INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Cerebellar Purkinje cells, the sole output of the cerebellar cortex, receive two major inputs: the climbing fiber input, which originates from the inferior olive (IO), and the granule cell input, which relays information from the mossy fibers. Each Purkinje cell is innervated by one climbing fiber, and a single IO spike generates a single "complex spike" in a Purkinje cell. In contrast, a Purkinje cell receives 200,000 granule cell inputs, which generate a high-frequency discharge ("simple spikes"). Because IO neurons fire at a very low frequency (at most 2 or 3 spikes within a second), there are comparatively few complex spikes. Although it is well known that simple spikes are involved in motor control, the role of complex spikes is more controversial (De Zeeuw et al. 1998; Simpson et al. 1996). The view that climbing fiber inputs function as detectors of control errors is well supported (e.g., Kitazawa et al. 1998; Kobayashi et al. 1998). However, it is still unclear whether climbing fibers exert a real-time influence on the targets of the Purkinje cells, and/or exert a short-term modulatory action on simple spike patterns, or induce long-lasting changes in the potency of granule cell-Purkinje cell synapses.

IO neurons exhibit a number of unique electrophysiological properties. First, IO cells respond to both injection of depolarizing currents and release from hyperpolarization by generating, respectively, dendritic and somatic spikes (Llinás and Yarom 1981a). Second, in vitro IO cells can exhibit 4- to 8-Hz rhythmic activity (Bal and McCormick 1997; Benardo and Foster 1986; Llinás and Yarom 1986). This rhythmic activity can either take the form of subthreshold sinusoid-like oscillations or rhythmic generation of sodium spikes. Furthermore these oscillations can occur either spontaneously [in ~10% of the cells recorded by Llinás and Yarom (1986); but Benardo and Foster (1986) reported a larger proportion of oscillating cells] or after harmaline or serotonin administration (Sugihara et al. 1995). Third, the cells are coupled by electrotonic gap junctions located on the dendrites (Llinás et al. 1974; Sotelo et al. 1974). Through the combined actions of the coupling between cells and the underlying rhythmicity of individual neurons, neighboring IO cells have been shown to fire with some rhythmicity in brain slices and acute preparations (Llinás and Yarom 1986; Sasaki et al. 1989). Yarom (1991) suggested that the IO is a network of damped oscillators that, when coupled, can generate sustained oscillations. However, the oscillatory behavior of IO cells in vivo is still under investigation because apparently random, rather than periodic, firing has been observed in behaving animals (Keating and Thach 1995). Thus it is probable that synaptic and modulatory inputs influence the oscillating behavior of single as well as group of coupled cells. Besides the very strong excitatory dendritic inputs (e.g., De Zeeuw et al. 1998), IO cells receive two types of inhibitory inputs: the first affects the excitability of the cell, whereas the second modulates the coupling strength between cells (Lang et al. 1996). Moreover, a strong serotonergic input to the IO from the midbrain raphe has been identified (Weiss and Pellet 1982).

Several studies have described the membrane currents of IO neurons and their influence on cell responses (Bal and McCormick 1997; Benardo and Foster 1986; Bleasel and Pettigrew 1994; Llinás and Yarom 1981b, 1986; Manor 1995; Manor et al. 1997; Ruigrok and Voogd 1985; Yarom and Llinás 1987). IO neurons exhibit two types of spatially distinct calcium currents: a low-threshold current (I_{Ca_l}) located in the soma and a high-threshold current (I_{Ca_h}) located in the dendrites (Llinás and Yarom 1981a,b). IO cells respond to both release from hyperpolarization and injection of depolarizing currents via these two calcium currents. The low-threshold calcium current has a window of conductance around the resting membrane potential, which causes the cell to be excited in response to hyperpolarizing current pulses. The high-threshold calcium current is noninactivating, which results in a prolonged plateau potential in response to depolarizing dendritic input. The calcium influx in the dendrites activates a calcium-dependent potassium current (I_{K_Ca}), which abruptly terminates the plateau potential after ~30 ms (Llinás and Yarom 1981a). The inactivation of I_{K_Ca} has a very long time constant (several hundred milliseconds), resulting in a long afterhyperpolarization (AHP). The AHP deinactivates the low-threshold calcium conductance and triggers a postinhibitory rebound. Furthermore, hyperpolarization activates an anomalous rectifying current (Bleasel and Pettigrew 1994; Yarom and Llinás 1987). Bal and McCormick (1997) recently argued that the h current (I_h), which originally was described in thalamocortical relay neurons (Huguenard and McCormick 1992), may account for many IO cell properties. They proposed that I_h, which is activated at hyperpolarized potentials, contributes to the presence, amplitude, and frequency of the subthreshold oscillations. Finally, like most other neurons, IO cells can generate somatic sodium spikes (with the sodium current I_Na) that are terminated by a delayed rectifier potassium current (I_{K_dr}). However, a single spike is generated during the plateau potential, indicating a strong refractoriness of the sodium current (Llinás and Yarom 1981a).

The existence of electrical coupling between IO cell dendrites make voltage-clamp recording of individual IO cells arduous (Manor 1995). It is thus difficult to analyze the contribution of individual currents to the response of olivary neurons. Conductance-based models of neurons, however, if sufficiently constrained by biological data, may provide a means to test different hypotheses by systematically changing their parameters (e.g., maximal conductances of ionic channels). In the present study, we extended the single compartment model developed by Manor et al. (1997), which included only I_{Ca_l} and a leak current, and built a two-compartment model of IO neurons, where the first compartment represented the soma and the second compartment represented the lumped dendrites. The dendritic compartment permits us to account for the known dendritic location of the high-threshold calcium current, the gap junctions, and the synaptic inputs. We constrained the parameters of our model with experimental data whenever possible and reproduced many experimentally observed findings.

First, we further explored how the modulation of individual current conductances leads to different asymptotic electrical behaviors in single olivary neurons in vitro. Specifically, we quantitatively tested the influence of the low threshold calcium current I_{Ca_l} and the h current I_h on single IO cell firing patterns. Consistent with previous experimental findings, we quantitatively demonstrated the major role of I_{Ca_l} in spiking excitability. In addition, in line with the study of Bal and McCormick (1997), we found that I_h had an important modulatory role in the spike generation and period of oscillations.

Then we examined the role of electrical coupling on the spiking behavior of IO cells. Because IO cells were discovered to be extensively coupled by gap junctions (Llinás et al. 1974; Sotelo et al. 1974), researchers have been looking for the functional roles of the electrical coupling. In most neurophysiological studies, electrical coupling is thought to be responsible for the synchronization of groups of neurons. However, in several mathematical studies of simple coupled neuron models, anti-phase as well as irregular spiking have been observed (Abarbanel et al. 1996; Kawato et al. 1979; Sherman and Rinzel 1992). Thus, although counterintuitive, electrical coupling theoretically could desynchronize rather than synchronize coupled cells. In the present study, we found that two coupled IO cells could have different spiking modes depending on the coupling strength, the hyperpolarization level, and the modulation of I_{Ca_l} and I_h currents: the two cells could be in-phase, phase-shifted, or anti-phase or could exhibit a complex, desynchronized spiking mode. We discuss the robustness and significance of the desynchronization for the in vivo functioning of the IO in light of the known effect of serotonin on IO neurons.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

General cell model

The electrotonic properties of the two-compartment model (Fig. 1A) were determined by two morphological parameters (Pinsky and Rinzel 1994): the ratio of the somatic area to total surface area (p) and the electrotonic coupling conductance between the two compartments (g_int). For each compartment, the membrane potential (V) changed according to
C_m is the membrane capacitance, I_app is the applied current (common to both compartments) and accounts for the ionic currents, the current flowing out into the other compartment, and the current flowing through the gap junctions (currents in µA/cm²).

View larger version (20K): [in this window] [in a new window]   Fig. 1. Model of inferior olive (IO) neurons. A: 2-compartment biophysical model of an IO cell. Somatic ionic currents include a low-threshold calcium inward current (ICa_l), an anomalous inward rectifier current (Ih), the Hodgkin-Huxley type inward sodium (INa) and outward delayed rectifier potassium (IK_dr) currents, the current flowing out into the dendritic compartment (Ids), and a leakage current (Ils). Dendritic currents include a high-threshold inward calcium current (ICa_h), an outward calcium-dependent potassium current (IK_Ca), the current flowing into the somatic compartment (Isd), and a leakage current (Ild). Because the dendritic potassium Current IK_Ca is calcium-dependent, Ca2+ concentration dynamics is included in the model. gint is the coupling conductance between the two compartments. B: 2 cells electrotonically coupled by a single hypothetical gap junction with maximal coupling conductance gc. Electrotonic coupling current between the cells is given by Ic.

Somatic currents

The somatic ionic currents included I_{Ca_l}, I_h, an inward sodium current (I_Na) and a delayed rectifier outward potassium current (I_{K_dr}), as well as the current flowing out into the dendrites (I_ds) and the leakage current (I_ls).¹
To model these currents, we used data obtained from IO cells whenever possible; but when not available, we used experimentally derived data or models of similar currents from other neural systems. Manor (1995) quantified the activation and inactivation functions of I_{Ca_l} in the IO cell (see APPENDIX). I_h was modeled in thalamic relay neurons by Huguenard and McCormick (1992). The equation for I_ds was obtained from Rush and Rinzel (1994), as was I_{K_dr}. Because a sodium current of the Hodgkin-Huxley type would generate several spikes during the plateau potential, whereas IO neurons normally only generate a single spike, we adapted a new type of I_Na with a long-lasting inactivation component. The somatic currents are given by





where k, l, q, h, and n are activation and inactivation variables, the kinetics of which are described in the APPENDIX; V_s is the membrane potential of the soma and V_d is the membrane potential of the dendrite. V_Ca, V_h, V_Na, and V_K are the reversal potentials of the sodium, potassium, calcium, and h currents, respectively (potentials in mV); g_{Ca_l}, g_h, g_Na, g_{K_dr}, and g_ls are maximal conductances (in mS/cm²).

Dendritic currents

The dendritic ionic currents include I_{Ca_h}, I_{K_Ca}, the current flowing into the somatic compartment (I_sd), a leakage current (I_ld), and the current flowing into other cells through electrical coupling (I_c). Thus the dendritic currents are given by
Traub et al. (1991) modeled I_{Ca_h}, I_{K_Ca}, and the dynamics of calcium concentration for CA3 neurons. Rush and Rinzel (1994) described the expression of I_sd for a two-compartment cell model. Finally, like all gap junctions between all mammalian cells thus far evaluated (Moreno et al. 1994), the conductance of I_c has been shown recently to be voltage dependent (Kogan and Yarom 1997). Precise voltage-dependency curves are not yet available for the gap junction protein connexin36, which is important for the IO (Condorelli et al. 1998). Therefore we adopted the general voltage-dependency model from Benett and Versedis (1992), and we set the parameters according to the activity of the gap junction protein connexin43 (Moreno et al. 1994). The number of gap junctions is presumably between 5 and 20 for any pair of IO cells (De Zeeuw et al. 1996). However, as we modeled only a single dendritic compartment per cell, an agglomerate of gap junctions between two cells can be represented by a single conductance, g_c.² Thus the dendritic ionic currents are given by




where r and s are activation and inactivation variables, respectively (see APPENDIX); f represents the transjunctional voltage dependence of the gap junction conductance; V_d is the dendritic membrane potential of the cell; V_de is the dendritic membrane potential of another IO cell in electrotonic contact with the cell; g_{Ca_h}, g_{K_Ca}, g_ld, and g_c are maximal conductances.

Modeling a "standard" cell

A neural model can have a wide variety of responses when large parameter ranges are considered (Rinzel and Ermentrout 1989). Our model of IO neurons, although simplified, contains a large number of parameters. Consequently whenever possible, physiologically plausible ranges of parameters are first taken into account (see Table 1). Moreover we use typical values for sodium, potassium, and calcium reversal potentials and membrane capacitance: V_Na = 55 mV, V_K = 75 mV, and V_Ca = 120 mV and C_m = 1 µF/cm^2, respectively. We took the reversal potential of the h current to be V_h = 43 mV, as reported by Huguenard and McCormick (1992). The remaining unknown parameters were investigated using the simulation software package XPPAUT (Ermentrout 1997) to integrate the equations of the system. We developed a "standard" cell that possessed most of the known electrophysiological properties of IO cells by taking a priori ranges of parameters into account when available.

Sensitivity analysis

We tracked the stationary and periodic solutions of the system with a change in a parameter (such as the input current) using the bifurcation analysis software AUTO (Doedel 1981). AUTO also allowed us to compute the frequency of oscillations, detect the bifurcation of solutions, and track the bifurcation points (e.g., the onset of oscillations) in two-dimensional parameter space. Because it was impossible to track the bifurcation of periodic orbits with this software (e.g., points for which 2 periodic solutions, such as subthreshold oscillations and spontaneous spiking, coalesce), we developed a custom simulator based on the integration simulation package CVODE (Cohen and Hindmarsh 1994). Thus we could explore spiking behaviors when one or two parameters were varied. By numerically integrating the model equations with different parameter values, we uncovered several asymptotic cell response domains: return to rest (R) after a period of transients, spontaneous subthreshold oscillations (SO), spontaneous somatic spiking (SS), spontaneous dendritic spiking (DS). In some cases, the cell could stay in different symptotic states depending on the initial conditions: the cell was then described bistable, and in this case the domains were overlapping.³ For instance, when the SO domain overlapped with SS, the cell behavior was denoted to be SS/SO bistable. Note that in the sensitivity analysis graphs, we plotted both the results obtained with AUTO (for the domain of oscillatory behavior emerging from rest) and the results obtained with our simulator (for the other domains).

Analyzing the responses of two coupled cells

We examined the behavior of two coupled cells when g_c varied (Fig. 1B). To describe the extent of synchronization between two identical cells that receive identical inputs, we computed the average "distance" D(, g_c), in millivolts, between the somatic membrane potentials (Abarbanel et al. 1996). This measure is essentially the root mean square difference between the somatic membrane potentials of the two cells, V_s1 and V_s2, with a time shift . The time shift between the activities of the neurons is necessary to "align" the neurons' responses. The average distance is mathematically defined by
where N is the total number of time steps (1 time step was 1 ms). As in Abarbanel et al., we plot D(_min, g_c), which is the distance for the time shift _min that yields the minimum value of D(, g_c), for fixed coupling. For instance, if the neurons are anti-phase, the distance D(_min, g_c) is zero and the time shift _min equals to half a period. Initially a perturbing stimulus input was given at t = 1 s in cell 1. The distance was then computed over 10 s, from t = 5 s to ignore the transients.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Standard cell

Using the following parameter values (conductances in mS/cm²), soma: g_Na = 70, g_{K_dr} = 18, g_{CA_l} = 1.0, and g_h = 1.5; dendrite: g_{Ca_h} = 4.0 and g_{K_Ca} = 35; leak: g_ls = g_ld = 0.015 and v_l = 10 mV; cell morphology: g_int = 0.13 and p = 0.20, data from our standard cell model were consistent with many reported experimental findings from actual IO neurons.

STEADY-STATE PROPERTIES. The membrane potential with no input current (I_app = 0 µA/cm²) was 57 mV, whereas the input resistance derived from the voltage-current curve was 36 M. When I_app = 5 µA/cm², the potential was 80.3 mV and the input resistance was 14 M, reflecting the effect of the h current. When I_app = +5 µA/cm², the membrane potential was 46 mV and the input resistance was only 10 M because of the effect of the delayed rectifier current. These steady-state values are in agreement with published experimental data (see for instance Table 1 in Llinás and Yarom 1981a; Fig. 1A in Yarom and Llinás 1987; Manor 1995). Note that 1 µA/cm² corresponds to 0.1 nA for a total cell surface of 10,000 µm².

DEPOLARIZING CURRENT INPUT. We injected depolarizing step currents of 50-ms duration, beginning at t = 200 ms in the cell at rest (Fig. 2). When the injected current intensity was >3 µA/cm², the cell first responded with a somatic sodium spike immediately followed by a dendritic calcium plateau potential (Fig. 2A), which lasted for ~35 ms. The calcium plateau potential, due to activation of I_{Ca_h}, created a large calcium ion influx in the dendrites, which in turn activated I_{K_Ca} (Fig. 2C). This eventually terminated the plateau potential and created a large, long-lasting AHP ~13 mV below the resting membrane potential. At the offset of the dendritic spike, I_{Ca_h}, I_{Ca_l}, and I_h are all completely inactivated (as are I_Na and I_{K_dr}). Thus the only nonzero current is I_{K_Ca.} Because this current is driven by relatively slow dendritic calcium dynamics and possesses a long inactivation time constant, the AHP is dominated by I_{K_Ca}. Note that even though the conductance of I_{K_Ca} is high, the amplitude of I_{K_Ca} is small during the AHP because the membrane potential is close to the potassium reversal potential. The AHP terminated sharply after ~300 ms due to rebound excitation, which is consistent with published data (see Fig. 7A in Llinás and Yarom 1981b). The time courses of the somatic currents (Fig. 2B) revealed that the AHP termination is initially due to the activation of I_h, which increases the membrane potential. Then when the soma is sufficiently depolarized, I_{Ca_l} starts to be activated, which results in the rebound of excitation.

View larger version (15K): [in this window] [in a new window]   Fig. 2. Responses of an IO cell model to a dendritic current pulse (intensity: 8 µA/cm2, duration: 50 ms). A: somatic membrane potential: the cell 1st responded with a somatic sodium spike immediately followed by a dendritic calcium plateau potential ensued by a long-lasting after-hyperpolarization, which is terminated by a rebound potential. B: somatic currents; C: dendritic currents (note difference in ordinate scale between B and C).

HYPERPOLARIZING CURRENT INPUT. We then studied the response of the standard cell model to a hyperpolarizing current pulse (t = 500 ms, duration = 100 ms, intensity = 1.5 µA/cm²) superimposed on different tonic hyperpolarizing currents (I_app). When I_app = 0 µA/cm², the activation of the low-threshold calcium conductance, as well as the recovery of I_h, during the hyperpolarizing step created a rebound sufficient to generate a sodium spike (Fig. 3A). This is consistent with previous experimental data (Bal and McCormick 1997; see their Fig. 2 for instance). I_{Ca_l} then became inactivated and I_{Ca_h} became activated to a moderate degree. The calcium influx in the dendritic compartment resulted in the I_{K_Ca} activation, creating an AHP. The cycle began again, and another spike was generated. After the second spike, however, I_{Ca_l} was no longer sufficiently activated to bring the membrane potential above threshold and only a few more subthreshold oscillations followed. Thus the standard cell was a damped oscillator, as reported by Lampl and Yarom (1997). When a tonic bias hyperpolarizing current I_app = 0.5 µA/cm² was maintained before the current pulse, the cell was SO (Fig. 3B). The cell fired a burst of six spikes after the pulse. When the bias was I_app = 1.0 µA/cm² (Fig. 3C), the cell generated two sodium spikes in response to the current pulse. Finally, when the bias was I_app = 1.5 µA/cm² (Fig. 3D), only one spike was generated and the oscillations were rapidly damped.

View larger version (27K): [in this window] [in a new window]   Fig. 3. Response of an IO cell model to a hyperpolarizing somatic current pulse (intensity: 1.5 µA/cm2, duration: 100 ms) at different states of tonic hyperpolarization. A: Iapp = 0.0; B: Iapp = 0.5; C: Iapp = 1.0; and D: Iapp = 1.5 µA/cm2. Cell is most excitable when Iapp = 0.5 µA/cm2 and subthreshold oscillations occur only for moderate values of the tonic hyperpolarizing current.

Parameter sensitivity analysis

BIFURCATION ANALYSIS. The steady-state cell responses at different hyperpolarized membrane potentials can be predicted from a bifurcation diagram plotting the minimum and maximum values of the somatic membrane potential for I_app. The cell had a stable steady state for all I_app values except between 1.17 µA/cm² and 0.37 µA/cm² (Fig. 4; 1.17 and 0.37 corresponded to the 2 Hopf bifurcation points marked a and b). Within this range, the cell exhibited subthreshold oscillations that reached a maximum amplitude of 9.8 mV for I_app = 0.85 µA/cm², and the frequency of these subthreshold oscillations was 5-7 Hz. These values are similar to those recorded in real IO neurons (Benardo and Foster 1986; Lampl and Yarom 1997; Llinás and Yarom 1986).

View larger version (13K): [in this window] [in a new window]   Fig. 4. Bifurcation diagram of the membrane potential Vs as a function of Iapp, which shows the behavior of the standard cell model as a function of the injected DC current. Cell has a stable steady state for all Iapp values except between the points marked a and b, where the cell exhibits SO. Thin line: stable steady state. Thin dotted line: unstable steady state. Thick line: stable oscillatory behavior. Hopf bifurcations occur at the thin and thick lines intersections. R, rest. SO, spontaneous subthreshold oscillations.

LOW-THRESHOLD CALCIUM CURRENT. Because the low-threshold calcium current has a central role in IO oscillatory behavior, we examined the variation of g_{Ca_l} in our model. We first analyzed the behavior of a cell with g_{Ca_l} = 1.2 mS/cm² and compared it to our standard cell (g_{Ca_l} = 1.0 mS/cm²). The relatively small increase in g_{Ca_l} resulted in a very different bifurcation diagram (Fig. 5). The cell had multiple stable behaviors, specifically, SS, SO, and R (an example of SS behavior can be seen in Fig. 7D). The spontaneous somatic spiking was between the points marked a and b in the figure, which correspond to a large range of input currents (1.48-0.19 µA/cm²). Between the points b and c (I_app between 0.23 and 0.19 µA/cm²), the cell had two stable states, either SS or R. This bistability has been observed previously (Bal and McCormick 1997 - see their Fig. 1B for instance). Between the points c and d, i.e., when I_app was between 0.73 and 0.23 µA/cm², the stable behaviors were either subthreshold oscillations or spontaneous somatic spiking. For the domain delimited by the points d and e (I_app between 1.30 and 0.73 µA/cm²), the only stable state was spontaneous somatic spiking. Finally, the points e and a (I_app was 1.47 and 1.30 µA/cm²) delimit another zone of bistability, with either subthreshold oscillations or spontaneous spiking.⁴ The frequencies of the spiking oscillations ranged from 3.8 to 9.4 Hz, which is similar to that previously recorded in IO neurons (4-8 Hz) (Bal and McCormick 1997). The frequencies decreased with stronger hyperpolarization, a result also consistent with previous observations (see Fig. 2A in Bal and McCormick 1997).

View larger version (18K): [in this window] [in a new window]   Fig. 5. Bifurcation diagram of the membrane potential Vs as a function of Iapp, which shows the behavior of a different cell in which gCa_l was changed to 1.2 mS/cm2 (for the standard cell, gCa_l = 1 mS/cm2). Cell now has multiple stable behaviors, namely spontaneous somatic spiking (SS), SO, and R. SS exists between the points a and b. Between the points b and c, the cell has two stable states, SS and R. Between the points c and d, the stable behaviors are either SO or SS. For the range delimited by the points d and e, the only stable state was SS. Finally, the points e and a delimit another bistability zone with either SO or SS. Thin line: stable steady state. Dotted thin line: unstable steady state. Thick lines: stable oscillatory behaviors. Open circles: unstable oscillatory trajectories that are not observable.

View larger version (13K): [in this window] [in a new window]   Fig. 6. Response domains when both gCa_l and Iapp are varied. Note how the low-threshold calcium current is crucial for subthreshold and spiking oscillations. In this figure and in Figs. 7 and 8, the smooth curves were obtained by tracking the bifurcation points with AUTO and the segmented curves were obtained by performing full two parameter searches (see METHODS).

h CURRENT. Previous in vitro experiments have shown that g_h is reduced with increasing concentrations of bath-applied Cs⁺ (Bal and McCormick 1997). To model the effects of Cs⁺, g_h was decreased from its standard value 1.5 mS/cm² (Fig. 7A) to 1.0, 0.5, and 0.0 mS/cm² (Fig. 7, B-D), while a tonic current of 0.5 µA/cm² was applied (note that a hyperpolarizing current pulse of 1.5 µA/cm² was injected at 500 ms for 100 ms to show bistability). The frequency decreased as the conductance value became smaller: it was ~8 Hz with g_h = 1.5 mS/cm^2, and it decreased to 6.2 Hz when I_h was fully suppressed (frequency analysis conducted with AUTO, data not shown). Similar results have been found in vitro (Bal and McCormick 1997; see their Fig. 7D). Note also that for g_h = 0.5 mS/cm² and g_h = 0.0 mS/cm², the hyperpolarizing pulse generates a dendritic spike. In these cases, as discussed by Yarom and Llinás (1987), somatic and dendritic compartments are effectively more strongly coupled because of higher input resistance: the sodium spike can invade the dendritic compartment enough to generate a dendritic spike.

View larger version (25K): [in this window] [in a new window]   Fig. 7. Effects of varying the conductance of the h current (gh) on cell responses. A: gh = 1.5 mS/cm2; B: gh = 1 mS/cm2; C: gh = 0.5 mS/cm2; D: gh = 0 mS/cm2. Iapp was held constant at 0.5 µA/cm2; and there is a pulse current injected at 500 ms (intensity: 1.5 µA/cm2, duration: 100 ms). Note the dendritic spike induced by the pulse current in both C and D. E: response domains as a function of gh and Iapp. Asterisk-labeled points, A, B, C, and D, correspond to conditions in A, B, C, and D.

Behavior of two coupled IO cells

RESPONSES TO A CURRENT PULSE WITH DIFFERENT TONIC CURRENTS. To investigate the effect of coupling on the oscillating behaviors of two cells, we examined the membrane potential of cell 1 as a function of both the tonic input current injected in cell 2 (I_app2), and the coupling conductance g_c (with I_app1 = 0 µA/cm²). The cells were "standard cells," except that we set g_{Ca_l} = 1.2 mS/cm² to increase the excitability of the cells. The two-parameter diagram (Fig. 8A) shows that for g_c < 0.18 mS/cm², there were two ranges of I_app2 with subthreshold oscillations. In Fig. 8A, to the right of the nearly vertical line (around I_app2 = 0.2 µA/cm²), and the area under the curved line, cell 1 was in a stable-steady state. In other regions, the cell oscillated.

View larger version (33K): [in this window] [in a new window]   Fig. 8. Oscillatory response of cell 1 as a function of the input current in cell 2, Iapp2, and gc (gCa_l = 1.2 mS/cm2 and Iapp1 = 0 µA/cm2). A: response domains. Note the existence of 2 domains where the cell did not oscillate. B-E: membrane potentials of the 2 cells were plotted for in both noncoupled (C and E) and coupled (B and D) case (gc = 0.05 mS/cm2) for 2 values of Iapp2 (Iapp2 = 1 µA/cm2 and Iapp2 = 4 µA/cm2). Subplot names B-E correspond to the 4 asterisks in A. Thin line: somatic membrane potential of cell 1. Thick line: somatic membrane potential of cell 2. B: Iapp2 = 4.0 µA/cm2, gc = 0.05 mS/cm2; C: Iapp2 = 4.0 µA/cm2, gc = 0.0 mS/cm2; D: Iapp2 = 1.0 µA/cm2, gc = 0.05 mS/cm2; E: Iapp2 = 1 0.0 µA/cm2, gc = 0.0 mS/cm2. Other details as in Fig. 3.

RESPONSES TO CURRENT PULSES IN SIMULATED HARMALINE APPLICATION CONDITIONS. The drug harmaline is known to induce very robust somatic spiking in IO cells via three pharmacological effects (Llinás and Yarom 1986): it increases the voltage sensitivity of the conductance of the low-threshold calcium current, it hyperpolarizes the cell, and it reduces the anomalous rectification. In simulations, we approximated the increase in voltage sensitivity of the I_{Ca_l} conductance by increasing g_{Ca_l}, the hyperpolarizing effect by injecting a tonic hyperpolarizing current in the soma, and the reduction in I_h by decreasing g_h. Thus we set g_{Ca_l} = 1.2 mS/cm², I_app1 = I_app2 = 0.8 µm/cm², and g_h = 0.7 mS/cm² (also we further increased g_Na to 80 mS/cm² to ensure robust spiking).⁵ The cells exhibited strong and robust spontaneous somatic spiking with this set of parameters.

View larger version (49K): [in this window] [in a new window]   Fig. 9. Responses of coupled spiking cells in 2 conditions. At t = 0.5 s, a depolarizing pulse was given to the dendrite of cell 1 for symmetry breaking. A: complex desynchronized firing patterns obtained for gc = 0.03 mS/cm2, gCa_l = 1.2 mS/cm2, gh = 0.7 mS/cm2, gNa = 80 mS/cm2, and Iapp1 = Iapp2 = 0.8 µm/cm2. B: anti-phasic spiking patterns for same parameters as in A, except gCa_l = 1.5 mS/cm2 and gc = 0.01 mS/cm2. For a better visualization of the desynchronization, we plotted cell 1's somatic membrane potential vs. cell 2's potential of for the complex firing pattern case (C) for the anti-phasic case (D).

MODULATION OF BOTH COUPLING STRENGTH AND IONIC CURRENTS IN SIMULATED HARMALINE APPLICATION. Here we show that, in addition to the coupling conductance, the maximal conductance values of both I_{Ca_l} and I_h have key roles in the appearance of complex, desynchronized firing patterns. In Fig. 10A, the distance D between the two cells is plotted as a function of both g_{Ca_l} and g_c. The contour D = 2 mV approximately corresponds to the appearance of a significant phase shift. The contour D = 8 mV corresponds to the induction of desynchronizing dendritic spikes by the current flowing through the gap junction (as in Fig. 9A).⁶ Very weak coupling (g_c < 0.003 mS/cm²) had no effect on the phase relationship of the spiking cells. For a large range of intermediate coupling values (g_c between 0.003 and 0.15 mS/cm²), spiking was complex. Only very large coupling conductances (g_c > 0.2 mS/cm²) brought the two cells into the in-phase mode. Because these large values are probably not biologically plausible (see DISCUSSION), complex firing behavior occurred for almost the whole physiological range of g_c. The range of g_{Ca_l} for which firing is complex was 1.15-1.4 mS/cm². This range should be compared with the spontaneous somatic spiking domain: if g_{Ca_l} was <0.9 mS/cm² only subthreshold oscillations were observed, and if g_{Ca_l} was >1.6 mS/cm², only spontaneous dendritic spiking were observed. Thus desynchronization occurred for more than a third of g_{Ca_l} values for which somatic spikes were observed.

View larger version (25K): [in this window] [in a new window]   Fig. 10. A: average distance between the 2 cells' membrane potentials as function of both gCa_l and gc. Contours where the distance D is 2 mV and 8 mV are shown (D = 2 mV approximately corresponds to the appearance of a significant phase shift; D = 8 mV corresponds to the induction of desynchronizing dendritic spikes by the current flowing through the gap junction). The point labeled "a" corresponds to the standard cell under harmaline condition (Fig. 9A) and the point labeled "b" to the anti-phase regime (Fig. 9B). Note how increasing both ionic and coupling conductances tends to desynchronize the cells. B: Distance between the two cells as function of both gh and gc. As above, very weak coupling had no effect, but for stronger coupling values firing became complex and the two cells were desynchronized.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

We propose a novel two-compartment biophysical model of IO neurons that reproduced many reported experimental results. The modeled neurons exhibited multiple firing patterns that depended on the level of depolarization or hyperpolarization, on the relative influence of the ionic currents I_{Ca_l} and I_h, and on the strength of the electrotonic coupling.

Comparison with related modeling studies

Two-compartment models similar to the one described here were shown to reproduce successfully many cellular responses of several neurons which contain active dendrites (Booth et al. 1997; Mainen and Sejnowski 1996; Pinsky and Rinzel 1994; Rush and Rinzel 1994). The second compartment, corresponding to the lumped dendrites, allowed us to segregate the soma from the dendritic location of the high-threshold calcium current, the gap junctions, and the synaptic inputs. The morphological parameters g_int and p were critical to the behaviors of the two-compartment model. When the coupling conductance between the two compartment g_int was set to large values (in effect lumping the somatic and dendritic compartments into 1 isopotential compartment), simulations showed that it was impossible to generate somatic spikes: the sodium spikes always triggered dendritic spikes. We found that, for proper functioning of the model g_int must be set to <0.2 mS/cm². Moreover, because it is reasonable to assume that intracellular coupling (g_int) is stronger than intercellular coupling (g_c), we have g_c < g_int < 0.2 mS/cm² (see Table 2 for plausible ranges of parameters inferred from simulations). In the present model, p = 0.2 was used for the ratio of the somatic area to the total surface area. In two compartment models, it is difficult, however, to relate directly p to the actual cell morphology due to the attenuation of the current flowing from distal dendrites to the soma (Pinsky and Rinzel 1994). Thus p should be larger than the actual morphological ratio (0.1-0.2) (Manor 1995).

In the present study, the calcium dynamics and the kinetics of the I_{Ca_h} and I_{K_Ca} currents were borrowed from Traub et al.'s (1991) CA3 hippocampal pyramidal neuron model. These kinetics lead to satisfactory performance of our IO model because most of the results of the present study are in agreement with experimental observations. It is possible, however, that the very low firing frequency of IO neurons in vivo is due to a longer lasting AHP than in the model (~350 ms when no tonic current is injected). When we used the following calcium dynamics equation (instead of Traub et al.'s)
then I_{K_Ca} was activated for a longer period and created an AHP that lasts ~700 ms (same experimental conditions as in Fig. 2). Thus the maximal firing rate of the cell would be ~1.4 spikes/s, in line with in vivo data (at most 2 or 3 spikes within a second).

To our knowledge, the model developed by Manor (1995) and Manor et al. (1997) is the only other published model of IO neurons. Their study focused on the subthreshold oscillations, so their model contained only I_{Ca_l} and a leakage current (because these 2 currents are sufficient to generate subthreshold oscillations). These authors notably showed that when two nonidentical cells were coupled, oscillations could be generated even though neither of the individual cells was a spontaneous oscillator. For example, the two cells shown in Fig. 8B did not oscillate when they were not coupled because the first cell was too depolarized and the second cell was too hyperpolarized. When coupled (with g_c = 0.05 mS/cm²), however, the two cells oscillated at the same frequency, but at different baselines and with a small phase shift. Moreover, if one cell started spiking, the oscillations in the other cell increased in amplitude.

Our model significantly extends the Manor et al. model, however, because we incorporated sodium spikes, dendritic spikes, and AHP. Thus we could investigate the conditions of IO spike generation when input current, electrical coupling strength, and the conductances of both I_{Ca_l} and I_h were varied. We further showed that a two-cell network model can exhibit anti-phase spiking (as previously found in simple models of coupled neurons) (e.g., Abarbanel et al. 1996; Kawato et al. 1979; Sherman and Rinzel 1992) and complex, desynchronized firing patterns (also observed in the model of Abarbanel et al. 1996). In the present model, the crucial and novel element is the triggering of dendritic spikes by electrical coupling leading to desynchronization. Our results suggest that this behavior is robust because it can be attained for a large range of coupling and ionic conductances.

Finally, our two-compartment model is complex enough to reproduce the essential properties of IO cells, such as the generation of dendritic and somatic spikes, but computationally simple enough to be later integrated into models of larger networks. Moreover, realistic, time varying inputs could be provided to the IO network by incorporating it into a complete sensorimotor neural system (see Schweighofer et al. 1998).

Ionic current modulationrhythmicity in single inferior olive cell

As previously described in experiments (Lampl and Yarom 1988; Llinás and Yarom 1986) and in the model of Manor et al. (1997), simulations showed that the low-threshold calcium current had a central role in generating rhythmic behavior. If the maximal conductance of I_{Ca_l} was below a threshold value, then spontaneous subthreshold oscillations did not occur regardless of the input current. Small changes in g_{Ca_l}, around the standard cell's parameter had dramatic effects on the cell excitability pattern (R, SO, or SS) as shown in Figs. 5 and 6. Depending on the brain slice, or even depending on the cell impaled, IO neurons recorded in vitro exhibit a variety of responses to the same stimulus, and some cells are more excitable than others. Our results show that even a relatively narrow distribution of g_{Ca_l} in real IO neurons can explain the diversity of responses encountered. Figures 4 and 5 suggest two new experiments to test further the effect of the low-threshold calcium current in single cells, i.e., cells grown in culture or isolated (this only will be feasible when specific gap junctions blockers become available). In these experiments, both the low-threshold calcium conductance and the injected tonic current are controlled independently. In the first experiment (Fig. 4), starting from rest at the depolarized level, a very slow continuous decrease in the injected tonic current first would produce subthreshold oscillations of increasing amplitudes, then oscillations of decreasing amplitudes, then rest again. In a second experiment (Fig. 5), if the maximal conductance g_{Ca_l} is increased (by harmaline for instance), the sequence of responses to decreasing injected current then would be: subthreshold oscillations of increasing amplitudes at slightly hyperpolarized membrane potential, then spontaneous somatic spiking, and finally rest again for large hyperpolarizing currents.

Our results indicate that the anomalous rectifier current I_h plays a major role in controlling the frequency of the oscillatory behavior and the generation of IO somatic spikes. This is consistent with the observations of Bal and McCormick (1997). I_{Ca_l} and I_h play antagonistic roles: while an increase in g_{Ca_l} makes the cell more excitable (Fig. 6), a decrease in g_h has similar effects (Fig. 7E). The effect of the anomalous rectifier could be tested by controlling simultaneously the conductance of I_h (with serotonin and/or Tris artificial cerebrospinal fluid) (see Bleasel and Pettigrew 1994) and the membrane potentials. For example, an experiment designed to follow a path at a fixed membrane potential similar to ABCD in Fig. 7 could be performed.

In the present model, the frequency of all oscillations are voltage dependent. In agreements with our model, the results of Llinás and Yarom (1986) and of Bal and McCormick (1997)see their Fig. 2seem to indicate that somatic spiking oscillations are voltage dependent. However, the spontaneous subthreshold oscillations, in slices where they occur, are voltage independent (Lampl and Yarom 1997; Llinás and Yarom 1986). Part of the discrepancy between our results and these experimental results may be due to network oscillations as opposed to cellular oscillations. Lampl and Yarom (1997) showed that the pattern of subthreshold oscillations is network specific rather than cell specific. Thus injection of current in one cell does not change its oscillatory frequency. Only when an experimental method is available to isolate a single IO cell, will it be possible to analyze the voltage dependency of subthreshold oscillations in single cells, as predicted here.

Influence of coupling and current modulationsynchrony and desynchrony of inferior olive cells

Even though gap junction coupling generally is thought to bring the membrane potential of the two coupled cells closer to each other, we found that, counterintuitively, increasing coupling strength can desynchronize the two spiking cells. If a somatic spike occurred in one cell, a dendritic spike in a coupled cell could be triggered by the current flowing through the gap junctions. Thus under certain conditions, it appears that electrical coupling may induce effects that are similar to those induced by chemical synapses, that is, they may trigger dendritic spikes. In addition, the long AHP after the dendritic spikes increased the complexity of the firing pattern. Desynchronization is robust because there were broad ranges of electrotonic coupling strengths and maximal conductances, g_{Ca_l} and g_h where desynchronization was observed. Moreover, we verified that desynchronization does not critically depend on the voltage dependency of the coupling conductance: Linear electrical coupling conductances also can desynchronize the cells (however, because of the stronger overall effect of linear coupling, the range of g_c where desynchronization was observed is smaller). Note that the occurrence of complex firing behavior by a reduction of g_h can be understood by the fact that a reduction of I_h decreases the electrotonic distance between the soma and the dendrites, which, in turn, increases the influenc