Lecture 1: Neurons Lecture 2: Coding with spikes Lecture 3: Tuning - PowerPoint PPT Presentation

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

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?

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. 2500 Hz 60dB 2000 Hz 50dB 1500 Hz 40dB 1000 Hz 30dB Tone (1000 ms) 1500 Hz Tone (1000 ms)

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?

A. Input Firing threshold Output B. Input Firing threshold Output C. Input Firing threshold Output

(I) (II) (III)

+ - + - (I) (II

- + (I) (I) (II) (II (III) + -

Air Odor

Firing rate 3 4 5 6 7 8 9 10 Number of carbons

http://www.youtube.com/watch?v=Cw5PKV9Rj3o&playnext=1&list=PLDB 130AF47B7A853C&feature=results_main

r max ) 2 ) max f s σ σ f − s ( 1 2 − exp( max r = ) s ( f

r 0 r max f(s) = r 0 + (r max -r 0 ) cos (s-s max )

− s s 1 primary motor cortex 2 = − max r f ( s ) exp( ( ) ) σ max 2 f olfactory bulb primary visual cortex Firing rate 3 4 5 6 7 8 9 10 f(s) = r 0 + (r max -r 0 ) cos (s-s max ) Number of carbons

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. 10 10 10 10 10 5 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 10 5 5 5 5

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. 6 20 10 10 10 10 10 5.5 5 5 5 10 15 15 15 10 4.5 15 4 5 10 15 20 15 10 3.5 5 10 15 15 15 10 3 10 5 10 10 10 10 10 2.5 2 5 10 5 5 5 5 1.5 1 5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

(b) How could the stimulus create a spiking response that is less than the spontaneous rate?

labeled line rate coding across fiber pattern

Exercise. Determine a “rule” constructed from these three tuning curves that would allow you to know intermediate light wave length.

369 270 225 180 135 90 45 0

90 o up x left = Const * cos ( α ) 0 α 0 o left 180 o right -180 -90 0 +90 +180 (left) (right) (up) x up = Const * sin ( α ) 0 x right = -Const * cos ( α ) 0

x = f( θ ) 90 o up I post = Σ w*x θ 0 o left 180 o right x post = I post w: synaptic weight or connection strength I post = w left *x left + x up *x up + w right *x right.

0.6*x left + 0.4*x up 0.6*x up + 0.4*x right 0.4*x up + 0.6*x right -180 -90 0 90 180 left up righ t

max at 0 max at 90 max ~40 0.6*x left + 0.4*x up -180 -90 0 90 180 left up righ t

x = f( θ ) 90 o up I post = Σ w*x θ 0 o left 180 o right x post = F(I post ) Linear threshold function: if I post > Θ x post = I post if I post <= Θ x post = 0

0.6*x left + 0.4*x 0.6*x up + 0.4*x right up 0.4*x up + 0.6*x right -180 -90 0 90 180 left up righ t

0.6*x left + 0.4*x 0.6*x up + 0.4*x right up 0.4*x up + 0.6*x right -180 -90 0 90 180 left up righ t

-180 -90 0 90 180 left up righ t -180 -90 0 90 180 left up righ t

90 o up X left = F(const*cos ( α ), θ ) α 0 o left 180 o right -180 -90 0 +90 +180 (left) (right) (up) X up = F(const*sin ( α ), θ ) X right = F(-const*cos ( α ), θ )

xup 0.6*xleft+0.4*xup xleft

90 o up X left = F(const*cos ( α ), θ ) θ 0 o left 180 o right -180 -90 0 +90 +180 (left) (right) (up) X up = F(const*sin ( α ), θ ) X right = F(-const*cos ( α ), θ )

x left 0.6 x left +0.4 x2 up x up 180 - 90 0 90 -

1 1 0 0 -1 -1 -200 -100 0 100 200 -200 -100 0 100 200 1 1 0 0 -1 -1 -200 -100 0 100 200 -200 -100 0 100 200 1 1 0 0.5 -1 0 -200 -100 0 100 200 -200 -100 0 100 200

Exercise: Draw the network and write the equations for a “push-pull” type computation.

-(-x up )) + 0.4*(x up -(-x left )) 0.6*(x left

amplitude 1.5 (0.6N1+0.4N2) amplitude 1

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.

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