Spiking Neurons Learning Phase Delays: How Mammals May Develop Auditory Time-Difference Sensitivity

Time differences between the two ears are an important cue for animals to azimuthally locate a sound source. The first binaural brainstem nucleus, in mammals the medial superior olive, is generally believed to perform the necessary computations. Its cells are sensitive to variations of interaural time differences of about $10\mu \mathrm{s}$. The classical explanation of such a neuronal time-difference tuning is based on the physical concept of delay lines. Recent data, however, are inconsistent with a temporal delay and rather favor a phase delay. By means of a biophysical model we show how spike-timing-dependent synaptic learning explains precise interplay of excitation and inhibition and, hence, accounts for a physical realization of a phase delay.

Figure 1

Tuning asymmetry of a gerbil. (a) A sound source (large circle) evokes high neuronal discharge rates in a brainstem nucleus MSO in the opposite (contralateral) brain hemisphere and low rates in the ipsilateral MSO at the same side of the source. (b) Corresponding ITD tuning curve from a gerbil; based on Fig. 3c of Brand et al. [2]. Tuning maxima are statistically significant [2] at contralaterally leading (positive) ITDs, mostly even outside the physically accessible ITD range (vertical dashed lines).

Figure 2

Developmental synaptic dynamics. (a) Learning windows [19] are ${\eta }^{(\pm )}{W}^{(\pm )}$ for excitatory (solid line) and inhibitory (dashed line) synapses, respectively; see main text. (b) Time course of synaptic strengths $J$ as a function of axonal delay $\Delta$ preceding a synapse. Initial excitatory (dots) and inhibitory (bars) synaptic weights are random. During the learning period, delays of excitatory connections become bilaterally restricted to an interval of only $250\mu \mathrm{s}$ (indicated by a thick horizontal bar). Contralateral inhibition becomes strong at delays slightly shorter than excitation. With $c_I^{(-)}=0.5$, ipsilateral inhibition initially ($\mathit{T}/30,\mathit{T}/6$) shows the same trend but finally at $\mathit{T}$ ends up temporally unstructured. (c1) The fully developed delay distributions from (b) account for an asymmetric tuning curve (triangles). The dotted lines indicate the physiological range of ITDs (here $\pm 0.12\mathrm{ms}$). For an explanation of other symbols, see main text. (c2) Tuning asymmetry can only be achieved with time constants ${\tau }^{(-)}⪅0.3\mathrm{ms}$. In order to preserve phase relationships, the inhibitory learning window was adapted to different values of ${\tau }^{(-)}$ so that ${\widehat{W}}^{(-)}\to {\widehat{ϵ}}_0^{(-)}{\widehat{W}}^{(-)}/{\widehat{ϵ}}^{(-)}$, with ${ϵ}_0^{(-)}$ denoting the inhibitory voltage response for ${\tau }_{(-)}=0.1\mathrm{ms}$. Learning with periodic input processes (open triangles) slightly alters the phase of the tuning curve but does not prevent asymmetry. Tuning curves are obtained with periodic 400 Hz input; cf., Fig. 3.

Figure 3

Frequency dependence of tuning curves. (a) Average tuning curves are obtained as best fits of a Gabor function to simulated firing rates. Tuning curves are normalized [4] so that their maximum is 1 and their minimum is 0. Tuning data are obtained through $1/f$-periodic input Poisson processes [5]. The absolute value of the Poisson intensity’s first harmonic divided by the harmonic of order zero is denoted by $r$ and taken to be $r(f)=1-0.1658(f/\mathrm{kHz}{)}^{1.219}$, a fit to experimental data [9]. The value of $r$ for the $(I,-)$ inputs is reduced by the factor $c_I^{(-)}=0.5$. Rates are averaged over 10 sec and mean input rates are 100 Hz. (b) The best ITDs (squares) are defined as the maxima of the tuning curves from (a). They monotonically decrease with stimulation frequency whereas the best IPDs (triangles, $\mathrm{IPD}=\mathrm{ITD}\times \mathrm{\text{frequency}}$) are approximately constant, which is in agreement with experiment [2, 4]. The solid line depicts the phase delay obtained from the linear theory.