Dissection and Reduction of a Modeled Bursting Neuron

Transcription

1 Journal of Computational Neuroscience 3, (1996) 1996 Kluwer Academic Publishers. Manufactured in The Netherlands. Dissection and Reduction of a Modeled Bursting Neuron R.J. BUTERA, JR. AND J.W. CLARK, JR. Dept. of Electrical and Computer Engineering, MS 366, Rice University, Houston, TX rbutera@ece.fice.edu J.H. BYRNE Dept. of Neurobiology and Anatom.3; University of Texas Medical School, Houston, TX jbyrne@nbal 9.med.uth.tmc.edu Received September 5, 1995; Revised February 1, 1996; Accepted February 13, 1996 Action Editor: John Rinzel Abstract. An 11-variable Hodgkin-Huxley type model of a bursting neuron was investigated using numerical bifurcation analysis and computer simulations. The results were applied to develop a reduced model of the underlying subthreshold oscillations (slow-wave) in membrane potential. Two different low-order models were developed: one 3-variable model, which mimicked the slow-wave of the full model in the absence of action potentials and a second 4-variable model, which included expressions accounting for the perturbational effects of action potentials on the slow-wave. The 4-variable model predicted more accurately the activity mode (bursting, beating, or silence) in response to application of extrinsic stimulus current or modulatory agents. The 4-variable model also possessed a phase-response curve that was very similar to that of the original 11-variable model. The results suggest that low-order models of bursting cells that do not consider the effects of action potentials may erroneously predict modes of activity and transient responses of the full model on which the reductions are based. These results also show that it is possible to develop low-order models that retain many of the characteristics of the activity of the higher-order system. Keywords: nonlinear dynamics, bifurcation, bursting, model reduction 1. Introduction The endogenous oscillations in the membrane potential of excitable membranes have been the subject of extensive analyses using mathematical modeling and computer simulation approaches. Among the most detailed models are those of cells whose oscillations in membrane potential exhibit bursting activity: a slow oscillation in membrane potential that alters between a silent state and a bursting state. The bursting state is characterized by the firing of a succession of action potentials. The most extensively modeled bursting cells are pancreatic fl-cells (Chay and Keizer, 1983; Chay, 1990; Keizer and Mangus, 1989; Sherman et al., 1988) and neuron R15 in Aplysia (Bertram, 1993; Butera et al., 1995; Canavier et al., 1991; Plant and Kim, 1976). Electrophysiological investigations over the past several decades have revealed a rich repertoire of biophysical mechanisms. Recent models simulate not only endogenous activity and changes in behavior produced by extrinsic currents, but also simulate changes in endogenous activity as the concentration of extrinsic modulatory agents are varied (Bertram, 1993; Butera et al., 1995; Keizer and Mangus, 1989). To achieve this level of accuracy and predictability the models

2 200 Butera, Clark and Byrne have become rather complex. Indeed, the most recent conductance-based model of neuron R15 (Butera et al., 1995) contains 12 state variables and over 50 parameters. Low-order models of the subthreshold oscillations that underly bursting behavior have frequently been employed to represent bursting cells in the analysis of small neural networks (Rowat and Selverston, 1993; Lofaro et al., 1994; Skinner et al., 1994). These models treat the membrane-potential waveform as an envelope that characterizes the activity of the neuron. This approach implicitly assumes that action potentials are not a significant factor in the subthreshold oscillations that underly bursting activity. Another approach is to average the effect of the firing of action potentials upon the slow variables that underly the burst cycle. This approach is numerically tedious unless the effects of fast dynamics of the action potential can be averaged by an explicit analytical calculation (Baer et al., 1995) o r a bifurcation analysis of the model allows one to apply analytically known spike-frequency characteristics (Ermentrout, 1994). None of the above techniques easily lends itself to the reduction of models of endogenously bursting cells. There are several complicating factors when dealing with bursting cells: (1) state variables operate on two time scales, one associated with the underlying subthreshold oscillation in membrane potential and one associated with the firing of action potentials; (2) the action potentials may perturb the slow state variables even while the slow variables dictate the nature of the firing of the action potentials; and (3) the mathematical expressions responsible for both the subthreshold oscillations and the firing of action potentials may share parameters, so that a change in parameters affects both the subthreshold oscillation and the character of the action potentials. In light of the issues outlined above, we wished to investigate two related questions. First, how well does the activity of low-order models of subthreshold oscillations predict the activity of higher-order systems that exhibit bursting, silent, and beating activity? Second, how can low-order models be extended so that they remain simple (and subject to analysis) while accurately predicting the activity of the higher-order system? In the present study we examined these questions using a model of Aplysia neuron R15 (Butera et al., 1995) as a representative model of a typical bursting cell. Reduced models of the isolated subthreshold oscillation (the SLOW system) and the isolated action potential subsystem (the FAST system) were derived. A form of geometric singular perturbation theory (Rinzel and Lee, 1987) was applied to the FAST system to examine the mathematical mechanism of bursting in the model. The effects of the FAST system upon the SLOW system were studied by employing simulations of the combined FAST and SLOW systems. The results of the mathematical analysis and simulations were applied to develop a low-order REDUCED model of bursting that retains the effects of the action potentials upon the slow variables of the system. Finally, the REDUCED and SLOW models were compared to assess how well each model predicts the activity of the original highorder model. 2. Model Review We used a recently developed model of neuron R15 in Aplysia (Butera et al., 1995), which consists of 12 state variables, including membrane potential, intracellular concentration of Ca 2+, intracellular concentration of came Ca~+-buffer occupancy, and eight voltage-dependent state-variables. The model adequately simulates bursting, beating, and silent activity as the stimulus current (Istim), the concentration of serotonin (5-HT), or the concentration of dopamine (DA) are modified. These parameters are hereafter referred to as the control parameters. The model also simulates the rather unique responses to current pulses in the presence of 5-HT or DA. In this study the complete model is referred to as the FULL system, which is identical to that published in Butera et al. (1995) except that a steady-state assumption is made for the concentration of camp ([camp] = 0), resulting in the following equation for the concentration of camp: where Kpde Kadc FS-HT CcAMP = (1) Vpde -- KadcfS-HT' C5_H T ) FS-HT ~ 1 -k- Kmod \'K5HT ~ C'--5-HT " This assumption reduces the FULL model to 11 state variables. Several changes in notation were made. For simplicity the variable c is used to represent the concentration of intracellular Ca 2+ in all mathematical expressions. The variables CcAMP, CDA, and C5-HT are used to represent the concentration of came DA, and 5-HT,

3 Dissection and Reduction of a Modeled Bursting Neuron 201 respectively. In addition, the term INS,Ca represents the portion of the INs current carried by Ca 2+ ions. The differential equations describing the activity of the FULL model are dv dt --(/spike q- /sub -- /stiin) Cm = (2) F INaCa(V, C) -- /Cap(C) 1 [ -IsffV, c, s) -/ca(v, d, f, c) dc [_ --INS,Ca(V, b, c) dt doc dt doc -n[b]i-- dt 2Voli F -- kvc(1 - Oc) - kroc /spike ~ INa(V, m, h) +/ca(v, d, f, c) + IK(V, n, l) + INs(V, b, c) /sub ~ ISI(g, C, S) "q- IR(V) q- IL(V ) -~ INaCa(V, C) q-/cap(c) + INaK(V), where V is the membrane potential, c is the concentration of intracellular Ca 2+, and Oc is the occupancy of the intracellular Ca 2+ buffer. The function /spike is defined as the sum of those currents that are active only during the firing of action potentials, whereas the function /sub is defined as the sum of those currents responsible for the subthreshold oscillations underlying bursting activity. (Refer to Butera et al., 1995, for a more detailed description of the role of individual currents and the associated parameters in the above equations.) The model also consists of eight Hodgkin- Huxley-type gating variables whose differential equations are of the form where dz dt z~(v) - ~z(v) z (3) (4) (5) Wanner, 1990). Calculation of the c and s nullclines lying upon the Vss solution surface was accomplished using PITCON (Rheinboldt and Burkardt, 1983a, b), a numerical continuation program that tracks the fixed solutions of a systems of equations as a free variable is varied. Computation of bifurcation diagrams, the two-parameter continuation of limit points, and the two parameter continuation of fixed orbits was performed using AUTO (Doedel, 1981), a bifurcation analysis package that solves for fixed and periodic solutions of a system of differential equations as one or two system parameters are varied. All time-dependent simulations were written in the C programming language except the numerical integration method, which was written in FORTRAN. All numerical analysis code (using routines of PITCON or AUTO) was written in FORTRAN. All code was run on Sun Microsystems Sparc-series workstations. 4. The FAST System In this decompositional analysis the model was divided into two subsystems: the FAST and SLOW systems. This division of the model was based upon the observation that the variables c (intracellular concentration of Ca 2+) and s (activation variable associated with the slow-inward current/st) are much slower than any of the other variables of the system. The FAST model consists of the equations of the FULL model with c and s treated as parameters of the system, rather than state variables. Thus, the differential equations for c, s and Oc (which is only dependent upon c) are removed, reducing the FULL model to the following differential equations: and dv --(/spike -]- /sub -- Istim) dt Cm (6) z ~ {h,m,d, f,s,b,n,l}. The gating variables are defined in Butera et al. (1995). The entire set of equations for the FULL model is given in Appendix A. where dz z~(v) - z dt rz(v) z ~ {h,m,d,f,b,n,l}. (7) 3. Computational Methods Temporal integration of all differential equations was accomplished using an implicit fifth-order Runga- Kutta method with variable step size (Hairer and Geometric singular perturbation theory (Rinzel, 1985; Rinzel and Lee, 1987) was employed to examine the mathematical structure that underlies bursting in the FULL model. This approach exploits the fact that SLOW variables c and s are much slower than

4 202 Butera, Clark and Byrne A = 125 nm stable equilbrium unstable equilibrium... stable periodic '~ 0- ~o L. ~ unstable periodic o Hopf bifurcation [] Saddle-node bifurcation I I -I 0 1 s activation c = 255 nm 40 ~2o ~."'~:~ g -2o ~ N -6o s activation Figure 1. Bifurcation diagram of FAST system as s is varied and c is fixed to 125 nm (A) or 255 nm (B) Periodic solution branches are indicated by the maximum (Vmax) and minimum (Vmin) values of the oscillation in V. The stable equilibrium solution branches are labeled EQ1 and EQ2. The stable periodic solution branches are labeled AP1 and AP2. The insets in Panel A illustrate 0.5 sec of periodic behavior on API and AP2 when s = 0.3. Vertical bar next to inset is a 40-mV scale. Shaded area indicates the physiological range ofs. the other variables in the FAST system. The variables c and s change on a time-scale of approximately 10 and 1 second, respectively, whereas the FAST variables responsible for the elicitation of action potentials change on a time-scale of milliseconds. By studying how the solution structure of the FAST system is altered as the SLOW variables are changed, we obtained an understanding of the mathematical mechanism underlying bursting in the FULL model. Using numerical bifurcation analysis software (Doedel, 1981) the solution structure of the FAST system was mapped out for a range of values in (c, s) parameter space. Portions of this solution are shown in the bifurcation diagrams of Fig. 1. Because there are two parameters one parameter (c) was fixed and the equilibrium and periodic solutions of the FAST system were calculated as s was varied. Figures 1A and B illustrate the solution structure of the FAST system for c = 125 nm and c = 255 nm, respectively. These values represent the minimal and maximal values of c encountered during the periodic burst cycle of the FULL system. An introduction to the concepts and terminology associated with bifurcation diagrams is provided in Appendix B. The shape of the steady-state solution manifolds illustrated in Figs. 1A and B are qualitatively similar. Although the dynamic range of s is only 0 to 1 (shaded range in both plots) it was necessary to vary s outside this range to understand the complete geometry of the solution structure. The stable solution manifolds are labeled for convenience. The most hyperpolarized stable equilibrium manifold is labeled EQ1. This manifold terminates at a limit point (a saddle-node bifurcation) where it joins an unstable equilibrium solution. The saddle-node bifurcation in Figs. 1A and B is a line in (c, s, V) space (see Fig. 2, line L). A second stable equilibrium manifold labeled EQ2 emanates from the Hopf bifurcation. However, the EQ2 manifold is of no importance during typical bursting activity of the model since its small basin of attraction is never passed through during the dynamics of normal bursting activity. The periodic solutions of the FAST system emanate from the Hopf bifurcation and are illustrated on the

5 Dissection and Reduction of a Modeled Bursting Neuron 203 ", jap1 40' v~ 20,,, \ -40, -6o, v~i. 10o c (nm) Figure 2. FULL system trajectory and partial bifurcation diagram of FAST system in (c, s, V) space. Solid mesh (thin solid lines representing c-isolines for c = {100, 200, 300) and s-isolines for s = {t3.2, 0.4, 0.6}) corresponds to the EQI manifold of the FAST system. The minimum and maximum of the periodic solution manifold AP1 at two values oft (dash-dotted lines) are also indicated. The saddle-node bifurcation L is indicated by the dashed line, and the c (dc/dt = 0) and s (ds/dt =- 0) nulmines are indicated by dotted lines. The FULL solution trajectory is represented by the thick solid line. bifurcation diagram of Fig. 1 by showing the minimum and maximum of the oscillation of membrane potential. This solution possesses three limit points, and at each limit point the system changes stability. The stable periodic solution branches are labeled AP1 and AP2 and are separated by an unstable solution branch. The solutions on branch AP1 approaches a homoclinic orbit (i.e., the period of the solution approaches infinity) as s approaches the saddle-node bifurcation. A similar saddle-node bifurcation has been demonstrated in analyses of other bursting systems (Rinzel and Lee, 1987; Rush and Rinzel, 1994). The periodic solutions AP1 and AP2 correspond to action potentials of the FAST system and possess distinctly different action potential characteristics. The insets of Fig. 1A show 0.5 sec of periodic activity on the stabie periodic manifolds AP1 and AP2 at s = 0.3. The bifurcation diagrams in Fig. 1 reveal that the FAST system possesses a rich repertoire of multistable behavior. At any particulax value of (c, s), there coexists a stable depolarized equilibrium solution (EQ2) and either a second stable hyperpolarized equilibrium solution (EQ1) or a stable periodic solution (AP1), depending on which side of the saddle-node bifurcation the point (c, s) is on. A second stable periodic solution (AP2) may also exist at certain values of (c, s). In the following analysis of the burst cycle, we are primarily concerned with the equilibrium solution EQ1 and the periodic solution AP1. Figure 2 illustrates the superposition of the solution trajectory of the FULL system in (c, s, V) space on top of the EQ1 and AP l manifolds from the bifurcation diagrams of Fig. 1 (AP2 is discussed below). The limit points comprising the saddle-node bifurcation are a line in (c, s, V) space labeled L. During the interburst interval of the burst cycle, the trajectory lies on the equilibrium manifold EQ1. The dynamics of the FULL system collapse to those of the SLOW system (nullclines of the SLOW system are superimposed on the EQ1 manifold in Fig. 2). As L is crossed, action potentials occur and the solution trajectory lies upon the AP1 manifold (dash-dotted lines, Fig. 2). These action potentials perturb the trajectory of the SLOW variables in (c, s) space, the significance of which will be investigated in the following section. At the cessation of bursting (action potential discharge), the solution trajectory of the FULL system again crosses L and relaxes back to the EQ1 manifold. Thus, the burst occurs as two SLOW variables (c, s) traverse the phase-space back and forth across the saddle-node bifurcation L.

6 - /stim) 204 Butera, Clark and Byrne Here L marks a transition between silent and periodic behavior of the FAST system. The saddle-node bifurcation L is the mathematical equivalent of the action potential threshold in (c, s) space. L is nearly isopotential in V and only varies from approximately -43 mv to mv. This solution geometry is qualitatively similar to that discussed in Section 5 of Rinzel and Lee (1987) and illustrated in Fig. 13 of Bertram et al. (1995). The periodic solutions AP1 and AP2 both exist in the FULL system. We ran simulations in which the FULL system was started with initial conditions on AP2. However, in each one of those simulations the solution trajectory "fell off" the edge of manifold AP2 onto AP1 (compare in Fig. lb). Thereafter, the burst cycle continued normally. We were not able to transiently perturb the solution trajectory onto AP2. This may be because any perturbation of a control parameter (such as/sam) perturbs not only the slow variables, but also the geometry of the FAST system. We find no numerical evidence that action potentials on AP2 play a physical role in the bursting activity of the model. 5. The SLOW System Previous studies have analyzed bursting systems by either eliminating the current responsible for the initiation of action potentials (Canavier et al., 1991; Plant and Kim, 1976) or by developing equations that represent only the slow-wave that underlies bursting (Kononenko, 1994; Plant and Kim, 1975). In both cases, the model in question was reduced to a secondorder system that was studied subsequently by nullcline analysis. In the present study, the SLOW system consists of the state variables that are responsible for the subthreshold oscillations that underly bursting activity. This approach is analogous to the first method described above in that all of the currents responsible for the action potential have been removed. However, we differ from Canavier et al. (1991) in our choice of state variables. Canavier et al. chose the two variables of the subthreshold oscillation to be V and c, with all other variables (including s) set to their steady-state values as a function of membrane potential. By ignoring the role of s, the slow-wave oscillation of the reduced model in Canavier et al. (1991) was that of a relaxation oscillator, with c as the slow variable and V as the fast variable. A comparison of the periodic solutions of the reduced and full model reveals that the solution trajectories of the two models differed, even in the interburst interval where action potentials are not elicited. Further investigations (Canavier et al., 1993) revealed s to also be a critical slow variable. As a result of these findings we chose c and s as the slow variables on which to base our reduction and made a quasi-steady-state assumption with regards to V (described below). The SLOW system was obtained by starting with the FULL system and reducing it in the following manner: 1. Eliminate each of the component currents of/spike (these currents are primarily active during the firing of action potentials), and 2. Assume that Ca2+-binding to the Ca2+-buffer (Oc) is nearly instantaneous (derivation given in Appendix C). These manipulations reduced the FULL system to the following three differential equations: ds soo(vss) - s dt dc dt "cs (Vss) INaCa(Vss, C) -- Isi(Vss, C, S) -- /Cap(C) 2Voli F Soc (c) - dvs~ --(/sub dt Cm ' where goc Sot(C) = 1 + nb[b]i (c + Koc) 2 (8) (9) (lo) and Koc is defined as kn/ku. The SLOW system is essentially of second-order if the state variable Vss is assumed to be at or near steadystate. This appears to be a valid assumption which we verified geometrically and numerically. Geometrically, the periodic solution of the SLOW system trajectory in (c, s, Vss) space was found to lie on a surface S (not shown) that represents the unique steady-state solution of Vs~ for any value of (c, s). The surface S is similar to the EQ1 manifold of the FAST system at subthreshold potentials (below--45 mv), and is geometrically similar to the surface shown in Fig. 2 of Rinzel and Lee (1987). Numerically, when the rate of the equation for Vss was increased significantly (by reducing CM by a factor of 1000), little change was observed in the solution trajectory, demonstrating that the equations for c and s are the rate-limiting equations of the system.

7 Dissection and Reduction of a Modeled Bursting Neuron 205 A 1 [=....~.eg... C 0.8 "~ L-..7 '... _ c (rim) (SLOW system) E,~ I~ Figure 3. Comparison of SLOW, FULL, and combined FAST-SLOW simulations. A. (c, s) phase-space comparison of solution trajectories. Nullclines of the SLOW system (~ , ~ = 0, dotted) and the saddle-node bifurcation L of the FAST system (dashed) are also shown. Simulations shown are the SLOW system (B), the FULL system (G), and different combinations of the combined FAST-SLOW simulation (C-F). See Table 1 and text for a description of the different FAST-SLOW interactions implemented in simulations D-G. B-G. Comparison of one period of membrane potential for the SLOW (13), FULL (G), and combined FAST-SLOW (C-F) simulations. In each simulation one period of the burst cycle is shown. 4 sec 6. Interaction of FAST and SLOW Processes The approach just described (in which the ionic currents responsible for the action potentials are eliminated) assumes that the perturbational effects of the FAST system upon the variables of the SLOW system are not significant--that is, the effects of action potentials can be ignored. However, a comparison of the temporal and phase-space trajectories of the periodic solution of the SLOW and FULL systems reveals that the FAST system exerts considerable influence on the slow variables. Figure 3 illustrates the (c, s) phase-space (Panel A) and temporal (in V) trajectories of the SLOW (Panel B) and FULL (Panel G) systems. The FULL trajectory in Panel A is a projection of the trajectory shown in Fig. 2 onto the (c, s) plane. The nullclines of tile SLOW system and the saddle-node bifurcation L of the FAST system are also shown in panel A. When the solution trajectory lies "below" L, the FULL and SLOW systems are governed by similar equations of motion (i.e., graphically the solution trajectories "obey" the SLOW system nullclines), This phase corresponds to the interburst hyperpolarization in the FULL system. However,

8 206 Butera, Clark and Byrne Table 1. Simulated effects of FAST system on SLOW system in Fig. 3. Simulation ds /dt dc/dt B s.~ (.%) -s (SLOW) ~s (vs,d C s~(v) -.,, rs (V) D s~ CVss) - s "rs(vss) E s~(v)-s ~(V) F s~ (v) - ~, ~N) G s~.(~ -s (FULL) r~(v) 1NaCa ( Vs~, c) -- IS I ( Vs,~. c.s) -- 1C~ p (c) 2V~Ii FSOc (c) 6~c~(Vs~.c) - tst(vs~,c,s) - IC~p (c) 2Voli F SOc (C) INaca(Vss,C ) -- lst(vss,c,s ) -/Cap(C) - lca(v,d,f,e) 2Voli FSOc (c) INa Ca(Vss,C) -- 1SI (Vss.c,s) --/CaP (c) -- lca0/.d.f,c) 2Voll FSOc (c) //'Ca Ca ( Vss, c) - lsi ( Vss, c,s ) -/Ca V (c) - lea (V,d. f, c) - INS,Ca (,b, ) 2Voll F So c (c) INa Ca(V,c) - Isl(V,c,s) -/Cap(C) - ICa (V,d,f.c) -INS,Ca (V,b, ) 2Voll FSOc (c) Differences between the simulated system and the SLOW system equations are highlighted in bold. when L is crossed and the solution trajectory crosses onto the AP1 manifold (Figs. 1 and 2), the firing of action potentials perturbs the intended trajectory of the SLOW system. In Fig. 3A, this is illustrated by the periodic perturbations of the FULL trajectory when the trajectory lies "above" L in (c, s) space. Computer simulations were used to artificially separate the FAST (action potential) and SLOW (subthreshold oscillations) systems of the model. The effects of the SLOW variables on the activity of the FAST system were studied in the analysis of the FAST system (e.g., Figs. 1 and 2), whereas in this section the effects of the FAST system upon the SLOW system are studied by simulating the FAST (Eqs. (6)-(7)) and SLOW (Eqs. (8)-(10)) systems concurrently. Our goal was to determine specific interactions that account for the primary effects of the FAST system upon the SLOW system. In the combined FAST-SLOW simulations we "turned on" individual dependencies of the SLOW system equations on the FAST variables as they exist in the FULL system. The results of the combined FAST- SLOW system simulations are illustrated in the phase plane in Fig, 3A and temporally (in V) in Figs. 3C-F. Panels B and G illustrate the periodic solutions of the SLOW and FULL systems, respectively. For each of the simulations in Panels C-G, a different functional dependency was made between the equations of the SLOW system and the state variables of the FAST system. Each of these functional dependencies is one that exists in the FULL system. These dependencies are summarized in Table 1. In Simulation C, the equation for ds/dt was modified to be dependent on V (of the FAST system) instead of the Vss that was used in the SLOW system. Each action potential perturbed the trajectory in the positive s direction, These perturbations increased the amount of time that the trajectory remained on the AP1 manifold (above L) and prolonged the active phase of the burst cycle. In Simulation D, the fast Ca 2+ current/ca (which is dependent on V, d, and f of the FAST system) was incorporated into the equation for dc/dt. Each action potential activated/ca and the resulting Ca 2+ influx perturbed the trajectory in the positive c direction. These Ca 2+ perturbations decreased the amount of time that the trajectory remained on the AP1 manifold because the trajectory was moved more rapidly towards L. At L the action potentials ceased firing. The results of Simulations C and D can be related to specific biophysical mechanisms within the model. It has been shown that the presence of a burst can be related to the magnitude of the subthreshold currents (Fig. 7, Canavier et al., 1991). In Simulation C, the perturbations of s by the action potentials increases the conductance of Isi, the inward current responsible for the slow-wave underlying the burst (see Canavier et al., 1991, for a review of the bursting mechanism). Thus a greater amount of inward current maintains the depolarization and prolongs the burst. In Simulation D, the perturbations of c by the action potentials inactivate Isb an inward current, and activates Ic~p, an outward current. This net-reduction in inward current eventually hyperpolarizes the cell below the threshold for the action potential and the burst is terminated. In Simulation E, the interactions in Simulations C and D were combined. The resulting solution trajectory was similar to that of the FULL system in both (c, s) space and temporally in V. The voltage,perturbations of s (C), which prolong the burst phase, and the Ca 2+influx with each action Potential (D), which shorten the burst phase, have a combined net effect ef shortening

9 Dissection and Reduction of a Modeled Bursting Neuron 207 the depolarized phase of the burst cycle (E) when compared to the SLOW system (B). Incorporation of additional interactions resulted in trajectories that even more closely approached that of the FULL system. For example, in Simulation F the current INS,Ca was incorporated into the material balance dc/dt in addition to the interactions considered in Simulation E. These simulations were a valuable tool for determining the effects of the action potentials upon the slow processes of the modeled bursting cell. The two interactions that accounted for a majority of the effects of the action potentials were (1) the voltage-dependency (or frequency-dependency) of s and (2) the effects of Ca2+-influx (through Ica and to a lesser extent INs.ca) on c. The voltage-dependent effects of the action potentials upon the subthreshold currents (/sub) did not appear to be significant. 7. Model Reduction The results of the analyses of the previous two sections were used to develop a low-order model of the cell that retains the essential features of the FULL system. The equations of the SLOW system were modified to account for the perturbational effects of the FAST system (i.e., the action potentials) on the SLOW variables. This modified system of equations is referred to as the REDUCED system. The SLOW and REDUCED systems are collectively referred to as the low-order systems. The modifications to the SLOW system were based on the results of the bifurcation analysis of the FAST system and the combined FAST-SLOW system simulations. From the bifurcation analysis, it was shown that the. saddle-node bifurcation L separates the (c, s) statespace into two regions. In the lower region the FAST system has a stable hyperpolarized equilibrium solution (the EQ1 manifold), and the dynamics of the FULL system reduce to the dynamics of the SLOW system. In the upper region the FAST system has a stable periodic solution (the AP1 manifold) that corresponds to the firing of action potentials. These action potentials perturb the trajectory of the slow variables c and s. Furthermore, the saddle-node bifurcation L is homoclinic (i.e., the period of action potentials approaches infinity as L is approached in (c, s) space). This homoclinicity is illustrated in Fig. 4A, which shows the solution sets of fixed periods of the FAST system on the API manifold for periods of 200 msec (5 Hz) and 333 msec (3 Hz) (short dashed lines). These solution sets were calculated using AUTO. The model was started with a given value of c and s and an initial periodic solution on AP1. AUTO was used to vary s and the period of A B 10 t.r If':.. 3 L i ~' ' ~6 e~ ID 4 / 0.I 0 1~ ;0 c (nm) o.b4 o.b6 o.bs 0'.1 Distance (unitless) Figure 4. Spike frequency is approximately a function of distance from L. A. Frequency isolines (dashed and dotted), burst trajectory, (solid) and saddle-node bifurcation L (long dashed) of the FULL and REDUCED systems in (c, s). Solutions of the FAST system on AP1 with fixed periods of 333 msec (3 Hz) and 200 msec (5 Hz) are also shown (short dashed). The period of oscillation approaches infinity as L is approached from above. Dotted line nearly identical to L is the y = mx + b approximation to L (L/) used in the REDUCED model. The other two dotted lines correspond to steady-state constant q solutions of the REDUCED model for q = 3 Hz and q = 5 Hz. B. Spike frequency as a function of distance from L. Horizontal lines correspond to distances in (c, s) from L I of fixed-period solutions of the FAST system (see text for details). Solid line is the steady-state frequency function q~ (D) used in the REDUCED model.

10 208 Butera, Clark and Byrne the solution on AP1 was monitored. Once the desired period was located, we used AUTO to perform a two parameter continuation of a fixed period to trace out a line in (c, s) that represents all solutions on AP1 with the same steady-state firing frequency. We found that the period of the orbit increases toward infinity as L is approached from the AP1 manifold (oscillations with periods as large as 8 sec were simulated). The homoclinic nature of L provided the first assumption used in developing the REDUCED model: the spike frequency of the FAST system is approximately a function of distance (in (c, s)) from L. Starting with the three differential equations for the SLOW system, a fourth state variable q was introduced that represents the spike-frequency of the action potentials of the FAST system. The differential equation for q is q~(d(c, s)) - q t) =, (11) rq(o(c, s)) where D(c, s) is a function representing a signed, scaled distance in (c, s) from U where U is a linear (y = mx + b) fit to L in (c, s) space. The function qoo(d) represents steady-state spike-frequency of the FAST system at a given point in (c, s) and was formulated as a function of distance from U. This formulation was based on the results of numerical simulations of spike-frequency versus distance from L' as illustrated in Fig. 4B. Numerically, the function q~(d) is a product of a tanh formulation to ensure that the function is positive when distance is positive and negative when distance is negative, and a Boltzmann-like fit to the data in Panel B. Positive values of q~ (D) represent steadystate spike frequency, but negative values are meaningless and correspond to the absence of action potentials. The horizontal lines in Fig. 4B illustrate the varying distances from L' of each fixed-period solution of the FAST system (e.g., the horizontal bar at 3 Hz shows the range of distances from U of the line representing the 3 Hz fixed-period solutions of the FAST system in Fig. 4A). The function qoo(d) is also illustrated in Fig. 4B. Plots of solutions to the REDUCED system at fixed values of q (dotted lines) are superimposed on the fixed-period solutions of the FAST system (dashed lines) in Fig. 4A. Although making q~ a function of distance from L is only a simple approximation (i.e., the fixed-period solution sets of the FAST system are not exactly parallel) there is reasonable agreement between the fixed-period solutions of the FAST system and the solutions of the REDUCED system at fixed values of q. The new variable q was formulated as a state variable instead of using the instantaneous expression qoo(d(c, s)) for the following reasons. First, the solutions of the FAST system on AP1 are steady-state periodic solutions. In the FULL (and FAST) system, there are two gating variables (b and l) that activate quickly but decay slowly and are active only during the firing of action potentials. Second, when the solution trajectory in (c, s) initially crosses from the EQ1 manifold to the AP1 manifold to begin the burst phase, a delay is encountered as the solution trajectory converges to the periodic AP1 manifold. To account for the time-dependency of the spike-frequency, we added the time-constant function rq (D). This function has a sigmoidal formulation that is step-like and saturates at 1.0 seconds when D(c, s) is positive and 0.1 milliseconds when D(c, s) is negative. The saturated value of 1 second is in the range of the values of l- b (V) and rl (V) of the FULL model at depolarized potentials, which are approximately 0.5 sec and 2.0 sec, respectively. The formulations for qo~(d), rq(d), and D(c, s) are given in Appendix D. The findings from the combined FAST-SLOW simulations provided the second assumption in developing the REDUCED model: the key mechanisms by which the FAST oscillations perturb the SLOW system variables are (1) the voltage dependency of s and (2) the Ca 2+-influx associated with each action potential. The equations for soo(v~s) and r.~.(vss) were reformulated to be dependent on q. This was achieved by formulating a function Vaep(q) that represents the average depolarizing effect of the action potentials on the membrane potential dependencies of s~(v~) and r,. (Vs~). Simulations were run comparing the average membrane potential over the time course of the oscillation of the FAST system (Vavg) with the steady-state membrane potential of the SLOW system (V~) as s was varied and c was fixed to 200 nm. This value of c lies in the middle of the dynamic range of c encountered during the burst Cycle. The steady-state spike-frequency of the FAST system was also measured. These results are shown in Fig. 5A (circles). A qualitative picture of the shape of Vs~ and V~vg and the calculation of Vdep is shown in the inset of Fig. 5A. The difference between Vss and V~vg varied from 4 to 6 mv as a function of spike-frequency. The function Vd~p(q) (solid line) provided a simple fit to the data and was formulated as a steep sigmoidal-function that is 0 for q < 0 and

11 Dissection and Reduction of a Modeled Bursting Neuron 209 A 5 O O O r~oooooooc ~ B 10 O oy /( Vdep = Vavg- Vss 6 N4 < 2 Frequency (Hz) Frequency (Hz) Figure 5. Perturbational effects of the FAST system on the SLOW variables quantified in terms of spike-frequency. A. Average depolarization of membrane potential (due to the FAST dynamics) as a function of spike-frequency. Circles indicate difference between average membrane potential of the oscillations of the FAST system (Vavg) and the steady-state membrane potential of the SLOW system (Vss) plotted versus the steady-state spike-frequency of the FAST system. Inset illustrates qualitatively the shape of Vss and Vavg. Shaded region represents Vdep. Data were obtained for both systems by fixing c to 200 nm and varying s. Solid line is a plot of Vdep versus q. B. Average calcium influx as a function of spike-frequency. Circles indicate average Ca 2+-flux due to/ca in FAST system versus spike-frequency of the FAST system. Data were obtained by fixing c to 200 nm and varying s. Solid line is a plot of ICa, aux versus q with c fixed to 200 nm. approaches 5.5 mv for q > 0. Specification of the function Vdep(q) allowed us to retain the original formulations for s~(v) and r, (V), while only modify the input to those functions. The new differential equation for s is given by where ds s~(vss + Vdep(q)) -- S -- =, (12) dt rs (Vr) V~ --- min(vss + Vdep(q), --25). The expression V~ was necessary in order to retain the original formulation for Vs. The function rs(v) begins to decrease at potentials positive to -25 mv. In the FULL model each action potential spends only a few milliseconds in this depolarized range. However, in the REDUCED model the membrane potential of the slow-wave oscillation for long bursts (such as those due to high concentrations of 5-HT) often spends several seconds at potentials above -25 mv. Therefore, the expression V~ was necessary to preserve the slow timescale of the s gating variable. The Ca2+-influx accompanying each action potentia] was modeled by the expression Ica~ux(C, q), which represents the average flux of Ca 2+ due to /Ca, and is given in Appendix D. This current increases with q and is inactivated by c in a manner identical to the Ca2+-dependent inactivation of/ca- To develop this expression, the average Ca 2+ flux due to/ca over one period of oscillation of the FAST system was measured as s was varied and c was fixed to 200 nm. The average value of Ica over one period of oscillation versus the frequency of oscillation of the FAST system is illustrated in Fig. 5B (circles). The function Icaflux is also illustrated (solid line). This function is a product of three terms: (1) a linear fit to the spike-frequency data of Fig. 5B, (2) the Ca2+-inactivation term present in the formulation of/ca of the FULL model, and (3) a steep sigmoidal function that ensures that Ica~ux is zero when q < 0. The error between the linear fit and the data at high frequencies is acceptable because during the burst cycle the higher frequencies are encountered only at high values of c, where Ca 2+ influx due to Ica is mostly inactivated. Incorporation of Icaflux into the model results in the following modified equation for c: 1NaCa(Vss, C) --/Cap(C) q) ] dc L -lsi(vss, c, s) + lea flux(c, dt 2VoliFSoc (c) (13) The REDUCED system was simulated and its periodic solution was compared with the solutions of

12 210 Butera, Clark and Byrne 0.8 i=0, l F--x 0.6 i.., "~ 04 -J' ~'~' E ",- -,."....'/ direction 0.2 /.. of motion ~._] 20 mv 2 sec oi q 200 ~ 250 J 300 c (nm) Figure 6. Solution trajectories of the SLOW (A), FULL (B), and REDUCED (C) systems. Trajectories are shown in (c, s) space, along with the SLOW system nullclines (dotted) and the saddle-node bifurcation L of the FAST system (dashed). Insets show 25 sec of activity. the SLOW and FULL systems. Figure 6 illustrates the (c, s) phase-space and temporal solutions for the three systems. The REDUCED system closely approximated the trajectory and period of the FULL system. The general effect of q in the REDUCED system (and the action potentials of the FULL system) is that the solution trajectory spends much less time in the active phase of the burst (i.e., the region of (c, s) phase-space in which action potentials occur) Effects of Control Parameters The effects of the controlparameters on the REDUCED model were also considered. The control parameters (Is~m, CS-HT,CDA) appear explicitly in the equations for the ionic currents of the SLOW system and need no modification. However, the control parameters also alter the solution structure of the FAST system. Istim appears explicitly in the equation for membrane potential, and CS-HT and CDA alter the conductances gr and gsi. Calculations using AUTO revealed that/stim and gr altered the slope and location of L in the FAST systems, and we quantified these changes in terms of the slope and y-intercept of U (ml and bl, respectively). Increasing gr caused a nearly proportionate increase in ml and a proportionate decrease in bl, so we formulated these changes in terms of a normalized conductance NR. Increasing (or decreasing) Istim caused an additive linear decrease (or increase) in both ml and bl. Based on these observations empirical expressions were developed for the effects of gr and/stirn on ml and bl. These expressions were incorporated into the distance function D(c, s), which is given in Appendix D. Insights into the relationship between gs~ and the solution structure of the FAST system were obtained by examining the equation for Isi (see Appendix A). Because s is a parameter of the FAST system and gsi is multiplied by s in the equation for Isb altering gsi simply scales the solution structure of the FAST system in the s direction. This scaling was incorporated into the distance equation D(c, s) by multiplying s by Nsb a normalized value of gsi. Thus, all the effects of the control parameters upon the FAST system were treated in the REDUCED system by altering the distance function D(c, s), which characterizes the distance measure (and subsequently the spike-frequency) in (c, s) from L'. 8. Model Evaluation The SLOW model does not incorporate the effects of action potentials on the underlying burst cycle, whereas the REDUCED model takes into account the net perturbational effects of the action potentials on the slow variables. The validity of each these assumptions was

13 Dissection and Reduction of a Modeled Bursting Neuron 211 assessed by comparing how well the SLOW (3 state variables) and REDUCED (4 state variables) systems predicted the activity of the FULL system (11 state variables) The low-order systems were compared to the FULL system according to three criteria: (1) predicted mode of activity (bursting, beating, silence) as control parameters are varied, (2) solution trajectories and period of oscillatory (bursting) solutions as control parameters are varied, and (3) response to transient input (brief current pulses) Activity Modes The ability of the low-order systems to predict the mode of activity of FULL system as individual control parameters were varied was investigated. Hyperpolarized stable solutions of the low-order systems were assumed to predict silent activity in the FULL system, oscillatory stable solutions of the low-order systems were assumed to predict bursting activity in the FULL system, and depolarized stable solutions of the low-order systems were assumed to predict beating activity in the FULL system. The extent of these activity modes as individual control parameters were varied was determined from bifurcation diagrams of the low-order systems (not shown) that were generated with AUTO. These results were confirmed with numerical simulations. The extent of the modes of activity of the FULL system as individual control parameters were varied was determined by numerous simulations of the FULL system over gradually varying parameter values. Figure 7 illustrates the modes of activity predicted by the low-order systems and the simulated activity of the FULL system as the parameters Istim (Panel A), concentration of DA (Panel B) and concentration of 5-HT (Panel C) were varied. The modes of activity exhibited by the FULL model as parameters are varied are consistent with the experimental data (Butera et al., 1995; Canavier et al., 1991). In all three systems the transitions between modes of activity were bistable, in that for each parameter there existed a range of values at the interface between two modes of activity where both modes coexist. The low-order systems differed however, in the location and extent of these bistable regions in parameter space. For all three control parameters (Panels A, B, and C) the low-order systems provided comparable predictions of the location of the bistable regions at the interface between the regions of silence and bursting activity. This is expected because the transition between oscillatory activity and silence occurs at hyperpolarized potentials where the fast processes are not active and the two low-order systems are dynamically similar. However, the low-order systems varied in their predictions of the size and location of the bistable regions at the bursting/beating interface (Panels A and C). As Istim was increased, the REDUCED system predicted a large bistable region that was not predicted by the SLOW system (Fig. 7A). Likewise, the RE- DUCED system was more accurate in predicting the size and location of the bursting/beating interface as the concentration of 5-HT was increased (Fig. 7C) Solution Trajectories The control parameters modulate the burst waveform of the FULL system in many ways (see Fig. 5 of Canavier et al., 1991, and Figs. 5 and 6 of Butera et al., 1995). Increasing Istim decreases length and depth of the interburst hyperpolarization while increasing burst length. Increasing concentrations of DA increase the length of the interburst hyperpolarization and shorten the burst. Low concentrations of 5-HT increase the length of the interburst hyperpolarization, whereas high concentrations of 5-HT increase the burst length. Thus the solution trajectories of the low-order systems were compared with the FULL system across a wide range of parameter values that represent the different forms of bursting. Representative samples of these comparisons are shown in Fig. 8. Each panel in Fig. 8 corresponds to a particular value of a control parameter. The top portion of each panel compares the system trajectories in (c, s) space, whereas the bottom portion compares the membrane potential waveforms of each system. Across the entire parameter space examined, the REDUCED system was much more accurate than the SLOW system at reproducing the dynamics of the FULL system in both the (c, s) phase plane and the waveforms of the underlying membrane potential Response to Transient Input Although it has been demonstrated that the REDUCED system is much more faithful than the SLOW system at reproducing the trajectory of the FULL system, one might question whether these differences are significant when cells of this type are driven by synaptic input. Phase-response curves (PRCs) were used to examine this issue quantitatively, as they provide

14 212 Butera, Clark and Byrne silence bursting beating A FULL ~ ~ ~ ~: ~ i ~ : ~ ~ ~ ".,,.. REDUCED iiiiii!!iiii!iii Ii ii!i!i~iii!ii!iii~!iiii~i! i!i:~i B SLOW FULL ij~ii~iiii {!~[i:1i{{:: i?!iiii ii:!i I I I I ISTIM (na) REDUCED N ~, i ~ C SLOW ~ FULL iiiiiiiiiiil i!ii:~i#i!ii!i~:i~/ri ii!:i~i~llii:!li/il )iigiii/~:i/i!!ii!iii:iii!i~il~ii~i~i~!i I I I I I [DA] (~xlvi) ~"~ ~ '~ i! ~ ~ ~o... ~ ~ I~,~i ~: " ~ ~ ~' ~', ~ii!ii!i/i/~!i~iil ii'~?~iil~i~ii:.i~'~iiiii:.iil,iil " ~ ~~ ~ ~ ~ ~,~.... ii... ~ ~, SLOW ililiill!,:, il,ii iii iiil iiiiiiii4 i, iliil ii!i I I I I I I I I I to0 120 [5-HT] (IxM) Figure 7. Modes of activity predicted by the SLOW and REDUCED models with that of the FULL model as the parameters /STIM (A), concentration of DA (B), and concentration of 5-HT (C) are varied. a measure of how a periodic system responds to transient input (Pavlidis, 1973; Pinsker, 1977). Because the SLOW system period was significantly different from that of the FULL system, we compared only the PRCs of the REDUCED and FULL systems. The PRCs for each system were generated by applying a transient hyperpolarizing current pulse stimulus (5 na, 250 msec) ts seconds into the burst cycle and recording the duration of the current burst period (P1). If the free-running period had a length of Po seconds, then if P1 - P0 is less than zero, the stimulus advanced the phase of the system. If P1 - P0 was greater than zero, the stimulus has delayed the phase of the system. The beginning of the cycle (t = 0) was defined as the point at which the membrane potential crosses -45 mv (dotted line in Fig. 9A) in a positive direction. A comparison of the PRCs for the FULL and REDUCED systems is illustrated in Fig. 9. Panel A illustrates the membrane potential waveforms for reference and Panel B illustrates the PRCs for each system. The inset in Panel B illustrates an experimentally obtained PRC (Pinsker, 1977). Because the experimental PRC is of a cell with a different burst period than that of the model, it is shown on a normalized time scale. The waveforms and PRCs of each model system were aligned so that the end of their periods coincided because the interburst phase of both models is nearly identical. The interburst phase was defined as the portion of the burst cycle where the membrane potential was below -45 mv. The PRCs of both are nearly identical during the interburst phase, which is expected since the interburst phase corresponds to the region in phase space where the FULL and REDUCED systems dynamically collapse to the SLOW system (i.e., absence of action potentials in the FULL system and q < 0 in the REDUCED system). During the active phase the REDUCED system has a similar PRC as the FULL system. This finding is expected, since a comparison of the

15 Dissection and Reduction of a Modeled Bursting Neuron 213 Control Is~M = -0.5 na Is~M = +1.0 na g 0.6 o.6 o,6 I I!!o, o, I 0.2 ~0.2 I P ~) c (nm) c (nm) c (nm) AIAL LJ!A_ R R R S ~ S S 100 ~M DA gm 5-HT 50 gm 5-HT 11 ~ 0-2 I ~ ~ ~0 1~0~0 c (nm) I 260 c (nm) ~0.6 t 1 5 o.21 F ~ i000 c (nm) R R S S S ~ - ~ ~ l ~ 40 mv 5s Figure 8. Solution trajectories of the FULL (F), REDUCED (R), and SLOW (S) systems for a range of values of the control parameters. The top portion of each panel illustrates the solution trajectories in (c, s) space. The bottom portion of each panel illustrates the oscillation in membrane potential versus time for each model. For Istirn = 1.0 na, the SLOW system remains at a stable depolarized equilibrium point. solution trajectories of the FULL and REDUCED system (Fig. 6) shows that both systems are always close to each other in (c, s) phase-space. Thus an applied hyperpolarizing stimulus should perturb each system in a similar manner. The responses of both the FULL and REDUCED systems are comparable to the data of Pinsker (1977). 9o Discussion 9.1. Experimental Evidence for FAST-SLOW Interactions Numerous experimental studies have demonstrated slow wave activity in R15 in the absence of action potentials (Junge and Stephens, 1973; Mathieu and Roberge, 1971; Strumwasser, 1971; Wilson, 1982). However, those studies did not focus on studying the interaction of the action potentials with the underlying slow wave. In the presence of tetrodotoxin (TTX), the slow-wave of R15 appears to have a more sinusoidal shape than that shown by our SLOW model (Junge and Stephens, 1973; Mathieu and Roberge, 1971; Strumwasser, 1971). Figure 3 of Mathieu and Roberge (1971) illustrates the effects of TTX on L6, another bursting neuron in the abdominal ganglion of Aplysia whose mechanism of bursting is believed to be similar to R15 (Alevizos et al., 1991). The underlying slow wave appears quite similar to that of our model.

16 214 Butera, Clark and Byrne A / B,-, 2 -.,--.,4_ O -2,~-4 i o... - o0.,.. _ +.0ot......,... + FL L _ ~_:Oo.~ L " " o REDUCED I 02 0t 0,, FULL I I I I I I r REDUCED I 0 L 2 I 4 q 6 I 8 10 I 12 t s (seconds) Figure 9. Phase-response characteristics of the FULL and REDUCED systems. A. Membrane potential waveforms for the two models. The FULL and REDUCED model waveforms are indicated by thin solid, and thick solid lines, respectively. Waveforms are aligned so that the end of their cycles coincide. The horizontal dotted line represents -45 mv, which is defined as the beginning of the burst cycle for both models. B. Phase-response curves (PRC) of the two models. PRCs are aligned so that they line up with the waveforms in panel A. Inset illustrates an experimentally obtained PRC from L3 in response to a 2 sec current pulse (magnitude unspecified). Data adapted from Fig. 9F of Pinsker (1977). Wilson (1982) found that phenobarbitol blocked action potentials in R15. His data are very similar to our simulations: after elimination of the action potentials, the period of the slow wave is prolonged due to an increase in the length of the depolarized phase of the slow wave. The hyperpolarizing phase of the burst cycle was identical before and after application of phenobarbitol, suggesting that phenobarbitol did not affect the subthreshold currents. This observation is consistent with our model analysis (Fig. 2), which suggests that the SLOW and FULL models are dynamically similar during the interburst interval of the burst cycle. In addition, the slow wave illustrated in Wilson (1982) is very close to that of our SLOW system. Thus it appears biologically plausible that the action potentials affect the underlying slow wave, an implicit assumption in our model Comparison to Earlier Analyses Our analysis of the mathematical mechanism underlying bursting revealed a solution structure that is consistent with analyses of other bursting models, specifically those described as parabolic bursters. The burst occurs as two slow variables (c and s) traverse the state space back and forth across a saddle-node bifurcation that marks a transition between silent and periodic (action potential) behavior of the fast subsystem. The parabolic nature of the burst is due to the homoclinicity of the saddle-bifurcation. That is, the beginning and end of the burst are close in state space to the saddlenode bifurcation, which corresponds to a higher period of action potentials. Our model is dynamically similar to the Ca-Ca model analyzed extensively in Rinzel and Lee (1987). In the classification scheme of bursting

17 Dissection and Reduction of a Modeled Bursting Neuron 215 models proposed by Rinzel (1987) and extended by Bertram et al. (1995) our model is classified as a Type II burster Model Reduction Methodologies Formal reduction schemes focus on minimizing the number of state variables while preserving the membrane potential waveform (e.g., Kepler et al., 1992). Our approach assumes that the system is operating on two different time scales and aims at preserving the dynamics of the slower time scale while retaining the "effects" of the faster time-scale. In models with a similar solution geometry as our fast model (i.e., the transition between silence and periodic behavior is a saddle-node bifurcation), it has been observed that if the saddle node bifurcation occurs at a parameter value p*, then for parameter values near p*, the spike frequency is proportional to ~/[p - P*I+ (Rinzel and Ermentrout, 1989). This approach was applied in Ermentrout (1994) to reduce a system of coupled neurons to functions of spikefrequency. We tried this approach with our FAST subsystem by redefining the steady-state spike frequency function in Eq. (11) to be q~(d(c, s)) = 30 sgn(d(c, s))v/[d(c, s)] (14) where D(c, s) is a measure of parameter distance, 30 is a proportionality constant that was selected to best fit the data of Fig. 4B, and the sgn function is to provide continuity at q = 0. With the above modification, the REDUCED model possessed solution trajectories and periods of oscillation that compared favorably with the FULL model. We did not use this approach because the expression that we developed to fit the data in Fig. 5B yielded a REDUCED model that compared even more favorably to the REDUCED model. However, the assumptions underlying Eq. (14) appear to apply to our model, and Eq. (14) suggests an even more compact form of the REDUCED model whose utility would be improved if a more reliable distance measure was developed. There are few examples of the application of formal averaging to the reduction of bursting models, although formal averaging has been applied to the analysis of such models (see the following section titled "Average Nullclines"). A formal analytical approach similar the the empirical-computational approach presented in this paper has recently appeared in the literature (Baer et al., 1995). These authors developed a minimal three- variable model of parabolic bursting consisting of two slow variables (x and y) and one fast variable (0). The fast variable did not represent membrane potential but instead was a phase variable representing the phase of the action potential. The differential equation of the fast variable was of the form O = 1 - cos(0) + A(x, y) (15) A(x, y) = tanh(ax - by + I), (16) where A (x, y) is an activation function quantifying the effects ofx and y on 0 and -1 < A(x, y) < 1. When -1 < A(x, y) < 0, Eq. (15) has a stable steady-state solution, while when 0 < A(x, y) < 1, Eq. (15) is always positive, representing the continuous firing of action potentials as 0 increases across multiples of 27r. As A(x, y) decreases towards 0, the period of 0 becomes infinite. The periodic behavior of Eq. (15) is similar to the FAST system of our model, and the function A(x, y) is analogous to our distance function D(c, s), which represents the signed distance from the saddle-node bifurcation L of our FAST system. When D(c, s) > 0, our FAST system exhibits periodic activity, and as D(c, s) approaches zero, due to the homoclinicity of L, the period of the oscillations of the FAST system approach infinity. When D(c, s) < 0, our FAST system is at a stable-steady state. Due to the simplicity of Eqs. (15) and (16), Baer et al. were able to apply formal averaging and develop analytical expressions of the effect of 0 on x and y to reduced their model to just the variables x and y. When A(x, y) < 0, an analytical steady-state solution to Eq. (15) exists (in terms of A(x, y)) and was substituted into the differential equations for x and y. When A(x, y) > 0, it was possible to analytically determine the period of 0 and use the results to obtain closed-form expressions of the average effect of 0 upon x and y in terms of A(x, y). The dynamics of their two-variable system compared quite favorably to their 3-variable model. Qualitatively, their 3-variable model was dynamically similar to the conductance based model of Rinzel and Lee (1987). The approach taken to reduce the phase model above eliminated FAST dynamics entirely and is analogous to what our REDUCED model would become if we eliminated the variable q and made c and s dependent on q~ (D). Reasons for not following this strategy were outlined in the Model Reduction section. However, we note that such a reduction does yield a 3-variable model

18 216 Butera, Clark and Byrne that is still more accurate than the SLOW system at producing dynamics comparable to the FULL system. Unfortunately, the approach taken by Baer et al. (1995) cannot be easily applied to conductance-based models. Most equation for dv/dt are a function of one or more currents with active nonlinear conductances that do not have analytic solutions. However, wherever the FAST system is parameterized by a single slow variable (Rinzel, 1985) or a single function of several slow variables, such as the conductance of /Ca in Rinzel and Lee (1987), it may be possible to apply formal numerical averaging of the fast subsytem to reduce the model to just the slow variables. This approach has been applied to a nonbursting metabolic model (Dvo~-~ik and Sigka, 1989). How do our expressions of the effect of the action potentials on the SLOW variables relate to formal averaging? Icaflux(q) represents the average of/ca over the time-course of the action potential. Assuming (from our FAST-SLOW simulations) that/ca is the only significant mechanism by which the action potentials perturb the slow variables, then our reduction is similar to averaging Eq. (2) over the action potential. However, the effect of the action potential on s were calculated in terms of Vavg (which is approximately equal to V + Vd~p(q)). Rather than explicitly averaging ds/dt over the time-course of an action potential, we substituted Vavg into soc (V) and vs (V). This method provided reasonable results because during repetitive firing the membrane potential spends most of its time between spikes in a narrow range of membrane potentials near Vavg. The use of Vavg may be applicable in reducing other periodic systems where the spike of the action potential occupies a small fraction of the period of the oscillation Average NuUclines Besides model reduction, averaging methods have also been employed in the analysis of bursting models. One such approach is called the method of "averaged nullclines" (Smolen et al., 1993; Baer et al., 1995). The general approach is to calculate the nulmines of the slow variables, including the effects of the fast processes averaged over one period of the fast oscillation. This method allows the calculation of nullclines across the entire phase space of the slow variables, not just in the region where the action potentials are not firing. Thus the average nullclines provide a graphical means of assessing the overall effects of how the periodic fast processes interact with the slow system variables. The geometry of the average nullclines and their intersection with the nullclines in the silent region of phase space offers clues to the dynamic nature of the interactions between the fast and slow processes (Baer et al., 1995; Bertram et al., 1995). Thus the method is complementary to our combined FAST-SLOW simulations, which are a method of parametrically searching for specific biophysical mechanisms (functional expressions) by which the FAST variables perturb the SLOW system Generalization of the Method The scheme presented in the present paper can be generalized to reduce certain classes of neuronal models to their essential components. Our approach does not have the rigor of a formal mathematical algorithm, and it requires a certain degree of knowledge of the solution structure of the model. This reduction is based on the assumption that there exist slow variables within the model that strongly regulate the model's activity. Working under this assumption, our scheme is as follows: 1. Identify the fast and slow variables within the model. 2. Examine the differential equations governing these variables. The fast system is the remaining equations when the slow variables are treated as parameters, rather than variables. The slow system is the differential equations of the slow variables with the functional dependencies on the fast variables eliminated. 3. Membrane potential, although a fast variable, will usually be a variable of the slow system as well due to the voltage-dependency of most formulations for ionic currents and gating variables. This can be treated by developing a second "dummy" variable for membrane potential (like V~s). In the differential equation for V~s, include only those ionic currents that are dependent on the SLOW variables and membrane potential. 4. Employ numerical bifurcation analysis to examine the solution structure of the fast system, treating the slow variables as parameters. Characterize the activity of the fast system as a function of the slow variables and use these results to develop a measure of average fast system activity (such as spike frequency) that is dependent on slow system variables.

19 Dissection and Reduction of a Modeled Bursting Neuron 217 Develop a state equation (such as q) describing this activity. 5. Employ combined fast-slow system simulations to identify key fast processes that affect the solution trajectory of the slow variables, Do this by selectively adding current formulations or dependencies on fast variables (such as membrane potential) to the differential equations of the slow system. 6. Once the key interactions are identified, characterize their net effect by developing formulations that are dependent on the variables developed in Step 4. In our model, the key interactions identified in Step 5 were the voltage perturbations of s and the Ca 2+influx via/ca. These two effects were quantified by the expressions Vdep(q) and Icaflux(C, q), which are dependent on q, the measure of fast system activity developed in Step Summary Our combined FAST-SLOW simulations represent a novel technique for identifying the mechanisms of the FAST subsystem that are important in perturbing the SLOW variables that underly the burst cycle. Likewise, the technique identifies the mechanisms that are not important. The primary effects of the action potentials on the SLOW variables were identified to be voltage-dependent perturbations of s and Ca2+-influx due to spike-activated inward Ca 2+ currents. The action potentials did not have a significant effect upon the SLOW system via the voltage-dependencies of the subthreshold currents of the SLOW system (other than s). Although our REDUCED model is based only on some simple approximations obtained via computer simulation and numerical bifurcation analysis, the REDUCED model performed surprisingly well in mimicking the activity of the FULL model. It is difficult to apply the software program AUTO to the analysis of periodic solutions of the FULL model due to the number of state variables and the two extremely different time-scales involved in bursting. However, given a strong degree of confidence in the predictions of the reduced model, we can reliably utilize AUTO to analyze the periodic solution structure of the REDUCED model and extrapolate these predictions to the activity of the FULL model. (The data for the REDUCED model in the bar charts of Fig. 7 were generated with this method.) Our comparison of the low-order models with the FULL model revealed that the REDUCED model was more accurate than the SLOW model at predicting the given mode(s) of activity of the FULL model at a given value of parameters. Furthermore, the periodic solutions of the REDUCED model (both the duration of the period and phase-space solution) were very close to those of the FULL model, while the periodic solutions of the SLOW system were significantly different. Given the similarities of the periodic solutions of the REDUCED and FULL model, we found that both models possessed similar phase-response curves; thus they respond in a similar fashion to transient input. The limitations of the SLOW model in (1) predicting activity mode(s) of the FULL model and (2) mimicking the phase-space trajectory and duration of period of the FULL model suggest that models of bursting cells that ignore the role of action-potential currents may not be valid approximations of the burst cycle. Although these conclusions are within the context of our own model, the insights are general enough that they may apply to other bursting systems which possess fast variables that significantly perturb slow variables (such as Ca 2+ influx via fast Ca 2+ channels in our model). However, our REDUCED model demonstrates that with the help of numerical analysis and model simulations, it is possible to minimally increase the order of the SLOW model while greatly enhancing its predictive power. Appendix A: Equations and Parameters of the FULL Model This appendix contains the equations and values of parameters for the FULL model. In these equations, time is in msec, current is in na, potential is in mv, and concentrations are in ram. Inward Currents INa: Fast Sodium Current INa = gnam3h( V -- ENa) m~(v) = rh = (m~(v) - m)/rm(v) exp(( V)/10.0) rm(v) = 1/(Am(V) "-k BIn(V)) Am(V) = 0.40(V + 6.0) 1 - exp((-v - 6.0)/4.09) Bm(V) = exp(( V)/4.01)

20 218 Butera, Clark and Byrne h = (h~(v) - h)/rh(v) 1 hoo(v) = 1 + exp((v )/3.0) Th(V) = 1/(Ah(V) -}- Bh(V)) Ah(V) = exp(( V)/25.0) Bh(V ) = 1 + exp(( V)/23.9) Ica: Fast Calcium Current d d~(v) rd(v) = 1/(Ad(V) -I- Bd(V)) Ad(V) Bd(V) f~(v) r~,(v) Af(V) Bj(V) 1 Ica = gca d2f(v - Eca) 1 + exp (~ =.~K] \ Dca ] = (d~(v) - d)/vd(v) exp((10.0- V)/3.8) (V ) 1 - exp((-v )/5.03) = 0.01 exp((25.0- V)/10.0) = (foo(v) - f)/ry(v) exp((v )/4.0) = 1/(Ay(V) + By(V)) = exp(( V)/7.57) exp(( V)/5.4) Isl" Slow Inward Calcium Current IsI = gsi(ccamp, CDA) s~(v) = X ( KsI,Ca )s(v-eca) ~ KSI,C a ~- C = (s~(v) - s)/rs(v) exp(( V)/ll.5) rs(v) = 1~(As(V) + Bs(V)) O.O014(V ) As(V) = 1 - exp((-v )/12.63) Bs (V) = exp(( V)/16.8) INS: Non-Specific Cation Current INS = gns c ) b(v - ENS) C d- KNS,Ca b = (b~(v) - b)/zb(v) 1 b~(v) = 1 + exp(( V)/3.0) vb(v) = 500 V Eca INs.Ca = 0. I97INs(V, b, c ) - - IL : Leakage Current Outward Currents Ix: Delayed Rectifier n~(v) = IK = g, Kn41(V -- EK) exp(( V)/3.0) ) V - ENS IL = gl(v - EL) h = (n~(v) - n)/~.(v) exp(( V)/14.46) rn(v) = 1/(A.(V) + B.(V)) (V ) A.(V) = 1 - exp((-v )/3.0) B.(V) = 0.04 exp(( V)/10.0) G(v) = i = (G(v) r/(v) = I)/-r1(v) 1 1 -I- exp(( V)/12.7) IR: Anomalous Rectifier IR exp(( V)/3.0) ) (V - EK ) gr(ccamp) 1 + exp ((V-e~-~5.3)ZF) Pumps and Exchangers c ) C + Kp,ca

21 Dissection and Reduction of a Modeled Bursting Neuron 219 Parameters INaK ~ [NaK F ( [Na], L\ [Na]i + Kp,Na,} x ( [K] ~ (1.5 + el;)~, )] \[K]o + Kp, K.] INaCa = KNaca(DFin - DFout)/S S DNaca([Ca]i[Na],~ + [Ca]o[Na] r) ((r-2)yvf) DFin --= [Na]r[Ca]o exp RT I DFout _ [Na]r [ Ca]i exp ( ( r - 2 ) (Y -_R771)VF) Internal Calcium Concentration INaCa -- /SI --/Ca -/CaP - INs,Ca 2Voli F --rib [B]i Oc Oc = kv[ca]i(1 - Oc) - kroc Membrane Potential /sub ~ ISI -t'- IR -~- IL -~- INaCa -~- /CaP q- INa /spike ~ INa + /Ca -1- IK + INS ~, = --(/spike q- /sub -- Istim) Cm Internal camp Concentration (steady-state) CcAMP ~ Kp dekadc F5-HT 1)pd e -- KadcF5_HT FS-HT = 1 + Kmod \ KSHT + CS-HT Modulatory Effects on Conductances KR mod.~ gr (CcAMP) = gr 1 "q" /KR.cAM~-- CcAMP"~ 1 + exp I, DReAM P ),] gsi (CcAMP, CDA) = gsi ( \ KDA CD '+ KDA "~,] K Imo_J x 1+ /KsI.~AMP-- CcAM~'~] l+exp\ Ds~aMp ]/ CM = 17.5 nf gna = 38 /zs gca /zS o~k = 70 #S gns = 0.2/zS gsi = 0.65/xS gr = 0.18 #S o~l = 0.075/xS ENa = 54 mv Eca = 65 mv EK = --77 mv ENS = --22 mv EL = 10.3 mv [Na]o = 500 mm [Na]i = 50 mm [K]o = 10 mm [Ca]o = 10 mm [B]i = x 10-3 mm /CaP = 7.0 na /=YAK = 5.9 na KNaCa = 0.01 DNaCa = 0.01 y = 0.5 r=4 R = 8,314 J/kg mol K F = 96,500 C/mol T = 295 K Z=2 Voli = 4.0 nl ku = 100 mm-lmsec -1 kr = msec -1 rib=4 KsI, ca = 25 x 10-6 mm Kys.ca = 150 x 10.6 mm Kp.ca = 350 x 10-6 mm Kp,Na = 5.46 mm Kp,K -~ mm Kca = mm

22 220 Butera, Clark and Byrne Dca : 0.15 x 10-3 mm KDA = 0.2 mm KSI,cAMP = mm DSI,cAMP = 0.35 X 10-3 mm KSI,mod = 5.5 KR,cAMP = 1 X 10-3 mm DR,cAMP = 0.4 X 10-3 mm KR,mod = 1.5 Kadc : mm/msec rode = mm/msec K5HT = mm Kpde : mm Kmod = 1.5 Appendix B: Bifurcation Diagrams Bifurcation diagrams are a graphical method of visualizing the steady-state solutions of a system of autonomous ordinary differential equations as a parameter is varied. We are concerned with systems of the form } = f(,/3), where ~ is the vector of state-variables and/3 is the vector of parameters. This appendix serves only as a brief introduction to the terms and concepts used in this paper; for greater explanation please refer to Hale and Kodak (1991). The parameter to be varied is assigned to the x-axis. In Fig. 1 we chose the parameter s. A measure of the state variables at a given solution is assigned to the y-axis. Because changes in any state variable effect a change in membrane potential, we chose to assign membrane potential to the y-axis. At any given value of s, one or more steady-state solutions to the FAST system exist. Each of these solutions is stable or unstable, and each of these solutions is at equilibrium or periodic. B.1. Equilibrium Solutions An equilibrium solution (also called a fixed-point) is a point in the state space such that the derivative of each state variable is zero (i.e., f(y,/3) = 0). An equilibrium solution may be stable or unstable. Given an initial condition that is sufficiently close to a stable equilibrium solution, the solution trajectory will converge to the equilibrium solution. A physiological example of a stable equilibrium solution is the resting potential of the Hodgkin-Huxley model (Hodgkin and Huxley, 1952). Given an initial condition that is sufficiently close to an unstable equilibrium solution, the solution trajectory will move away from the equilibrium solution. In the bifurcation diagrams of the FAST system illustrated in Fig. 1, stable and unstable equilibrium solutions are represented by solid and dotted lines, respectively. For example, when s = 0 in Fig. 1A, there is a stable fixed point solution at approximately-68 mv and another at approximately -16 mv. With the parameters s = 0 and c = 125 rim, any set of initial conditions in the FAST system will result in the solution converging to one of these two points. B.2. Periodic Solutions Steady-state solutions are not just limited to equilibrium solutions. For certain parameter ranges, stable and unstable periodic solutions may exist as well. If the solution trajectory of the system ~ = f(y,/3) is represented by ~b(t) where t is time, a periodic solution is one that satisfies ~(t) = ~(t + T), where T is the period of the oscillation. Given an initial condition sufficiently close to a stable periodic solution, the solution trajectory will converge to the orbit of the periodic system. A physiological example of a periodic equilibrium solution is the repetitive firing of action potentials that occurs when a constant depolarizing current is applied to the Hodgkin-Huxley model. Periodic solutions may also be unstable. While this may not be an intuitive concept, an unstable periodic solution is one that exists but has no region of attraction. When an unstable periodic solution is plotted in the phase space, it often defines boundaries of the regions of attraction of the stable solutions. In the bifurcation diagrams of the FAST system illustrated in Fig. 1, stable and unstable periodic solutions are represented by dotted and dash-dotted lines, respectively. It is not possible to represent the orbit of a periodic system on the bifurcation diagram, since only one state variable is plotted on the y-axis. Rather, periodic solutions are represented by two lines, one corresponding to the maximum of the membrane potential oscillation (Vmax) and the other to the minimum of the membrane potential oscillation (V~n). For example, in Fig. 1A when s = 0.3, there exist two stable periodic solutions, one with a V~n of about -45 mv and the other with a V~n of -28 mv (the insets show several periods of periodic activity in membrane potential).

23 Dissection and Reduction of a Modeled Bursting Neuron 221 B.3. Bifurcations A bifurcation is a point (or line) in parameter space where new steady-state solutions emerge and/or old steady-state solutions disappear. In our study we are concerned with two kinds of bifurcations: Hopf bifurcations and saddle-node bifurcations. Our description of these bifurcations is elementary. (See Hale and Kodak, 1991, for an introduction to the underlying mathematical foundations and Rinzel and Ermentrout, 1989, for a description of these bifurcations in the context of models of neural excitability.) In a general sense, a Hopf bifurcation is a point in parameter space at which a periodic solution is "born". Out of the Hopf bifurcation emerges a periodic solution with an initially very small amplitude. The amplitude of the oscillation typically grows as a parameter is changed past the point of the Hopf bifurcation. For example, in Fig. 1A the entire branch of periodic solutions originates at the Hopf bifurcation denoted by the open circle. In the Hodgkin-Huxley model, the transition between silence and spiking behavior as the stimulus current is varied is also a Hopf bifurcation. A saddle-node bifurcation is a point in parameter space where two equilibrium solutions, one stable and one unstable, coalesce and disappear, indicated by the box in Fig. 1A. When s is to the left of the saddle node bifurcation, there exist three equilibrium solutions (two stable and one unstable). As s is increased toward the saddle-node bifurcation, the two hyperpolarized equilibrium solutions approach one another and coalesce at the saddle-node bifurcation. In our FAST model (as well as other models of neural excitability) the saddle-node bifurcation also marks the boundary (in parameter space) of a periodic solution. This boundary is "homoclinic"--that is, the periodic solutions of the FAST system have a period approaching infinity. For example, as s approaches the saddle-node bifurcation from the positive direction in Fig. 1A the period of the solutions along AP1 increase toward infinity. The infinite solution corresponds to the solution trajectory of AP 1 getting "stuck" on the saddle-node bifurcation. Thus, the saddle-node bifurcation describes the interface in (c, s) parameter space between silent and spiking behavior. Thus it is effectively the equivalent of an action potential threshold in terms of the parameters c and s. Rinzel and Ermentrout (1989) performed an extensive analysis of the model of Morris and Lecar (1981). At one set of parameters, the transition between silence and spiking as the stimulus current is varied occurs via a Hopf bifurcation. At a different set of parameters, the solution structure of the model is altered, and the transition between silence and spiking occurs via a saddle-node bifurcation. In the latter case, the resulting bifurcation diagrams are qualitatively similar to those of Fig. 1A. B.4. Manifolds In discussing bifurcation diagrams, we will often use the phrase solution manifold. A solution manifold is an equilibrium solution that varies continuously as a parameter is varied. On a two-dimensional bifurcation diagram a solution manifold is typically represented by a line. For example, in Fig. 1A there is a solid black line that passes through the point (s = 0, V = -68 mv). This line is a stable equilibrium manifold and is labeled EQ1. On a three-dimensional bifurcation diagram, where two parameters are displayed, the same EQ 1 manifold is a surface represented by the solid mesh (each panel of Fig. 1 is a planar cross section through Fig. 2 at different values of c). Appendix C: Reduction of the Ca z+ Subsystem The rate of change of Ca 2+ in the FULL model is described by Eqs. (2) and (4), which represent the timerate of change of intracellular Ca 2+ (c) and occupancy of an intracellular Ca 2+ buffer (Oc). Assuming that Ca~+-binding to the buffer occurs on a much faster time-scale (milliseconds) than the rate of change of Ca 2+, buffer occupancy is approximated by a quasisteady-state assumption 0c = c (CI) c + Koc ' where Koc = kr/kv. However, Eq. (3) is dependent on doc/dt, not Oc. Differentiating Eq. (C1) in time, doc doc dc Koc dc.... (C2) dt dc dt (c-k- Koc) 2 de Eq. (C2) is substituted into Eq. (3). The dc/dt terms are regrouped and solved for, obtaining Eq. (9).

24 222 Butera, Clark and Byrne Appendix D: Equations and Parameters of the REDUCED Model This section contains the equations of the REDUCED model. All functions for gating variables and membrane currents are as previously defined unless otherwise noted. Normalized Conductance Changes Soc(C) = 1 + nb[b]i Koc = kn/ku ~= Membrane Potential Koc (c + Koc) 2 INaCa -- /CaP -- ISI -~ /Caflux 2VoliFS Oc (c) NsI(CcAMP, CDA) : Distance Function gsi (CcAMP, CDA) gsi(10-3, 0) 1 (gr(ceamp) ) NR (CcAMP) = ~ \ ~g ~ Parameters (/sub -- Istirn) Cm ml0 = bl0 = D(c, s) = s NsI( CcAMP, CDA) -- I O00m L C -- bl + 1 ml = NR( CcAMP)(mLO -- Istim/2.0) bl = (bl0 -- [stirn/25.0) / N R ( CcAMP) Spike Frequency Approximation qoo(d) = 12 tanh(100d) 1 + exp(-25(d )) rq(d) = 500(1 + tanh(500d)) gl = (q~(d) - q)/rq(d) Spike Frequency Activation Function 1 u(q) = =(1 + tanh(1000(q ))) z Modified s Activation Vdep(q) : 5.5u(q) Vr = min(vss + Vdep(q), --25) Modified Ca2 +-balance Ica flux = = (s~(vss -'F Vdep(q)) -- s)/'g(vr) 1.7u(q)q 1 + exp((c -- Kca)/Dca) Acknowledgments We wish to express our gratitude to Doug Baxter, Carmen Canavier, and Tony Difranceschi for helpful discussions. We thank the reviewers for suggestions which improved the content of the manuscript. This work was supported by Office of Naval Research grant N , NIMH Award K05 MH (JHB) and a Whitaker Foundation Graduate Fellowship in Biomedical Engineering (RJB). References Alevizos A, Skelton M, Weiss KR, Koester J (1991) A comparison of bursting neurons in Aplysia. Biological Bulletin 180: Baer SM, Rinzel J, Carrillo H (1995) Analysis of an autonomous phase model for neuronal parabolic bursting. Journal of Mathematical Biology 33: Bertram R (1993) A computational study of the effects of serotonin on a molluscan burster neuron. Biological Cybernetics 69: Bertram R, Butte MJ, Kiemel T, Sherman A (1995) Topological and phenomenological classification of bursting oscillations. Bulletin of Mathematical Biology 57: _ Butera RJ, Clark JW, Canavier CC, Baxter DA, Byrne JH (1995) Analysis of the effects of modulatory agents on a modeled bursting neuron: Dynamic interactions between voltage and calcium dependent systems. Journal of Computational Neuroscienc e 2: Canavier CC, Baxter DA, Clark JW, Byrne JH (1993) Nonlinear dynamics in a model neuron provide a novel mechanism for transient synaptic inputs to produce long-term alterations of post-synaptic activity. Journal of Neurophysiology 69:

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