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How is information about the stimulus represented in
the nervous system?!
Eric Young!
!
!
F. Rieke et al. Spikes MIT Press (1997). Especially chapter 2.!
Borst A, and Theunissen FE. Information theory and neural coding. Nature Neurosci
2: 947-957, 1999.!
I. Nelken et al. Encoding stimulus information by spike numbers and mean response
time in primary auditory cortex. J Comput Neurosci 19:199-221 (2005).!
The fundamental assumption is
that the representation is in
terms of spike times (as
opposed to subthreshold
potentials for example).
For the analysis, the spike train
is reduced to a series of time
points, the times at which action
potentials occur.
A basic analysis: how did the
neuron respond to the stimulus?
D.K. Ryugo
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There are two different problems:
(1)  characterizing the response to a given stimulus.
(2)  inferring the stimulus given a particular response.
Note
they
are
different!
Characterizing
the response,
given the
stimulus
Characterizing
the stimulus
given the
response
E(v|n)
E(n|v)
The response
(number of spikes in
0.2 s)
The stimulus (optical
flow velocity)
Rieke et al 1997
In principal, the two kinds of descriptions are interconvertible using Bayes’
theorem:
measure in an
experiment
P(v n) =
P(n, v)
P(n)
and so
P(v n) =
P(n v)P(v)
~ (const) P(n v) P(v)
P(n)
a normalizing constant
(given n), usually
ignored
There may be a problem estimating P(v), which is difficult to do meaningfully,
especially for natural stimuli. Thus we often work on the forward problem,
estimating the response given an arbitrary stimulus, and take various indirect
approaches to the reverse problem.
There is another problem: what is the appropriate variable to use for n, the
response? Number of spikes, spike timing, spike latency, population response or
synchrony, . . . . ? Mostly we ignore this problem in this lecture.
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For the system problem, estimating P(n|v), the ultimate goal is an accurate predictive
model, i.e. a model that predicts responses to an arbitrary stimulus. For the auditory
nerve, this has been accomplished, one example is below. For neurons in the CNS, the
problem is harder, but has been solved for special cases.
P(n|v)
Bruce et al. 2003
For the information problem, estimating P(v|n), various approaches have been taken.
One is to use the knowledge of P(n|v) gained from experimental studies and work
backward (e.g. maximum likelihood estimation). The examples below show responses
from which P(v|n) could be computed, by comparing the observed responses with these
expected responses.!
Rate!
For sound level, the response
grows with level and spreads
asymmetrically.!
!
!
!
!
!
For frequency, the response shifts
along the basilar membrane.!
Two kinds of analyses are commonly pursued:!
!
(1) Discrimination of stimuli, for which the information lies in the tails of the response!
!
(2) Identification of stimuli for which the information is in the shape or location of the
response curve.!
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Estimating the discriminability of two stimuli from the neural responses proceeds by
calculating the distribution of responses to the two stimuli P(n|v) from data (where n =
NT , the number of spikes); the stimuli v are noise (n) and tone plus noise (t). The
discrimination task is to detect the tone in the presence of the noise.
The task is solved (MLE) by setting a threshold C, above which a spike response Nt is
said to come from tone + noise. The probability of being correct is the shaded region
and the probability of false alarm is the dark region.
The difficulty of the task, i.e. the
quality of the representation can be
measured by the difference in the
means of the distributions divided by
the variance, called d’
d'=
µ N |t − µ N |n
σ n2 + σ t2
The representation of discrimination for a vowel using a rate code. Which auditory nerve
fibers signal the difference between two vowel spectra, based on discharge rate? The
answer is the fibers with BFs near the formants, where the vowels differ.
Stimulus
spectral
difference!
The spectra
of four test
stimuli
Rate
difference,
from P(n|v)!
Discriminability (MLE)!
d’ =
R(V4) - R(V0)
s2(V4) + s2(V0)
Note that one (well-chosen) neuron is
sufficient!!
Conley and Keilson 1995
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An example of identification of the stimulus (actually the location of the sound source)
based on the temporal pattern of neural responses in cortical neurons. Here neurons
code aspects of the stimulus by changing the overall temporal patterns of response in a
way that has little to do with the waveform or envelope of the stimulus.
Neurons in auditory
cortex show different
patterns of response
depending on sound
source direction.
A neural network model was
trained to compute source
azimuth from PSTs of neurons’
responses.
This worked pretty well.
(one neuron is sufficient,
or nearly so)
This is n, a vector of spike times.
Middlebrooks et al. 1998
A more direct approach to the information problem (estimating v from n) is to
compute quantitatively the information about the stimulus carried by the neural
responses.
Define information as the reduction in uncertainty about an event, where
uncertainty is measured by entropy:
?
Which stimulus was presented?
The possibilities and their a-priori probabilities
P1
P2
The uncertainty about the
stimulus or entropy H is
n
H = − ∑ Pi log 2 Pi
i =1
P3
P4
.
.
.
Then the information provided by the
spike trains R about the stimulus S is
the reduction in entropy:
MI = H (S) − H (S R)
or MI = H (R) − H (R S)
Pn
called the mutual information or MI.
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Made-up data from an experiment, shown as histograms of rate responses (R1-5) to
stimuli (S1-5). Note that the entropy H(R) of the sum of the responses is high, near
the maximum for a five-point distribution (log25 = 2.32). The entropies H(R|s) of the
responses to each stimulus are smaller. The difference between H(R) and the
average conditional entropy H(R|S) is the information about the stimuli carried in the
responses.
H(R|s)=
1.9
1.6
1.7
H(R|S)=1.56
1.5
1.0
MI = 2.30 – 1.56 = 0.74 bits
H(R)=2.30
An application of information analysis. Neurons often have to represent multiple
aspects of a stimulus simultaneously (its frequency, sound level, location in space,
etc.). How is this done? An approach to the analysis is to compute the information (as
MI) between two stimulus aspects and the neuron’s responses.!
sensitive
mainly to
ILD!
ILD
The stimulus set: 25 stimuli
varying simultaneously in two
parameters, 5 values each!
sensitive
to both
ILD and
SN!
SN
Chase and Young 2005
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It was found that neurons represent varying combinations of three different cues to
sound localization. The plots show the fraction of the total MI about the stimulus that is
available to encode either of two simultaneously-varying stimulus aspects for a group of
inferior-colliculus neurons. The extent of representation of a particular variable is quite
different across neurons.!
The diagonal lines show where the data points would be if the information provided by
the neurons were fully divided between the two variables on the axes. The data points
lie below the lines because not all of the response can be used, i.e. one response
variable can’t be exactly split between two stimulus variables. The difference is called
confounded information.
Chase and Young 2005
The stimulus estimation process
also involves the probability of the
stimulus,
P(v|n) = (const) P(n|v) P(v)
Neurons are sensitive to the local
P(v) in the sense that responses
are larger when a particular
stimulus is unexpected.
For example at right, both LFP
and multi-unit activity (and also
single units in other studies) give
larger responses to f1 (13.3 kHz)
when it is rare (“deviant”) than
when it is common and similarly
for f2 (19.2 kHz).
LFP
multi-units
Taaseh, Yaron, Nelken 2011
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Another example: trying to discover
what aspects of the stimulus the
neurons are responding to, when the
responses are very poorly time-locked. Responses in insula
versus A1 to a variety of
natural stimuli. The insula
responses bear only a
general relationship to the
stimuli.!
A monkey call
modified in
three ways!
Insula
neurons
provide more
information
about the real
calls than the
modified ones!
Remedios et al. 2009
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