Ghostbursting: A Novel Neuronal Burst Mechanism

Transcription

1 JCNS Ghostbursting: A Novel Neuronal Burst Mechanism Brent Doiron *, Carlo Laing *, Anré Longtin *, an Leonar Maler # * Physics Department, University of Ottawa, 150 Louis Pasteur, Ottawa, Ontario, Canaa K1N 6N5 # Department of Cellular an Molecular Meicine, University of Ottawa, 451 Smyth Roa, Ottawa, Canaa K1H 8M5 Wor Count Abstract: 225 Wor Count Introuction: 1013 Wor Count Discussion: 1844 Number of Text Pages: 28 Number of Figures: 13 Number of Tables: 1 Corresponing author: Brent Doiron Physics Department University of Ottawa Ottawa, Ontario Canaa K1N 6N5 Phone: (613) (8096) FAX: (613) [email protected] In Press, Journal of Computational Neuroscience Keywors: bursting, electric fish, compartmental moel, backpropagation, pyramial cell

2 Abstract Pyramial cells in the electrosensory lateral line lobe (ELL) of weakly electric fish have been observe to prouce high frequency burst ischarge with constant epolarizing current (Turner et al., 1994). We present a two-compartment moel of an ELL pyramial cell that prouces burst ischarges similar to those seen in experiments. The burst mechanism involves a slowly changing interaction between the somatic an enritic action potentials. Burst termination occurs when the trajectory of the system is reinjecte in phase space near the ghost of a sale-noe bifurcation of fixe points. The burst trajectory reinjection is stuie using quasi-static bifurcation theory which shows a perio oubling transition in the fast subsystem as the cause of burst termination. As the applie epolarization is increase, the moel exhibits first resting, then tonic firing, an finally chaotic bursting behaviour, in contrast with many other burst moels. The transition between tonic firing an burst firing is ue to a sale-noe bifurcation of limit cycles. Analysis of this bifurcation shows that the route to chaos in these neurons is type I intermittency, an we present experimental analysis of ELL pyramial cell burst trains which support this moel preiction. By varying parameters in a way that changes the positions of both sale-noe bifurcations in parameter space we prouce a wie gallery of burst patterns, which span a significant range of burst time scales. 1 Introuction Burst ischarge of action potentials is a istinct an complex class of neuron behaviour (Connors et al., 1982; McCormick et al., 1985; Connors an Gutnick, 1990). Burst responses show a large range of time scales an temporal patterns of activity. Many electrophysiological stuies of cortical neurons have ientifie cells that intrinsically burst at low frequencies (<20 Hz) (Blan an Colom, 1993; Steriae et al., 1993; Franceschetti et al., 1995). However, recent work in numerous systems has now ientifie the existence of chattering cells which show burst patterns in the high frequency γ range (>20 Hz) (Turner et al., 1994; Paré et al., 1995; Gray an McCormick, 1996; Steriae et al., 1998; Lemon an Turner, 2000; Brumburg et al., 2000). Also, the specific inter-spike interval (ISI) pattern within the active phase of bursting varies consierably across burst cell types. Certain bursting cells show a lengthening of ISIs as a burst evolves (e.g. pancreatic-β cells, Sherman et al., 1990), others a parabolic tren in the ISI pattern (e.g Aplysia R15 neuron; Aams 1985), an yet others show no change in the ISI uring a burst (e.g. thalamic reticular neuron; Pinault an Deschênes, 1992). This iversity of specific time scales an ISI patterns suggests that numerous istinct burst mechanisms exist. Knowlege of the burst mechanisms allows one to preict how the burst output may be moifie, or halte completely in 2

3 response to stimuli. This may have consequences for the information content of the cell s output (Lisman, 1997). Pyramial cells in the electrosensory lateral line lobe (ELL) of the weakly electric fish Apteronotus leptorhynchus have been shown to prouce either tonic firing an γ frequency sustaine burst patterns of action potential ischarge (Turner et al., 1994; Turner an Maler, 1999; Lemon an Turner, 2000). These seconary sensory neurons are responsible for transmitting information from populations of electroreceptor afferents that connect to their basal bushes (see Berman an Maler, 1999 an references therein). In vivo recorings from ELL pyramial cells have inicate that their bursts are correlate with certain relevant stimulus features, suggesting the possible importance of ELL bursts for feature etection (Gabianni et al., 1996; Metzner an Gabianni, 1998; Gabianni an Metzner, 1999). Thus, both the proximity of ELL pyramial cells to the sensory periphery, an the known relevance of their bursts to signal etection, suggest that stuies of ELL bursting may provie novel results concerning the role of burst output in sensory processing. Previous in vitro an in vivo experiments have focuse both upon specifying the mechanism for burst ischarge of ELL pyramial cells, an showing methos for the moulation of burst output (Turner et al., 1994, 1996; Turner an Maler,1999; Lemon an Turner, 2000; Bastian an Nguyenkim, 2001; Rashi et al., 2001). Lemon an Turner (2000) have shown that a frequency epenent or conitional action potential backpropagation along the proximal apical enrite unerlies both the evolution an termination of ELL burst output. Recently, through the construction an analysis of a etaile multicompartmental moel of an ELL pyramial cell, we have reprouce burst ischarges similar to those seen in experiment. This moel allowe us to make strong preictions about the characteristics of the various ionic channels that coul unerlie the burst mechanism (Doiron et al., 2001b). However, a eeper unerstaning of the ynamics of the ELL burst mechanism coul not be achieve ue to the high imensionality (312 compartments an 10 ionic currents) of the moel system. The analysis of bursting neurons using ynamical systems an bifurcation theory is well establishe (Rinzel, 1987; Rinzel an Ermentrout, 1989; Wang an Rinzel, 1995; Bertram et al., 1995; Hoppensteat an Izhikevich, 1997; Izhikevich, 2000; Golubitsky et al., 2001). These stuies have reuce complex neural behaviour to flows of low imensional nonlinear ynamical systems. In the same spirit, we present here a two-compartment reuction of our etaile ionic 3

4 moel of an ELL pyramial cell (Doiron et al., 2001a, 2001b). The reuce moel, referre to as the ghostburster (this term is explaine in the text), prouces burst ischarges similar to both the full moel an in vitro recorings of bursting ELL pyramial cells. This analysis supports our previous preictions on the sufficient ionic an morphological requirements of the ELL pyramial cell burst mechanism. In aition to this, the low imension of this moel allows for a etaile ynamical systems treatment of the burst mechanism. When applie epolarization is treate as a bifurcation parameter, the moel cell shows three istinct ynamical behaviours: resting with low intensity epolarizing current, tonic firing at intermeiate levels, an chaotic burst ischarge at high levels of epolarization. This is contrary to other burst mechanisms that show burst ischarge for low levels of epolarization an then transition to tonic firing as applie current is increase (Hayashi an Ishizuka, 1992; Gray an McCormick, 1996; Steriae et al., 1998; Wang, 1999). Both of the bifurcations separating the three ynamical behaviours of the ghostburster are shown to be sale-noe bifurcations of either fixe points (quiescent to tonic firing) or limit cycles (tonic firing to bursting). Treating our burst moel as a fast-slow burster (Rinzel, 1987; Rinzel an Ermentrout 1989, Wang an Rinzel, 1995; Izhikevich, 2000) an using quasi-static bifurcation analysis, we show that the burst termination is linke to a transition from perio-one to perio-two firing in the fast subsystem, causing the burst trajectory to be reinjecte near the ghost of the sale-noe bifurcation of fixe points. The time spent near the sale-noe etermines the inter-burst interval length. This concept of burst ischarge is quite ifferent from the two-bifurcation analysis use to unerstan most other burst moels (Rinzel, 1987; Rinzel an Ermentrout 1989,Wang an Rinzel, 1995; e Vries, 1998; Izhikevich, 2000; Golubitsky et al., 2001). Further analysis preicts that the route to chaos in transitioning from tonic to chaotic burst firing is through type I intermittency (Pomeau an Manneville, 1980). Comparisons of both moel an experimental ELL burst recoring ata supports this preiction. Furthermore, by changing the relative position of the two sale-noe bifurcations in a two-parameter bifurcation set, the time scales of both the burst an inter-burst perio can be chosen inepenently, allowing for wie variations in possible burst outputs. 4

5 2 Methos 2.1 ELL pyramial cell bursting Figure 1A shows in vitro recorings from the soma of a bursting ELL pyramial cell with a constant epolarizing input. The bursts comprise a sequence of action potentials, which appear on top of a slow epolarization of the subthreshol membrane potential. The epolarization causes the inter-spike-intervals (ISIs) to ecrease as the burst evolves. The ISI ecrease culminates in a high frequency spike oublet that triggers a relatively large after-hyperpolarization (AHP) labele a burst-ahp (bahp). The bahp causes a long ISI that separates the train of action potentials into bursts, two of which are shown in Figure 1A. The full characterization of the burst sequence has been presente in Lemon an Turner (2000). Immunohistochemical stuies of the apical enrites of ELL pyramial cells have inicate a patche istribution of soium channels along the first ~200 µm of the apical enrite (Turner et al., 1994). Figure 1B illustrates schematically such a Na + channel istribution over the enrite. The active enritic Na + allows for action potential backpropagation along the apical enrite, proucing a enritic spike response (Figure 1B; Turner et al., 1994). Na + or Ca 2+ meiate action potential backpropagation has been observe in several other central neurons (Turner et al., 1994; Stuart an Sakmann, 1994; for a review of active enrites see Stuart et al., 1997) an has been moele in many stuies (Traub et al., 1994; Mainen et al., 1995; Vetter et al., 2001; Doiron et al., 2001b). Action potential backpropagation prouces a somatic epolarizing after-potential (DAP) after the somatic spike, as shown in Figure 1B. The DAP is the result of a enritic reflection of the somatic action potential. This requires both a long enritic action potential half-with as compare to that of a somatic action potential, an a large somatic hyperpolarization succeeing an action potential. These two features allow for passive electrotonic current flow from the enrite to the soma subsequent to the somatic spike, yieling a DAP. Recent work has shown the necessity of spike backpropagation in ELL pyramial cells for burst ischarge (Turner et al., 1994; Turner an Maler, 1999; Lemon an Turner, 2000). These stuies blocke spike backpropagation by locally applying tetrooxin (TTX, a Na + channel blocker) to apical enrites of ELL pyramial cells, after which all bursting cease an only tonic firing persiste. Our previous moeling stuy (Doiron et al., 2001b), reprouce this result, since when active Na + conuctances were remove from all enritic compartments, similar results were 5

6 obtaine. However, in that stuy we moele the proximal apical enrite with ten compartments, five of which containe active spiking Na + channels. The large number of variables in such a moel is incompatible with the objectives of the present stuy. In light of this, an following previous moeling stuies involving action potential backpropagation (Pinsky an Rinzel, 1994; Bressloff, 1995; Mainen an Sejnowski, 1996; Lánsky an Roriguez, 1999; Wang, 1999; Booth an Bose, 2001), we investigate a two-compartment moel of an ELL pyramial cell, where one compartment represents the somatic region, an the secon the entire proximal apical enrite. Note that a two-compartment treatment of enritic action potential backpropagation is a simplification of the cable equation (Keener an Sney, 1998). However, in consieration of the goals of the present stuy, which require only DAP prouction, the two-compartment assumption is sufficient. 2.2 Two-Compartment Moel A schematic of our two-compartment moel of an ELL pyramial cell is shown in Figure 2, together with the active inwar an outwar currents that etermine the compartment membrane potentials. Both the soma an enrite contain fast inwar Na + currents, I Na,s an I Na,, an outwar elaye rectifying (Dr) K + currents, respectively I Dr,s an I Dr,. These currents are necessary to reprouce somatic action potentials, an proper spike backpropagation that yiels somatic DAPs. In aition, both the soma an enrite contain passive leak currents I leak. The membrane potentials V s (somatic) an V (enritic) are etermine through a moifie Hogkin/Huxley (1952) treatment of each compartment. The coupling between the compartments is assume to be through simple electrotonic iffusion giving currents from soma to enrite, I s/, or vice-versa, I /s. In total, the ynamical system comprises six nonlinear ifferential equations, Eq (1)-(6); henceforth, we will refer to Eq(1)-(6) as the ghostburster moel, an the justification for the name will be presente in the Results section. 6

7 Soma Vs 2 2 (1) = I S + g Na, s m, s ( Vs) (1 ns) ( VNa Vs ) + g Dr, s ns ( VK Vs) t gc + κ ( V Vs ) + gleak ( Vl Vs) (2) n t s n, s( Vs) = τ n, s n s Denrite V 2 2 (3) = g Na, m, ( V ) h ( VNa V ) + gdr, n p ( VK V ) t gc + ( V (1 κ) s V ) + g leak ( V l V ) (4) h t h, ( V ) = τ h, h (5) (6) n t p t n, ( V ) = τ n, p, ( V ) = τ p, n p Table I lists the values of all channel parameters use in the simulations. The soma is moele with two variables (see eq. (1) an (2)). The reuction from the classic four imensional Hogkin- Huxley moel is accomplishe by slaving I Na,s activation, m, s, to V s (i.e the Na + activation m s tracks V s instantaneously), an moeling its inactivation,h s, through I Dr,s activation, n s (we set h 1 s n s ). This secon approximation is a result of observing in our large compartmental moel (Doiron et al., 2001b) that h n 1uring spiking behaviour. Both of these approximations s + s 7

8 have been use in various other moels of spiking neurons (Keener an Sney, 1998). The enrite is moele with four variables (see eq. (3)-(6)). Similar to the treatment of I Na,s, we slave I Na, activation, m,, to V, but moel its inactivation with a separate ynamical variable h. Lemon an Turner (2000) have shown that the refractory perio of enritic action potentials is larger than that of somatic in ELL pyramial neurons. This result has previously been shown to be necessary for burst termination (Doiron et al., 2001b). To moel ifferential somatic/enritic refractory perio we have chosen τ h, to be longer than τ n,s (similar to our large compartmental moel (Doiron et al., 2001b)). This result has not been irectly verifie through immunohistochemical experiments of ELL pyramial cells Na + channels, thus, at present, this remains an assumption in our moel. The crucial element for the success of our moel in reproucing bursts is the treatment of I Dr,. Denritic recorings from bursting ELL pyramial cells show a slow, frequency epenent broaening of the action potential with as a burst evolves (Lemon an Turner, 2000). Such a cumulative increase in action potential with has been observe in other experimental preparations, an has been linke to a slow inactivation of rectifier-like K + channels (Alrich et al., 1979; Ma an Koester, 1996; Shao et al., 1999). In light of this, our previous stuy (Doiron et al., 2001b), moele the enritic K + responsible for spike rectification with both activation an inactivation variables. When the time constant governing the inactivation was relatively long (5 ms) compare with the time constants of the spiking currents (~ 1ms), the moel prouce a burst ischarge comparable to ELL pyramial cell burst recorings. Doiron et al. (2001b) also consiere other potential burst mechanisms, incluing slow activation of persistent soium, however, only slow inactivation of enritic K + prouce burst results comparable to experiment. At this time there is no irect evience for a cumulative inactivation of enritic K + channels in ELL pyramial cells, an these results remain a moel assumption. However, preliminary work suggests that the shaw-like AptKv3.3 channels may express such a slow inactivation (R.W. Turner personal communication); these channels have been shown to be highly expresse in the apical enrites of ELL pyramial cells (Rashi et al., 2001). In the present work, our ynamical system also moels enritic K + current, I Dr,, as having both activation, n, an slow inactivation p variables (see Eqs. (3),(5), an (6)). Slow inactivation of K + channels, although not a mechanism in contemporary burst moels, was propose by Carpenter (1979), in the early stages of mathematical treatment of bursting in excitable cells. We o not implement a similar slow 8

9 inactivation of somatic Dr,s since somatic spikes observe in bursting ELL pyramial cells o not exhibit broaening as the burst evolves (Lemon an Turner, 2000). The somatic-enritic interaction is moele as simple electrotonic iffusion with coupling coefficient g c, an scale by the ratio of somatic-to-total surface area κ. This form of coupling has been use in previous two-compartment neural moels (Mainen an Sejnowski, 1996; Wang 1999; Kepecs an Wang, 2000; Booth an Bose, 2001). I S represents either an applie or synaptic current flowing into the somatic compartment. In the present stuy I S is always constant in time, an will be use as a bifurcation parameter. Physiological justification for the parameter values given in Table I is presente in etail in Doiron et al. (2001b). Eqs (1) (6) are integrate by a 4 th orer Runge-Kutta algorithm with a fixe time step of t= s. 3 Results 3.1 Moel performance Figure 3A an 3B show simulation time series of V s an p, respectively, for the ghostburster with constant epolarization of I S = 9. We see a repetitive burst train similar to that shown in Figure 1A. Figure 4 compares the time series of V s an V for the ghostburster (bottom row) uring a single burst, to both a somatic an enritic burst from ELL pyramial cell recorings (top row), an the large compartmental moel presente in Doiron et al., (2001b) (mile row). All burst sequences are prouce with constant somatic epolarization. The somatic bursts all show the same characteristic growth in epolarization (DAP growth), an consequent ecreases in ISI leaing to the high frequency oublet. The enritic bursts all show that a enritic spike failure is associate with both oublet spiking an burst termination. The somatic AHPs in the simulation of the ghostburster o not show a graual epolarization uring the burst, as o both the AHPs in the ELL pyramial cell recorings an the large compartmental moel simulations. This is a minor iscrepancy, which is not relevant for the unerstaning of the burst mechanism. The mechanism involve in the burst sequences shown in Figures 3 an 4 has been explaine in etail (although not from a ynamical systems point of view) in past experimental an computational stuies (Lemon an Turner, 2000; Doiron et al., 2001b). We give a short overview of this explanation. Action potential backpropagation is the process of a somatic action potential actively propagating along the enrite ue to activation of enritic Na + cahnnels. Rapi 9

10 hyperpolarization of the somatic membrane, meiate by somatic potassium activation n s, allows electrotonic iffusion of the enritic action potential, creating a DAP in the somatic compartment. However, with repetitive spiking the enritic action potentials, shown by V, broaen in with an show a baseline summation (Figure 4). This is ue to the slow inactivation of I Dr,, meiate by p, as shown in Figure 3B. This further rives electrotonic iffusion of the enritic action potential back to the soma; consequently, the DAP at the soma grows, proucing an increase somatic epolarization as the burst evolves. This results in ecreasing somatic ISIs, as experimentally observe uring ELL burst output. This positive feeback loop between the soma an enrite finally prouces a high frequency spike oublet (Figure 4). Doublet ISIs are within the refractory perio of enritic spikes but not that of somatic spikes (Lemon an Turner, 2000). This causes the backpropagation of the secon somatic spike in the oublet to fail, ue to lack of recovery of I Na, from its inactivation, as shown in the enritic recorings (Figure 4). This backpropagation failure removes any DAP at the soma, uncovering a large bahp, an thus terminates the burst. This creates a long ISI, the inter-burst perio, which allows p an h to recover, in preparation for the next burst (see Figure 3B). 3.2 Bifurcation Analysis In the following sections we will use ynamical systems theory to explore various aspects of the ghostburster equations (Eq. (1)-(6)). An introuction to some of the concepts we will use can be foun for example in Strogatz (1994). An alternative explanation of the burst mechanism, given in physiological terms, was presente in Doiron et al. (2001b). Figure 5A gives the bifurcation iagram of h as compute from the ghostburster with I S treate as the bifurcation parameter. We chose I S since this is both an experimentally an physiologically relevant parameter to vary. Three istinct ynamical behaviours are observe. For I S < I S1 two fixe points exist; one stable, representing the resting state, an one unstable sale. When I S = I S1 the stable an unstable fixe points coalesce in a sale-noe bifurcation of fixe points on an invariant circle, after which a stable limit cycle exists. This is characteristic of Class I spike excitability (Ermentrout, 1996), of which the canonical moel is the well-stuie θ neuron (Hoppensteat an Izhikevich, 1997). For I S1 < I S <I S2 the stable limit cycle coexists with an unstable limit cycle. Both limit cycles coalesce at I S = I S2 in a sale-noe bifurcation of limit cycles. For I S > I S2 the moel ynamics, lacking any stable perioic limit cycle, evolve on a chaotic attractor giving bursting solutions as shown in Figures 3 an 4 (lower panel). As I S 10

11 increases further a perio oubling cascae out of chaos is observe, an a perio two solution exists for high I S. The importance of both of the sale-noe bifurcations will be explore in later sections. Figure 5B shows the observe spike ischarge frequencies, f ( 1/ISI), from the ghostburster as I S is varie over the same range as in Figure 5A. The rest state, I S < I S1, amits no firing, inicate by setting f = 0. For I S1 < I S < I S2 the stable limit cycle attractor prouces repetitive spike ischarge giving a single nonzero f value for each value of I S. f becomes arbitrarily small as I S approaches I S1 from above ue to the infinite perio bifurcation at I S1. However, for I S >I S2 the attractor prouces a varie ISI pattern, as shown in Figures 3 an 4. This involves a range of observe f values for a given fixe I S, ranging from ~ 100 Hz in the inter-burst interval to almost 700 Hz at the oublet firing. The burst regime, I S >I S2 oes amit winows of perioic behaviour. A particularly large winow of I S (13.13,13.73) shows a stable perio six solution which unergoes a perio oubling cascae into chaos as I S is ecrease. Finally, the perio oubling cascae out of chaos for I S >> I S2 is evient. Figure 5C shows the most positive Lyapunov exponent, λ, of the ghostburster as a function of I S. We see that λ < 0 for I S < I S1 because the only attractor is a stable fixe point. For I S1 <I S <I S2, λ =0 because the attractor is a stable limit cycle. Of particular interest is that λ is positive for a range of I S greater than I S2, inicating that the bursting is chaotic. The winows of perioic behaviour within the chaotic bursting are inicate by λ being zero (e.g. the large winow for I S (13.13,13.73)). For I S > 17.65, λ=0 because the ghostburster unergoes a perio oubling cascae out of chaos, resulting in a stable perio two solution. Figure 6 is a two parameter bifurcation set showing curves for both the sale-noe bifurcation of fixe points (SNFP) an of limit cycles (SNLC). The parameters are the applie current I S, alreay stuie in Figure 5, an g Dr, which controls the influence of the slow ynamical variable p (see Eq (3)). It is natural to choose g Dr, as the secon bifurcation parameter since the burst mechanism involves enritic backpropagation, which I Dr, regulates, an g Dr, can be experimentally ajuste by focal application of K + channel blockers to the apical enrites of ELL pyramial cells (Rashi et al., 2001). A vertical line in Figure 6 correspons to a bifurcation iagram similar to that presente in Figure 5A. The iagram in Figure 5A correspons to the rightmost value of g Dr, in Figure 6 (g Dr, =15). The intersection of the curves SNFP an SNLC with any vertical line gives the values I S1 an I S2 for that particular value of g Dr,. Thus, the curves 11

12 SNFP an SNLC partition parameter space into regions corresponing to quiescence, tonic firing, an chaotic bursting solutions of the ghostburster equations, as inicate in Figure 6. The curves intersect at a coimension-two bifurcation point corresponing to simultaneous fixe point an limit cycle sale-noe bifurcations. The curve to the left of the intersection point correspons to the coimension one SNFP curve; there is no stable perio-one limit cycle corresponing to tonic firing in this region. Figure 6 emonstrates that it is possible to make I S1 an I S2 arbitrarily close, by choosing g Dr, appropriately. This property will be of use later in the stuy. 3.3 The Burst Mechanism : Reconstructing the Burst Attractor. The ynamical system escribe by the ghostburster equations possesses two separate time scales. The time constants governing the active ionic channels n s, h, an n, are all ~ 1 ms, an the half with of the spike response of the membrane potentials V s an V are ~ 0.5 ms an 1.1 ms respectively. However, the time scale of p is characterize by τ p,, which is a factor of five times larger than any of the other time scales. Previous stuies of other burst moels have profite from a similar coexistence of at least two time scales of activity uring bursting (Rinzel, 1987; Rinzel an Ermentrout, 1989; Wang an Rinzel, 1995; Bertram et al., 1995; e Vries, 1998; Izhikevich, 2000; Golubitsky et al., 2001). This allowe for a separation of the full ynamical system into two smaller subsystems, one fast an one slow. We also treat our burst moel as a fast-slow burster. The natural variable separation is to group V s, n s, V, h, an n into a fast subsystem, enote by the vector x, while the slow subsystem consists solely of p. This gives the simplifie notation of our moel, x (7) = f ( x, p ) t (8) p t = p, ( x) τ p, p where f(x,p ) represents the right han sie of Eqs. (1)-(5) an Eq. (8) is simply Eq (6) restate. Since p changes on a slower time scale than x, we approximate p as constant, an use p as a bifurcation parameter of the fast subsystem (quasi-static approximation; see e.g. Hoppensteat an Izhikevich, 1997). We note that with p constant the fast subsystem (7) cannot prouce bursting comparable to that seen from ELL pyramial cells. Bursting requires the slow variable to moulate DAP growth ynamically (Doiron et al., 2001b). Treating p as a bifurcation parameter 12

13 will show how changes in p prouce the characteristics of ELL bursting through the bifurcation structure of the fast subsystem. Since p only irectly affects the fast subsystem through the ynamics of V (see eq. (3)) we choose V as a representative variable of the fast subsystem x. Figure 7A shows the local maxima of V on a perioic orbit as a function of p, while the fast subsystem is riven with I S = 9 > I S2. At a critical value of p, labele p 1, the fast subsystem goes through a transition from a perio-one to a perio- two limit cycle. This is shown by only one maximum in V for p > p 1, whereas there are two maxima for p < p 1. Figure 7B shows a time series of V (t) following the perio-one limit cycle when p = 0.13 > p 1, while Figure 7C shows the perio-two limit cycle when p = 0.08 < p 1. The secon enritic action potential in the perio-two orbit (Figure 7C) is of reuce amplitue; this correspons to the enritic failure observe in the full ynamical system (Eqs (7) an (8)) when p is low (see right column of Figure 4). The bifurcation iagram in Figure 7A may be thought of as a burst shell in a projection of phase space. The full burst ynamics will evolve upon the burst shell as p is moulate slowly by the fast subsystem. We therefore next aress the ynamics of p (t) uring the burst trajectory in the fast subsystem. Upon inspection of Figure 3B it is clear that there exists two oscillations in p (t), one fast oscillation occurring on the time scale of spikes, an the other on a much longer time scale, tracking the bursts. Figure 8 shows p (t) uring a burst solution of the full ynamical system. It is clear that the fast spike oscillations in p (t) are riven by the instantaneous value of V (t). This is ue to τ p being small enough to allow p (t) to be affecte by the spiking in the fast subsystem. In aition, there is a general ecrease in p (t) as the burst evolves, an an sharp increase in p (t) after the oublet ISI. The increase reinjects p (t) to a higher value allowing the burst oscillation to begin again. The perio of a burst oscillation encompasses several spikes, an thus cannot be analyze in terms of the instantaneous ynamics of the fast subsystem. Due to the separation of time scales, an the fact that p t epens only on V (eq. (6)), we expect that the burst oscillation epens on the average of V between consecutive spikes, efine as i (9) V = V t t i+ 1 i t ti ( t) t where t i is the time of the i th spike. We construct a iscrete function p~ 13

14 (10) ~ p = p ( V ), where p, ( ) is the infinite conuctance curve as in eq. (6). Figure 8 shows a sequence of p~ values constructe by using V from the burst solution of the full ynamical system. This sequence is plotte (soli circles) on top of the full p (t) ynamics uring the burst train. It is evient that the time sequence of p~ is of the same shape as the burst oscillation in p (t). This is evience that the slow burst oscillation can be analyze by consiering V. We now complete the burst shell by aing to Figure 7A the nullcline for p (from eq. (6)) as well as V compute for the stable perioic solutions of the fast subsystem. This is shown in Figure 9A. Note that as p ecreases through p 1, V ecreases by ~10 mv. This is ue to the enritic spike failure an subsequent long ISI occurring when p < p 1, both contributing to lower V on average (see Figure 7C). The p nullcline an V curves cross at p = p 2 < p 1. Since we have shown that the burst oscillation is sensitive to V, the crossing correspons to p t changing from negative to positive (see Figure 9D). A sale-noe bifurcation of fixe points occurs at p = p * for some p * > p 1 (ata not shown). This bifurcation is similar to the sale-noe bifurcation of fixe points in Figure 5A, where I S is the bifurcation parameter. This is expecte, since p is the coefficient to a hyperpolarizing ionic current (see eq. (3)), hence an increase in p is equivalent to a ecrease in epolarizing I S. Because of the sale-noe bifurcation at p = p *, the perio of the perio-one limit cycle scales as 1 for p near p * (Guckenheimer an Holmes, 1983). p p With the burst shell now fully constructe (Figure 9A) we place the full burst ynamics (eq. (7)-(8)) onto the shell. This is shown in Figure 9B. The irecte trajectory is the full six imensional burst trajectory projecte into the V - p subspace. As the burst evolves, p (t) ecreases from spike to spike in the burst. This causes the frequency of spike ischarge to increase ue to the graual shift away from the sale-noe bifurcation of fixe points at p = p *. However, once p (t) < p 1 the spike ynamics shift from perio-one spiking to perio-two spiking. This first prouces a high frequency spike oublet, which is then followe by a enritic potential 14

15 of reuce amplitue, causing V to ecrease. When p (t) < p 2, p t > 0 (see Figure 8D), an p (t) increases an is reinjecte to a higher value. The reinjection towars the ghost of the sale-noe bifurcation of fixe points causes the ISI (the inter-burst interval) to be long, since the velocity through phase space is lower in this region. Figure 9C shows the burst trajectory in the frequency omain. The perio oubling is evient at p = p 1 since two istinct frequencies are observe for p < p 1, corresponing to a perio two solution of the fast subsystem, whereas for p > p 1 only a perio one solution is foun. As p is reuce in the perio one regime (p > p 1 ) the frequency of the limit cycle increases, ue to the reuce effect of the hyperpolarizing current I Dr,. We superimpose the ISIs of the burst trajectory shown in Figure 9B on the frequency bifurcation iagram in Figure 9C. The sequence begins with a long ISI (numbere 1) with subsequent ISIs ecreasing, culminating with the short oublet ISI (numbere 5). The reinjection of p near p * occurs uring the next ISI (numbere 6). The reinjection causes this next ISI to be long; it separates the action potentials into bursts. Figure i 9D shows the average of the erivative of p, + 1 p 1 = p t, uring each ISI in t t t t i+ 1 i t ti the burst shown in Figures 9B, an 9C. Notice that p t is negative an ecreases as the burst evolves. This is because the ISI length reuces as the burst evolves, allowing the burst trajectory to spen less time in the region where p t > 0. However, a large fraction of the burst trajectory uring the inter-burst ISI (6) occurs in the region where p t > 0. Hence, the average p t is greater than zero for the inter-burst interval, proucing the reinjection of p (t) to higher values. Izhikevich (2000) has labele the burst mechanisms accoring to the bifurcations in the fast subsystem that occur in the transition from quiescence to limit cycle an vice versa. Even though there is never a true quiescent perio uring the burst phase trajectory, the inter-burst interval for our moel is etermine by the approach to an infinite perio bifurcation. This phenomenon is often labele as sensing the ghost of a bifurcation (Strogatz, 1994), an we naturally label the burst mechanism as ghostbursting. 15

16 The ghostburster system exhibits bursting, for some range of I S, only for 2< τ p <110 ms, with all other parameters as given in Table 1. The lower boun of τ p is ue to the fact that the inactivation of I Dr, must be cumulative in orer for there to be a reuction of the ISIs as the burst evolves. This requires a τ p larger than that of the ionic channels responsible for spike prouction (< 1 ms). The upper boun on τ p is also expecte since significant removal of p inactivation uring the inter-burst interval is necessary for another burst to occur. Too large a value of τ p will not allow sufficient recovery of I Dr, from inactivation an therefore bursting will not occur The Inter-burst Interval. By varying I S it is possible to set the inter-burst interval, T IB, to be ifferent lengths. This is because after the enrite has faile (removing the DAP at the soma) the time require to prouce an action potential in the somatic compartment (which is T IB ) is ictate almost solely by I S. The spike excitability of the somatic compartment is Type I (Ermentrout, 1996), as evient from the sale-noe bifurcation of fixe points at I S = I S1. As a consequence T IB is etermine from the well-known scaling law associate with sale-noe bifurcations on a circle (Guckenheimer an Holmes, 1983), (11) T IB ~ 1 I S I S1. Figure 10 shows the average inter-burst interval, <T IB >, as a function of I S - I S1 for the ghostburster with g Dr, = This value of g Dr, sets I S1 an I S2 close to one another (see Figure 6), allowing the system to burst with values of I S close to I S1. It is necessary to form an average ue to the chaotic nature of burst solutions. Nevertheless, <T IB > increases as I S approaches I S1, as suggeste by Eq. (11). A linear regression fit of 1/<T B > 2 against I S - I S1 gives a correlation coefficient of further verifying that Eq (11) hols. Figure 10 also shows ownwar ips in <T IB > that occur more frequently as I S - I S1 goes to zero. Time series of bursts with I S corresponing to the ips in <T IB > show scattere bursts with short inter-burst intervals that eviate from Eq (11), amongst bursts with longer inter-burst intervals, which fit the tren escribe by Eq (11). These scattere small values of T IB reuce <T IB > for these particular values of I S. These ips contribute to the eviation of the linear correlation coefficient cite above from 1. We o not stuy the ips further since the behaviour has yet to be observe experimentally. However, experimental measurements of ELL pyramial cell burst perio o inee show a lengthening of the perio as 16

17 the applie current is reuce (R.W. Turner, personal communication). This correspons to the general tren shown in Figure 10. Eq (11) an Figure 10 show that by choosing the moel parameters properly it is possible to regulate the effect of the ghost of the sale-noe bifurcation of fixe points on the burst solutions. We will show later how this property yiels great iversity of time scales of possible burst solutions of the ghostburster moel The Burst Interval Intermittency. Regions of chaotic an perioic behaviour exist in many burst moels (Chay an Rinzel, 1985; Terman, 1991; Terman, 1992; Hayashi an Ishizuka, 1992; Wang, 1993; Komenantov an Kononenko, 1998). The results of Figure 5 show that perioic spiking an chaotic bursting are also two istinct ynamical behaviours of the ghostburster. Moreover, the bifurcation parameter we have use to move between both ynamical regimes is the applie current I S which mimics an average synaptic input to the cell. This inicates that changing the magnitue of input to the cell may cause a transition from perioic spiking to chaotic bursting. In ELL pyramial cells a transition from tonic firing to highly variable bursting has been observe as applie epolarizing current is increase (Lemon an Turner, 2000; Bastian an Nguyenkim, 2001; Doiron an Turner, unpublishe results). It remains to be shown that the experimentally observe bursting is inee chaotic; preliminary results suggest that such an analysis is ifficult ue to non-stationarity in the ata (Doiron an Turner, unpublishe observations). Nonetheless, unerstaning the transitions or routes to chaos in the moel separating tonic an chaotic burst regimes is not only necessary for a complete escription of the ynamics of the moel, but also for characterizing the input-output relation of bursting ELL pyramial cells. Figure 5A shows that the transition from perioic spiking to chaotic bursting occurs at I S =I S2 when a stable limit cycle collies with an unstable perioic orbit in a sale-noe bifurcation of limit cycles. Since we are analyzing spiking behaviour on both sies of the bifurcation it is natural to consier the ISI return map for I S near I S2. We choose I S slightly larger than I S2 an plot in Figure 11A the ISI return map for a single burst sequence from the ghostburster (for I S1 <I S <I S2 the return map is a single point). We have labele the regions of interest in the Figure an explain each region in orer: (1) The burst begins here. (2) The ISI sequence approaches the iagonal. This prouces a clustering of points corresponing to the pseuo-perioic behaviour observe in the center of the burst. We refer to this region of the map as a trapping region. (3) The ISI 17

18 sequence leaves the trapping region with a ownwar tren. (4) The inter-burst interval involves a sharp transition from small ISI to large ISI. (5) The ISI sequence returns to the trapping region an another burst begins. The above escription inicates that the route to chaos is Type I intermittency (Manneville an Pomeau, 1980; Guckenheimer an Holmes, 1983). Intermittency involves seemingly perioic behaviour separate by brief excursions in phase space. The clustering of points in the ISI return map in the trapping region of Figure 11A (labele 2) is a manifestation of this apparent perioic firing. A trapping region is a characteristic feature of Type I intermittency an correspons to a sale-noe bifurcation of limit cycles in the return map, occurring specifically at I S =I S2 for the ghostburster equations. The escape an return to the trapping region (regions 3,4,5 in Figure 11A) are the brief excursions. These events correspon to the perio oubling transition an the cross of the V curve an p nullcline, in the fast subsystem, as explaine in Figure 9. Figure 11B shows the ISI return map for a moel burst of seven spikes an Figure 11C the same map for a seven spike burst recoring from an ELL pyramial cell. Both maps show the qualitative structure similar to in Figure 11A, incluing a clear escape from an reinjection into a trapping region near the iagonal. Interestingly, Wang (1993) has also observe Type I intermittency in the Hinmarsh- Rose moel). Since intermittent behaviour is connecte to a sale-noe bifurcation, the time spent in the trapping region T B, corresponing to the burst perio (the uration of the spikes in the cluster making up the burst), has a well efine scaling law (12) T B ~ 1 I S I S2 Similar to Figure 10 we consier the average of the burst perio, <T B >, because of the chaotic nature of the bursting. Figure 12 shows that <T B > asymptotes to infinity as I S approaches I S2. Linear regression fits to 1/<T B > 2 against I S - I S2 give a correlation coefficient of These results valiate Eq (12) for the ghostburster burst sequences. Again the eviation in the correlation coefficient from 1 is cause by slight ips in <T B >, similar to the ips observe in <T IB > (Figure 10). By choosing the quantity I S -I S2 we can obtain bursts with spike numbers comparable to experiment. 18

19 3.6 - Gallery of Bursts Eqs (11) an (12) give the inverse square root scaling relations of T B an T IB respectively. These results showe that T B is etermine by I S -I S2 an T IB by I S -I S1. Using this fact an the ability to vary the ifference between I S2 an I S1 (see Figure 6) we can prouce a wie array of burst patterns with iffering time scales. Figure 13A reprouces the (I S, g Dr, ) bifurcation set shown in Figure 6. The letters B-F mark (I S, g Dr, ) parameters use to prouce the spike trains shown in the associate panels B-F of Figure 13. Figure 13B uses (I S, g Dr, ) values such that the ghostburster is in the tonic firing regime. The burst trains shown in Figures 3 an 4 correspon to (I S, g Dr, ) values in the burst regime of Figure 6 which are not close to either of the SNFP or SNLC curves. An example of a burst train with such a parameter choice, is shown in Figure 13C. However, if we approach the SNLC curve but remain istant from the SNFP curve, we can increase T B by one orer of magnitue yet keep T IB the same. The burst train in Figure 13D shows an example of this. If we choose I S an g Dr, to be close to both the SNFP an SNLC curves we can now increase T IB as well (Figure 13E). The inter-burst perio T IB has now also increase ramatically from that shown in Figures 13C an D. Finally, for I S an g Dr, values to the left of the coimension two bifurcation point, burst sequences show only a perio two solution (Figure 13F). The burst sequences are no longer chaotic. This is to be expecte since there no longer is a sale-noe bifurcation of limit cycles, which gave rise to the intermittency route to chaos in the ghostburster equations (Figure 11). Approaching the SNFP curve allows for a large T IB, but the fact that only bursts of two spikes can appear forces T B to be small. The restriction of bursts to only oublets when g Dr, is small occurs because g Dr, is the coefficient to the hyperpolarizing K + current in the enrite (Eq (3)), an as such controls the effect of the DAP at the soma. For g Dr, to the left of the coimension-two bifurcation point, the first somatic spike in the oublet prouces a DAP of sufficient strength to cause the secon somatic spike which is within the refractory perio of the enrite. Thus enritic failure occurs after the first reflection an the burst contains only two spikes. 19

20 4 Discussion 4.1 Ghostbursting: a Novel Burst Mechanism We have introuce a two-compartment moel of bursting ELL pyramial cells, title the ghostburster. The moel is a significant reuction of a large multi-compartmental ionic moel of these cells (Doiron et al., 2001b). The large moel was motivate by the conitional backpropagation burst mechanism that has been experimentally characterize in ELL pyramial cells (Lemon an Turner, 2000). The results of Lemon an Turner (2000) an Doiron et al., (2001b), suggest that the ionic requirements necessary an sufficient to support bursting as observe in the ELL are 1) action potential backpropagation along the apical enrite sufficient to prouce somatic DAPs. 2) the refractory perio of enritic action potentials must be longer than that of the somatic potentials 3) slow inactivation of a enritic K + channel involve in repolarization. The fact that the ghostburster was esigne to contain only these three requirements, yet succees in proucing burst ischarge comparable to experiment, suggests that these three requirements capture the essential basis of the burst mechanism use in ELL pyramial cells. The simplicity of the ghostburster, as compare to the large compartmental moel, has allowe us to unerstan, from a ynamical systems perspective, the mechanism involve in this type of bursting. The ghostburster was analyze using a separation of the full ynamical system into fast an slow subspaces (Eq (7) an (8)), similar to the analysis of many other burst moels (Rinzel, 1987; Rinzel an Ermentrout, 1989; Wang an Rinzel, 1995; Bertram et al., 1995; Hoppensteat an Izhikevich, 1997; e Vries, 1998; Izhikevich, 2000; Golubitsky et al., 2001). Treating the slow ynamical variable p as a bifurcation parameter with respect to the fast subsystem allowe us to construct a burst shell upon which the full burst ynamics evolve. The shell shows that a transition from a perio-one limit cycle to a perio-two limit cycle occurs in the ynamics of the fast subsystem as p is reuce. The perio-two limit cycle causes a sharp reuction in <V > since the secon spike of the limit cycle is of reuce amplitue, ue to enritic refractoriness. The reuction in <V > causes the <V >(p ) curve to cross the p nullcline, an p (t)grows uring the secon ISI of the perio-two orbit. The growth in p (t) reinjects p (t) near a sale-noe bifurcation of fixe points occurring at high p. This passage near the ghost of the sale-noe bifurcation causes the ISI to be long, separating the action potentials into bursts. 20

21 Recently, Izhikevich (2000) has approache the classification of bursters from a combinatorial point of view. This has been successful in proucing a large number of new fastslow bursting mechanisms. One of these burst mechanisms has been recently observe in a biophysically plausible moel of bursting corticotroph cells of the pituitary (Shorten et al., 2000). In contrast, Golubitsky et al., (2001) (extening the work of Bertram et al. (1995), an e Vries (1998)) have classifie bursters in terms of the unfolings of high coimension bifurcations. Both these methos have use the implicit assumption that burst initiation an termination involve bifurcations from quiescence (or subthreshol oscillation) to limit-cycle an vice-versa. However, our burst mechanism oes not appear in any of the above classifications. This is because, the trajectories in the fast subsystem of the ghostburster are always following a limit-cycle, an are never in true quiescence, corresponing to a stable fixe point. The perio of the limit cycle changes ynamically because the slow subsystem is oscillating, forcing the fast system to sometimes pass near the ghost of an infinite perio bifurcation. Furthermore, in the ghostburster, burst termination is connecte with a bifurcation from a perio-one to a perio-two limit cycle in the fast subsystem. This is a novel concept, since burst termination in all other burst moels is connecte with a transition from a perio-one limit cycle to a stable fixe point in the fast subsystem (Izhikevich 2000; Golubitsky et al., 2001). Thus, while classifying burst phenomena through the bifurcations from quiescence to a perio-one limit cycle an vice-versa in the fast subsystem of a ynamical bursting moel has ha much success, our work requires an extension of the classification of bursting to inclue an alternative efinition of quiescence an a burst attractor which is compose of only perio-one an perio-two limit cycles with no stable fixe points. Rinzel (1987) shows that burst mechanisms with a one-imensional slow subsystem require bistability in the fast subsystem in orer to exhibit bursting. The slow subsystem of ghostbuster equations is one imensional, yet Figure 9 shows that the fast subsystem x is not bistable. This woul seem to be a contraiction; however, recall that as τ p approaches values that are similar to other bursting mechanisms, bursting is not observe. Thus our results o not contraict Rinzel s previous stuy, yet support a separate mechanism entirely. This illustrates a key istinction between the ghostburster an conventional bursting systems; the timescale of the slow variable has an upper boun in the ghostburster. The fast an slow timescales are sufficiently separate to allow us to successfully stuy the burst mechanism using a quasistatic approximation. 21

22 Thus ghostbursting, while istinct, oes share similarities with conventional burst mechanisms. Note that mechanisms exist similar to ghostbursting, which involve a slow passage phenomena (requiring sale-noe or homoclinic bifurcations), may exist, placing the ghostburster as only one in a family of new burst mechanisms. The ghostburster moel exhibits a threshol between tonic firing an bursting behaviour. Both Terman (1991,1992) an Wang (1993) have also ientifie threshols between these behaviours in the Hinmarsh-Rose moel an a moifie version of the Morris-Lecar equations, respectively. Both of these moels exhibite a homoclinic orbit in the fast subsystem as the spiking phase of a burst terminate. As a result, the bifurcations from continuous spiking to busting in the full ynamics were complicate. Wang observe a crises bifurcation at the transition (Grebogi et al., 1983), whereas Terman showe that a series of bifurcations occurs uring the transition, which coul be shown to exhibit ynamics similar to the Smale horseshoe map (Guckenheimer an Holmes, 1983). The sale-noe bifurcation of limit cycles that separates the two regimes in the Ghostburster moel is a great eal simpler than either of these bifurcations. However, interestingly Wang has shown that an intermittent route to chaos is also observe in the Hinmarsh-Rose moel as continuous spiking transitions into bursting, much like the Ghostburster system. The fact that the transition from tonic firing to bursting in the Ghostburster system occurs as epolarization is increase, is in contrast to both experimental an moeling results of other bursting cells (Terman 1992; Hayashi an Ishizuka, 1992; Wang 1993; Gray an McCormick, 1996; Steriae et al., 1998; Wang, 1999). However, since many experimental an moeling results, separate from ELL, show burst threshol behaviour, the concept of burst excitability may have broaer implications. To expan, the sale-noe bifurcation of limit cycles marking burst threshol can be compare to the sale-noe bifurcation of fixe points, which is connecte to the spike excitability of Type I membranes (Ermentrout, 1996; Hoppensteat an Izhikevich, 1997). The functional implication of a burst threshol have yet to be fully unerstoo, however recent work suggests that it may have important implications for both the signaling of inputs (Eugia et al., 2000) an iviing cell response into stimulus estimation (tonic firing) an signal etection (bursting) (Sherman, 2001). 22

23 4.2 Preictions for bursting in the ELL An integral part of the burst mechanism in ELL pyramial cells is the interaction between the soma an enrite through action potential backpropagation. One potential function of backpropagation is thought to be retrograe signaling to enritic synapses (Häusser et al., 2000). Further, a recent experimental stuy has shown that the coincience of action potential backpropagation an EPSPs prouce a significant amplification in membrane potential epolarization (Stuart an Häusser, 2001). These results may have consequences for both synaptic plasticity an enritic computation. Our results (an those of others, see Häusser et al., 2000 for a review) imply that backpropagation can also etermine action potential patterning. As mentione above, the ghostburster exhibits a threshol separating tonic firing an bursting as epolarization is increase. Similar behaviour has been observe in both in vitro an in vivo experimental recorings of ELL pyramial cells (Lemon an Turner, 2000; Bastian an Nguyenkim, 2001), an in our full compartmental moel simulations (ata not shown). A reuction of burst threshol was observe in ELL pyramial cells when TEA (K + channel blocker) was focally applie to the proximal apical enrite (Noonan et al., 2001; Rashi et al., 2001). Our work is consistent with this observation, since enritic TEA application is equivalent to a reuction in g Dr, conuctance in our moel. Figure 6 shows that as g Dr, is reuce burst threshol is lowere. Bursts, as oppose to iniviual spikes, have been suggeste to be a funamental unit of information (Lisman, 1997). In fact, Gabbiani et al., (1996) have correlate bursts from ELL pyramial cells with features in the stimulus riving the cell. Consiering these results, it is possible that the time scale of bursting, T (= T B + T IB ), coul be tune to sensory input, hence the ability of a bursting cell to alter T may improve its coing efficiency. A natural metho to alter T woul be to change the time constant(s), τ, that etermine the slow process of the burst mechanism (Giannakopoulous et al., 2000). Nevertheless, to achieve an orer of magnitue change in T requires a potentially large change in τ. Recently, Booth an Bose (2001) have shown, in a twocompartmental moel of a bursting CA3 pyramial cell, that the precise timing of inhibitory synaptic potentials can change the burst perio T. Their results have potential implications for the rate an temporal coing of hippocampal place cells. However, the ghostburster shows that both T B an T IB can be change by an orer of magnitue, but with only small changes in either epolarizing input an/or enritic K + conuctances (see Figure 13). Small changes in I S are 23

24 conceivable through realistic moulations of feeforwar an feeback input which occur uring electro-location an electro-communication in weakly electric fish (Heiligenburg, 1991). Changes in g Dr, can further occur through the phosphorylation of enritic K + channels, such as AptKv3.3 which has been shown to be abunant over the whole enritic tree of ELL pyramial cells (Rashi et al., 2001). Hence, the ghostbursting mechanism may offer ELL pyramial cells a viable metho by which to optimize sensory coing with regulate burst output. Further stuies, quantifying the information-theoretic relevance of bursting, are require to confirm these speculations. We conclue our stuy with a concrete preiction. Figures 10,12, an 13 show that the full burst perio T of ELL pyramial cells can be significantly ecrease as either epolarizing current (I S ) is increase or enritic K + conuctance (g Dr, ) is ecrease by a small amount. This preiction can be easily verifie by experimentally measuring T in bursting ELL pyramial cells for 1) step changes in I S, an 2) before an after TEA application to the apical enrites, which will change g Dr,. Moification of other ionic currents, persistent soium an somatic K + in particular, may also be use to create similar bifurcation sets as in Figure Acknowlegements We woul like to thank our colleague Ray W. Turner for the generous use of his ata an fruitful iscussions. Valuable insight on the analysis of our moel was provie by John Lewis, Kashayar Pakaman, Eugene Izhikevich, Gera DeVries, Maurice Chacron, an Egon Spengler. This research was supporte by operating grants from NSERC (B.D., A.L.), the OPREA (C.L.), an CIHR (L.M.). 24

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28 Traub R, Wong R, Miles R, an Michelson H (1994). A moel of a CA3 hippocampal neuron incorporating voltage-clamp ata on intrinsic conuctances. J. Neurophysiol. 66: Turner RW, Maler L, Deerinck T, Levinson SR, an Ellisman M (1994) TTX-sensitive enritic soium channels unerlie oscillatory ischarge in a vertebrate sensory neuron. J. Neurosci. 14: Turner RW, Plant J, an Maler L (1996) Oscillatory an burst ischarge across electrosensory topographic maps. J. Neurophysiol. 76: Turner RW, an Maler L (1999) Oscillatory an burst ischarge in the apteronoti electrosensory lateral line lobe. J. Exp. Biol. 202: Vetter P, Roth A, an Häusser M (2001) Propagation of action potentials in enrites epens on enritic morphology. J. Neurophysiol. 85: Wang X-J (1993). Genesis of bursting oscillations in the Hinmarsh-Rose moel an homoclinicity to a chaotic sale. Physica D 62: Wang X-J, an Rinzel J (1995). Oscillatory an bursting properties of neurons. In: e. Arbib MA. The Hanbook of Brain Theory an Neural Networks. Cambrige MA: MIT Press, pp Wang X-J (1999) Fast burst firing an short-term synaptic plasticity: a moel of neocortical chattering neurons. Neurosci. 89: Figure Legens FIG 1. ELL burst ischarge an enritic backpropagation. A. In vitro recoring of burst ischarge from the soma of an ELL pyramial cell with constant applie epolarizing current. Two bursts of action potentials are shown, each exhibiting a growing epolarization as the burst evolves, causing the ISI to ecrease; the burst ens with a high frequency oublet ISI. The oublet triggers a sharp removal of the epolarization, uncovering a prominent AHP, labele a burst-ahp. B. Active Na + conuctances are istribute along the soma an proximal apical enrite of ELL pyramial cells (left). Na + regions are inicate with vertical bars to the left of the schematic. Note that the istribution of enritic Na + is punctuate, giving regions of high Na + concentration (often referre to as hot spots ) separate by regions of passive enrite. The active enritic regions allow for backpropagation of a somatic action potential through a enritic action potential response, as seen from ELL recorings from both the soma an proximal (~ 150 µm) enrite (right). Somatic action potential rectification by K + currents an the broaer action potential in the enrite allow for electrotonic conuction of the enritic action potential to the soma, resulting in a DAP at the soma (bottom left). We thank R.W. Turner for generously proviing his ata for the figure. FIG 2. Schematic of two-compartment moel representation of an ELL pyramial cell. The ionic currents that influence both the somatic an enritic compartment potentials are inicate. Arrows which point into the compartment represent inwar Na + currents, whereas arrows pointing outwar represent K + currents (the specific currents are introuce in the text). The compartments are joine through an axial resistance, 1/g c, allowing current to be passe between the somatic an enritic compartments. 28

29 FIG 3. Moel bursting. A. Time series of the somatic potential V s uring burst output. B. Denritic I Dr, inactivation variable p uring the same burst simulation as in A. Note the cumulative (slow) inactivation as the burst evolves an the rapi recovery from inactivation uring the inter-burst perio. FIG 4. Moel performance. A single burst is obtaine from ELL pyramial cell recorings (top row; ata onate by R. W. Turner), full multi-compartmental moel simulations (mile row; simulation presente in Doiron et al., 2001b), an reuce two-compartment moel simulations (bottom row; eqs (1)-(6)). All bursts are prouce by applying constant epolarization to the soma (0.3 na top; 0.6 na mile; I s = 9, bottom). The columns show both somatic an enritic responses for each row. The reuce moel somatic spike train reprouces both the in vitro ata an full moel simulation spike trains by showing the growth of DAPs an reuction in ISI as the burst evolves. All somatic bursts are terminate with a large bahp, which is connecte to the enritic spike failure. FIG 5. A. Bifurcation iagram of the ghostburster equations (Eqs (1)-(6)) as a function of the bifurcation parameter I S. We choose h as the representative ynamic variable an plot h on the vertical axis. For I S < I S1 a stable fixe point (soli line) an a sale (ashe line) coexist. A sale-noe bifurcation of fixe points (SNFP) occurs at I S = I S1. For I S1 < I S < I S2 stable (fille circles) an unstable (open circles) limit cycles coexist, the maximum an minimum of which are plotte. A sale-noe bifurcation of limit cycles (SNLC) occurs at I S = I S2. For I S > I S2 a chaotic attractor exists; we show this by plotting the maximum an minimum of h for all ISIs that occur in a 1s simulation for fixe I S. A reverse perio oubling cascae out of chaos is observe for large I S. The software package AUTO (Doeel, 1981) was use to construct the leftmost part of the iagram. Chaotic states are shown by plotting the minimum an maximum of h for each ISI of a 1000 ms spike train. B. Instantaneous frequency (1/ISI) is plotte for 1000 ms simulations of the ghostburster moel for each increment in I S. The transitions from rest to tonic firing an tonic firing to chaotic bursting are clear. C. The maximum Lyapunov exponent λ as a function of I S. FIG 6. Two parameter bifurcation set. Both the sale-noe bifurcations of fixe points (SNFP) an limit cycles (SNLC) bifurcations were tracke, using AUTO (Doeel 1981) in the (I S, g Dr, ) subspace of parameter space. The curves partition the space into quiescence, tonic firing, an chaotic bursting regimes. FIG 7. A. Quasistatic bifurcation iagram. p is fixe as a bifurcation parameter while V is chosen as a representative variable from the fast subsystem x. The maxima in the enritic voltage ( V 2 V = 0 an < 0 ) are plotte for each value of p 2. At p = p 1 the maxima of V switch to t t two values, corresponing to the values taken uring each ISI of a perio-two solution. B. Time series of the enritic voltage, V (t), while p = 0.13 > p 1. The fast subsystem follows a perioone solution. C. Time series of the enritic voltage, V (t), while p = 0.08 < p 1. The fast subsystem follows a perio-two solution. A constant value of I S = 9 > I S1 is chosen for all simulations in A,B, an C. FIG 8. p (t) an p~ compute from integration of the ghostburster equations with I S =9 > I S2. Four bursts are shown with the corresponing time stampe spikes given above for reference. A Slow 29

30 burst oscillation in p (t) is observe. It is evient that the iscrete function p~ (soli circles) tracks the burst oscillation in p (t). p~ shows a monotonic ecrease throughout the burst until the interburst interval, at which point p~ is reinjecte to a higher value. The horizontal lines are the values p 1, corresponing to the perio oubling transition, an p 2, corresponing to the crossing of the nullcline curve with the <V > curve. The p (t) reinjection occurs after p (t)<p 2 as explaine in the text. p~ has been translate ownwar to lie on top of the p (t) time series. This is require because Eq (10) uses a unweighte average of V, given in Eq (9). This prouces a p~ series which occurs at higher values than p (t) because Eq (9) an (10) ignore the low pass characteristics of Eq (6). However, only the shape of p~ is of interest an this is not affecte by the ownwar translation. FIG 9. A. The bifurcation iagram of Figure 6A is re-plotte along with the p nullcline p, (V ) (ashe line labele N). Note that the p nullcline is inverte so as to give V 1, ( p ) = V1/2, p k p ln( ) 1 p. We plot the average of V over a whole perio of V,<V > (soli line), at a fixe p. Note the sharp ecline in <V > for p below p 1. B. The iagram in 9A is re-plotte with the labels remove. A single irecte burst trajectory projecte in the (V,p ) plane obtaine by integrating the full ynamical system (Eqs (1)-(6)) is plotte on top of the burst shell. C. All observe ischarge frequencies of the fast subsystem are plotte as a function of p. At p = p 1 a stable perio-one firing pattern of ~ 200 Hz changes to a perio-two solution with one ISI being ~ (700 Hz) -1 an the other ~ (100Hz) -1. The inverse of the ISIs of the single burst shown in Figure 9B are plotte as well. The ISIs are numbere from 1 (the first ISI) through to 5 p (oublet ISI) an 6 (inter-burst interval). D. The average of the erivative of p, t, is plotte for each ISI in the single burst shown in Figure 8B. Only the long inter-burst ISI has p t > 0,all other ISIs have p t < 0. A constant value of I S = 9 > I S1 is chosen for all simulations in A, B, C, an D. FIG 10. Inter-burst interval. <T IB > is plotte as a function of I S - I S1. The averaging was performe on 100 bursts prouce by the ghostburster equations at a specific I S. g Dr, was set to <T IB > shows a similar functional form to that escribe by Eq (11). The ips in <T IB > are iscusse in the text. FIG 11. Burst intermittency. A. The ISI return map for a single burst sequence with I S =6.587 an g Dr, =13 is shown (for these parameters I S1 =5.736 an I S2 =6.5775). The iagonal is plotte as well (ashe line). The labels (1)-(5) are explaine in the text. B. The ISI return map for a single burst sequence with I S =9 an g Dr, =15 as in Figure 3. C. The ISI return map for a single burst recoring from an ELL pyramial cell (Data courtesy of R.W. Turner). Compare with the moel burst sequence in B. FIG 12. Burst interval <T B > plotte as a function of I S - I S2. The averaging was performe on 100 bursts prouce by the ghostburster equations at a specific I S. g Dr, was set to <T B > shows a similar functional form to that escribe by Eq (12). 30

31 FIG 13. Burst Gallery. A. Reprouction of the two parameter bifurcation set shown in Figure 6. The letters B-F marke insie the Figure correspon to the (I S, g Dr, ) parameter values use to prouce panels B-F respectively. Examples of the inter-burst perio T IB an burst perio T B for each burst train are inicate (except for the tonic solution shown in B). The exact I S an g Dr, values use to prouce each spike train are as follows: B. I S =6.5, g Dr, =14 C. I S = 7.7, g Dr, =13; D. I S =7.6, g Dr, =14; E. I S =5.748, g Dr, =12.14; F. I S =5.75, g Dr, =11. The vertical mv scale bar in C applies to all panels, however, each panel has its own horizontal time scale bar. Table I Current g max V 1/2 K t I Na,s ( m ( V ) ) N/A, s s I Dr,s ( n )) s ( V s I Na, ( m ( V )/ h ( V ) ) 5-40/-52 5/-5 N/A /1, I Dr, ( n V ) / p ( V ) ) 15-40/-65 5/-6 0.9/5 ( TABLE I. Moel parameter values. The values correspon to the parameters introuce in eq (1)- (6). Each ionic current (I Na,s ; I Dr,s ; I Na, ; I Dr, ) is moele by a maximal conuctance g max (in units of ms/cm 2 ), sigmoial activation, an possibly inactivation, infinite conuctance curves involving both V 1/2 an k parameters V ) = 1, an a channel time constant τ (in units of m, s ( s ( Vs V1 / 2) / k 1+ e ms). Double entries x/y correspon to channels with both activation (x) an inactivation (y) respectively. If the activation time constant value is N/A then the channel activation tracks the membrane potential instantaneously. Other parameters values are; g c = 1, κ = 0.4, V Na = 40 mv, V K = mv, V leak = -70 mv, g leak = 0.18, an C m =1 µf/cm 2. These values compare in magnitue to those of other two-compartment moels (Pinsky an Rinzel, 1994; Mainen an Sejnowski, 1995). 31

32 Fig1 A Doublet 20 mv 10 ms Depolarization bahp I app B apical enrite 20 mv Denritic Na + 8 ms DAP 2 mv Somatic Na + soma 20 mv 5 ms 2 ms 32

33 Fig2 I Dr, I Na, Denrite I L I s/ I /s I Dr,s I Na,s Soma I L 33

34 Fig3 A 20 V s (mv) B p time (ms)

35 Fig4 Soma Denrite Data Denritic spike failure 5 mv DAP growth 10 mv 10 ms bahp 10 ms Large Moel Denritic spike failure 10 mv 10 ms DAP growth bahp 10 mv 10 ms Ghostburster Denritic spike failure DAP growth 10 mv 10 mv 10 ms bahp 10 ms 35

36 Fig5 36

37 Fig6 SNLC 8 Chaotic Bursting 7 I S Perioic Firing 6 SNFP Quiescence g Dr,

38 Fig7 A B 10 p = V (mv) C -10 p = p 1 20 mv ms p 38

39 Fig8 Burst Oscillation p 1 p ms 39

40 Fig9 A B 0 V,max 0 V (mv) p /t < p 2 p 1 p /t > 0 <V > N -60 C 0.10 p 0.15 D 0.10 p frequency (Hz) p 1 6 <p p /t> p ISI number

41 Fig <T IB > (ms) I S -I S

42 Fig11 A ISI i+1 (ms) B ISI i (ms) ISI i+1 (ms) C ISI i (ms) ISI i+1 (ms) ISI i (ms) 10 42

43 Fig <T B > (ms) I S -I S

44 Fig13 A I s F E C D B SNLC SNFP B g Dr, ms C T B T IB 40 mv D 25 ms T B T IB E 100 ms T B T IB F 200 ms T B T IB 50 ms 44