the lng time cnstant f m h means it des nt immediately respnd. Thus, I h has an enhanced pprtunity t deplarize the cell as it transitins t B because f the reduced inhibitin. Indeed, as we discuss later, if the ISIs d nt increase then the inhibited cell may never escape. Hill et al. demnstrated that the active neurn s ISIs must increase t a critical value in rder t release the inhibited cell [14]. The primary change fr I h seen in Figure 2 near the end f I is the reduced driving frce as the membrane ptential increases twards E h. In the same wrk, Hill et al. als calculated that increasing ḡ h in the mdel decreases the average ISIs in B [14]. In cnjunctin with the existence f a critical ISI fr release f the inhibited neurn, this suggests that I h might play a minr rle in setting this critical value, and therefre participate in the mechanism fr release as well as escape. Althugh the authrs did nt draw an explicit cnclusin frm these bservatins, we will directly address that rle thrugh Hypthesis 4, which is defined in the next sectin. Figure 2 indicates that I P and I L are the largest currents fr mst f I. I P slightly decreases ver the first 0.5s and then remains almst cnstant in relative magnitude fr the remainder f I. I CaF nly appears t be large enugh t be invlved in the dynamics within the final tw arriving spikes f I in the transitin t B. On average, I SynG changes n a much slwer time scale than actin ptential generatin, and seems t play a secndary rle at the beginning f I until I h becmes large. I SynS fluctuates at a fast time scale and creates the regular punctuatins in the graph as it transiently dminates all ther currents when it is active. Fr clarity in Figure 2, nly ne representative fluctuatin due t a single actin ptential is shwn. This transient effect f I SynS n the ther currents is due t the transient fluctuatin in the membrane ptential that it causes, which in turn briefly affects the driving frce terms f thse ther currents. I h is small at the beginning f I but becmes ne f the largest currents ( I h I P /2). Bradly speaking, I h deactivates during B and activates during I, but its influence peaks in the middle f I, apprximately 1s in advance f the nset f B fr the cannical parameter values. We will explre whether this is imprtant mechanistically, althugh it is already knwn frm varying τ h that the exact relative psitin f the peak in I is nt crucial t the mechanism f transitin t B [24].
6 Rbert Clewley a V 1, V 2 b A1 Q2 Ca gating variables m h c m CaF m CaS h CaS A2 Q1 h CaF time (s) Fig. 3 The mst relevant state variables ver ne scillatin perid fr the hybrid dynamical systems mdel M R at the cannical ḡ SynS = 150. a) Vltage traces (V 1 in black, bursting first, and V 2 in green). The sudden change in V 1 at t 4.5s is due t the discrete change in vectr field between I1 and I2 in neurn 1. b) m h (neurn 1 in black, unreduced neurn 2 in green), with the fur abstracted regimes labeled fr neurn 1. c) Ca gating variables fr neurn 1 nly: m CaF (slid black line), m CaS (slid red), h CaF (dashed black), h CaS (dashed red), and threshlds used fr defining m h dynamics in Eq. (2) are shwn by hrizntal blue bars. 4 Hyptheses We nw frmulate hyptheses frm the abve bservatins in the cntext f the five functinal criteria fr judging the existence and rbustness f the bursting rhythm, and the cntrl f the escape-release mechanism that transitins the system between bursting (B) and inhibited (I) states. The hyptheses allw us t prpse answers t the general questins R1 R4 psed abve, cncerning the rles f I h in the leech heartbeat HCO: H1: I h changes phasically ver ne cycle s that symmetry is maintained between the utput f the tw neurns (as they must generate symmetric and ppsite driving activity t mtr neurns n each side f the bdy). H2: I h is nly imprtant fr the rhythm s existence twards the middle f I. H3: I h must be present (but small) immediately after I fr rbust initializatin f B. H4: I h must be present (but small) later in B t ensure a slwer increase f ISIs and prevent premature release f the ther neurn. H5: I h des nt need t be smthly varying t functin in the cell. While being a natural cnsequence f the underlying chemical kinetics, the functinal requirements f I h in the wrking f the cell can be represented using simpler terms (e.g., as a piecewise cnstant functin). H6: The wrking levels f I h can be set accrding t an effectively discretized state f the system, nt requiring precise r cntinual feedback frm the membrane ptential. In particular, fur state-dependent
Inferring and quantifying a mechanism fr half-center bursting 7 regimes are sufficient, B1, B2, I1, and I2 based n using the magnitude f calcium current t split B and I int tw parts (f nn-equal duratin) each. H7: The escape-release mechanism f the cycle can be reprduced by a reduced mdel f the HCO that satisfies H1 H6. The final hypthesis, H7, is the largest in scpe and builds frm the previus hyptheses. It is presented mre explicitly as a hybrid dynamical system. 4.1 A reduced hybrid dynamical systems mdel We will test the abve hyptheses by building a frm f hybrid dynamical system mdel used in a previus analysis f a bursting neurn by Clewley et al. [5]. A hybrid dynamical system flexibly cmbines finite-state lgic fr simplified cmpnents with rdinary differential equatins [26]. As illustrated in Figure 1, we chse t reduce the HCO mdel M R using a piecewise cnstant dynamic prcess fr the dynamics f m h in neurn 1 nly, and is therwise defined in the same way as neurn 2, accrding t Sectin 2. We d nt reduce bth neurns in rder t mre strngly test the cmpatibility f the reductin in the cntext f the riginal circuit. A circuit where bth neurns are reduced sustains bursting scillatins (nt shwn), but it is harder t be certain that artifactual mechanisms are nt intrduced r riginal mechanisms nt lst when neither neurn pssesses all the riginal mdel equatins. The abve bservatins f tempral patterning in the current magnitudes ver ne cycle suggested fur regimes fr the reduced I h. Majr differences in the currents between B and I in this mdel have previusly been identified fr I CaS and I CaF [14], where they are large (small) in B (resp., I). A reductin using calcium threshlds t separate regimes fr these states f a bursting neurn mdel was used in [5], and a similar apprach is applicable here. Fr reference, the gating variables fr I CaS and I CaF are shwn ver ne cycle in Figure 3c. Tw f the distinct regimes crrespnd t beginning the bursting and inhibited states f the cycle, which we will refer t as B1 and I1, respectively. The nset f B1 can be defined in several ways, but a rbust chice is the crssing frm belw f a high threshld (0.96) n m CaF. The nset f I1 will be defined as the crssing frm abve f a lwer threshld (0.5) n m CaF. The ther tw regimes crrespnd t the later parts f B and I, which we will refer t as B2 and I2 respectively. The transitin t B2 is defined by the inactivatin h CaS falling thrugh a threshld, which we chse t be 0.2. The transitin t I2 is defined by h CaF reaching a lcal maximum during this state. The fur transitin pints are indicated in Figure 3c, but their exact psitins are nt crucial t the analysis that fllws. We can nw define the piecewise cnstant m h in M R as: m h,b1 when in B1, m m h (t) = h,b2 when in B2, m h,i1 when in I1, m h,i2 when in I2. The resulting set f states and transitins is summarized in Table 1, which will be referred t as the template f the bursting rhythm s mechanism (after [3]). As we fcus n I h nly, the template des nt exhaustively list all prperties f the system in each state. The deplarized and hyperplarized state cnditins n the membrane ptentials are nt quantified in terms f vltage dmains because there are n clear-cut vltage threshlds in the mdel that uniquely separate such dmains. Instead, changes in the gating variables fr I CaS and I CaF prvide a mre rbust means t distinguish B frm I. The template is an intermediate step in defining a hybrid dynamical system mdel that prepares us t test hypthesis H7. The template describes a cycle fr neurn 1, and requires an assumptin that neurn 2 underges the same pattern f state changes (in anti-phase). Therefre, this is a recursive definitin, and requires a self-cnsistency argument t validate it. As such, it is valuable t explicitly test the self-cnsistency thrugh a simulatin f a hybrid dynamical system. Here, we d nt make use f the increasing f ISIs in B2 as a defining characteristic, althugh this wuld be an interesting directin in which t develp the mdel. The values chsen here t define M R are m h,b1 = 0.35, m h,b2 = 0.2, m h,i1 = 0, and m h,i2 = 0.55, which rughly mimic the rise and fall f m h in M F. Fr these values, the HCO prduced rughly symmetric antiphase bursts at ḡ SynS = 150, a single cycle f which is shwn in Figure 3 fr sme f the key variables. m h,i1 was set at zer instead f a value clse t that seen in M F in I1 (such as 0.15) in rder t better test that I h is nly imprtant fr the rhythm s existence twards the middle f I (hypthesis H2). (2)
8 Rbert Clewley Table 1 The template descriptin f the escape-release mechanism hypthesis f bursting dynamics (H7) and the bursting rhythm relative t I h in terms f states and their allwed transitins. The template is a summary f bservatins in Sectin 3 and hyptheses H2 H4 (Sectin 4) and is the basis f frmally defining the hybrid dynamical system in Sectin 4.1. The first clumn indicates the discrete states f the hybrid dynamics, B1, B2, I1, I2, each separated by a rw defining the transitin between successive states. An upward (dwnward) arrw indicates an event threshld fr the indicated gating variable fr an increasing (resp., decreasing) transitin. The asterisk fr B2 indicates a recursive self-cnsistency assumptin described in the main text. State / Trans Characterizing descriptin Rle f I h / transitin in neurn 1 B1 Deplarized V 1 with regular spiking, hyperplarized V 2 Fr rbustness nly (H3) B1 B2 Dminance f I CaS reduces (ISIs begin t increase rapidly release f V 2 ) h CaS B2 Deplarized V 1 with regular spiking, hyperplarized V 2 Fr mdulatin f ISIs nly (H4) B2 I1 V 2 pr-actively begins spiking ( escape ) * m CaF I1 Hyperplarized V 1 with regular spiking in V 2 Nne I1 I2 V 1 becmes sufficiently plarized t de-inactivate I CaS and I CaF h CaF I2 Slwly deplarizing V 1 with V 2 regular spiking Causes deplarizing trend (H2) I2 B1 V 1 begins spiking, I CaS and I CaF activate strngly m CaF 4.2 Numerical methds and testing prtcl All ODEs in this study were slved using the adaptive time-step Radau integratr [13] using an abslute tlerance f 10-8 and a relative tlerance f 10-6. This slver was accessed via an interface with PyDSTl [6], an pen surce sftware envirnment develped by the authr fr dynamical systems mdeling. It prvides a cmmn set f cmpatible tls fr mdel specificatin, simulatin, and tlbxes fr applicatins. PyDSTl supprts simulatins with state- r time-dependent events and s can simulate hybrid dynamical systems f the kind described abve. Each threshld crssing needed fr the mechanism template is defined as a zercrssing event functin, detected t a tlerance f 10-4 with respect t functin values. Full cde used fr this wrk is available at http://www2.gsu.edu/~matrhc/hco.html. In this wrk, we vary the parameter ḡ SynS and cmpare the functinal metrics between stable bursting rhythms in mdels M F and M R. In sme test cases, we reprt a failure f the system t sustain a stable bursting rhythm. A 60s settle time was used t allw the system t apprach a stable rhythm befre making measurements. Hwever, it was bserved that a true limit cycle was nt eventually apprached by either mdel M F r M R (discussed further in Sectin 6). The cannical value f ḡ SynS frm the riginally published mdel (Hill et al. [14]) is ḡ SynS = 150, which was used as the starting pint fr develping the reduced mdel M R. In line with the physilgical ranges discussed in [20], ḡ SynS was varied between 50 and 300. The significance t H1 H7 f the cnstant values used t define m h will be explred thrugh an infrmal sensitivity analysis. The primary cnstant f imprtance fr m h in Eq. (2) was expected t be m h,i2, and is the nly ne that cannt be set t zer fr bursting t cntinue ver the range f ḡ SynS under the cnditins tested belw. In fact, when m h,b1 = m h,b2 = m h,i1 = 0, bursting typically ccurs in the netwrk as ḡ SynS is varied. As a result, these m h parameters will hencefrth be referred t as secndary. 5 Results 5.1 Perid, phase symmetry, and duty cycle The perid, phase symmetry, and duty cycle are fundamental measures f the tempral prperties f a halfcenter scillatr s rhythm. We define the cycle perid as T av = (T 1 + T 2 )/2, where T 1 (T 2 ) is the difference between tw successive burst nsets in neurn 1 (resp., 2). The phase symmetry in a cycle was measured by the difference in the timing f nsets t B in each cell, nrmalized by T av. Thus, a value f 0.5 indicates perfect symmetry between the nset f B in each cell as well as equal perids in their scillatin. The duty cycle was measured fr neurn 1 as the prprtin f the perid T 1 spent in the bursting state. Figure 4 cmpares these metrics fr HCO cycles between mdels M F and M R at the cannical parameters f Hill et al. [14], as a functin f ḡ SynS. The trend in perid is similar and the phase symmetry fr M R is, n average, within 2% f that fr M F. The duty cycles are similar in magnitude but are cnsistently lwer fr M F by n mre than 2%. These results supprt H1, H5 and H6, as bth the reduced and unreduced neurns behaved similarly when cupled tgether.
Inferring and quantifying a mechanism fr half-center bursting 9 5.1.1 Sensitivity analysis f m h If all values are set equal in Eq. (2) s that m h 0.45, the trend in perid is crrect fr ḡ SynS 225 but ffset by apprximately +1s, while at larger ḡ SynS neurn 1 remains disprprtinately lnger in B such that the perid increases greatly. The phase symmetry decreases frm 0.48 at ḡ SynS = 100 mntnically twards 0.1 as ḡ SynS 300, als reflecting that neurn 1 s bursting state becmes much lnger than that f its partner. The duty cycle increased almst linearly up t 0.85 as a functin f ḡ SynS. (The functinal cnsequence wuld be skewed timing and duratin in the activatin f mtr neurns driven frm this CPG circuit.) A bursting rhythm did nt exist at ḡ SynS = 300 fr these settings. These tests supprt hypthesis H1, that phasic changes in I h are necessary fr crrect and symmetric activity. These results are als cnsistent with H5 and H6, as bth the reduced and unreduced neurns behaved similarly when cupled tgether. When m h 0.45, the maximum hyperplarizatin f V 1 is much lwer. This is due t the cntinued presence f I h in I1 at a level much greater than is bserved in Figure 2 ver the first 1s. Cnsequently, the vltage-dependent de-inactivatins f I CaS and I CaF are lessened. Setting the secndary values m h,b1 = m h,b2 = 0 changed the perid by less than 0.1s at each ḡ SynS, except that stable rhythms failed t exist fr 3 f the 11 ḡ SynS values tested. This alteratin made n average change t the phase symmetry, and reduced the duty cycles t a value 0.5 ± 0.2 fr all ḡ SynS values except 50. These results are cnsistent with H2, in that I h is less imprtant during B. They als supprt H3 and H4, in that the presence f small amunts f I h in B helps t maintain a rbust rhythm. In every case f failure, neurn 1 enters B and fires ne (smetimes tw) actin ptential spikes. At this pint, well-timed inhibitry spikes riginating frm the verlap with neurn 2 at the end f its bursting state lead t an immediate suppressin f V 1 frm further spiking in this cycle. Subsequently, neurn 1 returns t a state f lw I CaS +I CaF, which had nt yet fully activated. This situatin is shwn in Figure 5 fr a failure at ḡ SynS = 150, where it is cmpared t the rbust dynamics at the cannical parameters. Recall that the highest level f m CaS de-inactivatin is nt reached until abut 3 spikes int B n a nrmal cycle (Figure 3c). Thus, even thugh the cell is already deplarized enugh t be spiking, the spiking state is fragile until I CaS increases t its full level. In the meantime, this test reveals that if m h activatin is present early in B, it allws the cell t recruit I h in respnse t inhibitin during the verlap f B and I in the tw neurns. Here, recruitment means that the hyperplarizing effect f the inhibitin frm the ther neurn increases the magnitude f the driving frce in I h, namely V 1 E h, thereby generating a deplarizing, restrative effect n the B state withut invlving a significant change in m h itself. When nly m h,b2 = 0 was changed frm the default values, there were n failures in stable rhythm generatin at the 11 ḡ SynS values tested. Frm these tests we cnclude that the failures all invlved the inhibited neurn failing t escape, and thus B was nt rbustly initialized in the absence f I h. This supprts H3, that I h is necessary t prvide this rbustness in B1. While the metrics shwed n significant change fr ḡ SynS 175, all shwed clse quantitative matching f the fluctuatins in M F fr larger ḡ SynS. This suggests that the reduced mdel fr I h wuld require greater refinement t explre a detailed quantitative cmparisn with M F. When m h,i1 was increased frm 0 t 0.2 the nly significant change in these metrics was a small imprvement in the agreement f perid between M F and M R at ḡ SynS 175. 5.2 Inter-spike intervals Hill et al. describe hw ISIs during a burst must increase t a critical value t release the ther neurn frm a suppressed state [14]. Here, we study the rle f the different tempral cmpnents f I h n ISIs in relatin t this part f the rhythm s mechanism. Figure 6 shws three representative sets f inter-spike intervals (ISIs) during the bursting state, as a functin f spike number. In all cases, the last ISI f the burst was 0.22s. The number f spikes are similar between mdels M F and M R, and the match in bth value and slpe is clse except fr a small step up in the values fr M F arund spikes 12 20. These steps are due t the abrupt change in m h between B1 and B2. Althugh quantitatively small, the difference in slpe between M F and M R indicates a greater mdulatry effect f I h n inter-spike timing during B than was assumed frm the nrmalized current magnitude values. Overall, these results supprt H1, H5 and H6, as bth the reduced and unreduced neurns behaved similarly when cupled tgether. The exact number f spikes per burst ccasinally varies by ±1 n sme cycles, due t the weakly chatic nature f the attractr (see Discussin).
10 Rbert Clewley Symmetry, duty cycle Perid (s) Maximal synaptic cnductance, gsyns Fig. 4 Perid, phase symmetry, and duty cycle metrics as a functin f ḡ SynS. Slid lines: mdel M R. Dashed lines: mdel M F. In the lwer panel, phase symmetry is shwn in black with val markers, and duty cycle is shwn in red with square markers. V 1, V 2 time (s) Fig. 5 Sensitivity t well-timed synaptic inputs at the end f the inhibited state. When m h,b1 = m h,b2 = 0,ḡ SynS = 150 in mdel M R, the (reduced) neurn 1 enters the B1 regime just befre the first vltage spike (V 1 in black) as a result f m h,b1 = 0. The slid lines shw that V 1 is suppressed after interacting with the final spike f the bursting state in neurn 2 (V 2 in green). Fr cmparisn, the successful cntinuatin f B fr V 1 when m h,b2 = 0.2 and m h,b1 = 0.35 (their default values) is shwn by the dashed lines. 5.2.1 Sensitivity analysis f m h Setting the secndary values m h,b1 = m h,b2 = 0 led t ISIs with the same increasing trend but a cnstant ffset f between +0.01 and +0.02s fr all ḡ SynS values tested, with a greater discrepancy as spike numbers increase. This discrepancy increased the verall number f spikes per burst by up t 5. When nly m h,b2 = 0 the initial ISIs clsely matched thse shwn in Figure 6, but after spike 20 the steeper upward trend persisted. The largest ISIs in these tests were all 0.22s. These results are cnsistent with H2, in that the fit between the ISIs f M R and M F is mstly due t the primary value m h,i2. The results als supprt H4, that the decline f I h during B plays a rle in cntrlling the increase f ISIs. As nted abve, setting m h 0.45 led t lnger bursting states in the reduced neurn with a smaller increase in ISI. In particular, fr ḡ SynS < 250 the maximum ISI was n mre than 0.03s smaller than that shwn in Figure 6, but fr larger ḡ SynS neurn 1 entered a tnic spiking mde with cnstant ISIs sufficiently small t keep neurn 2 suppressed indefinitely. These results supprt the suggestin by Hill et al. that ISIs in neurn 2 must increase t a critical value ( 0.22s) in rder t first release neurn 1 s that it may escape [14].
Inferring and quantifying a mechanism fr half-center bursting 11 a ISI (s) ISI (s) ISI (s) b c Spike number Fig. 6 Inter-spike intervals (ISIs) during the bursting state as a functin f spike number within a burst, cmpared between mdels M F (dashed lines) and M R (slid lines). a) ḡ SynS = 100, b) ḡ SynS = 200, c) ḡ SynS = 300. 5.3 Respnse t perturbatins Under typical circumstances, a CPG shuld cntinue t functin rbustly in spite f mild perturbatins t its state. Perturbatins may arise in the frm f small, nisy changes in synaptic timing r efficacy, r similar tempral variatins in ther parameters r state variables. In this sectin we cnsider the phasic dependence f this CPG t small perturbatins in vltage nly, and hw this changes as we adjust the reduced representatin f I h in M R. Our aim is t determine whether I h has a rle in creating rbustness by measuring the similarity in phase respnse between M F and M R. We cnsider the phase respnse f the circuit t weak current-pulse perturbatins I applied (t) applied t neurn 1, measured using the direct frm f phase respnse curve (PRC) defined in [11]. The pulses were applied at spike-triggered times alng the burst cycle, with a duratin f 0.3s that is lnger than any ISI within a burst. This minimizes artifactual differences between the mdels that arise frm a sensitivity in respnse t the perturbatin s precise timing relative t spikes [23]. Phase psitins ϕ were selected at the next clsest spike time t all spikes frm the bursting state f neurn 1 when it was in B, therwise frm neurn 2 (making a ttal f 64 74 perturbatins). Spike times were detected accurately during simulatin using a vltage threshld f -0.02V. The step pulse had amplitude 0.015pA, while its sign was negative (psitive) fr excitatry (resp., inhibitry) stimulatin. The chsen amplitude was small enugh s that inhibitin at the cannical parameter values wuld nt elicit catastrphic failure f B (transitining t I) and excitatin wuld nt elicit immediate entry t B frm I. (This wuld require additinal transitin rules in the mechanism template, e.g. fr B1 I1, and is beynd the scpe f this wrk.) Thus, these perturbatins are much weaker than thse naturally made thrugh synaptic cupling between the tw neurns (as discussed in Sectin 5.1.1). The phase difference ϕ n the next cycle (heren dented 1 PRC) and the subsequent cycle (2 PRC) in cmparisn t an unperturbed cycle was determined in the fllwing way. The next nsets t B were measured accurately using burst
12 Rbert Clewley nset events in the simulatins, and the difference between successive times t begin B were divided by the unperturbed cycle perid. ϕ > 0 (< 0) indicates an delay (resp., advance) in the start f the next cycle. Figs. 7 9 shw three representative sets f data as ḡ SynS was varied fr M R. B1 begins at ϕ = 0, and the ther transitin phases f the ther hybrid mdel are marked by small rectangles in the bttm panels f the figures. The PRCs indicate little sensitivity f the CPG fr mst phases except fr ϕ in the ranges frm 0.35 0.45 and 0.8 1.0, crrespnding t the latter parts f the bursting r inhibited states. At mst phases there is qualitative similarity in the respnses f mdels M F and M R fr all values f ḡ SynS, althugh M R shws a much smther respnse as ϕ varies. Particular lcal trends cmmn t bth mdels are indicated by red arrws in the figures. Generally, these results supprt H1, H5 and H6, as bth the reduced and unreduced neurns respnded similarly. Althugh the stimulus ccurs ver a duratin lnger than any ne ISI and averages sme effects f the discrete step change in m h between regimes in M R, there remains a marked accentuatin in the primary phase respnses f M R fr ϕ in the range frm 0.35 0.45 fr inhibitry perturbatins and ϕ > 0.8 fr excitatry perturbatins. This includes sme sign errrs in the 1 excitatry PRC f M R cmpared t that f M F, althugh these are largely mitigated by the beginning f the secnd cycle (shwn by the cmparisn f 2 PRCs). The discrepancy in 1 PRCs decreases with increasing ḡ SynS, suggesting that synaptic inhibitin can mask artifacts intrduced in the prcess f reductin t M R. The decrease in discrepancy was measured by cmparing lcal minima r maxima in the 1 PRCs in the afrementined ranges f ϕ. In the case f the excitatry PRCs, the lcal minima fr ϕ > 0.8 were measured. As ḡ SynS increased frm 100, thrugh 200, t 300, the difference in these lcal minima changed frm 0.056, thrugh 0.046, t 0.032, respectively. Fr the inhibitry PRCs, the sizes f the lcal minima immediately prir t the lcal maxima at ϕ 0.4, and the sizes f thse lcal maxima, were bth cmpared. Fr the same three ḡ SynS values in increasing rder, the difference in the minimum and maximum psitins given as pairs were (0.023, 0.049), (0.021, 0.044), and (0.001, 0.031), respectively. A surprising bservatin frm Figs. 7 9 is that there can be similar phase respnse effects in bth mdels frm bth excitatin and inhibitin arund certain phases, such as ϕ 0.4 in bth mdels and ϕ 0.9 in M F. A preliminary examinatin f the number f spikes in the bursts f each neurn immediately fllwing the perturbatin prvides sme insight as t the reasn. The 1 PRC measures the prprtinal change in the perturbed cycle s duratin cmpared t the unperturbed cycle. Fr this HCO, the increasing ISIs fllw a typical pattern (Fig. 6), and s we expect a carse crrespndence between the tempral duratin f the cycle and the ttal number f spikes per burst in the tw neurns befre the cycle repeats. Als, within the cycle including the perturbatin, a change in the number f spikes per burst in each neurn f the pair prvides mre detailed tempral infrmatin abut the circuit s respnse than the PRC. Fr instance, we cnsider ḡ SynS = 200 and ϕ = 0.4, fr which the unperturbed number f spikes per burst is 33. Fr the 1 excitatry PRC, the number f spikes in the perturbed burst f neurn 1 was 35, and the number fr the next burst f neurn 2 was 37. This is a ttal change f +6, resulting in ϕ 0.02. Fr inhibitin, the number f spikes in the crrespnding bursts were 30 and 51, respectively, giving a ttal change f +15, resulting in ϕ 0.04. Thus, in bth cases the immediate respnse was intuitive: excitatin led t a lnger B while inhibitin made it shrter. But in bth cases there appears t be a lnger-term cmpensatry mechanism that leads t a lnger I after B is either shrtened r lengthened, sufficient t create a lnger verall cycle. Similarly cunter-intuitive phase-dependent effects f excitatin and inhibitin have been bserved in islated bursting neurns by Sherwd and Guckenheimer [23], which were analyzed using gemetrical arguments based n a fast-slw decmpsitin [21]. In rder t minimize such effects here, the phases at which perturbatins were applied were chsen t crrespnd t the beginning f a spike. Nnetheless, ISIs increase during B s that the time at which a perturbatin ends relative t a later spike is nt cnsistent. Thus, the situatin may be similar t that in [23], in that it invlves small changes in perturbatins relative t spike times, which alter the number f spikes during B, and hence cause a ptentially large change in the duratin f B (see Fig. 5 fr a similar example). A full analysis f this situatin wuld invlve details f the fast sub-system fr spike generatin in additin t the lnger-term cmpensatry effects in the ther neurn. The time scale f spikes is faster than thse cnsidered in the present reductin, which des nt islate the spiking dynamics, and is beynd the scpe f this wrk. Additinally, the smther respnse f M R suggests that the detailed tempral dynamics f m h activatin cntributes t the small fluctuatins in ϕ thrugh small changes in the number f spikes per burst. Frm a quantitative perspective this may challenge the assumptin that smthness f I h is unnecessary (H5), but the
Inferring and quantifying a mechanism fr half-center bursting 13 1 2 V 1, V 2 Δϕ Δϕ V, V 1 eprc 2 eprc 1 iprc 2 iprc * * ϕ Fig. 7 Cmparisn f 1 and 2 excitatry (eprc) and inhibitry (iprc) phase respnse curves between mdels M F (blue vltage traces and PRCs) and M R (black vltage traces and PRCs) fr ḡ SynS = 100, as a functin f the phase f step-pulse perturbatin. All perturbatin phases are spike-triggered (see main text fr details). Fr reference, the tp panel shws vltage traces fr ne full cycle f each neurn in the HCO f M F, and the bttm panel shws ne full cycle fr M R (V 2 is always in green, bursting after neurn 1). V 1 always crrespnds t the perturbed neurn, V 2 t the unperturbed neurn. Rectangular markers in the bttm panel indicate transitin times between regimes fr M R. Red arrws indicate psitins f qualitatively similar trends in the respnses. Secndary PRCs fr M R are shwn with lnger dashes, and bth PRC traces fr M R are marked with an asterisk in a regin where they are clse t each ther. brader functinal cnsequence f such a fine-grained and weak difference is unclear, and shuld be tested against mre sphisticated metrics. 5.3.1 Sensitivity analysis f m h When the secndary values m h,b1 and m h,b2 were set t zer there were n differences in either the excitatry r inhibitry PRCs fr ḡ SynS = 100, 200, r 300 abve the backgrund level f fluctuatin (±0.02) seen in Figs. 7 9 fr M F. Recall that in Sectin 5.1 we fund that in this situatin rhythms failed t exist fr 3 f the 11 ḡ SynS values tested. Hwever, as fr thse metrics, when the rhythms d exist, the lack f sensitivity we find fr the PRCs here suggests n rle fr I h in the rbustness f rhythms t weak perturbatins during B1 r B2. Increasing m h,i1 frm zer made n significant change t any f the PRCs. The accentuated respnses fr ϕ 0.4 in M R invlve neurn 1 receiving a perturbatin during B2. Thus m h,b2 was varied individually t explre whether mdulatin f I h specifically arund the time f perturbatin
14 Rbert Clewley V 1, V 2 Δϕ Δϕ V 1, V 2 1 eprc 2 eprc 1 iprc 2 iprc * * ϕ Fig. 8 Cmparisn f 1 and 2 excitatry and inhibitry PRCs between mdels M F and M R fr ḡ SynS = 200. See legend fr Figure 7. culd cntrl the discrepancy in the phase respnses. In particular, an increase in m h,b2 culd be expected t decrease the ISIs during B2 and ensure greater dminatin f the bursting state by spiking currents s that perturbatins wuld have less effect. Hwever, setting m h,b2 t either 0.35 r 0 frm the default value f 0.2 made n significant difference t the PRCs abve the backgrund level f fluctuatin. The accentuated respnses fr ϕ > 0.8 in M R invlve neurn 1 receiving a perturbatin during I2. Thus m h,i2 was varied individually t explre this effect. Decreasing this value t 0.45 further accentuated the respnse by 2% fr all ḡ SynS values in the excitatry PRC (and als further accentuated the respnse at ϕ 0.45 in the inhibitry PRC). At ḡ SynS = 300 an additinal effect was a greater separatin between the 1 and 2 PRCs f 0.025. On the ther hand, when m h,i2 was increased t 0.6 (which is clse t the peak value fr m h fr M F at ḡ SynS = 300), the nly significant change was unifrmly greater separatin between all the 1 and 2 PRCs by 0.01. We can cnclude that the results f all the perturbatin tests d nt invalidate the hyptheses H1 H4 cncerning the phasic nature f I h, and that the PRC metric succeeds in demnstrating an imprtant functinal similarity f mdel M R with M F. In additin, the results f the sensitivity tests lead us t cnclude that there is n direct rle f I h at the time f perturbatins arriving at the end f either B r I. Instead, we may expect a delayed cnnectin between such perturbatins and the dynamics that alter the next cycle s phase. In additin, the similarity between effects f excitatin and inhibitin at the end f B r I can be cnjectured t invlve different cmpensatry mechanisms between the cupled neurns that take place ver an entire cycle perid.
Inferring and quantifying a mechanism fr half-center bursting 15 1 2 V 1, V 2 Δϕ Δϕ V, V 1 eprc 2 eprc 1 iprc 2 iprc * * ϕ Fig. 9 Cmparisn f 1 and 2 excitatry and inhibitry PRCs between mdels M F and M R fr ḡ SynS = 300. See legend fr Figure 7. 6 Discussin In this wrk we frmulated and tested simple hyptheses that elucidate the detailed rle f the hyperplarizatin-activated current I h in creating functinally rbust rhythms in a central pattern generatr (CPG) mdel f the leech heartbeat. This mdel uses a half-center scillatr (HCO) architecture f tw inhibitry interneurns. Previus investigatins f this CPG thrugh mdeling studies (e.g., [14]) and experimental studies (e.g., [24]) have yielded pssible mechanisms invlving I h in brad natural language terms. Here, ur primary gal was t take advantage f an existing mdel and previus studies f its prperties as an expsitry case study: we seek t validate the previus natural language descriptins thrugh a mre frmal apprach, and imprve upn their clarity, rbustness, and specificity. A secndary gal was t gain sme fresh insights int the mechanisms invlving I h that were previusly paque. 6.1 Insight int the escape-release mechanism T achieve ur primary gal, we demnstrated a cmputatinal prtcl fr the hypthesis-driven testing f a frmal descriptin f the rle f I h in an escape-release mechanism. This used a reductin t a hybrid dynamical systems mdel. The tests were made with respect t five metrics that quantify essential aspects f rhythmic behavir, including the nvel cnsideratin f phase respnse f this circuit t mild perturbatins.