Evoking prescribed spike times in stochastic neurons

Single cell stimulation in vivo is a powerful tool to investigate the properties of single neurons and their functionality in neural networks. We present a method to determine a cell-specific stimulus that reliably evokes a prescribed spike train with high temporal precision of action potentials. We test the performance of this stimulus in simulations for two different stochastic neuron models. For a broad range of parameters and a neuron firing with intermediate firing rates (20–40 Hz) the reliability in evoking the prescribed spike train is close to its theoretical maximum that is mainly determined by the level of intrinsic noise.

Figure 1

Three raster plots are given for different values of the prescribed rate $r_{†}$ and coefficient of variation $C_{V†}$. The prescribed spike times are marked above and below the respective raster plot (thick, red). The upper panel shows the raster plot for reference statistics $r_{†}=32.7\mathrm{Hz}=r_0$ and $C_{V†}=0.68=C_V^0$, yielding a coincidence between evoked and prescribed spike trains of ${\mathrm{\Gamma }}_{sd}=0.66$ [for this measure of similarity of spike trains, see Eq. (24)]. The middle panel is for $r_{†}=19.6\mathrm{Hz}<r_0$ and $C_{V†}=0.24<C_V^0$ and yields ${\mathrm{\Gamma }}_{sd}=0.62$. The lowest panel is for increased $r_{†}=45.8\mathrm{Hz}>r_0$ and increased $C_{V†}=1.15>C_V^0$ and has ${\mathrm{\Gamma }}_{sd}=0.55$.

Figure 2

Schematic representation of the procedure that generates prescribed spike trains by simulating a (one- or two-compartment) neuron model with colored Gaussian noise with a given cutoff frequency $f_c$. The green (dotted) lines indicate which parts of the procedure are influenced by model-specific properties (listed in the dashed ellipse). For a pseudo code of the procedure, see Sec. 2f.

Figure 3

One-compartment model, Eq. (1). Performance of the method in reproducing the prescribed statistics and spike times: (a) evoked rate $r$ over the prescribed rate $r_{†}$; (b) evoked $C_V$ over the prescribed $C_{V†}$; (c) intrinsic reliability, ${\mathrm{\Gamma }}_{ss}$ (middle, dashed curve), the similarity between evoked and prescribed spike trains, ${\mathrm{\Gamma }}_{sd}$ (lower curve), and the ratio of both (upper curve) as functions of the prescribed firing rate $r_{†}$; (d) same as in (c) but vs prescribed $C_{V†}$.

Figure 4

One-compartment model, Eq. (1), suprathreshold (mean driven) firing regime. Comparison between prescribed and evoked rate and coefficient of variation (a),(b) and between prescribed and evoked spike times (c),(d); power spectra of designed Gaussian stimuli (e)–(g). Thick lines within the surfaces indicate either $r_{†}=r_0$ or $C_{V†}=C_{V,0}$ and the intersection corresponds to the reference statistics $r_0, C_{V,0}$ in Fig. 1. (a) The evoked firing rate in the plane of prescribed rate $r_{†}$ and $C_{V†}$. (b) Same as (a) but with evoked $C_V$ on the $z$ axis. (c) Same as (a) but with the similarity between prescribed and evoked spike trains ${\mathrm{\Gamma }}_{sd}$ on the $z$ axis. (d) Same as (a) but the ratio of similarity and intrinsic reliability ${\mathrm{\Gamma }}_{sd}/{\mathrm{\Gamma }}_{ss}$ on the $z$ axis. (e)–(g) Power spectra of the evoked stimuli for different values of $r_{†}$ and $C_{V†}$ as indicated. Different rates are color-coded and marked by the vertical dashed lines at the respective frequency values; peak frequencies in (e), (f) increase with and appear close to the firing rate.

Figure 5

One-compartment model, Eq. (1), subthreshold (fluctuations driven) firing regime. Comparison between prescribed and evoked rate and coefficient of variation (a),(b) and between prescribed and evoked spike times (c),(d); power spectra of designed Gaussian stimuli (e)–(g). Thick lines within the surfaces indicate either $r_{†}=r_0$ or $C_{V†}=C_{V,0}$ and the intersection corresponds to the reference statistics $r_0, C_{V,0}$ in Fig. 1. (a) The evoked firing rate in the plane of prescribed rate $r_{†}$ and $C_{V†}$. (b) Same as (a) but with evoked $C_V$ on the $z$ axis. (c) Same as (a) but with the similarity between prescribed and evoked spike trains ${\mathrm{\Gamma }}_{sd}$ on the $z$ axis. (d) Same as (a) but the ratio of similarity and intrinsic reliability ${\mathrm{\Gamma }}_{sd}/{\mathrm{\Gamma }}_{ss}$ on the $z$ axis. (e)–(g) Power spectra of the evoked stimuli for different values of $r_{†}$ and $C_{V†}$ as indicated. Different rates are color coded and marked by the vertical dashed lines at the respective frequency values; peak frequencies in (e) increase with and appear close to the firing rate.

Figure 6

Susceptibility (absolute value) of the neuron models for the cases: One-compartment model in subthreshold regime (yellow, bottom line), One-compartment model in the suprathreshold regime (red, middle line), two-compartment model in the suprathreshold regime (blue, upper line).

Figure 7

The performance of the two-compartment EIF model plotted in the ($r_{†}, C_{V†}$) plane (a)–(d). A comparison between prescribed and evoked rate and coefficient of variation (a),(b) and between prescribed and evoked spike times (c),(d). The lines on the ground denote contour lines of equal height, the thick lines within the surfaces mark the cases where either $r_{†}=r_0$ or $C_{V†}=C_{V,0}$. (a) The evoked firing rate in the plane of prescribed rate $r_{†}$ and $C_{V†}$. (b) Same as (a) but with evoked $C_V$ on the $z$ axis. (c) Same as (a) but with the similarity between prescribed and evoked spike trains ${\mathrm{\Gamma }}_{sd}$ on the $z$ axis. (d) Same as (a) but the ratio of similarity and intrinsic reliability ${\mathrm{\Gamma }}_{sd}/{\mathrm{\Gamma }}_{ss}$ on the $z$ axis. (e)–(g) The power spectra of the stimuli for different $r_{†}$ and $C_{V†}$. Different rates are color coded and marked by the vertical dashed lines at the respective frequency values; peak frequencies in (e), (f) increase with and appear close to the firing rate.