Stochastic information transfer from cochlear implant electrodes to auditory nerve fibers

Cochlear implants, also called bionic ears, are implanted neural prostheses that can restore lost human hearing function by direct electrical stimulation of auditory nerve fibers. Previously, an information-theoretic framework for numerically estimating the optimal number of electrodes in cochlear implants has been devised. This approach relies on a model of stochastic action potential generation and a discrete memoryless channel model of the interface between the array of electrodes and the auditory nerve fibers. Using these models, the stochastic information transfer from cochlear implant electrodes to auditory nerve fibers is estimated from the mutual information between channel inputs (the locations of electrodes) and channel outputs (the set of electrode-activated nerve fibers). Here we describe a revised model of the channel output in the framework that avoids the side effects caused by an “ambiguity state” in the original model and also makes fewer assumptions about perceptual processing in the brain. A detailed comparison of how different assumptions on fibers and current spread modes impact on the information transfer in the original model and in the revised model is presented. We also mathematically derive an upper bound on the mutual information in the revised model, which becomes tighter as the number of electrodes increases. We found that the revised model leads to a significantly larger maximum mutual information and corresponding number of electrodes compared with the original model and conclude that the assumptions made in this part of the modeling framework are crucial to the model's overall utility.

Figure 1

Geometry of the modeling framework and the variables associated with the conversion from electrode current to channel output for both the original model and the revised model. (A) The geometry of the modeling framework. Both the original model and revised model share the same geometry of auditory nerve fiber locations and electrode locations. The $j\text{th}$ electrode produces current $C_{e,j}$ amperes at the normalized location $x_{e,j}$. This current decreases to $C_{f,i}$ amperes at the $i\text{th}$ fiber $x_{f,i}$, after attenuation with the normalized distance from the $j\text{th}$ electrode to the $i\text{th}$ fiber $d_{i,j}$. $Y_i$ represents whether a fiber produces an action potential. (B) The aggregate channel output in the original model. ${\varphi }_0,...,{\varphi }_M$ represent the boundaries locations of the intervals which are used to separate all fibers into $M$ intervals. $W_1,...,W_M$ outcomes the number of action potentials produced by fibers in each interval. (C) The observable channel output in the revised model. $Z_1$ is denoted as the normalized location of all active nerve fibers, in response to $C_{e,j}$. In the revised model, the fibers are not separated into $M$ intervals. [This figure is adapted from Fig. 1 in Ref. [12], © 2014 IEEE. Adapted, with permission, from Ref. [12].]

Figure 2

Full channel model from source to output for both original model and revised model. The channel input is the location of one of the $M$ electrodes. The output of noisy channel is vector $\mathbf{Y}$. For the original model, the channel output is converted to an observable channel output $Z$, via $f(·)$. For the revised model, the channel output is converted to an observable channel output $Z_1$, via $g(·)$. [This figure is adapted from Fig. 2 in Ref. [12], © 2014 IEEE. Adapted, with permission, from Ref. [12].]

Figure 3

Locations of fibers. Both uniform locations and nonuniform locations of fibers are shown. The dot-dash line shows the locations of uniformly distributed fibers, whereas the solid line shows the locations of nonuniformly distributed fibers.

Figure 4

Distribution of $Z_1$ for five different assumed locations of a single stimulating electrode, obtained by $10000$ iterations of Monte Carlo simulations, with a Gaussian distribution fit for each case. The number of surviving fibers is assumed to be $N=3000$ and they are uniformly distributed along the cochlea. Bipolar stimulation mode is used. Five cases are shown: (a) $x_{e,j}$ = 0, (b) $x_{e,j}$ = 0.25, (c) $x_{e,j}$ = 0.5, (d) $x_{e,j}$ = 0.75, (e) $x_{e,j}$ = 1.

Figure 5

The conditional mean (a) and the variance (b) of $Z_1$ given $X_e$ obtained numerically from $10000$ iterations of Monte Carlo simulations for each location of electrodes. The number of surviving fibers is assumed to be $N=3000$ and they are uniformly distributed along the cochlea. Bipolar stimulation mode is used.

Figure 6

Comparison between the exact mutual information for different nerve fiber survival rates and stimulation modes in the original model. Both uniformly and nonuniformly distributions of fibers are used. U. represents uniformly distributed fibers and NU. represents nonuniformly distributed fibers in this figure. Two cases are shown: (a) the bipolar stimulation mode and (b) the monopolar stimulation mode.

Figure 7

Comparison between the optimal number of electrodes for different stimulation modes in the original model. Both uniform (U.) and nonuniform (NU.) distributions of fibers are used.

Figure 8

Comparison between the mutual information for different nerve fiber survival rates and stimulation modes in the revised model. Both uniformly and nonuniformly distributions of fibers are used. U. represents uniformly distributed fibers and NU. represents nonuniformly distributed fibers in this figure. Two cases are shown: (a) the bipolar stimulation mode and (b) the monopolar stimulation mode.

Figure 9

Comparison between the optimal number of electrodes for different stimulation modes in the revised model. Both uniform (U.) and nonuniform (NU.) distributions of fibers are used.

Figure 10

Comparison between the density of $Z_1$ for different cases of number of electrodes. The density of $Z_1$ is obtained from numerical calculation. Four cases are shown: (a) M = 30, (b) M = 50, (c) M = 70, and (d) M = 100.

Figure 11

Comparison between exact mutual information and the upper bound of the mutual information. The number of surviving fibers is assumed to be $N=3000$ and they are uniformly distributed along the cochlea. The bipolar stimulation mode is used.