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9/23/2011
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L17. Neural processing in Linear Systems 2: Spatial Filtering
C. D. Hopkins Sept. 23, 2011
Crab cam (Barlow et al., 2001)
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self inhibition recurrent inhibition lateral inhibition
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Limulus
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Limulus eye: a filter cascade.
transduction and adaptation encoding
in
out
light to voltage
voltage to spike
rate
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light on
1
10-2
10-4
----1 second---------
transient
adapted
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Dynamic Response to Step Increase in Light Intensity
1) Light increment
2) Light decrement
adaptation
symmetrical (codes both)
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The Real World: An arbitrary stimulus
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an impulse an impulse response h(t)
IN OUT
nconvolutio
dtxhty
)()()(
k
knxkhny )()()(
x(t)
zero at t ~= 0 1 at t=0
response to impulse
For linear systems…….
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an impulse an impulse response h(t)
IN OUT
nconvolutio
dtxhty
)()()(
k
knxkhny )()()(
x(t)
zero at t ~= 0 1 at t=0
response to impulse
For linear systems…….
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an impulse an impulse response h(t)
IN OUT
nconvolutio
dtxhty
)()()(
k
knxkhny )()()(
x(t)
zero at t ~= 0 1 at t=0
response to impulse
For linear systems…….
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an impulse an impulse response h(t)
IN OUT
nconvolutio
dtxhty
)()()(
k
knxkhny )()()(
x(t)
zero at t ~= 0 1 at t=0
response to impulse
For linear systems…….
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Dynamic Response to Step Increase in Light Intensity
1) Start in constant, low level light.
Step increase in intensity for 2 sec.
Decrease back to previous level.
2) Decrement in light intensity generates the reverse (mirror image)
Time invariant, linear system.
Good fit to curve predicted from convolution
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Convolution Result Predicts Dynamic Responses to Steps
Three alternative methods for estimating h(t)
Response to impulse
Response to noise
Response to component sine waves
Ringach,D. and R. Shapley (2004) Cog. Sci. 28:147
The reverse correlation (“revcor”) method (de Boer)
Evans (1977)
Pickles (1988)
De Boer, E. (1967). Correlation studies applied to the frequency resolution of the cochlea. J. Audit. Res. 7, 209-217.
Spike Triggered Reverse Average • Determine the average (most likely) stimulus waveform preceding a spike.
• Measured by “spike-triggered averaging” with a white noise stimulus.
• Revcor functions of low-CF auditory-nerve fibers resemble the impulse response
of a bandpass filter centered at the CF.
• Fourier transforms of revcor functions match the tip of pure-tone tuning curves
over a wide range of noise levels.
• The revcor is an estimate of the crosscorrelation between stimulus and
response.
Reverse correlation and Wiener filters
• Given a linear system, the crosscorrelation of the response r(t) with a stationary, white noise input w(t) is proportional to the system’s impulse response h(t):
• The revcor is an estimate of the Wiener filter in the special case when r(t) consists of impulses (spikes).
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( ) ( )1
( ) ( ) ( ), with ( )T
h w t dw t r t dt h r tT
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Linear systems analysis of audition
response to click in auditory system
Calculate the Fourier Transform of the inpulse response to obtain the tuning curve of the auditory neuron.
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stimulus
spike histogram
stimulus after convolution with h(t)
h(t)
h(t) second cell
convolution (cell2)
deBoer, E, and H.R. de Jongh
Furthermore, convolution of stimulus with the impulse response predicts the spike density (post stimulus time histogram)
Bialek, W., Rieke, F., de Ruyter van Steveninck, R. R. and Warland, D. (1991). Reading a neural code. Science 252, 1854-7. Rieke, F., Warland, D., de Ruyter van Steveninck, R. and Bialek, W. (1997). Spikes: Exploring the Neural Code. Cambridge, Massachusetts: MIT Press.
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Using Sine Wave Stimuli
---Amplitude is multiplied by the gain
---Phase is delayed or advanced (add phase shift to sine wave)
Amplitude and phase are different for different frequencies.
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Do this for all relevant frequencies
Stimulate at all relevant frequencies with sinewave stimuli.
Measure gain and phase
Frequency
gain
phase
1
0
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0
m
k
k = spring constant (N/m) m = mass (kg) = frequency (radians/s)
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http://www.lon-capa.org/~mmp/applist/damped/d.htm
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Bode Plot
Gain vs. F
Phase vs. F
Frequency
gain
phase
1
0
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Separate Transfer Functions Data from Limulus (Knight et al., 1970)
Generator potential in response to sinusoidally modulated light.
Spike frequency in response to light (or to sinusoidally modulated current injection.
Spike frequency in response to modulated light.
in
out
light to voltage
voltage to spike
rate
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Separate Transfer Functions Data from Limulus (see Knight et al.)
Generator potential in response to sinusoidally modulated light.
Spike frequency in response to light (or to sinusoidally modulated current injection.
Spike frequency in response to modulated light.
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Cascade Filter
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Cascade Filter
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Gain and Phase for Limulus Eye generator potential in response to light
spike rate in response to injected current
spike rate in response to light (observed and predicted)
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Two Methods are Equivalent
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Filtering an Impulse Stimulus
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Arbitrary Stimulus Convert arbitrary stimulus waveform to sum of sines.
Calculate gain and phase shift for each frequency.
Sum up responses.
Compute predicted response.
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self inhibition recurrent inhibition lateral inhibition
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Open circles: spike frequency recorded from eccentric cell A, while A is given a step increase in light. Closed circles: constant illumination of ommatidium A while providing the step increase in light in B.
Lateral Inhibition
light increase in area B
Fahrenbach, W. H. (1985). Anatomical circuitry of lateral inhibition in the eye of the horseshoe crab, Limulus polyphemus. Proc R Soc Lond B Biol Sci 225, 219-49.
Lateral Inhibition also occurs in vertebrate retina Receptive Field of Mammalian Ganglion Cell (S. Kuffler, 1953)
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Lateral inhibition can be included in model
Linear cascade from one cell converts light to spike frequency.
Spikes from one cell inhibit neighbors (lateral inhibition).
Inhibition is mutual (varies with distance)
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Steady State Response
What is the response to a point of light.
Center (immediately over the eccentric cell): excitation.
Surround (adjacent areas): inhibition.
Barlow, R. B., Jr. (1969). Inhibitory fields in the Limulus lateral eye. J Gen Physiol 54, 383-96. 42
In two dimensions
A Mexican Hat.
Spatial impulse response.
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Lateral Inhibition Enhances Edges
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Prediction by Convolution impulse response step of light
result
convolution
Fahrenbach, W. H. (1985). Anatomical circuitry of lateral inhibition in the eye of the horseshoe crab, Limulus polyphemus. Proc R Soc Lond B Biol Sci 225, 219-49.
Parallel Processing in Retina
Wassle, Heinz (2004) Nat. Rev. Neurosci. 5: 747-57
1. rods 2 cones 3 horizontal 4 bipolar 5 amacrine 6 ganglion
cone triad
Salamander retina on electrode array. Meister M, Pine J, Baylor DA (1994) Multi-neuronal signals from the retina: acquisition and analysis. J Neurosci Methods 51: 95–106.
Tiger salamander
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Meister M, Pine J, Baylor DA (1994) Multi-neuronal signals from the retina: acquisition and analysis. J Neurosci Methods 51: 95–106.
stimulus
visualization
Record simultaneously responses from 61 electrode array.
Characterize receptive field (spatial and temporal) of each ganglion cell using flickering checkerboard.
For one ganglion cell, center circular spot on receptive field; add surround grating.
Contribution from On Bipolar cells: APB added to ringers prior to recording (blocks the metabotropic glutamate receptor, knocking out “on” pahtway.
Sharp electrodes for recording from amacrine cells.
Stimulus: circular spot, 800 microns diameter (slightly larger than RF. Surround flickering grating. Intensity changes every 30 ms, pseudorandom level variation. Grating flickers every .9 s.
A larger array for salamander studies: 512 electrodes
Buchen, L. (2008) From eye to sight. Symmetry magazine, 5(1), 2008.
http://www.symmetrymagazine.org/cms/?pid=1000591
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Lateral Inhibition
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Mexican Hat
0 0 -1 0 0
0 -1 -2 -1 0
-1 -2 16 -2 -1
0 -1 -2 -1 0
0 0 -1 0 0
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After convolution with mexican hat
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self inhibition recurrent inhibition lateral inhibition
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Lessons from Visual Coding
1. The goal: understand sensory coding. Vision: example of “frequency code”.
2. Visual processing includes: 1. transduction, 2. encoding
3. Adaptation can be thought of as self inhibition.
4. Most sensory neurons behave as temporal filters: adaptation (tonic vs. phasic)
5. Linear systems analysis can also be used to describe spatial effects such as lateral inhibition.
6. Convolution can be used to predict responses to arbitrary stimuli.
The end
