Accuracy of rate coding: When shorter time window and higher spontaneous activity help

It is widely accepted that neuronal firing rates contain a significant amount of information about the stimulus intensity. Nevertheless, theoretical studies on the coding accuracy inferred from the exact spike counting distributions are rare. We present an analysis based on the number of observed spikes assuming the stochastic perfect integrate-and-fire model with a change point, representing the stimulus onset, for which we calculate the corresponding Fisher information to investigate the accuracy of rate coding. We analyze the effect of changing the duration of the time window and the influence of several parameters of the model, in particular the level of the presynaptic spontaneous activity and the level of random fluctuation of the membrane potential, which can be interpreted as noise of the system. The results show that the Fisher information is nonmonotonic with respect to the length of the observation period. This counterintuitive result is caused by the discrete nature of the count of spikes. We observe also that the signal can be enhanced by noise, since the Fisher information is nonmonotonic with respect to the level of spontaneous activity and, in some cases, also with respect to the level of fluctuation of the membrane potential.

Figure 1

Schematic illustration of the studied model. Shown on top is a sample path of the membrane potential $X\left(t\right)$, described by a Wiener process. When $X\left(t\right)$ exceeds a constant threshold $B>0$, an action potential is generated. Then $X\left(t\right)$ is reset to 0 and its evolution starts anew. At time $t_0$, a stimulus is applied and the parameters of the process change from ${\mu }_0$ and ${\sigma }_0^2$ to $\mu \left(s\right)$ and ${\sigma }^2\left(s\right)$. The times from the stimulus onset to the first, second, third, etc., $n\text{th}$ evoked spike are denoted by $T_1,T_2,T_3,...,T_n$. Shown on the bottom is the corresponding counting process of evoked spikes $N\left(t^{*}\right)$, which gives the number of spikes in a time window $[t_0,t_0+t^{*}]$.

Figure 2

Fisher information $J\left(0\right)$ about the stimulus intensity $s=0$ based on the number of spikes $N\left(t^{*}\right)$ in a time window of length $t^{*}$. The observation time window starts with a spike and the initial value of the membrane potential at the beginning of the time window is thus $X_0=0$. In all cases, the diffusion coefficient is ${\sigma }^2\left(s\right)=k\mu \left(s\right)+m$. (a)–(c) Fisher information as a function of the spontaneous firing rate ${\mu }_0$ and the duration of the time window $t^{*}$ when $k=0.01$ and $m=0.5$. The Fisher information is nonmonotonic with respect to both ${\mu }_0$ and $t^{*}$. (d)–(f) Fisher information as a function of the spontaneous firing rate ${\mu }_0$ and the absolute coefficient $m$ of the diffusion parameter ${\sigma }^2\left(s\right)$ when $t^{*}=0.025\mathrm{s}$ and $k=0.01$. (g)–(i) Fisher information as a function of the spontaneous firing rate ${\mu }_0$ and the coefficient of proportionality $k$ of the diffusion parameter ${\sigma }^2\left(s\right)$ when $t^{*}=0.025\mathrm{s}$ and $m=0.5$. For some values of ${\mu }_0$ [dashed and dot-dashed lines in (e) and (f)], the Fisher information is initially increasing with respect to $k$ and $m$ and thus with respect to the fluctuation level of the membrane potential. The white dashed lines in (a), (d), and (g) mark the points satisfying $t^{*}=nB/\mu \left(s\right)$, which approximately correspond to the local maxima of $J\left(s\right)$ with respect to $t^{*}$, where $B=1$ is the membrane potential threshold and $n=1,2,\cdots$.

Figure 3

Fisher information $J\left(0\right)$ about the stimulus intensity $s=0$ when the time window starts at $t_0$ and the initial value of the membrane potential $X_0$ is thus random. In all cases, the coefficient diffusion is ${\sigma }^2\left(s\right)=k\mu \left(s\right)+m$. (a)–(c) Fisher information as a function of the spontaneous firing rate ${\mu }_0$ and the time window $t^{*}$ when $k=0.01$ and $m=0.5$. The Fisher information is nonmonotonic with respect to both ${\mu }_0$ and $t^{*}$. (d)–(f) Fisher information as a function of ${\mu }_0$ and $m$ when $t^{*}=0.025\mathrm{s}$ and $k=0.01$. (g)–(i) Fisher information as a function of ${\mu }_0$ and $k$ when $t^{*}=0.025\mathrm{s}$ and $m=0.5$. For any choice of ${\mu }_0$, the Fisher information is decreasing in $k$ and $m$. The white dashed lines in (a), (d), and (g) mark the points satisfying $t^{*}=nB/\mu \left(s\right)$, which approximately correspond to the local maxima of the Fisher information with respect to $t^{*}$, where $B=1$ is the membrane potential threshold and $n\in \mathbb{N}$.

Figure 4

Fisher information $J\left(0\right)$ about the stimulus intensity $s=0$ as a function of the spontaneous firing rate ${\mu }_0$ when the diffusion coefficient is independent of stimulation, namely, ${\sigma }^2\left(s\right)={\sigma }_0^2=2$. (a) Fisher information when the observation time window starts with a spike. (b) Fisher information when the observation window starts at $t_0$. Three lengths of the observation window are used: $t^{*}=0.007\mathrm{s}$ (red solid line), $t^{*}=0.02\mathrm{s}$ (blue dashed line), and $t^{*}=0.04\mathrm{s}$ (green dotted line). If the local extremes are ignored, the Fisher information has a mild overall increasing tendency in all the cases, suggesting that the decoding accuracy improves with increasing spontaneous firing rate ${\mu }_0$.

Figure 5

(a) Fisher information $J\left(0\right)$ about the stimulus intensity $s=0$ as a function of the length of the time window $t^{*}$ for the limit case of the zero diffusion coefficient ${\sigma }_0^2={\sigma }^2\left(s\right)=0$. The Fisher information goes to infinity for all $t^{*}$ satisfying $t^{*}=nB/\mu \left(s\right), n\in \{1,2,\cdots \}$. (b) Fisher information $J\left(0\right)$ when ${\sigma }_0^2>0$ and ${\sigma }^2\left(s\right)>0$. The local maxima are located approximately at $t^{*}=nB/\mu \left(s\right)$. (c) Probabilities of observing $n$ spikes $p_{N\left(t^{*}\right)}\left(n\right)=\mathbb{P}[N\left(t^{*}\right)=n]$ as functions of $t^{*}$ for the limit case ${\sigma }_0^2={\sigma }^2\left(s\right)=0$. At $t^{*}=nB/\mu \left(s\right)$, the probability of observing $n$ spikes is equal to 1 and all the other outcomes have zero probability. (d) Probabilities $p_{N\left(t^{*}\right)}\left(n\right)$ as functions of $t^{*}$ when ${\sigma }_0^2>0$ and ${\sigma }^2\left(s\right)>0$. At $t^{*}=nB/\mu \left(s\right)$, the probability of observing $n$ spikes is nearly one and decreases when the diffusion coefficient ${\sigma }^2\left(s\right)$ is increased. All figures were obtained for the spontaneous firing rate ${\mu }_0=50$ Hz.