On the precision of neural computation with interaural time differences in the medial superior olive

Petr Marsalek^{a,b,e}, (Marek Hajny)^{a,e}, Zbynek Bures^{c,d}

aInstitute of Pathological Physiology, First Medical Faculty, Charles University in Prague, U Nemocnice 5/478,

128 53, Praha 2, Czech Republic

bDepartment of Radioelectronics, Faculty of Electrical Engineering, Czech Technical University in Prague,

Technická 2/1902, 16627, Praha 6, Czech Republic cCzech Institute of Informatics, Robotics and Cybernetics, Czech Technical University in Prague, Jugoslávských partyzánů 4/1903, 166 36, Praha 6, Czech Republic dCollege of Polytechnics, Tolstého 16/1556, 586 01, Jihlava, Czech Republic eCorresponding authors: Petr.Marsalek@LF1.CUni.CZ, Marek.Hajny@LF1.CUni.CZ

Keywords: binaural hearing, cochlear implant, coincidence detection, ergodic hypothesis, ideal observer, interaural time difference, just noticeable difference, lateral and medial superior olive, neuronal arithmetic, psychophysics, sound localization.

Abstract

Abstract will be rewritten after the manuscript completion. Paper should explain anylytical MSO model, all figures and tables. We study a model of mammalian sound source localization in horizontal plane. Experiments on small rodents indicate that mammals use broadly tuned channels of azimuth for localization in horizontal plane. In mammals this neural computation is implemented by medial superior olive for low frequency sounds up to 1500 Hz. It has been shown previously that spike timing jitter, coincidence detection window length, sound frequency, among other input parameters, influence the output precision, measured by the just noticeable difference in output of the circuit. We use stochastic model with spiking neurons. We explore variations of the parameters mentioned above. We calculate properties of the model with the use of analytical methods. Predictions of this model have straightforward applications in testing and designing stimulation protocols used in cochlear implants.

Abbreviations and symbols

F_{S},sound frequency; F_{C}, critical sound frequency value; φ, sound phase; ILD, interaural level difference; ITD, interaural time difference; ISI, inter-spike interval; JND, just noticeable difference; LSO, lateral superior olive; K_{C}, K_{S}, A, B, C, proportionality constants; l, sound level; MSO, medial superior olive; r(V), r_{V}, VS, vector strength; R_{F}, firing rate; σ, standard deviation; T_{J}, timing jitter; t, time; T, T_{S}, sound period;

Preprint submitted to Brain Research April 18, 2019

1. Introduction

Mammalian sound localization circuits contain two nuclei in the auditory brainstem, the medial and the lateral superior olive (MSO and LSO). Neurons in these nuclei are the first binaural neurons in the auditory pathway, they are connected to both ears.

This article presents description of information encoding and neural computation in the MSO obtained mostly with analytical computations. Using the analytical tools we extend quantitative results obtained by Sanda and Marsalek (2012) only in simulations and connect them to Bures and Marsalek (2013) to arrive to unified description of neural circuits in the superior olive. We use this description to find maximum spike timing precision attained by the following two respective modalities (low and high frequency sound localization) of natural and electrical hearing.

Due to the physical nature of the binaural sound, the MSO neurons process spike timing differences in the range of tens of microseconds, μs. The MSO processes low frequency sounds, in human this is from 20 Hz to not more than 2 kHz. The LSO processes high frequency sounds, in human this is from 1 kHz up to 20 kHz. The overlapping region is known to have a sensitivity drop at 1.5 kHz Mills (1958).

Firing rate, first spike latency and individual spike timings are used in neural system coding, especially in the auditory pathway. The highest known spike timing precision that has been proven to be utilized in the mammalian auditory system is the motor response in a bat. In behavioral experiment, the big brown bat, Eptesicus fuscus catches tidbit larvae of the meal-worm beetle Tenebrio molitor. The motor precision is in the range of hundreds of nanoseconds, ns, and auditory separation of ultrasound echoes is in the order of microsecond units, 1 μs Simmons et al. (1998). Comparable performance of the human MSO is in the range of microseconds, 10 μs and can be improved after several hours of training.

Different neural mechanisms are employed in the two nuclei.

It has been reported that computation in the MSO is independent on sound intensity Grothe et al. (2010). With higher sound intensity, first spike latency is shortening. Relation of this dependence to ITD and ILD has been described, yet it is difficult to interpret. Michelet et al. (2012) have shown the latencies in experiment on the domestic cat, Felis silvestris.

The LSO and MSO extract location information with the use of different physical cues. The sensitivity of the system in dependence on the main sound frequency to sounds of different main frequencies should be different. In human, MSO is the larger of the two nuclei and contains approximately 15500 neurons, human LSO contains 5600 neurons.

We have previously studied parameter free models of human sound localization. Current development of binaural cochlear implant technology enables to stimulate auditory periphery with rich repertoire of patterns. Better understanding to the MSO mechanisms will help to improve electrical hearing.

Moving sound objects are localized using the ITD and ILD cues and in addition to this, Doppler effect pitch changes signal approaching and retreating objects, even though Doppler cues are less salient. Due to these phenomena, neural circuits are robust to pitch variation. It is therefore interesting to pose a question, whether wider pitch changes in a thought experiment in rescaling time will affect these neural mechanisms. Speeding up, or slowing down the system time can be extended to two octaves up and down the regular time, see Table 3.

The range of sound frequencies processed in the MSO circuit is limited to the low frequency band. In human, where audible sound frequencies range from 20 Hz to 20 kHz, this low frequency band spans from 20 Hz to 1.5 kHz. Moreover, in the marginal values of this interval, performance of sound azimuth discrimination is worse. It is generally agreed that the main reason, why the workings of the MSO circuit deteriorate towards higher frequency is lowering of the synchronization of spike trains in the circuit with the sound source phase. The synchronization between two corresponding series of point events can be expressed as a (discrete version of) vector strength, defined below in equation (1).

2. Methods 2.1. Preliminaries

Our procedures are mostly in using computational models, experimental data and statistical methods. Here we list several assumptions used in the paper.

(1) Single MSO neuron constants, see Toth and Marsalek (2015). (2) Synaptic machinery, see Toth and Marsalek (2015). (3) Ergodic assumption see Sanda and Marsalek (2012), Toth et al. (2018).

2.2. Vector strength

The vector strength is defined as it is used since Goldberg and Brown (1969). Let us have sample (spike) phases φ_{i}, i = 1, 2, …, N relative to phases of a given input master periodic function, which does not enter the formula. (Such function can be for example sound stimulus eliciting the spike train as a response.) (Discrete sum) vector strength of sample φ_{1}, …, φ_{N} is defined as

r_{V}(φ_{i}) = \frac{1}{N} \sqrt{(∑_{i=1}^{N} cos φ_{i})^{2} + (∑_{i=1}^{N} sin φ_{i})^{2}}.  (1)

2.3. Just noticeable differences of ITD

This subsection of Methods should be analogous to the corresponding section in Bures and Marsalek (2013)...

2.4. Model of the MSO neural circuit

To do: Describe model and calculations.

3. Results

We investigate how circuit output and overall performance depend on sound frequency and sound intensity. Figure 1 shows an example of typical vector strength lowering towards higher frequencies, as observed in the auditory nerve (AN). The prevailing majority of neurons in the auditory pathway has a vector strength spike train statistics sigmoidally dropping towards higher sound frequencies as it is in this example. In this figure, data recorded by Joris (1996) from the MSO in cat were fitted to the sigmoidal curve with the general formula of the Boltzmann function used in Marsalek and Lansky (2005). The curve fit of vector strength r_{V} in dependence on sound frequency F_{S} is:

r_{V} = (1 + exp(K_{S}F_{S} − K_{C}F_{C}))^{−1},  (2)

where K_{S} (sound coefficient) and K_{C} (critical coefficient), F_{C} (critical half frequency) are proportionality constants.

Figure 2 shows the best just noticeable difference (JND) obtained with the basic parameter set in dependence on the sound frequency. The quadratic curve fit of the JND denoted Δt_{JND} is:

Δt_{JND} = A(f_{S} − F_{C})^{2} + B,  (3)

where A and B are fitted constants and the other parameters and variables are as above. See also Zwislocki and Feldman (1956).

Figure 3 shows how ITD JND depends on timing jitter τ_{J}. There is one critical τ_{J} = 0.2 and one optimal value (normalized to τ_{J} = 1) of timing jitter related to the output JND and firing rate. Simulations show that with lowering timing jitter the circuit output is virtually more and more precise. The two curves fitted to the simulation are:

1. fit of exponential function to simulations:

Δt_{JND} = exp(A_{1}(τ_{J} − B_{1})) − C_{1},  (4)

where A_{1} = 1.9, B_{1} = 1.25 and C_{1} = 0.2 are fitted parameters. This is shown conveniently by the logarithmic y-axis in this figure.

2. Another fit, which also takes into account shot noise in lower jitter values, is to a quadratic function:

Δt_{JND} = A_{2}(τ_{J} − B_{2})^{2} + C_{2},  (5)

where A_{2} = 2.5, B_{2} = 1, C_{2} = 1. There is only one parameter (A) fitted to data, as the point (x, y) = (B_{2}, C_{2}) has been chosen to be a unit, this fit is a normalized fit of the model.

Logarithm of the simulated JND lowers with the exponential curve (4), which is concave function of jitter τ_{J} as the jitter gets lower. The trend towards higher accuracy diverges from the parabolic fit (5), when jitter reaches critical value between 0.2 ⪅ τ_{J} ⪅ 0.5 ms. Beyond that point towards the lower jitter values, the neural circuit cannot function properly, as too low jitter prevents the interaction of spikes from the left and right side within the coincidence detection mechanism.

Figure 4 shows the calibration curve. The rising slope of this curve is used as a readout function yielding th firing rate in dependency on the ITD which in turn signals the sound azimuth to the next nuclei up to the auditory pathway.

Comment

TO-DO - This Table X should be extended: more parameters and range of parameters...

4. Discussion 4.1. Things to do

First: Reproduce the curve in panel with sound frequency on x-axis and JND on the y-axis in Fig. 2.

Second: Present the curve from the right panel of Fig.4 from Sanda P., Jitter Effect on the Performance..., as part of Fig. 3.

Some other tentative questions: What is the highest slope of the ITD interpolation curve, such that it gives the resolution of 4^{∘} (angular degrees)?

(This we will do later: We also discuss the relation of the MSO neural circuit to its neighboring nucleus, the LSO, and its output.)

Cite: Mlynarski and Jost (2014)... Cite: Bures and Marsalek (2019)...

Notes

To Do: Add citations to discussion: Cite Franken et al. (2014), this is an experimental description of MSO workings

in line with our concepts and coincidence detection theories.

Cite Silver (2010), because this is a prospect of neuronal arithmetic. To do: check, whether both Smith et al. (1991) and Smith et al. (1993) are

relevant.

The ergodic paper of Toth et al. (2018) goes to Discussion. Another MSO model, already in 2005 is: Zhou et al. (2005). Cochlear implants and MSO: Kerber and Seeber (2012). Check Wu and Kelly (1992): WHY DO WE CITE THIS? Here are the three LSO experimental papers, which have detailed methodology

applicable to overlap of sound frequency ranges with the MSO Joris and Yin (1995); Joris (1996); Joris and Yin (1998).

Jeffress et al. (1962) goes instead of Jeffress, 1948. (No)Cite: Bures (2012) is neuronal arithmetic in Biological Cybernetics (No)Cite: Bures and Marsalek (2013) is “dual” paper in Brain Research Do we still need to cite this review: Grothe et al. (2010)?... Thee are the classics...: Goldberg and Brown (1969). Cochlear delays, very important: Joris et al. (2006). Yet another neuronal arithmetic: Koutsou et al. (2012). This is about other dimension; if we cite this, then we must discuss all the

three, Kopco et al. (2012). hall we do that?

Jitter and variability... : Kostal and Marsalek (2010). This is a chronological record of the neuronal arithmetic ideas, starting from:

Marsalek (2000), through Marsalek (2001) some more detailed account o the diversity of cell functions and shapes: Reed et al. (2002), through Marsalek and Kofranek (2004) to the Marsalek and Kofranek (2005); watch the years ...

Remains to be cited: no-cites of Smith et al. (1993), where we have by mistake fitted a curve without a-priory entering the known sound frequency.

Next Cariani (2011), Bouse et al. (2019), Bures (2014), Michelet et al. (2012), Srinivasan et al. (2018).

Other insider comments: Figure 4 should correct the omission in the paper by Sanda and Marsalek. When we cover everything else, we should add this “weeping of Jeremiah”: we urgently need binaural data/ psychophysics/ or electrophysiology, all the same.

Conclusions

This theoretical paper is a followup of sound localization precision descriptions in the MSO Sanda and Marsalek (2012) and in the LSO Bures and Marsalek (2013). Novel results are two: 1) analytical estimates of results obtained previously only by numerical simulation and 2) comparison of natural and electrical hearing modalities and demonstrating what is the maximum timing precision which can be attained by the two modalities.

Acknowledgments

L^{A}TEX compilation date is: April 18, 2019. TeX source size/ word count: > 875 lines 4100 (+key)words 29,500 Bytes. PM and MH were supported by: Charles University graduate students research program (acronym SVV) No. 260371/ 2017-2019. Special thanks to Pavel Sanda for permission to use his simulations.

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Figure 1: Vector strength of auditory nerve spike trains in dependence on sound frequency. X-axis shows fundamental sound frequency in Hz in a logarithmic scale and y-axis shows the vector strength. Even though is some nuclei up the auditory pathway the synchronization can be maintained towards higher frequencies than shown here, the decrease of the vector strength towards higher frequencies is a general property of all neurons in the auditory pathway.

Figure 2: The shortest ITD detected in the dependence on sound frequency. X-axis shows sound frequency in Hz in a logarithmic scale and y-axis the shortest JND in μs. This is a theoretical prediction based on the analytical model and basic parameter set used in previous simulations.

Figure 3: The JND values in basic parameter set in dependence on the jitter magnitude. This plot in semi-logarithmic y-scale shows just noticeable difference of interaural time difference depending on variation of the spike timing jitter. Jagged line: simulated data, solid line: an exponential fit to the simulations under the assumption of arbitrary time precision in the model circuit, line with circles: a quadratic function estimate of spike timing precision in a system with addition of noise.

Figure 4: Gaussian fit of the ITD to firing rate slope calibration curve. X-axis shows ITD in ms and y-axis shows corresponding firing rate in action potentials per second. Note that the curve peak is offset from the origin of coordinates at t_{ITD} = 0.

14

Parameter  Symbol Units Typical Value Ranges
Jitter  T_{J} = σ  ms  1  0.125 - 8
Window  w_{CD}  ms  0.6  0.15 - 1.5
Sound Frequency f  Hz  200  40-1600

Table 1: The basic set of parameters.

[a-top]  MSO/ low  MSO/ low  critical  LSO/ hi LSO/ hi
central frequency 0  0.250  0.500  1  2  4
range  0  0.175 - 0.350  0.350 - 700 0.7 - 1.4 1.4 - 2.8 2.8 - 5.6
N.H.  0  JND Bures Marsalek 2019  JND  JND  JND  JND
H.I.  0  cochlear model  JND  JND  JND  JND
E.H.  0 no jitter  literature  JND  JND  JND  JND
E.H. with jitter Laback  0 JND  JND  JND  JND
E.H. other  tuning  JND  J  0 ND  JND  JND

Table 2: Just noticeable differences in models of different operation regimes box a-top: mode/ frequency band [kHz]

Parameter  Symbol Units 4× Slower Values (man) 4× Faster  16× Faster
Sound frequency ranges f  Hz 17.5 - 110  70 - 440  280 - 1760 1120 - 7040 Hz
Window size  w_{CD}  μs  2400  600  150  37.5
Timing Jitter  T_{J}  ms  4  1  0.25  62.5 μs
Predicted JND  T  μs  40  10  2.5  N/ A
Predicted detection time^{∗} T  ms  2,600  650  162.5  N/ A

Table 3: TO DO add the last OVERLAPPING band 4480 - 28160 and make all a bit narrower to start at 20 Hz and end at 20000 Hz. This Table is direct scaling of parameters used in Sanda and Marsalek (2012). Make also MSO and LSO entries... Predicted values are: 1) just noticeable difference (JND) of the ITD 2) the time of observation by an ideal observer reading the information from one neuron.