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Sensation and Perception (I) (25 optimality (Ideal observer analysis #…
Sensation
and
Perception (I)
25
optimality
Ideal observer analysis
#
identify optimal algorithms for decoding neural signal
help pinpoint key bottlenecks to performance
provide tools to compare performance across different levels of a system
A theoretical
ideal observer
performs a given task in an optimal fashion given the available information and some specified constraints.
Notice
optimal performance is not equivalent to perfect performance
determine the optimal performance at one level of a system
Physical limits
noise due to temperature
Molecules undergo spontaneous thermally-driven changes in conformation.
Conformational changes induce noise into neural signals if the molecule transits between two different signaling states.
statistical variation
Actual stimulus reaching a detector fluctuate because of the division of physical stimuli into discrete units.
approach to systematizing the complexity of neural circuits
predictions based on optimization fail because
Capturing all constrains that contribute to circuit design with a tractable performance metric is effectively impossible.
The employed solutions of neural circuits are not optimal.
Neural circuits perform as well as they can given a set of operating conditions and hardware constrains.
26
Adaptation and
Natural Stimulus Statistics
:red_flag:Adpatation
Adaptive coding with
changing stimulus statistics
Linear-nonlinear cascade model
Nonlinear input-output mapping
nonlinear function itself may also vary with changes in stimulus statistics
change filter properties according to statistical context
Linear filtering
Sparse representation
stimuli are represented by a small number of neurons in the population that respond to certain spatial patterns.
Predictive coding
cancel out the correlations in the input and represent only "surprising" components of the stimulus.
Temporal decay
Explanations
\(f_{0}(I)\): input-output curve from the responses immediately following each change in input.
\(f_{\infty}(I)\): input-output curve from the steady state part of the response.
The firing rate relaxes between these two coding regimes according to the dynamics of an adaptation variable
A
.
\(f=f_{0}(I-A)\)
"capture rapid onset of the response"
\(\tau\frac{dA}{dt}=f^{-1}_{\infty}(f)+f^{-1}_{0}(f)-A\)
"the slow relaxation"
The transient responses are a strong response to
change
in stimulus.
Neural response incorporates history dependence and sensitivity to derivatives.
\(R(t)=g(\int_{0}^{\infty} d\tau f_{1}(\tau) s(t-\tau), \int_{0}^{\infty} d\tau f_{1}(\tau) s(t-\tau), ...)\)
Neural responses following a stimulus onset initially rapidly increase, then gradually decay over time.
Natural scenes and their challenges
Hypothesis
Neural systems should dynamically adjust their coding strategies to best represent statistics of stimuli that vary continuously in space and time.
Efficient coding
coding properties of neurons are adjusted to the distribution of possible stimuli as to maximize the neuron's
representational capacity
.
Structure at many scales
Natural inputs have a separate meaning between large and small scale features.
different coding schemes might be used at different time or space scales.
Overall self-similarity
Statistical composition of the scenes is relatively invariant to different scales.
Spatial frequency function of natural scenes behave as a power law.
Large variations
Neural response usually
increases with the intensity of the input.
:question:How to extract information about object identity invariant with intensity?
:question:how to represent such wide range of inputs without saturation?
modulations in stimulus amplitude are over many orders of magnitude in intensity.
28
Neural Computation
process of highly selective extraction of the important information from sensory inputs
Selectivity and invariance
The brain can construct circuits that make single neurons perfectly selective for a high-level abstract feature of the sensory input.
Mapping between abstract feature and sensory inputs is flexible under different tasks or contexts.
Selectivity and invariance prevalent for a wide range of neurons.
:red_flag:Receptive Fields
a summary of response
properties of a neuron
describe how stimuli at different locations and times combine to generate response of the neuron
a spatiotemporal RF \(L(x,y,t)\) is a function of space and time
LN model
Linear summation
at any give time, the system sums the intensity of the point-like stimuli in the recent past, weighted by the amplitude of the receptive field.
\(g(t)=\int_{x}\int_{y}\int^{t}_{t'=-\infty} S(x,y,t-t') L(x,y,t') dt'dydx\)
Nonlinear transform
nonlinear distortion function that turns \(g\) into the neuron's reponse.
\(r=N(g)=N(\boldsymbol{S\cdot L})\)
In practice, locations and times are in discrete form \(\vec{x_{i}}, \Delta t\)
so stimulus and RF could be prepresented as a vector.
\(\boldsymbol{S}=[S_{ij}]=[S(\vec{x_{i}}, t-j\Delta t)]\)
\(\boldsymbol{L}=[L_{ij}]=[L(\vec{x_{i}}, j\Delta t)]\)
then, \(g=\boldsymbol{S\cdot L}\)
Theory
Shapes of RF are designed to efficiently transmit visual information through the bottleneck of the optic nerve.
(Avoid redundancy.)
RFs remove the correlations in images.
The brain represents the stimulus using a large population of neurons that are each active only rarely and are statistically independent of each other.
This coding scheme highlights "suspicious coincidence" of firing in the population.
Limits of RF analysis
LN model for a given neuron often only accounts for a portion of its neural response.
especially when using natural (complex) stimuli
RFs are fickle and can change on a moment's notice.
Different environment might lead to different results for L and N in LN model.
The sensory computation expressed by LN model is rather primitive.
More than 20 types of RGCs perform quite sophisticated image computations.
RF analysis of this type does not help much in understanding computations such as those leading to face cells.
RF of any LN neurons has the same complexity as that of a sensory receptor and is no more helpful in explaining the advanced stimulus.
:red_flag:Beyond RFs
Extraclassical RF,
Surroud effect,
Contextual influences
RF along cannot account for sensory neuron responses.
Stimuli falling outside the RF might modulate the neuron's responses
Research approaches
Do experiment in a stimulus space of reduced dimensionality.
Maintain the full dimensionality of the stimulus space, but ask whether there exists a subset of these directions \(\boldsymbol{L_i}\) that can affect neural response.
\(r=N(g_1,...,g_n)=N(\boldsymbol{S\cdot L_1},...,\boldsymbol{S\cdot L_n})\)
Model the neural system as a cascade of LN stages leading up to the sensory neuron in interest.
Expanded
nonlinear representations
For each of the two stimulus axes, create
n
neurons that respond to only a small range along that axis.
Combine such neurons from each axis with a logical
AND
to create a large population of \(n^2\) neurons that each respond in a tiny region of the stimulus space.
Combine many of these neurons by a logical
OR
to create a neuron with a thin and convoluted response region.
29
Population Coding
Neural code
coding structure is different brain areas, individuals, species, etc.
Despite the differences, many aspects of the code are universal, and must be related.
Though single neurons exhibits feature selectivity, neural responses are not deterministic or reliable due to
*noise
".
:red_flag:
Noise
synaptic fluctuation
stochastic nature of ion channel opening
The mapping from stimulus to responses, thus, must be describe in terms of a probability distribution over the responses to each stimulus.
Encoding distribution
Because the system is probabilistic, mapping from stimuli to responses would be a prob. dist. of the response
r
given stimulus
s
, \(p(r|s)\).
Inverse distribution
mapping from responses to stimuli is also probabilistic, and the prob. dist. of the stimulus
s
given the response
r
is \(p(s|r)\).
sensor noise
:question:What do the neural responses convey about the the stimulus and the impact of noise?
Information theory
Single neurons level
Entropy \(H[p(x)] \)
Approach to quantify uncertainty
How much the neural response is "wasted" because of noise is the ratio of the information and the total response entropy.
\( {I(s;r)/H[p(r)]} \)
Mutual information
between stimulus and response is,
\( I(s;r)=H[p(s)]-Ave(H[p(s|r)])_r \), or
\( I(s;r)=H[p(r)]-Ave(H[p(r|s)])_s \)
Almost 50% of coding capacity was lost due to noise.
No assumptions about what features of the spiking patterns matter.
No assumptions of any particular kind of dependency.
Paired neurons level
Mutual information between the cell spiking patterns
\( I(r_1;r_2)=\sum_{r_1,r_2}p(r_1,r_2)log_2\frac{p(r_1,r_2)}{p(r_1)p(r_2)} \)
#
Mutual information between cells given a stimulus
\( I(r_1;r_2|s)=\sum_{r_1,r_2}p(r_1,r_2|s)log_2\frac{p(r_1,r_2)}{p(r_1|s)p(r_2|s)} \)
#
Synergy or redundancy of the code can be expressed as the difference between the info. conveyed together by two cells and the info. carried by each cell.
\(SR(r_1,r_2;s)=I(s;r_1,r_2)-(I(s;r_1)+I(s;r_2))\)
\(=I(r_1;r_2)-Ave(I(r_1;r_2|s))_s\)
Population level
Generalized linear model
Stimulus-dependent maximum entropy model
Population codes
If neurons overlap in terms of stimulus space, population coding implies higher fidelity of coding and the potential for noise correction.
If every neuron cover an almost unique par of stimulus space, population coding implies detailed coverage of the sensory space.
Must balance the richness of population codes with some form of overlap or dependency between cells.
Correlation between neurons
Signal correlation
Dependency is measured in terms of seeing joint activity patterns of the two cells \(p(r_1,r_2)\) to the independent prediction \(p(r_1)p(r_2)\)
Noise correlation
Dependency between cells is the relation between their activity patterns with respect to a specific stimulus.
\(p(r_1,r_2|s)\) and \(p(r_1|s)p(r_2|s) \)
Redundancy, independence, and Synergy
Independent coding
Info is carried uniquely by some cells but not by other.
Synergistic coding
info. is carried only by joint activity of cells.
Redundant coding
info. is over-represented by several cells.
Corr. between pairs of cells are typically weak.
Corr. of large groups of neurons
Corr. and redundancy are strong from collective effect of weak pairwise dependencies.
32
Effects of Stimulus and Neural Variability on Perceptual Performance
Most perceptual tasks are decision making in the presence of random variability.
Bayesian statistical decision theory
#
#
Goals
To derive ideal-observer models
As a framework for modeling perceptual performance
Specify the task
Specify the set of possible responses.
e.g. \(r = absent \ or \ present \)
Specify the goal of the task.
specify the
costs
and
benefits (utility)
\( \gamma (r,\omega) \)
Specify the set of possible stimuli.
the ground-truth (distal) stimuli \( \omega \)
the proximal stimuli \( s \)
Ground truth and input stimuli
Specify the statistical relationship
between the states of the world and the proximal stimuli.
Posterior prob.
\( p(\omega |s) \)
Usually, researchers specify the
stimulus likelihood distribution
\( p(s|\omega) \) and the
prior prob.
\(p(\omega)\), then use Bayes's rule to compute posterior prob.
Input data
There are properties of the perceptual system that are part of the specification of the input to the optimal Bayesian computations.
e.g. the optics of the eyes, tuning characteristics of neurons
The properties can be accounted in by a function \(g_{\theta}\) that map the input stimulus \(s\) onto input data \(z\).
\(z=g_{\theta}(s)\)
Bayes optimal response
Pick the response that maximize the utility, averaged over the posterior prob., given the input data.
\( r_{opt}(z)=\underset{r}{arg\ max}[\sum_{\omega}\gamma (r,\omega)p(\omega|z)]\)
The observer that makes the optimal response for a given task and constrained function is defined as the
ideal observer
.
Identification task
Maximum a posteriori (MAP) rule
\( r_{opt}=\underset{a_i}{arg\ max}[p(z|a_i)p(a_i)]\)
equal if post. dist. are unimodal and not skewed
Estimation task
Minimum mean
square error (MMSE)
\( \hat{\omega}_{opt} = \underset{\omega_i}{arg\ min}[\sum_{\omega} (\omega - \hat{\omega})^2 p(\omega |z)] \)
\(= \sum_{\omega} \omega p(\omega |z) = E{\omega |z} \)
large number of categories
30
Population-based
Representations
Approach
derived from proposals of David Marr
level 1
Define
population response vector
for a population of neurons.
The same set of neurons shows
response cloud
in the space due to noise.
The amount of info. available in the population to perform the task depends on the degree of non-overlapping between response clouds.
The shape of the decision boundary is the
population readout rules
for the task in this set of neurons.
level 2
Measure how well a population of neurons can solve the task.
Population performance depends on
info. conveyed by individual neurons
population size
number of different sets needed o be parsed.
neural correlations (noise or tuning corr.)
shape and position of the decision bound
level 3
#
establish reasonable neural computation models
Information should be transformed in a way that are gradually easier to access.
implicit info.
info. that is hard to extract directly, it requires a complex and non-linear readout.
Explicit info.
info. that is easily accessible, and requires only simple linear readout.