Dynamic threshold curves and response precision in forced excitable systems

TL;DR

The paper addresses how subthreshold periodic forcing and noise shape spike timing in excitable systems, with a focus on auditory-processing neurons. It introduces the dynamic threshold curve ($DTC$) and the dynamic threshold function $(t)$ to predict spike times as first-passage events to a time-varying boundary, showing that a simple Gaussian fluctuation model suffices to capture spike-phase distributions across Type 2 and Type 3 regimes. Through analyses of both nonlinear (V-U) and linear planar systems, and a homotopy between Type 2 and Type 3 dynamics, the work demonstrates that the shape of the $DTC$—not just local linearization—crucially governs phase-locking and response precision under periodic inputs. These findings offer a framework for predicting and tuning neural response precision in auditory and other excitable systems, with implications for understanding phase-locked spiking under realistic noisy, time-varying stimuli.

Abstract

We investigate here various properties of the responses of excitable systems subject to periodic forcing and noise. While the properties of intrinsic oscillators, subject to added periodic signals, are well understood, much less is known about the factors that determine the response precision of excitable units, intrinsically at rest, when activated by periodic forcing and stochastic noise. One motivation for considering this issue comes from the behavior of auditory neurons. These neurons reportedly have the ability to fire spikes in a precise range of phases in response to incoming sound waves, a behavior for which the mechanism is unknown. To account for such a response precision, we introduce the notion of dynamic threshold curve (DTC), which estimates at each time the effective likelihood that noise will subsequently generate a spike. The DTC effectively summarizes, in a single curve, a representation of the response precision of an excitable model, as we demonstrate by showing that the distribution of spike times produced in this setting is well captured by the first passage time of a simple, Gaussian stochastic process to the distance to the DTC. This result shows that peaks and troughs of the DTC, but also their slopes, convey fine information about spike timing in response to noise. In particular, it explains properties of Type 2 and Type 3 excitable cells studied previously and provides a framework to predict the DTC properties necessary to support the response precision of auditory neurons, as we illustrate in a well-established auditory neuron model.

Figures (13)

• Figure 1: Examples of one period of the superposition of two time-shifted (solid orange and dashed blue curves) rectified sinusoidal inputs (top) and tent inputs (bottom), with $t_0=1$ and various $\Delta T$ (top: $\Delta T=0, 2.25, 3.75, 5$; bottom: $\Delta T=0, 4, 7, 10$).
• Figure 1: Thresholds in the $V-U$ model. (A) Static threshold (purple) separating a spiking region (yellow, with typical trajectory in pink, cut for legibility) from a non-spiking region (cyan region, with typical trajectory in blue). Black: $V$-nullcline, Green: $U$-nullcline, black circle: globally stable equilibrium point. Black arrow: direction of perturbation. (B,C) Dynamical threshold in response to a superposition of rectified sine waves (plotted in C, top). (B) Phase plane solution and (C) voltage time course, with the unperturbed but forced trajectory (black) plotted along with trajectories resulting from subthreshold (light blue) and superthreshold (pink) perturbations along the $v$-direction. The DTC (blue in B) lies between those two trajectories. The static threshold is plotted as a purple curve in (B) for comparison.
• Figure 1: Type 2 neurons with noise. (A) 100 sample trajectories, some spiking (blue) and some not (red). (B) Time-averaged empirical covariance, $\tilde{C}(s)$ (left, with exponential fit in yellow) and instantaneous variance (right) computed from the non-spiking trajectories. (C) Spike time statistics for the $V-U$ model (blue histogram), together with the distributions of the first hitting times to the dynamic threshold function (purple) for the Ornstein-Uhlenbeck model (red), the white noise model (yellow), and the stationary white noise (green). All three models show a strong consistency.
• Figure 1: The relation between the coefficient matrix $M$ and phase-locking is non-trivial for a linear system. Families of deterministic trajectories, color coded by time within one input period (shared color bar), as (A) the real part of the eigenvalues of $M$, (B) the magnitude of the imaginary part of the eigenvalues of $M$, or (C) the $w$-nullcline slope is systematically varied. Note the use of different viewing angles in different panels, to highlight the structure present in each. (D) Heat map of RP as eigenvalues' real and imaginary parts are varied. (E) Slices through the RP heat map from (D), with real part fixed and imaginary part varied over the indicated range (red horizontal band at the top of D, red curve in E), or vice versa (blue vertical band in D, blue curve in E), or with both held fixed as the $w$-nullcline slope is varied (orange). The baseline parameter values for these simulations were $a=-0.4, b=1.0, c=-1.5, d=-0.2$.
• Figure 1: Surface of $\kappa(t)$ over one input period as $\lambda$ (homotopy parameter) is varied for $\Delta T=2$ (left), $6$ (center) and $12$ (right). The extremes of $\lambda=1$ (Type 2 case) and $\lambda=0$ (Type 3 case) have their $\kappa(t)$ outlined in red and cyan, respectively. The color-coding of the surface corresponds to height.

Theorems & Definitions (2)

• Definition 3.1
• Remark 3.2