Download Likelihood approaches to sensory coding in auditory cortex

INSTITUTE OF PHYSICS PUBLISHING
NETWORK: COMPUTATION IN NEURAL SYSTEMS
Network: Comput. Neural Syst. 14 (2003) 83–102
PII: S0954-898X(03)57828-9
Likelihood approaches to sensory coding in auditory
cortex
Rick L Jenison1 and Richard A Reale
Departments of Psychology and Physiology, Waisman Center University of Wisconsin-Madison,
1202 W Johnson Street, Madison, WI 53706, USA
E-mail: [email protected]
Received 19 March 2002, in final form 9 November 2002
Published 17 January 2003
Online at stacks.iop.org/Network/14/83
Abstract
Likelihood methods began their evolution in the early 1920s with R A Fisher,
and have developed into a rich framework for inferential statistics. This
framework offers tools for the analysis of the differential geometry of the full
likelihood function based on observed data. We examine likelihood functions
derived from inverse Gaussian (IG) probability density models of cortical
ensemble responses of single units. Specifically, we investigate the problem
of sound localization from the observation of an ensemble of neural responses
recorded from the primary (AI) field of the auditory cortex. The problem is
framed as a probabilistic inverse problem with multiple sources of ambiguity.
Observed and expected Fisher information are defined for the IG cortical
ensemble likelihood functions. Receptive field functions of multiple acoustic
parameters are constructed and linked to the IG density. The impact of
estimating multiple acoustic parameters related to the direction of a sound
is discussed, and the implications of eliminating nuisance parameters are
considered. We examine the degree of acuity afforded by a small ensemble of
cortical neurons for locating sounds in space, and show the predicted patterns
of estimation errors, which tend to follow psychophysical performance.
1. Introduction
The process by which neural responses are translated into a perceptual decision or estimate can
be viewed as a solution to a probabilistic inverse problem—that is, given a set of responses,
what is the most likely state of the environment that evoked that response? Historically,
Bayesians were the first to frame such problems in inferential statistics, and central to the
Bayesian approach is the use of axiomatic prior probabilities. One aim in statistical inference
is to infer something about the value of a parameter, which is unknown, based on observed
1
Author to whom any correspondence should be addressed.
0954-898X/03/010083+20$30.00
© 2003 IOP Publishing Ltd
Printed in the UK
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R L Jenison and R A Reale
data. Likelihood approaches, which address this aim, were introduced by Fisher (1925) within
the framework of estimation theory. Fisher believed that objective inference could be properly
based solely on the likelihood function (Efron 1998, Pawitan 2001). It has come to represent
more than just a method for generating point estimates, but rather a more general way of
reasoning objectively under conditions of uncertainty—a task the brain must continuously
undertake. But point estimates, in the form of maximum likelihood estimates, at the level of
the primary sensory cortices may very well not be the appropriate stage for making a decision
or perceptual estimate. Gold and Shadlen (2001) have recently suggested that log-likelihoods
may reflect the natural currency for combining sensory information from several sources, and
may provide a bridge between sensory and motor processing. Transformation of the entire
likelihood function, or the conveyance of the likelihood function, may be the more proper
way of thinking about the evaluation of information. Summarizing the likelihood function in
the form of a point estimate rarely conveys a sufficient representation of the entire function.
Alternative ways of simplifying a multiparameter log-likelihood function may involve the
removal of parameters that particular areas of cortex may be specialized for. An approach
within the formal likelihood framework can eliminate nuisance parameters by profiling or
integrating out parameters, and affords an analysis of the impact of parameter reduction on
information. Under some conditions, the log-likelihood function can be well approximated
using sequential orders of Taylor series expansions—often just a simple second-order quadratic
function is sufficient. In these cases the log-likelihood function can be summarized using just
the maximum likelihood estimate and the curvature about the maximum. It is this curvature
that defines the quantity referred to as the observed Fisher information.
In this paper we describe the application of likelihood approaches to the problem of sound
localization from the observation of an ensemble of neural responses recorded from the primary
auditory (AI) cortical field. The problem is framed as a probabilistic inverse problem with
multiple sources of ambiguity.
2. Auditory cortex
Understanding how the sensory cortices represent and convey information about the physical
world around us is one of the central questions facing the field of computational neuroscience.
Our specific interest in recent years has been the functional evaluation of ensembles of neurons
in auditory cortex in order to statistically characterize the information available for estimating
aspects of the spatial environment. Auditory cortex in the cat (figure 1) is located on the
exposed ectosylvian gyri and extends into the suprasylvian, the anterior ectosylvian and the
posterior ectosylvian sulci (Woolsey 1961, Merzenich et al 1975, Knight 1977, Reale and
Imig 1980). Superimposed lines delimit four auditory fields (anterior A, primary AI, posterior
P, ventroposterior VP) that are each distinguished by an orderly (tonotopic) and complete
representation of the audible frequency spectrum. A core of tonotopically organized fields is a
common finding in auditory cortex of mammals. Surrounding the four core fields is a peripheral
belt of auditory cortex in which complete tonotopic representations are generally not evident.
However, the belt region can be partitioned into a number of areas (e.g. second AII, ventral
V, temporal T and dorsoposterior DP) based on cytoarchitectonics, chemoarchitectonics,
connections and relative position with respect to the core fields. Neurons in the belt areas of the
cat, unlike those in the core fields, often respond poorly to pure tone stimuli and exhibit broad
or complex patterns of spectral tuning, even for tones near their threshold intensity (Merzenich
et al 1975, Reale and Imig 1980, Schreiner and Cynader 1984, Sutter and Schreiner 1991).
For these reasons, it is generally considered unlikely that cells in the core fields and belt
areas will process auditory information in a similar manner. Whether in core fields or in belt
Likelihood approaches to sensory coding in auditory cortex
85
DP
A
AI
AII
P
VP
T
Cat brain courtesy of Wally Welker - University of Wisconsin Comparative Mammalian Brain Collection
Figure 1. (a) Cat brain showing idealized boundaries of auditory cortical fields: anterior A, primary
AI, posterior P, ventroposterior VP, second AII, temporal T and dorsoposterior DP.
areas, neurons that possess complex spectro-temporal sensitivities are often hypothesized to be
candidates for specializations in function (Middlebrooks and Zook 1983, Sutter and Schreiner
1991, He et al 1997). In this regard we are particularly interested in the proposition that such
neurons may represent sound features along a few independent base dimensions that effectively
reduce the number of parameters that determine that representation (Schreiner 1998).
The directional sensitivity of auditory cortical neurons in the cat to free-field sounds has
been studied for nearly 30 years. Generally, due to physical limitations, only sensitivity to
changes in sound-source azimuth were investigated in detail (Eisenman 1974, Middlebrooks
and Pettigrew 1981, Middlebrooks et al 1994, Middlebrooks 1998, Imig et al 1990, Rajan
et al 1990, Eggermont and Mossop 1998, Xu et al 1998). With this approach the majority of
neurons can be classified as azimuthal sensitive. In this population, cells can have a contralateral
(most prevalent), ipsilateral, central, multipeaked or omnidirectional response preference. The
development of advanced signal processing techniques has permitted the use of virtual-space
stimuli, rather than free-field stimuli, to characterize spatial receptive fields (Brugge et al
1994). The observed shapes of virtual-space derived receptive fields are in close agreement
with those obtained in free-field studies. However, a major benefit of the virtual space approach
is that receptive fields were generally determined in both azimuth and elevation and with a
high spatial resolution (<5◦ ).
Our experimental approach depends on knowing first the relationship between spectral
transformation and sound-source direction for the ear of the cat. The technique originally used
for determining the free-field measurements has been described in detail in a previous paper
(Musicant et al 1990) from our laboratory. The simulation of these transformed sounds, as a
function of sound-source direction constitutes a virtual acoustic spherical space as shown in
figure 2. The virtual sound-source can be any waveform (e.g. wide- and narrow-band transients
or steady state noise, tones, speech, animal vocalizations) chosen to best suit the experimental
protocol and the response requirements of the neuron under study. Spatial receptive fields are
obtained by random presentation of sounds on the virtual sphere at various sound intensity
levels. We study the spatial receptive fields of auditory cortical neurons in cats under general
anaesthesia following well established protocols (Brugge et al 1994, Reale et al 1996, Jenison
et al 1998, 2001a). Tone burst stimuli delivered monaurally or binaurally are used to estimate
the best-frequency (BF) of neurons isolated along a given electrode penetration as described
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Figure 2. (a) Spherical auditory space synthesized using virtual space signal processing. The cat’s
nose is pointing to 0◦ azimuth, 0◦ elevation.
previously (Brugge et al 1994). Prior to single-unit recording we always derive a partial BF
map of the primary auditory (AI) cortical field. With such a map in hand, we are able to make
estimates of adjacent belt areas and direct electrode penetrations into acoustically responsive
areas.
2.1. Space–intensity receptive fields
The receptive field functions, rf (θ, φ, η), described in this paper were constructed to depend
on three acoustic parameters: azimuth θ , elevation φ and intensity η. Our prior functional
approximation work on spatial receptive fields used spherical basis functions (Jenison et al
1998, 2001a) with free parameters (wi j , κi j , βi j and αi j ) for fitting the basis functions. The
free parameters serve only to mathematically approximate the receptive field, where α and β
specify the placement of each basis function on the sphere in radians, κ specifies each width
and w each weight. The current extension of this approximation now includes an exponential
dependence on the intensity level, η, of the sound source defined as
rfi (θ, φ, η) =
M
wi j exp{log(η)κi j (sin φ sin βi j cos(θ − αi j ) + cos φ cos βi j )} + exp{ξi j η}
j =1
(1)
and a corresponding free fitting parameter ξ . Figure 3 illustrates space–intensity receptive fields
for three modelled units using a quartic-authalic mapping of first-spike latency (colour coded).
Here the empirically measured first-spike latency, averaged across all effective directions, and
the associated standard deviation are seen (full blue line) to increase with decreasing intensity
level. These empirically measured standard deviations contain both the unsystematic and the
High
Intensity
Likelihood approaches to sensory coding in auditory cortex
Short
Latency
Long
Latency
Measured
15
30
15
Modelled
25
35
45
40
50
35
60
55
45
70
65
55
75
75
0
Low
Intensity
0
10
20
30
40
50
Latency (ms)
60
70
10
20
30
40
Latency (ms)
50
60
0
10
20
30
40
50
60
Latency (ms)
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Figure 3. Auditory space–intensity receptive fields for three units recorded using virtual space techniques in primary auditory cortex. The spherical projections on
the left panels reflect the change in spatial sensitivity as a function of decreasing sound intensity. The cat’s nose points to the centre of the projection at (0, 0). The
left edge of the map corresponds to −180◦ and the right edge corresponds to +180◦ . The panel to the right of the spherical projections reflects the mean and standard
deviation of first-spike latency collapsed across all sound directions for each intensity level (dBA). The blue line corresponds to the measured recordings and the red
line reflects the modelled receptive fields. The standard deviation lines for the modelled conditions are shorter, as they reflect just the systematic variability due to
predicted direction selectivity at each intensity level. The measured standard deviations reflect both systematic and unsystematic (noise) variance.
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systematic variance of the response. The modelled results (full red line) are seen to capture the
increase in first-spike latency, averaged across directions, as a function of decreasing intensity.
The nonlinear dependence on η was introduced in order to modulate the width parameter κ of
the spherical basis functions and allow for the shape of the receptive field to depend on sound
intensity. Similarly, the nonlinear dependence of mean latency on intensity is provided by
introduction of the final term exp{ξi j η} in equation (1). Constrained optimization techniques
were used to fit the parameters wi j , κi j , βi j , αi j and ξi j of the M basis functions defined by
equation (1) to the dependent neural response of interest—which in this case was response
latency. The details of the approximation techniques can be found in Jenison et al (1998,
2001a). The linear combination of spherical basis functions effectively performs spatial lowpass filtering of the observed data. However, the placement and width free parameters allow
for varying degrees of smoothing (Jenison et al 1998).
The reliability of the receptive field function (equation (1)) in estimating the systematic
component of response variability is limited by the reliability of the fitting process itself.
There is no true solution of the receptive field function for a particular neuron, given a finite
input set consisting of measured response latencies together with their corresponding soundsource directions and intensity levels. A bootstrap of the receptive field function was used
to investigate the standard error of receptive field estimates for two typical AI neurons. The
total number of measured response latencies composing the input set for these neurons was
3900 and 3450, respectively. The sample and receptive field estimation process was repeated
40 times as prescribed by Efron and Tibshirani (1993). The results are shown in figure 4 for
the two neurons. The left-hand column maps the averaged bootstrapped receptive fields for
two intensity levels. The right-hand column maps the bootstrapped standard errors at each
direction. In general, regions with the shortest response latencies are also the regions that map
to the smallest standard deviations. These findings suggest that the function approximations
to the receptive fields shown in figure 3 were indeed valid estimates of systematic variability
in response latency that is due to both the dependence of latency on the direction of the sound
source and to the intensity level of the source.
Several observations can be made regarding the behaviour of the receptive field as a
function of direction and intensity. First, the units become more broadly tuned as the
sound source increases in intensity. Second, the average latency, collapsed across directions,
decreases as the sound source intensity increases, as does the variance of the latency to firstspike. Finally, the receptive fields are broadly tuned in general, an observation consistent
with previous investigations, but individual receptive fields are unique in their shape and their
evolution as a function of intensity. Given this obvious degree of ambiguity, the question is
then, how well can an ensemble of these units decode sound direction using first-spike latency
alone?
3. Decoding first-spike latency using likelihoods
3.1. First-spike latency
Recently, response latency has been shown to carry a significant proportion of the information
for the physical stimulus in addition to spike count (Heller et al 1995, Wiener and Richmond
1999, Petersen et al 2001). A common issue with respect to the use of latency as a neural
code for sound direction is the apparent lack of temporal marking for the onset of the stimulus.
This problem is not specific to directional coding in the auditory system, but is problematic for
decoding strategies that have been proposed in visual cortex as well (Gawne et al 1996). The
obvious resolution is an internally available global time marker, such as a collective oscillatory
Likelihood approaches to sensory coding in auditory cortex
A
89
BOOTSTRAP MEAN
BOOTSTRAP SE
40 dBA
60 dBA
12
22
B
0.1
0.5
0.1
0.5
30 dBA
60 dBA
12
16
Neuron #23 Response
Neuron #3 Response
Neuron #15 Response
Figure 4. Bootstrapped auditory space receptive field functions based on response latency from
two AI neurons. The intensity in dBA is shown above each row. Left column, map projection of
mean response latency obtained from 40 invocations of spherical modelling process. Right column,
map projection of standard error of modelling process. Colour bar annotations below each column
denote response latency and standard error limits, respectively.
Stimulus Onset
γ
10
11
12
13
14
Latency (ms)
Figure 5. Schematic of the response (first-spike latencies) from a population of neurons to a sound
in space. γ is the common latency across the ensemble of neurons.
pattern of activity (Hopfield 1995). An alternative resolution is that latencies relative to one
another in an ensemble of neurons could be decoded rather than absolute latency (Heil 1997,
Eggermont and Mossop 1998, Furukawa et al 2000, Jenison 2001). Recently we described a
formal method for decoding sound location from a population of cortical neurons in field AI of
the cat (figure 1) based on maximum likelihood estimation from absolute first-spike latencies
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(Jenison 1998). An extension to this approach is to replace the unknown stimulus onset by the
first observed spike in an ensemble cortical population (Jenison 2001). Under this scenario,
the observer measures subsequent evoked first-spikes within the ensemble relative to the first
poststimulus neural event in the ensemble (figure 5 sketches the general idea). Although still
based on parametric timing, this approach is related to the statistically nonparametric rankordered decoding described by Gautrais and Thorpe (1998) and reviewed by Thorpe et al
(2001). Using a relative referent, rather than absolute, implies that all of the ensemble neural
events are temporally shifted by some unknown common value relative to the physical onset
of the stimulus. The latency to the first-spike, x i reflects the timing of the i th neuron in the
ensemble relative to that of the first poststimulus event evoked by one of the neurons in the
ensemble. Any member of the ensemble could produce the first spike in the ensemble. By
definition the neuron with the shortest absolute latency will be assigned the value of 0. γ is
the common latency across the ensemble of neurons, and hence it reflects the parameter that
‘binds’ the ensemble responses to the stimulus onset. The statistical model of the relative
latency x i can be expressed in terms of the i th receptive field function of some parameter
vector , a common latency γ and error term εi as x i = rf i () − γ + εi , where the probability
density of the latency noise term εi is assumed to be distributed according to some parametric
probability density. Jenison (2001) showed that γ could be effectively eliminated from the
likelihood function by replacing γ with its maximum likelihood estimate at fixed values of .
This approach is referred to as profiling out a nuisance parameter and the resultant likelihood
function is commonly referred to as the profile likelihood (see Murphy and Van der Vaart
2000, for a review). This is an example of how formal likelihood approaches show promise in
providing insights into new decoding strategies, based on formal likelihood methods, which
may not have been obvious without these analysis tools.
3.2. Defining the proper probability density
Generalized linear models (GLIM) have served to unify the various classical inferential
statistical procedures since their introduction by Nelder and Wedderburn (1972). The
GLIM approach has also spawned the growth of alternatives to the multivariate normal
distribution. One example, the so-called dispersion model developed by Jorgensen and
colleagues (Jorgensen 1983, 1987), represents a class of probability densities that has
implications for spike train models. Specifically, we consider a subset known as reproductive
exponential dispersion models, or, as Jorgensen referred to them, the Tweedie densities
(Tweedie 1947, 1957). Tweedie densities are characterized by an index p, where E[x] = µ
and var[x] = µ p /λ. The indices p = [0, 1, 2, 3] correspond to the normal, Poisson, gamma,
and inverse Gaussian (IG), respectively (Jorgensen 1987, 1999). Jorgensen’s original paper
defined the power variance function for multivariate densities only under the assumption of
independent dimensions.
The normal and Poisson densities are well known in the neuroscience community. The
Poisson and its variants have been used extensively as point process and rate models. Less
familiar Tweedie densities are the IG and gamma, which were employed for inter-spike intervals
by (Tuckwell 1988, Levine 1991, Iyengar and Liao 1997). Most recently Barbieri et al
(2001) have modelled spike trains using these densities to address deficiencies in their earlier
Poisson models (Brown et al 1998). The IG distribution has a history dating back to 1915
when Schrödinger presented derivations of the density of the first passage time distribution
of Brownian motion with motion drift (Chhikara and Folks 1989, Seshadri 1999). Tweedie
(1941) coined the term ‘inverse Gaussian’ based on his observation that the cumulant generating
function of IG is the inverse of the cumulant generating function for the Gaussian. The IG
Likelihood approaches to sensory coding in auditory cortex
91
Figure 6. IG approximations to first-spike latency from 50 trials for two different spatial locations
(63◦ and −72◦ ). The histogram represents a probability of observing a particular latency. The
full curve corresponds to the IG probability density with fixed λ = 15 000 such that the variance
depends only on the mean rf (θ ). The predicted values (means) as a function of direction are
rf (63) = 11.2 and rf (−72) = 12.8.
density, with the linked receptive field function, is
−λ[x i + γ − rf i (θ, φ, η)]2
λ
p I G (x i | θ, φ, η) =
exp
.
2π[x i + γ ]3
2[x i + γ ]rf i (θ, φ, η)2
(2)
.
The mean corresponds to the receptive field rfi (θ, φ, η) and the variance is rfi (θ,φ,η)
λ
The parameter λ is a free parameter in the density function, and the variance depends on
λ. If λ can be reasonably fixed over an ensemble of neurons, then the variance depends only
on the mean. The parameter γ has typically been used as an additional degree of freedom for
purposes of improving the fit of univariate models. However, it can also serve as a common
ensemble latency similar to the manner in which it was employed as a binding parameter
for the multivariate Gaussian in Jenison (2001). Figure 6 shows examples of first-spike (50
trial) histograms corresponding to two different sound directions from a single unit in field AI.
These single direction examples were obtained from the receptive field data shown in figure 3.
Overlaid on top of the histogram is the corresponding IG density using the predicted mean,
with the parameter γ set to zero and λ set to 15 000 based on a reasonable approximation to the
entire ensemble data set. The IG density appears to do a good job of capturing the dependence
of first-spike variance as a function of the mean latency. Further goodness-of-fit analyses are
certainly necessary to access whether other members of the Tweedie densities might afford
better models.
3
3.3. Log-likelihood functions
Likelihood functions have been employed in point process analyses by Ogata and Akaike
(1982) and Brillinger (1988), and most recently by Kass and Ventura (2001) and Brown et al
(2001). Martignon et al (2000) have also examined ensemble interaction structure using both
likelihood and Bayesian techniques. A parametric likelihood function can be straightforwardly
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A
Intensity (dBA)
Low
Intensity
35 dBA
Latency (ms)
High
Intensity
Azimuth (deg)
Low
Likelihood
B
High
Likelihood
Intensity (dBA)
Low
Intensity
15 dBA
Latency (ms)
High
Intensity
Azimuth (deg)
Figure 7. Log-likelihood functions for a sound source at 45◦ azimuth 0◦ elevation at two intensities:
(A) 35 dBA and (B) 15 dBA. Black arrows denote maximum likelihood estimates given the
population of first-spike latencies shown in the insets.
constructed based on marginal probability density functions for each neuron in the population
under the assumption that the unsystematic variability (noise) is independent:
L(θ, φ, η) = p(
x |θ, φ, η) =
N
i=1
p(x i |θ, φ, η).
(3)
Likelihood approaches to sensory coding in auditory cortex
93
The log of the likelihood function, based on the IG (equation (2)), is
log L(θ, φ, η) =
N
N
N
λ
3
[x i + γ − rf i (θ, φ, η)]2
log
−
log[x i + γ ] − λ
.
2
2π
2 i=1
2[x i + γ ]rf i (θ, φ, η)2
i=1
(4)
An example of the log-likelihood surface defined by equation (4) (with an ensemble size
N = 26 and fixed elevation φ) is shown in figure 7. The full log-likelihood function
provides more than just the opportunity to estimate the true values of the parameters. The
differential geometry of the surface is of interest as well. In this example the mode of the loglikelihood surface would correspond to the maximum likelihood estimate of θ and η (azimuth
and intensity). Figure 7(a) shows the resulting log-likelihood surface for a sound source at
45◦ azimuth, 0◦ elevation presented at 35 dB down from a reference intensity level (dBA).
Following the onset of the sound stimulus, the first-spikes from an ensemble of 26 AI units2
(in a form shown in figure 5) are shown in the inset panel to the left of the log-likelihood surface.
This surface represents the log-likelihood for sound direction (azimuth) and intensity given the
ensemble response3. The maximum of the log-likelihood functions are easily observed in this
example, which signal the maximum likelihood estimates of [47◦ , 38 dBA]. The curvature
at the maximum of the log surface reflects the degree of observed Fisher information in the
ensemble for estimating the acoustic parameters of this particular sound source. For the more
intense sound source (figure 7(b)), the curvature about the maximum has decreased somewhat,
reflecting a decrease in observed Fisher information. In this case the maximum likelihood
estimates are [49◦ , 14 dBA]. The fact that a maximum exists, and the log-likelihood surface is
so well structured suggests that information for both parameters are conveyed in the population
of first-spike latencies.
The stochastic pattern of first-spike latencies shown in each inset, therefore, informs the
observer as to the direction of the sound source and, potentially of equal interest, the intensity
of the sound source.
4. Fisher information
We, as well as others, have investigated the consequences of broad receptive fields on
population coding using Fisher information and the Cramer–Rao lower bound (CRLB) under
the assumption of independent residual noise (Paradiso 1988, Seung and Sompolinsky 1993,
Jenison 1998) and correlated residual noise (Abbott and Dayan 1999,Gruner and Johnson 1999,
Jenison 2000, Sompolinsky et al 2001). Our prior work has shown that, under the assumption
of independence, even very broad and nonuniform spatial receptive fields in auditory cortex
can demonstrate observed psychophysical localization acuity with as few as ten cells in the
population (Jenison 1998). The CRLB is a lower bound on the variance, or the standard error,
of any unbiased estimator, and is derived from Fisher information with respect to a family of
parametric probability distributions. For a single parameter, the CRLB is inversely related to
Fisher information mathematically, and intuitively. As the magnitude of Fisher information
increases, we expect the estimation standard error to diminish. Although there may not exist an
algorithm that achieves the CRLB, a maximum likelihood estimator asymptotically approaches
2 The ensemble is constructed from 13 unique AI receptive field functions (predictions), with a mirrored set of
functions reflected about the midline, to produce a total of 26. The residual noise is not reflected and therefore is
considered here to be independent. The reason for the forced symmetry is to approximate the contributions from the
contralateral hemisphere.
3 The full log-likelihood function is actually a function of three acoustic parameters—azimuth, elevation and intensity.
For purposes of visualization, we assume for this example that elevation is known, i.e. 0◦ .
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R L Jenison and R A Reale
the CRLB. The Fisher information matrix for an ensemble of space–intensity receptive fields
and parameter vector = {θ, φ, η} is defined as follows:
∂ 2

∂ ∂ 
∂ ∂ E
E
E

∂θ
∂θ ∂φ
∂θ ∂η 
 ∂ 2
∂ ∂ 


∂ ∂ E
E

E
(5)
J =

∂φ ∂θ
∂φ
∂φ ∂η 


∂ ∂ ∂ 2 
 ∂ ∂ E
E
E
∂η ∂θ
∂η ∂φ
∂η
where the log-likelihood is denoted by = log L(θ, φ, η) and


σθ̂2
ρθ̂,φ̂ σθ̂ σφ̂ ρθ̂ ,η̂ σθ̂ ση̂

σφ̂2
ρφ̂,η̂ σφ̂ ση̂ 
K =  ρθ̂ ,φ̂ σθ̂ σφ̂
(6)

ρθ̂ ,η̂ σθ̂ ση̂
ρφ̂,η̂ σφ̂ ση̂
ση̂2
ˆ =
is the estimation covariance matrix for azimuth, elevation and intensity estimates {θ̂ , φ̂ and η̂}. K J −1 defines the relationship between the estimation covariance matrix
and the inverse of the Fisher information matrix. The inequality reflects the fact that the matrix
K − J −1 is nonnegative definite, and it follows that the diagonal of J −1 reflects the Cramer–
Rao lower bounds of estimate variability for θ̂ , φ̂, and η̂ (Blahut 1987, Cover and Thomas
1991). Under some regularity conditions, a maximum likelihood estimate of the parameters
is asymptotically distributed as a multivariate normal with vector mean and covariance
matrix J −1 (Lehmann and Casella 1998). The expected value for the azimuth element in the
Fisher information matrix under the assumption of the multivariate IG probability density with
independent noise is
2
N
[∂rf i (θ, φ, η)/∂θ]2
∂
log L(θ, φ, η) = λ
(7)
E
∂θ
rfi (θ, φ, η)3
i=1
where in the present case N is 26 units in the ensemble (see the appendix). Figure 8 illustrates
only two out of the nine cells in the K matrix for a maximum likelihood estimator—the square
root of the upper left cell σθ̂ and the correlation coefficient portion of the upper right cell ρθ̂ ,η̂ —
as a function of different azimuth directions and intensities. The lower bound on the standard
error of the estimate of azimuthal direction is σθ̂ given the ensemble of cortical responses, and
therefore can be interpreted as an analogue of a lower bound on acuity.
There are several interesting aspects of this error surface. First, there are several bands
of elevated standard error, particularly around azimuths4 of +50◦ and −50◦ . These azimuths
correspond to regions of maximum sound pressure amplification by the horn-like shape of
the cat’s pinnae, (Musicant et al 1990), and where the majority of neurons tend to exhibit
their shortest first-spike latencies at near-threshold intensities (Brugge et al 1996). However,
these regions of minimal response latency are also regions with near-zero response gradients,
∂rf i (θ, φ, η)/∂θ . Consequently, at these directions expected Fisher information deflates, with
a corresponding increase in the standard error.
Second, there is a general trend for regions of minimal standard error to occur at moderate
intensity levels; by comparison, the standard error is increased at higher and lower intensities.
Figure 8(b) illustrates this non-monotonic pattern by taking cross-sectional cuts across intensity
at three azimuthal directions. The explanation for the decrease in acuity at lower intensities
is that, as intensity decreases, first-spike latency becomes less precise. The opposite is true
at higher intensities. However the magnitude of the gradients, ∂rf i (θ, φ, η)/∂θ , decreases
4
The symmetry is the result of mirroring the receptive field functions as described previously.
Likelihood approaches to sensory coding in auditory cortex
95
A
Intensity (dBA)
Low
Intensity
σ∧
θ
deg
High
Intensity
Azimuth (deg)
B
σ∧
θ
deg
Intensity (dBA)
Intensity (dBA)
C
ρ∧
∧
θ ,η
Azimuth (deg)
Figure 8. (A) standard errors of the estimates σθ̂ as a function of sound direction (azimuth) and
intensity. Dotted vertical line: cross-section shown in (B) standard errors of the estimates σθ̂ as a
function of intensity at three azimuths: full: −48◦ , broken: −44◦ , dotted: −40◦ (C) correlation
coefficients of the estimates ρθ̂ ,η̂ .
96
R L Jenison and R A Reale
at high intensities, resulting in a relative deflation of expected Fisher information, and a
corresponding increase in the standard error. The explanation for this is revealed in figure 3
where tuning is shown to broaden with increasing intensity along with a corresponding decrease
in receptive field gradient at particular spatial directions. In summary there is a trade-off
between lower slopes of first-spike latency gradients at high intensity and higher variability
in first-spike latency at low intensity, both of which are reflected in the Fisher information
matrix.
The correlation coefficient of the estimates ρθ̂ ,η̂ (figure 8(C)) characterizes the degree of
linear dependence that is expected to occur when θ̂ and η̂ are jointly estimated by a maximum
likelihood estimator. The corresponding cell in the K matrix depends on the degree of crossinformation evident in the Fisher information matrix through the matrix inverse operation. If
the off-diagonal cells of the Fisher information matrix were all zero, then the K matrix would
necessarily be diagonal. This condition, known as an orthogonal Fisher information matrix,
would also imply that the parameters could be simply factored and treated as independent,
which has not been shown to be the case for any of the analyses that we have performed on
our recorded AI units.
5. Parameters of interest
5.1. Profile likelihood
The profile likelihood function for parameters of interest is obtained by substituting the
conditional maximum likelihood estimate for the so-called nuisance parameters into the
constructed likelihood function (Murphy and Van der Vaart 2000). The standard profile
likelihood function can be expressed simply as L p (θ, φ) = L(θ, φ, η̂) = maxη L(θ, φ, η).
So, for example, we may be interested in the profile likelihood construction when the sound
intensity η parameter is eliminated.
Although the profile likelihood is widely used for inference on the parameters of interest,
it is well known that the profile likelihood can provide less than satisfactory inference, and
yield biased and/or inconsistent results. Adjustments to the profile likelihood, with an eye
toward obtaining improved statistical inference on the parameters of interest, have generated
substantial interest. The work of Cox and Reid (1987, 1992, 1993), McCullagh and Tibshirani
(1990), Diciccio and Stern (1993, 1994), Barndorff-Nielsen (1994), Stern (1997) and Mukerjee
and Reid (1999) on adjustments to the profile likelihood has led to significant progress in
unbiased removal of nuisance parameters from likelihood function constructions.
5.2. Integrated likelihood
The integrated likelihood addresses some of the issues raised by the profile likelihood approach,
specifically where profile likelihood functions fail to capture the uncertainty present about
the maximum likelihood estimate of the nuisance parameter being eliminated. Profiling
out a nuisance parameter by maximization could be misleading if the region of nuisance
parameter maximization is atypical in particular regions of the whole likelihood surface (Berger
et al 1999). Integrating out the nuisance parameter is reminiscent of Bayesian approaches
(see Tierney et al 1989a, 1989b, Kass et al 1989, Diciccio et al 1997) and can be defined
using a uniform
prior p(η) over a reasonable range of intensities (10–60 dBA) as in the case
L U (θ, φ) = L(θ, φ, η) p(η) dη. Although a reasonable first choice, Berger et al (1999) have
raised potential caveats regarding the use of uniform priors that remain to been investigated.
One analytical approach to approximating this integral is the Laplace approximation, which is
Likelihood approaches to sensory coding in auditory cortex
A
97
B
60
Low
Intensity
-20
-40
Log-likelihood
Intensity (dBA)
50
40
30
-60
-80
-100
-120
20
High
Intensity 10
-140
-150
-100
-50
0
50
100
150
-150
Azimuth (deg)
-100
-50
0
50
100
150
Azimuth (deg)
Low
Likelihood
High
Likelihood
Figure 9. (A) Taylor series expanded intensity for the log-likelihood function shown in figure 7(A).
(B) Integrated log-likelihood function after removing the intensity parameter. The maximum
likelihood estimate is shown.
based on a second-order Taylor series expansion of the likelihood function about the maximum
likelihood estimate η̂:
2π
L(θ, φ, η) p(η) dη = L(θ, φ, η̂) p(η̂)
.
(8)
−∂ 2 log L(θ, φ, η̂)/∂η2
The denominator −∂ 2 log L(θ, φ, η̂)/∂η2 is the observed Fisher information I (η̂θ,φ ). This
approximation is useful so long as a unique maximum exists, I (η̂θ,φ ) = 0 and the log-likelihood
function is approximately quadratic about the maximum (Pawitan 2001). Although it is implied
that the integral is computed over the whole line, it may be sufficiently accurate as long as the
mass of the approximating function is within the limits of the integration (Goutis and Casella
1999). This result can be shown to be roughly equivalent to an adjusted profile likelihood
described by Cox and Reid (1987). The importance of these analytical approaches is that they
may offer new theoretical ways of thinking about perceptual invariance within the framework
of cortical ensemble likelihood functions. For example, a population of neurons that remain
invariant for a particular parameter over different levels of intensity may reflect this process of
intensity integration.
The log-likelihood function for direction (azimuth) and intensity, shown in figure 7, can be
second-order Taylor series expanded across intensity, conditioned on direction (figure 9(A)).
The resulting surface reflects the fitting of a quadratic to the maximum of the log-likelihood
function of intensity at each direction. This now allows the intensity dimension to be integrated
out using equation (8) to yield the so-called integrated log-likelihood function shown in
figure 9(B). Following this transformation, the maximum of the resulting integrated loglikelihood can be determined, and the point estimate used for locating the sound source.
Furthermore, the curvature at the maximum conveys the amount of available observed
information, which should more accurately account for the intensity uncertainty that has been
marginalized.
Removal of one dimension may seem counter to the premise that perceptual decisions
based on the likelihood function should be deferred as long as possible. However, there
are advantages to the statistical removal of dimensions at downstream stages of information
processing. We are agnostic, at this point, as to which dimensions would be eliminated—
pursuant to this example it could be either intensity or direction. Given this perspective,
98
R L Jenison and R A Reale
although a unique global maximum exists, it can happen that multiple modes appear in the
log-likelihood function. Given an inaccurate estimate of intensity, a maximum could be
chosen corresponding to a direction of near 90◦ , rather than near the veridical estimate of
45◦ . Integrating out the intensity dimension reveals a single maximum near the true direction.
Integrated likelihood incorporates the added uncertainty introduced by the estimation of the
eliminated parameter, such that the resulting curvature reflects the deflation of information
resulting from the intensity parameter. The use of the profile or integrated likelihood is
aligned with the psychophysical judgment scenario where, for example, the observer is
asked to estimate sound direction at different sound intensity levels. This is typically how
behavioural localization studies are performed—direction estimation accuracy is of interest,
not the accuracy of estimating the sound level. We are not suggesting that the intensity
information is eliminated, just that different cortical regions may evaluate the likelihood
functions differently, depending on the particular role of the cortical area.
6. Implications for cortical field processing
Profile likelihood or integrated likelihood methods can be used to analytically eliminate
parameters that are hypothesized to be redundant and thus allow us to make specific predictions
of reduced likelihood functions. Profile likelihood methods represent the simplest approach
to the removal of parameters as an approximation to marginal likelihood functions. Both
approaches require a maximum likelihood estimate, which is often intractable or non-unique.
Although there are numerical approaches to the derivation of estimators, analytical solutions
can offer deeper insights into the consequences of multiplexing multiple parameter sources of
information and the consequences of reductions in dimensionality. There is recent evidence for
separate populations at the level of primary auditory cortex that segregate coding schemes of
time-varying sounds into synchronous versus rate based codes (Liu et al 2001), and that appear
to respond invariantly across ranges of intensity, suggesting a form of profiled or integrated
likelihood. Convergent horizontal connections between AI subregions have recently been
revealed, which suggest the existence of intrinsic modularity within cortical fields (Schreiner
et al 2000, Read et al 2001). These anatomically and functionally segregated subregions may
reflect a form of integrated likelihood where specific dimensions have been eliminated.
Fisher information, and particularly its inverse, can serve as a link between neural
population decoding performance and psychophysical performance. Recent evidence supports
the observations described by our results, predicting that direction estimation acuity declines
at low intensities (Zu and Recanzone 2001), improves at moderate intensities and declines at
higher intensities (MacPherson and Middlebrooks 2000). The likelihood analyses presented
here assumed the residual noise to be independent. Our recent findings, and others,
challenge the common interpretation that correlated noise between information channels
always compromises ideal performance (Snippe and Koenderink 1992, Dan et al 1998, Oram
et al 1998, Abbott and Dayan 1999, Gruner and Johnson 1999, Panzeri et al 1999a, 1999b,
Jenison 2000), thus suggesting a need for further inquiry into the dependence structure of the
noise within the population. The need is further driven by recent evidence that the structure
of the ensemble covariance matrix is related to auditory cortical receptive field properties
(Eggermont 1994, Eggermont and Smith 1995, Brosch and Schreiner 1999). Therefore, an
important extension to the likelihood analyses developed here using non-Gaussian densities
is to introduce a flexible covariance structure to describe the noise dependence. However,
this extension is nontrivial, particularly one that leaves the resulting marginal distributions of
alternative densities. We are currently pursuing ways to couple the Tweedie densities in order
to construct flexible multivariate densities and their likelihood functions.
Likelihood approaches to sensory coding in auditory cortex
99
Acknowledgment
This work was supported by National Institutes of Health Grants DC-03554, DC-00116, and
HD-03352.
Appendix
There are two ways to evaluate expected Fisher information. One approach is to derive the
expected value directly using integration by parts, which is the more general approach. The
alternative approach is to utilize the score function, which for θ is defined as
∂
log L(θ, φ, η).
(9)
∂θ
Under the regularity conditions met by the exponential family, the expected value of the score
function is 0, therefore
2
∂
2
varS(θ, φ, η) = E S(θ, φ, η) = E
log L(θ, φ, η) .
(10)
∂θ
S(θ, φ, η) =
So the variance of the score function is equal to the expected Fisher information. The score
function for the multivariate IG assuming independence (equation (4)) and linked to the space–
intensity receptive fields is
S(θ, φ, η) = λ
N
([x i + γ ] − rfi (θ, φ, η)) ∂
rfi (θ, φ, η).
rfi (θ, φ, η)3
∂θ
i=1
(11)
In our application E{[x i + γ ]} = rf i (θ, φ, η). The expectation can be distributed and, under
the assumption of independence, the cross-terms removed to yield
2
∂
N
rf (θ, φ, η)
2
2
∂θ i
E S(θ, φ, η) = λ
E [xi +γ ] ([x i + γ ] − rf i (θ, φ, η))2 .
(12)
3
rf
(θ,
φ,
η)
i
i=1
The final term is just the variance of [x i + γ ] so
2
∂
N
rf
(θ,
φ,
η)
rfi (θ, φ, η)3
i
∂θ
E S(θ, φ, η)2 = λ2
rf i (θ, φ, η)3
λ
i=1
then
E
∂
log L(θ, φ, η)
∂θ
2
=λ
N
[∂rf i (θ, φ, η)/∂θ]2
i=1
rfi (θ, φ, η)3
.
(13)
(14)
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