Information filtering in resonant neurons

Abstract

Neuronal information transmission is fre- quency specific. In single cells, a band-pass like frequency preference can arise from the subthreshold dynamics of the membrane potential, shaped by properties of the cell’s membrane and its ionic channels. In these cases, a cell is termed resonant and its membrane impedance spectrum exhibits a peak at non-vanishing frequencies. Here, we show that this frequency selectivity of neuronal response amplitudes need not translate into a similar frequency selectivity of information transfer. In particular, neurons with resonant but linear subthreshold voltage dynamics (without threshold) do not show a resonance of information transfer at the level of subthreshold voltage; the corresponding coherence has low-pass characteristics. Interestingly, we find that when combined with nonlinearities, subthreshold resonances do shape the frequency dependence of coherence and the peak in the subthreshold impedance translates to a peak in the coherence function. In other words, the nonlinearity inherent to spike generation allows a subthreshold impedance resonance to shape a resonance of voltage-based information transfer. We demonstrate such nonlinearity-mediated band-pass filtering of information at frequencies close to the subthreshold impedance resonance in three different model systems: the resonate-and-fire model, the conductance-based Morris-Lecar model, and linear resonant dynamics combined with a simple static nonlinearity. In the spiking neuron models, the band-pass filtering is most pronounced for low firing rates and a high variability of interspike intervals, similar to the spiking statistics observed in vivo. We show that band-pass filtering is achieved by reducing information transfer over low-frequency components and, consequently, comes along with an overall reduction of information rate. Our work highlights the crucial role of nonlinearities for the frequency dependence of neuronal information transmission.

Appendix A

1.1 A.1 Gradual transitions from resonators to integrators

Here, we present our numerical scheme to differentiate gradually between resonators and integrators. This allows us to study the relationship between the power filter properties and the information filter properties of these neuron models. The question arises whether there is a relation between impedance and coherence quality. Therefore we have to vary the impedance quality with respect to certain constraints. These constraints are experimentally motivated and should hold under variation of the impedance quality

• a) resonance frequency is fixed (5 Hz, 9.5 Hz, 15 Hz),

• b) natural frequency is smaller than resonance frequency (see Fig. 10 in Erchova et al. 2004), here: $\frac {f_{\mathrm nat}}{f_{\text {res}}}\approx 0.8$,

• c) band-with of the impedance resonance should be proportional to the resonance freq. (see Fig. 7B in Erchova et al. 2004)

• d) impedance quality should be varied in a systematic way,

We implemented a numerical scheme which solves a problem under these constraints. It turned out that the resistance of the inductor R _L is a suitable parameter to vary the impedance quality with the above described constraints. Suitable means that the experimentally observed parameter set of the stellate cell should lie on our parameterization manifold (see Fig. 17).

Fig. 17

Parameter sets yielding the same resonance frequency of the impedance in the linear, subthreshold dynamics. a Variation of impedance quality. By fixing the resonance frequency and changing the resistance of the inductor R _L one can systematically vary the impedance quality. This way gradual transitions from resonators to integrators (including the physiological parameter set of the stellate cell, red data point) were obtained. b–d Lines of equal resonance frequency in subspaces of parameters of the subthreshold dynamics of the RF model (resistance R _L versus damping factor, resistance R, and inductance L, respectively). e Impedance quality decreases with an increasing damping factor

Figure 17a–b shows how the increase of the resistance R _L increases monotonically the impedance quality Q _Z and changes the damping factor ζ, the membrane resistance R, and the inductance L.

We display in Fig. 17e the expected relationship between the damping factor ζ and the impedance quality: with increasing damping factor the impedance quality decreases. These approximately 75.000 parameter sets are then used to simulate the resonate-and-fire neuron model within different excitable regimes (variation of DC input I ₀ at fixed D, D _OU, V _{re s e t} , τ _abs) at different impedance resonance frequencies (see Figs. 9–11) in order to reveal the dependence of the filtering properties between power filtering, described by the impedance function and information filtering, described by the coherence function.