There is a lot of inertia behind a full-force study of the cochlear amplifier in mammals as of late. Optical coherence tomography (OCT) has given researchers the ability to measure sub-nanometer displacement in vivo, leading to a number of exciting developments in the field in recent years.
Here at Columbia's Fowler Lab, we have discovered some surprising recovery properties for the cochlear amplifier. Our general approach has been to measure vibrations in vivo to gain a base case, then perform an injection or profusion to change the properties of the cochlea, then continue to monitor the vibrations as the cochleas electrical, chemical and mechanical properties change.
Of course, "measure vibrations" is an imprecise term - the cochlea contains multiple membranes and regions which vibrate differently. The stapes, oval window and round window (RW), for example, vibrate mostly as a faithful reconstruction of the sound stimulus, while the basilar membrane (BM) and organ of corti (OOC) vibrate differently as a result of varying width and electromechanical feedback.
The OOC, resting atop the BM, is the region of most interest as it contains the sensory inner hair cells (IHC) and the electromotile outer hair cells (OHC). The OOC is the active component which makes in vivo research necessary. Vibration of the BM and OOC is, as a result, what we are most interested in, but where? The OOC and BM are three-dimensional, they wrap longitudinally around the cochlea, but also expand radially across the width of the cross-sections, as well as having some axial thickness.
The theory of cochlear amplifications centers mostly around the longitudinal behavior of the BM's vibration. This makes perfect sense, as it is along this longitudinal direction that frequency is spatially encoded. However, there are a number of reasons that the axial and radial components of BM motion may be of interest to researchers as we move towards a firmer understanding of the cochlear amplifier.
One is as a guiding tool. Marcel Van Der Heijden et al showed in 2018 that regions of the BM nearest the OHC have the greatest vibration amplitudes when compared to other points along the BM and OOC in a given cross section. To achieve maximum signal strength, and thus SNR, one would think it best to sample near this region of greatest vibration.
The other is as a discriminatory tool. We have found that OHC vibrations and BM vibrations can have different recovery properties at different points along the BM - it is valuable to not only measure vibrations within the OHC region, but also in the BM for some experiments as how they differ can indicate which structures involved in cochlear amplification may be failing.
In OCT, an axial scan or A-Scan gives intensities along a line through a tissue. In our experiments, A-scans are taken through the round window membrane, and they correspond to axial lines through a basal cross-section of the cochlea. The beam of the OCT system can be swept to give a B-Scan, which is composed of a number of A-scans stacked side-by-side. This gives a two-dimensional image of a cross-section of the cochlea taken so that the x-axis is the radial direction and the y-axis is the axial direction. Different viewing angles could achieve images wherein the x-axis where longitudinal rather than radial.
To achieve vibrometry, we look at A-scans as a function of time, or an M-Scan. The linerate of an OCT system is, functionally, a sampling rate for A-scan data. Our Thorlabs Telesto system has a 100kHz linerate, so we take 100,000 A-scans per second. By Nyquist's theorem, we can resolve vibrations at 50kHz and lower, which is reasonable - the basal turn of a gerbil's cochlea, for example, will vibrate at around 20-30kHz.
An M-scan gives vibrometry data along an entire axial line through the cochlea, and in a good image, we can distinguish between the OHC region and a section of the BM. We also can always measure vibration at the RW which is a useful control, as it is not affected by the cochlear amplifier significantly. Our recent experiments have shown that decrease in endocochlear potential leads to a loss and subsequent recovery of the OHC and BM motion, but interestingly these regions recover at different time scales.
While we do know that the BM has radial phase and amplitude variations, and Van Der Heijden et al have showed what these look like in a healthy gerbil cochlea, we do not know how these respond to varying endocochlear potential, or any other modification made to the cochlea. Limitation of the study of cochlear amplification to one dimension is most expedient for data acquisition, but may not tell the whole story.
Furthermore a deeper understanding of radial behavior in healthy and unhealthy cochleas could be very important for the development of cochlear models in the future. As finite element models have become quite popular, it is not clear if their radial behavior is matched to physiology. This could be of value, as it may illuminate further why the BM and OHC respond differently to certain changes.
Unfortunately, the study of radial behavior is not simple. Mathematically, we have gone from a signal A(y,t) in space in time to B(x,y,t). Naturally, y must be sampled as specified by the OCT system's photodetectors - we do not control how many pixels are in an A-scan. Similarly, t is sampled at the linerate of the OCT.
But x is an option - we can choose how to sample our data in the radial direction. We will be creating a spatial sampling lattice for our signal which is evenly spaced in y but is not constrained on spacing in x. Note also that due to our linerate and the sampling theorem, we cannot sample multiple x coordinates at once. We must instead take samples at one x coordinate and then move to another. This creates a number of problems regarding reconstruction.
These issues, although seemingly striking a mortal blow to this entire concept, are not insurmountable, I believe. To get around each, however, we must make a simplifying assumption.
For sampling in x, we should appeal to the Petersen-Middleton theorem for lattice sampling. Petersen-Middleton is essentially an analogue to the Nyquist theorem for spatial-domain sampling: where Nyquist requires a bandlimited signal to be sampled at a sufficiently high rate in time, Petersen-Middleton requires a "wavenumber-limited" signal to be sampled at a sufficiently dense rate in space. In this case, "wavenumber-limited" means that there are no abrupt changes in the signal's variation in the axial direction. This, from Van Der Heijden's results, seems true, although it must be quantified.
Given Petersen-Middleton, if we choose too coarse a sampling in the axial direction, we may not get a proper representation of the signal in space. However, a sampling that is too dense may be so time consuming to complete and process that it precludes the study from being viable. Thus, a smart choice of lattice must be made.
One additional complication which could prove fruitful is non-uniform sampling. In non-uniform sampling, regions in which the signal changes faster can be sampled more densely, while time can be saved in more constant regions by sampling more sparsely. Van Der Heijden sees a faster change in amplitude occurring near the OHC region, and thus we can choose to sample more densely in this region while making our samples in the BM far from the OOC sparse. With an intelligently chosen lattice, the process of taking two-dimensional recording should be time-effective and computationally tractable.
However we still have an issue of time - most generally, lattice sampling theory is built off of the understanding that each point in the lattice is being sampled at the same point in time. Here, we are not sampling each point at once, but rather are sampling one x-coordinate at a time. Is this even valid at all?
Well if we consider the frequency domain representation of the vibrations at a point on a lattice, assuming the response is stationary, we will see the exact same Fourier transform no matter what time the recording is taken at. This is to say that we can perfectly reconstruct the signal at each x-coordinate in the lattice up to some timing offset. How do we account for this timing offset, however? As it could be exacerbated by a natural phase offset between different x-coordinates? Well, we can use spatial relationships between points in the lattice to reconcile these timing and phase offsets. This is, in some sense, built in to the lattice sampling theorem, as a consequence of the signal being wavenumber-limited in space.
Our only assumptions then are wavenumber-limitation and stationarity. As for the first, any issue of spatial aliasing can be solved by denser and denser sampling. However, the more one samples, the longer it will take to collect all x-coordinate data, and thus the more likely the physiology may change in a real experiment. That is to say that the more one samples, the less stationary the signal becomes over the sampling period. Stationarity is only a viable assumption if sampling occurs "quickly", where "quick" is with respect to the recovery time scales of mechanisms within the cochlea. For example, in recent experiments where we vary endocochlear potential, "quickly" means less than one minute or so.
These warring constraints will depend on the physical mechanism being observed - in a healthy or dead cochlea, the signal is entirely stationary, and sampling can take as long as necessary. However, they all boil down to the intelligent choice of a lattice. It will only be through attempting two-dimensional data acquisition in vivo that we can learn whether or not this method is viable, but it is certainly an exciting prospect.
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