Subthreshold membrane-potential resonances shape spike-train patterns in the entorhinal cortex

Abstract

Many neurons exhibit subthreshold membrane-potential resonances, such that the largest voltage responses occur at preferred stimulation frequencies. Because subthreshold resonances are known to influence the rhythmic activity at the network level, it is vital to understand how they affect spike generation on the single-cell level. We therefore investigated both resonant and nonresonant neurons of rat entorhinal cortex. A minimal resonate-and-fire type model based on measured physiological parameters captures fundamental properties of neuronal firing statistics surprisingly well and helps to shed light on the mechanisms that shape spike patterns: 1) subthreshold resonance together with a spike-induced reset of subthreshold oscillations leads to spike clustering and 2) spike-induced dynamics influence the fine structure of interspike interval (ISI) distributions and are responsible for ISI correlations appearing at higher firing rates (> or =3 Hz). Both mechanisms are likely to account for the specific discharge characteristics of various cell types.

FIG. 1.

Schematic representation of the subthreshold model part and interspike interval (ISI) distributions obtained with the renewal model. A: the subthreshold part of the model extends the electrical circuit analogy of Erchova et al. (2004) and Schreiber et al. (2004a). Specifically, the previously constant conductance of the middle parallel branch is now replaced by a time-dependent stochastic conductance to capture cell-intrinsic noise. As stated, the parameters γ, δ, R₀, and V_r can be expressed in terms of the circuit parameters. The parameter Q denotes the noise strength of conductance fluctuations: Q ∼ Np(1 − p)/τ, where N is the number of channels, p is the opening probability, and τ is the correlation time of conductance noise. For details, see Verechtchaguina et al. (2007). B: two cumulative ISI distributions obtained with the renewal model (in the Stratonovich approximation). The analytical curves are obtained for two parameter sets that differ in the subthreshold resonance frequency f_res and, consequently, also in the frequency of subthreshold membrane potential oscillations f_osc (top: f_osc = 10 Hz; bottom: f_osc = 5 Hz); see Eq. 4. The distance between the 1st and 2nd peaks of the ISI densities (insets) is equal to the period of subthreshold oscillations and thus also approximately equal to the inverse of the subthreshold resonance frequency.

FIG. 2.

Subthreshold dynamics and firing patterns of stellate and pyramidal cells in the rat entorhinal cortex (left and right columns, respectively). A and B: impedance-amplitude profiles (main panels) as calculated from voltage responses (insets) to time-varying oscillatory (ZAP) currents with amplitude I₁ = 100 pA. The stellate cell has a clear resonance maximum at about 7 Hz; the pyramidal cell is nonresonant (gray: experiment; black: model fit). C and D: autocorrelation functions (main panels) of subthreshold voltage responses (insets) to constant current injections. Rhythmic components of the membrane-potential oscillations cause the trough and side peaks in the autocorrelation function of the stellate cell (I₀ = 250 pA). In the pyramidal cell (I₀ = 300 pA), on the other hand, the autocorrelation function decays monotonically, as expected for nonrhythmic fluctuations. E and F: spike-train responses to constant depolarizing currents in the stellate (I₀ = 300, 400, and 650 pA) and pyramidal cell (I₀ = 600, 700, and 900 pA). The stellate, but not the pyramidal cell, generates clustered action potentials. G and H: ISI statistics. Cumulative ISI distributions P(T) are shown for 3 different activity levels (with firing rates of 1.5, 5.1, and 7.5 Hz for the stellate cell; 1.3, 3.9, and 10.4 Hz for the pyramidal cell, respectively). Probability densities ℱ(T) are depicted in the insets for the intermediate firing rates. In the stellate cell, P(T) exhibits a plateau structure. Consequently, ℱ(T) is multimodal. For all firing rates the sharp rise in P(T) occurs at about the same T. This interval corresponds to the dominant ISI within a spike cluster. For the pyramidal cell, the ISI probability density is unimodal and, correspondingly, P(T) has a single inflection point.

FIG. 3.

Resonate-and-fire model. Voltage dependence of subthreshold model parameters, as obtained from physiological data. The 4 rows represent the four parameters (C, /2π, γ/C, D), shown for 4 stellate and 4 pyramidal cells (left and right columns, respectively). Different colors correspond to different neurons; the 2 cells presented in Fig. 1 are marked in black. A and B: capacity C, derived from fitting the experimental impedance function with the theoretical curve obtained from the resonate-and-fire model. C and D: eigenfrequency /2π, obtained from fitting the experimental impedance and autocorrelation functions with the model. E and F: damping coefficient γ/C, calculated from the experimental impedance and autocorrelation functions. G and H: noise intensity D, obtained from the experimental autocorrelation function. As shown by these data, the noise intensity D strongly depends on the level of membrane depolarization in both cell classes. In addition, in stellate cells the damping coefficient γ/C decreases rapidly with increasing membrane depolarization.

FIG. 4.

ISI distributions. Predictions from the resonate-and-fire model are depicted as black lines, experimental results as thick gray lines, and thin gray lines mark the boundaries of a Kolmogorov–Smirnov test with significance level 5%. Insets depict the corresponding ISI densities. A: stellate cell, same as in Fig. 1. Two model variants with a different SD of the voltage fluctuations are shown: σ_x = 5.6 mV as solid line and σ_x = 1.6 mV as dashed line. The first variant provides the better fit to the measured data but its voltage fluctuations are much larger than experimental values (σ_x ≈ 2.5 mV). The second variant shows the best fit obtained with an upper variance bound of 4.0 mV. B: pyramidal cell, same as in Fig. 1. The best approximation, shown in black, closely matches the experimental ISI distribution. The parameters of the underlying resonate-and-fire models are specified in Table 1.

FIG. 5.

Second-order ISI statistics for stellate (left) and pyramidal cells (right). Both neurons are the same neurons as in Fig. 1. A and B: probability density of the ISI return map. For stellate cells, spike clusters are reflected in the 2-arm structure with a pronounced peak. In contrast, the ISI return map of the pyramidal cell has a cloudlike structure, as expected for unclustered spike patterns. C and D: density of the ISI correlation maps. Both cell types exhibit negative correlations between adjacent ISIs. E and F: serial correlation coefficients (SCCs) ρ_k with lags k = 1, … , 20 for 2 different average firing rates (stellate cell: 1.5 and 5.1 Hz; pyramidal cell: 1.3 and 3.9 Hz). The insets depict SCCs scaled by the respective mean ISI to represent the duration of correlations in seconds. SCCs of both stellate and pyramidal cells show similar negative correlations at short lags for sufficiently high firing rates. Independent of the cell type, the correlation time is about 300 ms.

FIG. 6.

Nonrenewal spike generation. Left column: example of a nonrenewal spike train generated by the extended resonate-and-fire model (bottom) and the corresponding time evolution of the probability to skip a spike p_s(t) (top). In this model, the voltage variable may briefly exceed the threshold for spike generation without producing a spike, resulting in a “silent overshoot.” Right column: experimental data from a stellate cell. Multiple silent overshoots are visible in this recording.

FIG. 7.

Nonrenewal resonate-and-fire model explains ISI distributions and ISI correlations. Left panels: stellate cell; right panels: pyramidal cell (same cells as in Fig. 1). Experimental ISI statistics are shown in gray, modeling data in black. A and B: ISI distributions (main panels) and corresponding ISI densities (insets). C–F: serial correlation coefficients ρ_k. r_m specifies the mean firing rate in the model. C and E correspond to A and B, respectively. The values of the fit parameters are given in Table 1. D and F illustrate that the strong reduction of ISI correlations at lower firing rates (compare with C and E) can be reproduced in the model [r_m = 1.6 Hz for the stellate cell (D) with weaker noise D = 0.003 nA² Hz, larger friction γ/C = 0.63 × 10² Hz and other parameters as in A; and r_m = 1.3 Hz for the pyramidal cell (F) with weaker noise D = 0.003 nA² Hz and other parameters as in B]. As shown by these results, even simple nonrenewal resonate-and-fire models can explain rather complex spike-train data.

FIG. 8.

Relation between the frequency of subthreshold resonance and the inverse distance between the first and second peaks in the ISI density. Each data point corresponds to a different stellate cell, which are known to have resonances in the frequency range of 5–15 Hz. The horizontal and vertical bars mark the minimal and maximal observed values, as described in detail in methods. The arrow points to the cell presented in Figs. 2–7. Our model framework predicts that for every cell the inverse differences between the two ISI peaks are very similar to the natural oscillation frequency near the firing threshold. This frequency, in turn, has been shown to be close to but smaller than the subthreshold resonance frequency (compare f_osc and f_res, Eq. 4), as discussed in methods and Erchova et al. (2004). It follows that the resonance frequency should slightly exceed the inverse ISI-peak difference so that the data points should be above but close to the dashed identity line, as confirmed by the population data.