Учебники / Hearing - From Sensory Processing to Perception Kollmeier 2007

The idea that is developed in this chapter provides a possible solution to this problem. We demonstrate here that the huge number of noise sources can actually reduce the complexity of the system behaviour.

The “central limit theorem” states that any sum of many independent identically distributed random variables will be distributed normally (independent of the original distribution). The according theorem for probability distributions for our argument was formulated by Fischer and Tippett (1928) as the “central limit theorem of extreme value distributions”: the extreme values of many independent identically distributed random variables will take the form of an “extreme value distribution” (EVD) (independent of the original distribution). The EVD is a valid model under the following conditions: The membrane potential is affected in a complicated manner by many random processes. The event when the membrane potential exceeds the threshold can be seen as “extreme”, when the threshold is high compared to the noisy resting potential. This can be shown by intracellular recordings (Calvin and Stevens 1967). Furthermore, the resting potential and the underlying probability density function for spike generating must be stationary. To apply the EV theorem we do not have to know or assume any (physiologically motivated) models of the underlying spike generating process.

The resulting spike generating process then takes on the form of a Markov process in which the probability of the next spike explicitly depends on its immediate predecessor but is conditionally independent of all other preceding spikes (Tuckwell 1988). The shape of the PDF of such a process is determined by the EVD.

2Methods

All physiological recordings were obtained from the cochlear nucleus of the anaesthetized guinea pig (Cavia porcellus). The procedures are described in detail in Bleeck et al. (2006).

The normalised cumulative first order interval distributions of half the number of all intervals between stimulus onset and offset was fitted by a maximum likelihood method with the generalised extreme value distribution:

where τ is the time since the last spike (ISI), a the its location, b the scale

and ζ the shape of the distribution. The shape parameter ζ governs the tail behaviour of the distribution, and the sub-families defined by ζ>0, ζ→0 and ζ <0 correspond, respectively, to the Fréchet, Gumbel and Weibull families, whose cumulative distributions are shown in Fig. 1. The quality of the

Fig. 1 The three families of extreme value distributions that constitute the generalized extreme value distribution. Note the different tail behaviour

fitting procedure was tested by performing a Kolgomorov-Smirnov test with a confidence interval of 5%. The other half of the intervals (that were not used for the fit) were used as the comparison distribution with the resulting fitting function.

3Results

In total, 1867 ISIHs from 877 units were investigated. A subset of these (323 ISIHs from 184 units) were also classified by one of the authors (IMW) using conventional criteria such as PSTH shape and regularity analysis (Young et al. 1988). The eight classification categories were: sustained chopper (CS), transient chopper (CT), wide chopper (CW), primary-like (PL), primary-like with notch (PN), onset (ON), low frequency (LF) and unusual (UN). Figure 2 shows data of six example units. The panels for each unit show a) the Post Stimulus Time Histogram (PSTH), b) the first order interval distribution (FOID) – the black line represents the GEV-fit to the data, c) the normalised

cumulative FOID and the fit, d) the hazard function PSpike=F′/(1−F) which is the probability that a spike happens after that interval, and the calculated

hazard function (calculated by the fitting parameters), and e) reconstruction of the PSTH using the hazard function. Neuronal responses to pure tones were simulated for 250 sweeps. At every point in time the probability that a spike happens is calculated as a Markov process given the values of µ, σ, ζ and the time of the last spike.

Fig. 2 PSTH (time in ms), FOID, recovery function, hazard functions (time interval in ms) and reconstructions (time in ms) for six example neurons of different types. “Chop S” unit (characterized by very regular firing) has a relatively sharp FOID. In effect its recovery function is steep. “Chop T” unit (less regular firing pattern then Chop S) has a less steep recovery function. The hazard functions is complex. “Primary like” (similar to an auditory nerve fibre) and “Primary like with Notch” (characterized by a distinct notch after the initial peak) units show slower recovery and almost linear Hazard functions. Their FOID is very broad. “Onset” unit (has a low sustained rate) have a delayed recovery function and a hazard function that is slowly rising in the beginning and the very steep. “Unusual” units have different characteristics that prevent them from being classified as one of the others. The example shown here has recovery properties that places it between Chopper and Onset

3.2The Neurons Journey Through Parameter Space

During the course of 50 ms stimulus the neurons behaviour and hence the fitting parameter changes due to adaptive processes (Fairhall et al. 2001). Not only the ISIs during the stimulus but also the latency of the first spike after stimulus onset and the firing patterns without a stimulus (spontaneous activity) can be described adequately using a GEV-fit. The number of neurons that can be fitted successfully as such are comparable to the interstimulus fit. The firing probability of a neuron during a stimulus (and also as a function of stimulus parameter like level) can therefore be traced in the parameter space spanned by the fitting parameters (data now shown).

4Discussion

Since the coding capacity of a spike train may be reflected in both the profile and the asymmetric dispersion of the ISIH, an information approach can be used to overcome the limitation of traditional second moment statistics. Recently, inter-spike intervals have formed the basis for quantifying the coding capacity of spike trains. Numerical methods have been developed based on logarithmic (Bhumbra et al. 2004) and linear (Reeke and Coop 2004) intervals. These measures suffer, however, from the absence of a generally accepted form of the distribution and must be calculated numerically for every case. It would therefore be extremely useful to have a general model to describe ISIHs as this allows direct access to the information content of spike trains.

Distinct cell types in the VCN can be associated with almost unique temporal response patterns (Rhode et al. 1983). However, we show here a method to represent neuronal responses in the VCN within overlapping parameter areas. The continuous filling of the space in the parameter map indicates that the different neuron classes have overlapping features. Therefore we argue here that it might be useful to reconsider unit classification as continuous in the descriptive parameter space rather then in distinct classes.

5Summary

The automatic algorithm allows classification of a neuron with good accuracy. Most of the errors made by the algorithm are actually between unit types that are similar by visual inspection as well. Additionally the automatic classifier is faster: Usually 40–100 intervals are enough to get a stable classification (data not shown). Analysed conventionally, many units are classified as unusual because they do not fit completely into the classic classification scheme. The

GEV fitting reveals that many of these neurons are actually highly regular in terms of their firing patterns and might well described in a continuous parameter space.

The spike probability at each time can be described by the hazard function as a function of the two main parameters of the ISI-distribution. The change of the ISI-distribution can be visualised as a journey through parameter space.

The advantage of using the extreme value distribution is that the outcome is determined by the statistical laws of big numbers independent of the actual distributions and properties of the underlying processes. It is nevertheless still possible to infer certain neuronal characteristics from the results. For instance the shape ζ is mainly constant for processes like amplitude changes and adaptation (data not shown). The location α and the scale β on the other hand are determined by the stimulus: the louder the stimulus, the smaller α, the longer its duration, the bigger β.

Acknowledgements. Supported by a grant to the second author from the BBSRC

References

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Bleeck S, Sayles M, Ingham NJ, Winter IM (2006) The time course of recovery from suppression and facilitation from single units in the mammalian cochlear nucleus. Hear Res 212:176–184 Calvin WH, Stevens CF (1967) Synaptic noise as a source of variability in the interval between

action potentials. Science 155(764):842–844

Fairhall AL, Lewen GD, Bialek W, de Ruyter Van Steveninck RR (2001) Efficiency and ambiguity in an adaptive neural code. Nature 412:787–792

Fisher RA, Tippett LHC (1928) Limiting forms of the frequency distribution of the largest or smallest member of a sample. Proc Cambridge Philos Soc 24:180–190

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relationship between unresolved harmonics sufficiently to abolish temporal responses to envelope periodicity. It is suggested that, under reverberant conditions, temporal information is limited to frequency channels containing resolved harmonics.

2Methods

Detailed methods describing the surgical approach and recording techniques can be found elsewhere (e.g. Bleeck et al. 2006) and will be briefly described below.

2.1Physiology and Recording

The data reported in this chapter were recorded from pigmented guinea pigs anaesthetised with urethane (1.0 g/kg i.p.). Supplementary analgesia was provided by Hypnorm (fentanyl/fluanisone) (1.0 ml/kg i.m.). Additional doses of urethane and Hypnorm were given when required. The surgical preparation and stimulus presentation took place in a sound attenuating chamber (IAC). All animals were tracheotomised and core temperature was maintained at 38ºC with a thermostatically controlled heating blanket. The electrocardiogram and respiration rate were monitored throughout and on signs of suppressed respiration the animal was artificially ventilated.

Recordings were made using tungsten-in-glass microelectrodes (Merrill and Ainsworth 1972). Broadband noise was used as a search stimulus. Upon isolation of a single unit, estimates of best frequency (BF) and threshold were made using audio-visual criteria. Single units were classified according to peri-stimulus time histogram shape in response to suprathreshold BF tonebursts, interspike interval distribution and discharge regularity.

2.2Complex Stimuli

The stimuli were linearly swept harmonic complex tones summed in cosine phase (Fig. 1). The complexes were constructed from harmonics 1–20 with starting F0 frequencies spaced at 1/3 octave intervals between 100 and 400 Hz and ending F0 frequencies one octave above that at the start of the sweep. Sweep duration was 500 ms.

Each sweep was convolved with impulse responses provided to us by Dr Tony Watkins (University of Reading). These were recorded at 0.32, 0.63, 1.25, 2.5, 5.0 and 10.0 m from a sound source in a corridor measuring approximately 2 m wide, 35 m long and with a ceiling height of 3.4 m and are the same impulse responses as used in previous psychophysical studies of

Fig. 1 Waveforms and spectrograms of harmonic complex sweeps with F0 100–200 Hz

reverberation (Watkins 2005). In this chapter we show only responses to the dry condition and the 32, 125, and 500 cm distance conditions. All stimuli were normalised after convolution for equal rms voltage and were presented monaurally with typically 25 repetitions and a 1 s repetition period. All stimuli were gated on and off with a 1 ms cos2 ramp.