The application of suprathreshold stochastic

Fluctuation and Noise Letters Vol. 0, No. 0 (2001) 000–000 c World Scientific Publishing Company °

The application of suprathreshold stochastic resonance to cochlear implant coding

N. G. Stocks and D. Allingham School of Engineering, University of Warwick Coventry CV4 7AL, UK R. P. Morse MacKay Institute of Communication and Neuroscience, University of Keele Keele, ST5 5BG, UK Received (received date) Revised (revised date) Accepted (accepted date) In this paper we explore the possibility of using a recently discovered form of stochastic resonance - termed suprathreshold stochastic resonance - to improve speech comprehension in patients fitted with cochlear implants. A leaky-integrate-and-fire (LIF) neurone is used to model cochlear nerve activity when subject to electrical stimulation. This model, in principle, captures key aspects of temporal coding in analogue cochlear implants. Estimates for the information transmitted in a population of nerve fibres is obtained as a function of internal (neuronal) noise level. We conclude that SSR does indeed provide a possible mechanism by which information transmission along the cochlear nerve can be improved - and thus may well lead to improved speech comprehension. Keywords: suprathreshold stochastic resonance, auditory neurones, cochlear implants, information transmission, noise

1.

Introduction

In a recent series of papers [1–6] a novel form of stochastic resonance(SR) - termed suprathreshold stochastic resonance(SSR)- was introduced and discussed. Similar to ‘conventional’ SR [7], the SSR effect gives rise to a nonmonotonic dependence of the transinformation (transmitted information) on noise intensity - thus, maximum transinformation occurs at a non-zero level of noise intensity. However, SSR has an important difference, unlike conventional SR it can occur for signal strengths that are above threshold. Indeed, the SSR effect is optimised precisely when the signals are suprathreshold [1,2]. It was for this reason the effect was termed suprathreshold SR [1]. The fact that SSR occurs for suprathreshold signal strengths gives it potential advantages of conventional SR; SSR gives rise to much larger noise-induced

The application of suprathreshold stochastic resonance to chclear implant coding

information gains than can be obtained with conventional SR and it does not rely on the signal being weak. SSR occurs in parallel arrays of threshold elements (it cannot occur in a single element) like that shown in Figure(1). Each unit in the array is subject to the common input signal s(t) but evolves under the influence of its own internal (independent) noise source. The SSR effects was first studied in an array comprising of simple threshold devices (comparators) [1, 2, 6] but has more recently been applied to neural populations [3–5] where each unit in the array was modelled as a FitzHugh-Nagumo neurone [8]. It has been demonstrated that global noise-induced information gains can be obtained regardless of the input signal magnitude or the value of the common threshold. In short, in a population of similar neurones, the information can always be maximized by an optimal level of internal noise [3]. It is the possible exploitation of this idea that we explore in the context of cochlear implants.

U1 U2 U3

Σ

Y’(t)

Y(t)

. . .

S(t)

UN Fig 1. Parallel array of N identical ‘units’denoted U1 to UN . Each unit has a common input signal s(t) but evolves under the influence its own internal noise source. The noise intensities of each unit is assumed to be the same. The response y(t) is obtained by summing the units’ reponses and low-pass filtering.

In many profoundly deaf people the hair cells of the inner ear that transduce mechanical sound vibration into electrical signals are damaged or missing [10,11]. A Cochlear implant can often be surgically implanted into the cochlea to replace the function of the hair cells [12–14] and partially restore hearing. Cochlear implants work by direct electrical stimulation of the cochlearnerve fibres. The purpose of the implant is to evoke, with electrical stimulation, a similar pattern of nerve activity to that expected in a healthy ear when subject to acoustic stimulation. A schematic diagram of a cochlear implant is shown in Figure(2). They consist of a number of electrodes (typically about twenty) that each stimulate a large group of nerve fibres. In principle, it is possible to excite the whole population with a common signal just as in Figure(1). However, in practice, this is not done. It is usual to first pass the signal through a bank of passband filters (channels) - each channel having a different centre frequency. The output from each channel is then connected to just one electrode. This processes is undertaken to reflect the different tuning

Stocks, Allingham and Morse

characteristics of the hair cells and hence mimic the effect of ‘place coding’ in the ear. However, the array in Figure(1) can be used to model the cochlear implant in two possible ways. First, each electrode stimulates a (sub)population of nerve fibres with the same signal. In this case the parallel array models information propagation in this subpopulation. Second, in principle, it is possible to connect more than one electrode to each filter channel - thus again stimulating a population with the same signal. Thus, in effect, we are considering information transmission in only a subpopulation of the total number of nerve fibres that the cochlear implant stimulates. We make the initial simplifying assumption that all neurones are identical.

Cochlear nerve

Apex

Cochlea

Base

Preprocessing

Low frequency passband

Postprocessing

Mid frequency passband

Postprocessing

High frequency passband

Postprocessing

Microphone

Reference electrode (outside cochlea)

Fig 2. Schematic diagram of a cochlear implant.

To understand how noise may be used to improve speech comprehension in cochlear implants it is important to first understand what gives rise to the SSR effect. The SSR effect, over the range of noise intensities that displays an increasing transinformation, can be understood as follows; if there is no internal noise, and each unit is identical, then the response of all the units will also be identical (because they each receive the same input signal). The implication of this in informationtheoretic terms is (rather obviously) that each unit will carry identical information about the signal and, hence, the total information transmitted by the array will be simply equal to the information transmitted by any one unit. However, this is not the case if internal noise is taken into account. The presence of the noise results in a desynchronization [2] of the individual responses that can lead to an increase in the output entropy1 of the array. The increase in output entropy occurs because 1 The output entropy is simply the total information (of signal and noise) and should not be confused with the transinformation which is the information at the output about the signal alone.


the noise introduces a degree of independence in the information transmitted by individual units by randomising slightly the firing times of the units. This increase in output entropy gives rise to a net increase in the transinformation (because the output entropy rises faster than the information lost due to noise). These information gains are only observed over a limited range of noise intensity. At sufficiently larger noise the situation reverses and the information loss due to noise increases faster than the output entropy - this leads to a reduction in the transinformation. It is important to note that the net global gain in information [3] is obtained by summing information over all units - noise lowers the amount of information carried by each individual unit (assumming the signal is suprathreshold). Indeed, the SSR effect has other non-intuitive features. For example, although the transinformation goes through a maximum with increasing noise, no such maximum is observed in the output signal-to-noise ratio(SNR). The SNR decreases monotonically with noise [2]. This is another major difference in the phenomenology of SSR effect compared to conventional SR. This difference has implications for signal detection - it means that SSR cannot be used to improve detection in a hypothesis testing sense (i.e. whether a signal present) but it can be used to improve the fidelity of the transmitted signal. It is the latter of these tasks that is important for cochlear implants. The idea that noise may be useful as a means of desynchronizing neural responses across a population has already begun to receive attention in the context of cochlear implant coding. This is because it is known that conventional cochlear implants produce temporal cochlear nerve activity that differs significantly from that observed in a healthy ear. The main difference is that the cochlear implant produces abnormally high levels of across-fiber synchrony. Work by Morse and Evans [15] demonstrated that, because of high syncronization, the temporal pattern of nerve discharges evoked by conventional analogue cochlear implants would not be expected to convey an essential cue for both vowel identification and the identification of some consonants. This abnormally high level of cross-fiber synchronization has also been noted in other studies using periodic stimuli [16–20]. It has been postulated that the abnormally high level of synchronization is due to lower than normal neuronal noise level. In a healthy ear the hair cells appear to be a major generator of noise - the noise originates from Brownian motion and from synaptic connections to the cochlear nerve fibres. Therefore, in cochlear implant patients, the absence of hair cells is likely to lead to a lower than normal stochastic nerve activity. This postulate is backed up by studies that have demonstrated that the healthy mammalian cochlea has a significant amount of spontaneous activity (neural firing) in quiet [21] that is largely not observed if the hair cells and presynaptic connections are damaged or absent [9, 22]. This has led to the suggestion that noise should be re-injected back into the deafened ear to reduce the across-fibre synchrony. [15, 20, 23]. Indeed, the addition of noise has been shown to enhance the coding of vowel formant cues [15] reduce across-fibre synchrony [20] and lead to improved sensitivity to envelope modulation [23]. However, it has not yet been established that internal noise can actually lead to enhanced information transmission. It would appear that SSR offers a mechanism by which this can be achieved. Indeed, cochlear implants are normally operated at suprathreshold stimulus intensities - a parameter range that precludes the occurrence of conventional


SR [1, 2] but not SSR. The key finding in this paper is that SSR does appear to offer a means of improving information transmission in cochlear implants. 2.

Model of electrically stimulated auditory neurones

We use a leaky-integrate-and-fire (LIF) [8] model to capture the behaviour of cochlear nerve fibres subject to elecrical stimulation. The LIF model has been developed directly from physiological studies of the electrically stimulated sciatic nerve of the toad Xenopus laevis. The use of the toad sciatic nerve as a physiological model for the mammalian cochlear nerve will be addressed in detail elsewhere [24] as will the detailed comparison between the LIF model and the sciatic nerve data [25]. Here we simply present the model and show some representative comparisons between the LIF model and physiological data. Each unit in the array was modelled using the LIF neurone τm x˙i = −xi + s(t) +

√

2Dξi (t)

(1)

hξi (t)ξj (t0 )i = δij δ(t − t0 ); hξi i = 0 where xi represents the membrane voltage of the ith neurone, τm is the membrane time constant, D is the common noise intensity, s(t) is the input signal, δij is the Kronecker delta and i = 1...N , where N is the total number of neurones in the array. Each neurone evolves under the influence of its own independent noise source ξi (t). In the physiological experiments this noise term originates from the fluctuations in the opening and closing of the ion channels (membrane noise) but in this study we treat it as an adjustable parameter. The membrane voltage is allowed to evolve according to (1) until it reaches the common threshold value θ, at which point a spike (modelled as a delta function) is deemed to have occurred and the membrane voltage is reset to zero. An absolute refractory period, τabs , is then imposed during which time any further threshold crossings are ignored. After a duration τabs the neurone then enters a relative refractory period, τrel , where the threshold is increased. This behviour is modelled by multiplying the signal and noise in (1) by the factor (1 − exp(−t/τrel )). The spike trains from all units are then summed to give y 0 (t) and the final response y(t) is obtained from the convolution Z ∞ y(t) = y 0 (s)f (t − s)ds (2) −∞

where, ½ f (t) =

A(1 − exp(−t/τm )) if t < Tp , A(1 − exp(−Tp /τm )) exp(−(t − Tp )/τm ) if t ≥ Tp .

(3)

Convolving with the function f (t) is equivalent to assuming the original action potential has a rectangular pulse of width Tp and amplitude A that is then passed through a single-pole low-pass filter with a 3bB cut-off frequency of ω = 1/τm . In physical terms, the total response y(t) can be viewed as the membrane voltage of a summing neurone (with membrane time constant τm ) to which all the other


neurones are connected (although in this picture synaptic filtering effects have been neglected). The filter output provides an estimate of the input signal - it removes the high frequencies that are largely dominated by noise. The value of Tp = 2ms was chosen to maximise the transinformation. To demonstrate that (1) captures the important dynamics of electrically stimulated nerve fibres, we show a comparison between the results obtained from the LIF model and results from physiological experiment. Figure (3) shows results for the discharge rates (spikes per second) of a single fibre (from the sciatic nerve of the toad) when subject to a sinusoidal current for three different frequencies of the driving current. The results are plotted against α - the stimulus-amplituderelative-to-threshold ratio expressed in decibels; 0dB corresponds to the case when the signal is just sufficient (in the absence of noise) to cause the neurone to spike. The parameters τm , τabs , τrel were adjusted to give the best fit to the data by eye. The value of D was obtained from measurements of the relative spread (RS) [26,28]. Excellent agreement between the model and experiments is observed for stimulus frequencies of 50Hz and 100Hz, although some discrepancy is seen at 200Hz. However, even at 200Hz the model captures the important qualitative features such as the small step at a discharge rate of approximately 100Hz.

Discharge rate (Hz)

200

150

data 50 Hz model 50 Hz data 100 Hz model 100 Hz data 200 Hz model 200 Hz

100

50

0 -10

-5

0

5

10

15

20

25

30

α (dB)

Fig 3. Discharge rates vs signal-to-threshold ratio in decibels. Three sets of curves are shown, representing the response to a sinusoidal signal with a frequency of 50 Hz, 100 Hz or 200 Hz. The results from the LIF model were obtained using the parameter values τ = 4.8ms, τabs = 3.3ms, τrel = 5.5ms and D = 2.5 × 10−6 . All these values are consistent within the know physiological values [24]. The experimental data was obtained from the same neurone.

The membrane noise level is estimated in the experiment by measuring the RS. This is measured by repeatedly applying a single pulse of fixed width to obtain the probability of firing. This process is repeated for various pulse heights and the results plotted as in Figure(4). An integrated Gaussian (error function) with mean µf it and standard deviation σf it is then fitted to the data. The relative spread is then defined to be [26, 28], RS =

σf it µf it

(4)


The noise intensity in the model (1) can then be adjusted to give the same RS value. A family of curves is produced by varying the noise intensity D, this enables a table (table 1) to be constructed that can be used to map the experimentally measured values of RS into an equivalent noise intensity. 1

Probability of firing

0.8 0.6 Noise intensity 1e-6 1.5e-6 2e-6 5e-6 1e-5 2e-5 data

0.4 0.2 0 -1.5

-1

-0.5

0

0.5

1

1.5

α (dB)

Fig 4. Probability of firing vs the signal-to-threshold ratio in decibels (with the mean value µf it subtracted from the abscissa value), for various noise intensities. The value of 1.5 × 10−6 matches the relative spread as calculated from experimental data. Parameters for fitting an error function to each of these curves are shown in Table 1. A pulse width of 500µs was used.

Table 1. Relative spread and parameters of fit for error function, for the discharge rate curves in Figure 3.

Noise intensity 5e-4 2e-4 1e-4 5e-5 2e-5 1e-5 5e-6 2e-6 1.5e-6 1e-6

3.

µf it 0.875 0.963 0.976 0.988 0.997 1.001 1.002 1.003 1.003 1.003

σf it 1.6387 0.6045 0.3189 0.1930 0.1074 0.0721 0.0498 0.0310 0.0268 0.0218

Relative spread 1.873 0.6277 0.3267 0.1953 0.1077 0.0720 0.0497 0.0309 0.0267 0.0217

The transinformation

The global information transmission through the array is characterised by the transinformation I, which, for a continuous channel, is defined as [27], Z I

=

∞

H(y) − H(y|s) = − −∞

Py (y) log2 Py (y)dy


−

µ Z −

Z

∞

¶

∞

Ps (s)ds

−∞

−∞

P (y|s) log2 P (y|s)dy

(5)

where H(y) is the information content (or entropy) of y(t) and H(y|s) can be interpreted as the amount of encoded information lost in the transmission of the signal. Py (y) and Ps (s) are the probability density functions (pdfs) of y(t) and s(t) respectively and P (y|s) is the conditional pdf. All the relevant distributions can easily be obtained from simulation of model(1)-(3). In the absence of noise H(y|s) = 0 and hence I = H(y) and for sufficiently large noise H(y|s) = H(y) and hence I = 0. Therefore, I can only increase in the presence of noise if H(y) increases faster (with increasing noise) than H(y|s). This is precisely what happens in SSR [2].

(a) 2.5

I (bits)

2

1 neurone 4 neurones 8 neurones 12 neurones 16 neurones

1.5 1 0.5 0 -50

-40

-30

-20

-10 σ (dB)

0

10

20

(b) 2.5

I (bits)

2 1.5 1 0.5

1 neurone 4 neurones 8 neurones 12 neurones 16 neurones

0 -70

-60

-50

-40

-30

-20

-10

0

10

20

σ (dB)

Fig 5. Average mutual information vs noise intensity for arrays of neurones of various sizes, for a signal strength (a) 8 and (b) 30 decibels above threshold. The signal strengths were chosen to coincide with those typically employed in cochlear implants. The arrow indicates the approximate average value of internal noise in the experimental data.

4.

Results

All results presented in the section were obtained using an input signal that was Gaussianly distributed and exponentially correlated with a correlation time of 0.2s. The 0dB signal strength relative to threshold was defined as θ = 2σs where σs is the signal strength measured at the output of one of the neurones i.e. it is the standard


(a) -4 dB +0 dB +4 dB +8 dB +12 dB +16 dB +20 dB +24 dB +28 dB

2.5

I (bits)

2 1.5 1 0.5 0 1e-08

1e-06

1e-04

1e-02 D

1e-00

1e+02

(b) -4 dB +0 dB +4 dB +8 dB +12 dB +16 dB +20 dB +24 dB +28 dB

2.5

I (bits)

2 1.5 1 0.5 0 -50

-40

-30

-20

-10

0

10

20

30

40

50

σ (dB)

Fig 6. Average mutual information vs noise intensity for an array of 16 neurones for a variety of signal strengths (shown in the legend). The arrow indicates the approximate average value of internal noise in the experimental data. In (b) the results have been replotted against scaled noise intensity σ.

deviation of the membrane voltage xi in the absence of noise. With this definition, at 0dB, the system (in the absence of noise) spends 99% of its time below threshold and is, thus, predominantly subthreshold. In general the signal intensity in decibels, SdB , is found from the formula, SdB = 20 log

2σs θ

In Figure(5)the transinformation against scaled noise intensity, σ, is plotted for two different signal strengths and various N . The scaled noise intensity is defined as σ = σn2 /σs2 where, σn2 = D/τm , is the variance of the membrane voltage xi measured in the absence of the signal (i.e. variance due to noise). Consequently, σ is equal to the inverse of the SNR measured relative to the output. The results presented were obtained with θ = 1. Clearly, all curves in Figure(5) (except N = 1) show a non-monotonic dependence on noise intensity and thus display an SR type effect. The height of the maximum is seen to increase as the number of neurones in the arrays is increased. Note that the optimal noise intensity (position of maximum) is located in the range −10dB < σ < 0dB - this implies that the information is maximised when the


variance of the noise is of the same order as the variance of the signal. All these observations are consistent with the SSR effect. The arrow on the abscissa indicates the estimated value of the noise intensity based on the physiological experiments. Clearly this lies well below (by more than two orders of magnitude) the optimal noise intensity and, therefore, suggests that improved information transmission in electrically stimulated neural populations could be achieved by increasing the level of independent noise at each neurone. In Figure(6) the transinformation is plotted for a number of different signal strengths. The signal strengths were chosen to represent the approximate values used in cochlear implants. Considering Figure(6a) first, it is observed that the height of the maximum increases with signal strength i.e. the more suprathreshold the signal the greater the information transfer. It is also evident that the optimal noise intensity increases with increasing signal strength. Again these observation are consistent with the SSR effect [1–4]. The data in Figure(6b) is identical to that shown in Figure(6a) except it has been plotted against the scaled noise intensity σ. This rescaling again demonstrates that the optimal noise intensity occurs when the signal and noise intensities are of similar size - i.e. when the output SNR is in the range 0 − 10dB (recall σ=1/SNR.) 5.

Discussion and Conclusions

The results shown in the previous section clearly indicate that SSR can be observed in a model population of electrically stimulated cochlear nerves. This gives hope that SSR could be exploited in cochlear implants to improve speech comprehension. It was also observed that the noise intensity estimated from the physiological studies [24] fell two to three orders of magnitude below that required to optimise information transmission via SSR. Although the level of noise observed in the cochlear nerve is twice that observed in the sciatic nerve [24] this level of noise is still well below the optimal noise intensity. This suggests that deafened cochlear nerve fibres are sufficiently noise free that the addition of independent noise to cochlear implant electrodes could lead to improved information transmission along the auditory nerve. However, this statement needs additional qualification. First, our model assumes that all neurones are identical - clearly this is not the case in reality. Nevertheless, given that there are approximately 30,000 cochlear nerve fibres in the human ear [29], small subpopulations of fibres with similar characteristics would be expected. Hence, our results may well be applicable to these subpopulations. A second, related, point is it that our model does not bandpass the signal before application to the neurones and therefore fails to capture the tuning characteristics (place-coding effects) of the auditory system. However, this fact opens up some interesting questions. Current cochlear implants filter the signal through a bank of bandpass filters before applying the filtered signal to the electrodes of the cochlear implant (see Figure 2). Each electrode receives a filtered signal with a different centre frequency. This is to mimic the situation where each neurone is connected to a hair cell tuned to a different frequency. Our results suggest that this may not be the most efficient design strategy. Clearly, benefit can be obtained by connecting the same filtered signal (i.e. with the same centre frequency) to different electrodes.


Providing the electrodes stimulate neurones with similar properties then SSR could be employed to enhance information transmission. The question as to how filterbanks should be split between different electrodes would be something that needs further consideration. Finally, we note that the SSR effects we observe are not small. At a signal intensity of 8dB the transinformation improved by 1-bit when the noise was increased from the physiologically estimated value to the optimal value. An increase of 1-bit is equivalent to a doubling of the accuracy to which the transmitted signal is known. The results also suggest that the effect does not need large populations for it to be useful - even a population of four neurones can display a 40% gain in information. Therefore, the implementation of such a coding scheme is well within current cochlear implant technology. Acknowledgements This work was in part funded by the MRC (grant G0001114) and the EPSRC (grant GR/35650/01). References [1] N G Stocks, Suprathreshold stochastic resonance in multilevel threshold systems, Phys. Rev. Lett. 84 (2000) 2310-2314. [2] N G Stocks, Information transmission in parallel arrays of threshold elements: suprathreshold stochastic resonance, Phys. Rev. E 63 art. num. 041114 (2001) 1-11. [3] N G Stocks and R Mannella, Generic noise-enhanced coding in neuronal arrays, Phys. Rev. E 64 art. num. 030902 (2001) 1-4. [4] N G Stocks and R Mannella, Suprathreshold stochastic resonance in a neuronal network model: A possible strategy for sensory coding, in Future Directions for Intelligent systems and Information Sciences, ed. N. Kosabov (Physica-Verlag, 2000) 236-246. [5] N G Stocks, Optimising information transmission in model neuronal ensembles, in in Stochastic Processes in Physics, Chemistry and Biology, ed. Jan A. Freund and Thorsten Poschel, Lecture Notes in Physics, LNP 557, (Springer-Verlag, Berlin Heidelberg, 2000) 150-159. [6] N G Stocks, Suprathreshold stochastic resonance: an exact result for uniformly distributed signal and noise, Phys. Lett. A 279(2001) 308-312. [7] L. Gammaitoni, P. Hanggi, P. Jung and F. Marchesoni, Stochastic Resonance, Rev. of Mod. Phys. 70,(1998)223-287; Proceedings of the International Workshop on Fluctuations in Physics and Biology: Stochastic Resonance, Signal Processing and Related Phenomena, ed. Mannella, R. et al, Nuovo Cimento Soc. Ital. Fis. 17D, (1995). [8] H. C. Tuckwell, Introduction to Theoretical Neurobiology: Volume 2, Nonlinear and stochastic theories, (CUP, Cambridge 1988) [9] N. Y. S. Kiang, E C Moxon and R. A. Levine, Auditory nerve activity in cats with normal and abnormal cochleae, in Sensorineural Hearing Loss, eds. G. E W. Wolstenholme and J. Knight, (Churchill, London, 1970) 241-273. [10] R. Hinjosa and J. R. Lindsay, Profound deafness: associated sensory and neural dgeneration, Archives of Otolaryngology 106 (1980) 193-209. [11] J. B. Nadol, Jr., Histopathology of human aminoglycoside ototoxicity, in Aminoglycoside ototoxicity eds. S. A. Lerner, G. J. Matz and J. E. Hawkins (Little and Brown and Co., Boston, 1981) 409-434.


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