Abstract
The hyperpolarizationactivated cation current I_{h} exhibits a steep gradient of channel density in dendrites of pyramidal neurons, which is associated with location independence of temporal summation of EPSPs at the soma. In striking contrast, here we show by using dendritic patchclamp recordings that in cerebellar Purkinje cells, the principal neurons of the cerebellar cortex, I_{h} exhibits a uniform dendritic density, while location independence of EPSP summation is observed. Using compartmental modeling in realistic and simplified dendritic geometries, we demonstrate that the dendritic distribution of I_{h} only weakly affects the degree of temporal summation at the soma, while having an impact at the dendritic input location. We further analyze the effect of I_{h} on temporal summation using cable theory and derive bounds for temporal summation for any spatial distribution of I_{h}. We show that the total number of I_{h} channels, not their distribution, governs the degree of temporal summation of EPSPs. Our findings explain the effect of I_{h} on EPSP shape and temporal summation, and suggest that neurons are provided with two independent degrees of freedom for different functions: the total amount of I_{h} (controlling the degree of temporal summation of dendritic inputs at the soma) and the dendritic spatial distribution of I_{h} (regulating local dendritic processing).
Introduction
The hyperpolarizationactivated cation current (I_{h}) is expressed in many neurons in the CNS. I_{h} is one of only a few voltagedependent currents activated at resting membrane potential. The combination of its “inverted” voltagedependent activation properties and its suprathreshold reversal potential gives I_{h} a unique signature in its contribution to neuronal excitability. I_{h} has been ascribed a plethora of functions in neurons (Robinson and Siegelbaum, 2003), probably the most recognized being its role in synaptic integration (Magee, 1999; Williams and Stuart, 2000; Berger et al., 2001), where I_{h} has been shown to shorten the width of postsynaptic potentials (PSPs) and dampen the summation of trains of inputs. The magnitude of this effect is distance dependent: I_{h} has a large effect on PSPs originating in the distal dendrites and a smaller effect on proximal inputs. Thus, regardless of their origin, all PSPs arriving at the soma have similar widths, and thus I_{h} is said to normalize the broadening effect of dendritic filtering on synaptic input.
Previous studies using dendritic cellattached patchclamp recordings have shown that, in hippocampal CA1 and layer 5 neocortical pyramidal neurons, I_{h} shows a pronounced increase in density along the dendrites (Magee, 1998; Williams and Stuart, 2000; Berger et al., 2001; Kole et al., 2006). These gradients have recently been confirmed on the ultrastructural level using EM immunogold techniques (Lörincz et al., 2002; Notomi and Shigemoto, 2004). This striking nonuniform distribution has led to the assumption that there is a tight link between the role of I_{h} in temporal summation and its high density in distal dendritic branches.
Purkinje cells exhibit a significant I_{h} (Crepel and PenitSoria, 1986; Roth and Häusser, 2001; Williams et al., 2002), which is linked to HCN1 expression (Nolan et al., 2003). Given the differences in dendritic structure and excitability between Purkinje cells and pyramidal neurons, it is important to establish whether a dendritic gradient of I_{h} also exists in Purkinje cells, and how the I_{h} distribution influences synaptic integration in these neurons.
We used cellattached patchclamp recordings to map the distribution and characterize the kinetics of I_{h} in Purkinje neurons, and demonstrated that I_{h} exhibits a uniform dendritic density. We used simultaneous wholecell somatic and dendritic recordings to explore the effect of the current on single and multiple synaptic inputs, and found that, as in pyramidal neurons, I_{h} shortens EPSP duration and minimizes EPSP summation. Despite the conductance change accompanying activation of I_{h}, its effect on synaptic integration was linear. Analyzing this linearity shows that it is crucial for understanding how I_{h} contributes to normalization of summation. To dissect the contribution of I_{h} to the integration of synaptic inputs, we used detailed and simplified compartmental models. Our simulations reveal that normalization of EPSP summation can be achieved with uniform as well as nonuniform distributions of I_{h}. We also define the boundaries of maximum and minimum summation for practically any spatial distribution of I_{h}. These findings provide a deeper understanding of the contribution of I_{h} to the regulation of the input–output functions of neurons.
Materials and Methods
Slice preparation and electrophysiology.
All experiments were performed using acute cerebellar slices prepared using standard techniques in accordance with institutional and national regulations. Sprague Dawley rats [postnatal day 18 (P18) to P25] were anesthetized with isoflurane (Abbott Labs, Kent, UK) and decapitated. At this age, HCN1 expression in the cerebellum as detected by immunoblotting has reached a steady state (Lujan et al., 2005). The cerebellum was surgically dissected and cut sagittally at a slice thickness of 250 μm on a vibratome (VT1000S; Leica Microsystems, Nussloch, Germany). Before recording slices were incubated for 30–45 min at 32°C in external solution bubbled with 95% O_{2} and 5% CO_{2}. The standard external recording solution contained the following (in mm): 125 NaCl, 2.5 KCl, 2 CaCl_{2}, 1 MgCl_{2}, 25 NaHCO_{3}, 1.25 NaH_{2}PO_{4}, and 25 dglucose.
Neurons were visualized with a 60× water immersion objective (LUMPlanFI/IR; Olympus, Tokyo, Japan) using infrared (IR)differential interference contrast optics on an Olympus microscope (BX51WI) equipped with an IR camera (C240007; Hamamatsu, Tokyo, Japan) connected to a video monitor. Pipettes were prepared from thickwalled filamented borosilicate glass capillaries (Harvard Apparatus, Kent, UK) pulled on a twostage puller (model PC10; Narishige, Tokyo, Japan) with tip resistances between 4 and 6 MΩ and between 7 and 9 MΩ for wholecell somatic and dendritic recording, respectively. In all cellattached experiments, the pipette resistance was standardized at 10 MΩ to minimize variations in patch area. The composition of the internal solution was as follows (in mm): 130 methanesulfonic acid (Fluka, Buchs, Switzerland), 10 HEPES, 7 KCl, 0.05 EGTA (Fluka), 2 Na_{2}ATP, 2 MgATP, 0.5 Na_{2}GTP, and 4% biocytin, KOH titrated to pH 7.2. Cellattached pipettes were filled with high potassium solution of the following composition (in mm): 120 KCl, 2 CaCl_{2}, 1 MgCl_{2}, 10 HEPES, 20 tetraethylammoniumCl, 5 4aminopyridine, 1 BaCl_{2}, 0.02 CdCl_{2}, and 200 nm TTX (Tocris Cookson, Bristol, UK). The osmolarity was adjusted to 290 mOsm with dglucose, and the pH was titrated to 7.4 with KOH. The I_{h} blocker 4(NethylNphenylamino)1,2dimethyl6(methylamino)pyridinum chloride (ZD 7288) (Tocris) was dissolved in distilled water to a stock concentration of 10 mm and diluted in the external solution to a final concentration of 40 μm. To prevent spontaneous firing during cellattached recording, 200 nm TTX was added to the external solution.
Recordings were performed using a Multiclamp 700A amplifier (Molecular Devices, Palo Alto, CA) connected to a Macintosh computer via an ITC18 board (Instrutech, Port Washington, NY). Data were acquired with Axograph (version 4.9). The currentclamp signal was Bessel filtered at 6 kHz and sampled at 20 kHz. Wholecell dendritic recordings were discarded if the access increased >50 MΩ. PSPs were mimicked by injecting doubleexponential current waveforms (τ_{rise}, 0.3 ms; τ_{decay}, 3 ms). All voltage recordings shown are the average of between 20 and 25 single traces. The cellattached voltageclamp recordings were Bessel filtered at 1–2 kHz and acquired at 10 kHz. Leak subtraction of cellattached data were done either online by running a P/N leak subtraction protocol in Axograph, or offline. The liquid junction potential (+9.3 mV and +3 mV for the wholecell and the cellattached solutions, respectively) was not corrected for. For cellattached recordings, the resting membrane potential was determined by recording the membrane potential immediately after breakin (−51.8 ± 0.8 mV; n = 40; in the presence of TTX). All data analysis was performed in Igor Pro (Wavemetrics, Lake Oswego, OR) using Neuromatic (courtesy of Jason Rothman, University College London, London, UK). The rise time of simulated EPSPs (sEPSPs) was measured between 20 and 80% of the peak amplitude. The integral of single sEPSPs is calculated as the area, normalized to peak amplitude, from the beginning of the voltage response to 312.5 ms after.
Computer modeling.
All simulations were performed using NEURON, version 5.7 (Hines and Carnevale, 1997). The cylinder model had the following passive properties: R_{m} = 20,000 Ωcm^{2}, R_{i} = 200 Ωcm, C_{m} = 1 μF/cm^{2}, whereas dendritic trees in Figure 4 had R_{m} = 50,000 Ωcm^{2}, R_{i} = 70 Ωcm, C_{m} = 0.85 μF/cm^{2}. These values were corrected for spines by scaling membrane conductance (both passive and g_{h}) as well as C_{m} by a factor of 2. For simulations using realistic dendritic morphologies, we used reconstructions of layer 5 pyramidal cells, Purkinje cells, and CA1 pyramidal cells, taken from the Dendritica software package (www.dendrites.org) (Vetter et al., 2001). The cells that are presented in Figure 4 are “Rapp_cell3” (Purkinje), “913c.hoc” (CA1), and “p21.hoc” (pyramidal). For each cell, we divided the compartments to 10 groups according to their distance from the soma, and selected at random 10 locations from each group to obtain a total of 100 random locations spanning the whole tree. We computed the summation for each one of the locations for each of the distributions and for each density. To obtain a quantitative measure of the degree of normalization, we computed the SD of the summation as a function of distance over the 100 locations. For each cell and for every density, we also simulated the case in which g_{h} follows a linear gradient with distance from the soma (Fig. 4D). The slope of the gradient was computed such that the integral of g_{h} over the whole dendrite (including spine correction) will be identical with its value in the corresponding uniform case. Similarly for the exponential distribution, we followed the formula described by Kole et al. (2006) and normalized the conductance to have the same average g_{h} over the tree. Kinetics of input currents were identical with those used in the experiments. We used fixed time steps with dt = 0.1 ms, because using finer time steps did not produce any apparent difference. In simulations in which the total density of I_{h} was changed, this manipulation led to changes in the resting membrane potential (V_{rest}). This change in V_{rest} makes it more difficult to compare different cases. To keep the dynamics comparable between similar conditions, we computed for each I_{h} density the value of E_{leak} that keeps V_{rest} at −70 mV when I_{h} is uniformly distributed. Changing the spatial distribution of I_{h} also affects V_{rest} at each compartment. For reasons of simplicity and analytical tractability, in the simulations involving cylinders (Figs. 5, 6), we thus changed E_{leak} in each compartment, such that even when I_{h} is not uniform, V_{rest} is still uniform. Note, however, that in Figure 4, once E_{leak} was computed for the uniform distribution for a given g_{h} density, no additional adjustments were made for the nonuniform case, which means that the resting potential in different compartments was different, with differences reaching 10 mV in the case of layer 5 pyramidal neurons in Figure 4.
The nonlinear kinetic model of I_{h} is given by the following set of equations: where I_{h} is a current density in units of milliamperes per square centimeter, V is the membrane potential in millivolts, E_{h} = −34.4 mV is the reversal potential of I_{h}, ḡ_{h} is the maximal conductance (millisiemens per square centimeter), and q is the time and voltagedependent activation variable, dictating the active proportion of the total conductance. q_{∞} sets the steady state of the activation variable according to the sigmoidal function, and τ_{h} is the voltagedependent time constant in milliseconds with which q approaches q_{∞}. All of the kinetic parameters were obtained by fitting the data from cellattached recordings shown in Figure 1(see also supplemental Fig. 1, available at www.jneurosci.org as supplemental material). Note that, in contrast to Hodgkin–Huxley type models, here q_{∞} and τ_{h} are independent. Moreover, we used a single function to describe τ_{h}, which fits both the activation and deactivation time constant obtained from the data. In the simulation using reconstructions of CA1 pyramidal neurons and layer 5 pyramidal neurons, τ_{h} was sped up by a factor of 4, because these cell types exhibit approximately three to five times faster I_{h} activation kinetics compared with Purkinje cells, depending on membrane potential (Magee, 1998; Williams and Stuart, 2000; Berger et al., 2001). The simulations in Figure 4 were tested with both fast and slow I_{h}, giving no significant difference in the results (data not shown). This factor of 4 acceleration in the kinetics was also used in all simulations involving cylinders.
We also derived a linearized model of I_{h} using a quasiactive description from the above nonlinear model (Koch, 1984; Cox and Griffith, 2001). In this model (supplemental Figs. 1, 2, available at www.jneurosci.org as supplemental material), the activation variable is voltage dependent, but in a linear manner. Assuming V_{0} is the resting membrane potential, then I_{h} is given by the following equations: where q_{∞} and τ_{h} are identical with the previous model, q′_{∞}(V_{0}) is the derivative of the sigmoidal activation function q_{∞} at V_{0} given by the following: Note that in the first equation the first component describes the standing current attributable to the open conductance at V_{0}, whereas the second part is the linear component of the voltage dependence of the current. Also important to note is that the time constant of the activation variable is fixed to its value at the holding potential.
Results
Kinetics and distribution of the I_{h} current in Purkinje cells
We determined the kinetics and distribution of I_{h} in Purkinje cells using cellattached patchclamp recordings from the soma and dendrites of neurons visualized in cerebellar slices. Figure 1 shows an example of the properties of I_{h} in dendritic cellattached recordings. The current activates with hyperpolarization beyond −60 mV and has a midpoint of activation at −91 mV (Fig. 1A2) with steady state attained within 25–300 ms depending on the degree of hyperpolarization (Fig. 1A3). The time constants of activation and deactivation were determined using single exponential fits and were slower than those of I_{h} in pyramidal neurons, which activates three to five times faster when measured under similar conditions (Magee, 1998; Williams and Stuart, 2000; Berger et al., 2001).
To determine the spatial distribution of I_{h}, cellattached recordings were made using pipettes of the same tip size (estimated by pipette resistance) at different distances from the soma along the dendrite. The steadystate current measured at −142 mV is plotted as a function of the distance from the soma in Figure 1B. The apparent I_{h} current density was highly variable from patch to patch, both for somatic and dendritic patches. Mean current amplitude was 9.5 ± 1.2 pA (n = 53) at the dendritic recording sites and 1.2 ± 0.3 pA (n = 23) at the soma. A linear regression to the dendritic data points demonstrates that I_{h} has a uniform dendritic distribution (slope, −3 pA/100 μm; not significantly different from zero slope, p > 0.2).
Effect of I_{h} on PSP waveforms
To determine how I_{h} in Purkinje cells shapes synaptic potentials, we injected synaptic current waveforms via a dendritic wholecell recording and measured the resulting sEPSPs at the soma. We used the I_{h} blocker ZD 7288 (BoSmith et al., 1993; Gasparini and DiFrancesco, 1997), which blocks the hyperpolarizing “sag” and I_{h} in Purkinje cells (Roth and Häusser, 2001; Williams et al., 2002), to identify the contribution of I_{h}. After washin of ZD 7288, Purkinje neurons hyperpolarize, indicating the presence of an I_{h} conductance “at rest.” To study the effect of I_{h} on the shape of a synaptic potential in isolation of its effect on spontaneous firing (Häusser and Clark, 1997; Williams et al., 2002), and to achieve comparable conditions in control and in ZD 7288, neurons were held at −70 mV using holding current. Figure 2A shows an example of somatic sEPSPs resulting from dendritic waveform injection (134 μm from the soma; see inset) in control conditions and after washin of ZD 7288. As expected, the peak amplitude of the propagated signals is attenuated because of dendritic filtering (Fig. 2A, top vs bottom panel). Superposition of the control and ZD 7288 traces for both somatic and dendritic current injections (Fig. 2A, right panels) shows that blocking I_{h} has little effect on sEPSP rise time and peak, but slows the decay and blocks the undershoot of the sEPSP. This results in increased halfwidths both for the somatic and the propagated sEPSP (an effect that is also revealed in the integral of the sEPSP) (Fig. 2D). The effect of input location on sEPSP width recorded at the soma is illustrated by overlaying sEPSPs from somatic and dendritic injections in control and in ZD 7288 conditions (Fig. 2B). This comparison reveals that the halfwidth of the propagated sEPSP is only slightly larger than that of the somatically generated sEPSP, both with and without I_{h}, indicating that I_{h} does not have a strong distancedependent effect on somatic sEPSP halfwidth.
We examined how the effect of I_{h} on somatic sEPSP shape depends on dendritic location by injecting the synaptic current waveform at different distances from the soma. As expected from dendritic filtering, the 20–80% rise times of somatic sEPSPs depended on their dendritic origin (Fig. 2C) (slope_{ctrl} = 2.2 ± 0.2 ms/100 μm; slope_{ZD} = 2.5 ± 0.2 ms/100 μm; p < 0.05 compared with zero slope). In contrast, there was no significant location dependence of somatically recorded sEPSP integrals for dendritic inputs at different distances from the soma (Fig. 2D) (slope_{ctrl} = 5.7 ± 7.1 mV · ms/100 μm; p > 0.05 compared with zero slope). Application of ZD 7288 significantly increased somatic sEPSP integrals (31 ± 4 mV · ms in control; 122 ± 5 mV · ms in ZD 7288; mean ± SEM of all inputs; p < 0.05). However, there was also no dendritic location dependence in the presence of ZD 7288 (Fig. 2D) (slope_{ZD} = 1.4 ± 9.7 mV · ms/100 μm; p > 0.05 compared with zero slope).
We next explored the effect of I_{h} on summation of multiple sEPSPs by injecting trains of five EPSCs at 50 Hz at various somatic and dendritic locations while recording the somatic sEPSP (Fig. 3A). Temporal summation was defined as the increase in sEPSP peak resulting from the fifth relative to the first current waveform injection (Magee, 1999). In control conditions, somatic sEPSP summation did not exhibit significant location dependence for dendritic inputs at different distances from the soma (Fig. 3C) (slope_{ctrl} = 21 ± 19%/100 μm; p > 0.05 compared with zero slope). The degree of sEPSP summation was significantly enhanced in ZD 7288 compared with control (Fig. 3A). In addition to producing an upward shift on the yaxis (Fig. 3C) (shift_{sEPSP} = 134 ± 48%; p < 0.05), ZD 7288 also increased the slope of the relationship between summation and dendritic location of inputs (Fig. 3C) (slope_{ZD} = 46 ± 14%/100 μm; p < 0.05 compared with zero slope). Thus, although summation is location independent in control, block of I_{h} reveals slight location dependence, indicating that I_{h} helps to generate normalization of temporal summation in Purkinje cells. Similar results were found with respect to injections of single and trains of hyperpolarizing sIPSP waveforms (data not shown).
To study how the effect of I_{h} develops during an input train, we compared single and trains of sEPSPs in the same cell (Fig. 3D). The linear sum of five single sEPSPs at 50 Hz was calculated offline (Fig. 3D, linear sum), and perfectly matched the recorded response to a “real” train of EPSCs (Fig. 3D, overlay). Thus, I_{h} has the same impact on the first sEPSP as it has on the subsequent sEPSPs in a train. This result was consistent over a wide range of amplitudes of the injected current and also for negative waveform injections (data not shown). The linear prediction deviated from the recorded response only when action potentials were generated in response to the current injection. To examine whether I_{h} acts in this linear manner at all dendritic locations, we calculated the linear sum of single sEPSPs recorded at different locations (Fig. 2D) and compared them to summation of real trains in Figure 3C. The overlap of the linear prediction (dashed line) with real data demonstrates that the linear effect of I_{h} is location independent (Fig. 3C). The relationship between summation and sEPSP amplitude was examined by injecting trains of EPSC waveforms of increasingly larger amplitudes (0.2–2 nA) (Fig. 3D). We found that summation scales linearly for a broad range of first peak amplitudes (0–8 mV; n = 3). Essentially, this observed linear feature of the I_{h} current implies that, when analyzing dendritic or somatic inputs of different amplitudes, they can be scaled without changing the actual shape of the response, and therefore the normalization of temporal summation (which is a relative measure) is independent of the actual sEPSP peak or the amplitude of the injected current.
A detailed analysis of the linearity is provided in supplemental Figures 1 and 2 (available at www.jneurosci.org as supplemental material), which present a model in which the nonlinear activation curve of I_{h} is replaced with a linear one, and the voltagedependent activation/deactivation time constants are replaced with a constant value. The I_{h} conductance in this model is therefore voltage dependent, but in a linear way, and thus despite any transient change in conductance it imposes, the integration process remains linear. This model provides a surprisingly good approximation to the behavior of the current, and reproduces the essential features of the experimental data. As we show below, this linearity proves to be a crucial feature for understanding how I_{h} contributes to normalization of temporal summation.
Modeling the effect of I_{h} density and distribution in realistic dendritic geometries
The fact that a uniform dendritic distribution of I_{h} still leads to normalization of summation is puzzling, as is the linear effect of I_{h} on temporal integration despite its highly nonlinear activation curve. To explore these issues, we used a theoretical approach to allow us to systematically vary the properties of I_{h} and its distribution in different dendritic geometries. This allows us to assess the contribution of I_{h} to shaping synaptic potentials and temporal integration in a more systematic manner than is possible using an experimental approach (Magee, 1999; Williams and Stuart, 2000, 2003). We started by using simulations in detailed compartmental models of different cell types (Purkinje neurons as well as neocortical and hippocampal pyramidal neurons) with different densities and distributions of I_{h} (Fig. 4). These simulations show that, for each of these morphologies, the average density of I_{h} is a far more important determinant of the degree of summation than its dendritic distribution. When the density of I_{h} increases, in all morphologies and spatial distributions the profiles of summation as a function of distance follow the same trend. At low I_{h} densities, summation increases with distance from the soma; then at some intermediate I_{h} density normalization of temporal summation is achieved, and finally for higher densities distal inputs summate less than proximal ones. Remarkably, normalization is achievable for both uniform and nonuniform I_{h} distributions in all cell types for a specific density. Figure 4G demonstrates that this is a general phenomenon (tested on 12 cells). In other words, a dendritic gradient of I_{h} is not a necessary condition for normalization of temporal summation in neurons.
It is interesting to note that the Purkinje cells in the simulations show, in agreement with the experimental results, that even in the passive case summation is not strongly dependent on distance from the soma. Nevertheless, for some Purkinje cells, there is a slope in the profile of summation for proximal inputs (Fig. 4F, top, green). The length and steepness of this slope is dependent on the specific morphology of the cell (and especially on the “bottleneck” connecting the dendritic tree to the soma). This feature is not obviously apparent in the experimental results (Fig. 3C), presumably because the population data are based on averaging results from different cells with different structures. Figure 4 also demonstrates that the longer the cell the bigger the effect of the spatial distribution of I_{h} on the temporal summation profile (compare different profiles for Purkinje cells and layer 5 pyramidal neurons). The fact that Purkinje cells are electrotonically short partially explains why in these simulations they are insensitive to the spatial distribution of I_{h}. Nevertheless, the effect of the distribution of I_{h} on temporal summation is weak even in extremely long cells. This phenomenon is analyzed in detail in supplemental Figures 3 and 4 (available at www.jneurosci.org as supplemental material).
Dissecting the mechanisms underlying temporal summation normalization by I_{h}
To understand the mechanisms underlying the contribution of I_{h} to normalization of temporal summation, we turned to a simple model consisting of a single cylindrical dendrite. This reduced model proved capable of capturing all the phenomena found both in our experiments and in detailed models of dendritic neurons (Fig. 4; compare with supplemental Fig. 4, available at www.jneurosci.org as supplemental material). The contribution of I_{h} to shaping synaptic potentials and temporal integration can be broken down into four steps (Fig. 5). In step 1, we begin by modeling the condition when I_{h} is absent (i.e., where summation is determined only by the passive properties of the dendritic tree). Next, we add g_{h} to the model in two steps (see Materials and Methods): (1) the fraction of g_{h} that is open in the steady state at the resting potential (step 2; “static g_{h}”), and (2) the transient deactivation on depolarization (step 3). The static g_{h} condition is also a passive cylinder, but with a leakier membrane. The additional conductance has a negligible effect on the peak of the sEPSP (Hubbard et al., 1969), but shortens the membrane time constant, resulting in a shorter sEPSP halfwidth. Concomitantly, temporal summation is significantly reduced (see pink arrow) because of the slow time course of the “static shunt” (step 2; dashed pink trace; see figure legend), which leads to significantly more summation of the static shunt compared with the peak of the sEPSP. Interestingly, the effective linearity of I_{h} (Fig. 3 C–E; supplemental Figs. 1, 2, available at www.jneurosci.org as supplemental material) permits us to likewise use the difference between the sEPSPs in the two different conditions (step 3; “hidden sag”) to show that the effect of the transient deactivation of g_{h} (representing a change in conductance) is qualitatively similar to that of the static shunt. This result holds both for the kinetics measured in Purkinje cells as well as when the activation and deactivation time constants are sped up fourfold (as in the figure) (see also Materials and Methods).
In step 4, we examine how the effect of I_{h} depends on input location. The peak of the propagated sEPSP attenuates strongly for the distal input (left panel; compare with proximal). At the same time, the undershoot after the sEPSP is identical for both proximal and distal inputs (left panel; overlapping “undershoot”). In other words, the peak amplitude and the sag of the propagated distal sEPSP are affected differentially by cable filtering. To compare the relative contribution of the peak and sag for temporal summation, we normalize the proximal input to the peak of the propagated sEPSP. This scaling reveals a smaller relative undershoot of the proximal sEPSP than that of the propagated one (left panel; dashed line). Indeed both the static shunt and hidden sag of the distal input are larger than that of the normalized proximal input (compare middle panels). Although the difference in the undershoot is small for individual sEPSPs, being slow the static shunt and hidden sag summate more than the sEPSP peaks (see steps 2 and 3). Therefore, by the fifth input, the difference in the contribution of the sag of the distal input and the proximal input becomes large. In this specific example, it is precisely large enough to compensate for the difference in temporal summation induced by passive cable filtering (i.e., complete locationindependent normalization; right panel). In conclusion, the differential filtering of the sag compared with the peak, in conjunction with the linearity of I_{h}, is essential for normalization of temporal summation.
Analyzing how the dendritic distribution of I_{h} affects temporal summation
The model in Figure 5 explains how normalization can be achieved in an electrotonic structure with a uniform distribution of I_{h}. How does the dendritic distribution of I_{h} affect temporal summation? In Figure 6, we use the cylinder model to compare the effect of two spatial distributions of I_{h} on temporal summation: a uniform distribution (as in Fig. 5) and a nonuniform distribution, in which the same total amount of g_{h} is concentrated in the final 100 μm of the cylinder (Fig. 6A1, blue). Both of these distributions yield qualitative normalization of temporal summation (Fig. 6A2), with slightly different spatial profiles. The deactivation of g_{h} (the hidden sag) is responsible for the difference between these profiles as they are very similar when only the static g_{h} is present (Fig. 6A3) (London et al., 1999). We thus compare the properties of the hidden sag in the different distributions in Figure 6B. Local responses (i.e., injection and recording at the same site) both proximal (Fig. 6B1) and distal (Fig. 6B2) are sensitive to the spatial distribution of I_{h}, such that the amplitude of the hidden sags is larger when the current is injected where g_{h} is higher (Fig. 6B2, dashed blue trace), and vice versa. However, this local sensitivity of the hidden sag to the I_{h} distribution disappears when considering the propagated signals (Fig. 6B3). The linearity of I_{h} permits us to consider the hidden sag as a signal that propagates and is filtered by the cable. For the distal current injection, the propagated hidden sags of the two distributions are very similar when “recorded” at the proximal site (this is in contrast to the difference in the local hidden sags) (Fig. 6, compare B2 and B3, bottom panel). This shows that, although the spatial distribution of I_{h} has an effect on the local signals, its effect on the propagated ones is very small. Figure 6, C1 and C2, shows this as a function of input location.
In Figure 6C3, we compare the attenuation of the sEPSP peak and the hidden sag (solid vs dashed lines) as a function of input location for the two distributions. This highlights the differential filtering in which the peak attenuates more than the sag (Fig. 6B) because the hidden sag is a slow signal and is therefore less filtered by the cable. In conclusion, the differential filtering of the sag compared with the sEPSP peak in the presence of I_{h} exists for both uniform and nonuniform distributions of I_{h}, providing both distributions with the ability to achieve normalization of temporal summation.
Interestingly, both distributions have identical degrees of temporal summation when the current is injected at the most distal location (Fig. 6A2, arrows). This is a consequence of the identical propagated hidden sags (Fig. 6B3,C2, arrow) resulting from a larger local hidden sag in the nonuniform case, which attenuates more than that of the uniform case. How general is this finding? Figure 7 shows the amplitude of the propagated hidden sag (Fig. 7B) and the degree of temporal summation (Fig. 7C) for seven spatial distributions of g_{h}. Practically all distributions show a temporal summation profile that is only weakly dependent on input location when compared with the passive case. Furthermore, there is a onetoone relationship between the profile of the propagated hidden sag and the profile of temporal summation. Although some variability exists between the different conditions, we can derive bounds for the propagated sag (and hence temporal summation) for most input locations (supplemental Fig. 3, available at www.jneurosci.org as supplemental material). These findings confirm that the effect of I_{h} on temporal summation depends only weakly on its dendritic distribution.
Discussion
The strikingly nonuniform dendritic distribution of I_{h} in pyramidal cells has attracted widespread attention and remains the only example of a nonuniform dendritic voltagegated channel distribution that has both been characterized functionally and confirmed at the EM level. We showed that a nonuniform distribution of I_{h} is not a general feature of principal neurons in the CNS. Furthermore, we demonstrate that there is no causal link between the nonuniform distribution of dendritic I_{h} and normalization of temporal summation. Rather, our simulations demonstrate that the total density of I_{h} determines input summation. These findings place important constraints on our view of how the distribution and properties of I_{h} regulate synaptic integration.
Functional properties and dendritic distribution of I_{h} in Purkinje cells
Our experiments provide the first direct information regarding the properties and distribution of I_{h} in dendrites of Purkinje neurons. Cellattached recordings revealed that the basic voltagedependent properties of I_{h} in Purkinje cells are similar to those of I_{h} in other principal neurons. The halfactivation voltage we found is within the range found in other studies (Magee, 1998; Berger et al., 2001; Roth and Häusser, 2001; Kole et al., 2006). Although the activation curve predicts only a relatively small fraction of I_{h} active during spontaneous firing in Purkinje cells, the halfactivation voltage of HCN channels is sensitive to both extracellular potassium (Azene et al., 2003) and intracellular cAMP (Wang et al., 2002). The effective activation curve in Purkinje cells under physiological conditions is therefore likely to be depolarized compared with those obtained in cellattached experiments, and hence a significant fraction of g_{h} should be open under resting conditions in Purkinje cells, consistent with the hyperpolarizations observed with ZD 7288 application (Williams et al., 2002).
The kinetics of I_{h} in Purkinje cells are slower than for I_{h} in pyramidal neurons, with activation time constants being ∼300 versus ∼50 ms, respectively, at −80 mV (Magee, 1998; Williams and Stuart, 2000; Berger et al., 2001), but are still within the range associated with expression of the fast subtype of HCN channels, HCN1. This is in agreement with finding that the HCN1 subunit accounts for the majority of the I_{h} conductance of Purkinje neurons (Nolan et al., 2003). The discrepancy between the kinetics of I_{h} we measured and those in pyramidal neurons suggests that other, slower HCN subunits may also be expressed in Purkinje neurons (Notomi and Shigemoto, 2004). HCN1 and HNC2 subunits can combine to form stable channel complexes with heteromeric assembly, giving an I_{h} current with slower kinetics than that of homomeric HCN1 channels (Ulens and Tytgat, 2001). Alternatively, accessory subunits or modulation of HCN1 channels may contribute to changes in the kinetics of the current in Purkinje cells (Yu et al., 2001; Decher et al., 2003).
We showed that I_{h} exhibits a uniform distribution in the dendrites of Purkinje cells, in striking contrast to the principal neurons of the cortex and hippocampus. (Magee, 1998; Williams and Stuart, 2000; Berger et al., 2001; Kole et al., 2006). Consistent with our findings, HCN1 protein expression in Purkinje cells is low with no expression gradient detectable in dendritic shafts and spines with immunogold labeling (Lujan et al., 2005).
Our results suggests that the gradient of I_{h} density found in pyramidal neurons is not a general rule across all neuronal types, but rather may be a specific feature of these cells. Interestingly, recent experiments on pyramidallike principal neurons in the stratum radiatum region of hippocampus have revealed a reverse I_{h} gradient with higher somatic than dendritic expression (Bullis et al., 2007).
I_{h} and temporal summation in Purkinje cells
Despite the impressive dendrites of Purkinje cells, their unusual geometry, consisting of short spiny branchlets connected to a thick trunk, results in the halfwidth of dendritic EPSPs measured at the soma being mostly location independent, even in the passive case (Roth and Häusser, 2001; Vetter et al., 2001). Thus, Purkinje cell dendrites act to “selfnormalize” the EPSP shape, such that temporal summation of dendritic inputs is relatively location independent (Figs. 2⇑–4).
We demonstrate that I_{h} shortens the time course of EPSPs in Purkinje neurons, reducing their halfwidth (with little effect on peak amplitude), in agreement with results in pyramidal neurons (Magee, 1998; Williams and Stuart, 2000; Berger et al., 2001). Using a model we show that this effect of I_{h} on EPSP shape can be well described by a twostep process. First, the effect of the static “resting” conductance of I_{h} shortens the membrane time constant. Second, the depolarization deactivates I_{h} channels because of the voltage dependence of I_{h}. Because I_{h} is an inward cationic current, this has a hyperpolarizing effect and thus accelerates the decaying phase of the EPSP. Although the EPSP undershoot is the “signature” of I_{h}, most of this effect takes place during the EPSP depolarized phase, hence the term “hidden sag” (Fig. 5). Because the time course of the static shunt and the hidden sag are slow and tend to summate more than the peaks of the relatively fast transient EPSP, the effect of I_{h} on temporal summation increases during an EPSP train.
We show that, surprisingly, the I_{h} response to multiple asynchronous inputs scales linearly and is identical with superposition of the individual inputs. Each input in the train has an equal contribution to the reduction of summation generated by I_{h}. Our simulations confirm that this linearity holds true for a wide range of resting potentials and activation–deactivation time constants (supplemental Figs. 1, 2, available at www.jneurosci.org as supplemental material). This linear property only breaks down when the depolarization is large enough to activate other currents that elicit highly nonlinear responses such as dendritic spikes or axonal action potentials.
How does I_{h} normalize temporal summation?
Our simulations reveal that the EPSP sag originating in the dendrites attenuates less than the EPSP peak. This differential cable filtering is the key feature enabling normalization. For distal inputs, the relative effect of the sag on temporal summation is larger than for proximal ones. Upregulating the density of g_{h} can increase the amplitude of the propagated sag from distal locations sufficiently to compensate for the increase in temporal summation caused by passive dendritic filtering, and thus generate normalization of temporal summation. The differential filtering exists for all spatial distributions of the g_{h} conductance, indicating that normalization of temporal summation can be achieved with uniform as well as nonuniform distributions of g_{h}. This challenges the commonly held view that the dendritic gradient of I_{h} in pyramidal cells and normalization of temporal summation are causally related. Our findings are further supported by recent experiments showing that one can achieve normalization of summation even with a reverse somatodendritic gradient of I_{h} (Bullis et al., 2007).
Our dissection of the mechanism by which I_{h} normalizes temporal summation takes advantage of the surprising linearity of I_{h} observed in our experiments. The boundaries on temporal summation obtained in Figure 7 and supplemental Figure 3 (available at www.jneurosci.org as supplemental material) rely on the properties (and especially the reciprocity) of transfer resistance in linear systems (Koch, 2004). Importantly, this reciprocity has been observed in a study of I_{h} in dendrites of layer 5 pyramidal neurons (Berger et al., 2001).
What is the functional role of the dendritic I_{h} distribution?
Both the local hidden sag and the static shunt are larger in the dendrites for models exhibiting a dendritic I_{h} gradient compared with the uniform case (Fig. 6B2). This suggests that spatial gradients of I_{h} will have a strong effect on local synaptic integration in distal dendrites. Indeed, several experimental studies have shown that I_{h} in neocortical and CA1 pyramidal cells can affect local integration in distal dendrites, where it reduces the effective membrane time constant and EPSP summation (Magee, 1999; Poolos et al., 2002), while also lowering the threshold for initiation of dendritic Ca^{2+} spikes (Berger et al., 2003). However, it remains to be demonstrated directly (e.g., using molecular manipulations) that such effects are causally dependent on expression of an I_{h} gradient.
Our results provide a new framework for understanding how the local and global effects of I_{h} on neuronal excitability are integrated. The fact that a nonuniform I_{h} distribution can influence local integration in dendrites while keeping temporal summation normalization at the soma unchanged permits a high degree of flexibility. In particular, our results suggest that a neuron may regulate the degree of temporal summation by controlling the total number of I_{h} channels, while at the same time local excitability can be determined by regulating the gradient of I_{h} expression. Indeed, recent studies show that both global and local regulation of I_{h} can be observed after synaptic plasticity (Wang et al., 2003; Fan et al., 2005; Xu et al., 2006). Regulation of I_{h} expression may be under homeostatic control (Marder and Goaillard, 2006; Rabinowitch and Segev, 2006) to maintain independent operation of these effects, such that upregulation of I_{h} in a specific dendritic subregion will be accompanied by a corresponding downregulation of I_{h} in other subregions, allowing the total amount of I_{h} and the degree of normalization of temporal summation to be preserved. Resolving how these effects of I_{h} are “read out” by the neuron to in turn regulate its distribution and thus tune neuronal excitability should lead to a deeper understanding of the origin, maintenance, and functional relevance of nonuniform channel distributions in dendrites.
Footnotes

This work was supported by grants from the Wellcome Trust, the Medical Research Council, and the Gatsby Foundation. K.A. was supported by the Lundbeckfonden, and M.L. was supported by a Human Frontier Science Program LongTerm Fellowship. We thank Arnd Roth, Andreas Schaefer, Julian Jack, and Steve Cox for helpful discussions, and Nelson Spruston, Muki Rapp, and Beverley Clark for contributing neuronal reconstructions.
 Correspondence should be addressed to Michael Häusser, Wolfson Institute for Biomedical Research, University College London, Gower Street, London WC1E 6BT, UK. m.hausser{at}ucl.ac.uk
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