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Institute for Theoretical Biology, Department of Biology, Humboldt University Berlin, 10115 Berlin, Germany
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Johannes Müller Institute for Physiology, Charité, Humboldt University Berlin, 10117 Berlin, Germany
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Abstract |
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 Top Abstract IntroductionMethodsResultsAppendixReferences |
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(Received 14 June 2004;
accepted after revision 21 July 2004;
first published online 29 July 2004)
Corresponding author I. Erchova: Unité des neurosciences Intégratives et Computationelles, Institut de Neurobiologie Alfred Fessard, Bâtiment 33, 1, Avenu de la Terrasse, 91198 Gif-sur-Yvette, France. Email: irina.erchova{at}iaf.cnrs-gif.fr
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Introduction |
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TopAbstractIntroductionMethodsResultsAppendixReferences |
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MPOs are closely related to resonant behaviour (Lampl & Yarom, 1997; Hutcheon & Yarom, 2000). Resonance properties have been studied by injecting sinusoidal currents with fixed frequencies (Falk & Fatt, 1964; Cole, 1968; Nelson & Lux, 1970; Leung, 1998) or currents whose frequency varies monotonically in time. The latter ‘impedance amplitude profile’ (ZAP) functions allow one to rapidly characterize the resonance properties (Gimbarzevsky et al. 1984; Puil et al. 1986; Hutcheon & Yarom, 2000). Using this method, it has been shown that neocortical neurones (Jansen & Karnup, 1994; Gutfreund et al. 1995; Hutcheon et al. 1996a) and neurones in the trigeminal root ganglion (Puil et al. 1987), mediodorsal thalamus (Puil et al. 1994) and medial geniculate body (Tennigkeit et al. 1997) exhibit an electrical resonance while other neurones act as low-pass filters (Gimbarzevsky et al. 1984). Resonance has also been observed in stellate cells of the entorhinal cortex (Haas & White, 2002), but a systematic study of the resonance properties in entorhinal cortex including a comparison of the main cell classes has not yet been performed. This is, however, of general interest as different cell types within the entorhinal cortex (Van der Linden & Lopes da Silva, 1998) have different targets in the hippocampus. Stellate cells project to the dentate gyrus (Dugladze et al. 2001) while layer III cells project directly to area CA1 (Boulton et al. 1992; Empson et al. 1995). Both cell types have been shown to transfer synaptic inputs in a frequency-dependent manner, with EC layer III cells being most effective at low frequencies (up to 10 Hz) and stellate cells effective at higher (above 5 Hz) frequencies (Gloveli et al. 1997b). We therefore investigated the resonance properties of EC layer II and III cells and their relation to intrinsic subthreshold membrane potential oscillations. Our data analysis is based on a minimal mathematical model which facilitates the qualitative understanding as well as the quantitative comparison of the observed phenomena. This phenomenological description captures the data without any assumption about the intracellular mechanism responsible for the resonance and can thus be applied to any cell type. The model also provides a simple explanation why, contrary to heuristic reasoning, the resonance frequency of a given cell can be larger than its oscillation frequency.
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Methods |
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TopAbstractIntroductionMethodsResultsAppendixReferences |
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21°C). Slices for electrophysiological studies were transferred, one at a time, to an interface recording chamber and perfused with aCSF (1.6 ml min–1) at 35°C. Electrophysiology
Intracellular recordings were obtained using sharp micropipettes filled with 2 M potassium acetate containing 1% biocytin (75–85 M
) and an intracellular recording amplifier (Neuro Data model IR-183, New York, USA). All recordings were carried out in current-clamp mode; the series resistance was compensated through bridge balance under visual guidance and verified several times during the recording session; and capacitance transients were removed using the capacitance compensation circuit of the amplifier. The recorded data were low-pass filtered at 3 kHz and digitized by an IO card (DAQCard-AI-16E-4, National Instruments Inc., TX, USA) at a sampling rate of 8 kHz. To block GABAA, GABAB and ionotropic glutamate receptors, the following drugs were added in all experiments to the aCSF (µM): 30 6-cyano-7-nitroquinoxaline-2,3-dione (CNQX), 60 DL-2-amino-5-phosphonopentanoic acid (APV), 5 bicucculine (all from Sigma-Aldrich, Deisenhofen, Germany) and 1 CGP55845A: 3-N-[1-(s)-(3,4-dichlorophenyl)-ethyl]amino-2-(s)-hydroxypropyl-P-benzyl-phosphinic acid (CGP), a GABAB blocker, kind gift from Novartis, Basel, Switzerland).
Stimulus generation
For stimulus generation and data acquisition the program Labview (National Instruments Inc., TX, USA) was used. Hyper- and depolarizing current steps with a duration of 400 and 500 ms were applied to obtain information about the input resistance and general cell properties. To study subthreshold membrane potential oscillations (MPOs), depolarizing currents that lasted for 1 or 10 s were injected such that cells were just below firing threshold.
To obtain impedance amplitude profiles (ZAPs), a frequency-modulated sinusoidal current whose frequency f(t) increased linearly in time (Gimbarzevsky et al. 1984; Puil et al. 1986; Hutcheon & Yarom, 2000) was applied:
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| (1) |
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| (2) |
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Histology
After filling the recorded neurones with biocytin, the slices were incubated and stored in 4% paraformaldehyde in 0.1 M phosphate buffer. Shortly before staining, the slices were left in the 30% sucrose solution for 30 min. For staining, the slices were cut at 50 µm on a cryotom (Leika Jung RM 2035, Leitz, Nussloch, Germany). After washing the slices three times in 0.1M phosphate buffer, they were incubated overnight in 0.1 M phosphate buffer containing 1% Triton X-100 (Sigma-Aldrich, Deisenhofen, Germany) and 0.1% avidin Alexa Flour 488 conjuate (AFK, Molecular Probes A-21370, Leijden, the Netherlands). Then the slices were washed again three times with 0.1 M phosphate buffer, mounted on coated object slides, dried overnight at room temperature, dehydrated in 70%, 80% and 96% ethanol and in isopropanol, and glass-covered using 50 ml ProlongAnti-fade Kit (Molecular Probes P-7481, Leijden, the Netherlands) to preserve fluorescent colours. Pictures of stained cells were made using a confocal laser scanning microscope (Leitz, Nussloch, Germany). Cell were identified on the basis of their laminar position and shape.
Analysis of electrical resonance
The frequency-dependent impedance (or ‘transfer function’) was calculated from the ZAP recordings as the ratio between the Fast Fourier Transform (FFT) of the output (measured voltage V) and the input (applied current I):
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| (3) |
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| (4) |
Transfer functions obtained with the ZAP method (eqns (3) and (4)) were fitted with the theoretical impedance–frequency function Ztheory(f) given in eqn (A9), using an iterative algorithm and least-square methods. The phenomenological model (Fig. 2A) underlying the function Ztheory(f) is introduced in the section ‘Mathematical model’, its response properties are discussed in the Appendix. After a satisfactory fit (residual errors vary less than 0.001 per iteration step) was obtained, the model parameters were estimated. To characterize the shape of the impedance profiles, several phenomenological parameters were calculated too (see also Fig. 2B):
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From a purely mathematical point of view, any Q-value larger than unity indicates a resonance. However, only resonances with sufficiently large Q-values will surpass the intrinsic noise level and thus be of biological relevance. With respect to a functional classification, cells with Q-values between, say, 1.0 and 1.1 can therefore hardly be considered as resonant; on the other hand, resonance curves with Q-values above 1.3 clearly indicate resonant behaviour. As no rigorous functional definition of a ‘best’ boundary value between the two dynamical regimes is possible, we took an intermediate value and considered cells with a Q-value greater than 1.2 as resonant cells. Similarly, cells with a Q-value of less than 1.2 and a D-value of less than 0.8 are referred to as low-pass filters whose attenuation properties are characterized by fHD if this value can be determined.
Analysis of sag potentials
Upon depolarizing or hyperpolarizing current injections a cell might generate a ‘sag potential’, i.e. an overshooting deflection of the membrane potential. From a mathematical point of view, two subtypes can be distinguished (see also Fig. 8 and the Appendix): sag potentials where a damped oscillation with successively decreasing extrema follows the initial deflection (scenario A) and sag potentials with only a single overshoot (scenario B-I). The transition between the two regimes is continuous; due to stochastic components of the intrinsic dynamics, the damped oscillation pattern of scenario A may not even be visible in a recording. To characterize the sag potentials, we fit the appropriate solution of the biophysical model, given by eqn (A13) (scenario A) or eqn (A14) (scenario B-I).
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Several complementary methods were used to analyse the spectrum and coherence of the MPOs (see Fig. 9 for an example). An analysis based on Windowed Fourier Transforms (WFT) was performed with a running window of length 950 ms and window overlaps of 500 ms using the Welch method (a non-parametric periodogram estimate based on splitting the time series in overlapping segments multiplied by data windows, and on the ensemble average of periodograms computed separately in each data window). Each window was smoothed using a Hanning function to decrease aliasing effects. This method allows one to reduce noise components from the spectrum of Fast Fourier Transform (Lyons, 1998) but introduces an error in the frequency resolution. The WFT is useful for extracting local frequency information, but is rather inaccurate for time–frequency localization, as it imposes a time window and aliases high- and low-frequency components that do not fall within the frequency range of the window.
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Power spectra were estimated from both methods using data samples that lasted for 10 s. The peak frequency was taken as a measure for the dominant frequency and the full width at half height was taken as a measure for the coherence of the oscillations. In addition, an auto-correlation analysis was performed to analyse local oscillatory properties. Auto-correlograms (bin width 1.25 ms) were calculated from the same data sets. The time interval between the central and first side peak was used as an estimate for the dominant frequency. The ratio between the magnitudes of the first and second side peaks will be called ‘relative decay’ and served as a further measure for the internal coherence of the oscillations.
The validity and resolution errors of all methods were tested on simulated data traces (sinusoidal 8 Hz oscillations with a duration of 10 s and an amplitude of 2 mV). The frequency estimation from the WFT was 7.99 Hz; the full width at half height was 1.5 Hz. The wavelet analysis led to a frequency estimate of 7.86 Hz, with a full width at half height of 1.6 Hz. The auto-correlation analysis resulted in a frequency of 8.0 Hz and a relative decay of 0.98. All three methods closely coincide with respect to the dominant frequency. While the auto-correlation analysis seemed to be the most precise for the simulated data this method is sensitive to noise in the recorded data and may fail for neurones showing MPOs with low coherence. We therefore always compared the results of all three methods and then computed average values.
Mathematical model
The measured neurophysiological data were analysed within a quantitative framework. We focused on a minimal phenomenological model capable of capturing the essence of the observed neural properties. Using this approach we can precisely determine the differences between the studied cell types, investigate the relation between subthreshold oscillations and resonance phenomena, and make quantitative predictions about responses to time-varying inputs.
The least complicated model accounting for the data is illustrated in Fig. 2A in terms of its equivalent RLC circuit. We chose this intuitive interpretation of the neural dynamics throughout the paper because of its simplicity and direct relation with previous approaches based on the analogy with electrical circuits; see, for example, Koch (1984) or Hutcheon & Yarom (2000). Alternatively, the model can be interpreted as a systematic and linearized reduction of Hodgkin-Huxley type dynamics, as has been shown by various authors; see, for example, Koch (1999) or Richardson et al. (2003).
The model is linear, assumes an isopotential neurone, and consists of two parallel branches. The first branch is characterized by a resistance R in parallel with a capacitance C and mimics the dynamics of a leaky-integrator model neurone. The second branch consists of a resistance RL in series with an inductance L and endows the model with the general dynamical properties of delayed rectifying currents. The model's effective input resistance 
is given by = RRL/(R + RL).
Note that the parameters R, C, RL and L are phenomenological parameters that depend on the state (e.g. holding potential) of a neurone. In particular, the capacitance C is an effective quantity that will in general differ from the membrane capacitance (Mauro et al. 1970). It is therefore of particular interest to investigate which parameters change as, for example, the holding potential is varied, and how they change under such variations. These data can help to constrain later detailed biophysical models. In addition, they illustrate the limitations of an overly simplified view that interprets the four parameters as fixed cell properties.
The time evolution of the model is given by two coupled ordinary first-order differential equations:
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| (5) |
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This minimal model is deterministic and does not cover the stochastic fluctuations seen in measurements of intrinsic subthreshold oscillations. These fluctuations represent a dynamic balance between damped oscillations of the membrane potential and intrinsic excitations caused by random opening and closing of ionic channels (see, e.g. the review by White et al. 2000). However, such intrinsic noise sources can be easily incorporated into the model. In the spirit of quasi-active neurone models with stochastic components (Steinmetz et al. 2000), we assume that channel noise generates a stochastic intrinsic current Iint(t). Due to the voltage dependence of the channel kinetics, the statistical properties of Iint(t) will in general depend on the mean membrane potential.
The intrinsic current has to be added to the externally applied current Iext(t) so that the total current I(t) becomes:
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The model can be used in these two very different experimental conditions and is thus well suited to quantify salient characteristics of autonomous subthreshold oscillations as well as responses to time-dependent inputs, such as resonance phenomena and sag potentials, within a single mathematical framework.
Statistics
Properties of the identified cells will be characterized by their mean values and standard error of the mean. Resonance and oscillation parameters are given as mean values and standard deviation. Groups were compared using Student's t tests. The significance of a correlation coefficient r was also calculated with Student's t test, t = r(n – 2)1/2(1 – r2)–1/2. n is the number of samples. P-values were obtained from standard tables with degree of freedom d = n – 2.
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Results |
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Membrane potential resonance
Stable recordings were obtained from 67 entorhinal cortex cells in layers II and III. Within this population, 46 cells were identified as stellate cells based on a sag potential occurring both during hyper- and depolarizing current injection, a membrane time constant between 6 and 9 ms, and a resting membrane potential between –58 and –67 mV (Alonso & Klink, 1993; Jones, 1994); 21 of these cells were also morphologically identified as stellate cells (see, for example, Fig. 1A). The cells that were not classified as stellate cells include eight pyramidal cells (see also Fig. 3A) of which five were also morphologically identified. The remaining 13 cells did not show a sag potential and/or had slower membrane time constants and will be referred to as ‘other cells’ (see also Fig. 3B). The data on spike amplitudes, input resistances, resting potentials and time constants are summarized in Table 1. All cells had a membrane potential below –58 mV, an input resistance larger than 20 M and an overshooting action potential.
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To quantify the resonance properties of a given cell, the impedance-frequency function Ztheory (eqn (A9)) was fitted to the experimental data. In terms of the model framework (Fig. 2), this provides a compact four-dimensional description (C, L, R and RL) of the resonance behaviour. Average parameters for different cell classes (at resting membrane potential) are summarized at Table 2. As discussed in Methods and the Appendix, these parameters are phenomenological; in particular, they may depend on the holding potential. For the sample cell of Fig. 1A, the membrane impedance Z is shown in Fig. 1C as a function of the stimulus frequency. The cell has a pronounced resonance (Q = 1.8) at a frequency of 8.9 Hz, with a half-band width HB of 10.0 Hz, and a high-frequency decay D of 1.01 (see Methods for the definition of these quantities). The impedance profile of a characteristic layer III pyramidal cell (Fig. 3A) is shown in Fig. 3C. With a Q-value of 1.08, the resonance of this cell is too weak to surpass the noise level under realistic conditions. For the typical multipolar cell depicted in Fig. 3B, the same is true (Q = 1.03) as evident from Fig. 3D.
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Among the other cells, two neurones exhibited a weak resonance with resonance frequencies of 7.4 Hz (Q = 1.28) and 6.0 Hz (Q = 1.23). We also found five cells whose impedance varied only a little between 1 and 20 Hz. Three of these cells displayed membrane potential fluctuations of up to 2 mV but a dominant frequency could not be determined because of very low coherence of the fluctuations. All other cells (n = 11) showed low-pass filter properties with impedance values decreasing at 20 Hz to less than 60% of the value measured by a constant current of the same amplitude. Two of these neurones exhibited a 50% decrease in their impedance already at 10 Hz, five at frequencies below 15 Hz and four other cells at frequencies between 15 and 20 Hz. Some of these cells had a small impedance maximum at frequencies below 3 Hz with Q-values of less than 1.1. A typical example is the multipolar cell of Fig. 3B and D.
Voltage dependence of the electrical resonance and sag potentials
Neural dynamics are governed by voltage-dependent conductances. A more accurate model description of the resonance properties would therefore involve voltage-dependent parameters. The required resonance measurements at different values of the membrane potential are shown in Fig. 5A. These measurements were carried out in seven similar cells at several individually adjusted levels of the membrane potential for each cell, from a slightly hyperpolarized level to the maximum possible subthreshold depolarization. For each resonance measurement, a separate fit to the model (eqn (A1)) was carried out. The changes of the various parameters as the membrane potential was varied are presented in Fig. 5B–F. Definitions of the parameters are given in Method and the Appendix. Surprisingly, both the resonance frequency fres and the sharpness Q remain almost constant for each investigated cell (Fig. 5B and C). The input resistance and natural frequency fnat increase slightly upon depolarization (Fig. 5D and E) whereas the decay factor 
decreases (Fig. 5F).
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should decrease with increasing membrane potential and vanish at the firing threshold. Similar, the natural oscillation frequency fnat should approach the resonance frequency fres as the cell is more and more depolarized and become equal to fres at the firing threshold (see Appendix). Both predictions are in agreement with the experimental data as shown by Fig. 5F and by a comparison of Fig. 5E with Fig. 5B. When the ZAP current was injected at more depolarized membrane potentials action potentials were induced (Fig. 5A, top trace, left panel). Nevertheless the impedance profile which now also reflects the generation of action potentials still shows a peak near the subthreshold resonance peak (Fig. 5A, right panels). The frequency f(t) of the ZAP-function input (eqn (2)) increases linearly in time (eqn (3)) so that equal frequency bandwidths are covered in equal time intervals. Taking advantage of this property, we quantified the suprathreshold neural response within four adjacent frequency bands, each with a width of 5 Hz. The cell shown in Fig. 5A generated a total of 229 spikes in the 10 trials: 14 (6.1%) in the frequency range from 0 to 5 Hz, 115 (50.2%) in the 5–10 Hz range, 96 (42.0%) in the 10–15 Hz range, and 4 (1.7%) in the 15–20 Hz range. Across all investigated stellate cells these ratios were: 3.3%, 48.1%, 43% and 5.6% for the four frequency bands, respectively. Notably the cell with a resonance frequency around 17 Hz (shown by the plus marks in Fig. 5B–E) did not spike at all at frequencies below 5 Hz and produced 13.0%, 45.1% and 41.9% spikes in the remaining three consecutive frequency bands. Together these findings suggest a close relation between the subthreshold properties and the suprathreshold firing characteristics.
Evidently, a minimal isopotential model that is based on parameters obtained at one single level of the membrane potential cannot fully describe the entire neural subthreshold behaviour, e.g. sag potentials (Van der Linden & Lopes da Silva, 1998). Nevertheless, we tried to predict their size and time course, based on model parameters obtained at the level of resting membrane potential, as shown in Fig. 5G. As the model is linear without voltage-dependent dynamics, deviations of the measured sag potentials from the model predictions are therefore clear indicators for such dynamics.
Close inspection of Fig. 5G reveals that for depolarizing step currents, the overshooting transients are indeed larger than the theoretical predictions, although the cell's long-term behaviour is in excellent agreement with the model at all tested depolarization levels. For hyperpolarizing step currents, on the other hand, the entire voltage response confirms the model predictions apart from an overall scaling factor. These findings imply that currents with different dynamics are activated depending on the level of the membrane potential. In the hyperpolarized regime, the observed phenomena can be explained by a rapidly activated and long-lasting current. Such dynamics have been reported for the Ih current (see, for example, Dickson et al. 2000a). In the depolarized regime, however, the voltage-dependent conductance quickly inactivates after the initial transient. This may be explained by a de-activation of the hypothetical Ih current or an activation of a slow delayed-rectifier potassium current, such as IKS, IM, or ID. All these currents have been previously described in stellate cells (Eder et al. 1991; White et al. 1993; Eder & Heinemann, 1994, 1996; Richter et al. 2000; Shalinsky et al. 2002).
The filter characteristics of layer III pyramidal cells remained about the same when the membrane potential was varied, similar to the situation in stellate cells (Fig. 6). The input resistance of the pyramidal cells increased with increasing membrane potential (Fig. 6D), again in accordance with the results from stellate cells (Fig. 5C). Upon supra-threshold depolarization the cell of Fig. 6A fired predominantly at frequencies around the modest maximum of the resonance profile. The cell shown in Fig. 6A produced 9–15 spikes in each of the 10 repetitions, and a total of 131 spikes: 94 (71.8%) in the frequency range between 0 and 5 Hz and 37 (28.2%) in the 5–10 Hz range. Across all investigated cells this ratio was 86.3%, and 13.7% for these two frequency bands. Notably, none of the investigated cells spiked when presented with ZAP frequencies above 10 Hz. Thus, as for stellate cells, subthreshold properties strongly influence the firing pattern above but close to the firing threshold.
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and ß (see Appendix), that allow one to characterize the neural responses within a two-dimensional graphical representation (Fig. 8). Based on the model framework, damped oscillations are expected in a certain parameter region, which we call region A (see Appendix and Fig. 8). Sag potentials should exist in region A (multiple overshoots) as well as in region B-I (single overshoots); see eqns (A13) and (A14), respectively. Cells that fall into region B-II should not exhibit sag potentials but rather approach the new holding potential in a monotone fashion. Transitions between the expected phenomena in the different regimes are smooth, as also demonstrated by the schematic examples in the right panels of Fig. 8. Furthermore, secondary and subsequent overshoots (in parameter regime A) may hardly be visible in recorded data as the oscillation amplitude rapidly decreases for realistic values of the decay factor . Finally, it is important to realize that the boundary between the domain where sag potentials occur (regions A and B-I) and where no sag potentials occur (B-II), does not coincide with the curve that separates band-pass behaviour (Q > 1, above the dashed line in Fig. 8) from low-pass behaviour (Q = 1, below the dashed line) in resonance experiments, cf. eqn (A21).
To test how these predictions relate to the observed data, the parameters and ß (eqns (A15) and (A16)) were calculated for all neurones from the resonance experiments. Putting each cell at its proper position in the /ß-plane (Fig. 8) demonstrates that the different cell classes cluster in different regions of this plane. All stellate cells (filled circles) fall into region A, and far from the boundary with region B-II. Based on data from the resonance experiment only, the model thus predicts that stellate cells exhibit prominent sag potentials, as indeed observed (Fig. 4B).
All but one EC layer III pyramidal cell are below or very near the dashed line, in accordance with Fig. 4G. Some of these cells do fall into region A, but close to the boundary with region B-II. The expected weak sag potentials were not observed. This can be attributed to the difficulty of clearly identifying sag potentials from voltage traces in this transition zone. The remaining ‘other cells’ did show weak sag potentials, again as predicted by the model.
Membrane potential oscillations
Upon depolarizing current injection to membrane potential values near the firing threshold, spontaneous membrane potential oscillations (MPOs) were displayed by all stellate cells tested for this behaviour. In preliminary experiments we had found that the generation of an action potential generally resulted in subsequent MPOs with larger amplitudes than those evoked after depolarization that do not cause action potentials. To avoid any spurious effects our data analysis is therefore exclusively based on recordings that did not contain any action potentials.
Typical MPOs are depicted in Fig. 9A. They were generated by the sample stellate cell of Fig. 1A in response to a constant depolarizing current injection of 140 pA. A characteristic feature of the observed subthreshold activity is its irregularity compared to an ideal periodic oscillation. This is also evident from the power spectral analysis which generally resulted in broad frequency distributions. For the particular cell shown in Fig. 9A, the peak frequency is 8.9 Hz (Fig. 9B). The full width at half-maximum is 3.5 Hz (from 7.4 to 10.9 Hz). An auto-correlation analysis (Fig. 9C) provides information on both the dominant frequency and the coherence of the oscillations. In this particular cell, the dominant frequency is 9.5 Hz, close to the value obtained with the power spectral analysis. The coherence of the MPOs, defined as the ratio of the second to the first side peak in the auto-correlation function, is 0.22. This ather low value is in accordance with the cell's power spectrum and typical for the whole cell population (data not shown). A wavelet analysis (Fig. 9D) reveals the time-resolved frequency distribution of the membrane potential fluctuations. The time course is characterized by a waxing and waning of different frequency components within the MPOs, in agreement with the low MPO coherence of this cell. The temporal average of the wavelet spectrum from Fig. 9D is depicted in Fig. 9E. Its peak frequency of 8.7 Hz is close to the values obtained from the power spectrum and auto-correlation analysis. To reduce spurious effects due to possible artefacts of each individual method, oscillation frequencies fosc were always determined as the average obtained from the three methods. For all but three cells the three different measures agree well (Fig. 9F–H) so that we may use average values to reduce potential numerical artifacts. A summary of the oscillation frequencies fosc from all analysed stellate cells (n = 30) is shown in the inset of Fig. 10. The distribution of measured fosc values extends from 3 to 14 Hz with a mean value of 9.0 Hz and a standard deviation of 2.2 Hz.
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Within the model framework, these findings are readily explained. According to eqn (8), the power spectrum of the observed MPOs is equal to the product of the intrinsic noise power spectrum and the squared impedance profile. This profile is known from the resonance experiments (Fig. 1C). We did not attempt to measure the detailed frequency dependence of the noise spectrum as we were primarily interested in the relative positions of the maxima of |V|(f) and |Z|(f). However, it is well known that channel noise generates a rather smooth spectrum that gradually falls off with increasing frequency f (Hille, 1992; White et al. 1998). Equation (8) then predicts that fosc, i.e. the frequency at which |V|(f) has its maximum, is smaller than fres, i.e. the frequency at which |Z|(f) has its maximum. This prediction is in full agreement with the measured data for every single neurone (Fig. 10). The difference between fres and fosc varies from cell to cell, which may be due to slight variations of the intrinsic noise spectrum. Care should be taken, however: membrane potential oscillations were measured near threshold whereas the resonance profiles were derived from data close to the resting potential. Thus |V|(f) and |Z|(f) correspond to different neural states. However, as shown in Fig. 5B, the resonance frequency is almost independent of the membrane potential so that fres and fosc may, indeed, be directly compared.
Let us finally turn to the oscillatory properties of non-stellate neurones: pyramidal cells did not exhibit MPOs, as described earlier for layer III projection cells by Gloveli et al. (1997a, 1999) and Van der Linden & Lopes da Silva (1998). We also found five ‘other’ cells whose impedance varied only a little between 1 and 20 Hz, and three of these cells displayed fluctuations of membrane potential of up to 2 mV. However, a dominant frequency could not be determined reliably because of very low coherence of the membrane potential fluctuations. Therefore, within the studied cell classes, pronounced membrane potential oscillations seem to be a characteristic feature of stellate cells.
Discussion
Our findings about the intrinsic dynamics of different neurones within the superficial layers of the entorhinal cortex can be summarized as follows: EC layer II stellate cells exhibit both a prominent peak of their frequency-resolved membrane impedance and noise-driven subthreshold membrane potential oscillations in the upper theta frequency range. Contrary to this situation, non-oscillatory EC layer III cells (pyramidal neurones and ‘other cells’) have at most a small impedance maximum at a frequency below the theta range or display only pure low-pass filter characteristics. In accordance with the theoretical predictions, all resonant cells react with a prominent sag potential upon both depolarizing and hyperpolarizing current injections and all oscillatory cells exhibit resonance. The frequency of near-threshold MPOs in stellate cells is similar but always lower than their resonance frequency, measured at the resting potential. Extending and complementing previous findings, these data suggest that different cell types in the entorhinal cortex vary significantly with respect to their integrative properties in the temporal domain. As EC layer II stellate cells provide the major input to the dentate gyrus while EC layer III pyramidal cells mainly project directly to area CA1, our results also suggest that time-structured information is transmitted in a differential and frequency selective way to different down-stream regions of the hippocampal formation.
Membrane potential oscillations
Previously, MPOs had been reported for stellate cells of layer II of the entorhinal cortex (Alonso & Llinas, 1989; Van der Linden & Lopes da Silva, 1998; Dickson et al. 1997, 2000a), for deep-layer projection cells of the entorhinal cortex (Schmitz et al. 1998; Dugladze et al. 2001; Gloveli et al. 2001) and for cells of the perirhinal cortex (Bilkey & Heinemann, 1999). Oscillatory properties of CA1 pyramidal cells have also been observed upon depolarization to values very close to or above firing threshold (Buhl et al. 1996; Pike et al. 2000). In addition, many cortical pyramidal cells display membrane potential oscillations when depolarized or exposed to carbachol (Metherate et al. 1992; Amitai, 1994). Frequencies range from 4 to 16 Hz and can thus be classified as theta activity according to the rodent literature (for a review see, for example, Gottesmann, 1992a,b). It has also been suggested that such MPOs are involved in the generation of network oscillations in the same frequency range (Alonso & Llinas, 1989). Conductance-based models suggest that MPOs can depend on a variety of ionic currents determined by the cell type (Wang, 1993; Desmaisons et al. 1999). MPOs in stellate cells have been analysed by Dickson et al. (2000a,b). The model proposed by these authors suggests that the interplay between a persistent sodium current and a hyperpolarization-activated cationic current (Ih) is responsible for the observed MPOs. Within the authors' deterministic framework, MPOs are then interpreted as limit-cycle oscillations of a non-linear dynamical system. Based on the observation of highly regular MPOs in systems such as the inferior olive, this view has also been put forward by various other authors (see, e.g. Lampl & Yarom, 1997; Hutcheon & Yarom, 2000).
In accordance with previous investigations (White et al. 1998, 2000), the irregularity and low coherence of the subthreshold oscillations observed in our study requires a different explanation and suggests that intrinsic noise sources play a paramount role for the generation of MPOs in the entorhinal cortex. This precludes the use of deterministic models to describe MPOs and alters their mechanistic interpretation. In fact, our data indicate that MPOs are not caused by an instability of the equilibrium state of a low-dimensional deterministic system but rather by stochastic forces that are most likely to be due to channel noise. Thus although the same or similar ionic currents may be involved as in the picture of Lampl & Yarom (1997), Dickson et al. (2000a, b) or Hutcheon & Yarom (2000), these currents do not cause deterministic periodic oscillations but exhibit stochastic fluctuations. The striking difference between the entorhinal cortex and inferior olive may partly be due to the self-entrainment caused by electric couplings in the inferior olive.
The primary physiological function of intrinsic membrane potential oscillations might be an augmentation of synaptic inputs that are in synchrony with ongoing MPOs (Volgushev et al. 1998). This is particularly important when cells are depolarized for prolonged periods of time to membrane potentials near their firing threshold, for example during cholinergic input. But intrinsic oscillations may also support network oscillations, particularly if mechanisms exist which synchronize the individual MPOs in neural ensembles. This could be accomplished by a phase-resetting mechanism, for example by synchronous inhibitory synaptic input to a group of cells or by electrical coupling between those cells. In our histological preparations we rarely observed dye coupling between stellate cells; a more focused study would be required to identify mechanisms by which MPOs could be synchronized. By interaction with interneurones, MPOs in a given set of cells might also be translated into superimposed higher frequencies, as suggested by findings of Gloveli et al. (1999). In this study, carbachol application to deep layer cells in the EC induced rhythmic synaptic potentials in superficial entorhinal cortex cells containing both theta and gamma frequencies.
Resonance properties
In many early studies of single-neurone properties only stationary input resistances were determined. However, once researchers started to expose cells to oscillating current injections it became evident that the cell impedance may strongly depend on the stimulation frequency (Falk & Fatt, 1964; Cole, 1968; Mauro et al. 1970; Nelson & Lux, 1970). A number of cell types in the mediodorsal thalamus (Puil et al. 1994), the inferior olive (Lampl & Yarom, 1997) or the entorhinal cortex (Haas & White, 2002) were found to display resonant properties. With improved understanding of the underlying cellular physiology, it was then realized that MPOs and neural resonance are closely related phenomena, as reviewed by Hutcheon & Yarom (2000).
As shown by our study, MPOs do occur in neurones that are intrinsically resonant while resonance itself is not sufficient for MPO generation. This finding implies that one has to carefully distinguish between the conditions under which neurones exhibit resonance and those for subthreshold oscillations; even in an intrinsically resonant cell MPOs are simply not possible if the intrinsic noise level is so low that there are not enough ionic current sources to trigger measurable voltage fluctuations. Together with the irregular nature of MPOs, these findings underscore the importance of stochastic descriptions of subthreshold phenomena (for a review, see White et al. 2000). As demonstrated by our modelling results, a simple two-dimensional linear model can account quantitatively for the observed phenomena.
The cell classes investigated in this study differ widely in their resonance behaviour. Stellate cells from layer II of the entorhinal cortex possess a resonance with an impedance increase between 20% and more than 100% at the resonance frequency while for other cell types the impedance function either has only a small peak or simply decays with increasing frequency. Pyramidal neurones of EC layer III exhibit clear low-pass filter properties and thus differ significantly from the stellate cell population (t = 7.6, d = 41, P < 0.001). The cut-off frequency of the low-pass filter was always less than 5 Hz with a half-decay frequency of less than 20 Hz. This implies that EC layer III pyramidal cells integrate synaptic input best for frequencies below 5 Hz while stellate cells integrate input best in the frequency range from 5 to 15 Hz. Note also that our heuristic resonance criterion (Q 
1.2) nicely reflects the dichotomy seen in the population data (cf. Fig. 3B and G).
The frequency-dependent information transfer between two neurones can be strongly influenced by the synaptic properties of the presynaptic cell (see, e.g. Markram et al. 1997). As demonstrated by the large differences of the resonance properties of EC layer III pyramidal cells and EC layer II stellate cells, respectively, the integrative properties of the postsynaptic cell may be an equally important factor.
In the current study we did not make an attempt to identify ionic currents responsible for the resonance behaviour through pharmacological methods. Possible candidates for these currents are slowly activated currents that oppose changes of the membrane voltage. In CA1 pyramidal neurones two different resonance currents (IM and IH) have been identified as acting upon depolarization and hyperpolarization (Hu et al. 2002). Those currents have also been identified in stellate cells. Simulations have shown that a model that includes IM, IH and INap in addition to the classical Hodgkin-Huxley currents INa, IK and IL (Hodgkin & Huxley, 1952) provides a good description of the observed resonance characteristic for an average stellate cell. Nevertheless, the other voltage-dependent currents observed in stellate cells (Eder et al. 1991; White et al. 1993; Eder & Heinemann, 1994, 1996; Bruehl & Wadman, 1999; Magistretti & Alonso, 1999; Richter et al. 2000; Shalinsky et al. 2002) may also play a role in precisely tuning the cell frequency profile and may also help to explain the large variability of the individual cell frequency preferences. As ion channels may be differentially distributed on dendritic and other cellular compartments, differences in resonance properties are expected not only between individual cells but also between structural regions within the same cell. Indeed, when comparing dendritic and somatic recordings of cortical pyramidal cells, significant differences in resonance behaviour were observed by Ulrich (2002).
We have found a number of EC cells whose impedance was approximately constant over the whole investigated frequency range. We have not identified these ‘other’ cells. It could be that they are GABAergic cells. Furthermore, we cannot exclude the possibility that some of the recordings may have resulted from measurements in dendrites.
From single-cell to network oscillations
Average impedance profiles for the two identified cell groups are depicted in Fig. 7C. On average, EC layer III pyramidal cells integrate low-frequency inputs best in the range below 5 Hz, while stellate cells are more responsive to inputs at higher frequencies. When we applied a ZAP input at strongly depolarized membrane potentials, action potentials were generated at the frequencies close to the resonance frequency. This is in agreement with previous studies on frequency preferences in the entorhinal cortex. For example, during repetitive synaptic subthreshold stimulation with frequencies below 5 Hz, the pathway from the EC layer II to the dentate gyrus remains quiet and is preferentially activated with frequencies above 5 Hz (Gloveli et al. 1997b). In contrast, EC layer III cells projecting to the subiculum and CA1 area respond preferentially to low stimulus frequencies (below 10 Hz) and are strongly inhibited when stimulated with higher frequencies (Heinemann et al. 2000). Both cell classes possess a common range of frequencies between 5 and 10 Hz where integration of synaptic inputs is similarly effective. It would be interesting to investigate how this finding is related to the ease of theta-rhythm induction in the hippocampal formation and to study the implications of impedance resonance for synaptic plasticity.
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