How does maintenance of network activity depend on endogenous dynamics of. Network homeostasis and endogenous neuronal dynamics

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1 TITE How does maintenance of network activity depend on endoenous dynamics of isolated neurons? Andrey V. Olyper and Ronald. Calabrese Department of Bioloy, Emory University, Atlanta, Georia SHORT TITE Network omeostasis and endoenous neuronal dynamics Correspondence to: Andrey V. Olyper Department of Bioloy, Emory University 1510 Clifton Road N.E. Atlanta, GA pone: (404) fax: (404) /27

2 Network omeostasis and endoenous neuronal dynamics Abstract Robust activity of some networks, suc as central pattern enerators, suests te existence of pysioloical mecanisms wic maintain te most important caracteristics, for example, te period and spike frequency of te pattern. Watever tese mecanisms are, tey cane te appropriate model parameters to or alon te isomanifolds on wic te caracteristics of te pattern are constant, wile teir sensitivities to parameters may be different. Settin synaptic connections to zero at te points of isomanifolds allows for dissectin te maintenance mecanisms into components involvin synaptic transmission and components involvin intrinsic currents. Te pysioloical meanin of te intrinsic current canes mit be revealed by analysis of teir impact on endoenous neuronal dynamics. Here, we sout answers to te followin two questions: 1) Do parameter variations in insensitive directions, i.e. alon isomanifolds, cane endoenous dynamics of te network neurons? 2) Do sensitive and insensitive directions for network pattern caracteristics depend on endoenous dynamics of te network neurons? We considered a leec eartbeat alf-center oscillator model network, and analyzed isomanifolds on wic te burst period and/or spike frequency of te model are constant. Based on our analysis, we ypotesize tat te dependence on endoenous dynamics of te isolated neurons is te stroner te more caracteristics of te pattern ave to be maintained. We also found tat in eneral, te network was more flexible wen it consisted of endoenously tonically spikin rater tan burstin or silent neurons. Finally, we discuss pysioloical implications of our findins.

3 Olyper & Calabrese 1 Introduction Neuronal networks must function reliably yet be flexible enou to cane teir activity smootly or even switc from one functional reime to anoter wen necessary. Tese contrastin properties can be produced by a variety of intrinsic and extrinsic mecanisms, suc as omeostatic reulation and neuromodulation (Calabrese, 1998; Marder & Tirumalai, 2002; Turriiano & Nelson, 2004; Marder & Goaillard, 2006). In particular, as suested by Goldman et al. (2001), tere mit be mecanisms tat prepare a neuron for te subsequent action of neuromodulators by canin sensitivities of te activity caracteristics to parameter canes. Te aim of te present study was to use te concepts of sensitive and insensitive directions for te analysis of te mecanisms reulatin network activity. Network connections modify te dynamical repertoire of a neuron considerably. For example, in central pattern enerators suc as leec eart timin network and te Pre-Bötziner complex, essential to te eneration of respiratory rytm in mammals, neurons in te network burst in a muc wider rane of parameters tan wen isolated (Butera et al, 1999; Cymbalyuk et al., 2002). In our study, we explored te relations between te sensitivities of te network activity caracteristics and dynamics of isolated neurons. We considered two specific questions. 1) Do parameter variations in insensitive directions cane te dynamics of isolated network neurons? 2) Do sensitive and insensitive directions for network pattern caracteristics depend on endoenous dynamics of te network neurons? We based our analysis on te notion of an isomanifold, a manifold in te parameter space on wic functional caracteristics of te neuronal system are constant (Olyper & Calabrese, 2007). Insensitive directions at a point are tanent to te isomanifold at te point. In tese directions none of te caracteristics, constant on te isomanifold, cane. Te most sensitive 3/27

4 Network omeostasis and endoenous neuronal dynamics direction for a functional caracteristic at a point is, by definition, te direction of te caracteristic s radient at tat point. FIG.1. Electrical activity of a leec eartbeat alf-center oscillator. (A) Simultaneous intracellular recordins from two participatin leec eart interneurons, HNs, in te tird anlion. (B) Simulations of te alf-center oscillator model, used in te study, for canonical values of parameters. Burst period, T, defined ere as te time interval between te middle spike of one burst and te middle spike of te next burst, is a pysioloically important caracteristic of te network; it is equal to te period of te leec eartbeat. We considered a network, consistin of two reciprocally inibitory neurons tat paces te leec eartbeat ( Kristan, Calabrese, & Friesen, 2005). Te network produces a caracteristic pattern of activity, alf-center oscillations, wit te two neurons burstin alternately (Fi. 1). We considered a well-developed model of tis alf-center oscillator, HCO, described in Hill et al. (2001). Our focus was on te isomanifolds on wic te burst period and spike frequency - te most important functional caracteristics of te network were constant wile parameters

5 Olyper & Calabrese critically affectin tese caracteristics, varied. As was sown in our earlier experiments and simulations (Hill et al., 2001; Cymbalyuk et al., 2002; Sorensen et al., 2004; Olyper, Cymbalyuk, & Calabrese, 2006) tose parameters are te maximal conductances of te yperpolarization-activated, spike-mediated synaptic, and leak currents, and te rate of inactivation of a low-tresold slowly inactivatin calcium current. To explore te interplay between dynamics of te isolated network neurons and te mecanisms, maintainin te burst period and spike frequency, we analyzed te isomanifolds for te networks composed of endoenously burstin, silent, or tonically spikin neurons. We found tat insensitive directions and sensitivities in te HCO model do depend on endoenous dynamics of te network neurons. On te basis of our analysis, we ypotesize tat tis dependence is te stroner te more caracteristics of te pattern are maintained. We also found tat in eneral, a network of endoenously tonically spikin neurons is more flexible tan a network of endoenously burstin or silent neurons. Finally, we discuss pysioloical implications of our analysis. 2 Metods and Teory 2.1 Te model We used te Hill et al. (2001) model of te HCO based on te Hodkin-Huxley formalism, wit five inward and tree outward voltae-dependent currents. Amon te model parameters, as in Olyper, Cymbalyuk, & Calabrese (2006) and Olyper & Calabrese (2007), we considered a scalin factor, η, for te inactivation time constant of a slowly inactivatin lowtresold calcium current, I CaS. By definition, η reater (less) tan one slows down (speeds up) te inactivation of I CaS. Wen a parameter in te model was varied, it was varied in bot neurons simultaneously. As in Olyper & Calabrese (2007), te Hill et al. (2001) model was 5/27

6 Network omeostasis and endoenous neuronal dynamics slitly modified to make it smoot wit respect to te variables and parameters. Namely, in te expression for te total calcium current max( 0, x ) was substituted by f ( x) x wit a smoot simoid function f (x). Te effect of te neuropeptide FMRFamide was modeled by an additional K + current described in (Nadim & Calabrese, 1997; Hill et al., 2001). 2.2 Details of te simulations To quantify te system s activity for a particular set of parameters, te model was simulated for 150 seconds. Te first 50 seconds of te simulation were considered as an interval sufficient for stabilizin te pattern and were discarded from analysis. To determine stable reimes of isolated neurons, te model was simulated for 700 seconds. Te initial conditions for all simulations were te same. Simulations were performed wit te Matlab (MatWorks, Natick, MA) solver ode15s wit te absolute and relative tolerances and 10 respectively. Te Matlab code is available at ttp://calabreselx.bioloy.emory.edu/pub/hc.zip. In our analysis, we focused on te burst period and spike frequency as te most important functional caracteristics of te network activity. In te livin system, te burst period sets te eartbeat period and te spike frequency determines te level of inibition bot witin te HCO and between te oscillator interneurons and teir motor neuron tarets (Hill et al., 2001). Te followin definitions were used. Te period is te time between te middle spikes of consecutive bursts. Te spike frequency is a number of spikes in a burst divided by te burst duration. Burst duration is a time interval between te first and te last spike of a burst. A sensitivity of a caracteristic wit respect to a parameter is a cane, in percent, of a caracteristic caused by a one percent cane of te parameter. Sensitivities were considered to be proportional to partial derivatives; see details in Olyper & Calabrese (2007).

7 Olyper & Calabrese To find te isocurve, on wic te period T of te model is equal to frequency F is equal to * T, and te spike * F, we used te Matlab (MatWorks, Natick, MA) function fminsearc to minimize + * 2 * ( T T ) ( F F ) 2. We stopped te minimization wen bot T and F were less tan 0.5% different from teir taret values. Initial approximations for new points were based on previously found points of te isocurves. New points were calculated wit te step of equal to 30 ns. In some cases, te continuation of an isocurve to certain values of was impossible. We cecked tis by calculatin te period and te spike frequency at a fine rid in a lare domain around te last point found on te isocurve. To build te isoperiod and isofrequency curves in te plane (, E ) we used te Matlab function contour. 2.3 Teory Our main idea is tat watever te bioloical mecanisms maintainin te activity pattern of a network are, teir effect can be modeled as movin parameters of te network back to te isomanifold on wic te caracteristics of te pattern ave taret values. Te sensitivities of te caracteristics to parameter canes from teir values on te isomanifolds can vary alon te isomanifolds. Wit decreased/increased sensitivities te network is more robust/flexible to subsequent modulations. By studyin wat co-variations of parameters form tese isomanifolds and ow sensitivities of te activity pattern caracteristics cane at te points of te isomanifold, one can terefore better understand mecanisms reulatin network activity. Te subspace of parameters in wic te isomanifolds were considered in tis study, included te maximal conductances of te spike-mediated, currents. Settin, and raded, SynG, synaptic and SynG to zero allowed us to dissect te maintenance mecanisms into components involvin synaptic transmission and components involvin intrinsic currents. 7/27

8 Network omeostasis and endoenous neuronal dynamics 3 Results 3.1 Maintenance of burst period and of spike frequency in te face of variation of leak current parameters conductance, First, we found isoperiod and isofrequency curves in te plane of te maximal, and te reversal potential, E, of te leak current. Cymbalyuk et al. (2002) determined te domain in te plane (, E ) were te HCO model exibits alf-center oscillations, and subdomains were te neurons burst, tonically spike, or are silent, wen isolated. In te plane (, E ), insensitive directions for te burst period and spike frequency are directions, tanent to te isoperiod and isofrequency curves respectively. Sensitive directions are te directions ortoonal to te isocurves. For te period, insensitive and sensitive directions were similar in te wole burstin area. Te isofrequency curves bend visibly at te border between tonic spikin and silence. In oter words, maintainin te burst period or its efficient cane require similar co-variation of and E for all tree types of endoenous neuronal dynamics. Te maintenance of spike frequency or its efficient cane require similar co-variation of and E in te HCOs consistin of burstin and tonically spikin neurons, but different co-variation in te case of silent neurons. Te isoperiod and isofrequency curves ad different tanent slopes almost everywere. Consequently, it is impossible to vary and E, and preserve bot te period and frequency. Indeed, to et a one-dimensional isomanifold on wic two caracteristics are maintained, at least tree parameters ave to be co-reulated; two of tem, like and E, are not sufficient (Olyper & Calabrese, 2007). In te next simulations, we cose tese tree parameters to be

9 Olyper & Calabrese,, and η because we sowed previously tat tese parameters effectively control te period of te network (Nadim & Calabrese, 1997; Hill et al., 2001; Sorensen et al., 2004; Olyper, Cymbalyuk, & Calabrese, 2006). 3.2 Isocurves of constant burst period and spike frequency in te space of, η, and In (Olyper & Calabrese, 2007), we found te isocurve in te space of, η, and for te HCO model composed of burstin neurons wit = 9. 9 ns and E = mv. On tat isocurve, te burst period was equal to 7.91 s and te frequency was equal to 8.82 Hz. Here we compared tis isocurve wit te isocurves for te HCO models wit tonically spikin or silent neurons. Usin te results of Cymbalyuk et al. (2002) we set = 8. 0 ns and E = 60 mv, and = 9. 9 ns and E = mv (cf. Fi.2) for tese models respectively. Te oter parameters of te models at te initial point wit = 150 ns ad canonical values, in particular = 4 ns and η = 1. In wat follows, we refer to tese models as consistin of oriinally tonically spikin or silent neurons. It was not a priory clear if te variations of parameters alon te isocurves would conserve oriinal types of endoenous neuronal dynamics. 9/27

10 Network omeostasis and endoenous neuronal dynamics FIG. 2. Isoperiod and isofrequency curves of burst activity in te leec eartbeat HCO model in te plane of te leak current parameters. (A) Isoperiod curves for te period. (B) Isofrequency curves for te intraburst spike frequency. Te tick black curve circumscribes te domain were te model exibits alf-center oscillations. Witin te burstin area, tere are subdomains were isolated model interneurons spike tonically (lit ray), burst (dark ray), or are silent (wite). A tin stripe between intrinsic burstin and quiescence is a subdomain of multistability (cf. Cymbalyuk et al., 2001). Te taret values of te burst period, T, and spike frequency, F, were taken from te HCO simulations for = 150 ns: T = 8.58 s and F = Hz for tonically spikin neurons, and T = 7.99 s and F = 8.33 Hz for silent neurons. Te cosen values of te burst period and spike frequency, tou arbitrary, lie witin te ranes s and Hz observed in te experiment by Cymbalyuk et al. (2002). Te isocurves for tese models are sown in Fi. 3.

11 Olyper & Calabrese In wat follows tese isocurves are referred to as N-isocurves wit N standin for te coice of te natural values of T and F maintained at tese isocurves. FIG. 3. Te isocurves of te constant period and spike frequency of te HCO model for tree networks wit te different taret values of te period and spike frequency, and for = 150 ns, consistin of neurons wit tree different types of endoenous dynamics, N- isocurves. For = 150 ns, te HCO models were different only in te parameters of te leak current, (, E ), wic were (9.9 ns, mv) for burstin neurons, solid curve, (8 ns, -60 mv) for tonically spikin neurons, dased curve, and (9.9 ns, mv) for silent neurons, dotted curve. All te oter parameters ad canonical values. Te taret values of te period and frequency were 7.91 sec and 8.82 Hz (burstin neurons, B ), 7.99 sec and 8.33 Hz (tonically spikin neurons, T ), 8.58 sec Hz (silent neurons, S ). (A) Te N-isocurves in te space of,, and η. Note, tat all tree isocurves cross at te same point, correspondin to te canonical values = 150 ns, = 4 ns, and η = 1. (B) Planar projections of te isocurves. Note, tat te dependence between and η was qualitatively te same for all te tree types of networks: η decreased proportionally to te increase of. Note also a wide rane of values wic could be tolerated by te network of tonically spikin neurons. Tis rane was muc wider tan for te networks of burstin and silent neurons. 11/27

12 Network omeostasis and endoenous neuronal dynamics N-isocurves demonstrated tat te tree HCO models sustained different variations of parameters. Especially distinctive were te ranes of and tonically spikin neurons te networks sustained lare variations of oriinally silent neurons, and η could compensate variations of and. Wit oriinally burstin, wile wit in te rane ns only (Fi. 3B, left). On te oter and, te networks of oriinally burstin and oriinally silent neurons sustained only small variations of (Fi. 3B, rit). N-isocurves also revealed tat in te tree network models te co-reulation implied a similar dependence of η on : reater values of required almost proportionally smaller values of η (Fi. 3B, middle). Could te taret values for period and spike frequency influence te rane of inibition for wic te network models wit oriinally tonic spikers and silent neurons can maintain tese values? To answer tis question we cose te taret values for tese networks to be te same as for te network of oriinally burstin neurons, i.e., T=7.91 s and F = 8.82 Hz. In wat follows te resultin isocurves (Fi.4) are referred to as C-isocurves wit C standin for te canonical taret values of T and F.

13 Olyper & Calabrese FIG. 4. Te isocurves of te constant period and spike frequency of te HCO model for tree networks wit te same taret values of te period and spike frequency, C-isocurves. Te network models were different in te parameters of te leak current, (, E ), wic were (9.9 ns, mv), solid curve, (8 ns, -60 mv), dased curve, and (9.9 ns, mv), dotted curve. As in Fi.3, tese tree pairs of parameters corresponded to tree types of endoenous neuronal dynamics, burstin, tonic spikin, and silence, provided all te oter parameters of te model ad canonical values. Te taret values of te period and frequency were 7.91 sec and 8.82 Hz for te all tree networks. (A) Te C-isocurves in te space of,, and η. Note, tat te tree isocurves do not cross at te point, correspondin to te canonical values = 150 ns, = 4 ns, and η = 1. Te HCOs of tonically spikin neurons and silent neurons could ave te same values of period and frequency for te canonical values of,, and η as te HCO composed of bursters. In fact, in te case of tonically spikin neurons, tere were no values of, and η for wic te HCO model could ave taret values of te period and burst frequency for = 150 ns. (B) Planar projections of te isocurves. Note, tat te dependence between and η was qualitatively te same for all te tree types of networks: η decreased proportionally to te increase of. However, te dependence between and η for te tonically spikin neurons sifted to smaller values of η. Note tat a rane of wic could be tolerated by te network of tonically spikin neurons, was smaller in tis case compared to Fi.3. 13/27

14 Network omeostasis and endoenous neuronal dynamics Te period and spike frequency of te network models wit oriinally silent and tonically spikin neurons, as expected, did not ave te taret values of T and F for te canonical values of parameters. For = 150 ns, te network of oriinally silent neurons ad te taret values of T and F wen and η were equal to 6.13 ns and 1.18 respectively. In te case of te oriinally tonically spikin neurons and = 150 ns, tere were no values of and η for wic te HCO model ad te taret values of T and F. Te ranes of inibition for wic te network could maintain te canonical period and spike frequency by co-variation of and η caned differently dependin on te oriinal type of neuronal dynamics. In te network model wit oriinally silent neurons, te rane of rew to ns. In te network model wit oriinally tonically spikin neurons, it srank to ns. Te proportionality between and η remained. 3.3 Endoenous neuronal dynamics on te isocurves of constant burst period and spike frequency To furter caracterize te mecanisms of activity pattern maintenance, we explored weter maintenance of period and spike frequency on te isocurves was acieved by canin te oriinal neuronal dynamics. Te simulations of te isolated model neurons ( = 0 ns) wit, and η at te points alon te isocurves ave te followin results. Oriinally tonically spikin neurons exibited tonic spikin at all te points alon te N- and C-isocurves. Despite considerable canes of and η alon te isocurves, te frequency of tonic spikin varied only a little: Hz for te N-isocurve and Hz for te C-isocurve.

15 Olyper & Calabrese Oriinally silent neurons remained silent at most of te points of te isocurves. On te N-isocurve, te isolated neurons were silent at points wit in te rane of ns, includin = 4 ns, η = 1, and = 150 ns were tey were silent by definition. At two oter points of te isocurve wit equal 30 ns or 60 ns, te isolated neurons turned into bursters, and ad a period of 7.41 s and 7.19 s and a frequency of 8.43 Hz and 8.42 Hz respectively. In te case of te C-isocurve, oriinally silent neurons turned into bursters for rane ns. For oter points, wit be silent. Wen in te in te rane ns, te neurons remained to was in te rane ns, te burst period of te isolated neurons monotonically decreased from 7.91 s to 6.28 s; te spike frequency almost monotonically decreased from 9.11 Hz to 7.47 Hz. Finally, we studied te dynamics in HCOs wit oriinally burstin neurons. As we sowed in (Olyper & Calabrese, 2007), te endoenous burstin of te isolated neurons was observed at te points of te isocurve. In te present study, we found tat te period and spike frequency in te isolated burstin neurons decreased wit increasin compensatin inibition. Te period almost linearly decreased from 8.07 s to 5.90 s wit increasin. For all te points wit 90 ns, te period was less tan 7.91 s. Te spike frequency at tose points monotonically decreased from 6.90 to 6.26 Hz. 3.4 Isocurves of constant burst period and spike frequency in te space of, η, and for different values of 15/27

16 Network omeostasis and endoenous neuronal dynamics Te previous set of simulations sowed wat canes in individual neurons allow te HCO to maintain te period and spike frequency despite te canes of reciprocal inibition. Are tese canes different in te presence of a neuromodulator? We modeled te effect of te endoenous neuropeptide FMRFamide by addin a slowly activatin and deactivatin outward current I (Nadim & Calabrese, 1997; Hill et al.,2001) wit te maximum conductance = 2.5, 5, 7.5, 10, 15, and 20 ns. Te isocurves for te HCO model consistin of oriinally burstin neurons and different values of are sown in Fiure 5. As in previous simulations of oriinally burstin neurons = 9. 9 ns and E = mv. Te taret values were 7.91 s for te burst period, and 8.82 Hz for te spike frequency. In te absence of compensatin covariation of and η, i.e., in te network wit te canonical values of = 4 ns and η = 1, wen = 20nS and = 360 ns te period was equal to 6.45 s and te spike frequency was equal to 8.87 Hz. Wit te increase of te taret values of te period and spike frequency could be maintained only wit te increasinly stron inibition: 270 ns for = 15 ns, and 360 ns for = 20nS. Te increase of decreased te rane of and simultaneously increased te ranes of and η tat could co-vary wit. In particular, for = 0 ns, te ranes of and η producin te maintenance of te burst period and spike frequency were 1.37 ns and 0.33 respectively, wile for = 10 ns te ranes were 5.98 ns and 1.54.

17 Olyper & Calabrese FIG. 5. Te isocurves of te constant burst period and frequency of te HCO model consistin of endoenous bursters for different values of. Te maintained period and frequency on eac isocurve were te same, and were calculated for te canonical set of parameters wit te parameters of te leak current, (, E ) correspondin to endoenous burstin: (9.9 ns, mv). was varied from 0 ns (te litest curve) to 20 ns (te darkest curve) includin te followin values: 0, 2.5, 5, 7.5, 10, 15, 20 ns. (A) Te isocurves in te space of,, and η. Note tat wit te increase of above 10 ns te network model could assume te taret values of te period and frequency only for i values of different viewpoint. Note te inverse order of te. (B) Te same as (A) but from a axis. All te isocurves are restricted to almost parallel planes. (C) Planar projections of te isocurves. Note, tat te dependence between and η was qualitatively te same for all te values of : η decreased proportionally to te increase of. However, te dependence between and η sifted to reater values of η wit te increase of. All te isocurves lie witin almost parallel planes (Fi.5B, Table 1). Tis means tat tere is a stron and similar linear relationsip between te tree parameters for all te values of 17/27

18 Network omeostasis and endoenous neuronal dynamics considered. For example, for =10 ns te relationsip as te form η = 0. Importantly, tese relationsips eld even for reater tan or equal to 15 ns were tere was no apparent linear dependence between η. and (ns) A B C E E E E TABE 1. Planar fittin of te isocurves for HCOs composed of bursters. Fittin plane equation: η = A + B C. + Wen isolated, te neurons fired in bursts at all te points of te isocurves for equal to 0 and 5 ns. Wen was equal to 10 ns, te neurons fired in bursts everywere but at te point wit = 420 ns were te isolated model neuron exibited bistability: one of te isolated neurons fired in bursts wile te oter was silent. Wen = 15 ns and = 420 ns bot isolated neurons were silent. Finally, wen = 20 ns bot isolated neurons were silent at all four points of te isocurve. For all isocurves, te period of te isolated neurons mostly decreased wit te increase of te inibition and was smaller tan te taret value, T = 7.91 s for

19 Olyper & Calabrese te HCO everywere wit te exception of two points wit = 0 ns and equal to 30 or 60 ns. At tese points te period was equal to 7.97 s and 8.07 s respectively. In te livin system, parmacoloically isolated interneurons indeed ave a sorter period tan te period of an intact HCO (Cymbalyuk et al., 2002). 3.5 Period sensitivities to, η, and spike frequency, and on te isocurves of constant burst period Sensitivity alon isocurves caracterizes te period s flexibility to parameter canes from teir values on te isocurves. For te isocurves wit, = 0 by sensitivity we mean partial derivatives of te period wit respect to sensitivities of te period wit respect to eac of te parameters. Our observations sow (Fi.6) tat te,, η, and were qualitatively te same wen te network consisted of oriinally burstin or oriinally silent neurons. Te beavior of sensitivities did not cane qualitatively wen te FMRFamide activated current, I, was added to te model wit endoenously burstin neurons. Te period sensitivity to network. decreased wit increasin, indicatin saturation to s impact on te Te period sensitivities for te HCO models wit oriinally tonically spikin neurons were quite different. Tey were considerably larer and not monotonous. Tis observation implies tat a HCOs consistin of tonic spikers are more flexible tan HCOs consistin of busters or silent neurons: small canes of te parameters cause larer canes in te period. 19/27

20 Network omeostasis and endoenous neuronal dynamics FIG. 6. Te sensitivities of te HCO period to te HCO period to,,, η, and. (A) Te sensitivities of, η, and at te calculated points of te isocurves. Te types of te curves are te same as in Fis Namely, te sensitivities for te model wit = 0, 5, 10, 15, 20 ns are in solid ray wit darker colors for reater values of. Te networks wit te oriinally tonically spikin neurons are dased curves wile te networks wit te oriinally silent neurons are dotted curves, wit ray standin for te N-isocurves (cf. Fi.3), and black standin for te C-isocurves (cf. Fi.4); see te definitions of te N- and C-isocurves in te text. (B) Te sensitivity of te HCO period to ( T ) for = 8, 9.9, and 12.7 ns wit all te oter parameters avin canonical values, includin = 0. Isolated model neurons are endoenously silent (circles), burst (squares), spike tonically (trianles), or bistable (ollow square and circle) (cf. Fi.2). depends on We developed tis result furter by studyin ow te HCO s period sensitivity to E. To tis end, T was calculated, as a function of E, for tree values of

21 Olyper & Calabrese to explore domains wit low ( = 8 ns, [ 70mV, 55mV ]), moderate ( = 9. 9 ns, E E [ 66mV, 55.5mV ]), and stron ( =12. 7 ns, [ 61.5mV, 57.5mV ]) dependence of E T on E (cf. Fi.2); all te oter parameters ad canonical values, includin = 0. Te results sow tat in te first two domains tere is an increase of T wit te increase of E in endoenously tonically spikin neurons (Fi.6B). 4 Discussion Robust activity of some networks, suc as central pattern enerators, in te face of varyin parameters (Marder & Goaillard, 2006) suests te existence of pysioloical mecanisms wic maintain te most important caracteristics, for example te period and spike frequency of te pattern. Watever te bioloical mecanisms maintainin network activity are, tey cane te appropriate model parameters to or alon te isomanifolds on wic te caracteristics of te pattern are constant, wile teir sensitivities to parameters may be different. By studyin te properties of tese isomanifolds, one can terefore understand te mecanisms maintainin te pattern of te network activity in terms of co-variations of te model parameters. Te pysioloical meanin of tese co-variations can at least partially be interpreted by te analysis of teir impact on endoenous neuronal dynamics. In tis study, we sout answers to te followin two questions: 1) Do parameter variations in insensitive directions, i.e. alon isomanifolds, cane endoenous dynamics of te network neurons? 2) Do sensitive and insensitive directions for network pattern caracteristics depend on endoenous dynamics of te network neurons? For a leec eartbeat alf-center oscillator model network we sow tat te answers to tese questions can be positive or neative dependin, in particular, on weter bot te burst 21/27

22 Network omeostasis and endoenous neuronal dynamics period and spike frequency, or only one of tese caracteristics is maintained. In accord wit intuition te dependence on endoenous dynamics of te isolated neurons was te stroner te more caracteristics of te pattern ave to be maintained. We also found tat in eneral, wit endoenously tonically spikin neurons HCOs are more flexible tan wit endoenously burstin or silent neurons. We came to tis conclusion by considerin variations of te several parameters wic accordin to our earlier experimental and modelin studies predominantly control te burst period and spike frequency in leec eart interneurons (Hill et al., 2001; Cymbalyuk et al., 2002; Sorensen et al., 2004; Olyper, Cymbalyuk, & Calabrese, 2006). It is difficult to state definitively wat activity synaptically isolated leec eart interneurons manifest and tus weter tey exploit suc flexibility. Wit intracellular recordin (sarp electrodes) sufficient leak is introduced so tat tese neurons always spike tonically in te i concentrations of bicuculline needed to block inibitory synaptic transmission between tem ( mm), but suc neurons burst normally in HCOs after removal of te bicuculline (Cymbalyuk et al., 2002). Wit extracellular recordin (suction electrodes), synaptic isolation by 1.0 mm bicuculline results in reular burstin tat sometimes is interrupted by sort bouts of tonic spikin (Cymbalyuk et al., 2002). Synaptic isolation by 0.5 mm bicuculline results in irreular burstin often interrupted by tonic spikin for many seconds and burstin is sped and reatly reularized by myomodulin (Tobin & Calabrese, 2005). Tus at te i concentrations needed in te leec, bicuculline itself appears to promote burst capabilities. Wit extracellular recordin and synaptic isolation by eliminatin inibitory inputs wit yperpolarizin voltae clamp of te presynaptic neuron, eart interneuron exibit very slow irreular burstin and burstin is reatly sped and reularized by myomodulin (Tobin & Calabrese, 2005).

23 Olyper & Calabrese 4.1 Maintenance of te burst period or spike frequency in te face of leak current variations is not specific for te type of endoenous neuronal dynamics Te observation tat for a wide rane of te period and spike frequency values, te isoperiod and isofrequency curves crossed te borders between te subdomains of te plane, E ) wit different neuronal dynamics suests tat mecanisms maintainin eiter te ( period or te spike frequency of HCOs are not associated wit dynamics of isolated neurons and can easily cane te latter. Can te leak current of te livin neurons of a eartbeat HCO be selectively altered to ceck te predictions of te model? Tere are modulators tat reulate te leak current parameters toeter wit oter currents. For example, Tobin & Calabrese (2005) sowed recently tat an endoenous leec peptide myomodulin (Wan, Price, & Saley, 1998) decreases Na/K pump current and increases cane of. A cane of Na/K pump current is equivalent to te E (Tobin & Calabrese, 2006). Hence, it is possible tat in te experiment, a selective alterin of te leak current mit be acieved by a specific combination of myomodulin wit Cs + (Masino & Calabrese, 2002) to partially block myomodulin. to compensate its increase caused by 4.2 Maintenance of te burst period and spike frequency in te face of,, and η in te network of endoenously burstin neurons in presence of a neuropeptide FMRFamide We modeled te effect of a neuropeptide FMRFamide by addin an FMRFamide-activated K + current, I, to te model. We found an almost linear relationsip between η for all te isocurves correspondin to te different values of,, and. Te relationsip eld even 23/27

24 Network omeostasis and endoenous neuronal dynamics for reater tan or equal to 15 ns were tere was no linear dependence between and η. Te isolated neurons were bursters at all points of te isocurves. 4.3 Maintenance of te burst period and spike frequency in te face of,, and η variations depends on te type of endoenous neuronal dynamics Te relations between te period sensitivity to parameters and te types of endoenous neuronal dynamics, sown in Fi.6, suest a number of testable predictions expandin our understandin of possible interactions between myomodulin and FMRFamide. In particular, i period sensitivity to for HCOs wit oriinally tonically spikin neurons compared to te networks wit oriinally burstin or oriinally silent neurons (Fi. 6A, bottom rit) allows us to make te followin prediction. Because an application of myomodulin increases reularity of burstin in isolated neurons (in particular by eliminatin te intervals of tonic spikin (Tobin and Calabrese, 2005), ten myomodulin application sould decrease te sensitivity of te HCO period to FMRFamide. Furter analysis is required for understandin te detailed mecanism of te FMRFamide and myomodulin interaction. Tis interaction can be quiet complex iven tat te effect of myomodulin on leec eart interneurons may not be restricted to te decrease of Na/K pump current and increase of found by Tobin and Calabrese (2005). Te approac, wic we developed in te present study, is based on a eneral idea of dissectin te mecanisms maintainin network activity into te components involvin synaptic transmission and components involvin intrinsic neuronal ionic currents. Tis approac sould be useful for understandin omeostatic reulation in various networks, especially central pattern enerators, for wic te caracteristics of te pattern to be maintained are often quite obvious. In particular, te analysis of te synaptic (Soto-Trevino et al., 2001) and intrinsic current

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