Natural Sound Statistics and Divisive Normalization in the Auditory System
Authors: Odelia Schwartz, Eero P. Simoncelli
Published in: Advances in Neural Information Processing Systems 13  (NIPS 2000) (Main Conference Track), 2000

Abstract: We  explore  the  statistical  properties  of  natural  sound  stimuli  pre(cid:173) processed  with  a  bank  of linear  filters.  The  responses  of such  filters  exhibit a striking form of statistical dependency, in  which the  response  variance of each filter grows with the response amplitude of filters tuned  for  nearby  frequencies.  These  dependencies  may  be  substantially  re(cid:173) duced  using  an  operation  known  as  divisive  normalization,  in  which  the  response  of each  filter  is  divided  by  a  weighted  sum  of the  recti(cid:173) fied responses of other filters.  The weights may  be chosen to  maximize  the independence of the normalized responses for  an ensemble of natu(cid:173) ral  sounds.  We  demonstrate that the resulting model  accounts for  non(cid:173) linearities in the response characteristics of the auditory nerve, by com(cid:173) paring model simulations to electrophysiological recordings. In previous  work  (NIPS,  1998)  we  demonstrated  that  an  analogous  model derived  from the statistics of natural images accounts for non-linear properties of  neurons in primary visual cortex.  Thus, divisive normalization appears to  be a generic mechanism for eliminating a type of statistical dependency  that is prevalent in natural signals of different modalities. 
Signals in the real world are highly structured.  For example, natural sounds typically con(cid:173) tain both harmonic and rythmic structure. It is reasonable to assume that biological auditory  systems are designed to represent these structures in an efficient manner [e.g.,  1,2]. Specif(cid:173) ically, Barlow hypothesized that a role of early sensory processing is to remove redundancy  in the sensory input, resulting in a set of neural responses that are statistically independent. 
Experimentally, one can test this hypothesis by examining the statistical properties of neural  responses under natural stimulation conditions [e.g.,  3,4], or the statistical dependency of  pairs (or groups) of neural responses.  Due to  their technical difficulty, such  multi-cellular  experiments are only recently becoming possible, and the earliest reports in vision appear  consistent with  the hypothesis [e.g.,  5].  An  alternative approach, which we  follow here,  is  to  develop a neural  model from  the statistics  of natural  signals  and show that response  properties of this model are similar to those of biological sensory neurons. 
A number of researchers have derived linear filter models using statistical criterion. For vi(cid:173) sual images, this results in  linear filters  localized in frequency, orientation and phase [6, 7]. 
Similar work in audition has yielded filters localized in frequency and phase [8].  Although  these linear models provide an important starting point for neural modeling, sensory neu(cid:173) rons  are highly  nonlinear.  In  addition,  the  statistical properties of natural signals  are  too  complex to expect a linear transformation to result in an independent set of components. 
Recent results  indicate  that nonlinear gain control plays  an important role in neural pro(cid:173) cessing.  Ruderman and Bialek  [9] have shown that division by a local estimate of standard  deviation can increase the entropy of responses of center-surround filters to natural images.  Such  a model is consistent with  the properties of neurons in the retina and lateral genicu(cid:173) late nucleus.  Heeger and colleagues have shown that the  nonlinear behaviors of neurons  in primary visual cortex may  be described using a form of gain control known as  divisive  normalization [10], in which the response of a linear kernel is rectified and divided by the  sum  of other rectified kernel  responses  and  a constant.  We  have  recently  shown that the  responses  of oriented linear filters  exhibit nonlinear statistical dependencies that may  be  substantially reduced using  a  variant  of this  model,  in  which the  normalization signal  is  computed from a weighted sum of other rectified kernel responses [11,  12]. The resulting  model, with weighting parameters determined from image statistics, accounts qualitatively  for physiological nonlinearities observed in primary visual cortex. 
In this paper, we demonstrate that the responses of bandpass linear filters  to natural sounds  exhibit  striking  statistical  dependencies,  analogous  to  those  found  in  visual  images.  A  divisive  normalization procedure can  substantially remove these dependencies.  We  show  that this model, with parameters optimized for a collection of natural sounds, can account  for nonlinear behaviors of neurons at the level of the auditory nerve. Specifically, we show  that:  1) the shape offrequency tuning curves varies with sound pressure level, even though  the underlying linear filters are fixed; and 2) superposition of a non-optimal tone suppresses  the response of a linear filter in a divisive fashion, and the amount of suppression depends  on the distance between the frequency of the tone and the preferred frequency of the filter. 
1  Empirical observations of natural sound statistics 
The basic  statistical  properties of natural  sounds,  as  observed through a linear filter,  have  been previously documented by  Attias  [13].  In particular, he showed that,  as  with visual  images, the spectral energy falls roughly according to a power law, and that the histograms  of filter responses are more kurtotic than  a Gaussian  (i.e.,  they have  a sharp peak at zero,  and very long tails). 
Here we examine the joint statistical properties of a pair of linear filters  tuned for nearby  temporal frequencies.  We choose a fixed  set of filters  that have been widely used in  mod(cid:173) eling  the peripheral auditory system [14].  Figure 1 shows joint histograms of the instanta(cid:173) neous responses of a particular pair of linear filters  to five different types of natural sound,  and white noise.  First note that the responses are approximately decorrelated: the expected  value of the y-axis value is roughly zero for all values of the x-axis variable. The responses  are  not,  however,  statistically  independent:  the  width  of the  distribution  of responses  of  one  filter  increases  with  the response  amplitude  of the  other filter.  If the  two  responses  were  statistically  independent,  then  the response of the  first  filter  should not provide any  information about the distribution of responses of the other filter.  We have found that this  type  of variance  dependency  (sometimes  accompanied by  linear correlation) occurs  in  a  wide range of natural  sounds, ranging from animal  sounds  to  music.  We  emphasize that  this dependency is  a property of natural sounds, and is not due purely to our choice of lin(cid:173) ear filters.  For example, no such dependency is  observed when the input consists of white  noise (see Fig.  1). 
The  strength  of this  dependency  varies  for  different  pairs  of linear  filters .  In  addition,  we  see  this  type  of dependency between instantaneous responses  of a single  filter  at two