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Lecture 1: Neurons Lecture 2: Coding with spikes Lecture 3: Tuning curves and receptive fields Learning objectives: To gain a basic understanding of how neurons represent the environment
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- 1. Name some of the parameters that one can extract from a neural
spike train in order to test for a correlation with a given stimulus quality (like amplitude).
- 2. Can you describe (or draw) what a rasterplot looks like and why it
is useful?
- 3. What is an interspike-interval distribution?
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- 5. Look at the following spike trains depicting a neuron’s responses
to tones of a) increasing frequency and b) increasing loudness. Find the spike train parameter(s) that vary with frequency and/or loudness and draw a response-curve.
Tone (1000 ms)
1000 Hz 1500 Hz 2000 Hz 2500 Hz
1500 Hz Tone (1000 ms)
30dB 40dB 50dB 60dB
SLIDE 4 1. Describe the elements of the McCulloch Pitts neuron. How do they correspond to elements in real neurons?
- 2. Which characteristics of real neurons are not taken into account in McCulloch
Pitts neurons?
- 3. Very briefly describe how the characteristics mentioned in (2) can be taken into
account using integrate-and-fire or leaky integrate-and-fire neurons.
- 4. Which characteristics of real neurons can you think of that leaky integrate-and-
fire neurons do not model?
- 5. If one does not want to explicitly model action potential generation using Na+ and
K+ channels, what is a good alternative? How is a refractory period modeled in that case? How can noise be introduced in these simulations?
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Input Output
B.
Input Output Firing threshold Input Output
C.
Firing threshold Firing threshold
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(I) (II) (III)
SLIDE 9 (I) (II) (III)
+
(I) (II
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Air Odor
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Firing rate
Number of carbons 3 4 5 6 7 8 9 10
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http://www.youtube.com/watch?v=Cw5PKV9Rj3o&playnext=1&list=PLDB 130AF47B7A853C&feature=results_main
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) ) s ( 2 1 exp( ) s ( f
2 f max max
s r σ
− − =
rmax σf
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f(s) = r0 + (rmax-r0) cos (s-smax) rmax r0
SLIDE 22 f(s) = r0 + (rmax-r0) cos (s-smax)
) ) s ( 2 1 exp( ) s ( f
2 f max max
s r σ
− − =
Firing rate Number of carbons 3 4 5 6 7 8 9 10
primary motor cortex
primary visual cortex
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- Exercise. a) Construct a tuning curve for the following experiment. You are
recording from a visual neuron on the thalamus. The cell has a spontaneous firing rate of 10 Hz (spontaneous firing rate is how much the cell fires when NO stimulus is applied). You are moving the stimulus on a 6x6 grid and record the following average numbers of spikes at each location. Its not easy to draw such a 3-dimensional tuning curve, so be creative.
5 5 5 5 10 5 10 10 10 10 10 5 10 15 15 15 10 5 10 15 20 15 10 5 10 15 15 15 10 5 10 10 10 10 10 5
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- Exercise. a) Construct a tuning curve for the following experiment. You are
recording from a visual neuron on the thalamus. The cell has a spontaneous firing rate of 10 Hz (spontaneous firing rate is how much the cell fires when NO stimulus is applied). You are moving the stimulus on a 6x6 grid and record the following average numbers of spikes at each location. Its not easy to draw such a 3-dimensional tuning curve, so be creative.
5 5 5 5 10 5 10 10 10 10 10 5 10 15 15 15 10 5 10 15 20 15 10 5 10 15 15 15 10 5 10 10 10 10 10 5
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 5 10 15 20
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(b) How could the stimulus create a spiking response that is less than the spontaneous rate?
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labeled line rate coding across fiber pattern
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- Exercise. Determine a “rule” constructed from these three tuning curves
that would allow you to know intermediate light wave length.
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SLIDE 31 45 90 135 180 225 270 369
SLIDE 32 180o right 90o up 0o left α
(left) (right) (up)
x left = Const * cos ( α) x up = Const * sin ( α) x right = -Const * cos (α)
SLIDE 33 180o right 90o up 0o left θ
x = f( θ) I post = Σ w*x
w: synaptic weight or connection strength
x post = I
post
Ipost = wleft*xleft + xup*xup + wright*xright.
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180 left up righ t 0.6*x left + 0.4*xup 0.4*x up + 0.6*xright 0.6*x up + 0.4*xright
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left up righ t 0.6*xleft+ 0.4*xup
max at 0 max at 90 max ~40
SLIDE 36 180o right 90o up 0o left θ
x = f(θ) Ipost = Σ w*x
x post = F(I post)
Linear threshold function: if Ipost > Θ x post = Ipost if Ipost <= Θ x post = 0
SLIDE 37 0.6*x
left + 0.4*x up
0.4*x
up + 0.6*x right
0.6*x
up + 0.4*x right
180 left up righ t
SLIDE 38 0.6*x
left + 0.4*x up
0.4*x
up + 0.6*x right
0.6*x
up + 0.4*x right
180 left up righ t
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180 left up righ t
180 left up righ t
SLIDE 40 180o right 90o up 0o left α
(left) (right) (up)
Xleft = F(const*cos (α), θ) Xup = F(const*sin (α), θ) Xright = F(-const*cos (α), θ)
SLIDE 41 xleft 0.6*xleft+0.4*xup xup
SLIDE 42 180o right 90o up 0o left θ
(left) (right) (up)
Xleft = F(const*cos (α), θ) Xup = F(const*sin (α), θ) Xright = F(-const*cos (α), θ)
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xleft 0.6 xleft+0.4 x2
up xup
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100 200
1
100 200
1
100 200
1
100 200
1
100 200
1
100 200 0.5 1
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Exercise: Draw the network and write the equations for a “push-pull” type computation.
SLIDE 46 0.6*(xleft
- (-xup)) + 0.4*(xup-(-xleft))
SLIDE 47 amplitude 1.5 (0.6N1+0.4N2) amplitude 1
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- Exercise. (a) Create a network that can resolve 9 different colors
from the three color tuning curves. Write down the equations and define what colors would be approximately resolved.
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Real-time control of a robot arm using simultaneously recorded neurons in the motor cortex John K. Chapin1, Karen A. Moxon1, Ronald S. Markowitz1 and Miguel A. L. Nicolelis2
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