Dynamics and diversity in interneurons: a model exploration with slowly inactivating potassium currents

PII: S 0 3 0 6 - 4 5 2 2 ( 0 2 ) 0 0 1 6 8 - 9

Neuroscience Vol. 113, No. 1, pp. 193^203, 2002 D 2002 IBRO. Published by Elsevier Science Ltd All rights reserved. Printed in Great Britain 0306-4522 / 02 $22.00+0.00

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DYNAMICS AND DIVERSITY IN INTERNEURONS: A MODEL EXPLORATION WITH SLOWLY INACTIVATING POTASSIUM CURRENTS F. SARAGAa;b and F. K. SKINNERa;b
a

Toronto Western Research Institute, University Health Network, Departments of Medicine (Neurology) and Physiology, 399 Bathurst Street, MP 13-308, Toronto, ON, Canada M5T 2S8 b

Institute of Biomaterials and Biomedical Engineering, University of Toronto, Toronto, ON, Canada

Abstract;Recent experimental and model work indicates that slowly inactivating potassium currents might play critical roles in generating population rhythms. In particular, slow ( 6 1^4 Hz) rhythms recorded in the hippocampus correlate with oscillatory behaviors in interneurons in this frequency range. Limiting the ion channels to the traditional Hodgkin^ Huxley sodium and potassium currents, a persistent sodium current, and a slowly inactivating potassium current, we explore the role of slowly inactivating conductances in a multi-compartmental interneuronal model. We ¢nd a rich repertoire of tonic and bursting behaviors depending on the distribution, density and kinetics of this conductance. Speci¢cally, burst frequencies of appropriate frequencies could be obtained for certain distributions and kinetics of this conductance. Robust (with respect to injected currents) regimes of tonic ¢ring and bursting behaviors are uncovered. In addition, we ¢nd a bistable tonic ¢ring pattern that depends on the slowly inactivating potassium current. Therefore, this work shows ways in which di¡erent channel distributions and heterogeneities could produce variable signal outputs. We suggest that an understanding of the dynamical pro¢les of inhibitory neurons based on the density and distribution of their currents is helpful in dissecting out the complex roles played by this heterogeneous group of cells. D 2002 IBRO. Published by Elsevier Science Ltd. All rights reserved. Key words: hippocampus, bursting, bistability, mathematical model, heterogeneity, slow rhythm.

sion of metabotropic glutamate receptors has been found in distinct hippocampal interneurons (van Hooft et al., 2000). As indicated by these and other authors (e.g. Maccaferri et al., 2000; Parra et al., 1998; Cobb et al., 1995), such di¡erences suggest that functionally distinct roles might be played by interneurons. How might we understand this seemingly specialized group of neurons? The dynamical output of a neuron is dictated by incoming synaptic input as well as its morphological structure with its given density and distribution of currents (Mainen and Sejnowski, 1996). Although all the biophysical details of hippocampal interneurons are not presently known, there is much evidence for the di¡erential distribution of currents in these cells (McBain and Fisahn, 2001; Parra et al., 1998; Zhang and McBain, 1995b), including the presence of active dendrites (Martina et al., 2000), that could contribute to their heterogeneous natures. Indeed, a recognized di¡erence between interneurons and principal neurons is in the nature of their intrinsic channel proteins (e.g. Martina and Jonas, 1997; Zhang and McBain, 1995a, b). Voltage-gated channels have di¡erent kinetic properties, densities and distributions in interneurons (McBain and Fisahn, 2001), all of which would contribute to their varied and heterogeneous response to input and thus their role in determining network output. Experimental and computational studies suggest that the role of slowly inactivating potassium currents should

Interneurons or Q-aminobutyric acid (GABA)-ergic inhibitory cells are recognized as being essential units in cortical functioning (Buzsa¤ki and Chrobak, 1995; Je¡erys et al., 1996), perhaps being responsible for setting the various rhythmic frequencies observed in electroencephalogram (EEG) recordings that are characteristic of normal and pathological states (Niedermeyer and Lopes da Silva, 1993). Much work now focuses on interneurons (see recent review of McBain and Fisahn, 2001) and it is apparent that we are dealing with a highly diverse population of neurons in terms of morphology, neuromodulators and electrophysiological responses. It is not enough to consider these cells as simply providers of inhibition, but rather as determinants of brain output. For example, Buzsa¤ki and colleagues have shown that interneurons are di¡erentially active and correlate di¡erently with various behavioral states (Csicsvari et al., 1999). At a di¡erent level of investigation, Gupta et al. (2000) have shown that organizing principles for GABAergic interneurons and synapses might be possible based on their synaptic dynamics, and di¡erential expres-

*Corresponding author. Tel. : +1-416-6035800 ext. 5107; fax: +1416-6035745. E-mail address: [email protected] (F. K. Skinner). Abbreviations : EEG, electroencephalogram; GABA, Q-aminobutyric acid ; HH, Hodgkin^Huxley. 193

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be explored. In recent experimental work, slow ( 6 1^4 Hz) ¢eld rhythms were found to occur spontaneously or be induced in rodent hippocampus (Zhang et al., 1998; Wu et al., 2002). It was shown that these rhythms were inhibitory in nature, possibly due to the innervation of a synchronized GABAergic interneuronal population onto the principal cells, since rhythmic discharges in interneurons were correlated with the rhythm (Zhang et al., 1998). We previously used a computational approach to try to understand how interneuronal networks could produce these observed slow rhythms (Skinner et al., 1999). We found that inclusion of a slowly inactivating potassium current allowed the physiological data to be reproduced, in particular, to produce slow bursting behavior in the interneurons. However, in that paper, the focus was on why both inhibitory and gap-junctional coupling might be essential for the rhythm, and only single-compartment models were used. In another modeling study, slowly inactivating potassium currents were used in single-compartment models to suggest an ionic basis for 40-Hz oscillations, and the characteristic behavior of mixedmode bursting in these models was introduced (Wang, 1993). Even more recently, the contribution of slowly inactivating potassium currents was examined in a model of cellular short-term memory (Delord et al., 2000). In an attempt to penetrate the complex issue of interneuronal heterogeneity, we present a focused study of the contribution of slowly inactivating potassium currents to interneuronal dynamics. There is much evidence for slowly inactivating potassium currents of various and di¡erent biophysical characteristics in neocortical and hippocampal neurons (Du et al., 1996; Grissmer et al., 1994; Joho et al., 1999; Lenz et al., 1994; Lu«thi et al., 1996; Storm, 1988). Such works, together with results from previous computational and experimental studies, described above, suggest functionally distinct roles for interneurons. In this work, we examine single- and multi-compartment models of interneurons containing slowly inactivating currents with di¡erent channel kinetic characteristics.

EXPERIMENTAL PROCEDURES

Single-compartment interneuron model This model consists of a single-compartment cell and is created using the software package NEURON (Hines and Carnevale, 1997). The ion channels are limited to the traditional Hodgkin^Huxley (HH) sodium, INa , and potassium, IK , currents, a persistent sodium current, INap , and a slowly inactivating potassium current, IKsi .

db ¼ ðbr 3bÞ=d b dt

ð3Þ

dh ¼ ð P ðK h ð13hÞ3 L h hÞ dt

ð4Þ

dn ¼ P ðK n ð13nÞ3 L n nÞ ð5Þ dt 2 2 where C = 1 WF/cm , Iext (WA/cm ) is the injected current, V is the voltage and t is the time. Parameters for the HH and leak currents are taken from Wang and Buzsa¤ki (1996) where they describe hippocampal interneurons. The leak current has a conductance gL = 0.1 mS/cm2 and a reversal potential VL = 365 mV. The sodium channel has a maximal conductance gNa = 35 mS/cm2 and a reversal potential VNa = 55 mV. The sodium channel activation variable m, is considered fast and can be replaced by its steady-state value mr = Km /(Km +Lm ), where Km = 30.1(V+35)/{exp[30.1(V+35)]31} and Lm = 4 exp(3(V+ 60)/18); h is the sodium channel inactivation where Kh = 0.07 exp[3(V+58)/20] and Lh = 1/{exp[30.1(V+28)]+1}. The delayed recti¢er potassium channel has a maximal conductance gK = 9 mS/cm2 and a reversal potential VK = 390 mV. n is the activation variable where Kn = 30.01(V+34)/{exp[30.1(V+34)]3 1} and Ln = 0.125 exp[3(V+44)/80]; P = 5. INap and IKsi are adapted from Skinner et al. (1999). The maximal conductances of these currents are gNap = 0.1 mS/cm2 and gKsi = 20 mS/cm2 . The activation kinetics of the INap channel are considered fast and can be described by their steady-state behavior pr = 1/{1+exp[3(V+51)/5]}. The general form of the steady-state activation and inactivation for the IKsi channel are ar = 1/{1+exp[3(V+V1=2 act: )/5]} and br = 1/{1+exp[(V+ V1=2 inact: )/6]}. V1=2 act: and V1=2 inact: are the half-activation and half-inactivation values that are varied when considering di¡erent biophysical characteristics. Three di¡erent voltage ranges are considered (355/385 mV; 335/365 mV; 0/330 mV) (see Fig. 1). Unless stated otherwise, the activation and inactivation time constants for IKsi are 5 and 1500 ms, respectively, and variations are considered. Multi-compartment interneuron model This interneuron model consists of 12 compartments. Passive properties are chosen to approximate an oriens/alveus (O/A) hippocampal interneuron. The surface area for this model is taken from the literature (Morin et al., 1996) and the model consists of an axon, a soma and a dendrite (10 segments). The total surface area is 2.21U1034 cm2 . The input resistance of the cell is V290 M6 which approximately matches values measured experimentally (Morin et al., 1996). The calculated membrane time constant is V6 ms which also approximates experimental data (Lacaille et al., 1987; Lacaille and Williams, 1990). Although the passive properties are chosen to approximate an O/A hippocampal interneuron, the use of a highly simpli¢ed morphology implies that an exact comparison with speci¢c hippocampal interneurons should not be done at this stage. Rather, we use this multi-compartment interneuron model to investigate dynamic consequences of neuronal models with extended morphologies. The equations governing the currents used in this model are the same as those described above. Conductances are the same unless otherwise stated. Various distributions of currents are used (see Results). Simulations are performed with a 25-Ws time-step. Unless stated otherwise, initial membrane voltage (IMV) is set to 360 mV. Simulations are allowed to run until a stable ¢ring pattern is attained, i.e. transients have passed.

RESULTS

ð1Þ da ¼ ðar 3aÞ=d a dt

ð2Þ

In Tables 1 and 2, we present a summary of the behaviors observed in the single- and multi-compartment mod-

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Fig. 1. Steady-state activation and inactivation curves. (A) 8/m/- - -, hyperpolarized/middle/depolarized ranges for activation and inactivation curves of the IKsi current, respectively. (B) 8, the steady-state activation of the persistent sodium current INap .

els of the interneuron. Explanations and descriptions are given for various patterns and, in particular, a bistable tonic ¢ring pattern is uncovered for a given kinetic description and distribution of the slowly inactivating potassium current. Finally, given the correlation of the interneuronal signal with a slow population response, we describe the frequency dependence of the signal in this context. Single-compartment model With the inclusion of all four channels and with IKsi active at the hyperpolarized range the model produces a bursting signal (see Fig. 2A1 , A2 , top trace). As described in our previous model (Skinner et al., 1999), it is the interplay between IKsi and INap that allows bursting to occur. The slowly inactivating potassium current is necessary for the model neuron to produce a bursting signal. To contribute to the signal it must be su⁄ciently deinactivated. During the burst, the outward current, IKsi , increases with each spike while INap , an inward current, remains relatively the same (Fig. 2A2 , bottom trace). This eventually leads to the termination of the burst. During the interburst interval, the inward INap increases faster relative to IKsi , which leads to the initiation of the burst. Figure 2B1 , B2 , and B3 shows the response to the cell with injected current changes. The frequency within a burst increases linearly with the amount of current

Table 1. Single-compartment behaviors with variations in IKsi characteristics V1=2 act: /V1=2 inact:

Firing patterns

Spiking frequency

355/385 mV 335/365 mV 0/330 mV

bursting, 0.13^0.62 Hz bursting, 1^3.33 Hz tonic ¢ring

36.9^65.5 Hz 15.3^24.7 Hz 2.4^243 Hz

Summary of ¢ring patterns and frequencies for the single-compartment model with di¡erent activation/inactivation ranges for the IKsi current. V1=2 act: , half-activation voltage, V1=2 inact: , halfinactivation voltage.

injected into the cell (Fig. 2B1 , 8). The burst frequency, however, rises to a peak frequency of 0.62 Hz at Iext = 2.03 WA/cm2 and then gradually decreases with added injected current (Fig. 2B3 , 8). Finally, with large enough injected current, s 3.75 WA/cm2 , the cell saturates to a steady-state value at a depolarized level. Two dynamic processes are occuring concurrently as the amount of injected current is raised. The interburst time period decreases (Fig. 2B2 , 8, thinner line) and the duration of the burst increases (Fig. 2B2 , 8, thicker line). As current is injected incrementally, the resting membrane voltage is pushed closer to the spike threshold voltage which allows the cell to ¢re sooner. This therefore decreases the interburst time period. However, IKsi is a potassium channel that, when open, works to keep the membrane voltage at a hyperpolarized value. As we inject steady depolarizing current, more of the IKsi channels become inactivated, thereby allowing the membrane voltage to be more depolarized and the spiking to continue for a longer time period. As seen in Fig. 2B2 (8), for values of Iext 6 2.03 WA/cm2 , the interburst time period decreases at a faster rate than the increase rate of the burst duration. This results in a rise in the burst frequency. For Iext s 2.03 WA/cm2 the two rates are such that the interburst time period decreases at slower rate than the increase rate of the burst duration resulting in an overall decrease in burst frequency (Fig. 2B3 , 8). When the activation range for IKsi is set to the middle range, the cell still produces a bursting signal but this time, distinct subthreshold oscillations are seen in the interburst periods similar to that observed by Wang (1993) (data not shown). With IKsi active in the middle voltage range, the overall resting membrane voltage is more depolarized which allows the subthreshold oscillations to appear. As more current is injected, the bursts, whose frequency ranges from V1 to 3.33 Hz (Table 1), give way to tonic ¢ring as more of the IKsi channels become inactivated. The burst frequency is higher here, than when the IKsi is active in the more hyperpolarized range, because the burst duration is shorter and does not change signi¢cantly as more current is injected. At the same time, the interburst interval decreases with depola-

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F. Saraga and F. K. Skinner Table 2. Multi-compartment behaviors with variations in IKsi characteristics and location

V1=2 act: /V1=2 inact:

Case 1 S/A: INa , IK , INap , IKsi D: passive

Case 2 Case 3 Case 4 Case 5 S/A: INa , IK , INap , IKsi S/A: INa , IK , INap S/A: INa , IK , INap , IKsi S/A: INa , IK , INap D : INap , IKsi D: INap , IKsi D: INa , IK , INap , IKsi D: INa , IK , INap , IKsi

355/385 mV (hyperpolarized range) 335/365 mV (middle range) 0/330 mV (depolarized range)

bursting, 0.34^0.58 Hz

bursting, 0.3^0.64 Hz

bursting, 0.79^2.23 Hz

bursting, 1.3^2.6 Hz

bistable tonic ¢ring tonic ¢ring

tonic ¢ring

tonic ¢ring

tonic ¢ring

steady state

steady state

robust bursting, 1.06 Hz robust tonic ¢ring

robust bursting, 1 Hz tonic ¢ring

Summary of ¢ring patterns and frequencies for the multi-compartment model with di¡erent activation/inactivation ranges for the IKsi current. V1=2 act: = half-activation voltage, V1=2 inact: = half-inactivation voltage, S/A = soma/axon, D = dendrite. The term robust is used here to describe no change in the frequency with respect to injected current.

rization. The number of spikes per burst remains within a narrow range of 3^5 for all current injection values. This is in part due to the activation curves of IKsi and INap being relatively close which implies that as the resting membrane voltage is depolarized, the same proportion of channels for the two currents will be active. This shows the intricate balance possible between INap and IKsi to keep the number of spikes per burst relatively constant. The average spike frequency within the bursts ranges from V15.3 to 24.7 Hz (Table 1). Finally, when the depolarized activation range is incorporated for IKsi , the signal changes to spontaneous tonic ¢ring due to the highly depolarized membrane voltage (data not shown). The spontaneous ¢ring frequency of the neuron is V22.6 Hz and this frequency increases linearly with injected current, spanning from V2.4 to 243 Hz for injected current values of 30.01^0.7 nA (Table 1). In this voltage range, the slowly inactivating current does not contribute signi¢cantly to the output signal. Multi-compartment models Interneurons are not point processes, but have a rich morphology with channel proteins distributed along its surface. What channels and how they are distributed in the cell contributes to the various outputs that the cell could produce. We consider a simple 12-compartment representation of a CA1 interneuron in hippocampus with somatic, axonal and dendritic compartments, and ¢ve di¡erent distributions of currents (see Table 2). For all cases, the kinetics of the slowly inactivating potassium current are modi¢ed so as to allow the channel to be active at three di¡erent voltage ranges (see Fig. 1). Not surprisingly, the multi-compartment model is able to exhibit a wider range of dynamic behaviors as compared to the single-compartment model (see Tables 1 and 2). These behaviors include not only bursting and tonic ¢ring, but also bistability and robustness in ¢ring frequency with respect to injected currents, and steadystate or non-¢ring regimes. In other words, a distributed morphology expands the ¢ring pattern repertoire. Burst frequency ranges decrease and non-¢ring modes emerge in the extended morphology models. With the dendrite containing only a leak current and the soma/

axon containing all four channels described earlier (case 1), the cell displays similar ¢ring patterns and ranges of frequencies as those seen when only INap and IKsi are included in the dendrite (case 2). Moreover, these patterns are comparable to the single-compartment model situation with some di¡erences, detailed below. As in the single-compartment model, the multi-compartment model also produces bursting. Figure 2A3 (top trace) shows the bursting signal obtained in the multicompartment model when the slowly inactivating potassium current is active in the hyperpolarized range. All four currents are placed in the soma and axon while the dendrite only contains IKsi and INap (case 2) (Fig. 2A1 ). In this situation, the average spike frequency increases relatively linearly for lower injected current values, although with a more gradual slope than seen in the single compartment (Fig. 2B1 , - - -). The burst frequency rises to a peak of 0.635 Hz at Iext = 0.5 WA/cm2 and then decreases. Figure 2B2 shows the interburst (- - -, thinner line) and burst duration (- - -, thicker line) time periods. Similar reasoning to that used for the single-compartment model applies here to explain the rise and fall of the burst frequency seen in Fig. 2B3 (- - -). In general, the explanation underlying this bursting behavior is similar to that described for the single-compartment model (i.e. a balance between INap and IKsi , but note that the range of burst durations and interburst intervals has decreased due to the extended morphology). The e¡ect of the distributed morphology is also apparent in the decreased heights of the somatic spikes. They start at a more depolarized level and do not reach as high a depolarized level compared to the single-compartment case. In essence, this is due to shunting e¡ects of the axonal and dendritic compartments. Figure 2A3 (bottom trace) shows how the more depolarized dendrite (dashdot line) and hyperpolarized axon (gray line) contribute to the signal measured in the soma (black line). With IKsi active in the middle range and the same distribution of currents as described directly above (case 2), the signal is similar to that seen for the singlecompartment model. Subthreshold oscillations are again present during the interburst periods (data not shown). An interesting situation of robust bursting arises if now the dendritic compartments also include the HH sodium and potassium currents and IKsi is active in

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Multi-compartment interneuron model outputs

Fig. 2. Single-compartment and multi-compartment models can produce a bursting signal with IKsi active in the hyperpolarized range. (A1 ) Schematics of single-compartment and multi-compartment models with HH currents, persistent sodium current, and slowly inactivating potassium current shown placed in the compartments. (A2 ) Top trace, voltage recording from single-compartment model. Iext = 2.12 WA/cm2 , burst frequency = 0.60 Hz. Bottom trace, IKsi (thicker line) and INap (thinner line) current recordings. (A3 ) Top trace, voltage recordings from the soma (black line), axon (gray line), and dendrite (dashdot line) of the multi-compartment model. Iext = 0.14 WA/cm2 , burst frequency = 0.59 Hz. Bottom trace, individual action potentials shown on expanded time scale. (B1 ) Average spike frequency versus injected current, single-compartment model (8) and multi-compartment model (- - -). (B2 ) Burst/interburst duration versus current, single-compartment model (8), multicompartment model (- - -); thicker lines refer to burst duration, thinner lines refer to interburst durations. (B3 ) Burst frequency versus current, single-compartment model (8) and multi-compartment model (- - -).

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Fig. 3. Multi-compartment model displays a robust burst signal whose frequency is independent of the injected current amount when IKsi is active in the middle range and the HH currents are allowed in the dendrite. (A) Schematic of multicompartment model with distribution of channels shown. (B) Voltage recording from the soma. Iext = 1 WA/cm2 , burst frequency = 1.05 Hz. (C) IKsi (thicker line) and HH (thinner lines) current recordings from the soma.

the middle range. This includes cases 4 and 5, and case 4 is depicted in Fig. 3A. In this situation the burst frequency does not change with injected current, but remains the same at about 1 Hz (Fig. 3B). However, the amplitude of the burst decreases with the depolarizing e¡ect of more injected current. In these situations, the interplay between the HH currents (Fig. 3C, thin lines) and the IKsi current (Fig. 3C, thick line) becomes more important. The inactivation time constant of the IKsi channel, which is very slow (1500 ms), plays a critical role in establishing the burst frequency to be around 1 Hz since enough IKsi channels must be inactivated to allow the membrane to depolarize and reach spike threshold. For example, when the inactivation time constant is decreased from the original 1500 ms to 1200 ms, a 20% decrease, the burst frequency increases by V30%. The robustness of the burst frequency with respect to injected current amounts, is a result of the intricate balance and interplay of the intrinsic currents that make up the signal. This robust output is seen only in the extended multi-compartment model, suggesting that morphology may play an active role in the generation or stabilization of signals (Saraga and Skinner, 2002). Using our multi-compartment model we ¢nd that it is possible to obtain situations where the cell does not ¢re, but produces a steady-state signal regardless of the amount of injected current. This occurs when IKsi is active in the hyperpolarized range and the dendritic compartments include all four currents (cases 4 and 5). In these cases, IKsi is the dominant current keeping the membrane at a hyperpolarized potential and preventing

it from spiking at lower injected current values. As the cell becomes more depolarized with larger injected current values, IKsi contributes less to preventing the cell from spiking, but then the HH sodium current is too inactivated to allow a spike to occur. If this were a single-compartment case, bursting would occur, but the distribution morphology of the multi-compartment model produces enough of a shunting to prevent the cell from spiking. Burst signal dependence on conductance and time constants of the slowly inactivating potassium current. There are four distributions for which the model cell produces a bursting signal. For each of these cases, the IKsi conductance changes to either half or double the original value keeping all other parameters the same. These changes result in a loss of the bursting signal. Given that the burst behavior requires a balance between INap and IKsi currents, and we did not change INap , this is not completely surprising. If there is too little IKsi current, the HH currents will dominate and the cell will ¢re regularly. If there is too much of the current, it will hold the membrane at a hyperpolarized value and therefore not allow the cell to spike. However, it is interesting to note that the range for gKsi , where bursting is possible, is similar in all four cases examined (average: 16^26 mS/ cm2 ). This suggests that there may be optimal conductance values for bursting behavior to occur. It is clear that the presence of bursting in our compartmental models depends on appropriate balances between intrinsic currents (INap , IKsi ). Besides variations

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Fig. 4. Robust bursting is replaced by subthreshold oscillations in the multi-compartment model when the IKsi activation time constant da is reduced from 5 to 3 ms. (A) Voltage recording from the soma. (B) IKsi (thicker line) and INap (thinner line) current recordings from the soma.

in activation and inactivation voltage ranges, di¡erent activation and inactivation time constants for slowly inactivating potassium channels have been reported in the literature. The activation, da , and inactivation, db , time constants of the IKsi are modi¢ed to see the e¡ect on bursting frequency. In general, shorter time constants increase burst frequencies and longer time constants decrease burst frequencies. For example, when da or db are shortened from their original values, 5 and 1500 ms, to 3 and 1200 ms, respectively, burst frequency increases by V11^55% for the same amount of injected current. When da or db are lengthened to 8 and 1800 ms, respectively, burst frequency decreases V8^59%. However, these changes lead to di¡erent non-linear interactions and in some cases, bursting can be lost completely. For example, the robust bursting seen in cases 4 and 5 (middle range) are lost and replaced by subthreshold oscillations (Fig. 4A) when the activation time constant, da is reduced to 3 ms (all other parameters remaining the same). This shortening of the activation time constant allows the IKsi current to activate more quickly and become the dominant hyperpolarizing current (Fig. 4B). This strong pull towards a negative voltage is o¡set partly by the HH currents, causing the subthreshold oscillations. Tonic ¢ring and bistability in the extended model. The cell regularly ¢res when the IKsi current is active in the depolarized range for all cases. For case 4, we see a robust tonic ¢ring that is independent of the amount of injected current imposed on the cell (Fig. 5A). The HH currents are the main contributors to the signal without much in£uence from IKsi while INap channels remain open longer at more depolarized values and therefore work to keep the membrane voltage at a

more depolarized level to in£uence the shape of the signal (Fig. 5A, B). The cell also produces a tonically ¢ring signal when IKsi is restricted to the dendrites (case 3: hyperpolarized and middle range). In these cases, without the HH currents included in the dendrites, IKsi is not large enough to counter the persistent sodium current in order to produce a bursting signal, but it is large enough to have some in£uence over the frequency of the ¢ring resulting in bistable tonic ¢ring (see below). A bistable tonic ¢ring signal emerges when IKsi is used with the hyperpolarized activation/inactivation range (case 3: hyperpolarized range). This bistability can be uncovered by initiating the membrane at di¡erent voltages (Fig. 6). If the membrane voltage is initiated at a depolarized value for which IKsi is mostly inactivated ( s 355 mV), the dendrite is seen to oscillate with a small amplitude around a depolarized value (V320 mV) (Fig. 6B1 dendrite: dash-dot line). Figure 6B2 shows how INap in the dendrite (thinner line) is much larger than IKsi (thicker line) resulting in depolarization of the dendrite. If the voltage is initiated at a su⁄ciently hyperpolarized voltage ( 6 355 mV) which allows IKsi to be active (due to removal of inactivation), then the signal in the dendrite is seen to oscillate around a hyperpolarized voltage (V365 mV) (Fig. 6C1 dendrite: dash-dot line). Figure 6C2 shows how IKsi (thicker line) dominates over the INap current (thinner line) resulting in the hyperpolarization of the dendrite. This more negative dendrite acts as a shunt on the soma thereby slowing the frequency measured there. Figure 6A shows the frequency response of the neuron in these two di¡erent stable states. With suf¢cient injected current, the two states merge to become one state. This happens when the membrane is su⁄ciently depolarized so that the IKsi channels become completely inactivated and the dendrite is no longer

Fig. 5. When all four channels are placed in all the compartments and IKsi is active in the depolarized range, the model produces a robust tonic ¢ring pattern. Iext = 0.045 WA/cm2 (black lines), Iext = 0.223 WA/cm2 (gray lines). (A) Voltage recordings from the soma. (B) Current recordings from the soma. Top trace, IKsi ; bottom trace, INap .

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held at a hyperpolarized value. At that point, the bistable behavior is lost and only one stable behavior is produced. This monostable signal output in the soma, axon and dendrite resembles that seen in Fig. 6B1 .

DISCUSSION

Synchronized output from populations of interneurons are known to sculpt pyramidal cell behavior, which is then seen as oscillations of di¡erent frequencies at the EEG level. GABAergic interneurons in the hippocampus exert strong inhibition onto excitatory pyramidal cells via dendritic and somatic synapses. Interneurons have extensive axon arbors and a single interneuron can make contact with over 1000 pyramidal cells so that an individual interneuron can phase-lock the ¢ring of multiple pyramidal cells, by synchronously depressing their

activities (Buhl et al., 1994; Cobb et al., 1995; Freund and Buzsa¤ki, 1996; Csicsvari et al., 1998; Sik et al., 1995). Therefore, individual interneurons capable of producing signals such as bistable tonic ¢ring or bursting (see Sik et al. (1995) for in vivo bursting behavior of interneurons), can strongly in£uence the generation of di¡erent rhythmic outputs at the network level. In this paper, we have focused on how the dynamic output produced by a multi-compartment model of a hippocampal interneuron is in£uenced by slowly inactivating potassium currents of di¡erent properties. Properties of slowly inactivating potassium currents Experimental work shows that hippocampal interneurons have di¡erential distributions of currents, in particular potassium channels. Potassium channels contribute to the diversity of spiking patterns that are seen in neu-

Fig. 6. IKsi restricted to the dendrites of the multi-compartment model produces a bistable tonic ¢ring output that is dependent on the IMV. (A) Frequency of the tonic ¢ring versus injected current ; 8, IMV set to 345 mV; - - -, IMV set to 365 mV. (B1 ) Voltage recording from soma (black line), axon (gray line), and dendrite (dash-dot line) when the membrane was initialized at 345 mV. (B2 ) IKsi (thicker line) and INap (thinner line) recordings from the dendrite. (C1 ) Voltage recordings from soma (black line), axon (gray line), and dendrite (dash-dot line) when the membrane was initialized at 365 mV. (C2 ) IKsi (thicker line) and INap (thinner line) recordings from the dendrite.

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rons. They help to shape the postsynaptic response, determine the rate of spike repolarization, and allow for adaptation of repetitive ¢ring (Storm, 1993). Specifically, evidence for slowly inactivating potassium currents in interneurons exists (Chikwendu and McBain, 1996; Du et al., 1996; Parra et al., 1998; Zhang and McBain, 1995b). There are at least three di¡erent slowly inactivating potassium currents that have been found in neurons in the hippocampus: ID (Storm, 1988), IKðslowÞ (Lu«thi et al., 1996), and Kv3.1 (Massengill et al., 1997; Joho et al., 1999; Grissmer et al., 1994). Although they are all slowly inactivating currents they di¡er with respect to their absolute sensitivities to tetraethyl-ammonium (4-AP), activation ranges, sensitivities of 4-aminopyridine (TEA), and rates of removal of inactivation. As well, some of these channels have been found in speci¢c interneuron types. The protein Kv3.1b, which makes up part of the Kv3.1 channel, is expressed in parvalbumincontaining interneurons, but is not found in somatostatin-positive interneurons in the hippocampus (Du et al., 1996). Kv3.1 is associated with short-duration action potentials, fast after-hyperpolarization, brief refractory periods, and high-frequency ¢ring (Joho et al., 1999). It requires membrane voltages of s 310 mV to be activated and seems to contain a component that never demonstrates time-dependent inactivation (Du et al., 1996). ID and IKðslowÞ both activate at a more negative membrane voltage, 355 and 365 mV, respectively (Storm, 1993; Lu«thi et al., 1996), and inactivate much more slowly, ID : 754 ms and IKðslowÞ : 7.5 s (Lu«thi et al., 1996). These currents are associated with a delay in the onset of discharge as well as participating in spike repolarization. Neurons that contain the ID current were found to discharge in groups of spikes (Parra et al., 1998) implying that this current may be necessary for a bursting behavior. Recently, a slowly deinactivating potassium current has been found in GABAergic neurons of the amygdala (Royer et al., 2000). It allows the cell to produce a bistable ¢ring pattern that is dependent on the membrane voltage similar to that seen in our model. The ability of the cell to regularly ¢re or remain quiet depends on the degree of activation of this slowly deinactivating current. Related model studies Wang (1993) was the ¢rst to consider the contributions of a slowly inactivating potassium current by incorporating a biophysical model into a pyramidal neuron model in cat sensorimotor cortex. He found that the interplay between a persistent sodium current and the slowly inactivating potassium current was able to produce 10^50-Hz intrinsic oscillations, suggesting a cellular origin for this neuronal rhythmicity. Wang also noted an interesting characteristic displayed by his single-compartment model cells: mixed-mode bursting in which clusters of spikes alternated with small subthreshold oscillations. Given the similarity of our mechanisms, we also observed this mixed-mode bursting in our single-compartment as well as our multi-compartment model cells. However, in our study we focused on slowly inactivating

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potassium currents in di¡erent ranges of voltage activation and used multi-compartment models and di¡erential distributions of currents. We previously considered two-cell networks of interneurons that were coupled by mutual inhibition and gap junctions. In that study, we also only used single-compartment models and our individual cell models did not exhibit bursting activity individually with the choice of parameters. The HH sodium current (based on Wang, 1993) had slightly more depolarized threshold than what was used here and a slowly inactivating potassium current active in the hyperpolarized range was used. The focus there was on network aspects of having both gap junctions and GABAergic synapses to maintain rhythmic behavior. A bistable ¢ring pattern as obtained in our model has been seen in other models. Booth and Rinzel (1995) found a dendritic origin for bistability in ¢ring frequency in a two-compartment motoneuron model. Their model did not contain the slowly inactivating current, but instead contained a dendritic non-inactivating calcium L-type current which was responsible for the bistability by allowing the dendrite to generate plateau potentials. This bistability was dependent on the coupling strength between the soma and dendrite. Their plateau potentials resemble the depolarized dendritic oscillations observed in our model (case 3, hyperpolarized range). Similarly, plateau or depolarized potentials in the dendrite resulted in higher frequencies measured in the soma. Network dynamics, information coding and functional aspects Given the diversity of interneurons present in brain, it is important to ask precise questions about the role of speci¢c inhibitory interneurons in generating brain output (McBain and Fisahn, 2001). We know that interneurons exhibit a varied dynamic pro¢le (Parra et al., 1998; Freund and Buzsa¤ki, 1996), which naturally depends on the distribution and characteristics of channels on its surface. How information is coded in the brain is not well understood (Rieke et al., 1997), but it clearly depends on the speci¢c dynamic patterns produced by neurons. One typically invokes rate or temporal coding strategies, but it is also possible that a bursting signal could be conveying a particular type of information distinct from isolated spikes (Lisman, 1997). In addition, the brain may use more sophisticated ways of conveying information such as periodically changing excitabilities of pyramidal cells due to di¡erent dynamical activities in interneurons as suggested by Buzsa¤ki and colleagues (Csicsvari et al., 1999). Whichever situation may be correct, it is reasonable to consider that an understanding of interneuronal dynamics will give us insight into functional aspects of brain behavior given the critical role interneuronal networks are believed to play (Buzsa¤ki and Chrobak, 1995; McBain and Fisahn, 2001). From several theoretical works, it is clear that network behavior is critically dependent on the intrinsic properties of its components (Marder and Calabrese, 1996;

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F. Saraga and F. K. Skinner

Skinner et al., 1994; Wang and Rinzel, 1992; White et al., 1998). For example, seminal work by Wang and Rinzel (1992) showed that synchrony was possible in purely inhibitory networks if there was an appropriate balance between intrinsic and synaptic properties. Pinsky and Rinzel (1994) pioneered the use of a ‘ping-pong’ model of an individual neuron to exhibit bursting. This model had two compartments to separate somatic and dendritic components which included active components. They also considered larger networks of these cells, showing how characteristics of the individual cell a¡ected network behavior. Summary, limitations and closing Our initial motivation for this work came from earlier modeling work of a slowly inactivating potassium current providing a mechanism by which an experimentally observed slow rhythm could be produced (Skinner et al., 1999), the di¡erent observed characteristics of slowly inactivating potassium channels, and the emergence of works describing detailed di¡erences in interneurons (McBain and Fisahn, 2001). We found several combinations of channel distributions that allow for the dynamics of bursting in the observed slow-frequency (approximately 1 Hz) range. Speci¢cally, these results show us what types of kinetics are necessary for the slowly inactivating potassium channel in order to obtain the desired signal, and what sorts of channel distributions might facilitate this output. It is di⁄cult to determine distributions of channel types located in dendrites and so suggestions of distribution pro¢les are particularly helpful. Although we have not done exhaustive parameter var-

iations, we have clearly illustrated how the dynamic range of behaviors that can be expressed by this ‘simple’ interneuronal model can be understood when the location and properties of a particular current are varied. We have now formulated a more detailed multi-compartmental model of a particular O/A hippocampal interneuron for further investigation (Saraga et al., 2000; Saraga, 2001). Traub (1995) and Traub and Miles (1995) were the ¢rst to develop multi-compartment models of interneurons in which active dendrites and electrical coupling was examined. However, the focus there was not on the mechanisms by which intrinsic properties might contribute to the variety of possible dynamical outputs in interneurons. In closing, this work suggests that it might be possible to link an understanding of interneuronal heterogeneity to its dynamics which are less diverse than its various currents and distributions, its neuromodulatory responses, its morphologies and biochemical sensitivities. These dynamics, in turn, would dictate network output and thus possible functional and behavioral aspects. As suggested previously and emphasized recently (Zhang and McBain, 1995a; McBain and Fisahn, 2001), this work illustrates why detailed investigations of the underlying currents (and other properties) need to be undertaken to get a handle on the complex roles of interneurons.

Acknowledgements+This work was supported by CFI/ORDCF, MRC/CIHR and DCIEM of Canada. F.K.S. is an MRC Scholar and a CFI Researcher. F.S. was supported by OGSST and NSERC student awards.

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