Data Analysis

Thresholds to binaural, ipsilateral, and contralateral single electric pulses were quantified by the minimum level of the search stimulus that evoked significantly pulse-locked firing (Hancock et al. 2012). At each level and for each of the three pulses, a peristimulus time histogram (PSTH) was computed using 0.1-ms bins. Confidence bounds were assigned to identify significant peaks in the PSTH. Specifically, synthetic random spike trains were used to generate an additional 1000 PSTHs, where the number of synthetic spikes was equal to the number of spikes in the actual neural response. The confidence bound on each bin in the neural PSTH was the 99th percentile of the synthetic PSTHs. A phase-locked response was deemed to exist when two consecutive bins of the neural PSTH exceeded the 99 % confidence bounds. Interpolation was used to find the lowest level where this criterion was met.

Firing rate was computed as a function of envelope shape (burst width and repetition rate) by counting spikes over the 288-ms stimulus duration, excluding the first period of the repetition rate to avoid onset effects. Firing rates were normalized between the minimum and maximum rates across envelope shapes (in practice, the minimum was almost always zero) and visualized using heat maps (e.g., Fig. ​Fig.2b).2b). To improve the clarity of the visualization relative to the coarse sampling of the parameter space, the normalized firing rates were interpolated on a 4× finer scale (griddata, MATLAB). The interpolation does not change the value of the original firing rates; it serves only to guide the eye, in the same manner as the line connecting data points on a line plot.

Responses of one IC neuron as a function of envelope burst width and modulation rate. a Dot rasters show spike times relative to stimulus onset. Alternating colors distinguish blocks of trials at different burst widths. Ticks at bottom of each column mark the start of each modulation cycle. b Corresponding heat map showing normalized firing rate as a function of envelope shape. Solid line: contour enclosing firing rates ≥75 % of maximum. Circle: centroid of 75 % contour. Blue inset: best envelope shape (waveform corresponding to centroid). Gray: waveforms illustrating subset of envelope shapes on perimeter of heat map.

The contour corresponding to 75 % of the maximum firing rate was computed (contour, MATLAB) and used to characterize the region of the envelope shape parameter space to which the neuron was best tuned. The burst width/repetition rate combination at the centroid of this region was taken as the “best envelope.” The centroid is a less noisy and less discretized measure of central tendency than simply taking the point with the maximum firing rate.

The variability across neurons with respect to the most effective envelope shape was summarized by plotting best envelope shape (i.e., the centroid of each 75 % contour) in the burst width-repetition rate space. A k-means clustering analysis (kmeans, MATLAB) was applied to this scatterplot to objectively segregate neurons into groups based on most effective envelope shape. Centroids are initially grouped at random into k clusters. The k-means procedure then iteratively re-clusters the centroids to minimize the summed Euclidean distance between each centroid and its cluster centroid. The entire clustering procedure was repeated 10 times with different random initial clusters to increase the likelihood of finding the global maximum. The number of clusters k is an independent variable that was systematically varied from 1 to 10.

Period histograms were constructed by binning neural spike times modulo the period of the envelope repetition rate. For ease of comparison, the time axes were rotated to center each envelope burst within the repetition rate cycle. We subtracted the first spike latency to a single pulse from each spike time so that the period histogram indicates more directly the phase of the envelope responsible for triggering spikes.

The first spike latency was estimated for each neuron from its responses to the search stimulus (single bilateral, contralateral, and ipsilateral pulses varied in level). Specifically, the latency was computed individually from the responses to single bilateral, contralateral, and ipsilateral pulses measured at the same levels as used to acquire the envelope shape data and the level 1 dB higher. The overall first spike latency was taken as the mean of those six latencies. In rare cases when this method did not provide a reliable estimate (11/121 units), the first spike latency was instead estimated from the envelope data set itself, specifically the response to the 4-ms burst width (shortest burst width) presented at the lowest repetition rate. In those cases, it was assumed that the first spike was elicited by the peak of the burst which occurs 2 ms after stimulus onset, and this additional delay was subtracted from the computed latency.

Phase locking was quantified using vector strength (Goldberg and Brown 1969). The preferred or “best phase” was computed as a fraction of the repetition rate period in the usual way, but then scaled relative to the burst width to allow direct comparison of response phase across envelope parameters. Thus, best phases of 0, 0.5, and 1 correspond to the beginning, peak, and end of the burst, even if the burst is followed by a silent interval. Best phases <0 or >1 indicate spikes elicited during the silent portion of the repetition rate cycle, when present.

A simple phenomenological model of envelope coding involving an interplay between excitation and feedforward inhibition was fit to the neural responses. It closely follows the model of Smith and Delgutte (2008) for interaural time difference coding of amplitude-modulated CI stimulation, except here it is diotic in form, rather than binaural. The model is also similar to the SFIE model of Nelson and Carney (2004) used to describe response of IC neurons to AM in NH animals. To compute model responses, a threshold was applied to the pulse train stimuli used in the experiment which were then convolved with a positive-going alpha function simulating an excitatory postsynaptic potential (EPSP) with strength α and time constant τ:

where u(t) is the unit step function. The threshold was fixed at −3 dB re: peak to produce stimulus levels comparable to the experimental data. Feedforward inhibition was implemented by convolving the excitatory waveform with a negative-going alpha function:

where the term t _d indicates a synaptic delay. The excitatory and inhibitory waveforms were then summed and half-wave rectified, yielding an instantaneous probability of firing which was integrated over time to compute a mean firing rate. This was repeated for every envelope in the parameter space (Fig. ​(Fig.1)1) to generate data sets in the same form as the experimental data.

The model nominally has five free parameters: α _EPSP, α _IPSP, τ _EPSP, τ _IPSP, and t _d. However, the excitatory and inhibitory strengths, α _EPSP and α _IPSP, trade directly for one another, so the excitatory strength was fixed to 1. To a lesser extent, the inhibitory strength also trades with the synaptic delay (not shown), so t _d was fixed to 1 ms. The remaining three free parameters (α _IPSP, τ _EPSP, and τ _IPSP) were fit to experimental data sets using a gradient descent algorithm (lsqcurvefit, MATLAB).