Bifurcation enhances temporal information encoding in the olfactory periphery

Abstract

Living systems continually respond to signals from the surrounding environment. Survival requires that their responses adapt quickly and robustly to the changes in the environment. One particularly challenging example is olfactory navigation in turbulent plumes, where animals experience highly intermittent odor signals while odor concentration varies over many length- and timescales. Here, we show theoretically that Drosophila olfactory receptor neurons (ORNs) can exploit proximity to a bifurcation point of their firing dynamics to reliably extract information about the timing and intensity of fluctuations in the odor signal, which have been shown to be critical for odor-guided navigation. Close to the bifurcation, the system is intrinsically invariant to signal variance, and information about the timing, duration, and intensity of odor fluctuations is transferred efficiently. Importantly, we find that proximity to the bifurcation is maintained by mean adaptation alone and therefore does not require any additional feedback mechanism or fine-tuning. Using a biophysical model with calcium-based feedback, we demonstrate that this mechanism can explain the measured adaptation characteristics of Drosophila ORNs.

FIG. 1.

Encoding the intensity and timing of signal fluctuations. (a) Flies navigate odor plumes using information about the intensity and timing of the odor fluctuations they encounter. (b) The distributions of odor stimulus with low (blue) and high (red) variances (top) and the corresponding ORN firing rate distributions (bottom) [29]. The inset shows changes in ORN gain in response to changes in the stimulus variance. (c) The response dynamics of a neuron spiking above and below the firing threshold. Top: Input current over time for a current temporarily above (right, grey) and below (left, black) the firing threshold. Middle: corresponding membrane voltage over time. Bottom: Average firing rate r as a function of a fixed input current. The simulated neuron is quiescent below the threshold current $I_{\text{b}}$, while the firing rate suddenly and monotonically rises with the current above the threshold. (d) For dynamic (non-static) currents, the dose-response curve from (c) may not apply. Top: a static current (black) and a current with a brief dip below the firing threshold (red). Middle and bottom plots: corresponding neuron membrane voltages for two currents. For the current with a dip, the neuron crosses into a non-firing regime and causes a brief delay in spikes, compared to the response to the static current (dotted line and arrow). This delay will manifest as a reduction – but not zeroing – of the firing rate, obscuring the fact that the neuron is crossing into a quiescent state.

FIG. 2.

Dose-response curves of conductance-based neuron models are intrinsically adaptive (invariant) to input signal variance regardless of the bifurcation type. (a) A schematic showing how input current is translated to firing rate in a conductance-based neuron model. Here, the input current $I\left(t\right)$ is modeled as an O-U process with input correlation timescale ${\tau }_s=200ms$, mean $\mu =4.54pA$, and standard deviation $\sigma =0.1$. The firing rates are calculated by binarizing and filtering the spikes with an exponential filter $f$ with firing rate filter timescale ${\tau }_r$. This model has ${\text{Na}}^{+}$ and ${\text{K}}^{+}$ channels driving action potentials and fires when the injected current is above $I_{\text{b}}=4.54pA$. (b) Dose-response curves of the firing rate as a function of the input current difference from the firing threshold ($I_{\text{b}}=4.54pA$) obtained from the Na+K model with an SNIC bifurcation. The different curves correspond to increasing fluctuations with size $\sigma$, ranging from $\sigma =0.04pA$ (light red) to $1.6pA$ (dark red). (c) When rescaled by the magnitude of $\sigma$, the dose-response curves collapse to a single curve, implying that the system inversely adapts its gain with $\sigma$. (d) Dose-response curves from a neuron model with a Hopf subcritical bifurcation, where the firing rate is discontinuous at the threshold current $I_{\text{b}}=101pA$. (e) The same plot with the x axis rescaled by $\sigma$, as in (c).

FIG. 3.

Bifurcation-induced variance adaptation maintains coding capacity. (a) Mutual information between the distribution of input current $I$ (modeled as an O-U process with mean $\mu$, standard deviation $\sigma$, and correlation timescale ${\tau }_{s}s=500ms$) and the firing rate $r$ as a function of the strength of the fluctuations in the input current (MI, black). Green: same but only for currents below the threshold ($I<I_{\text{b}}$, MI-). MI increases monotonically with the signal variance. Meanwhile, MI- remains largely independent of the signal variance over two orders of magnitude. The firing rate filter timescale is ${\tau }_r=55ms$. (b) MI- as a function of mean $\mu$ and standard deviation $\sigma$ of the input current. The blue box corresponds to $\mu ~I_{\text{b}}=4.54pA$. (c) MI- as a function of $\sigma$ and ${\tau }_r$. MI- is maximized when ${\tau }_r=40ms$, which is ~ 1/10 of the input ${\tau }_s$ (blue dashed line). (d) MI- as a function of ${\tau }_s$ and ${\tau }_r$ shows that information transfer is bounded by the constraint ${\tau }_r\lesssim 1/5{\tau }_s$ and large enough input timescale ${\tau }_s$ to cover the inter-spike interval. Note the change in color scales for (b-d).

FIG. 4.

The coding capacity of signal timing is elevated when ORN is proximal to the bifurcation point. (a) A schematic illustrating different temporal statistics that ORNs may encode for odor-guided navigation. Odor encounters and odor blanks correspond to the stimulus crossing the firing threshold from below/above. The timings between subsequent encounters or blanks are defined as $\Delta t_{\text{ON}}$ and $\Delta t_{\text{OFF}}$, respectively. The duration of odor encounters $t_{\text{dur}}^{\text{enc}}$ and blanks $t_{\text{dur}}^{\text{bnk}}$ denote the time from an encounter to a blank or a blank to an encounter, respectively. (b) MI between the distribution of encounter times $\Delta t_{\text{ON}}$ and the firing rate $r$ as a function of signal mean $\mu$ and standard deviation $\sigma$. MI increases and saturates as encounter frequency increases, which depends on the relative values of $\mu$ and $\sigma$. When $\mu$ is close to $I_{\text{b}}=4.54pA$ (blue box), information about $\Delta t_{\text{ON}}$ is encoded over the widest range of $\sigma$. (c-e) Same as (b) for the distributions of blank times $\Delta t_{\text{OFF}}$, encounter duration $t_{\text{dur}}^{\text{enc}}$, and blank duration $t_{\text{dur}}^{\text{bnk}}$. Unlike other temporal cues, information about encounter duration becomes small when $\mu <I_{\text{b}}$. Note the changes in color scales for different temporal statistics.

FIG. 5.

Overview of the biophysical model of ORNs with calcium-mediated mean adaptation. (a) Mean adaptation in Drosophila ORNs is mediated by cytoplasmic calcium level. Odorant drives response by binding to olfactory receptors (OrX). Active receptors drive calcium influx along with sodium ions, which, in return, regulates the receptor activity through a calmodulin-based process or calcium-dependent phosphatase-based dephosphorylation that targets Orco, decreasing the receptor activity. Meanwhile, cytoplasmic calcium ions are sequestered in the mitochondria or pumped out, slowly recovering the sensitivity towards a fixed steady state under a continuous odor input. (b) The firing rate adapts to about $~30Hz$ independent of the background odor concentration, as shown experimentally in ORN ab3A in response to ethyl acetate. To leverage bifurcation-induced benefits, we chose a steady state firing rate that corresponds to a signal slightly above the bifurcation point. Left: an odor signal trace where the mean concentration changes by an order of magnitude from 4.5 to 45 A.U. but with the same variance. Right: while the increase in odorant concentration creates a transient increase in firing rate, the ORN quickly adapts to the same average firing rate of $~30Hz$.

FIG. 6.

Mean adaptation maintains robust bifurcation-induced information transmission across stimulus backgrounds. (a) Left: the lag between the input signal and the firing rate is quantified from the peak in the cross-correlation between the two time traces. Right: differences in firing lag ($\Delta \text{Lag}$) with respect to the lag at $\mu =4pA$ (right, black line) as a function of the mean signal intensity $\mu$ for Na+K (green) and Na+K+Ca (red) models. The input signal standard deviation is $\sigma =0.4$. The firing lag does not increase significantly with the mean signal intensity. $\rho$ and $p$ denote the Spearman correlation coefficient and the corresponding $p$-value. (b) MI- from the Na+K+Ca model using O-U odor signals with $\mu$ ranging from 4 to 15 A.U. and $\sigma$ ranging from 0.01 to 3.16. Once rescaled by the signal mean, the curves collapse (right). (c-e) MI between the firing rate and the distributions of encounter times $\Delta t_{\text{ON}}$, blank times $\Delta t_{\text{OFF}}$, and encounter duration $t_{\text{dur}}^{\text{enc}}$, as a function of $\mu$ and $\sigma$ given the firing rate threshold of $~30Hz$.