Modulation of potassium conductances optimizes fidelity of auditory information

Significance

The numbers and types of potassium channels in a neuron determine its firing patterns and sensitivity to synaptic inputs. Using the auditory system as a model sensory system, the present work shows that a rapid change in levels and characteristics of potassium current in response to an incoming stimulus allows a neuron to transmit the maximal amount of information as the intensity of the stimulus is increased. This is independent of the firing rate of the neuron but depends on the number of neurons in the ensemble that encode the stimulus. The findings provide an explanation for previously documented changes in potassium currents in response to altered sensory inputs.

Abstract

Potassium channels in auditory neurons are rapidly modified by changes in the auditory environment. In response to elevated auditory stimulation, short-term mechanisms such as protein phosphorylation and longer-term mechanisms such as accelerated channel synthesis increase the amplitude of currents that promote high-frequency firing. It has been suggested that this allows neurons to fire at high rates in response to high sound levels. We have carried out simple simulations of the response to postsynaptic neurons to patterns of neurotransmitter release triggered by auditory stimuli. These demonstrate that the amplitudes of potassium currents required for optimal encoding of a low-amplitude auditory signal differ from those for louder sounds. Specifically, the cross-correlation of the output of a neuron with an auditory stimulus is improved by increasing potassium currents as sound amplitude increases. Temporal fidelity for low-frequency stimuli is improved by increasing potassium currents that activate at negative potentials, while that for high-frequency stimuli requires increases in currents that activate at positive membrane potentials. These effects are independent of the firing rate. Moreover, levels of potassium currents that maximize the fidelity of the output of an ensemble of neurons differ from those that maximize fidelity for a single neuron. This suggests that the modulatory mechanisms must coordinate channel activity in groups of neurons or an entire nucleus. The simulations provide an explanation for the modulation of the intrinsic excitability of auditory brainstem neurons by changes in environmental sound levels, and the results may extend to information processing in other neural systems.

Neurons in the spiral ganglion and in auditory brainstem nuclei are required to encode all aspects of a sound stimulus, allowing the nervous system to detect very small differences in sound stimuli, such as the discrimination of a regional accent or microsecond differences in time of arrival at the two ears (1). All of this information must be encoded in the timing of action potentials (APs) in neurons at the earliest stages of processing. The exact pattern of action potential firing is determined not only by the timing and amplitude of synaptic inputs coming from inner cochlear hair cells or neurons in other auditory pathways but also by the intrinsic excitability of the postsynaptic neurons (2–4).

The specific way that a neuron responds to a synaptic input is determined by the subset of ion channels it expresses. Ion channels selective for sodium, calcium, and chloride ions each contribute to the intrinsic excitability of a neuron (5). The greatest diversity of responses is, however, provided by the types and levels of potassium currents. There are over 70 genes that encode different pore-forming (α-subunit) members of the potassium channel superfamily (6). Most of these can give rise to several different channel subunits by alternate splicing of their messenger RNA (mRNA). A functional channel is typically composed of a tetramer of α-subunits that may be the products of the same or different genes. Finally, the characteristics of potassium channels can be shaped by interaction with auxiliary subunits that modify aspects such as gating and voltage dependence as well as cellular localization.

One essential aspect of a voltage-dependent potassium channel that determines how its current influences firing patterns is the voltage range at which it activates. Potassium currents that are closed near the resting potential and that open only at potentials that are reached during an action potential have been termed “high-voltage-activated” I_KH currents (also termed HVA currents). An increase in I_KH currents increases neuronal excitability by speeding repolarization of action potentials, which reduces the inactivation of sodium channels (7). The prototypical channels that underlie I_KH currents belong to the Kv3 family. These are highly expressed in neurons capable of firing at high rates, including the majority of neurons in auditory brainstem nuclei and the spiral ganglion (8–12). Potassium currents that activate at much more negative potentials, at or near the resting potential, have been termed “low-voltage-activated” I_KL currents (also termed LVA currents). A major class of channels that underlie I_KL currents in neurons is the Kv1 voltage-dependent family, although other types of potassium channels also contribute (7). In auditory neurons, including spiral ganglion neurons (13), these I_KL currents are essential for maintaining temporal accuracy in response to precisely timed synaptic inputs (14–22).

A variety of experiments have revealed that auditory stimulation alters the intrinsic excitability of auditory neurons by rapid short-term mechanisms such as phosphorylation of potassium channels and by longer-term mechanisms that change the rate of synthesis of new channels. For example, physiological increases in the amplitude of environmental sounds, lasting 5 min or less, produce increases in K3.1b channel activity in vivo by direct dephosphorylation of the channel protein (23, 24). Auditory stimulation at physiological levels lasting tens of minutes increases the synthesis of such channels and alters their tonotopic distribution across auditory nuclei (25–27). We have now carried out numerical simulations of the response of neurons to synaptic inputs evoked by sounds of different frequencies and amplitudes. We show that optimal encoding of information is not correlated with the number of evoked action potentials but depends on levels of potassium current. As the amplitude of a sound stimulus is raised, an increase in either I_KH or I_KL, or in both types of currents, is required to extract the maximal temporal information from the sound-generated synaptic train. The precise level of current that is required also depends on the number of neurons in the ensemble that relays the incoming signal.

Results

Description of Model.

A simple computational model was used to covert an auditory stimulus into trains of action potentials in groups of neurons (Fig. 1). The synaptic outputs from such neurons were then combined, and the maximal value of the cross-correlation between this combined synaptic output and the original signal (Xcorr) was used to quantify the degree to which the firing patterns are able to represent the stimulus. For simplicity, Fig. 1 depicts neurotransmitter release from a cochlear inner hair cell, and subsequent postsynaptic responses are depicted in type 1 spiral ganglion neurons because these are the very first steps in auditory processing, and the results are therefore relevant to these neurons. Because similar channels are expressed in many types of auditory brainstem neurons, as well as other neurons in other systems, the general principles are likely to apply to preservation of information by synaptic transmission through subsequent auditory relay stations. Because the major goal of the model was to generate different patterns of spike outputs based on levels of I_KH and I_KL currents, no attempt was made to incorporate all known conductances in auditory neurons or to model spatial aspects, e.g., the localization of different channel types to subcellular compartments such as somata, dendrites, or the axon hillock.

Fig. 1.

Outline of computations used in this study. Steps 1 and 2: A sigmoid function was used to convert a stimulus corresponding to a sound into a time series of probability of neurotransmitter release from a model sensory cell. Step 3: A random number generator was then used to generate a time series of synaptic currents. Step 4: The synaptic currents were applied to a single-compartment postsynaptic neuron, and the voltage response of the neurons was calculated. Step 5A: The action potentials in the neuron then evoked postsynaptic currents in a hypothetical follower cell. Step 5B: The postsynaptic currents generated as in step 5A were calculated for 30 neurons and combined into a single cumulative output. A cross-correlation was then carried out between the original stimulus and cumulative output (Step 6). The maximal value of this was labeled Xcorr. See text and Materials and Methods for more details.

The individual steps in the computation are shown in steps 1-6 in Fig. 1. A sigmoid function was first used to convert each value of the sound stimulus y(t) into a probability of neurotransmitter release Pr(t) (Fig. 1, steps 1 and 2). Because neurons are not capable of locking their action potentials to each wave of high-frequency stimuli (>2 kHz), simple pulses of high-frequency stimuli were represented as square wave pulses. The high-frequency component of more complex stimuli was rectified to a DC analog signal that follows the envelope of the stimulus, matching the response of cochlear hair cells to high-frequency stimuli (28) (Materials and Methods). A random number generator was then used to generate a time series of synaptic currents (Fig. 1, step 3) that was applied to a single-compartment postsynaptic neuron. The voltage response of the postsynaptic neurons V(t) to this synaptic input (Fig. 1, step 4) was then calculated using a neuronal model that incorporated a voltage-dependent Na⁺ current I_Na, a voltage-independent leak current I_leak, and two different K⁺ currents, I_KH and I_KL. The latter represent components of voltage-dependent K⁺ currents that activate at positive and negative membrane potentials, respectively. The levels of conductance for I_KH and I_KL are given by the parameters g_KH and g_KL. The values for all of the parameters were based on parameters used in previous simulations (24, 29–33) with minor adjustments to match characteristics to those recorded in spiral ganglion neurons (9). Full description of the parameters is provided in the Materials and Methods section.

A pattern of firing was calculated for the same sound stimulus for each postsynaptic neuron in an ensemble of 1 to 30 identical neurons. The pattern of synaptic inputs g_syn(t) was calculated independently for each neuron, consistent with the neurons receiving inputs from different presynaptic active zones (34). To provide a measure of the combined output of the ensemble, a second set of synaptic conductances was calculated, in which the occurrence of an action potential in each neuron triggered an exponentially decaying function (comparable to a hypothetical synaptic conductance, Fig. 1, step 5A). These functions were then summed for the output of all of the neurons in the ensemble (Fig. 1, step 5B). Cross-correlation was then calculated between this cumulative neuronal output and the sound stimulus y(t) after subjecting the latter to half-wave rectification to remove negative values. The maximal positive value of the cross-correlation function, Xcorr, was then used as a measure of fidelity of coupling between the stimulus and the neuronal firing patterns (Fig. 1, step 6). For most of the simulations, we used ensembles of 1 to 30 neurons, and the simulations were carried out 3 to 10 times to provide SEs for values of Xcorr and the numbers of evoked action potentials.

Representation of Input Stimulus Is Maximal with a Fixed Level of Sodium Current.

We first examined the effect of increasing the voltage-dependent Na⁺ conductance (gNa) while maintaining a fixed K⁺ conductance. We simulated the model for 80 ms with a variety of stimuli. These included low-frequency sinusoidal stimuli (100 Hz), for which neurons are capable of locking an action potential to each wave, and pulse stimuli corresponding to much higher-frequency stimuli, for which phase locking is not possible (Fig. 2 A and B). We also tested different amplitudes of these stimuli (Fig. 2 C and D). In addition, we tested stimuli in which amplitude varied throughout the stimulus. Typical results are shown in Fig. 2 for low-frequency (400 Hz, Fig. 2E) and high-frequency (2 kHz, Fig. 2F) sine wave stimuli applied as a ramp from zero to full amplitude. These stimuli were applied to neurons with a fixed value of either a high-threshold K⁺ conductance (g_KH, left panels) or a low-threshold K⁺ conductance (g_KL, right panels).

Fig. 2.

Plots of the effect of increasing voltage-dependent Na⁺ current (gNa) on mean numbers of action potentials (APs) and on values of Xcorr evoked by stimuli to neurons containing either a high-voltage-activated K⁺ current (g_KH = 0.15 µS and g_KL = 0 µS, left graphs in each panel) or a low-voltage-activated K⁺ current (g_KH = 0 µS and g_KL = 0.02 µS, right graphs in each panel). (A) Responses to a 250-ms stimulus containing a 190-ms 100-Hz sinusoidal waveform (normalized amplitude = 1.0) beginning 30 ms after the onset of the stimulus. The ensemble comprised 30 neurons, and graphs show values ± SEM for three simulations. Top plots show number of APs evoked, and lower plots show corresponding values of Xcorr. (B) As for A but with a 190-ms square pulse waveform. (C and D) As for A and B, respectively, but responses are shown for lower-amplitude waveforms (normalized amplitude = 0.2). (E) Responses to a 100-ms stimulus with 400-Hz ramp lasting 80 ms beginning 10 ms after the stimulus onset and reaching a normalized amplitude of 1.0 at 90 ms. The ensemble comprised 1 neuron, and graphs show values ± SEM for 10 simulations (F) As for E but with a 2.0-kHz ramp. Red bars highlight the region 0.3 to 0.4 µS for gNa close to the maximal evoked values for Xcorr for the majority of conditions.

For these neurons with fixed levels of K⁺ conductance, gNa was progressively increased starting at a value for which no neuronal action potentials could be evoked. In all cases, independent of the stimulus or the type of K⁺ conductance, the number of action potentials that were evoked by the stimulus increased monotonically with gNa (Fig. 2 A–F, Top). The maximal correlation (Xcorr) between the sound stimulus and the output for these same conditions is shown in the lower panels. In all cases, Xcorr reached a maximum at an intermediate value for gNa and subsequently declined as action potential firing increased. In the majority of combinations of stimuli and K⁺ currents, peak values of Xcorr occurred when gNa was within the range of 0.3 to 0.4 µS (pink bar, Fig. 2) or close to that range. A value of 0.35 µS gNa was therefore used in all further simulations.

Fidelity of Information Transfer Depends on Potassium Current Amplitude.

With the fixed value of gNa, we next manipulated levels of potassium conductance to test their effects on Xcorr. We first tested several different conductances with varying midpoints of activation and slope factors. These included g_KH and g_KL and others termed g_KA1, g_KA2, and g_KA3 (Fig. 3A and SI Appendix, Fig. S1). The response of an ensemble of 30 neurons, each of which expressed only one of these conductances, was tested using a 250-ms sound containing either a simple low-frequency (100 Hz) or high-frequency (800 Hz) sine wave applied for 190 ms beginning 30 ms after the onset of the stimulus (Fig. 3B). These stimuli were applied either at full amplitude or at two reduced amplitudes (with maximal values of y(t) of 1, 0.2, and 0.05, respectively). Fig. 3B shows an example of how Xcorr and the number of evoked action potentials change with increases in g_KA3 for an 800-Hz stimulus at each amplitude. In each case, a maximal value of Xcorr was attained at a fixed level of g_KA3. The optimal level of K⁺ conductance, however, increased as the amplitude of the stimulus was raised (gray, blue, and pink bars in Fig. 3 B, Upper). Moreover, this optimal level of g_KA3 was not related to the evoked firing rate (Fig. 3 B, Lower). The finding that, as the amplitude of the stimulus is increased, an increase in K⁺ conductance is required to attain the highest levels of Xcorr was confirmed for each of the conductances tested, including g_KH and g_KL (SI Appendix, Fig. S1 and Fig. 3 C–G).

Fig. 3.

Fidelity of transmission of higher-amplitude stimuli is enhanced by increasing potassium conductance. (A) Steady-state activation curves for five different conductances with varying midpoints of activation (g_KL, g_KA1, g_KA2, g_KA3, and g_KH SI Appendix, Fig. S1) (B) Plots of the effect of increasing levels of g_KA3 in an ensemble of 30 neurons with no other K⁺ conductance. Top shows the stimulus (250 ms duration with a 190-ms 800-Hz sinusoidal waveform beginning 30 ms after the stimulus onset). Center shows values of Xcorr (± SEM for three simulations) in response to this stimulus applied at three different intensities (0.05, 0.2, and 1.0 of full amplitude). Gray, blue, and red bars highlight the maximal evoked values for Xcorr for the three intensities. Bottom shows the corresponding numbers of action potentials evoked in single neurons of the ensemble. (C–G) Two-dimensional plots show the effects on increasing g_KL (Y axis) and g_KH (X axis) on numbers of action potentials evoked by stimuli (grayscale arrays) and Xcorr (pseudocolor arrays). Results are shown for 250-ms stimuli containing a 190-ms 50-Hz (C) 100-Hz (D), 200-Hz (E), and 800-Hz (F) sinusoidal waveforms or a square pulse (G). The amplitude of the applied stimuli is the lowest for the top row plots (0.05 amplitude of the maximal stimulus amplitude), intermediate in the center rows (0.2 amplitude), and full amplitude in the bottom set of rows. Combinations of g_KH and g_KL that provide close to the maximal values of Xcorr for each type of stimulus are outlined in red.

Combinations of g_KH and g_KL Provide Optimal Fidelity at Different Sound Levels.

Plots such as those in Fig. 3B demonstrate that changes in the fidelity with which a signal is processed by an ensemble of neurons depends on the level of neuronal voltage-activated K⁺ conductance and that maximal fidelity does not always occur when the firing rate is the highest. Because g_KH and g_KL are generally coexpressed in auditory neurons but are regulated differentially and have very different effects on firing patterns, we carried out similar calculations for all combinations of g_KH and g_KL for 50, 100, 200, and 800 Hz and square wave stimuli at three different amplitudes (1, 0.2, and 0.05). To simplify presentation of the results, we converted Xcorr values for each calculation into pseudocolors, with yellow representing the highest values and blue representing low values. These were then plotted in two dimensions with g_KH and g_KL as the x and y axes, respectively (Fig. 3 C–G). The numbers of action potentials evoked in each condition were converted to a gray scale and plotted similarly.

The Fig. 3C shows the rate of firing evoked by 50-Hz stimulation, as well as the corresponding values of Xcorr for all combinations of g_KH and g_KL. For the low-amplitude stimulus, maximal firing is obtained with a low value of g_KH with no g_KL conductance. This combination of conductances does not, however, provide the maximal match of the stimulus to the combined output. Instead, high values of Xcorr require a slightly higher level of either g_KH or g_KL, resulting in a yellow “band” of optimal conductances. The highest value of Xcorr is indicated with a red square. As the amplitude of the stimulus is increased to the intermediate level, maximal firing is still evoked with low g_KH and no g_KL (Fig. 3 C, Center). The band of optimal Xcorr values, however, is moved to slightly higher levels of both types of conductance. Optimal Xcorr occurs for an increase in low-voltage-activated g_KL and no high-voltage-activated g_KH_.

When the 50-Hz stimulus is increased to full amplitude, maximal firing is still obtained with a low value of g_KH (Fig. 3 C, Bottom). As expected, Xcorr values are increased by the higher-amplitude stimulus. Optimal correlation of the combined output to the stimulus, however, requires much higher values of either g_KH or g_KL than those for the lower-amplitude stimuli. Increases in low-threshold g_KL provide the major improvement in temporal fidelity for this 50-Hz stimulus.

The Fig. 3 D and E show that maximizing the values of Xcorr for 100-Hz and 200-Hz stimuli requires increasing levels of potassium channel conductance with stimulus amplitude. The yellow bands of optimal Xcorr values shift downward and to the right as the stimulus amplitude is increased. As with the 50-Hz stimulus, an increase in g_KL alone is most effective in raising Xcorr values. These results are consistent with previous experimental and computational work, which has demonstrated that the ability of neurons to lock action potentials to lower rates of stimulation (up to 100 to 200 Hz) is largely insensitive to the presence or absence of g_KH (33, 35).

A slightly different response to increased stimulus amplitude is observed for the higher-frequency stimulus (800 Hz) and to the square pulse, which represents the physiological response to even higher frequencies (Fig. 3 F and G). Maximal firing was still obtained with a low value of g_KH with no g_KL conductance. As with the lower-frequency stimuli, this did not provide maximal Xcorr values, which occurred with a range of combinations of g_KH and g_KL that reduced firing below its maximal value. As the amplitude of the stimulus is increased, higher levels of potassium conductance are required to maximize Xcorr. In the case of these stimuli, however, selective increases in the high-voltage-activated g_KH conductance, rather than g_KL, provided the best correlation. In summary, increases in the amplitude of both the low- and high-frequency stimuli required increases in K⁺ conductance to maximize Xcorr values, but for low-frequency stimuli, increasing g_KL is more effective, while for the high-frequency stimuli, optimizing Xcorr is achieved by increasing high-voltage-activated g_KH conductance.

Increasing K⁺ Conductances Optimizes Fidelity of Transmission at Higher Amplitudes of Complex Sounds.

Real sounds contain rapidly changing frequencies. Moreover, cochlear hair cells and spiral ganglion neurons, as well as neurons in auditory brainstem nuclei, respond to a relatively wide range of frequencies centered around their characteristic frequency. We therefore repeated the simulations of how changes in g_KH and g_KL influence the response of model neurons using two recorded complex sounds at high and low amplitudes. We first stimulated the 30-neuron network with the word “Art”, which lasts 275 ms. A sonogram indicates that the majority of this stimulus contains primarily low frequencies (<800 Hz, Fig. 4A). The second stimulus was the word “Hiss”, which lasts 847 ms and almost exclusively contains high frequencies (>2,000 Hz, Fig. 4B).

Fig. 4.

Effects of changing levels of potassium conductance on the fidelity of transmission of stimuli corresponding to the sounds Art and Hiss in ensembles of 30 neurons. (A and B) Waveforms and sonograms for Art and Hiss, respectively. (C and D) Two-dimensional plots show the effects on increasing g_KL (Y axis) and g_KH (X axis) on numbers of action potentials evoked by stimuli (grayscale arrays) and Xcorr (pseudocolor arrays) for Art and Hiss, respectively. The amplitude of the applied stimuli is the lowest for the top row plots (0.05 amplitude of the maximal stimulus amplitude), intermediate in the center rows (0.2 amplitude), and full amplitude in the bottom set of rows. Combinations of g_KH and g_KL that provide close to the maximal values of Xcorr for each type of stimulus are outlined in red. (E) Bar graphs showing the value of R_HL, which reflects the balance between g_KH and g_KL conductances for the maximal Xcorr values in each of the plots in Figs. 3 C–G and 4 C and G. Negative values indicating higher contributions of g_KL are shown in blue, and positive values indicating higher g_KH are shown in red.

For both Art and Hiss, the levels of both g_KH and g_KL potassium conductances required to optimize Xcorr increased with stimulus amplitude. Consistent with the effects of stimulation with simple sinusoidal and square wave stimuli, the highest levels of Xcorr for the high amplitude of the lower-frequency Art stimulus occurred with high levels of g_KL (Fig. 4C), but for the high-frequency Hiss stimulus, the greatest fidelity occurred with selective increases in g_KH.

To quantify the relative contributions of changes in g_KH and g_KL to optimizing fidelity for low- and high-frequency stimuli, we calculated a parameter R_HL that provides a measure of the balance between g_KH and g_KL when the maximal Xcorr values are attained (Materials and Methods). R_HL varies between −1 and +1, becoming −1 or +1 when the maximal value of Xcorr is attained with either g_KL or g_KH alone, respectively. Combinations of g_KH and g_KL give intermediate values. The values of R_HL for each of the stimuli in Figs. 3 C–G and 4 C and D are shown in Fig. 4E. It is evident that optimal fidelity for the stimuli with lower-frequency components (50, 100, and 200 Hz and Art) is achieved by increasing the relative contribution of g_KL, whereas for higher-frequency stimuli (800 Hz, pulse, and Hiss), increases in g_KH are required.

Optimal Levels of K⁺ Conductances in a Multineuron Network Differ from Those That Optimize the Firing of a Single Neuron.

The majority of the simulations presented above calculated the summed output of a 30-neuron ensemble for comparisons with the input signal. The output of the ensemble provides a much better fit to the original signal than could be attained with a single neuron. Fig. 5 shows plots of Xcorr as a function of the number of neurons (from 1 to 30) with only a g_KH or g_KL conductance. These correspond to the optimal value of g_KH or g_KL alone for the high-amplitude stimuli in Figs. 4 C–G and 5 C and D. Consistent with the conclusions of the previous paragraph, a g_KL conductance alone provides higher Xcorr values for low-frequency stimulation (50 Hz), while g_KH provides higher Xcorr values for high-frequency stimuli (800 Hz, Hiss, and pulse) for all ensemble sizes. As a consequence, the balance of g_KH or g_KL minimizes the number of neurons necessary to transmit a given amount of information.

Fig. 5.

Fidelity of output depends on the number of neurons in the ensemble. Dependence of Xcorr on the number of neurons in the ensemble (from 1 to 30) for the 50-, 100-, 200-, Art, 800-Hz, and Hiss stimuli of the Figs. 3 and 4. Neurons had only a single K⁺ conductance [either g_KL (black plots) or g_KH (red plots)]. Values of g_KL or g_KH were those corresponding to the highest values of Xcorr for the 30-neuron simulations with either conductance alone in Figs. 3 and 4.

As detailed in Figs. 3 and 4, the highest levels of Xcorr are generally attained with a mix of g_KH and g_KL. Even though the values for Xcorr are much lower for the output of a single neuron, it is possible that the set of conductances that generate the optimal values for Xcorr could be the same for the output of an ensemble or a single neuron. Alternatively, the output of an ensemble could encode additional features of the input not predicted in the response of single neurons. To answer this question, we determined the optimal g_KH/g_KL combinations for a single neuron in response to the 100-Hz, square pulse, Art and Hiss stimuli and then compared the results to those with the 30-neuron ensemble.

The Fig. 6 A–D shows the results for the high-amplitude 100-Hz stimulus. The number of action potentials evoked in any neuron depend on g_KH and g_KL but not on the number of neurons in the ensemble (Fig. 6A). A plot of how Xcorr varies with the number of neurons using the optimal g_KH/g_KL levels calculated for one neuron or 30 neurons, however, confirms that the values of g_KH and g_KL that provide the optimal Xcorr values for small numbers of neurons do not optimize the response for larger ensembles of 25 to 30 neurons (Fig. 6B). The pseudocolor plots of Xcorr also clearly show that the range of values of g_KH and g_KL that generate the optimal values is quite different for the output of one neuron (Fig. 6C) or 30 neurons (Fig. 6D). For one neuron, optimal fidelity is attained with a high level of g_KH (red square on pseudocolor plot, Fig. 6C). The traces shown below the plot demonstrate that this set of parameters allows the neuron to fire in response to every single positive wave of the stimulus and to phase-lock precisely to the stimulus. In contrast, when the set of parameters corresponding to the optimal values of the 30-neuron ensemble are applied to the single neuron (white square, Fig. 6C), the neuron fails to respond to some of the individual waves, and the timing of each response with respect to the stimulus is more varied. Fig. 6D shows the same data as in Fig. 3D for 30 neurons but with a more compressed color range for clarity. It also shows traces in which the combined output of 30 neurons (blue shading) is superimposed over the stimulus (red shading). The parameters that provide the more varied response with failures for a single neuron now generate an output (blue) that more faithfully matches the entire positive range of the 100-Hz stimulus (red). In contrast, the parameter set that generates more faithful phase-locking for a single neuron (white box, Fig. 6C) generates waves that are shorter in duration than the original stimulus, as evident by the red shading at the end of each wave (lowest trace, Fig. 6D).

Fig. 6.

Different combinations of currents provide optimal fidelity of output for single neurons vs. ensembles of 30 neurons. (A) Schematic of 100-Hz stimulus with time course of rectified signal shown in red (Left). Right shows two-dimensional plot of effects of increasing g_KL (Y axis) and g_KH (X axis) on numbers of action potentials evoked by 100-Hz stimulus (as in Fig. 3D). (B) Bottom plot shows dependence of Xcorr on the number of neurons in the ensemble for the 100-Hz stimulus. Black plots show data for the combination of g_KL and g_KH providing the highest Xcorr value for a single neuron, while red plots show data for g_KL/g_KH, giving the highest Xcorr for the 30-neuron ensemble. Two graphs at the top show expanded plots for 1 to 10 neurons (Left) and 21 to 30 neurons (Right) to illustrate the deviation of the two curves. (C) Array (Top) shows effects of g_KL and g_KH on Xcorr calculated for the output of a single neuron stimulated with a full-amplitude 100-Hz stimulus. The red box indicates the parameter set that defines the highest value of Xcorr for the single neuron. The white box indicates the parameter set that corresponds to the highest value of Xcorr calculated for an ensemble of 30 neurons. Lower traces show the cumulative output of the neuron (black trace with blue fill) superimposed over the rectified stimulus (red) for 75 ms after the onset of the 100-Hz stimulus. Traces are shown for parameters corresponding to the white box (Top trace) or the red box (Bottom trace). (D) As for C but with Xcorr calculated for the output of an ensemble of 30 neurons. The red box indicates the parameter set that defines the highest value of Xcorr for 30 neurons. The white box indicates parameters that correspond to the highest Xcorr for one neuron. Lower traces show the cumulative outputs (black trace with blue fill) superimposed over the rectified stimulus (red) for 35 ms at the onset of the 100-Hz stimulus and correspond to red box parameters (Top trace) or white box parameters (Bottom trace). (E–H) As for A–D but for a square pulse stimulus (0.2 amplitude as in Fig.3 G, Center).

The same calculation of the difference in outputs of a single neuron and the 30-neuron ensemble was also made for a square pulse, which mimics the response to a sustained high-frequency sound (>2 kHz) (Fig. 6 E–H). In this case, there was much less difference between the levels of g_KH and g_KL that maximize Xcorr. Specifically, high values of g_KH promote sustained firing during the pulse itself both for the single neuron and the ensemble.

Finally, we compared the correlation between the output of a single neuron with that of the 30-neuron ensemble for the complex low-frequency Art stimulus and the high-frequency Hiss stimulus (Fig. 7). For the full-amplitude Art stimulus, the parameters required for the highest values of Xcorr were quite distinct for small numbers of neurons and the 30-neuron ensemble (Fig. 7 A–D). The superimposed traces of outputs and the Art stimulus (shown on two different times scales for 30 neurons in Fig. 7D) clearly show that the optimal g_KH and g_KL values for 30 neurons provide a better match to both the timing and changes in amplitude of the stimulus. In contrast, the optimal set of parameters for the single neuron primarily increase the number of action potentials that occur during the stimulus (Fig. 7 C, Bottom Left traces).

Fig. 7.

Levels of current that optimally encode complex stimuli differ for single neurons and ensembles of 30 neurons. (A) Time course of partly rectified Art word stimulus and two-dimensional plot of effects of increasing g_KH and g_KL on numbers of action potentials evoked by the full-amplitude stimulus (as in Fig. 4C). (B) Plots of dependence of Xcorr on the number of neurons in the ensemble for the Art stimulus. Details as in Fig. 6B. (C and D) Pseudocolor arrays showing values of Xcorr calculated for the output of a single neuron or a 30-neuron ensemble in response to the Art stimulus. Red boxes indicate the highest values of Xcorr for each condition. White boxes in the plots for the high-amplitude stimulus indicate the parameters that correspond to the highest value of Xcorr in the other condition (one neuron or 30 neurons). Traces at Bottom show the cumulative outputs (black trace with blue fill) superimposed over the rectified stimulus (red) for the boxed parameter sets in the Xcorr arrays. For the output of a single neuron, the full trace is shown (275 ms, Left traces). For the output of 30 neurons, two sets of traces are shown for each condition. Traces at Left show 35 ms of response at the onset of the stimulus, while traces at Right show the full trace. (E–H) As in A–D but for the higher-frequency Hiss stimulus (as in Fig. 4B). Traces at bottom correspond to the full 847-ms stimulus.

For the higher-frequency Hiss stimulus (amplitude 0.2), there was very little difference between the g_KH and g_KL combinations that yielded the highest Xcorr for a single neuron and the 30-neuron ensemble (Fig. 7 D–G). The greater values of g_KH that are required for the Hiss response allow the neurons to fire more rapidly during this sustained high-frequency stimulus. Thus, the results of the Art and Hiss simulations in Fig. 7 generally match those observed for the much simpler low-frequency 100 Hz and “higher-frequency” square pulse in Fig. 6.

Discussion

The simple computations in this manuscript focused on the effects of changing two major classes of potassium channel on the response of neurons receiving synaptic inputs triggered by sound stimuli. The major conclusion is that increases in neuronal K⁺ currents promote fidelity of information transmission as signal intensity is increased. As with real neurons, responses were calculated in the presence of a low level of random excitatory postsynaptic potentials driven by spontaneous transmitter release from hair cells. The simulations indicate that low-amplitude sounds are most reliably encoded in firing patterns of neurons with lower levels of potassium conductance. For low-frequency stimuli, increases in low-voltage-activated K⁺ currents preferentially improve the correlation of sounds with the neuronal output. In contrast, for high-frequency sounds, increases in high-voltage-activated currents, which enable rapid firing, are specifically required as sound amplitudes are increased.

While the present work focused only on K⁺ currents that activate at different potentials, it is evident that other types of channels, as well as structural and morphological aspects of neurons, can be modulated by inputs from sensory organs, as well as by higher neural pathways. Thus, in real neurons, optimal adjustments to incoming stimuli almost certainly involve changes in these other factors. For example, modulation of HCN nonselective cation channels may exert a strong influence on the firing of mouse spiral ganglion neurons (36). In addition, although we maintained a fixed sodium conductance for most simulations, sodium currents are modified by neurotransmitters (37) and by deafness (38) in some neurons. Consistent with experimental observations (39), we found that increasing gNa beyond the value that yields maximal Xcorr led to a progressive reduction in temporal precision (Fig. 2). In our simulations of neurons with g_KH currents, the rate of increase in the firing rate with gNa was slowed beyond this point. In some neurons, the firing rate actually declines with increasing gNa above that required for optimal temporal fidelity (39). This was not seen in our simulations probably because somatic sodium currents were not differentiated from those at the axonal initial segment in the one-compartment model.

Because the patterns of synaptic inputs generated in the model used a very simplified model of cochlear inner hair cells, the most obvious implication of these results is for responses of type I neurons of the spiral ganglion. Proteomic, genetic, and electrophysiological studies have provided a wealth of information on the potassium channel subunits that are expressed in spiral ganglion neurons (9, 11, 15, 40–42), and their intrinsic excitability is modulated by second messengers (43). The classes of channels providing g_KH- and g_KL-type currents in spiral ganglion neurons are very similar to those expressed in their first- and second-order postsynaptic target neurons in the cochlear nucleus and related auditory brainstem nuclei. Because of their experimental tractability, however, most studies of how potassium channels are regulated by auditory stimulation have been carried out using neurons in the auditory brainstem such as those of the medial nucleus of the trapezoid body (MNTB). Nevertheless, because the same channels are expressed in these different nuclei, it is likely that similar principles apply to their regulation.

A wide variety of studies have documented that the amplitude of potassium currents in auditory neurons is modified by acoustic stimulation in vivo and by different patterns of synaptic stimulation in vitro. These modifications take place over many different time spans. Rapid modification of potassium currents, occurring within seconds or minutes of stimulation, occurs in response to changes in the level of acoustic stimulation and is mediated by changes in the activity of protein kinases/phosphatases that can modify the channel itself (23, 24, 44, 45). Acoustic stimulation over tens of minutes also alters the rate of synthesis of potassium channel subunits (25, 26, 46). Finally, loss of auditory inputs, caused, for example, by cochlear ablation, genetic mutations that result in deafness, or age-related hearing loss in certain strains of mice, leads to major changes in overall expression levels of potassium channels (38, 47–51).

Rapid changes in potassium currents of auditory neurons in response to acoustic or synaptic stimulation have been documented for the high-voltage-activated Kv3.1b channel in the anteroventral cochlear nucleus and the MNTB. The g_KH conductance in the present simulations was based on the kinetic properties and voltage dependence of Kv3.1b. Higher levels of auditory stimulation in vivo (white noise clicks at 70 dB), or higher rates of stimulation (400 to 600 Hz) of brain slices in vitro, result in the dephosphorylation of a serine residue in the cytoplasmic C-terminal domain of the channel, leading to increased Kv3.1b current (23, 24). In the absence of stimulation, phosphorylation of this residue, which suppresses Kv3.1b current, is mediated by several isozymes of protein kinase C (23). Stimulation of brain slices at much lower rates (100 Hz) suppresses Kv3.1b current by a mechanism mediated by the release of nitric oxide (44).

Another channel whose amplitude in auditory brainstem neurons is modulated by stimulation is Kv2.2 (45). This channel activates at a more negative potential than Kv3.1b but at more positive potentials than other channels generally considered to constitute g_KL. Accordingly, Kv2.2 currents are increased by low rates of stimulation (10 Hz) (45). Channels that activate at negative potentials and are expressed in auditory brainstem neurons or spiral ganglion neurons include Kv1.1, Kv1.2, Kv1.3, Kv1.6, Kv11, K_Na1.1, and several K_2P subunits (29, 47, 52–55). While changes in expression of many of these channels occur in deafness, little is known about rapid acute changes in response to auditory stimulation. Nevertheless, based on studies of these subunits in expression systems, all have the potential to be modified rapidly by mechanisms such as phosphorylation (reviewed in ref. 7).

In auditory nuclei, g_KH and g_KL channels are generally expressed along a tonotopic gradient. For example, the highest levels of Kv3.1b are found in neurons in the medial aspect of brainstem nuclei, which preferentially respond to high-frequency sounds (26, 51, 56, 57). In contrast, Kv1 family channel subunits, as well as Kv2.2 channels, are expressed in the opposite direction, with the highest levels in the low-frequency lateral regions of these nuclei (9, 50, 52, 57–63). One potential interpretation of these findings is that neurons that respond to higher-frequency sounds, and that have high levels of g_KH and low levels of g_KL, may be able to fire at higher rates than their low-frequency counterparts. There is, however, no evidence that these neurons do fire at higher rates in vivo. Instead, our simulations suggest that this pattern of channel expression optimizes their firing patterns to match the envelope of high-frequency stimuli.

The channels that underlie g_KH and g_KL are highly expressed in diverse vertebrate organisms with a similar cellular and molecular organization of the auditory transduction cascade but with distinct sound sensitivities. While some species, such as mice, cannot detect low-frequency sounds, they respond to high-frequency sounds that are amplitude modulated at low rates. Thus, similar principles for channel modulation may apply in all species, although specific conductances and modulation mechanisms may differ. For example, rapidly inactivating Kv4 channels are absent in most peripheral and brainstem auditory neurons, although they determine spike latency and firing frequency in many other neurons. Nevertheless, MNTB neurons in mouse, but not rat, have Kv4.3 A-type K⁺ currents (64). These are largely inactivated at the resting potential but may potentially modulate low-fidelity transmission in this high-frequency hearing species.

Finally, the results suggest that mechanisms that adjust potassium conductances must act coordinately on groups of neurons rather than optimizing the response of each neuron to the incoming stimuli. This is consistent with the existence of mechanisms that coordinates the activity of nearby neurons, even when some of those neurons have not been activated synaptically. These mechanisms include the release of the volume transmitter nitric oxide that spreads laterally within a nucleus during low-frequency stimulation (44, 45), coordinated transcription of genes in small clusters of auditory neurons mediated by CREB (cyclic AMP/Ca²⁺ response element–binding protein) (51), the existence of recurrent collateral axons that projects back to an auditory nucleus (65, 66), and local interactions mediated by glial cells (67, 68). A key aim for the future will be to determine the molecular and systems mechanisms that coordinate changes in ion conductances to allow an animal to discriminate minute differences in auditory stimuli.

Materials and Methods

We first constructed a simple model to trigger a series of excitatory synaptic inputs from an auditory stimulus. This was based on changes in neurotransmitter release from a model inner hair cell. Simple stimuli such as single-frequency pulses or ramps were generated synthetically using an intersample interval of 22.67 µs (44.1-kHz sampling). Monaural sound stimuli in the wav format were converted to a similar stimulus train (Fig. 1). The maximal amplitude of each stimulus was normalized to a value of 1 and then reduced as appropriate to mimic stimuli of lower amplitudes. A sigmoidal function (Fig. 1, steps 1 and 2) was then generated to convert each value of a stimulus y(t) to the relative probability of release S(y) of discrete quanta of neurotransmitter from the hair cell.

$$S=\frac{1}{1+e^{-a(y-c)}}$$ [1]

where a = 8.0 and c = 0.3 (Fig. 1, step 2).

At each discrete sampling interval, S(y) was used to determine the timing of the release of a quantum of neurotransmitter. A quantum of transmitter was released at each interval that matched the condition b. S(y) > r, where r is a random number 0 < r < 1. With b = 0.05 and an intersample interval of 22.67 µs, this condition provided a spontaneous rate of release of 140 quanta/second when y(t) = 0 (i.e., in silence).

This transformation generated a digital time series R, with the same duration T as the sound stimulus, containing P nonzero elements, each of which had a value of 1 and occurred at a time t_P, representing the occurrence of the release of a quantum of neurotransmitter. The digital time series was then converted into train of postsynaptic conductances g_syn(t) in which each of the P positive values in R triggered a conductance of the form

$${\mathrm{g}}_{\mathrm{synP}}\left(\mathrm{t}\right) = 0.002 \exp \left({\mathrm{t}}_{\mathrm{p}}-\mathrm{t}\right)/\mathrm{\tau })$$

with τ = 0.5 ms.

The synaptic conductances evoked by each release event were then summed to provide the full synaptic input into the postsynaptic cell.

$$g_{syn}\left(t\right)={\displaystyle \sum _{n=0}^{\mathbb{N}}{g}_{syn}{}_{\text{P}}\left(\text{t}\right)},$$

as illustrated in Fig. 1, step 3.

The voltage response (V) to this synaptic input was then calculated for neurons using models similar to those described previously (24, 29–33) (Fig. 1, step4). Specifically,

$$C\frac{dv}{dt}=-I_{\mathit{Na}}-I_K{H}_{.}-I_{K}L-I_{leak+}{I}_{\mathit{syn}}\left(t\right),$$

where C represents cell capacitance, I_Na represents Na⁺ current, and I_KH and I_KL represent components of voltage-dependent K⁺ currents that activate at positive and negative membrane potentials, respectively. I_Leak is the leak current, and I_syn is the current evoked by the synaptic input.

The capacitance C of each model neuron was 0.01 nF. Equations for I_Na, I_KH, I_KL, and I_L were identical to those in ref. 33 with

$$l_{Na} = g_{Na}{m}^{3}h\mathit{ }\left(\mathrm{V}-50\right)\mathit{,}$$

$$l_{k}H=g_{k}H n^3\mathrm{p}\mathit{ }\left(\mathrm{V}+80\right)\mathit{,}$$

$$l_{k}L=g_{k}L lr\mathit{ }\left(\mathrm{V}+80\right)\mathit{,}$$

$$l_{leak}=g_{leak}\mathit{ }\left(\mathrm{V}+63\right)\mathit{,}$$

and

$$l_{syn}=g_{syn}\left(\mathrm{t}\right) \left(\mathrm{V}\right).$$

The evolution of the variables m, h, n, l, and r was given by equations of the form

$$\frac{dj}{dt}={\alpha }_j(1-j)-{\beta }_j,$$

where

$${\alpha }_j=k_{\alpha j}exp\left({\eta }_{\alpha j}V\right),$$

$${\beta }_j=k_{\beta j}exp\left({\eta }_{\beta j}V\right),$$

and j = m, h, n, l, and r.

The values of g_Na, g_KH and g_KL were varied as described in the text. Other kinetic parameters for voltage-dependent Na⁺ current were k_αm = 76.4 ms⁻¹, ηαm = 0.037 mV⁻¹, k_βm = 6.93 ms⁻¹, η_βm = −0.043 mV⁻¹, and k_αh = 0.000135 ms⁻¹, η_αh = −0.1216 mV⁻¹, k_βh = 2.0 ms⁻¹, and η_βh = 0.0384 mV⁻¹. For the I_KH current, k_αn = 0.5438 ms⁻¹, η_αn = 0.04 mV⁻¹, k_βn = 0.3946 ms⁻¹, η_βn = 0 mV⁻¹, k_αp = 0.03426 ms⁻¹, η_αp = −0.1942 mV⁻¹, k_βp = 1.8705 ms⁻¹, and η_βp = 0.0058 mV⁻¹. For the I_KL potassium current, k_αl = 1.2 ms⁻¹, η_αl = 0.03512 mV⁻¹, k_βl = 0.2248 ms⁻¹, η_βl = −0.0319 mV⁻¹, k_αr = 0.04378 ms⁻¹, η_αr = −0.005312 mV⁻¹, k_βr = 0.0562 ms⁻¹, and η_βr = 0.0047 mV⁻¹. For the conductances, g_KA1, g_KA2, and g_KA3 (Fig. 3 A and B and SI Appendix, Fig. S1), the parameters were as for I_KH, with the following changes. For g_KA1, I_KA1 = g_KA1 np (V + 80), with other parameters unchanged. For g_KA2, I_KA2 = g_KA2 n²p (V + 80) with k_αn = 0.7794, η_αn = 0.0222, k_βn = 0.248, and η_βn = −0.0378. For g_KA3, I_KA3 = g_KA3 np (V + 80), k_αn = 0.6375, η_αn = −0.0304, k_βn = 0.878, and η_βn= −0.0704. Note that while a Hodgkin–Huxley formalism was used for the I_KH current, the repolarization of action potentials by Kv3.1, which contributes to I_KH in many neurons, is dominated by a resurgent K⁺ current produced by channel opening from a nonconducting state during recovery from brief depolarizations (69). To compensate for this, the value of g_KH was simply increased in the present simulations.

Firing patterns in response to a single stimulus pattern y(t) were calculated for ensembles of N identical neurons with patterns of synaptic input g_syn(t) calculated independently for each neuron, consistent with each an independent presynaptic active zone for each connection. To provide a measure of the combined output of the ensemble, the output of each neuron was first converted to a digital time series O, with the same duration T as the sound stimulus, containing P nonzero elements, each of which had a value of 1 and occurred at a time t_A corresponding to an action potential (determined by a positive crossing of V(t) at 0 mV) (Fig. 1, step 5A). These were in turn converted into a second set of conductances, g_outputA (t).

$${\mathrm{g}}_{outputA}\left(\mathrm{t}\right)=\exp \left({\mathrm{t}}_{\mathrm{A}}-\mathrm{t}\right)/\mathrm{\tau })$$

with τ = 0.02 ms. These were then summed to provide the full synaptic output for each neuron (Fig. 1, step 5A).

$$g_{output}\left(\mathrm{t}\right)=\sum _{n=0}^P{g}_{outpu{t}_{\mathrm{A}}}\left(\mathrm{t}\right).$$

The individual g_output (t) traces for each of the N neurons were then summed linearly and normalized to a maximal value of 1 to generate G_output(t), the final output of the ensemble (Fig. 1, step 5B).

To quantify the similarity between G_output(t) and the original sound stimulus y(t), the stimulus y(t) was first transformed to a time series y’(t) with all positive values. This was accomplished by subjecting y(t) to half-wave rectification by setting all negative values of y equal to zero and then normalizing the resultant time series to have a maximal value of 1. For every combination of channel parameters and y’(t), the similarity between the stimulus and neuronal firing pattern was quantified as the Xcorr, the maximal value of the cross-correlation function of G_output(t) and y’(t) (Fig. 1, step 6).

The protocol above was used for simple sinusoidal stimuli, ramps of sinusoidal stimuli, square waves, and for the more complex stimuli corresponding to the words Art and Hiss. Because the latter stimuli contain a mixture of low- and high-frequency components, however, an additional transformation of these two auditory stimuli was also carried out to transform the high-frequency components into a sustained depolarization of hair cells, as is the case for real cochlear inner hair cells (28). The stimulus y(t) was first rectified using the sigmoidal function in equation (1), low-pass filtered at 500 Hz, and then normalized to a maximal value of 1.0, 0.2, or 0.05 corresponding to high-intermediate and low-amplitude stimuli, respectively (compare Fig. 4A with Fig. 4C, and Fig. 4B with Fig. 4D). Qualitatively similar results were obtained using both the unrectified and rectified stimuli.

To illustrate the relative contributions of g_KH and g_KL in optimizing Xcorr for each stimulus, we calculated the parameter R_HL, given by the equation

$$R_{\mathrm{HL}}=g_{k}H/g_{H}max-g_{k}L/g_{L}max,$$

where g_Hmax and g_Lmax represent the maximal values of g_KH and g_KL in the two-dimensional plots in Figs. 3 and 4 (0.6 and 0.04 µS, respectively).

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

All study data are included in the article and/or SI Appendix.