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Tuning tuning curves
So far: Receptive fieldsRepresentation of stimuliPopulation vectors
Today: Contrast enhancment, cortical processing
0 45 90 135 180 225 270 369
Firin
g fr
eque
ncy
N1 N2
N3N4 N5
smax(N1) = 40o
smax(N2) = 110o
smax(N3) = 135o
smax(N4) = 180o
smax(N5) = 230o
s1
10
90o
x
y
0o 180oN1
N2 N3
N4
N5
90o
x
y
0o 180o
N2 N3
90o
x
y
0o 180o
N2
N3N2+N3
0 45 90 135 180 225 270 369
Firin
g fr
eque
ncy
N1 N2
N3N4 N5
smax(N1) = 40o
smax(N2) = 110o
smax(N3) = 135o
smax(N4) = 180o
smax(N5) = 230o
s1
10
90o
x
y
0o 180oN1
N2 N3
N4
N5
90o
x
y
0o 180o
N2 N3
90o
x
y
0o 180o
N2
N3N2+N3
Winner take all
In many sensory systems, the tuning curves of neurons are not identical to the receptive fields projected to these neurons by sensory neurons. They may be more narrow, include inhibitory parts of the curve, they may be wider or more separated from each other. There are a number of processes that “tune” tuning curves, these include interactions between neurons such as inhibition, excitation and feedback interactions. As we noted before, sensory receptive fields are often broad and relatively non-specific, for example frequency tuning curves in the auditory nerve can span a large range of frequencies.
Frequency (100 Hz)1 2 3 4 5 6 7 8 9 10
Firin
g ra
te (H
z)100
80
60
40
20
0Stimulus
Frequency (100 Hz)1 2 3 4 5 6 7 8 9 10
Firi
ng r
ate
(Hz)
100
80
60
40
20
0 Stimulus
N1 = 20-50 = -30 (=0)N2 = 50 – 20 – 10 = 20N3 = 10-50 = -40 (=0)
Exercise: Assuming linear interactions and all synaptic weights being zero, construct the approximate resulting receptive fields for these neurons and this network:
frequency
100 200 300 400 500 600
100
80
60
40
20
0
Excitatory neurons N1, N2, N3
N1
N2 N3
frequency
100 200 300 400 500 600
100
80
60
40
20
0
Inhibitory neurons I1, I2, I3
N1
N2 N3
for all neurons: x = infor all synapses: w=-1
outp
ut fr
eque
ncy
outp
ut fr
eque
ncy
Firin
g ra
te
Stimulus 1 Stimulus 2 Stimulus 2
Exercise: Look at the recordings below. Think about what you could learn from these and what additional information you would need to get useful information from this experiment.
Because of broad receptive fields and tuning curves, neural circuits are thought to enhance “contrast” or increase the difference between sensory stimuli in order to make them more easily recognizable, their features more salient, more distinguishable from each other.
Exercise: (a) You have a number of chairs and a number of tables. List features these have in common. Now list features that differentiate them. Write a list of yes no questions that would allow you to decide (i) if an object does belong to either category and (ii) if it is a chair or a table. Now find some examples that would not easily be classified. Create a neural network with a layer of feature detectors (respond to a specific feature), a layer of inhibitory neurons and one or two more layers of neurons including a layer of output neurons. At the output, you want to know if the object you detect is a chair or a table. Think about which features you want to suppress (inhibit) and which you want to have compete against each other.
Exercise: (a) You have a number of chairs and a number of tables.
List features these have in common.
Now list features that differentiate them.
Write a list of yes no questions that would allow you to decide (i) if an object does belong to either category and (ii) if it is a chair or a table.
Now find some examples that would not easily be classified. Create a neural network with a layer of feature detectors (respond to a specific feature), a layer of inhibitory neurons and one or two more layers of neurons including a layer of output neurons. At the output, you want to know if the object you detect is a chair or a table. Think about which features you want to suppress (inhibit) and which you want to have compete against each other.
#of carbons
act ivity
U3
U4
(4)CHO
U3
U4
(5)CHO
U3
(6)CHO
U3
U4
U3
U4
U3
U4
(5)CHO-(4)CHO (6)CHO-(5)CHO (4)CHO-(6)CHO
Distance: 2)43( UUD −=
U3
U4
U3
U4
U3
U4
(4)CHO
(5)CHO-(4)CHO (6)CHO-(5)CHO (4)CHO-(6)CHO
Dot product AB = A * B * cos(α)
(5)-(4) < (4)-(5) < (5)-(6)
#of carbons
act ivity
U3
U4
(4)CHO
U3
U4
(5)CHO
U3
(6)CHO
U3
U4
U3
U4
U3
U4
(5)CHO-(4)CHO (6)CHO-(5)CHO (4)CHO-(6)CHO
Distance: 2)43( UUD −=
U3
U4
#of carbons
act ivity
U3
U4
(4)CHO
U3
U4
(5)CHO
U3
(6)CHO
U3
U4
U3
U4
U3
U4
(5)CHO-(4)CHO (6)CHO-(5)CHO (4)CHO-(6)CHO
Distance: 2)43( UUD −=
U3
U4
U3
U4
U3
U4
(4)CHO
(5)CHO-(4)CHO (6)CHO-(5)CHO (4)CHO-(6)CHO
Dot product AB = A * B * cos(α)
(5)-(4) < (4)-(5) < (5)-(6)
How to compare vectors
#of carbons
act ivity
U3
U4
U3
U4
U3
(6)CHO
U3
U4
U3
U4
U3
U4
(5)CHO-(4)CHO (6)CHO-(5)CHO (4)CHO-(6)CHO
Distance: 2)43( UUD −=
U3
U4
U3
U4
U3
U4
(4)CHO
(5)CHO-(4)CHO (6)CHO-(5)CHO (4)CHO-(6)CHO
Dot product AB = A * B * cos(α)
(5)-(4) < (4)-(5) < (5)-(6)
Exercise: What happens to the distance measure and dot product measure if the vectors are “normalized” first (this means they all have length 1.0 and span the unit circle).
Photoreceptors in eye Auditory receptors in cochlea
Olfactory receptors in nose
Other retinal neurons Brain stem neurons Olfactory bulb
LGN in thalamus MGN in thalamus ?
Primary visual cortex Primary auditory cortex
Olfactory cortex
Secondary visual cortex
Secondary auditory cortex
Hippocampus
Association cortex
AcetylcholineNoradrenalineSerontonineDopaminePeptides ....
I. I. Molecular LayerMolecular Layer
II. II. External Granular LayerExternal Granular Layer
III.III. External Pyramidal LayerExternal Pyramidal Layer
Line of Line of KaesKaes--BechterewBechterew
IV.IV. Internal Granular LayerInternal Granular Layer
Outer band of Outer band of BaillargerBaillarger
-- Line of Line of GennariGennari in area 17 in area 17
V. V. Internal Pyramidal LayerInternal Pyramidal Layer
Giant pyramidal cell of BetzGiant pyramidal cell of Betz
Inner Band of Inner Band of BaillargerBaillarger
VI. VI. Polymorphic LayerPolymorphic Layer
Golgi Golgi NisslNissl WeigertWeigert
Inputs Outputs
Cell body
Pyramidal cell
Piriform cortex circuitry
Haberly, L.B. Chem. Senses, 10: 219 -38 (1985)
P P P
Ia
Ib
II
III
Afferent Input from OB mitral cells (LOT)
AssociationFibers
afferent input from olfactory bulb
association fibersfrom other pyramidalcells
cell body layer
deep interneuronsP
output
Piriform cortex circuitry
Haberly, L.B. Chem. Senses, 10: 219 -38 (1985)
P P P
Ia
Ib
II
III
Afferent Input from OB mitral cells (LOT)
AssociationFibers
afferent input from olfactory bulb
association fibersfrom other pyramidalcells
cell body layer
deep interneuronsP
output
Piriform cortex circuitry
Haberly, L.B. Chem. Senses, 10: 219 -38 (1985)
P P P
Ia
Ib
II
III
Afferent Input from OB mitral cells (LOT)
AssociationFibers
FF
FB
afferent input from olfactory bulb
association fibersfrom other pyramidalcells
cell body layer
deep interneuronsP
Feedforwardinterneurons
Feedback interneurons
output
Piriform cortex circuitry
Haberly, L.B. Chem. Senses, 10: 219 -38 (1985)
P P P
Ia
Ib
II
III
Afferent Input from OB mitral cells (LOT)
AssociationFibers
FF
FB
afferent input from olfactory bulb
association fibersfrom other pyramidalcells
cell body layer
deep interneuronsP
Feedforwardinterneurons
Feedback interneurons
output
Neuromodulatory inputsOther association fiber inputs
Layer II (cell bodies)
Layer Ib
Layer Ia
Stimulus : citronellal
activity patternacross mitral cells
activity patternacrosspyramidal cells
Stimulus : citronellal
activity patternacross mitral cells
activity patternacross pyramidal cells
feedforward inhibition
Stimulus : citronellal
activity patternacross mitral cells
activity patternacross pyramidal cells
feedforward inhibition
association fibers between pyramidal cells
xsj = cos(α-offsetj) when cos(α-offsetj) >= θandXsj = 0.0 when cos(α-offsetj) < θ
where offsetj = 0, -30, -60, -90 and -120.
Sensory neurons
Local interneurons
Pyramidal cells
α
Ipj = xsj; Iij = xsjxpj = Ipj and xij = Iij if I > θ and xpj = 0 and xij = 0 if I < θ.
Sensory neurons
Local interneurons
Pyramidal cells
α
Exercise. Lets work through an example[1] to get used to writing and reading equations. 1) We will start by constructing a network of neurons in motor cortex receiving inputs from presynapticneurons s with the following tuning curves as a function of the angle α of arm movement and α threshold θ:xsj = F(cos(α−offseti), θ) where offsetj = 0, -30, -60, -90 and -120.
Now, we will create 10 postsynaptic neurons (5 excitatory pyramidal cells and 5 inhibitory local interneurons), each receiving presynaptic neurons, assuming all synaptic weights w = 1:Ipj = xsj; Iinj = xsj with Ipj being the input to pyramidal cell j and Iij the input to interneuron j. These are linear threshold neurons with continuous output for now, so xpj = Ipj and xij = Iinj if I > θ and x = 0 and xij = 0 if I < θ. In vector notation:
[1] Notations: I: input to a neuron; x: output from a neuron; j: neuron index; θ: threshold; α: movement angle; s: sensory neuron; p: pyramidal neuron; in: interneuron.
Ipj = xsj – xi(j-1) - xi(j+1)
Sensory neurons
Local interneurons
Pyramidal cells
α
Sensory neurons
Local interneurons
Pyramidal cells
α
Ipj = xsj – xi(j-1) - xi(j+1) + 0.5*xpj
Ipj = xsj – xi(j-1) - xi(j+1) + 0.5*F(Ipj, Θ2)
Sensory neurons
Local interneurons
Pyramidal cells
α
Sensory neurons
Local interneurons
Pyramidal cells
α
Ipj = xsj – xi(j-1) - xi(j+1) + 0.5*F(Ipj, Θ2) + SUMk=j (wjk*xpk)
prob (xjp = 1.0) = F (Ijp, θ)
I
X
θ
X=F(I, θ) linear threshold functionX = 0 if I <= θX = I if I > θ
prob (xjp = 1.0) = F (Ijp, θ)
I
X
θ
X=F(I, θ) linear threshold functionX = 0 if I <= θX = I if I > θ
I
Prob (X=1)
θ
Prob (x=1) = 0 if I <= θProb (x=1) = I if I > θ
prob (xjp = 1.0) = F (Ijp, θ)
I
Prob (X=1)
θ
Prob (x=1) = 0 if I <= θProb (x=1) = I if I > θ
time
Θ = 0.0; I = 0.1 -> p = 0.1
On average, one spike every 10 time steps
time
Θ = 0.0; I = 0.9 -> p = 0.9
On average, nine spikes every 10 time steps
Sensory neurons
Local interneurons
Pyramidal cells
α
α = -100 α = 50