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Hotchkiss Brain Institute, University of Calgary, Calgary, Alberta, Canada
Submitted 19 March 2007; accepted in final form 30 April 2007
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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, 2007; Izhikevich et al. 2003; Rinzel and Ermentrout 1998). Consequently, gaining a biophysical and dynamical systems perspective of intrinsic neuronal firing behavior is important. From both an electrophysiological and dynamical systems perspective cerebellar Purkinje cells possess complex intrinsic firing behaviors, which include bistability, hysteresis, and plateau potentials (Hounsgaard and Midtgaard 1988; Llinas and Sugimori 1980b; Loewenstein et al. 2005). Given that Purkinje cells integrate numerous synaptic inputs and are the sole output of the cerebellar cortex, an understanding of intrinsic Purkinje cell dynamics is critical to comprehending cerebellar function.
Previous work in vitro has identified numerous intrinsic electrophysiological behaviors of Purkinje cells that are conserved across different species (Hounsgaard and Midtgaard 1988; Llinas and Sugimori 1980a,b; Loewenstein et al. 2005; Williams et al. 2002). Properties that have received particular attention are bistability and the generation of plateau potentials. Recent work has shown that intrinsic membrane bistability allows climbing fiber (CF) input to transition the cell between a rest and firing state (Loewenstein et al. 2005; McKay et al. 2007; Williams et al. 2002). Central to this are plateau potentials that allow an excitatory stimulus to generate a depolarization that outlasts the original stimulus (Llinas and Sugimori 1980a). Other characteristic features of Purkinje cells are a linear frequency–current (F–I) relationship with a minimum firing frequency between 40 and 70 Hz and the ability to sustain tonic firing frequencies in excess of 150 Hz (Khaliq et al. 2003; Llinas and Sugimori 1980b; Martina et al. 2003; McKay and Turner 2004, 2005; Williams et al. 2002). Electrophysiological studies have identified many of the ion conductances that contribute to these different aspects of Purkinje cell output (Hounsgaard and Midtgaard 1988; Llinas and Sugimori 1980a,b; Raman and Bean 1999a,b).
Previous modeling studies in Purkinje cells have emphasized how the distribution of ion channels or morphological features of the dendritic tree shape synaptic integration (De Schutter and Bower 1994a,b; Santamaria and Bower 2005; Santamaria et al. 2002) or spike backpropagation (Vetter et al. 2001), or were used to explain a specific property of Purkinje cell firing (Genet and Delord 2002; Khaliq et al. 2003; Loewenstein et al. 2005). We sought a more general firing model based on an understanding from dynamical systems theory that could account for as many properties of Purkinje cells as possible within a realistic biophysical framework. Using a combined electrophysiological, modeling, and dynamical systems analysis we show that Purkinje cell firing dynamics are consistent with a system undergoing a saddle-node bifurcation in the transition from rest to firing and a saddle homoclinic bifurcation from firing to rest [for a review of these concepts and bifurcation theory as applied to intrinsic neuronal biophysics see Izhikevich (2007) or Rinzel and Ermentrout (1998)]. Within this framework we illustrate that numerous aspects of Purkinje cell firing are fundamentally related to the underlying bifurcation structure and further reveal previously unrecognized firing properties of these neurons.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Sprague–Dawley rats [postnatal day 17 to 25] were obtained from Charles River Laboratories (Quebec, Canada) and procedures were conducted according to guidelines approved by the local Animal Care Committee and the Canadian Council for Animal Care.
Electrophysiology
All chemicals were obtained from Sigma (St. Louis, MO) unless otherwise noted. Whole cell current and voltage-clamp recordings were obtained using a 700A amplifier (Axon Instruments) and data collected with pCLAMP 8.1 software (Axon Instruments). Tissue slices of cerebellum were prepared as detailed in McKay and Turner (2005). Briefly, artificial cerebrospinal fluid (aCSF) was composed of (in mM): NaCl (125), KCl (3.25), CaCl2 (1.5), MgCl2 (1.5), NaHCO3 (25), and D-glucose (25) preoxygenated by carbogen (95% O2-5% CO2) gas. Rats were anesthetized with sodium pentobarbital (MTC Pharmaceuticals, Cambridge, Ontario, Canada). Parasagittal tissue slices (300-µm thickness) were cut and subsequently transferred to the recording chamber of a Zeiss Axioskop FS-2 microscope and maintained at 34°C as a submerged preparation. Neurons were visualized using differential interference contrast optics and infrared light transmission (DIC-IR). All current-clamp recordings were carried out in synaptic blockers that were bath applied after obtaining the initial seal: picrotoxin (50 µM; Tocris Cookson, Ellisville, MO) for 
-aminobutyric acid type A (GABAA) receptors, DL-2-amino-5-phosphonopentanoic acid (D-AP5, 25 µM; Tocris Cookson) for N-methyl-D-aspartate receptors, 6,7-dinitroquinoxolinedione (DNQX, 10 µM; Tocris Cookson) for 
-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptors, and (2S)-3-[[(1S)-1-(3,4-dichlorophenyl)ethyl]amino-2-hydroxypropyl] (phenylmethyl)phosphinic acid (CGP 55845, 1 µM; Tocris Cookson) for GABAB receptors. Pipettes had a resistance of 5–8 M
with access resistance of 6–15 M (80–90% compensation in voltage clamp). The internal solution for current-clamp recordings consisted of (in mM): K-gluconate (130), EGTA (0.1), HEPES (10), NaCl (7), MgCl2 (0.3), di-tris-creatine phosphate (5), Tris-ATP (2), and Na-GTP (0.5) (pH 7.3 with KOH). For outside-out voltage-clamp recordings of K+ current the internal solution consisted of (in mM): KCl (140), EGTA (5), HEPES (10), and MgCl2 (2.5), with CsCl (2) in the external bath to block the hyperpolarizing activated cation current IH.
Antidromic stimulations were conducted with a concentric bipolar electrode placed in the white matter region of the folia in the presence of synaptic blockers. Stimuli were delivered by a stimulus isolation unit (DS2 isolation stimulator, Digitimer, Hertfordshire, UK).
Analysis of current-clamp data was accomplished using custom software written in MatLab 7.1 (The MathWorks, Natick, MA). Spike threshold was determined through analysis of the voltage derivative. A calculated junction potential of 11 mV was subtracted from all current-clamp recordings. In voltage-clamp recordings the junction potential was not subtracted because the value was small (
2 mV) for internal KCl.
ZAP protocol
A ZAP protocol consisted of a time-varying current stimulus of 6 s duration generated by the following equations
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Simulations
Simulations were constructed in MatLab 7.1 using a fourth-order Runge–Kutta algorithm with a time step (dt) of 0.001 ms. Models were formulated using Hodgkin–Huxley formalism. Bifurcation and phase-plane analyses were carried out in XPPAUT v 5.85 (Ermentrout 2002).
Five-equation model
Somatic voltage
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Dendritic voltage
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Na+ activation
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The h variable represents a merged refractory variable (Na+ inactivation and K+ activation).
IH activation
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Slow K+ activation
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Testing model robustness
To test for the robustness of the five-equation model individual parameters (maximum conductances, coupling coefficient between compartments, and time constants) were independently varied by as much as 20% from the default values. Under all cases the model retained the key characteristics described in this study provided that the bias current (IE) was adjusted to compensate for small changes in threshold. Properties tested include a substantial bistable range (>0.15 µA/cm2), the ability to generate a long delay to first spike relative to the subsequent ISI, the generation of a DAP, and the ability for a unipolar input to switch between a rest and firing state.
Reduced two-equation model
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Na+activation
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K+ activation
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C = 1.5 µF/cm2, gNa = 20 mS/cm2, gK = 4.2 mS/cm2, and gleak = 0.05 mS/cm2.
For all simulations reversal values were set to: ENa+ = 45 mV, EK+ = –95 mV, EIH = –20 mV, and Eleak = –77 mV.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Understanding the dynamics governing the transition from rest to firing is critical because it will largely set the dynamical framework from which many properties of Purkinje cell firing may be accounted for. We started by considering the type of bifurcation underlying Purkinje cell spike threshold. In brief, a bifurcation can often consist of a qualitative change in the system dynamics (e.g., transition from rest to firing) that involves the creation or elimination of fixed points or limit cycles. A neuronal resting state is an example of a stable fixed point and a spiking state a limit cycle (Izhikevich 2000, 2007; Rinzel and Ermentrout 1998; Strogatz 1994). The type of bifurcation that the Purkinje cell undergoes at threshold will largely determine the potential nature of a plateau potential and bistability (Izhikevich 2000, 2007; Rinzel and Ermentrout 1998).
The two principal types of bifurcation governing thresholds in neurons are a Hopf bifurcation and a saddle-node bifurcation (Izhikevich 2000, 2007; Rinzel and Ermentrout 1998). An important criteria for identifying the type of bifurcation underlying spike threshold are steady-state current–voltage (I–V) relationship and the voltage trajectory in the subthreshold range (Izhikevich 2000, 2007; Rinzel and Ermentrout 1998). These two factors can be used to differentiate whether a cell undergoes a saddle-node bifurcation of fixed points or a Hopf bifurcation when driving current is the sole bifurcation parameter. In essence, the presence of a highly nonlinear I–V relationship with an abrupt increase in membrane voltage deflections (resistance) near threshold is consistent with a saddle-node bifurcation. The ability to generate long delays to first spike is also consistent with a saddle-node bifurcation. Finally, an approach to spike threshold with no fast subthreshold or damped oscillations in membrane voltage, combined with long first delays to the first spike, are also consistent with a saddle-node bifurcation (Izhikevich 2000, 2007). Thus we evaluated the I–V relationship and subthreshold voltage trajectory to distinguish which bifurcation was consistent with the activity of Purkinje cells.
To evaluate the steady-state I–V relationship the membrane potential was held near –71 mV and stepped to voltages ranging from –74 to –64 mV in 10 pA increments for 350 ms. Membrane voltage was measured at 275 ms, which provided sufficient time for the IH-mediated sag in the voltage response to equilibrate. As previously shown (Llinas and Sugimori 1980a,b), Purkinje cells showed a highly nonlinear I–V relationship for depolarizations greater than –70 mV (Fig. 1, A and B). Application of TTX resulted in a linear I–V relationship, indicating a steady-state activation of Na+ current that increased the slope of the I–V relationship and thus the input resistance in this voltage region (Fig. 1, A and B). This occurs because a steady-state negative conductance acts to increase the amplitude of positive and negative voltage deflections by increasing the inward current with depolarization (amplifying a depolarization) and decreasing inward current with hyperpolarization (amplifying a hyperpolarization) (Stafstrom et al. 1982). From a dynamical systems perspective the rapid increase in membrane input resistance can represent a nonmonotonic relationship in the steady-state I–V relationship. This nonmonotonicity is critical for producing the fold associated with the coalescing of a node and saddle point, and thus a saddle-node bifurcation (Izhikevich 2000, 2007; Rinzel and Ermentrout 1998). Purkinje cells thus satisfy the criterion of exhibiting a nonlinear steady-state I–V relationship with a dramatic increase in input resistance near threshold necessary for a saddle-node bifurcation of fixed points.
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; McKay and Turner 2004). Previous work in mouse Purkinje cells has shown evidence for a subthreshold inactivating K+ current (Sacco and Tempia 2002). These experiments, however, were carried out in postnatal day 3 to 9 animals. Their results are thus consistent with immunohistochemical data that indicate only a transient early developmental expression of channels contributing to the A-type current (Hsu et al. 2003). Thus our data support previous conclusions that A-type currents are absent in mature rat Purkinje cells.
As a final test for the type of bifurcation governing threshold dynamics in Purkinje cells we applied a ZAP protocol. A ZAP protocol provides a simple method to assess the subthreshold impedance of neurons by stimulating the neuron with sinusoidal current with a slowly increasing frequency (Hutcheon and Yarom 2000). Neurons that undergo a Hopf bifurcation have a dampening conductance active below threshold that acts similar to an inductor (Izhikevich 2007). Consequently, when combined with the membrane resistance and capacitance, neurons undergoing a Hopf bifurcation resonate at intermediate frequencies of subthreshold stimulation, which produces a band-pass impedance profile. Conversely, neurons undergoing a saddle-node bifurcation have no resonance behavior and thus have a low-pass impedance profile associated with the membrane resistance and capacitance (Izhikevich 2007). As shown in Fig. 1D, application of a ZAP protocol produced no resonance behavior in the Purkinje cell membrane voltage response and also produced a monotonic membrane impedance profile (n = 8).
In summary, these simple experiments provide evidence that Purkinje cell spike threshold dynamics are consistent with a saddle-node bifurcation of fixed points. They also confirm that a steady-state Na+ current is the principal determinant of the nonlinearities in the subthreshold range of the I–V curve. Thus as the membrane voltage approaches threshold, more net inward current with a negative conductance is activated. Consequently, from a dynamical systems perspective any model of Purkinje cell firing and bistability must incorporate a saddle-node bifurcation of fixed points to account for the physiological properties apparent during the transition from rest to firing.
Purkinje cells respond nonmonotonically to climbing fiber–like stimuli
Previous work has established that Purkinje cells have the ability to switch from rest to firing as well as from firing to rest in response to a strong unipolar current input (Fig. 2 A) (Loewenstein et al. 2005). Consequently, bistability in Purkinje cells occurs between a fixed point and a limit cycle. This differs from some neurons that have been shown to be bistable in which the two stable states are represented by a hyperpolarized or depolarized resting state (Crunelli et al. 2005; Williams et al. 1997). Physiologically such a unipolar input can be provided by climbing fiber (CF) activation, which induces a strong synaptic current lasting about 15 ms (Schmolesky et al. 2002). Therefore in the next set of experiments we focused on the intrinsic dynamics of Purkinje cell firing in the context of postsynaptic CF-like stimuli. In particular, we were interested in whether the membrane voltage response of Purkinje cells to CF-like stimuli could further constrain our identification of dynamics underlying spike firing (Fig. 1).
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A simple explanation for the nonmonotonicity in the relationship between CF-like stimulus magnitude and state transition probability is that stronger stimulation recruits a greater amount of K+ current activation or Na+ current inactivation. This in turn brings about a failure to transition to the firing state and a rapid return to the rest state. In fact, this can be observed in the rate of repolarization after different magnitudes of CF-like stimuli. To test this we held cells at a slightly more negative membrane potential (–74 mV) so that the cell did not transition to the firing state and the rate of repolarization after stimulation could be observed. We used three different strengths of CF-like stimuli (1, 2, and 5 nA). As expected, the membrane potential returned more quickly to baseline voltage after stronger stimulation (Fig. 2D), which suggested the recruitment of a K+ current or inactivation of Na+ current.
An interesting aspect of Purkinje cell state transitions is that the same magnitude depolarizing pulse can induce transitions both from rest to firing and from firing back to rest (Fig. 2A) (Loewenstein et al. 2005). The ability for such a unipolar and fixed input (i.e., complex spike depolarization) to toggle the cell between firing and rest states may also be accounted for in terms of the relative activation of K+ and Na+ currents. Thus the firing state recruits more net K+ current or Na+ inactivation than at rest, allowing a CF-like stimulus to switch the cell to the rest state. If this scenario is valid it would be expected that the rate of repolarization after a square-step depolarization that induced a moderate rate of firing would be greater than a similar step that is slightly subthreshold because, during the firing state, more K+ current activation or Na+ inactivation would be present. Once again we held cells at a sufficiently negative membrane voltage to avoid state transitions and observed the rate of repolarization after the step pulse. We used a 200 ms step depolarization of either 100 pA (subthreshold) or 175 pA (suprathreshold and firing at 80 Hz) from a holding potential of –75 mV. As shown in Fig. 2, E and F, the rate of repolarization after a step depolarization was faster if the cell membrane potential returned to baseline from a firing state rather than a depolarized but subthreshold state.
Because of the absence of spiking in the presence of TTX we could measure the I–V relationship over a wider range of membrane voltages. In particular, we were interested in the conductance behavior at increasingly depolarized membrane voltages. We reasoned that if increasing depolarization induces a faster rate of repolarization to the rest state, then the I–V relationship should indicate a strong saturation (decrease in input resistance) at more depolarized membrane potentials. Furthermore, if the rate of repolarization in the presence of TTX is greater when preceded by a stronger depolarization it indicates that depolarization activates a K+ conductance that facilitates repolarization on cessation of the current step. This prediction was borne out in the steady-state I–V relationship of Purkinje cells, which began to saturate at membrane voltages greater than –50 mV in the presence of TTX (Fig. 2, G and H). This was also accompanied by an increase in the rate of repolarization after cessation of stronger depolarizing steps (Fig. 2G). These results are consistent with an increased activation of K+ current during the firing state that deactivates slowly enough to influence the rate of repolarization at the end of the current step. Thus the same depolarizing stimulus that switched a cell to the firing state can be used to switch the cell back to the rest state through the additional activation of K+ current.
A model of Purkinje cells indicates the dynamical basis of bistability
To gain greater insight into Purkinje cell firing dynamics we constructed a firing model. We used the data on threshold dynamics, which indicated the presence of a saddle-node bifurcation, and the membrane voltage response to CF-like stimuli as a general framework for the model's dynamics and biophysics. We also considered experimental results from previous electrophysiological studies of Purkinje cells. We started by incorporating the essential elements required to reproduce basic properties of Purkinje cell behavior, including a somatic and dendritic compartment and conductances required to generate spike firing.
Previous work has shown that a portion of Purkinje cells can produce a three-stage trimodal firing pattern that arises when cells fire at sufficiently high frequencies (McKay and Turner 2004; Womack and Khodakhah 2002). In our experiments we found that approximately half of Purkinje cells at the age examined could develop a trimodal firing pattern. Recent in vitro work, however, reveals that reintroducing low-frequency (1 Hz) CF activation or strong depolarization inhibits trimodal firing and stabilizes Na+ spike output at a new lower-frequency level (McKay et al. 2007), whereas the removal of CF inputs in vivo leads to trimodal-like activity (Cerminara and Rawson 2004). This suggests that the tonic firing behavior is most likely the normal firing mode for Purkinje cells in vivo. For these reasons we did not incorporate trimodal firing dynamics in our model.
We first considered the addition of a dendritic compartment given the important role this can have in establishing spike firing behavior (Doiron et al. 2001; Fernandez et al. 2005; Mainen and Sejnowski 1996). It has been established that Purkinje cell dendrites do not generate active regenerative events (Na+ or Ca2+ spikes) during tonic repetitive Na+ spiking at the soma (McKay and Turner 2004; Stuart and Hausser 1994). Consequently, the charging of the dendritic compartment draws current away from the active zone in the soma/axon hillock, raising current threshold at the active zone. Once charged, however, the relative absence of fast voltage-gated Na+ and K+ conductances in dendrites prevents the dendritic membrane voltage from equilibrating at the same time as the somatic compartment. This causes the dendritic membrane potential to remain positive relative to the somatic compartment for a substantial period of time during an ISI. This behavior is in fact consistent with dual-patch recordings from Purkinje cell soma and dendrite, where dendritic membrane voltage remains at a more depolarized level during the somatic afterhyperpolarization (AHP) (McKay and Turner 2004; Stuart and Hausser 1994). This property has also been recognized as important in previous models of Purkinje cells (Jaeger et al. 1997). To simplify the model as well as facilitate analysis we used a reduction approach developed by Pinsky and Rinzel (1994). With this reduction the two compartments influence each other through a current term that is proportional to the difference in voltage between the two compartments and scaled by a term representing the resistance between the compartments (Pinsky and Rinzel 1994). In our model we used a coupling resistance between the two compartments that reproduced the delay and voltage drop observed in dual soma-dendritic recordings (see METHODS) (McKay and Turner 2004; Stuart and Hausser 1994).
The somatic compartment contains a transient Na+ current and a noninactivating K+ current. The dendritic compartment contains a slow K+ current subsequently described. To reduce the dimensionality of the model we set Na+ activation to equilibrate instantaneously with voltage and described Na+ inactivation and K+ activation as h and 1 – h, respectively (Rinzel 1985). Furthermore, to take into account the fast refractory dynamics observed in Purkinje cells, we made the time constant of the refractory variable (h) relatively fast (0.1–0.9 ms) to permit the model to fire at frequencies >150 Hz. This is consistent with previous work in Purkinje cells that established that fast activating and deactivating K+ currents as well as a resurgent Na+ current enable high-frequency firing by contributing to a greater net depolarization between spikes (Khaliq et al. 2003; Martina et al. 2003; McKay and Turner 2004; Raman and Bean 1999a,b, 2001; Swensen and Bean 2003). More recent work has shown that Purkinje cells express exceptionally fast K+ currents with activation and deactivation time constants ranging from 0.2 to 2.5 ms when measured at room temperature (Martina et al. 2007). Finally, we selected a leak conductance density that produced an approximate membrane time constant of 90 ms, which is within the range measured in previous studies (Rapp et al. 1994; Roth and Hausser 2001).
The TTX-sensitive Na+ current that contributes to a nonlinear increase in membrane voltage at depolarized potentials (Fig. 1B) may arise through a steady-state Na+ window current or a persistent (noninactivating) Na+ current (Stafstrom et al. 1982). To incorporate this in the model we set Na+ conductance to have a window current by providing an overlap in the steady-state activation and inactivation variables. The model also includes a greater Na+ current density relative to K+ and leak current. In doing so we ensure that more net Na+ current activates at subthreshold membrane voltages (–70 to –65 mV) than K+ or leak current, which reproduces the nonlinearity in voltage response in the subthreshold region (Fig. 3 B). A persistent Na+ current is not included because the Na+ window current is sufficient to reproduce the experimentally observed I–V relationship. In Fig. 3A we have calculated the steady-state I–V relationship as a function of membrane voltage for the entire model—both the total current and each individual active conductance. As expected, the steady-state I–V relationship for the total current is nonmonotonic with a fold region corresponding to the increase in input resistance during the approach to threshold (Fig. 3A). Consequently, the steady-state I–V relationship for the model as a function of current is nonlinear with increasing input resistance at depolarized membrane potentials similar to that observed in Purkinje cells (cf. Figs. 1B and 3B).
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). However, we did not model Ca2+-dependent K+ currents explicitly because of the increase in dimensionality this approach requires and the lack of experimental data on the specific biophysics governing intracellular Ca2+ dynamics in Purkinje cells. This does not affect the fundamental formulation of the model because, from a dynamical systems perspective, it is not critical whether the K+ conductances are voltage-gated indirectly through voltage-gated Ca2+ channels or directly through an intrinsic voltage sensor. Finally, we incorporated IH into the model using parameters established by previous experimental studies (Roth and Hausser 2001; Williams et al. 2002), which reproduced the typical sag in the membrane voltage associated with hyperpolarization (Fig. 3B). To establish the performance of our model to Purkinje cells we began by comparing the responses evoked by step depolarizations of 250 ms. Purkinje cells were held at –74 mV and depolarized with a maximum of 200 pA (Fig. 4 Ai). In comparison, our model is able to reproduce the major characteristics associated with Purkinje cell depolarization in the subthreshold range, including a long first spike latency relative to the subsequent ISI and an absence of damped oscillatory activity (Fig. 4Aii). The model also accurately reproduces the slow decaying plateau potential subsequent to cessation of the depolarizing current (Fig. 4A, i and ii). Like the Purkinje cell, the decay rate of the plateau potential is faster after greater depolarization and firing frequency (Fig. 4A, i and ii).
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In terms of the bistable dynamics the model is also successful in reproducing the nonmonotonic relationship between CF-like stimulation strength and state transition probability. Consequently, an intermediate stimulus of 15 ms and 3 µA/cm2 is successful in mediating state transitions from rest to firing, whereas weaker (1.4 µA/cm2) or stronger (7 µA/cm2) stimuli are unsuccessful (Fig. 4B). Note that a weak stimulus (brief subthreshold current step) generates a substantial plateau potential lasting >100 ms (Fig. 4B). This is a result of the model having a saddle-node bifurcation and the corresponding steady-state Na+ conductance in the subthreshold range required to establish the bifurcation.
Like the cell, the model is able to switch from rest to firing and from firing to rest using a strong unidirectional input (15 ms and 4.2 µA/cm2) (Fig. 4C). This behavior in the model depends critically on the presence of the slow K+ current. Thus in the presence of slow K+ current, stimuli of intermediate strength from the rest state can transition the model to the firing state (Fig. 4C). Stimuli of the same strength delivered from the firing state recruits a greater amount of the slow K+ current, which then mediates the repolarization back to the rest state (Fig. 4C). By comparison, without slow K+ current a depolarizing input can transition the model only from rest to firing (not shown). Note also in Fig. 4C that after the initial transition from rest to spiking the onset of tonic spike firing is delayed immediately after the CF-like stimulus. A similar observation has been made in the Purkinje cell by both ourselves and others (Fig. 2A) (Hounsgaard and Midtgaard 1989; Williams et al. 2002). In the model this is caused by an increased activation of the slow K+ current after the first CF-like stimulus, which is sufficient to delay the onset of tonic spike firing (Fig. 4C).
To understand the dynamical and biophysical basis of bistability in the model we began by performing a single-parameter bifurcation using driving current as the bifurcation parameter. As expected, with increasing driving current the node (rest state) and saddle point coalesce at the fold (Fig. 5 A) in a saddle-node bifurcation of fixed points. With the elimination of the rest state the system jumps onto a limit cycle attractor. Note, however, that the limit cycle exists before the coalescing of the fixed points. Within a range of driving current (Fig. 5Aii) the rest state is only locally stable with perturbations (such as a CF-like stimulus) capable of transitioning the system from the node to the limit cycle. If the system is already on the limit cycle, however, it remains on a periodic solution past the fold region where both a node and saddle point exist. Analysis of the frequency–current (F–I) relationship of the model indicates a discontinuity associated with the bistable region if the system is driven from rest to spike firing (Fig. 5B). Thus the lowest possible firing frequency attainable in the model if the system is driven from rest to firing is about 43 Hz. This is consistent with previous work, which has shown that the F–I relationship of Purkinje cells is linear and discontinuous if measured using square-wave current steps that evoke spike output ranging from threshold to high-frequency firing (Llinas and Sugimori 1980b; McKay and Turner 2005; Williams et al. 2002). If the system is driven from spike firing to rest, however, the model F–I relationship is continuous, albeit with a very high gain in the low-frequency range. Furthermore, our analysis indicates that bistability in the model is limited to the low-frequency and higher-gain region of the F–I relationship.
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; Rinzel and Ermentrout 1998). Also consistent with this bifurcation is the discontinuity in the F–I relationship when transitioning from rest to firing. Finally, a saddle homoclinic bifurcation is consistent with a bistable system that produces long first spike latencies in the absence of fast subthreshold oscillations (Izhikevich 2000, 2007; Rinzel and Ermentrout 1998). In essence a model with a saddle homoclinic bifurcation can be viewed as a system with an extremely high gain compared with one with a saddle node on an invariant cycle bifurcation. The high gain is caused by a large amount of positive feedback after a spike. If the feedback is sufficiently high (biophysically this can be generated by factors contributing to a DAP or inhibiting an AHP) any input that crosses threshold and generates a spike will cause strong positive feedback driving an additional spike and thus lead to a permanent spiking state. Under these conditions the system will be bistable and undergo a saddle homoclinic bifurcation. The saddle homoclinic bifurcation will occur when the system is taken from firing to rest with negative driving current (the bifurcation parameter). Furthermore, the gain of the F–I relationship in the transition from firing to rest will be extremely high.
Given that a saddle homoclinic bifurcation can be produced in a two-equation firing model (Izhikevich 2000, 2007; Rinzel and Ermentrout 1998), we proceeded to investigate whether this type of reduced model and bifurcation could account for the general behavior of Purkinje cell firing and CF-induced state transitions.
A saddle homoclinic bifurcation with a nonmonotonic separatrix can account for Purkinje cell dynamics
We started by constructing a model of two dynamical variables undergoing a saddle homoclinic bifurcation of the limit cycle. This permits phase-plane analysis such that the stable manifold of the saddle point associated with spike threshold can be visualized. Note that this is not possible with the five-dimensional model. The phase-plane analysis also allows one to more easily discern the key biophysical features required to establish a saddle homoclinic bifurcation. In addition, it allows visualization of a separatrix, which is often associated with a bistable system. Consequently, the key requirements that control the presence and shape of the separatrix can be considered.
The two-equation model contained only a Na+, K+, and leak conductance. To retain the basic biophysics of spike firing using only two equations we let Na+ conductance activation equilibrate instantaneously with membrane voltage and modeled Na+ inactivation and K+ activation using the same variable (Rinzel 1985). Consequently, the two dynamical variables are membrane voltage and refractoriness (representing both K+ activation and Na+ inactivation). Previous work has established that a saddle homoclinic bifurcation and the resulting bistability can be achieved using a refractory time constant that is small relative to the membrane time constant (Izhikevich 2000, 2007; Rinzel and Ermentrout 1998). We manipulated leak conductance density (to adjust the membrane time constant) and the refractory time constant to establish a saddle homoclinic bifurcation. We set the default value for the refractory time constant at 0.6 ms. Although this value suggests a very fast K+ current, recent voltage-clamp data from Purkinje cell nucleated patches has measured and confirmed the presence of a fast and prominent K+ conductance (Martina et al. 2007).
In Fig. 6 A we show the phase-plane projection of the model for three different time constants of the refractory variable. When the model has a large time constant for the refractory variable (1.2 ms) the transition from the limit cycle to the stable fixed point is mediated by a saddle node on an invariant cycle bifurcation. As a consequence the system has no bistable dynamics and the stable manifold (dashed line) of the saddle point connects to the unstable point on its right (Fig. 6Ai). As the refractory time constant is reduced to an intermediate value (0.6 ms), however, the system undergoes a saddle homoclinic bifurcation of the limit cycle. Under this condition the stable manifold becomes a separatrix, which separates two regions in phase space that correspond to the two possible steady-state solutions (rest and firing) of the model (Fig. 6Aii). A separatrix can be viewed as a line in phase space corresponding to a demarcation of a hard threshold. Thus any perturbation that crosses the separatrix in a bistable system will result in a transition to the firing or rest state depending on the current state of the model. In a nonbistable system undergoing a saddle-node bifurcation the stable manifold simply demarcates the firing threshold of the model. Note that with a saddle homoclinic bifurcation a single spike is capable of generating a large depolarizing afterpotential (DAP) as a consequence of the fast refractory dynamics (Fig. 6B). Also important is the shape of the separatrix. Because the separatrix is nonmonotonic, such that it surrounds the limit cycle, a unipolar input can be used to switch between the stable point and the limit cycle (Fig. 6C). For this reason intermediate inputs can switch the model from the rest state to the firing state, whereas excessively strong inputs fail because they cross the separatrix twice, bypassing the firing state and returning the system to the rest state. This provides a dynamical systems explanation of the restricted range of CF-like stimuli that are able to induce state transitions in the bistable region. If the refractory time constant is further reduced (0.2 ms) the model retains the saddle homoclinic bifurcation and bistability but the separatrix becomes monotonic (Fig. 6Aiii). Thus switching from the rest state to the limit cycle can be done with a positive current input, whereas switching from the limit cycle to the rest state requires a negative current input. Note that when the model has a saddle homoclinic bifurcation it generates a long first spike latency relative to the subsequent ISI (Fig. 6E). This is a consequence of the fast refractory dynamics, which actively shorten the ISI relative to the first spike latency, and is consistent with our previous interpretation of the five-equation model.
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Biophysical basis for bistability in the five-equation model
In the five-equation model dynamics similar to those present in the two-equation model should establish bistability, although additional factors will be important. The two-equation model, however, predicts that the presence of conductances that slow membrane repolarization and increase the membrane time constant will establish a saddle homoclinic bifurcation and bistability. Thus we predict that removing the dendritic compartment and reducing Na+ conductance will eliminate bistability by removing a spike-dependent depolarizing factor (dendritic feedback) and decreasing the membrane time constant (Na+ conductance removal). Alternatively, increasing the K+ conductance and leak conductance will eliminate bistability by producing a strong and long-duration hyperpolarization after a spike and decreasing the membrane time constant (arising from the increased leak conductance).
To test these predictions in the five-equation model we proceeded to manipulate the components predicted to establish a saddle homoclinic bifurcation and control the bistable range. We performed a bifurcation analysis under each manipulation to evaluate overall system dynamics. We considered four factors: the dendritic compartment, refractory dynamics, leak conductance, and the Na+ window current. For each parameter we considered the scenario that would be predicted to reduce the bistable range. Thus we removed the dendritic compartment, slowed the refractory variable, increased somatic leak conductance, and reduced the Na+ window current (through a –6 mV shift in the steady-state inactivation parameter). As shown in Fig. 7, A–E each of the manipulations reduces or eliminates the bistable range of the model (indicated by a reduced overlap between the limit cycle and node across different current values). Under the control condition (Figs. 5 and 7A) the bistable range occurs over 0.26 µA/cm2 of driving current. Under each of the four manipulations the bistable range is either eliminated or reduced to <0.03 µA/cm2 range of driving current (Fig. 7, B–E).
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The five-equation model can account for previous experimental results
As a test of the validity of our model we sought to examine whether it could reproduce previous experimental results related to bistability that remain unexplained. In particular, we were interested in the effects of changing the density of Na+ current on plateau potentials and the density of IH on the bistability range.
Previous work has established that a small reduction of Na+ conductance with TTX in Purkinje cells can reduce the duration of spiking plateau potentials with no effect on spike parameters (Williams et al. 2002). We proceeded to reduce the Na+ conductance density (gNamax) and test the ability of the model to generate plateau potentials in response to a brief current step. We note that in the model the plateau potential is generated by the same factors that sustain the DAP. For these simulations the model was held at –73 mV with 0.023 µA/cm2. As shown in Fig. 7, a small reduction in gNamax that reduces the DAP shortens the duration of the spiking plateau potential (Fig. 7G, i and ii). Because the plateau potential is caused by a balance of inward and outward currents, a small change in gNamax has a dramatic effect on its duration, eventually eliminating the ability to generate a plateau potential and bistability. In contrast, during the spike upstroke membrane permeability is dominated by Na+ conductance. As a result, a small change in gNamax has no effect on spike parameters (Fig. 7Gii). Thus our model can accurately reproduce these experimental observations within the biophysical context described previously to establish a saddle homoclinic bifurcation.
We next examined the effects of IH on the bistable range of the model. The effect of IH on bistability remains somewhat controversial. Experimental results indicate that removal of IH with pharmacological blockers or modulators does not inhibit bistability (Williams et al. 2002). In fact, the overall effect of IH block is to increase the observation of bistability in Purkinje cells (Williams et al. 2002). Modeling data from a more recent study, however, place IH as the critical conductance underlying bistability in Purkinje cells (Loewenstein et al. 2005). Given that only modeling data suggest a critical role for IH in sustaining bistability (Loewenstein et al. 2005), we carried out experiments to examine the effects of removing IH with 20 µM 4-(N-ethyl-N-phenylamino)-1,2-dimethyl-6-(methylamino) pyridinium chloride (ZD7288) on bistability. To test for bistability cells were held at –68 mV and given two 15 ms duration 2 nA current steps of positive and then negative polarity spaced 500 ms apart (Fig. 8 Ai). In all cells tested (n = 8) blockade of IH did not prevent state transitions between rest and firing (Fig. 8Ai). Thus bistability was not blocked by the removal of IH, confirming previous results by Williams et al. (2002). An effective block of IH was confirmed by delivering a negative 2 nA current pulse (15 ms), which under control conditions produced a voltage overshoot and dampening (indicating deactivation of IH) that was abolished in the presence of 20 µM ZD7288 (Fig. 8Aii).
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Although IH might be expected to increase the bistable range of the neuron by contributing to the DAP (by rebounding membrane voltage during the spike AHP), its major contribution to membrane dynamics comes about through a decrease in input resistance and the membrane time constant. This occurs because the time constant of IH activation and deactivation is too large for any dynamic contribution during a single spike cycle. Consequently, only the steady-state activation of
