Minimal conductance-based model of auditory coincidence detector neurons

Abstract

Sound localization is a fundamental sensory function of a wide variety of animals. The interaural time difference (ITD), an important cue for sound localization, is computed in the auditory brainstem. In our previous modeling study, we introduced a two-compartment Hodgkin-Huxley type model to investigate how cellular and synaptic specializations may contribute to precise ITD computation of the barn owl's auditory coincidence detector neuron. Although our model successfully reproduced fundamental physiological properties observed in vivo, it was unsuitable for mathematical analyses and large scale simulations because of a number of nonlinear variables. In the present study, we reduce our former model into three types of conductance-based integrate-and-fire (IF) models. We test their electrophysiological properties using data from published in vivo and in vitro studies. Their robustness to parameter changes and computational efficiencies are also examined. Our numerical results suggest that the single-compartment active IF model is superior to other reduced models in terms of physiological reproducibility and computational performance. This model will allow future theoretical studies that use more rigorous mathematical analysis and network simulations.

Fig 1. NL models.

(A) Circuit of the conductance-based two-compartment HH-type "active Na" model [13,29]. The somatic compartment contains leak and K_LVA currents whereas the nodal compartment has high voltage activated potassium (K_HVA) and Na currents (required for spike generation) in addition to leak and K_LVA. (B) Circuit of the "non-spiking" conductance-based single-compartment model with leak and K_LVA currents. This model was designed for simulating subthreshold membrane responses [13,32]. (C) Membrane impedance of the somatic compartment of the active Na model. (D) Membrane impedance of the nodal compartment of the active Na model. (C-D) In the "RC only" and "RC + K_LVA" conditions, g _axon was fixed to zero (i.e., no axonal current) to isolate the compartment. In the "RC only" and "RC + axon" conditions, g _KLVA of the compartment was fixed to zero. Membrane potential was fixed at -61 mV. (E_1-3) Two-compartment active IF model: (E₁) circuit diagram, (E₂) average spike shape at the soma, and (E₃) membrane responses to DC step current injection. Spike amplitude H = 9.7 mV. Half-amplitude spike width W = 0.3 ms. (F_1-3) Single-compartment active IF model: (F₁) circuit diagram, (F₂) average spike shape, and (F₃) membrane responses to DC step current injection. Spike amplitude H = 9.8 mV. Half-amplitude spike width W = 0.3 ms. (G_1-3) Single-compartment active IF model: (G₁) circuit diagram, (G₂) average spike shape, and (G₃) membrane responses to DC step current injection. Spike amplitude H = 9.8 mV. Half-amplitude spike width W = 0.3 ms. (E₁, F₁, G₁) Θ denotes the IF spike generator. (E₃, F₃, G₃) Holding potential = -60 mV.

Fig 2. Membrane potential traces.

(A) In vivo intracellular recording from a barn owl's NL neuron (data taken from [13]). Tonal stimulus at the best frequency (3600 Hz) was presented to the both ears by changing ITDs. (B) Simulated membrane potential of the two-compartment active IF model. (C) Simulated membrane potential of the single-compartment active IF model. (D) Simulated membrane potential of the single-compartment passive IF model. (B-D) Simulated binaural inputs are injected to the model neurons (see Materials and Methods for detail). "In-phase" means that the oscillation peaks of both inputs coincide resulted in the largest oscillation amplitudes. "Out-of-phase" means that the oscillating bilateral synaptic inputs are cancelled with each other to minimize the signal component at f _stim = 4000 Hz [29].

Fig 3. Spiking properties of the reduced models.

(A_1-3) Two-compartment active IF model. (B_1-3) Single-compartment active IF model. (C_1-3) Single-compartment passive IF model. First row (A₁,B₁,C₁): simulated spike rates plotted against the phase difference δ of the bilateral inputs. Peaks appear at δ = 0 or ±2π (in-phase rate) whereas troughs appear at ±π (out-of-phase rate). Second row (A₂,B₂,C₂): simulated spike rates plotted against the threshold of the IF unit. Refractory period was fixed to 0.9 (ms). Modulation depth (gray lines) is defined as the difference between the in-phase and out-of-phase rates (black lines). Dotted lines show the criterion of 180 spikes/sec for the modulation depth. Third row (A₃,B₃,C₃): simulated spike rates plotted against the length of the refractory period. Threshold was fixed to (A₃) -56.7, (B₃) -58.3, and (C₃) -58.6 mV, respectively. Shaded area of each panel in the first and third rows shows the range of typical discharge rates of the owl's NL neuron (see Materials and Methods for the definition).

Fig 4. Model efficiency and reliability.

(A) Relative integration time of the active Na (upward triangles), two-compartment active IF (downward triangles), single-compartment active IF (circles), non-spiking (diamonds), and single-compartment passive IF (squares) models. Single-compartment active IF model with Δt = 10.0 μs (small arrow) was used as the reference point (i.e., relative integration time = 1.0). Open symbols denote numerical unreliability with corresponding panels indicated by the small letters. (B) Unreliability of the two-compartment active IF model. A large time step (Δt = 3 μs, thin black line) leads to an erroneous oscillation in the nodal potential induced by spike currents. Inset shows a magnified view of the spike-induced unreliability, which does not occur with a sufficiently small time step (e.g., Δt = 1 μs, gray line). (C) Unreliability of the single-compartment active IF model. Too large a time step (Δt = 20 μs in this example) leads to an imprecise computation of the somatic shape. (D) Unreliability of the single-compartment passive IF model. Too large a time step (Δt = 20 μs in this example) leads to an imprecise computation of the spike shape. Small arrows in C and D show the timings when the simulated traces with Δt = 20 μs started to diverge from the traces with Δt = 1 μs by more than 0.5 mV.