Transcript Document

LECTURE 9
Neural coding (2)
I.
Introduction
− Topographic Maps in Cortex
− Synesthesia
− Firing rates and tuning curves
II. The nature of neural code
− Rate coding or temporal coding?
(Barn owl auditory system, place cells,
and grid cells)
− Population code
• Population correlation code:
(Synchrony and oscillations)
• Population code with statistically
independent neurons
• Rate coding:
Information is encoded in the firing rate
• Temporal coding:
Precise spike timing is a significant element in neural encoding
• The debate between rate and temporal coding dominates
discussions about the nature of the neural code.
The decoding cue: the time difference
between a sound reaches the two ears
(the order of 0.1ms).
Coincidence detector: the neuron will
only be active when the inputs from two
ears are received simultaneously.
Accuracy 1 degree
Temporal precision <5us
Jeffress model (Jeffress, 1948)
Remarkably enough, such a coincidence detector circuit was found
four decades later by Carr and Konishi (1990) in the nucleus
laminaris of the barn owl.
It gives, however, no indication of how the precision of a few
microseconds is finally achieved.
Temporal precision is less than 5μs even though the membrane
time constant and synaptic time constant are in the range of
100−1000 microseconds. How is it reached within this circuit?
Interaural intensity differences (for high frequency sounds
(wavelength smaller than the head)
Interaural phase differences (for low frequency sounds)
Delay tuning in barn owl auditory system
Place cells in rat hippocampal pyramidal cells
1 mV
200 ms
Two types of theta: I and II
(Skaggs et al. 1996)
Examples of raw and filtered EEG. Filter bandpass: 1-100 Hz (A); 6-10 Hz (B)
Place fields of place cells
1.
2.
Finding place cells by O’Keefe and Dostrovsky (1971)
Finding theta phase precession by O’Keefe and Recce (1993)
Theta phase precession in hippocampal
pyramidal cells
0o/360o phase
270o phase
(Huxter, Burgess, and O’Keefe 2003)
Theta phase precession in a place cell
(Huxter, Burgess, and O’Keefe 2003)
Theta phase
precession
in a place cell
• Theta rhythm 7-12 Hz.
• The spikes of the place cell
gradually and monotonically
advances to earlier phase
relative to hippocampal theta
rhythm as the rat traverses
along the cell’s place field
(Mehta, Lee and Wilson 2002)
Place cell coding in hippocampal pyramidal cells
－ Hippocampal place cells code the spatial position of the
animal both by their firing rate and the precise timing of their
firings.
－ A variety of different models have been developed to account
for mechanisms underlying both unimodal firing profile and
theta phase precession.
Argument focuses on:
1) whether phase precession emerges in hippocampus itself or
is inherited from upstream brain areas (Current evidences
point to the latter);
2) whether dual coding is independent or inseparable (It
remains unclear now).
Evidence 1: Phase
precession is
preserved after
stimulation-induced
perturbation
(Zugaro, Monconduit & Buzsáki 2005)
One class of models predict that if one or both oscillators are
reset, the resuming spike-phase relationship should be strongly
altered by the perturbation.
Thus, a simple two-oscillator model in which at least one
oscillator is within the hippocampus (as opposed to the
entorhinal cortex) cannot account for the present observations
Evidence 2: Phase precession in grid cells
Fyhn, M., Molden, S., Witter, M. P., Moser, E. I. & Moser, M. B. Spatial
representation in the entorhinal cortex. Science 305, 1258–1264
(2004)
1.0m
0.3m
0.5m
1.0m
In the superficial layers of the
dorsocaudal region of the
medial entorhinal cortex
(dMEC)
Firing fields of 3 simultaneously recorded cells (30 min
running) (Hafting et al. 2005, Nature)
• Spacing: 39 – 73 cm across different cells of different rats
• standard deviation of spacing within a grid: 3.2 cm averaged across cells
Population data for hippocampus-independent phase precession in
entorhinal grid cells (Hafting et al. 2008, Nature)
Persistence of phase precession after hippocampal
inactivation in layer II cells recorded before (c) and
after (d) inactivation (Hafting et al. 2008, Nature)
In summary,
－Phase precession is expressed independently of the
hippocampus in spatially modulated grid cells in layer II of
medial entorhinal cortex, one synapse upstream of the
hippocampus.
－Phase precession is apparent in nearly all principal cells in
layer II but only sparsely in layer III. The precession in
layer II is not blocked by inactivation of the hippocampus,
suggesting that the phase advance is generated in the grid cell
network
－The results point to possible mechanisms for grid formation
and raise the possibility that hippocampal phase precession is
inherited from entorhinal cortex.
How to distinguish between rate and temporal
coding in practice?
When precise spike timing or high-frequency firing-rate
fluctuations are found to carry information, the neural code is
often identified as a temporal code.
The temporal structure of a spike train or firing rate is
determined both by the dynamics of the stimulus and by the
nature of the neural encoding process.
The interplay between stimulus and encoding dynamics
makes the identification of a temporal code difficult.
An MT neuron
responded to the same
moving random dot
stimulus with the
varied motion
coherence
Another proposal is to use the
stimulus, rather than the
response, to establish what
makes a temporal code. In this
case, a temporal code is
defined as one in which
information is carried by
details of spike timing on a
scale shorter than the fastest
time characterizing variations
of the stimulus.
c=1
c=0.5
c= 0
(Bair and Kock 1996)
I.
Introduction
− Topographic Maps in Cortex
− Synesthesia
− Firing rates and tuning curves
II. The nature of neural code
− Rate coding or temporal coding?
(Barn owl auditory system, place cells,
and grid cells)
− Population code
• Population correlation code:
(Synchrony and oscillations)
• Population code with statistically
independent neurons
How is a stimulus encoded by neural activities?
(Do you remember the tuning curve?)
• The discussion to this point has focused on information carried by
single neurons, but information is typically encoded by neuronal
populations
• Encoding by the most active neuron sounds reasonably if there is
no noise, but it does not work in practice because of large
fluctuations in neural activities. Basically many nervous systems
use large numbers of neurons to encode information..
Population coding
When we study population coding, we must consider whether
individual neurons act independently, or whether correlations
between different neurons carry additional information.
Synchronous firing of two or more neurons is one mechanism for
conveying information in a population correlation code.
Synchrony and oscillations
A theory of perception --- the temporal binding.
This model assumes that neural synchrony with precision in the
millisecond range is crucial for object representation, response
selection, attention and sensorimotor integration
It defines dynamic functional
relations between neurons in
distributed sensorimotor
networks, i.e., neurons that
respond to the same sensory
object may fire in temporal
synchrony
(Engel, Fries and Singer 2001)
An example: bistability
Bistability: Two interpretations are possible of this figure
(Engel, Fries and Singer 2001)
In this case, the temporal binding model predicts that neurons
should dynamically switch between assemblies and, hence, that
temporal correlations should differ for the two perceptual states
Four visual cortical neurons with
receptive fields over these four
image components: the grouping
which changes from one precept
to another.
(Engel, Fries and Singer 2001)
Neurons 1 & 2 should
synchronize if the respective
contours are apart of the
one background face; and
for neurons 3 & 4 for the
candlestick.
When the image is
segmented into two
opposing faces, the
temporal coalition
switches to synchrony
between 1-3 and 2- 4
respectively
(Engel, Fries and Singer 2001)
I.
Introduction
− Topographic Maps in Cortex
− Synesthesia
− Firing rates and tuning curves
II. The nature of neural code
− Rate coding or temporal coding?
(Barn owl auditory system, place cells,
and grid cells)
− Population code
• Population correlation code:
(Synchrony and oscillations)
• Population code with statistically
independent neurons
• The biggest advantage of Population code is the ability to
average out noises in individual neurons if they are
independent.
• Our target is to learn how a continuously moving direction is
decoded by a population of neurons. Firstly, we show two
systems: the cercal system of cricket and M1 cortex of the
monkey.
Population coding in the cercal system of
cricket by a small number of neurons
Crickets have two projections
sticking out their posterior end:
cerci. Each cercus is covered
with small innervated hairs.
Thousands of these primary sensory
neurons send axons to a set of
interneurons that relay the sensory
information to the rest of the cricket’s
nervous system. No single interneuron of
the cercal system responds to all wind
directions, and multiple interneurons
respond to any given wind direction.
An interpretation from the view of statistical
inference
• Neural decoding is essentially a statistical inference process,
that is, to infer the stimulus value based on the observation of
data.
• Consider S represents the stimulus, b the neural response,
R the noisy data.
• Two phases in neural coding:
－The encoding phase:
S  b
－The decoding/inference phase:
R  b
• Noise is ubiquitous in neural systems.
• Statistical inferential sensitivity: how robust is the inferred
result with respect to noise?
Tuning curves for the four low-velocity interneurons of the
cricket cercal system plotted as a function of the wind direction.
rmax ≈ 40 Hz. Wind speed is constant.
(Theunissen and Miller 1991)
At low wind velocities, information about wind direction is
encoded by just four interneurons. The tuning curve for
interneuron a:
Decoding the cercal system by employing the close relationship
between the representation of wind direction and a Cartesian
coordinate system.
This vector is known as the
population vector, and the
associated decoding method is
called the vector method.
(Dayan and Abbott 2001)
Decoding arm movement
direction in M1 cortex of
the monkey by population
vector method
Recordings from the primary motor cortex of a monkey performing an
arm reaching task
• Noises always exist
• If the preferred directions point uniformly in all directions and
the number of neurons N is sufficiently large, the population
vector:
N

f ( s) 
v pop   (
) a ca
Compare it with
a 1 rmax

v : thedecoded movingdirection

ca : thepreferred- directionvect orof theath neuron

ra ca : thecontributon
i from theath neuron
rmax : themaximumaveragefiring rat e
Comparison of
population vectors
with actual arm
movement
directions
(Dayan and Abbott 2001)
• The neural system reads out the moving direction by the
average of preferred stimuli of all active neurons weighted by
their activities. This sounds reasonable since more active
neurons, whose preferred stimuli are more likely close to the
true stimulus, and hence should contribute more on the final vote.
• Population vector demonstrated that information can be accurately
represented by the joint activities of a population of neurons in a
noise environment..
• The idea of population coding is also found in the representation of
moving direction in other parts of cortex, and the representation of
other stimuli, such as the orientation of object and the spatial
location.
Up to now, we have considered the decoding of a direction angle.
We now turn to the more general case of decoding an arbitrary
continuous stimulus parameter.
We need use Maximum Likelihood Inference (MLI) or Bayesian
inference.
An array of N neurons with preferred stimulus values distributed
uniformly across the full range of possible stimulus values
An array of Gaussian tuning curves spanning stimulus values
from -5 to 5
Tuning curves give the mean firing rates of the neurons across
multiple trials. In any single trial, measured firing rates will vary
from their mean values. To implement the MLI approach, we
need to know the conditional firing-rate probability density p[r|s]
that describes this variability:
1. ra = na/T: the firing rate of neuron a:
T: the trial duration:
2. Homogeneous Poisson model.
3. p[ra|s]: the probability of stimulus s evoking na = raT
spikes, when the average firing rate is ra = fa(s)
Do you remember Poisson distribution?
The probability that any sequence of n spikes occurs within a trial
of duration T obey the Poisson distribution:
If we assume that each neuron fires independently, the firing-rate
probability for the population is the product of the individual
probabilities,
The assumption of independence simplifies the calculations
considerably.
To apply the MLI estimation algorithm, we only need to consider
the terms in P[r|s] that depend on s. It is convenient to take its
logarithm and write
The MLI estimated stimulus, sMLI, is the stimulus that
maximizes the righthand side of above equation.
On the biological plausibility of a decoding strategy
• MLI, though very accurate, is often too complicated to be
implemented in neural architecture, especially, when noises are
correlated.
• Population vector, may appears to be simple to computers (just
some addition and times operations), is not guaranteed to be also
simple in the view of neural systems (e.g., how to carry out these
additions and times is not obvious). Moreover, in some noise
correlation structures, population vector can be very inefficient.
• In the below we will show that template-matching can be
naturally achieved in neural systems through the idea of
continuous attractor.
Attractor Computation
Attractor: a steady state of a neural ensemble memorizes a
stimulus value
Information retrieval: a noisy input will be attracted to a
steady state of the system
Discrete versus continuous attractor:
The properties of continuous attractors
－ Continuous attractors allows the system to change status
smoothly, following a fixed path. This property (not shared by
discrete attractors) is crucial for the system to seamlessly track
the smooth change of stimulus
－Continuous attractor seems to be most suitable for representing
continuous stimulus such as the moving direction, but may also
works well for encoding discrete objects if there is a continuous
underlying feature linking all these objects
Neural implementation of template-matching
in continuous attractor neural networks
1 The steady states of the network must have the same shape of
tuning function in order to generate the template.
2 When no stimulus exists, the network should be neutrally
stable on a line attractor, parameterized by all possible
stimulus values. This enables the network to be ready to
decode (match) any stimulus value that may arise.
3 An external input that contains the stimulus information drives
the template (the steady state of the system) to the position
that has the maximum overlap with the noisy population
activity (this is the final execution of template-matching).
Homework
1. What does it mean by temporal coding？Give
examples.
2. What is the basic idea of Population Vector Method
and Maximum Likelihood Inference coding?