How Close Should the Outer Hair Cell RC Roll-Off Frequency Be to the Characteristic Frequency?

AbstractRecent experiments have shown a much larger conductance in outer hair cells, the central components of the mammalian cochlear amplifier. The report used only the cell's linear capacitance, which together with increased conductance, raised the cell's RC corner frequency so that voltage-dependent motility was better able to amplify high-frequency sounds. We construct transfer functions for a simple model of a high characteristic frequency (CF) local cochlear resonance. These show that voltage roll-off does not occur above the RC corner. Instead, it is countered by high-pass filtering that is intrinsic to the mammal's electromechanical resonance. Thus, the RC corner of a short outer hair cell used for high-frequency amplification does not have to be close to the CF, but depending on the drag, raised only above 0.1 CF. This high-pass filter, built in to the mammalian amplifier, allows for sharp frequency selectivity at very high CF

Fast Negative Feedback Enables Mammalian Auditory Nerve Fibers to Encode a Wide Dynamic Range of Sound Intensities

Mammalian auditory nerve fibers (ANF) are remarkable for being able to encode a 40 dB, or hundred fold, range of sound pressure levels into their firing rate. Most of the fibers are very sensitive and raise their quiescent spike rate by a small amount for a faint sound at auditory threshold. Then as the sound intensity is increased, they slowly increase their spike rate, with some fibers going up as high as,300 Hz. In this way mammals are able to combine sensitivity and wide dynamic range. They are also able to discern sounds embedded within background noise. ANF receive efferent feedback, which suggests that the fibers are readjusted according to the background noise in order to maximize the information content of their auditory spike trains. Inner hair cells activate currents in the unmyelinated distal dendrites of ANF where sound intensity is rate-coded into action potentials. We model this spike generator compartment as an attenuator that employs fast negative feedback. Input current induces rapid and proportional leak currents. This way ANF are able to have a linear frequency to input current (f-I) curve that has a wide dynamic range. The ANF spike generator remains very sensitive to threshold currents, but efferent feedback is able to lower its gain in response to noise

Auditory nerve spike generator modeled as a variable attenuator based on a saddle node on invariant circle bifurcation.

Mammalian inner hair cells transduce the sound waves amplified by the cochlear amplifier (CA) into a graded neurotransmitter release that activates channels on auditory nerve fibers (ANF). These synaptic channels then charge its dendritic spike generator. While the outer hair cells of the CA employ positive feedback, poising on Andronov-Hopf type instabilities which make them extremely sensitive to faint sounds and make CA output strongly nonlinear, the ANF appears to be based on different principles and a different type of dynamical instability. Its spike generator "digitizes" CA output into trains of action potentials and behaves as a linear filter, rate-coding sound intensity across a wide dynamic range. Here we model the spike generator as a 3 dimensional version of a saddle node on invariant circle (SNIC) bifurcation. The generic 2d SNIC increases its spike rate as the square root of the input current above its spiking threshold. We add negative feedback in the form of a low voltage-threshold potassium conductance that slows down the generator's rate of increase of its spike rate. A Poisson random source simulates an inner hair cell, outputting a series of noisy periodic current pulses to the model ANF whose spikes phase lock to these pulses and have a linear frequency to current relation with a wide dynamic range. Also, the spike generator compartment has a cholinergic feedback connection from the olive and experiments show that such feedback is able to alter the amount of H conductance inside the generator compartment. We show that an olive able to decrease H would be able to shift the spike generator's dynamic range to higher sound intensities. In a quiet environment by increasing H the olive would be able to make spike trains similar to those caused by synaptic input

For comparison a spike train is shown from a biophysically-based 10 compartment model of the ANF dendritc spike generator.

<p>The simplified 3d SNIC model is a reduction of this more detailed model in which AMPA current is input into compartment 1, leak conductances are equally distributed throughout the 10 compartments, and the spike machine (sodium and delayed rectifier conductances) placed into compartment 10. The characteristic frequency of the noisy AMPA current pulses is 440 Hz (black). Fast negative feedback from low threshold voltage-gated potassium conductance spread throughout the 10 compartments extends the generator’s dynamic range to about 38 dB. The sample spike train is driven by an average of 1 vesicle released by the IHC per sound wave cycle and includes vesicle release noise, with some of the vesicles released by the simulated IHC having an incorrect phase. 1 mV of Gaussian voltage noise is included. For low noise the action potentials phase lock to the AMPA current pulses. By comparing the attenuation of the back propagating action potentials in compartment 1 (orange) to the generator’s action potentials in compartment 10 (blue) the model dendrite can be seen to have an electrical length of about 0.22 space constant for small input current. The dendrite “lengthens” to 0.26 space constant for large input by turning on more low threshold potassium conductance which increases its attenuation (a static potential is <i>e</i>-fold damped in 1 space constant). The epsps from compartment 1 (orange) are similarly attenuated after arriving at the generator compartment (blue).</p

Evidence of a Hopf Bifurcation in Frog Hair Cells

The membrane potential of hair cells in the low-frequency hearing organ of the bullfrog, the amphibian papilla, sinusoidally oscillates at small amplitude in the absence of acoustical input. We stimulate the cell with a series of periodic currents close to this natural frequency and observe that its current-to-voltage transfer function is compressively nonlinear, having a large gain for small stimuli and a smaller gain for larger currents. Along with the spontaneous oscillation, this implies that the cell is poised close to a dynamical instability such as a Hopf bifurcation, because distant from the instability the transfer function becomes linear. The cell’s frequency selectivity is enhanced for small stimuli. Simulations show that the cell’s membrane capacitance is effectively reduced due to a current gain provided by this dynamical instability. We propose that the Hopf resonance is widely used by transducer cells on the sensory periphery to achieve small-signal amplification.This work was supported by National Institutes of Health Grant DC00241 to Dr. A. J. Hudspeth. During the conduct of this research M.O. was an Associate of Howard Hughes Medical Institute.Peer reviewe

Spike trains, K+ currents and Ca++ concentrations from a model of the distal dendrite, spike generator region of an auditory nerve fiber (ANF).

<p>The model employs fast negative feedback to stretch its dynamic range. A. Spike train from a low threshold ANF with 5 pA quiescent AMPA input current. It has a 10 Hz spontaneous rate that is increased to 28 Hz during a 10 pA input that simulates a near threshold tone burst between 300 and 500 msec. B. Outward Shaker K+ current that helps to shape the spike train in part A. Note that Shaker current decreases between spikes. C. Outward K+ leak current for part A. Leak current increases in between spikes. D. Dendritic Ca++ concentration for part A. The second messenger Ca++ was used to initiate negative feedback, turning on K+ leak almost immediately and Shaker with a ∼10 msec delay. E. 100 pA input obtains a 73 Hz spike rate, where the faster rate near start of the pulse is due to the short time delay between input current and outward Shaker current. A part of the Shaker current is due to fast negative feedback that is enabled by Ca++. Also, slower LOCS feedback is assumed to partly turn off H current. To simulate LOCS feedback ∼2 pA of H current was turned off at 500 msec, and this acts to halve the firing rate. F. Outward Shaker current for part E. G. Outward K+ leak current for part E. H. Ca++ concentration for part E. I. Same model without any LOCS feedback is driven by a succession of increasing inputs: 5, 6, 10, 20, 30, 40, 60, 80, 100, 200, 300, 400 and 500 pA. The model fiber has a 40 dB input dynamic range with a maximum spike rate ∼290 Hz. Note that for larger input currents spike amplitude drops due to decreased dendrite impedance. These smaller action potentials could later be enlarged by spike repeaters after the nerve is myelinated <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0032384#pone.0032384-Hossain1" target="_blank">[13]</a>. J. High threshold version of the model ANF has a 0 Hz rate for 38 pA input and is driven by the current sequence 38, 39, 40, 50 and 38 pA.</p

Spike rate vs. <i>N</i>, the average number of vesicles released by the IHC per sound wave cycle for the 3d SNIC model.

<p>The size of the data points shows the standard error in the spike rate, which increases with <i>N</i> (red; <i>gKl</i> = 8, <i>gL</i> = 5, <i>gH</i> = 1). The plot is similar to spike frequency vs. input current and is made linear across approximately a 48 dB range (from <i>N</i>∼1/8 to <i>N</i>∼32) by fast negative feedback from the low threshold potassium conductance <i>gKl</i>. For small input the small standard error allows the <i>N</i>  = 0, 1/8, 1/4 excitations to be resolved from each other, while for large inputs the accompanying large standard error limits the ability to resolve intensities above <i>N</i> = 32. Turning off inward leak H current moves dynamic range to higher sound intensities (black; <i>gKl</i> = 8, <i>gL</i> = 5, <i>gH</i> = 0; ∼27 dB range from <i>N</i> = 2 to 46). H was turned off as if it were under the control of cholinergic feedback from the olive. Here the standard error starts out large for the small spike rates just above the bifurcation (<i>N</i> = 2), decreases away from the bifurcation and then increases for large inputs. For comparison we show the loss of dynamic range and nonlinear rise of the spike rate due to turning off the low threshold potassium conductance (green; <i>gKl</i> = 0, <i>gL</i> = 5, <i>gH</i> = 1). This curve was extended to negative <i>N</i> to show the square root shape in the rise of the spike rate above the now 2d SNIC bifurcation. Fluctuations in the size of the standard errors occurred because spike rates and standard errors were calculated over short time intervals, trying to reproduce the approximately one quarter second that the auditory pathway has to resolve the intensities of specific formant frequencies in order to be able to identify a particular vowel sound in real time. At a given frequency that is sensed by a particular IHC small numbers of ANF rapidly and accurately determine the intensity of the tone, each using only a small number of spikes.</p

Low threshold model now includes 300 microvolt RMS Johnson voltage noise.

<p>A. 100 pA input starting at 0.3 sec obtains a 75 Hz spike rate that is a similar rate to its no noise version in 2 E. Also, similar to 2 E, the spike rate is reduced to 42 Hz by simulating LOCS feedback as turning off H current, starting at 0.6 sec. B. Same low threshold model and noise. Two noisy spike trains each driven by a near threshold 10 pA input current between 0.4 and 0.6 sec (red trace displaced upward by 3 mV). C. Post stimulus time histogram (PSTH) for 30 data sets under the same noise conditions and driven by a 10 pA input between 200 and 400 msec (shown are the cumulative numbers of spikes in sixty 10 msec wide bins). Note the on transient between approximately 200 and 260 msec, the off transient between about 400 and 440 msec and the rate coding of a near threshold current input that occurs between about 260 and 400 msec.</p

ANF spike generator modeled as a 3d version of a SNIC bifurcation.

<p>A. Spiking trajectories are shown for small (<i>N</i>  = 2 blue) and large (<i>N</i>  = 20 green) noisy periodic input currents. For comparison a trajectory driven only by increasing H conductance by 50% is shown (orange). <b><i>v</i></b> is membrane potential, <b><i>n</i></b> is the open probability of the high threshold voltage-gated potassium conductance (usually referred to as the delayed rectifier) and <b><i>n_l</i></b> the open probability of the low threshold voltage-gated potassium conductance (negative feedback that slows the rate of spike rate increase). B. Spike trains for the same small input (blue) and large input (green) cases. Synaptic AMPA input currents (black) that correspond to an average of <i>N</i> vesicles released by the inner hair cell per sound wave cycle are given by a Poisson random variable. H current driven spike train is in orange. Experiments show that H can be altered by cholinergic feedback from the olive via a second messenger pathway <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0045326#pone.0045326-Yi1" target="_blank">[10]</a>. Feedback control of H would be able to generate spike trains similar to those caused by synaptic input.</p