The Effects of Background Noise on a Biophysical Model of Olfactory Bulb Mitral Cells

Abstract

The spiking activity of mitral cells (MC) in the olfactory bulb is a key attribute in olfactory sensory information processing to downstream cortical areas. A more detailed understanding of the modulation of MC spike statistics could shed light on mechanistic studies of olfactory bulb circuits and olfactory coding. We study the spike response of a recently developed single-compartment biophysical MC model containing seven known ionic currents and calcium dynamics subject to constant current input with background white noise. We observe rich spiking dynamics even with constant current input, including multimodal peaks in the interspike interval distribution (ISI). Although weak-to-moderate background noise for a fixed current input does not change the firing rate much, the spike dynamics can change dramatically, exhibiting non-monotonic spike variability not commonly observed in standard neuron models. We explain these dynamics with a phenomenological model of the ISI probability density function. Our study clarifies some of the complexities of MC spiking dynamics.

Appendices

Appendix A Viability of Alternative Approaches

A common approach to reduce the number of state variables in describing (regular) spiking is to apply a phase reduction, which seems appealing because the results hold for relatively weak noise forcing where firing rates do not vary much. This motivated us to analyze the bifurcation between quiescence and spiking (using XPP-AUTO (Ermentrout 2002; Doedel 1981)), in the noiseless case (Fig. 6a). We see that the stable rest state loses stability via a saddle-node on invariant circle (SNIC) bifurcation as the applied current increases. However, we do not find actual periodic solutions for relevant firing rates with which to calculate commonly used entities for analysis like the phase-resetting curve (PRC). Curiously, the bifurcation diagram shows stable period solutions (green dots) that are interlaced with unstable periodic solutions. Figure 6b shows that even with $I_{\text {app}}=144\,\mu \text {A/cm}^2$ where the diagram (Fig. 6a) suggests there is a well-behaved periodic solution, the voltage (and other variables) are not strictly periodic. Note that with $I=144$, the firing rates are large (91.13 Hz) and not physiologically relevant. Also, for $I=144$ although the PRC is numerically calculable (Fig. 6c), the rather large negative region does not resemble canonical PRCs associated with a SNIC (Rinzel and Ermentrout 1989; Ermentrout 1996).

Fig. 6

Dynamics of noiseless MC model: phase reduction assumptions violated. a Bifurcation diagram of voltage varying I shows SNIC at onset of spiking. b The voltage traces in Fig.2a are not strictly periodic, even for an ideal well-behaved I value. For unrealistically high firing rates ($I=144\,\mu \text {A/cm}^2$, firing rate: 91.13 Hz), we see the system is not periodic despite the bifurcation diagram suggesting it should be. The infinitesimal phase-resetting curve can be numerically calculated with XPP (Ermentrout 2002); notice that the negative region is rather large, not resembling canonical PRCs synonymous with a SNIC (Rinzel and Ermentrout 1989; Ermentrout 1996). This all suggests phase reduction descriptions would likely be inadequate to capture the observed phenomena in the regimes we are interested in (i.e., smaller I with physiological firing rates) (Color figure online)

Note that in a standard phase-reduced scalar model (Ermentrout and Terman 2010):

$$\begin{aligned} \frac{d\Theta }{dt} = \omega + \frac{\sigma ^2}{2}\Delta '(\Theta )\Delta (\Theta )+\sigma \Delta (\Theta )\xi (t) \end{aligned}$$

(A1)

the ISI density can be approximated (assuming weak noise) via (Ly and Ermentrout 2011):

$$\begin{aligned} f_{ISI}(t) \approx \frac{1}{\sigma \sqrt{2\pi \int _0^t \Delta ^2(s)\,ds}}\exp \left( -\frac{\left( t-\frac{1}{\omega }+\frac{\sigma ^2\Delta ^2(t)}{4} \right) ^2}{2\sigma ^2\int _0^t\Delta ^2(s)\,dx} \right) . \end{aligned}$$

(A2)

Since the formula closely resembles a normal distribution, the $\sigma _{ISI}$ generally increases as $\sigma \nearrow$. Even though there is multiplicative noise: $\sigma \Delta (\Theta )$, these models would not help describe observations in Fig. 3.

Another common approach to model analysis with weak noise is use a potential well: $\frac{dV}{dt}=-U'(V)+\sigma \xi (t)$, where U is the potential function, either principally derived from the system (simple) or ad hoc (high-dimensional). For example, for the leaky-integrate-and-fire model, $U(V) \propto \frac{1}{2}(V-V_L)^2$ and V has a stable fixed point at the global minimum $V=V_L$. This approach was pioneered in physics by Kramers (1940) and has been applied to several neural models where ‘exiting’ from the potential well from crossing a threshold is spiking. The rate of spiking is often $\propto e^{-U/\sigma ^2}$, which is not directly related to spiking variability. The signal-to-noise ratio (SNR) in these systems, and in other applications of stochastic resonance, often have a maximal SNR value for an intermediate level of noise (Gammaitoni et al. 1998). However, this dynamic is associated with a minimum variability value in the denominator (ignoring the dynamics of the signal in the numerator) of SNR, rather than a maximal spiking variability for intermediate input noise level, as we have observed in the MC model.

Whether the potential well or ‘Arrhenius escape’ approach by Nesse et al. (2008) for a low-dimensional Fitzhugh–Nagumo model (with 1 activity variable x and a set of identical adaptation variables H endowed with multiple timescales that depend on x) could be successfully applied to our MC model is an open question. Nesse et al. (2008) exploited a separation of timescales, the slow variable was fixed and the mean first passage time (or escape) T could be calculated (in the fast variable) and set to the inverse of the mean firing rate: $\lambda (H(t))=1/T(H(t))$. The ISI density is approximated with:

$$\begin{aligned} \rho (t) = \lambda (H(t)) e^{-\int _0^t \lambda (H(t'))\,dt'} \end{aligned}$$

we see how the slow variation in H affects $\rho (t)$. This framework successfully described the non-monotonic spiking dynamics (in the CV at least) in their model. As previously mentioned in the Discussion, an analogous approach would require identifying all of the effective timescales in our 13 variable model and having a significant separation of timescales when the neuron is excitable. Even if the slow variables are frozen, one would still have to calculate the mean first passage time with the remaining fast variables, which is generally not feasible unless the resulting dimension is small. Solving for the mean first passage time requires solving an ODE system derived from the backward Fokker–Planck equation, a PDE with the number of dimensions equal to the number of state variables (Gardiner 1985; Risken 1989). The viability and the accuracy of this approach for capturing our results is an open question but beyond the scope of this current study.

Appendix B Other Spike Statistics

Fig. 7

The autocorrelation function (Eq (B3)) and power spectrum (Eq (B4)) of the MC model. Three values of input current: $I=120\,\mu \text {A}/\text {cm}^2$ in a–c, $I=130\,\mu \text {A}/\text {cm}^2$ in d–f, $I=140\,\mu \text {A}/\text {cm}^2$ in g–i, with each panel showing the effects of increasing input noise ${\tilde{\sigma }}$. The effects of input noise are nonlinear and highly dependent on I (Color figure online)

We have focused on the ISI distribution, but there are other commonly used entities to characterize neural spike trains. For instance, the autocorrelation function (ACF) and power spectrum (P), defined below, are commonly used and can be unrelated to the ISI, in particular, when the system does not reset after a spike. Letting R(t) denote the spike train consisting of 0’s and 1’s, the (normalized) autocorrelation function is:

$$\begin{aligned} ACF(\tau ) = \Big ( {\mathbb {E}}_t \left[ R(t+\tau )R(\tau ) \right] - {\mathbb {E}}_t[R(t)]^2 \Big ) \Big /ACF(0) \end{aligned}$$

(B3)

and the power spectrum:

$$\begin{aligned} P(\omega ) = \Big ( \Big \vert \int ACF(t) e^{-i2\pi \omega t} \,dt \Big \vert ^2 \Big ) \Big / P(0) \end{aligned}$$

(B4)

Figure 7 shows these entities for the biophysical MC model with various applied current and input noise values.

With $I=120\,\mu \text {A}/\text {cm}^2$, $ACF(\tau )$ is relatively flat for these input noise values, while $P(\omega )$ changes from having peaks at regularly spaced intervals with no noise (black) to being relatively flat with ${\tilde{\sigma }}>0$. With $I=130\,\mu \text {A}/\text {cm}^2$, $ACF(\tau )$ has local maxima at irregularly spaced $\tau$ with no noise (black) that flatten out as input noise increases; the $P(\omega )$ is similar to $I=120,\mu \text {A}/\text {cm}^2$ but the curves have smaller values compared to $I=120$. With $I=140\,\mu \text {A}/\text {cm}^2$, $ACF(\tau )$ indicates relatively regular spiking, although increased input noise shifts and broadens the peaks (same for $P(\omega )$). The flatter ACF with $I=120$ compared to larger I indicates that the spiking has less temporal regularity, which is not surprising. For a given value of I (i.e., a row in Fig. 7), increasing input noise flattens the $ACF(\tau )$ and shifts and/or diminishes peaks (local max), and for $P(\omega )$ input noise can broaden/shift/diminish peaks. Overall, the effects of input noise are nonlinear and highly dependent on I.

The ACF and P were plotted using built-in functions in MATLAB, and there appears to be slight numerical round-off errors in the ACF, e.g., between the peaks in Fig. 7g, $ACF\approx 0$, and likely in the P. Nevertheless, these plots given insight to some dynamics of various spike trains.