Active dendrites and spike propagation in multicompartment models of oriens-lacunosum/moleculare hippocampal interneurons

Authors

  • F. Saraga,

    Corresponding author
    1. Toronto Western Research Institute, University Health Network, Toronto, Ontario, Canada M5T 2S8
    2. Departments of Physiology, University of Toronto, Toronto, Ontario, Canada M5T 2S8
    • Corresponding author
      F. Saraga: Toronto Western Research Institute, University Health Network, 399 Bathurst Street, MP13-308, Toronto, Ontario, Canada M5T 2S8. Email: fernanda.saraga@utoronto.ca

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  • C. P. Wu,

    1. Toronto Western Research Institute, University Health Network, Toronto, Ontario, Canada M5T 2S8
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  • L. Zhang,

    1. Toronto Western Research Institute, University Health Network, Toronto, Ontario, Canada M5T 2S8
    2. Departments of Medicine (Neurology), University of Toronto, Toronto, Ontario, Canada M5T 2S8
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  • F. K. Skinner

    1. Toronto Western Research Institute, University Health Network, Toronto, Ontario, Canada M5T 2S8
    2. Departments of Medicine (Neurology), University of Toronto, Toronto, Ontario, Canada M5T 2S8
    3. Departments of Physiology, University of Toronto, Toronto, Ontario, Canada M5T 2S8
    4. Departments of Institute of Biomaterials and Biomedical Engineering, University of Toronto, Toronto, Ontario, Canada M5T 2S8
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Abstract

It is well known that interneurons are heterogeneous in their morphologies, biophysical properties, pharmacological sensitivities and electrophysiological responses, but it is unknown how best to understand this diversity. Given their critical roles in shaping brain output, it is important to try to understand the functionality of their computational characteristics. To do this, we focus on specific interneuron subtypes. In particular, it has recently been shown that long-term potentiation is induced specifically on oriens-lacunosum/moleculare (O-LM) interneurons in hippocampus CA1 and that the same cells contain the highest density of dendritic sodium and potassium conductances measured to date. We have created multi-compartment models of an O-LM hippocampal interneuron using passive properties, channel kinetics, densities and distributions specific to this cell type, and explored its signalling characteristics. We found that spike initiation depends on both location and amount of input, as well as the intrinsic properties of the interneuron. Distal synaptic input always produces strong back-propagating spikes whereas proximal input could produce both forward- and back-propagating spikes depending on the input strength. We speculate that the highly active dendrites of these interneurons endow them with a specialized function within the hippocampal circuitry by allowing them to regulate direct and indirect signalling pathways within the hippocampus.

A fundamental issue in neuroscience is to understand how synaptic and intrinsic properties interact and integrate to produce neuronal output that is appropriate to its functional role. Interest in GABAergic cortical and hippocampal interneurons has grown in recent years due to the discoveries that these cells do not just provide simple inhibition, but also have a large impact on network dynamics and population signal generation (Ylinen et al. 1995; Buzsáki, 2001; Wu et al. 2002). Although interneurons only make up about 10-20 % of the neurons in the neocortex or hippocampus, they are an extremely heterogeneous population and it is unclear how best to classify and thus understand their diverse profiles. These differences in morphologies of axonal and dendritic arbors, electrophysiological responses, ion channel distribution and kinetics, neuromodulatory responses and neurochemical content suggest functionally distinct roles for interneurons (Freund & Buzsáki, 1996; Parra et al. 1998; McBain & Fisahn, 2001).

To understand these functional roles, we need to focus on the specific characteristics of different interneurons. For example, it has recently been shown that long-term potentiation (LTP) can be induced specifically on the stratum oriens-lacunosum/moleculare (O-LM) interneuron (Perez et al. 2001). The O-LM interneuron is located in hippocampus CA1 and its cell body and dendritic tree lie horizontally in the oriens stratum while the axon arborizes in the lacunosum/moleculare strata. Furthermore, sodium channel density measured in the dendrites of hippocampal O-LM interneurons is almost double that found in CA1 pyramidal cells and almost triple that found in neocortical neurons (Martina et al. 2000; Migliore & Shepherd, 2002). Considering the distinctness of the input and output orientation of this cell and its feedback inhibition of pyramidal neurons in the CA1 circuitry, what might the functional implications of these highly active dendrites be? The role of O-LM interneurons in CA1 network activities remains to be clearly defined.

Given how widespread the dendrites of interneurons are in the local circuitry of the cortex and hippocampus, the dendritic arbors of interneurons cannot be ignored. However, recording from dendrites of pyramidal cells is difficult, and even more so for interneurons. Thus, the creation and use of multi-compartment models is required. Indeed, our understanding of dendritic function has been greatly enhanced by theoretical and experimental interactions (Segev et al. 1995; Segev & London, 2000). Multi-compartment models of neocortical and hippocampal pyramidal cells and motoneurons have been created to explore the issues of spike initiation and back-propagating signals (Warman et al. 1994; Migliore et al. 1995; Paréet al. 1998; Lüscher & Larkum, 1998; Stuart & Spruston, 1998). Although there are some interneuron models available, these are limited to single-compartment (Skinner et al. 1994, 1999; Wang & Buzsáki, 1996) or general multi-compartment models (Traub & Miles, 1995; Emri et al. 2001; Saraga & Skinner, 2002). To date, there are no multi-compartment models of any interneuron subtypes that incorporate intrinsic properties (passive and active) specific to them.

Much experimental work has been performed on the O-LM interneuron (Lacaille et al. 1987; Lacaille & Williams, 1990; Zhang & McBain, 1995a,b; Maccaferri & McBain, 1996a; Martina et al. 2000). This, coupled with the seemingly special characteristics of the OL-M interneuron of LTP induction and active dendrites, make it a prime candidate for the building of multi-compartment models and thus for allowing investigations of its computational characteristics and potential functional roles. In this study we have created a detailed multi-compartment model of an O-LM hippocampal interneuron which includes an appropriate morphology, experimentally measured ion channel distributions and densities, channel kinetics and passive properties. Investigating this model provides us with an understanding of how this interneuronal subtype integrates its intrinsic and synaptic properties and suggestions of the functionality of its highly active dendrites.

METHODS

Model neuron

Morphology

The modelled cell is an O-LM interneuron recorded from the hippocampus of wild-type c57bl6 mouse (16 days old) (see Fig. 1A). In a whole hippocampus preparation (modified from Khalilov et al. 1997), perfused in vitro at room temperature (21-22°C) the interneuron was whole-cell-dialysed with a patch pipette solution containing 0.5 % neurobiotin (Vector Labs, Burlington, ON, Canada). The whole hippocampus was kept in the recording chamber and perfused for another 30-40 min to wash out any leaked neurobiotin and to allow intracellular distribution of neurobiotin in the recorded cell. The whole hippocampus was then fixed overnight in 4 % paraformaldehyde in 0.1 m phosphate buffer and resectioned to 100 μm. The sections were treated with an avidin-biotin-peroxidase complex (ABC Kit, Vector Labs) and rinsed and reacted with diaminobenzidine tetrahydrochloride and H2O2. The sections were mounted on glass slides. The prepared slide was examined under a × 60 objective lens (Nikon Eclipse E800 light microscope) and photographed using a CCD camera connected to a computer. Images were saved as TIFF files using the software package Image Pro Plus 4.1. Measurements of diameter and length were taken from the images using the software package Scion Image (http://www.scioncorp.com/). The 196-compartment model has eight main dendrites stemming from the soma which subdivide further into other branches (see Fig. 1B). The diameters of the dendrites span from 1.38 μm for the proximal dendrites to 0.61 μm for the distal dendrites. The entire dendritic tree lies within a radius of ≈300 μm from the soma. The surface area of this model is 9.884 × 10−5 cm2. The morphology, ion channels and synapses were modelled using the software package Neuron version 5.1 (Hines & Carnevale, 1997) running on a Linux OS. Despite the large variability among individual interneuronal groups, there appears to be distinct dendritic and cable properties leading to characteristic patterns of signal propagation (Emri et al. 2001). Therefore, the modelling of one neuron is not unreasonable and the conclusions found here can represent general properties for this interneuronal subpopulation.

Figure 1.

Oriens-lacunosum/moleculare (O-LM) hippocampal interneuron

A, sketch of the O-LM interneuron that was reconstructed from superposition of four 100 μm slices that contained the cell. The soma and dendrites of this cell are oriented horizontally in the stratum oriens of the hippocampus. View seen here is from above the hippocampus onto the oriens layer. B, 196-compartment model created using the software package Neuron (Hines & Carnevale, 1997) based on the measured diameters of the dendrites and soma from the cell in A.

Passive properties

Various values for the passive properties were explored within physiologically relevant ranges (membrane capacitance (Cm), 0.5-1.5 μF cm−2; axial resistivity (Ra), 100-300 Ω cm; leak conductance (gL), 10-50 μS cm−2; leak reversal potential (EL), −80 to −50 mV (Mainen & Sejnowski 1998) in order to match the experimentally measured input resistance (Rin) and membrane time constant (τm) in O-LM interneurons (Morin et al. 1996)). The following values were used for the passive properties: Cm, 1.3 μF cm−2; Ra, 150 Ω cm; gL, 50 μS cm−2; EL, −70 mV. These choices result in Rin of 215 MΩ and τm of 23.3 ms for the model interneuron, which match the values for these parameters of Morin et al. (1996).

Ion channels

Ion channels for the O-LM interneuron model were taken from the literature specific to this cell type. They include the traditional Hodgkin-Huxley (HH) sodium (INa) and delayed-rectifier potassium (IK) channels, the transient potassium channel (IA) and the hyperpolarization-activated channel (Ih). The somatic (s) and dendritic (d) compartments of the model interneuron obey the following current balance equations:

display math(1)
display math(2)

where C is the membrane capacitance (see above), Iext is the external or imposed current, Vs and Vd are the somatic and dendritic voltages, respectively, and t is the time. In eqn (2), the dendritic compartment is shown with only HH (INa and IK) channels, but other combinations were explored (see Results). In the Appendix, details of the channel kinetic equations and parameter values are given. Figure 2 illustrates the activation/inactivation curves and time constant curves for all currents used.

Figure 2.

Activation and inactivation curves and their time constants

Plot of the activation and inactivation curves (A, C, E and G) and their time constants (B, D, F and H) for the model ion channels. Equations for these curves and definitions for the symbols are given in the Appendix. Continuous lines, somatic channels; dotted lines, dendritic channels.

Model axon

A simple representation of an initial segment and axon was included in some simulations. The dimensions for these sections were taken from Traub & Miles (1995) (initial segment: length, 75 μm; diameter, 2 μm; axon: length, 75 μm, diameter, 1 μm). Passive properties were kept identical to the rest of the model neuron. Experimentally it has been shown that most axons do not stem from the soma body, but rather from a dendrite up to 110 μm away from the soma (Martina et al. 2000). We thus chose our model axon to stem from a dendrite ≈40 μm from the soma (see Fig. 8). The axon contains both sodium and potassium channels with kinetics as described previously for the somatic channels. Various conductance values for the axonal channels were explored (see Results).

Figure 8.

Models with added axonal segment

Schematic diagram shows the multi-compartment model with the addition of a 150 μm initial segment and axon in green. A, Case 2 model (no IA in the dendrites). The conductance values for the sodium and potassium current are 1.5 times the conductance values used in the soma (gNa,axon, 160.5 pS μm−2; gK,axon, 478.5 pS μm−2) Left-hand panel, a long/small depolarizing current (100 ms, 0.1 nA) is injected into the soma. The axon-bearing dendrite (in red) spikes first followed by the soma (in black) and the axon-lacking dendrite (in blue) with 0.01 ms delay between each spike peak. Right-hand panel, a short/large depolarizing current (0.1 ms, 10 nA) is injected into the soma. There is a reversal of spike order with the axon-lacking dendrite spiking first followed by the soma and the axon-bearing dendrite with 0.01 ms delay between each spike peak. A hyperpolarizing sustained current of −0.05 nA was injected into the soma to suppress spontaneous firing. B, Case 3 proximal model (27 % IA in the proximal dendrites). Left-hand panel, a long/small depolarizing current (100 ms, 0.2 nA) is injected into the soma. The axon-bearing dendrite (in red) spikes first followed by the soma (in black) and the axon-lacking dendrite (in blue) with 0.01 ms delay between each spike peak. Right panel, a short/large depolarizing current (0.1 ms, 12 nA) is injected into the soma. There is a reversal of spike order with the axon-lacking dendrite spiking first followed 0.03 ms later by both the soma and the axon-bearing dendrite simultaneously. The large and small insets show the spike peak and upstrokes, respectively, on an expanded time scale.

Model synapse

Some simulations included excitatory synapses that are based on simple first-order kinetics and are taken from models for AMPA synapses already created by Destexhe et al. (1994a,b).

display math(3)
display math(4)

where α is 1.1 × 106m−1 s−1, β is 190 s−1 and EAMPA is 0 mV. s is the synaptic activation variable and the transmitter concentration (T) is 1 mm. A presynaptic spike generator included in the Neuron package then imposes a transmitter release frequency onto the postsynaptic cell.

Simulations were performed with a 25 μs time step. Smaller time steps were tested to ensure numerical accuracy. Initial membrane voltage was set to −60 mV for all cases. Simulations were allowed to run until a steady firing pattern or steady-state voltage was attained.

RESULTS

Multi-compartment models capture the essence of the O-LM interneuron

We investigated distributions of channels that consist of four channel types (INa, IK, IA and Ih). Although sodium and potassium conductances have been measured in the dendrites of O-LM cells, other details are unclear. Furthermore, what and how many distal ion channels are present is unknown. We explored various dendritic channel distributions and densities. We found a particular combination that best matches the experimental data, but that does not include IA in the dendrites. Since there is experimental evidence for IA in the dendrites, we further explored a number of dendritic IA scenarios. We then explored issues of signal initiation and propagation with our multi-compartment models.

Exploring different dendritic channel distributions

We explored eight combinations of dendritic channels (see Table 1). These different cases include passive dendrites, the HH currents, the HH currents combined with IA or Ih alone, IA and Ih together or graded versions of IA and Ih based on channel densities in pyramidal cells (Hoffman et al. 1997; Magee, 1998, 1999). The graded Ih and IA channels increase in density every 100 μm along the dendritic tree by 37 pS μm−2 and 110 pS μm−2, respectively (see Appendix, Table 3). Ion channel content and density in the soma was kept constant for all simulations and contained INa,s, IK,s, Ih and IA with densities as given in the Appendix.

Table 1. Dendritic channel combinations
CaseDendritic channels
  1. The soma contained INa,s, IK,s, IA, Ih for all combinations tested. The kinetics and conductances for these channels are described in the Appendix. Only Cases 2,3 and 7 (shown in bold) could produce repetitive firing in response to tonic input. The remaining cases produced only transient firing responses.

1Passive
2 I Na,d, IK,d
3 I Na,d, IK,d, IA
4 I Na,d, IK,d, IA(graded)
5 I Na,d, IK,d, Ih
6 I Na,d, IK,d, Ih (graded)
7 I Na,d, IK,d, IA, Ih
8 I Na,d, IK,d, IA(graded), Ih(graded)
Table 3. I A and Ih channel conductances based on distances from soma
Channel conductance< 100 μm100–200 μm200–300 μm
  1. Values were taken from Hoffman et al. (1997) and Magee (1998, 1999).

g A 165 PS μm−2275 PS μm−2385 PS μm−2
g h 13.85 PS μm−251 PS μm−288 PS μm−2

Of all combinations of channels explored, only three produce repetitive firing when tonic current is injected into the soma (see Table 1). One is Case 2 (see Fig. 3), with INa,d and IK,d in the dendrites, while the other two include adding IA alone (Case 3) or both IA and Ih (Case 7) to the dendritic HH channels. Furthermore, the latter two distributions of channels do not produce spontaneous firing (i.e. with no injected current). The remaining five cases result in either a steady voltage output for all levels of injected current, or show a transient spiking response that is not sustained.

Figure 3.

Characteristic electrophysiological response of O-LM interneurons

A, experimentally measured voltage response of an O-LM interneuron to 0.1 nA of injected current in the soma. (Reprinted with permission from Martina et al. (2000)Science287, 295-300. Copyright 2000 of American Association for the Advancement of Science.) B, voltage response of model interneuron to 0.1 nA of injected current into soma. C, frequency-response of model interneuron to different injected current values into the soma. In ascending order, the current pulses of 200 ms are −0.5, −0.3, −0.1 and 0.1 nA. A sustained current of −0.05 nA for 500 ms was applied before the current pulses to suppress the spontaneous firing of the model interneuron. The model interneuron spontaneously fires (0 nA of current) at ≈12 Hz. The characteristic ‘sag’ due to the activation of the Ih channel can be seen for negative current pulses. D, frequency-response curve for model interneuron to somatically injected current of different values. This model interneuron corresponds to Case 2 in Table 1.

The distribution that best matches the electrophysiological data is Case 2. This is the distribution of channels that consists of all four ion channel types (INa,s, IK,s, IA and Ih) in the soma, while the dendrites have two channel types (INa,d and IK,d) (see Fig. 2). Our model interneuron spontaneously fires at ≈12 Hz with no injected current (Fig. 3D). Although not a general property of interneurons, O-LM interneurons are known to spontaneously fire at ≈5-20 Hz (Lacaille et al. 1987; Maccaferri & McBain, 1996a; Ali & Thomson, 1998). The model interneuron continues to match experimental recordings when 0.1 nA of sustained current is injected into the soma. The response of the model is very similar in shape, frequency and amplitude to action potentials measured experimentally in this cell type (see Fig. 3A and B). Figure 3C shows how the model replicates the ‘sag’ in voltage at negative current injection values that is characteristic for the O-LM interneuron (Maccaferri & McBain, 1996a). This ‘sag’ is due to the activation of the Ih channel. Figure 3D shows the frequency-response curve for the model interneuron. This response curve closely matches the response curve found experimentally by Maccaferri & McBain for this cell (see Fig. 1C in Maccaferri & McBain, 1996a).

The hyperpolarization-activated inward current Ih is found specifically in O-LM interneurons (Maccaferri & McBain, 1996a). Even though the conductance value for the Ih current is small, this current does affect the spiking response of the model interneuron. In particular, it counters the hyperpolarizing effect of the IA channel. If Ih is removed from the model entirely, and all other parameters are kept the same, the cell cannot fire spontaneously. It has been suggested that Ih plays a role in pacemaker potentials (Maccaferri & McBain, 1996a) and therefore is an important channel for the spontaneous firing of a cell.

While it is satisfying to determine a model that closely matches the experimental data, this does not mean that our Case 2 model is the only model choice for an O-LM cell. Rather, we would say that with the available experimental data, some essence of the O-LM cell is captured with this Case 2 model. At this point, it is important to note that even though available experimental data were used, several details are unknown. For example, although sodium and potassium conductances have been measured in the dendrites of O-LM interneurons, details of their time constants and voltage dependencies are unknown. We took these values from elsewhere. In addition, other ion channels may exist in the dendrites. This may partially explain why our Case 2 model is able to replicate O-LM cell behaviour (see Fig. 3) even though IA is not present in the dendrites. In other words, the particular kinetic details taken from non-O-LM cell data may compensate for the lack of IA in the dendrites such that a close match of the electrophysiological data occurs. However, because of the known presence of dendritic IA channels we considered this issue further.

Exploring dendritic IA

In exploring dendritic IA, the conductance value chosen for this current was based on results from Martina et al. (2000) where the TEA-sensitive and TEA-resistant components of the potassium current were teased apart. The TEA-resistant component was ≈35 % of the total potassium current and thus consists of IA channels. In a more recent study, Lien et al. (2002) found that the IA current was ≈19 % of the total potassium current. We explored the firing characteristics of model cells that contained varying densities of dendritic IA channels. Figure 4 shows the frequency-response curves for model cells that contain varying densities of IA channels. These cases include Case 2 (0 % IA in the dendrites), 19 % IA (92 pS μm−2), 27 % IA (130 pS μm−2), 35 % IA (165 pS μm−2) and 27 % IA only in the proximal dendrites (< 100 μm from the soma). The latter case is considered because Martina et al. (2000) measured out to ≈100 μm from the soma and the choice of 27 % is an average of the smaller and larger measured conductance values. We will refer to this specialized case as Case 3 proximal when using it to explore issues later on.

Figure 4.

Frequency-response curves for Case 2 and multiple Case 3 models

Details of models are shown in Table 1. Tonic current (as shown on the x-axis) is injected into the soma. For all models, the soma contains INa, IK, IA and Ih. Only Case 2 (black trace), in which the dendritic channels are restricted to INa and IK, produces the appropriate electrophysiological response of the O-LM interneuron. The four grey traces represent various Case 3 models (in which the dendrites contain INa, IK and IA). From left to right the models contained the following IA densities; 27 % (130 pS μm−2) in the proximal dendrites (< 100 μm from the soma), and 19 % (92 pS μm−2), 27 % (130 pS μm−2) and 35 % (165 pS μm−2) throughout the dendrites.

The presence of dendritic IA changes, affects the frequency-response curves as shown in Fig. 4. However, the cell is still unable to fire spontaneously as found experimentally. If IA is removed from the model entirely, the cell spontaneously fires at approximately double (≈20.6 Hz) the firing frequency of Case 2 (≈12 Hz) in which IA is present only in the soma. IA is a hyperpolarizing potassium current that is activated by depolarization and delays the onset of action potentials. Its exclusion from the model allows the membrane voltage to remain more depolarized and therefore the cell can fire at higher frequencies. In summary, it is unclear what is the appropriate amount of IA to use as well as how to appropriately distribute it on the dendrites.

Exploring different HH channel densities

The ability of our model interneuron to fire spontaneously and at a given frequency depends on the dendritic conductances. If, for example, the sodium conductance from a CA1 pyramidal cell is used instead of that found for the O-LM interneuron, approximately half the sodium conductance in interneurons, the model interneuron does not produce sustained firing but instead shows an oscillatory transient response followed by a steady-state membrane voltage (Fig. 5A). Although channel densities were measured in the dendrites, they were restricted to proximal locations. Our Case 2 model choice assumes that the channel conductances are the same proximally and distally. However, they could be different. Figure 5B shows the firing response of the model interneuron with different distal (> 100 μm) dendritic densities. The distal densities are increased from 52 % to 150 % (Fig. 5Bi-iii) of the proximal dendritic densities. As expected, the firing characteristics change. In particular, the firing frequencies decrease as the distal dendrites become less active. However, without data to the contrary, we kept distal and proximal dendritic conductance densities the same.

Figure 5.

Varying dendritic conductances in model interneuron

A, voltage recordings from soma of Case 2 model interneuron in response to 0.1 nA of sustained injected current. Thin line, 100 % of dendritic Na+ and K+ conductances (117 and 230 pS/μm2, respectively). Thicker line, 66 % of dendritic Na+ and K+ conductances. Thickest line, 50 % of dendritic Na+ and K+ conductances. The model interneuron cannot sustain repetitive firing of action potentials when the dendritic channel conductances are less than 75 % of the maximal values given above for the thin line. B, voltage traces show the spontaneous firing of the model interneuron when the distal dendritic Na+ and K+ conductances (> 100 μm from soma) are modified from 100 % of their values (ii, 12 Hz), to 150 % (i, 18 Hz) and 52 % (iii, 9.4 Hz).

These parameter manipulations illustrate the sensitivity of the response of the model cell to intrinsic property changes and how the intricate balance of its channels affect the response. In summary, appropriate electrophysiological responses of O-LM interneurons are obtained with our Case 2 model, namely INa,s, IK,s, IA and Ih in the soma and INa,d and IK,d in the dendrites. We used this model together with another case involving dendritic IA, Case 3 proximal, to explore further issues of signal initiation and propagation.

Signal attenuation in model interneuron

Integration of synaptic inputs within a cell depends on its electrotonic characteristics. Therefore, using the Electrotonic Workbench tools (Carnevale et al. 1997) within the Neuron program we explored the electrotonic architecture of the model interneuron. We looked at the attenuation both into and out of the soma using the Vin and Vout transforms for Case 2 (see Fig. 6). The membrane voltage was initialized at −60 mV. The Vout transform shows the influence of somatic potentials on the dendrites, while the Vin transform shows the effect of dendritic inputs at the soma (Carnevale et al. 1997). Figure 6 shows V(measure)/V(inject), the ratio of the voltage measured and the voltage injected, as a function of physical distance from the soma for Vout (top) and Vin (bottom). In the Vin transform, the branches are much steeper than the Vout. This implies a much greater attenuation of signal when the input is coming from the dendrite to the soma, than when the signal is moving from the soma to the dendrites. The Vout transform in Fig. 6 shows us that back-propagating signals have the potential to be strong, i.e. only have a small amplitude decrease. If 1 mV is injected into the soma, we would measure 0.76 mV in the most distal dendrite (≈320 μm from the soma) as indicated by the arrow in the upper graph of Fig. 6 (Vout). This asymmetry in voltage attenuation is known from classical studies with passive dendrites by Rall & Rinzel (1973).

Figure 6.

Graphs showing the attenuation of voltage as a function of the anatomical distance along the dendrites from the soma for the Vout and Vin transforms of the model interneuron

The attenuation is measured as V(measure)/V(inject). The Vout transform shows the attenuation of a 1 mV direct current (DC) signal when the input site is the soma, as measured in the dendrites. The Vin transform shows the attenuation of a 1 mV DC signal when the input sites are the dendrites, as measured in the soma. Each line on the graph represents a physical distance along a dendritic branch. For example, the arrow on the model (inset) shows the location of a distal dendritic branch ≈320 μm from the soma. The arrows on the graphs show the attenuation of voltage along that dendritic branch when the input site is the soma (Vout) and when the input site is the dendritic branch itself (Vin). With the input at the soma, the voltage measured at the distal dendrite would be 0.76 mV (see Vout). With the input at the distal dendrite, the voltage measured in the soma would be 0.18 mV (see Vin). This shows the ability of this cell type to have strong back-propagating signals.

Dendrites as frequency filters

Using our Case 2 model we examined how localized depolarization affects signal output in this model cell. Tonic current is injected into the soma, a proximal dendrite (150 μm from the soma), or a distal dendrite (326 μm from the soma) for different current values (see Fig. 7). As the location of input moves distally, the firing frequency range (as measured in the soma) decreases and the injected current range increases. Outside this range, the cell does not fire and the membrane voltage reaches a steady-state value. Too little current results in a hyperpolarized steady-state value while too much current saturates the cell and results in a depolarized steady-state value for the membrane voltage. Input into a distal dendrite will produce only a narrow range of firing frequencies in the soma regardless of the strength of that input (see Fig. 7). Note that the injected current values reflect an absolute amount. Therefore, this result can be explained if one considers the relative current to surface area ratio as well as the amount of connectivity of a particular membrane section to surrounding sections. The soma is centrally located and connected to all the main branches of the dendritic tree. Therefore, if the soma becomes saturated with injected current (i.e. the membrane voltage is depolarized to a level where the Na+ channels are essentially all inactivated), the whole cell is essentially saturated as well. On the other hand, if a distal section of a dendrite becomes saturated, the effect on the rest of the tree will not be as large as a result of its smaller surface area and since it is connected only to one or two close branches. Due to its smaller surface area, a distal dendrite will require less absolute current to saturate than the soma. Increasing the absolute current beyond this dendritic saturation point has little effect on the overall response of the cell. In addition, the attenuation of inputs from the distal dendrites is much larger than the attenuation of inputs from the soma as seen in Fig. 6. As a result, the frequency, as measured in the soma, increases more gradually but for a wider range of currents as compared to direct input into the soma which has a steeper increase of frequencies for a smaller range of currents (see Fig. 7).

Figure 7.

Frequency-response curves produced by three different locations of tonic current injection sites in the Case 2 model interneuron: soma, proximal dendrite (150 μm from the soma), and distal dendrite (326 μm from the soma)

Locations of injection sites are shown with circles in inset. Frequency is measured at the soma. Distal inputs produce smaller frequency ranges, compared to more proximal inputs. Due to the central location of the soma, when the soma becomes saturated, the entire dendritic tree is essentially saturated. If a distal dendritic branch becomes saturated, its low connectivity to other branches combined with the larger attenuation of voltage from dendritic inputs seen in Fig. 6, results in a much smaller influence over the rest of the dendritic tree.

If instead of tonic injected current we impose presynaptic spike trains, we find that dendrites continue to act as frequency filters. We see in Fig. 7 that there is a specific frequency range that can be measured in the soma for tonic inputs depending on their location, therefore we would expect similar distinct ranges for presynaptic spike trains imposed at varying locations along the dendritic tree. A presynaptic release frequency of 40 Hz placed at the soma, 150 μm along a dendrite and 326 μm along the dendrite, will result in postsynaptic frequencies of 24.53, 20.04 and 14.58 Hz, respectively, as measured at the soma (data not shown). These simulations were performed without the addition of a hyperpolarizing current to suppress spontaneous firing. Distal input trains result in lower postsynaptic frequencies measured in the soma as compared to proximal input trains. The general attenuation in output frequency that was seen in the case of tonic current injection persists when presynaptic spike trains are imposed instead.

Even though the dendrites in this model are active and capable of generating and sustaining action potentials, we observed a filtering effect that is location-dependent. A presynaptic spiking input that is placed in the distal dendrites results in lower measured frequency in the soma when compared to proximal input sites. Theoretical studies suggest that to counter the location-dependent response there is an increase in the number of excitatory receptors at the distal dendrites (Cook & Johnston, 1999; Magee & Cook, 2000). Experimentally, O-LM interneurons are found to have approximately double the number of excitatory inputs in the distal dendrites as in the proximal dendrites (Martina et al. 2000). However, the number and location of synapses that are active during physiological conditions remains to be explored.

Addition of an axon to the interneuron model

Although the soma and dendrites of the O-LM interneuron are restricted to the oriens/alveus layer of the hippocampus, the axon is known to traverse through several layers and arborize within the lacunosum/ moleculare layer where it synapses onto the distal dendrites of pyramidal cells. Due to technical complications in measuring from such thin processes, there are no experimental data on the channel content and kinetics for axons in O-LM interneurons or any other interneuron subtype making it difficult to determine what parameter choices should be made in including an axonal compartment to a multi-compartment interneuron model. However, because of experiments by Martina et al. (2000) showing differences in dendritic signal initiation on axon-bearing dendrites and axon-lacking dendrites, we include an axon (see Methods) to perform simulations that can be compared with their experimental data.

In general, the experimental data showed that the initiation site for action potentials shifted from somatic to dendritic depending on the length and intensity of the stimulus (as measured by the peaks). Long but small inputs (100 ms, 0.1 nA) resulted in action potential initiation on axon-bearing dendrites followed by the soma and axon-lacking dendrites. On the other hand, short but large inputs (0.1 ms, 9 nA) resulted in a reversal of the ordering (axon-lacking dendrites followed by soma and axon-bearing dendrites) (see Fig. 3 in Martina et al. 2000). We can replicate these results in our model interneuron if we make appropriate restrictions on the density of axonal sodium and potassium channels. Initially, axonal conductance values for the HH channels were chosen from the model data of Traub & Miles (1995) but the reversal of spike initiation order in response to short, high intensity stimulus as found experimentally could not be reproduced. Consider first the model interneuron described by Case 2 (without dendritic IA). If we constrain the axonal sodium and potassium channel densites to ≈1.5 times the somatic channel densities we are able to match the experimental findings of Martina et al. (2000). Figure 8 shows the voltage recordings from an axon-bearing dendrite (red trace), an axon-lacking dendrite (blue trace) and the soma (black trace) from our multi-compartmental model in response to both the long/small input (Fig. 8A left-hand panel) and the short/large input (Fig. 8A right-hand panel). If the density of sodium and potassium channels is increased beyond 1.5 times that in the soma, we cannot obtain a reversal in the spike order in response to the short/large input. In particular, the axon-bearing dendrite will always spike before the soma regardless of the input protocol.

Now let us consider a model interneuron with dendritic IA, Case 3 proximal (with IA only in the proximal dendrites; see Fig. 4). Again we can match the experimental findings of Martina et al. (2000) concerning a reversal of spike peaks depending on the input. However, the amount of injected current required to elict an action potential is now slightly greater due to the presence of dendritic IA. Also the required axonal sodium and potassium channel densities necessary to produce the reversal in spike peaks is different from ≈1.6 times that in the soma. If the density of channels in the axon is smaller or larger than this value, we find that the model cannot produce the reversal in spikes due to the different inputs. The timing between spike peaks in the soma, axon-bearing dendrite and axon-lacking dendrite for both Case 2 (described above) and Case 3 proximal, is consistent with what was experimentally measured by Martina et al. (2000).

Although we can match the reversal of spike peaks in response to different input stimuli, we are not able to match the spike upstrokes (see smaller insets in Fig. 8). This is partially due to our minimal representation of the axon (see Methods) since we find that varying the length of the axon changes the shape of the action potential, as measured in the axon-bearing dendrite (data not shown). The inappropriate ordering of the upstrokes may also be due to different channel densities in proximal and distal locations.

Spike initiation and propagation

We now turn to a consideration of spike initiation and propagation. We note that it is difficult to examine this issue from a purely experimental viewpoint since recording and stimulating from distal locations along the dendritic tree is challenging. To explore spike initiation and propagation in the Case 2 model interneuron, we injected −0.05 nA of tonic current into the soma to suppress the spontaneous firing of the cell. We found that the location for spike initiation depends on the location and amount of input and intrinsic properties. We note that the dendritic membrane is more excitable than the somatic membrane in our model due to its lack (or smaller amount) of the hyperpolarizing current IA, in addition to the higher HH conductances measured in the dendrites by Martina et al. (2000). This is clear if we simply compare our chosen biophysical characteristics in dendritic and somatic compartments. A one-compartment model cell (diameter, 2 μm; length, 10 μm) that contains INa,d and IK,d spontaneously fires at ≈11 Hz. The same one-compartment model adjusted to contain INa,s, IK,s, IA and Ih does not spontaneously fire and requires an injected current in order to produce repetitive firing (data not shown). The existence of these highly excitable dendrites suggests that dendritically initiated action potentials are possible.

Action potential propagation velocity

The conduction velocity is determined from the physical distance between the soma and a specific dendritic segment, and the time difference between the peaks of the action potentials evoked by a stimulus. Due to the highly active dendrites in this model, the conduction velocity of action potential propagation is not a constant. The conduction velocity measured at a dendritic segment ≈38 μm or ≈182 μm from the soma is 0.76 m s−1 or 2.43 m s−1, respectively (Case 2 model). Martina et al. (2000) measured a mean conduction velocity of 0.91 ± 0.13 m s−1 for O-LM interneurons, but their dendritic recordings were only out to ≈100 μm from the soma due to the technical difficulty of recordings from distal dendrites, and most of their measurements were around 60 μm. Therefore we compared the conduction velocity measured at this distance in the model. The model has a conduction velocity of 1.3 m s−1 at a distance of 60 μm from the soma which is slightly faster than the experimental measurement.

Since the density of channels in the distal dendrites (>100 μm) is unknown, we examined how variations in the distal densities would affect the conduction velocity. As the distal dendritic density increases from 52 % to 150 % of the proximal dendrite density, the conduction velocity increases from 0.4 to 6.1 m s−1 as measured 60 μm from the soma. This suggests that the conduction velocity, as measured in the proximal dendrites, can be used to predict distal dendritic channel densities. To match the conduction velocity as measured by Martina et al. (2000), we need the density of channels in the distal dendrites to be ≈87 % of that in the proximal dendrites. This is a simplistic approximation which does not take into account the possibility of a graded density of channels along the dendrites, but it does give us an idea of what distal channel densities are possible based on measurements made at proximal dendrites. As experimental instruments and techniques improve, the measurement of velocity of propagation at more distal dendrites will either confirm or refute the prediction of the model that conduction velocity is not a constant.

Proximal vs. distal inputs

Simulations were conducted with AMPA synaptic inputs in order to simulate the excitatory input this cell would receive in a network. Several experimental sources were used to determine appropriate single channel conductance, receptor number per synapse and synapse density along the cell body (Koh et al. 1995; Nusser et al. 1998; Martina et al. 2000). We found that a spike can be evoked with only 0.25 % of the maximal conductance for AMPA synapses with the distribution of channels found by Martina et al. (2000). This distribution consists of having approximately twice the AMPA conductance in the distal dendrites (72.3 mS cm−2 for dendritic segments further than 85 μm from the soma) as in the proximal dendrites (33.9 mS cm−2 for segments closer than 85 μm). Since background activity clearly will affect this (Destexhe & Paré, 1999; Rudolph & Destexhe, 2001), the situation quickly becomes complex and we did not explore this further here. In addition, we found that multiple inputs are more efficient than single inputs in generating spikes (see Supplementary material).

We examined the response of Case 2 model to proximal inputs as compared to distal inputs. This has not been explored experimentally. A small number of distal inputs (total maximal conductance, gmax, 111.4 nS) result in somatically initiated, back-propagating signals (see Fig. 9A top panel), whereas a small number of proximal inputs (total gmax, 69.4 nS) produce dendritically initiated forward-propagating action potentials (Fig. 9B top panel). If the synaptic inputs are small (gmax < 12 nS), a somatically initiated spike (which then back-propagates) can also be seen in the proximal input case (data not shown). The dendritically initiated spikes seen in Fig. 9B (top panel), are a result of the lower threshold for spike initiation in the dendrites as compared to the soma. The smallest or most distal dendrites require the least amount of current to reach threshold due to their small surface area. Therefore, a dendritic spike will be initiated distally and the action potential will forward-propagate towards the soma as seen in Fig. 9B (top panel). The somatically initiated spike seen in Fig. 9A (top panel), is due to the saturation of the dendrite with relatively little current as discussed earlier. This saturation results in a depolarization that is passed down the dendritic tree to the soma where a spike initiates and is actively back-propagated (see Fig. 9A top panel). Note that there is little attenuation of the back-propagating spike from the soma indicative of the strong back-propagating signals possible in this interneuron model with highly active dendrites.

Figure 9.

Forward- and back-propagating action potentials depend on location of input

A and B, left-hand panels show the locations of the three synaptic inputs used in each simulation. Right-hand panels illustrate voltage recording sites on the model interneuron: the soma, a proximal dendrite (100 μm from the soma), and a distal dendrite (250 μm from the soma). Middle panels show the voltage recordings from the three locations shown on the right in response to simultaneous activation of three synaptic inputs with input locations as shown on the left (top and bottom panels refer to Case 2 and Case 3 proximal, respectively). A, total gmax of the three distal inputs is 111.4 and 334.2 nS (top and bottom middle panels, respectively). Location of these inputs is ≈150 μm from the soma. Distal inputs produce somatically initiated back-propagating signals for both model cases. The second spike produced shows only a small attenuation of amplitude, implying that this cell type can produce strong back-propagating signals. B, total gmax of the three proximal inputs is 69.4 nS (for both top and bottom panels). Location of these inputs is ≈10 μm from the soma. Proximal inputs produce dendritically initiated, forward-propagating signals for Case 2 (top panel) due to the highly excitable dendrites which have a lower threshold for spike initiation than the soma. For Case 3 proximal (bottom panel) only the second spike is dendritically initiated. Insets show expanded time scale of second spike. See Results for further details.

Similar results are obtained for Case 3 proximal. We used the same locations of proximal and distal inputs. Using the distal synaptic locations we needed three times the total conductance used for Case 2 to elicit the two spikes (Fig. 9A bottom panel). These spikes are initiated in the soma and then back-propagate towards the distal dendrites. Using the same total conductance for the proximal inputs as described for Case 2, we can elicit two spikes. The dendritically initiated second spike is very similar to its counterpart in Case 2 (see Fig. 9B bottom panel). The overall results of spike initiation due to proximal or distal synaptic inputs are consistent for both our Case 2 model and for a model which includes dendritic IA channels (Case 3 proximal).

If the dendrites are made less active (75 %, 60 % etc. of the maximal HH conductances), the spike amplitude decreases until a dendritic action potential (and thus somatic action potential) is no longer possible at about 50 % of maximal HH conductance values for the distal input values used (Case 2; data not shown). Similar results are obtained if tonic current is used instead of synaptic input. A tonic input into a distal dendrite will always evoke a spike that is first seen in the soma and which then back-propagates into the dendritic tree. A tonic input into the soma can produce either an antidromic (back-propagating) spike that originates in the soma or an orthodromic (forward-propagating) spike that originates in the distal dendrite depending on the amount of current injected (Saraga & Skinner, 2003). We tested our ‘spike initiation due to proximal or distal synaptic input’ results in a Case 2 model that contained an axon with 1.5 times the somatic sodium and potassium channels. We obtained similar results (spike ordering and initiation) to those obtained in our axon-lacking model cell of Case 2 (see Fig. 9) as described above.

DISCUSSION

Using an appropriate morphology and experimentally measured passive properties, channel distributions and channel kinetics, we created multi-compartment models that capture the essence of O-LM interneurons in hippocampus CA1. With these multi-compartment models, we investigated signal initiation and propagation profiles of O-LM interneurons. With their highly excitable dendrites we found that this cell type allows action potential initiation in either the soma or the dendrites depending on the location and strength of the input to the cell. Distal inputs result in somatically initiated spikes while proximal inputs produce somatically initiated spikes if the inputs are small or dendritically initiated spikes if the inputs are large. Although highly excitable, dendrites act as frequency filters with distal inputs having a lesser effect on the somatic response as compared to proximal inputs.

Active dendrites and back-propagating spikes

Neurons with active dendrites allow dendritic action potentials to be produced. While the back-propagation of action potentials (initiated in the axon hillock) into the dendritic arbor has been clearly shown, the degree and efficacy of back-propagating spikes depends on the dendritic morphology and density of voltage-gated channels (Stuart et al. 1997; Golding et al. 2001; Vetter et al. 2001). Back-propagating action potentials may play a role in the induction of synaptic plasticity as seen in layer V neocortical pyramidal neurons (Markram et al. 1997) and hippocampal pyramidal neurons (Magee & Johnston, 1997). For our O-LM model interneuron, we found that distal inputs always generated antidromic spikes. These back-propagating signals show little attenuation (see Fig. 9A) giving the dendrites of O-LM interneurons a strong antidromic signal that may have functional implications for this cell type within the neuronal network. Antidromic signal propagation provides a retrograde signal to the dendritic tree indicating the level of neuronal output. This could provide an associative link between presynaptic excitation and postsynaptic response. These signals could also reset dendritic membrane potentials which would have important implications for the precise timing of subsequent action potentials and spike-timing-dependent plasticity (Abbott & Nelson, 2000).

Functional aspects of O-LM interneurons

Due in part to the heterogeneous nature of interneurons, evidence for plasticity in interneurons has been unclear (Maccaferri & McBain, 1996b; McBain et al. 1999). However, recent work shows that a Hebbian form of long-term potentiation (LTP) could be induced specifically in vertical and horizontal interneurons of stratum oriens but not in interneurons of radiatum/lacunosum-moleculare (r/l-m) (Perez et al. 2001). Given that O-LM interneurons are mostly involved in feedback inhibition of CA1 pyramidal cells, whereas the interneurons of r/l-m are involved in feedforward inhibition, LTP in interneurons may be related to their interconnections within the hippocampal circuitry of neurons. Do the highly active dendrites of O-LM interneurons that we have modelled confer any particular functional aspects?

There are two major signalling pathways within the hippocampus, and it has been suggested that the O-LM interneurons may provide a switch mechanism allowing one path to be dominant over the other under certain circumstances (Blasco-Ibáñez & Freund, 1995; Maccaferri & McBain, 1995). The direct path is from the entorhinal cortex via the temporoammonic path to the CA1 pyramidal cells. The indirect path is via the well-known trisynaptic loop (entorhinal cortex → dentate gyrus → CA3 pyramidal cells → CA1 pyramidal cells). The CA1 pyramidal cells therefore receive excitatory input on the distal dendrites directly from the entorhinal cortex while the proximal dendrites receive excitatory inputs from the CA3 cells via the Schaffer collaterals. Since the O-LM interneurons also synapse onto the distal dendrites of the CA1 pyramidal cells, these interneurons can regulate the amount of input coming directly from the entorhinal cortex.

From a behavioural perspective, sharp waves (SPWs) and ripples occur during slow wave sleep, awake immobility and consummatory behaviours, whereas theta/gamma rhythms occur during arousal and exploration states (Buzsáki et al. 1983, 1992; Buzsáki, 1986; Leung, 1998). Functionally, SPWs are associated with memory consolidation and theta/gamma rhythms with learning processes (Lisman, 1999; Draguhn et al. 2000). If the CA1 pyramidal cells are highly excited, as during the occurrence of SPWs due to massive firing of CA3 pyramidal cells (Ylinen et al. 1995), then the O-LM interneurons will be excited which in turn will inhibit the entorhinal inputs to the CA1 pyramidal cells. This then allows the trisynaptic path to be dominant. During theta/gamma rhythms, where the direct path plays a larger role (Buzsáki, 2002), a decreased response of the CA1 pyramidal cells (relative to during SPW occurrence) will lead to disinhibition so that the direct path can dominate. Our model results here suggest that this switch between direct and indirect pathways is strengthened by the ability of these cells to exhibit strong (i.e. little attenuation) back-propagating signals by virtue of their highly active dendrites. This possibly represents a specialized function of the computational capabilities of this type of interneuron. The ability of these particular interneurons to express LTP (Perez et al. 2001) further strengthens this distinct functional role. Given that the number of excitatory synapses is approximately double in the distal dendrites compared to the proximal dendrites in O-LM interneurons (Martina et al. 2000), it is possible that the distal inputs have a significant importance to the cell. We have shown that distal inputs result in strong back-propagating action potentials in these cells with their highly active dendrites. These back-propagating signals may contribute to the LTP seen in the interneurons which would result in a stronger inhibition of the direct entorhinal inputs to the CA1 pyramidal cells during SPWs. Therefore, the role of the O-LM interneurons within the network is crucial and their highly excitable dendrites would seem to provide the appropriate characteristics for this interneuron type to have this distinct role and positioning in the neuronal network.

It is also interesting to note that although the majority of excitatory input to the O-LM interneurons is via the CA1 pyramidal cell collaterals (Blasco-Ibáñez & Freund, 1995), the CA3 pyramidal cell axons are also known to innervate the oriens and alveus strata (Li et al. 1994). If these excitatory inputs are at distinct locations relative to the CA1 pyramidal inputs, then the forward-propagating and back-propagating signals seen in the model under varying locations of input could be utilized by CA1 and CA3 pyramidal cell populations in order to manipulate the signal output of the O-LM interneuron. More experimental work needs to be done to distinguish the exact origins of the excitatory inputs to the O-LM interneuron and where these inputs synapse onto the dendritic tree.

Limitations of the model

Although the passive properties of the model were made to match experimental values, other issues, such as non-uniformity, were not considered at this stage due to the lack of experimental data. There is variability in experimental values of the passive properties for the O-LM interneuron and we chose to match the input resistance and membrane time constant to the most recently measured values for this cell type. With these chosen passive properties, appropriate morphology and experimentally found channel distributions and kinetics, the model interneuron responds very much like a real O-LM cell in response to steady injected current (see Fig. 3). Given this, more detailed studies of the passive parameters were not attempted at this time.

Hippocampal interneurons have extensive axonal arborizations. Although the soma and dendritic tree of the O-LM interneuron are contained within the oriens alveus layers, the axon is known to extend deep into the lacunosum/moleculare region where it synapses onto the distal dendrites of the pyramidal neurons. The classical view of spike initiation in a neuron involves a summation of postsynaptic potentials (PSPs) within the axon hillock resulting in action potential generation, which then propagates down the axon and in some cases, back-propagates through the soma and dendrites. There are several ideas as to why axons have a low threshold for spike initiation. It has been predicted from modelling studies that Na+ channel density in the initial segment may be 20-1000 times that in the soma or dendrites (Mainen et al. 1995; Rapp et al. 1996). It has also been suggested that Na+ channel density only increases about 2- to 3-fold but that channel kinetics differ between somatic and axonal sodium channels resulting in a lower threshold for action potential initiation in the axon (Colbert & Pan, 2002). The ion channel content and distribution within the axons of O-LM interneurons are not known, making it unclear how best to incorporate an axon into a multi-compartment model. However, to match the experimental findings of Martina et al. (2000) regarding dendritically initiated spikes in axon-bearing and axon-lacking dendrites we included a minimal representation for an axon. It was interesting to find that their results could be matched if the axonal sodium and potassium densities were constrained to ≈1.5-1.6 times the somatic densities. This allowed us to make preliminary predictions concerning the ion channel densities in the axon of the O-LM interneuron. However, even though our model data matched the experimental data regarding the spike peaks, they did not for the spike upstrokes. Preliminary simulations suggest that the spike upstroke depends on the length of the axon. We did not thoroughly explore the model (including varying the passive properties of the axon and extending the axonal segment) at this time for reasons outlined above.

We found that the distribution of channels that produced the most appropriate electrophysiological responses for this cell type was Case 2, which does not include dendritic IA channels. Martina et al. (2000) measured dendritic IA channels out to 100 μm from the soma. We did find that our Case 3 proximal model (which included only proximally located IA channels) produced a frequency-response curve that was close to our Case 2 model, although this model cell did not spontaneously fire. Unlike CA1 pyramidal neurons, which have an increase in the density of the transient potassium current towards the distal dendrites, it might be that O-LM cells have a decrease in this IA density towards the distal dendrites, or only proximally located IA channels. There is also experimental evidence that the activation of protein kinase A (PKA) or protein kinase C (PKC) causes a downregulation of the IA channel (Hoffmann & Johnston, 1998; Johnston et al. 2003). Specifically, phosphorylation of the IA channels results in a rightward (depolarized) shift of ≈15 mV in the activation curve for this channel. This modulation of the IA channels results in a significant increase in the amplitude of back-propagating signals in CA1 pyramidal neurons (Hoffman & Johnston, 1998). Therefore, our results of signal initiation and propagation using Case 2 and Case 3 proximal models may be indicative of PKA (activated via dopamine, noradrenaline (norepinephrine) or serotonin) or PKC (activated via muscarinic receptors) phosphorylation of IA. From both experimental and computational perspectives, further explorations of dendritic IA are warranted.

Predictions, suggestions and concluding remarks

Several suggestions and predictions arise from our work. The major prediction regards the different response of the model cell to proximal and distal inputs, that is, proximal inputs can result in either forward- or back-propagating spikes, whereas distal inputs always produce back-propagating spikes. It should be possible to test directly the prediction of switching from back- to forward-propagating spikes with increasing proximal input strength. We speculate that there might be distinct innervation locations, via CA1 and CA3 excitatory input for example, of this cell type worth examining. One might further look for differences in excitatory receptor kinetic characteristics in distal and proximal locations. Martina et al. (2000) have already noted a difference in excitatory synaptic density, and differences in the kinetics of AMPA synapses in interneurons and pyramidal cells in neocortex and dentate gyrus are known (Geiger et al. 1997; Angulo et al. 1999). Our model results predict that such differences would have the potential to control the ability of the O-LM cell to produce forward- or back-propagating spikes.

Another interesting idea that results from our model observations is to use spike ordering and action potential conduction velocities as a way to determine conductance densities in axonal and distal dendritic compartments which are technically difficult to obtain. Specifically, we found that when we included a minimal representation for an axon, severe constraints on the axonal conductance densities were required to match the spike ordering observed in experiments. Conduction velocities measured in proximal dendritic regions are very sensitive to changes in distal dendritic conductance densities. For example, an approximate 3-fold increase in the distal dendritic density gives rise to about a 15-fold increase in conduction velocity measured proximally in the model. This suggests that conduction velocity measurements in cells with active dendrites may provide an indirect, yet sensitive, measure of dendritic channel conductance densities.

Although predictive in nature, our models come with a degree of uncertainty which, as experiments become more sophisticated and detailed, will lessen. These uncertainties include not having passive properties and electrophysiological properties for the particular cell modelled, channel densities both in the distal dendrites and axonal segments, and knowing the contribution of various channels under different physiological conditions. Finally, it is always important to obtain experimental data that completely describe the characteristics of the particular ion channel. For example, although somatic and dendritic channel conductance densities were measured for O-LM cells, the sodium inactivation and voltage dependencies of the time constants were not. Such completeness becomes more important as additional channel types are included in a model cell thus affecting the intricate balance of the channel dynamics, which determine the cell's output.

Understanding the fundamental roles of dendrites is difficult as one must not only know their intrinsic characteristics that confer their computational capabilities, but also what networks the particular neurons are involved with and to what end. Only in rare cases can one understand dendritic computations fundamentally (Agmon-Snir et al. 1998). There are always further refinements that can be made to a multi-compartment model as data becomes available. For example, Lien et al. (2002) have recently measured two delayed rectifier K+ currents and one transient K+ current in O-LM interneurons. However, the key aspect is to understand mechanistically the dynamics of signal initiation and propagation. We feel that our O-LM models in their present form have shown the complex and exciting role of active dendrites in the overall activity of a neuron. Here we suggest that the highly active dendrites of O-LM hippocampal interneurons that give them the ability to produce non-attenuating back-propagating spikes to distal input might be a critical requirement for them to act as gatekeepers between two different signalling pathways.

Appendix

The sodium current, INa, is described by:

display math((A1))
display math((A2))
display math((A3))

where the conductance values for the sodium current in the soma and dendrite are 107 and 117 pS μm−2, respectively, ENa is 90 mV, as taken from (Martina et al. 2000), and m and h are the activation and inactivation variables, respectively. Using the Hodgkin-Huxley formalism, we found appropriate forward (αm and αh) and backward (βm and βh) rate constants that fitted the steady-state activation and inactivation curves for sodium in the soma and the dendrites as found experimentally by (Martina & Jonas, 1997; Martina et al. 2000). The steady-state activation/inactivation curve is described by (m, h)αm,h/(αm,hm,h).

display math((A4))
display math((A5))
display math((A6))
display math((A7))

These forward and backward rate constants allow us to calculate activation and inactivation time constants which are of the form τm,h= 1/(αm,hm,h) (see Fig. 2A and B).

The potassium current, IK, is described by:

display math((A8))
display math((A9))

where the conductance values for the potassium current in the soma and dendrite are 319 and 230 pS μm−2, respectively, EK is −100 mV as taken from Martina et al. (2000), and n is the activation variable for this channel. Using the general form of the rate constants from Warman et al. (1994) to describe the delayed-rectifier potassium current, we fitted the steady-state activation curves for the soma and the dendrites described by Martina et al. (2000). The steady-state activation curve is described by nn/(αn+βn). The values of αn and βn that are found to best fit the steady-state curves are:

display math((A10))
display math((A11))

The transient potassium current, IA, is described by:

display math((A12))
display math((A13))
display math((A14))

where the conductance is 165 pS μm−2 as taken from Martina et al. (2000) and a and b are the activation and inactivation variables, respectively. The steady-state activation and inactivation curves are taken from Zhang & McBain (1995a): a= 1/(1 + exp{-(V+ 14)/16.6}) and b= 1/(1 + exp{-(V+ 71)/7.3}). The time constant for activation is voltage-independent and set to 5 ms (Zhang & McBain, 1995a). The time constant for inactivation is voltage-dependent and of the form τb= 1/(αbb) where the rate constants are modified from the equations describing the inactivation rate constants used by Warman et al. (1994) (see Fig. 2E and F).

display math((A14))

The non-specific cation channel, Ih, is described by:

display math((A15))
display math((A16))

This channel activates upon hyperpolarization and has been found in the soma of O-LM interneurons (Maccaferri & McBain, 1996a), although no conductance values have been measured. The conductance value for this channel of 13.85 pS μm−2 is therefore taken from CA1 pyramidal cells (Magee, 1998). The reversal potential (Eh) of −32.9 mV, is taken from Maccaferri & McBain (1996a) and r is the activation variable for this channel. The steady-state activation curve used is from Maccaferri & McBain (1996a) where r= 1/(1 + exp{(V+ 84)/10.2}). The activation time constant, τr, is modified from the one given by Huguenard & McCormick (1992) who described this channel in a thalamic cell model, so as to fit the data given by Maccaferri & McBain (1996a) where τr= 1/{exp(-17.9-0.116 V) + exp(-1.84 + 0.09V) + 100; See Fig. 2G and H. Table 2 provides the conductance values used in the simulations unless stated otherwise.

Table 2. Ion channel conductances for sodium and potassium in the soma (INa.s.IK.s) and in the dendrites (INa,d,IK,d), the transient potassium current (IA) and the hyperpolarization-activated current (Ih)
Ion channelConductance value
  1. Values were taken from Martina et al. (2000) and Magee (1998).

g Na,s 107 PS μm−2
g Na,d 117 PS μm−2
g K,s 319 PS μm−2
g K,d 230 PS μm−2
g A 165 PS μm−2
g h 13.85 PS μm−2

Acknowledgements

This work was supported by the Canadian Institutes of Health Research. F.K.S. is an MRC Scholar and a CFI Researcher. F.S. was supported by OGSST and NSERC student awards. We would like to thank M. Hines for help with the software Neuron and M. Martina and P. Jonas for explaining figures and sharing data from their publication. We would also like to thank the anonymous reviewers for their several helpful suggestions which we feel have significantly improved the manuscript.

Supplementary material

The online version of this paper can be found at:
    DOI: 10.1113/jphysiol.2003.046177
and contains material entitled:
Effect of multiple inputs on somatic action potentials

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