Predicting the Timing of Spikes Evoked by Tactile Stimulation of the Hand

Abstract

What does the hand tell the brain? Tactile stimulation of the hand evokes remarkably precise patterns of neural activity, suggesting that the timing of individual spikes may convey information. However, many aspects of the transformation of mechanical deformations of the skin into spike trains remain unknown. Here we describe an integrate-and-fire model that accurately predicts the timing of individual spikes evoked by arbitrary mechanical vibrations in three types of mechanoreceptive afferent fibers that innervate the hand. The model accounts for most known properties of the three fiber types, including rectification, frequency-sensitivity, and patterns of spike entrainment as a function of stimulus frequency. These results not only shed light on the mechanisms of mechanotransduction but can be used to provide realistic tactile feedback in upper-limb neuroprostheses.

INTRODUCTION

Cutaneous cues play a critical role in the dexterous manipulation of objects. First, signals from mechanoreceptive afferents that innervate the skin convey information about the location and timing of contact of an object on the skin (Johannson and Birznieks 2004; Ochoa and Torebjörk 1983; Wheat et al. 1995 and about the forces it exerts on the skin when it is grasped (Goodwin and Wheat 2004; Knibestöl 1973, 1975; Macefield et al. 1996; Muniak et al. 2007). Cutaneous afferents also signal when our grip on an object is slipping (Johansson and Westling 1984). Although this information is sometimes (though not always) available visually, visual signals are slower and less reliable and require greater concentration to guide movement (Ghez et al. 1995). Second, cutaneous afferents convey information about object properties—such as form and surface texture—that can help guide object manipulation and recognition (Johansson and Flanagan 2009; Johnson 2001).

Tactile information from the hand is conveyed to the brain by four types of mechanoreceptive nerve fibers, each sensitive to different aspects of skin deformation (Bolanowski et al. 1988; Hsiao and Bensmaia 2008; Johansson 1978; Johnson 2001; Talbot et al. 1968). In the hand of the macaque monkey, innervated by three of these fiber types (Pare et al. 2002), slowly adapting type 1 (SA1) fibers are sensitive to coarse spatial structure, rapidly adapting (RA) fibers to motion, and Pacinian (PC) fibers to surface microgeometry. Although mechanoreceptive responses have been extensively characterized using a variety of stimuli, no quantitative framework exists to predict their responses to novel stimuli. Such a model would reduce the need to measure the peripheral representation of stimuli when seeking to characterize how these are processed in the brain.

Indeed recent studies have shown that the spatial properties of tactile objects undergo a nonlinear transformation at the periphery (Dandekar and Srinivasan 1998; Phillips and Johnson 1981; Sripati et al. 2006), and it is this transformed representation of the stimulus to which the brain has access. However, very little is known about how temporal properties of tactile stimuli influence the neuronal response. For instance, it is not clear whether SA1 fibers simply respond to the depth to which a probe is indented into the skin or if they are also sensitive to the rate at which the probe is indented. Even if the stimulus quantity were known, how the precise pattern of spikes is generated based on this input remains to be elucidated. Here we describe a framework to address both issues. First, we recorded the patterns of spikes elicited by a wide variety of vibrations delivered to the skin by a punctate probe with no spatial elaboration. We then used a small subset of these data to fit an integrate-and-fire model. The resulting model can predict the timing of individual spikes evoked in all three fiber types by arbitrary tactile vibrations applied to the skin and accounts for all the well-known properties of afferent responses to vibrations. Second, by comparing model predictions based on different physical quantities or their combinations, we identify the stimulus quantities that drive transduction in each fiber type. Finally, we discuss how this temporal model, combined with a spatial model developed previously (Sripati et al. 2006), can be used to elucidate the representation of complex tactile stimuli varying in both spatial and temporal properties.

METHODS

The methods and stimuli are described in detail in a previous paper (Muniak et al. 2007) and so are just summarized here.

Neurophysiology

All experimental protocols complied with the guidelines of the Johns Hopkins University Animal Care and Use Committee and the National Institutes of Health Guide for the Care and Use of Laboratory Animals. Single unit recordings were made from the ulnar and median nerves of four anesthetized Macaque monkeys (Macaca mulatta) using standard methods (Talbot et al. 1968). Standard procedures were used to classify mechanoreceptive afferents according to their responses to step indentations and vibratory stimulation (Freeman and Johnson 1982; Talbot et al. 1968). An afferent was classified as SA1 if it produced sustained firing in response to a step indentation. It was classified as RA if it had a small receptive field and responded only to the onset and offset of an indentation. It was classified as PC if it was vigorously activated by air blown gently over the hand, it was activated by transmitted vibrations produced by tapping on the hand restraint, and its receptive field was large. The point of maximum sensitivity of the afferent (or hotspot) was located on the skin using a handheld probe and then marked with a felt-point pen. The stimulator probe was centered on the hotspot of the afferent to the extent possible (PC fibers do not have clear hotspots). The tip of the probe was fixed with cyanoacrylate glue to the skin at its resting position, i.e., with no preindentation.

Apparatus

A custom-made Chubbuck motor (Chubbuck 1966), driven by a servo-controlled amplifier and equipped with a high precision linear variable displacement transducer (LVDT) with sub-micron-resolution, was used to deliver tactile stimuli. The input voltage to the amplifier, under computer control, was generated using a digital to analog card (PCI-6229, National Instruments, Austin, TX; output rate = 20 kHz). The Chubbuck displacement sensor was calibrated using an Optodyne laser interferometer (Optodyne LDS 1000, Compton, CA), capable of measuring absolute displacement to sub-micron resolution: the actual position, as measured by the interferometer, was regressed onto the output of the position sensor. The contactor consisted of a steel-tipped stylus fixed to the table of the Chubbuck. The stylus was 175 mm long with a radius of 12.5 mm and a weight of 12.5 g. The tip of the stylus was flat and had a diameter of 1 mm (cf. Freeman and Johnson 1982).

Stimuli

SINUSOIDS.

Sinusoids were presented at 1, 5, 10, 20, 30, 40, 50, and 100 Hz. At 1 Hz, amplitudes ranged from 5 to 360 μm, zero-to-peak; at 10 Hz, from 2.5 to 130 μm; at 100 Hz, from 0.5 to 130 μm. The stimulus duration was either five stimulus cycles or 0.1 s, whichever was longer. At each frequency, amplitudes were incremented in equal logarithmic steps over the range. In total, 120 different sinusoids were presented.

DIHARMONIC STIMULI.

Diharmonic stimuli were specified using the following expression

$$x\left(t\right)=A_1\text{sin}\left({\omega }_{1}t\right)+A_2\sin \left({\omega }_{2}t+\varphi \right)$$ (1)

where A₁ and A₂ are the amplitudes of the low- and high-frequency components, respectively, ω₁ and ω₂ are the two frequencies (ω₁ < ω₂), and ϕ is the phase of the high-frequency component relative to that of the low-frequency component. Component amplitudes ranged from 2 to 125 μm if their frequency was <100 Hz; from 2 to 100 μm at 100 and 125 Hz. The phase ϕ took on one of four values: 0, π/2, π, or 3π/2. The stimulus duration was either five cycles of the low-frequency component or 0.1 s, whichever was longer. The amplitudes of the two frequency components were incremented in equal logarithmic steps over their respective ranges. In total, 240 different diharmonic stimuli were presented.

TRIHARMONIC STIMULI.

Triharmonic stimuli were specified using the following expression

$$x\left(t\right)=A_1\text{sin}\left({\omega }_{1}t\right)+A_2\text{sin}\left({\omega }_{2}t+{\varphi }_2\right)+A_3\text{sin}\left({\omega }_{3}t+{\varphi }_3\right)$$ (2)

where A₁, A₂, and A₃ are the amplitudes of three sinusoidal components, respectively, ω₁, ω₂, and ω₃ are their frequencies (ω₁ < ω₂ < ω₃), and ϕ₂ and ϕ₃ are the phases of the two higher frequency components (with matching subscripts). Component amplitudes ranged from 2 to 125 μm if their frequency was <100 Hz and from 2 to 100 μm at 100 and 125 Hz. The phases ϕ₂ and ϕ₃ took on one of two values: 0 or π (phase values were parametrically combined). The stimulus duration was either five cycles of the low-frequency component or 0.1s, whichever was longer. The amplitudes of the three frequency components were incremented in equal logarithmic steps over their respective ranges. In total, 40 different triharmonic stimuli were presented.

BAND-PASS NOISE STIMULI.

Stimuli were generated by first creating wide-band noise, then band-pass filtering it to the specified frequency range using a finite impulse response filter (fir1 function, MATLAB, Natick MA). The low cut-off frequency was 5 Hz and the high cut-off frequencies were 10, 25, 50, and 100 Hz. Each noise stimulus was scaled to a set of predetermined RMS amplitudes, namely 0.5, 1, 5, 10, and 50 μm. The duration of all noise stimuli was 1 s. Each stimulus was preceded and followed by a period of no stimulation to reduce the effects of vibratory adaptation (Bensmaia et al. 2005; Leung et al. 2005). In total, 20 different noise stimuli were presented.

The interstimulus interval was 100 ms for sinusoidal, diharmonic, and triharmonic stimuli and 1 s for noise stimuli.

Stimulus selection

For model fitting, we excluded stimuli that elicited weak or irregular responses with no repeatable spike timing. For RA and PC fibers, we also limited our analysis to low-intensity stimuli. For PC fibers, this restriction was implemented because PC fibers saturate (see Supplemental Figs. S4 and S5¹), a feature that cannot be captured by a model the input filters of which are linear. For RA fibers, we used a restricted range of stimuli because RA fibers receive input from multiple Meissner corpuscles (Pare et al. 2002). As stimulus intensity increases, an increasing number of spatially displaced Meissner corpuscles are recruited. As a result, the spiking activity measured in the afferent fiber reflects input from multiple receptors, a characteristic that cannot be captured using a single-compartment model. We verified that the model fit with low-intensity stimuli could not account for responses to high-intensity stimuli (Supplemental Fig. S5B) and models fit to high-intensity yielded poor fits as expected if the responses reflect the activity of multiple receptors. Overall ∼20% the stimuli were excluded because they were intense enough to fall within the saturation range for PC fibers or within the multiple-receptor range for RA fibers (no such exclusion was necessary for SA1 afferents). Note, however, that the range of intensities over which afferent responses are linear and determined by a single receptor spans a large portion of the range of natural tactile experience.

Selection of candidate physical modalities

Of the 15 possible combinations of input variables, we examined the 10 contiguous combinations (shown in Fig. 6) and excluded 5 others (position-acceleration, position-jerk, velocity-jerk, position-velocity-jerk, and position-acceleration-jerk). We reasoned that physical systems are usually driven by contiguous combinations of temporal derivatives rather than noncontiguous combinations. However, this restriction in our analysis carries no loss of generality because if a neuron was sensitive to (say) position and acceleration, its responses would be fit best by a model comprising position, velocity, and acceleration with predominant contributions from position and acceleration but not velocity. However, we did not observe such instances. Instead most responses were best fit by a single input variable (RA neurons) or contiguous pairs of variables (SA1 neurons). Even in the case of PC neurons, the responses of which were fit by position, velocity, and acceleration, we found significant contributions of all three inputs, suggesting that all three variables contributed significantly at different input frequencies (Fig. 7B).

Fig. 6.

Model fit. Mean spike distance (D_spike) per spike for each of the 10 models for each afferent type for the test data, pooled across the population of neurons sampled. A: for SA1 afferents, the best model was one with position and velocity as inputs. This model provided equivalently good fits as the full model (p, v, a, j) as well as one of the two triplet models (p, v, a and v, a, j) but comprised far fewer parameters. B: for RA afferents, a model with only velocity as input was as good as any other model. C: for PC afferents, the position, velocity, and acceleration model was as good as or better than the other models.

Fig. 7.

Contribution to the stimulus induced currents. Current contribution of position (blue), velocity (green) and acceleration (red) for SA1 (A) and PC (B) afferents.

Calculation of derivatives

We used observed probe position signals to calculate the indentation velocity, acceleration, and jerk. We preprocessed the position (indentation) signal to mitigate the noise introduced by taking the derivatives. We used the Loess quadratic smoothing method (smooth function, MATLAB, Natick MA) on the raw data (with a 20-kHz sampling rate) before performing the derivative. We also zero padded the end of the signal after each derivative operation to maintain a constant number of samples. We then resampled the signals to 1 kHz for the SA1 and RA data and 4 kHz for the PC data. These resampled signals were used as input to the model.

Integrate-and-fire model

We implemented an integrate-and-fire (IF) model that replicates the properties of all three types of mechanoreceptive afferents and attempts to capture the underlying biophysical mechanisms. The model can be driven by four inputs—indentation depth, velocity, acceleration, and jerk. Note that the force exerted on the surface of the skin by a punctate probe is approximately linear with the depth to which it indents it over a range of depths (Pawluk and Howe 1999), so force and indentation are interchangeable for the purposes of the model. Indentation velocity, acceleration, and jerk were calculated as the first, second, and third derivatives of the time-varying probe position signal. Because our method for estimating the model parameters only allows for linear transformations of the inputs, we allowed for rectification by separating each input into its positive and negative components (i.e., indentations and retractions in the case of position). This allowed us, for example, to represent responses to probe indentation but not retraction (i.e., half-wave rectification) by multiplying the negative component of the probe position by zero (a linear filter). In general, this partitioning allows for full-wave, half-wave, or partial rectification of the input signal using appropriate linear transformations.

In the model, inputs are filtered separately and then summed to form the input to a leaky IF mechanism. The IF mechanism is a reasonable approximation to the membrane potential dynamics preceding a spike (Koch 1999). The summed input to the IF mechanism is treated as a postsynaptic current that drives the membrane potential in the soma. The membrane potential is also affected by a leak current with a time constant τ_m that drives it toward a resting potential V_r. When the membrane potential reaches a voltage threshold V_T (set to 1), a spike is registered and the membrane potential is reset to zero. To mimic the short refractory period following a spike, a postspike inhibitory current is injected into the neuron. The total postspike inhibitory current at any given time is the sum of the postspike currents due to all previous spikes. To mimic the intrinsic variability in current due to ion channel stochasticity, a white noise current is injected with a specified variance. This is a common assumption in models of variable spiking in neurons (Koch 1999; Tuckwell 1988) and makes the estimation of model parameters analytically tractable (Paninski et al. 2004).

The equations governing the integrate-and-fire mechanism are as follows. The membrane potential, V, is given by

$$dV=\left(-\frac{\left(V-V_r\right)}{{\tau }_{\text{m}}}+I_{\text{in}}\left(t\right)+I_{\text{ps}}\left(t\right)\right)dt+W_t$$ (3)

where I_ps(t), the accumulated postspike inhibitory current, is given by

$$I_{\text{ps}}\left(t\right)={\displaystyle \sum _{j=0}^{s-1}h\left(t-t_j\right)}$$ (4)

where h is the waveform of a single postspike inhibitory current (Fig. 12C), s is the number of past spikes, and t_j is the time of the jth spike. Thus the postspike current at time t represents the accumulation of the postspike currents produced by all previous spikes. I_in(t) is the input current, given by

$$I_{\mathrm{in}}\left(t\right)={\displaystyle \sum _i{H}_i^{+}\left(t\right)*y_i^{+}\left(t\right)+H_i^{-}\left(t\right)*y_i^{-}\left(t\right)}$$ (5)

where y_i⁺(t) and y_i⁻(t) are the positive and negative components of the ith derivative of time-varying indentation depth, H_i⁺(t) and H_i⁻(t) are the corresponding linear filters, and the summation is performed for the variables hypothesized to drive the afferent response. W_t is Gaussian noise with zero mean and variance σ². We implemented the model using the discrete time version of the preceding equations with a 1-ms sampling time for SA1 and RA afferents. Because PC afferents are sensitive to higher frequencies, we used a sampling time of 0.25 ms for PC data.

Fig. 12.

Model parameters and properties. A: correlation between indentation and retraction filters for SA1 (blue), RA (red), and PC (green) fibers. The indentation and retraction filters tended to be correlated for all fiber types and all stimulus quantities (position, velocity, acceleration). B:

$$\text{rectification index=1−}\frac{4}{\pi }{\tan }^{-1}\left(\frac{{\displaystyle \sum _t\left|H_i\left(t\right)-H_r\left(t\right)\right|}}{{\displaystyle \sum _t\left|H_i\left(t\right)+H_r\left(t\right)\right|}}\right)$$

where H_i(f) and H_r(f) are the indentation and retraction filters. A rectification index of −1 indicates no rectification; 0 half-wave rectification; 1 full-wave rectification. Afferents tended to exhibit rectification properties intermediate between half- and full-wave. C: mean postspike inhibitory currents. SA1 and RA currents were similar, whereas PC currents exhibited a considerably faster time course. D: mean membrane time constants, τ_m, for the three afferent types. SA1 and RA afferents yielded similar membrane time constants, whereas those obtained from PC afferents were considerably faster.

Model parameters

The free parameters in the model are: the membrane time constant (τ_m), the resting potential (V_r), the SD of the noise (σ), the postspike inhibitory current (6 values specifying the shape of the filter in terms of weights associated with a raised cosine basis set), and the pair of filters corresponding to the positive and negative components of each input variable (60 values for SA1 and RA and 120 for PC afferents, specifying the shape of each filter as a function of number of samples). Thus the number of free parameters in the model can range from a minimum of 129 for SA1 and RA and 249 for PC afferents (when a single physical variable, e.g., position, is used as input) to a maximum of 489 for SA1 and RA and 969 for PC afferents (when all physical variables i.e., probe position, velocity, acceleration, and jerk are used as input).

The large number of parameters in the model might raise concerns that there are not enough data to adequately constrain the model. We consider this unlikely for two reasons: first, if the model was indeed overfit to the data, it would not be able to generalize well to novel stimuli. We found that the model fit to a subset of data was able to predict responses to a wide variety of vibratory stimuli to which it had not been previously exposed (see results). Second, the bulk of the model parameters was subject to a smoothness constraint (described in Regularization of input filters), which substantially reduces the effective number of degrees of freedom in the model.

Basis set for the postspike inhibitory current and the input filter

In our preliminary fits, we found that the postspike inhibitory current had a highly stereotyped shape across neurons. We therefore found it convenient to represent this current using a compact basis set, namely raised cosine functions (Pillow et al. 2005). The postspike inhibitory current can then be specified in terms of only six parameters, each specifying the weight to be assigned to a particular raised cosine function. In contrast to the stereotyped nature of the postspike current, we found the input filters to be highly variable and could not be represented in a compact fashion. We therefore represented each filter using 60 samples for SA1 and RA fibers and 120 for PC fibers specifying the filter magnitude for each time sample. In other words, the basis set for the input filters were 60 delta functions (120 for PCs), each with a peak at a particular time sample. During the fitting process, we imposed smoothness constraints to reduce noise in the estimated filter shapes due to over-fitting (see following text).

Estimation of model parameters

Because the IF mechanism is nonlinear, it is extremely challenging to estimate the underlying model parameters using conventional least-squares algorithms. Recent studies have gained traction by reformulating this problem into a form amenable to optimization based on maximizing likelihood (Paninski 2004; Paninski et al. 2004; Pillow et al. 2005). The essential insight behind this approach is that the probability of observing a spike at a particular time t_j depends only on the input current and voltage dynamics since the spike at t_j-1 and on residual postspike inhibitory currents. Thus the probability of observing an entire set of spikes at times t₁, t₂, …, t_N can be calculated simply as the product of the individual probabilities. To calculate the probability of a spike at t_j given the preceding spike at t_j−1, the approach assumes that Gaussian noise is added to the membrane potential. In this case, this probability is given as

$$p\left(\text{spike at }{t}_j|\text{spike at }{t}_{j-1}\right)={\displaystyle \underset{V\in C_j}{\int }G\left[V\left(t\right)|y,\theta \right]}$$ (6)

where C_j is the set of possible voltage trajectories that can give rise to the inter-spike interval j, G[V(t)|y, θ] is the Gaussian density function of the voltage trajectory over this interval (from Eq. 3), y is the stimulus, and θ represents the parameters of the IF model. The mean and covariance of G can be easily calculated given the input currents and the model parameters (Paninski et al. 2004; Pillow et al. 2005). The probability of observing a given sequence of spikes at times t₁, t₂, …, t_N can then be calculated as the product of the individual probabilities, each given by Eq. 6. We can then estimate the parameters that are most likely to have given rise to the set of observed spikes. The problem of finding these maximum likelihood parameters is convex and therefore can be solved using standard convex optimization techniques (Paninski et al. 2004; Pillow et al. 2005). See in the following text for further details.

Regularization of input filters

Because a large number of parameters are used to specify the input filters, it is possible that certain parameters in the filter (say, the filter magnitude at time t = 60 ms) are of little consequence to the quality of fit and remain unconstrained. This can result in a filter waveform with irregular shapes. To avoid this problem, we introduced a regularization term (to penalize large variations in the filter values) into the cost function (E) to be minimized

$$E={\displaystyle \sum _i{w}_i\left[\lambda {\displaystyle \sum _{l=1}^q{\left(d_l^i\right)}^2+\delta {\displaystyle \sum _{l=2}^q{\left(d_l^i-d_{l-1}^i\right)}^2}}\right]-{\displaystyle \sum _{j=2}^n\text{log}\hspace{1em}}p\left(t_j|t_{j-1}\right)}$$ (7)

where t_jis the jth spike time, n is the number of spikes, i represents each input filter, q is the length of the input filter, d_lⁱ is the value of the ith filter at position l, and w_i > 0, λ > 0, and δ > 0 are positive weights. λ and δ determine the relative importance of the regularization terms compared with the log-likelihood terms, and w_i is the penalty for each filter, chosen such that all filters are penalized to the same degree during regularization. In particular, because the coefficients for each filter will depend on the magnitude of its input, the penalty w_i was chosen to be proportional to the square of the sum of the absolute input signal to ensure fair weighting across different modalities (because the d_lⁱ terms were squared in the regularization); w_is were normalized by the maximum.

We used an adaptive algorithm to set the values of λ and δ. The key idea is to regularly update λ and δ but only allow them to decrease or remain constant when the magnitude of the penalty is comparable to that of the negative log-likelihood cost. To this end, we started with small input filters, as an initial guess, such that the model did not generate any spikes. We then selected λ and δ such that the regularization term was 10% of total number of observed spikes. We set the initial ratio between λ and δ to a small value (0.05) and began the least-squares optimization. As the optimization proceeded, the filters became larger and the resulting models generated spikes. After 10 iterations of optimization, the regularization term became >10% of the total number of observed spikes (as the filters got large), so we adjusted λ and δ again such that the regularization term was once again 10% of the number of observed spikes. However, we updated λ and δ only if their new value was smaller than before. They were not allowed to increase because, otherwise, the smoothness penalty eventually dominated and filters began to decrease in magnitude. After several updates (typically 3 or 4), λ and δ stopped decreasing and remained approximately constant, indicating that both the regularization and log-likelihood term are balanced in the optimization process. This process of quadratic regularization is very similar to ridge regression and decreases the effective degrees of freedom of the linear filter, reducing the risk of overfitting.

Estimation of latency

The response latency of each afferent fiber is mostly determined by the recording site relative to the stimulation site and varies from 3 to 8 ms. We did not explicitly measure the latency to fit the model. Rather we let the response latency emerge from fitting process. The only restriction was that the filter value within the first 1 ms was set to be zero to impose a causal relationship between stimulus and response. The latency can be estimated as the position of the first major peak of the filter.

Comparing observed and predicted spikes

We fit the model using diharmonic stimuli and compared the various models statistically using the responses to triharmonic and noise stimuli. We used a measure of distance between two spike trains that takes into account the potential discrepancy both in the number of spikes and in the relative timing of spikes (Victor and Purpura 1996). D_spike increases as two spike trains become increasingly dissimilar; the more dissimilar the two trains are, the greater the cost to transform one into the other: the cost associated with adding or subtracting one spike is 1, and the cost for shifting a spike by 1 ms is a free parameter, c. To compare these values across stimuli, we calculated D_spike (using the method described in Victor and Purpura 1996) and divided it by the number of observed spikes. All values of D_spike reported here represent the distance in milliseconds per spike. The temporal jitter reported in Fig. 5 is the mean distance between matching spikes divided by the cost for shifting a spike.

Fig. 5.

Discrepancy between model predictions and data. Coarse structure (top) and fine temporal structure (bottom) across the population of neurons: SA1 (A), RA (B), and PC (C). The scatter plot in each panel represents the relationship between the firing rate predicted by the model and the firing rate observed in the data. The bars indicate the jitter in spike times between spike trains elicited by repeated presentations of the stimulus (obs vs. obs) and the jitter in spike times between model predictions and observed spikes (pred vs. obs). Data from the diharmonic training set is shown in blue, from the triharmonic test set in red, and from the noise test set in green.

Neurons may vary in the precision of their response to repeated stimuli, so we reasoned that the cost parameter c used in the calculation of D_spike should vary across neurons. Accordingly, for the diharmonic stimuli used to fit the model, we searched for the value of c that minimized the distance between spike trains elicited by repeated presentations of the same stimulus (within-stimulus D_spike) and maximized the distance between spike trains corresponding to distinct stimuli (between-stimulus D_spike). The within-stimulus D_spike for a given stimulus was calculated as the sum of D_spike of every possible pair of five repeats, and the between-stimulus D_spike of a pair of stimuli was calculated as the sum of D_spike of every possible pair of repeats between the five spike trains elicited by one stimulus and the fives spike trains of the other. The optimal value of c ranged from 0.2 to 0.6 across neurons and was typically around 0.4.

Simulation of afferent responses to scanned letters

To simulate afferent responses to scanned letters, we first computed, using a model of skin mechanics (Sripati et al. 2006), the instantaneous strain produced at the location of the hypothetical receptor at a series of time intervals (in increments of 1 ms) spanning the stimulus. For SA1 fibers, we used maximum tensile strain; for RA fibers, we used change in receptor surface area (cf. Sripati et al. 2006). We then used maximum tensile strain and its first derivative (with respect to time) as an input to the SA1 model and the derivative of change in receptor surface area (with respect to time) for the RA model. Given that local strain is linearly related to the depth of the indentation at the skin's surface (for a punctate probe), we could straightforwardly scale the strain and use it as input to the model. PC responses were not simulated as these fibers are not sensitive to the spatial properties of the stimulus. These simulated results were compared with neural data collected under corresponding conditions (Phillips et al. 1988).

RESULTS

We recorded the responses evoked in five SA1, five RA, and five PC fibers from four macaque monkeys by dynamic vibratory stimuli delivered through a flat probe glued to the glabrous skin of the hand. The stimuli consisted of sinusoids, diharmonic, triharmonic, and band-pass noise stimuli whose frequencies spanned the range from 0 to 125 Hz (limited by the bandwidth of the stimulator) and amplitudes ranged from 15 to 200 μm.

We drew extensively from previous studies of mechanoreceptive fiber responses to develop a model that predicts individual spikes elicited by these vibratory stimuli. First, the differential response properties of the three fiber types suggest the possible stimulus quantities that might drive transduction (Burgess et al. 1983; Knibestöl and Vallbo 1970; Looft 1996; Vega-Bermudez and Johnson 1999). For instance, SA1 fibers produce a tonic response to a sustained indentation that increases with the depth of indentation, suggesting that they may be sensitive to the position of the skin relative to its resting position (along the axis perpendicular to its surface, see Vega-Bermudez and Johnson 1999). However, they may also be sensitive to the rate at which the skin is indented as evidenced by an increased response when the skin is indented rapidly (Burgess et al. 1983; Knibestöl and Vallbo 1970). The transient responses of RA and PC fibers to indentations (and their lack of a sustained response) suggest that they are driven by changes in indentation, i.e., to the indentation velocity, acceleration, and/or jerk (the 1st, 2nd, and 3rd derivatives of indentation depth, respectively) (Lowenstein and Skalak 1966; Pubols 1980). Second, the frequency sensitivity of the three fibers suggests that the inputs to the model must be temporally filtered (e.g., see Makous et al. 1995). Third, the sensitivity of some fibers to indentations but not retractions (and that of other fibers to both) suggests that these inputs undergo half- or full-wave rectification (Bolanowski and Zwiskocki 1984; Whitsel et al. 2000) (see Supplemental Fig. S2). Finally, spike production is driven by nonlinear biophysical mechanisms and is usually followed by a refractory period. The results of previous studies suggest that these nonlinear mechanisms may be adequately described as an integrate-and-fire mechanism (Bensmaia 2002; Freeman and Johnson 1982; Slavik and Bell 1995).

The nature of the neuronal response to even a simple stimulus is strongly influenced by the stimulus quantity that drives transduction. To illustrate this principle using a simple example, we simulated the responses of two neurons to a ramp stimulus (Fig. 1B). The simulated SA1 fiber, being sensitive to position and velocity, responds to both the rising and sustained phases of the stimulus. In contrast, the simulated RA fiber, being sensitive to full-wave rectified velocity, responds to both the transient phases of the stimulus but not to its sustained phase. We reasoned that these differences would be even more pronounced in the responses to more complex vibratory stimuli, and as a result, models based on different physical quantities would yield divergent predictions.

Fig. 1.

Model description. A: Summary of the model used to fit the neuronal data. The stimulus, specified as a time-varying indentation, is converted into position (p), velocity (v), acceleration (a) and jerk (j) signals. Each signal is separated into its positive (—) and negative (- - -) components and filtered using a linear filter. This allows for rectification and frequency sensitivity to be accounted for using linear transformations. The temporal filtering is likely the product of the viscous properties of the skin and of the transducer mechanisms. Finally, the transformed inputs are summed and form the current input to a leaky, noisy integrate-and-fire (LNIF) model. In the LNIF model, a spike is produced whenever the input currents cause the membrane potential to depolarize beyond a threshold. After the production of a spike, a postspike inhibitory current is injected to mimic refractoriness. We tested models with the following input combinations: p, v, a, j, pv, va, aj, pva, vaj, pvaj. B: responses of a simulated slowly adapting type 1 (SA1) fiber, sensitive to both stimulus position and velocity, and of an rapidly adapting (RA) fiber, sensitive to velocity alone. Note that the firing rate of the SA1 afferent is higher during the on ramp than it is during the holding phase of the stimulus. Were this fiber sensitive to position alone, its firing rate would be lower during the on ramp. The RA fiber fires only during the on and off ramps of the stimulus. Were this fiber sensitive to positive velocities alone (half-wave rectification), it would fire during the on ramp but not during the off ramp.

Because the exact physical quantity that drives transduction in a given fiber is unknown, we assumed that the observed spikes may be governed by the time-varying indentation depth (position), its velocity, acceleration or jerk—or even a combination of these four quantities—a total of 10 models for each neuron (we only tested “contiguous” combinations of temporal derivatives—see methods). The full model of transduction is illustrated in Fig. 1A. Each input (position, velocity, acceleration, or jerk) was separated into its positive and negative components, which are filtered and summed across inputs to drive an integrate-and-fire mechanism (see methods). We estimated parameters for models with each of the 10 combinations of position, velocity, acceleration, and jerk as input (listed in Fig. 6). We then selected the model that yielded the best performance, or, if several models were equivalent, we selected the model with the fewest inputs (see following text).

We estimated model parameters using neuronal responses to diharmonic stimuli because these constituted our largest stimulus set. The observed responses of a single SA1 fiber to multiple presentations of a single diharmonic stimulus and model predictions obtained using the best model are shown in Fig. 2A. The model was able to capture the highly repeatable and precise temporal pattern of spikes evoked by the stimuli. If the model truly captured the underlying transduction process, it would generalize readily to novel stimuli. We therefore used the parameters estimated from responses to diharmonic stimuli (Fig. 2A) to predict responses to novel stimuli. We first examined model predictions of responses to a related class of triharmonic stimuli consisting of three superimposed sinusoids at different frequencies (Fig. 2B) and found that the model predicted the timing of individual spikes elicited by these novel stimuli with a high degree of accuracy. As a further challenge to the model, we tested its ability to predict responses to band-pass noise stimuli, which consisted of arbitrary combinations of frequencies, to many of which the model was never previously exposed. Even in this case, the spikes predicted by the model precisely matched the observed counterparts (Fig. 2C). We repeated this procedure for RA (Fig. 3) and PC fibers (Fig. 4) and obtained similar results.

Fig. 2.

Model predictions for an SA1 afferent. Measured and predicted afferent responses to a subset of training stimuli (A, diharmonic stimuli), triharmonic test stimuli (B), and a noise test stimulus (C). The black traces show the stimulus position as a function of time, red rasters show the responses of the afferent to 5 repeated presentations of the stimulus, and the blue rasters show the responses predicted by the model.

Fig. 3.

Model predictions for an RA afferent. Conventions as in Fig. 2.

Fig. 4.

Model predictions for a Pacinian (PC) afferent. Conventions as in Fig. 2.

The spikes predicted by the model could differ from the observed pattern of spikes in two ways: first, model predictions may differ in coarse structure from the measured responses—i.e., the model may “miss” some spikes (see Fig. 2A and 4B, e.g.) or predict extra spikes (see Fig. 4C). Second, the model predictions may diverge from the data in their fine structure—i.e., the times of spikes predicted by the model may not precisely match the times of the spikes in the data. We examined the agreement between the model and the data at the level of both coarse and fine structure. To determine the agreement in coarse structure between model predictions and data, we plotted the predicted firing rate against the observed firing rate (Fig. 5, scatter plots) for each set of stimuli. These plots reveal a strong and significant correlation (all correlations were significant at P < 0.01). We conclude that the model captures the coarse structure of the response in all three fiber types.

To determine the agreement between the model and the data at the level of fine structure, we calculated the temporal jitter between matching spikes in the model and the data (see methods). We compared this temporal jitter to the baseline jitter observed between spike trains evoked by repeated presentations of the same stimulus (Fig. 5, bar plots). The plots reveal that the temporal jitter between the model and the observed spikes was slightly higher but comparable to the temporal jitter across repeats. Although the difference was statistically significant (P < 0.01), the temporal jitter between observed and predicted spikes never exceeded 1.2 ms per spike with respect to the jitter across repeats. Thus the temporal jitter was within the limits of the temporal resolution of the simulation (1 ms). We conclude that the model captures the fine temporal structure of the response in all three fiber types, in most cases with submillisecond accuracy. That the largest discrepancies were observed for PC afferents is to be expected. The model's relative inaccuracy in predicting PC responses can be, in part, attributed to inaccuracies in the characterization of the input stimulus. Indeed PC afferents are exquisitely sensitive to high-frequency vibrations on the order of hundreds of nanometers and our stimulator transducer was only accurate on the order of about half a micrometer.

We tested the performance of the model using 10 different combinations of position, velocity, acceleration, and jerk with the aim of identifying the combination that best predicted the observed data. To compare model performance, we calculated a measure of dissimilarity that takes into account differences in the coarse and fine temporal structure across two spike trains. This measure, spike distance (D_spike) (Victor and Purpura 1996), represents the cost required to transform one spike train into another by adding or moving spikes and increases with dissimilarity between the two spike trains (see methods). For a model driven by a particular combination of physical inputs, we calculated the distance between the observed and predicted spike trains across all neurons and stimuli. We compared the performance of models driven by two distinct sets of physical inputs (e.g., position and velocity vs. velocity and acceleration) by performing a pair-wise Wilcoxon's signed-rank test on these distances.

We first confirmed the validity of this approach using simulated data. Specifically, we verified that a simulated neuron driven by a known set of physical variables could indeed be identified as being driven by this set of variables as distinct from other possible combinations (see Supplemental Fig. S1). Next we examined the degree to which models based on different combinations of inputs were able to predict the observed spike trains. The responses of the three fiber types were best predicted by different combinations of inputs (Fig. 6). Specifically, SA1 responses were predicted as well using position and velocity as input as they were using any other combination of inputs (Fig. 6A); RA responses were predicted as well using only velocity as they were using any other combination of inputs (B); the best model to predict PC responses included position, velocity and acceleration (C; also see Supplemental Fig. S3). We conclude, then that SA1 responses are determined by the indentation depth and velocity, RA fibers are sensitive to indentation velocity alone, and PC fibers are sensitive to indentation position, velocity, and acceleration.

Because the best model for SA1 and PC fibers include two or more physical quantities, we investigated the degree to which each physical quantity contributes to the stimulus-induced current as a function of stimulus frequency (Fig. 7). At low frequencies, position-induced currents dominated the response for both SA1 and PC fibers. For SA1 fibers, the velocity signals became increasingly dominant as stimulus frequency increased. For PC fibers, velocity signals peaked at intermediate frequencies, and stimulus-induced currents were determined primarily by acceleration at the highest frequencies. Thus stimulus position, velocity, and acceleration contribute differently to the neuronal response depending on vibratory frequency.

If the models truly captured the underlying transduction process, they would also qualitatively capture the well-documented response properties of mechanoreceptive fibers. For instance, mechanoreceptive afferent responses to sinusoidal stimuli exhibit stereotyped relationships with vibratory amplitude (Freeman and Johnson 1982; Johnson 1974; Muniak et al. 2007). At weak amplitudes, no response is elicited. As the vibratory intensity is increased, spikes are evoked on some cycles. Over a range of amplitudes, the average number of impulses per cycle grows linearly with increasing amplitude until one spike is evoked on each vibratory cycle. As the intensity is further increased, the impulse phase occurs earlier in the stimulus cycle, but the firing remains constant at one spike per cycle. Additional increases in intensity yield a second impulse on some cycles, and the average firing rate again increases linearly with increased amplitude. Beyond the intensity at which two impulses are elicited on every cycle, there is phase advancement and stabilization as there is with the first impulse. At still higher intensities, a third impulse occurs. Figure 8 shows typical predicted and measured phase- and rate-intensity functions for one fiber of each type. As can be seen from the figure, predicted responses exhibit phase advance and entrainment plateaus as do observed responses.

Fig. 8.

Rate-intensity (top) and phase-intensity (bottom) functions for a typical SA1 (A), RA (B), and PC (C) afferent, stimulated with sinusoids varying over a range of intensities at 10, 30, and 40 Hz, respectively. The predicted rate and impulse phase (—) matched their observed counterparts (●) for all 3 afferent types. At low intensities, the phase of the spikes was bimodal (i.e., spikes alternated between 2 phases within stimulus cycles), and the mean of each mode is shown.

The three mechanoreceptive fiber types also exhibit varying sensitivity to sinusoidal stimulation as a function of stimulation frequency. SA1 fibers are most sensitive at low frequencies, RA fibers at intermediate frequencies, and PC fibers at high frequencies. Afferent sensitivity can be measured using two quantities: the absolute threshold, the minimum amplitude to evoke a single spike in the fiber, and the entrainment threshold, the minimum amplitude required to evoke one spike per stimulus cycle. Figure 9 shows frequency sensitivity functions derived from the model together with data from previous studies: SA1 fibers peak in sensitivity at around 10 Hz, RA at around 40 Hz, and PC sensitivity increases over the range of frequencies tested. Furthermore, the absolute and entrainment thresholds diverge with increasing frequency for all three fiber types. These threshold-frequency functions are strikingly similar to those measured in previous studies (Freeman and Johnson 1982; Muniak et al. 2007; Talbot et al. 1968). Furthermore the model captures afferent responses to step indentations: SA1 afferents fire throughout the indentation whereas RA and PC fibers only fire at the onset and offset of the indentation (see Fig. 1 for SA1 and RA example).

Fig. 9.

Threshold frequency functions: SA1 (A), RA (B), and PC (C) afferents threshold frequency functions predicted by the model (dashed). Absolute threshold (red) represents the minimum indentation amplitude required to generate a single spike, and the entrainment threshold (blue) represents the minimum amplitude necessary to generate a spike on every cycle. Observed thresholds (solid) were extracted from previous neurophysiological studies (Freeman and Johnson 1982; Muniak et al. 2007) and were shifted along the vertical axis to compensate for differences in adaptation levels (Bensmaia et al. 2005) and for differences in sensitivity across afferents of a given type (Freeman and Johnson 1982).

In a previous study, we showed that afferent responses to statically indented spatial patterns are linearly related to the strain experienced by the associated mechanoreceptors (Sripati et al. 2006). Here we have shown that the responses to vibratory stimuli can be predicted based on indentation depth and its derivatives. The two models can thus be straightforwardly combined to predict the responses to arbitrary spatiotemporal stimuli impinging on the skin. First, the strain at the receptor site can be obtained using the previously developed skin mechanics model at each time point of the stimulus. This time-varying strain signal and/or its time derivatives can be used as inputs to the model described here, assuming that tangential viscoelastic forces exerted on the skin do not fundamentally affect the neuronal response. The skin mechanics model accounts for the static properties of the skin and mechanoreceptors while the present model accommodates their elastic properties. As an illustration of how the combined model would work, we used this approach to simulate the responses of an SA1 and an RA afferent to embossed letters scanned across their receptive fields on the skin (Fig. 10), replicating results from studies by Phillips, Johnson, and Hsiao (Johnson and Phillips 1988; Phillips et al. 1988). In addition, this simulation was designed to show how simulated afferents would respond to slowly changing stimuli, which is directly represented by the skin-strain pattern (compare letter “E” to see the difference, between SA1 and RA, of their responses to prolonged indentation). The responses of the model differ in their fine temporal details but are qualitatively similar to the responses reported in these studies. Indeed the responses of the simulated SA1 afferent capture the spatial configuration of the letters. Similarly, the responses of the simulated RA afferent capture certain features; for instance the leading edges are enhanced, whereas the middle portions of the letter and falling edges are obscured. Only the coarse features of the stimulus are captured in the spatial structure of the RA response as has been found previously. The crudeness of the RA representation is due to the increased spatial filtering effected by these afferents (as modeled using skin mechanics) and to the fact that they respond to changes in strain, which further obscures the central portion of the letters where strain changes little.

Fig. 10.

A simulation of neurophysiological experiments conducted by Phillips, Johnson, and Hsiao showing the representation of embossed letters in SA1 (A) and RA (B) afferent responses. Gray-scale images: strain profile; Top row of spike rasters: model prediction; bottom row of spike rasters: Data reproduced from Phillips, Johnson and Hsiao (1988). Each letter (height = 7 mm) is horizontally scanned across the receptive field of the simulated afferent at 50 successive positions, separated by 200 μm. Tissue strain, computed using a continuum mechanics model, along with its 1st time-derivative, is used as input to the model. The strain profiles were computed using the mean parameters for SA1 and RA fibers obtained in a previous study (Sripati et al. 2006) in which we concluded that SA1 and RA fibers respond to different strain components. The structure of the spatial event plot (as they are called) is similar to that observed in the study. The similarity is striking given that the simulations were derived from data obtained from different afferents than those the actual data of which are shown here. Scale bar = 200 ms.

Note that the differences at the level of fine temporal structure between the model predictions and previously recorded data may arise from intrinsic variability across neurons (the model and data come from different neurons) or from an aspect of tactile stimulation absent in the model (i.e., it does not account for forces parallel to the skin). Testing these possibilities will require neuronal recordings using static, dynamic, and spatiotemporal stimuli on the skin.

DISCUSSION

Here we describe a model that predicts the precise patterns of spikes elicited in the three types of mechanoreceptive fibers (SA1, RA, PC) that convey tactile information. Building on previous attempts to model transduction in primate glabrous skin using an IF framework (Bensmaia 2002; Freeman and Johnson 1982; Slavik and Bell 1995), the model captures the response properties of mechanoreceptive afferents across a wide range of vibratory frequencies and predicts the timing of individual spikes with millisecond accuracy. Furthermore by comparing model performance using combinations of position, velocity, acceleration, and jerk, we were able to identify the stimulus quantities that drive transduction in each fiber type. SA1 afferents were sensitive to position and velocity, RA afferents to velocity alone, and PC afferents to position, velocity, and acceleration.

The major limitation of the model in its present form is its inability to account for nonlinearities in the response. The model was able to accurately predict SA1 responses across stimuli of all amplitudes because the responses of these afferents are strikingly linear (Johnson 2001). However, the model predictions deviated from the observed responses of RA and PC afferents at large amplitudes (Supplemental Fig. S5). This dependence on stimulus amplitude suggests a response nonlinearity for PC and RA afferents. This is consonant with the finding that RA fibers receive input from multiple Meissner corpuscles and PC afferents begin to saturate in their response for indentations as small as 125 μm (Vega-Bermudez and Johnson 1999). The model in its present form consists of linear filters that preclude saturation. This limitation of the model can be addressed by using existing approaches to estimate input nonlinearities (Ahrens et al. 2008). A second limitation of the model is its inability to account for response adaptation. Afferents have been shown to become desensitized when they are subjected to supraliminal stimulation over an extended period of time (Bensmaia et al. 2005; Leung et al. 2005). This form of slow adaptation seems to stem from a progressive increase in the spike initiation threshold that is independent of spiking rate. While the model in its current form does not adapt, adaptation could be incorporated by adding a term that modifies the threshold of the receptors as currents flow into the cell. A recently developed IF model (Mihalas and Niebur 2008) includes a variable threshold that can be sensitive to input current and could thus be applied to mechanotransduction once a method has been devised to efficiently derive its parameters. Finally, afferent responses in this study were obtained using stimuli with frequency content restricted to 1–125 Hz, a range of frequencies commonly encountered during natural tactile exploration. Although SA1 and RA afferents become increasingly insensitive beyond this range, PC afferents in fact increase in sensitivity with increases in frequency up to 250–300 Hz (Freeman and Johnson 1982; Muniak et al. 2007; Talbot et al. 1968). The present model could straightforwardly be extended to accommodate higher-frequency stimuli by estimating parameters using data obtained with a stimulator with a wider bandwidth.

Although our approach is similar to other efforts to predict individual spikes in the mystacial follicles (Mitchinson et al. 2008) and thalamus of rodents (Petersen et al. 2008), in visual neurons (Keat et al. 2001; Pillow et al. 2005), and in motor neurons (Truccolo et al. 2005), we have used the model to overcome a particular challenge of the primate somatosensory system, namely that the details of the transduction process remain unknown. The striking degree of success of the model suggests that the processes underlying mechanotransduction, in essence, involve filtering, rectification, and a process well approximated by an IF mechanism. The model also allows us to identify the physical quantities that drive afferent responses (position and velocity for SA1, velocity for RA, and position, velocity, and acceleration for PC). Precisely how these mechanisms map onto the structure of mechanoreceptors is unclear. Sensitivity to position may easily arise from the strain experienced by the receptor, which is linearly related to the force (and therefore to indentation depth) (Sripati et al. 2006). Sensitivity to velocity or acceleration may arise from the dynamics of receptor deformation or through biophysical mechanisms that make the neuron sensitive to changes in strain rather than to strain itself (Loewenstein and Mendelson 1965). Rectification may arise from differential sensitivity to probe indentations compared with probe retractions (see following text). To clarify these issues will require further structural and functional studies of mechanoreceptors.

Although the model itself is agnostic about the actual transduction process, it provides several additional insights into the mechanisms underlying transduction. First, SA1 and RA filters are low-pass filters and PC filters are band-pass filters (Fig. 11) as might be expected given the threshold-frequency characteristic of the afferents. The low-pass filtering is likely the product of the viscous properties of the skin, which reduce the impact of high-frequency vibrations, and of the transducer mechanisms themselves.

Fig. 11.

Mean filters in the frequency domain for SA1 (A), RA (B), and PC (C) neurons. Filters corresponding to position (blue) and velocity (green) and acceleration (red). The width of the shaded bands denote the SE at each frequency. Filters exhibit low- or band-pass characteristics with different frequency cut-offs. Note that although they respond to static indentations, SA1 afferents have high thresholds for these stimuli (Johnson 2001), a fact reflected in the low power of the filter at 0 Hz. For typical filter waveforms in the time domain, see Supplemental Fig. S5.

Second, the linear filters corresponding to indentation and retraction were highly correlated (Fig. 12A and Supplemental Fig. S6), suggesting that the stimulus is filtered prior to rectification rather than vice versa. Third, we found that the majority of afferents of all types were less sensitive to stimulus retraction than to indentation (Fig. 12B). This partial rectification may arise due to the tendency of the skin to experience less strain when the probe is retracted compared with when the probe is indented (Pubols 1980). Fourth, the membrane time constants (Fig. 12C) and the time courses of the postspike inhibitory currents (Fig. 12D) were similar for SA1 and RA afferents but were considerably shorter for PC fibers. This distinction may help explain why PC afferents are sensitive to high frequencies compared with their SA1 and RA counterparts. Finally, the success of the linear IF model in predicting the timing of spikes validates its two major assumptions: that the complex biophysical mechanisms mediating spike production can be approximated by a fixed threshold and that the mechanisms of voltage integration below the spiking threshold can be reasonably approximated by a linear integrator. Violation of the second assumption would lead to poor model predictions of low-amplitude stimuli (for which subthreshold dynamics might exert greater influence), but we observed no such trend.

The proposed model constitutes a first step in understanding transduction in primate glabrous skin. It accounts for most of the known temporal properties of mechanoreceptive afferents. The framework also allows for systematic exploration of further extensions to the model, both to accommodate response properties such as adaptation and saturation as well as to include integration of spatial patterns into the skin. Ultimately, this model can be used as a research tool to characterize the peripheral representation of arbitrary spatiotemporal stimuli and, in some cases, reduce the need to perform neurophysiological experiments on the peripheral nerve. Finally, the model may have an important practical application. With the development of highly sophisticated upper-limb neuroprostheses equipped with force sensors, realistic sensory feedback is not only feasible but also of the utmost importance. Using the model described here, force signals from the neuroprosthesis can be converted into the spike trains that would have been evoked in the intact limb. These spike trains can then be effected into the residual nerve through electrical stimulation delivered using chronically implanted multi-electrode arrays (Kim et al. 2009).

GRANTS

This work funded by National Institutes of Health Grants NS-18787 and EY-018620 and Defense Advanced Research Projects Agency Grant N66001-06-C-8005 and by the Samsung Scholarship Foundation.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).