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Σ
tone airpuff eyeblink
Σ
tone airpuff eyeblink time airpuff tone weight airpuffeyeblink weight
Hebbian learning rule Wij (t) = *xj*xi Wij (t) = F(xj, xi, , t, - - PowerPoint PPT Presentation
Hebbian learning rule Wij (t) = *xj*xi Wij (t) = F(xj, xi, , t, - - PowerPoint PPT Presentation
Hebbian learning rule Wij (t) = *xj*xi Wij (t) = F(xj, xi, , t, ) Time? airpuff airpuff weight tone eyeblink tone eyeblink airpuffeyeblink weight tone airpuff time What is an association? Pavlov: Eye blink:
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What is an association? Pavlov: Eye blink: Other:
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Linear associator: Notations and vectors
- 1. One postsynaptic neuron:
- 2. Many postsynaptic neurons:
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xi = Ii xi = Σwij xj Δwij = γ * gi * fj
Write out for 3 pre and 3 post neurons
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⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = fn . . fj 2 f 1 f f ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = gn . . gi 2 g 1 g g
Δwij = γ * gi * fj Pre post
[ ]
fn ... fi ,.. 2 f , 1 f gn . . gj 2 g 1 g ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = γ =
f
T
g w
⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = = = = = = = = fngn wnn ... gnf2 w2n 1 gnf 1 wn 1 f 2 g 21 w g1fn w1n ... g1f2 w12 1 f 1 g 11 w w
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layer f layer g f1 f2 fj fn g1 g2 gi gn weight matrix w w11 w12 w1n g1 = x(f1) w11 + x(f2) w12 ... x(fj) w1j ... + x(fn) w1n gi = x(f1) wi1 + x(f2) wi2 ... x(fj) wij ... + x(fn) win etc ..
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Olfactory system Auditory system Grandmother baking cookies and talking
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Olfactory system Auditory system Grandmother baking cookies and talking
Δwij = γ * gi * fj.
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Olfactory system Auditory system Cookies
x(gi) = Σ wij * x(fj)
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Olfactory system Auditory system Bread
x(gi) = Σ wij * x(fj)
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Olfactory system Auditory system Cookies
x(gi) = Σ wij * x(fj)
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Olfactory system Auditory system Aunt cooking stew and talking
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Olfactory system Auditory system Aunt baking cookies and talking
Grandmother’s voice recalled!
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Calculate example here
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input from olfactory bulb feeback interactions (association fibers intrinsic cnnection)
- utput
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input from olfactory bulb feeback interactions (association fibers intrinsic cnnection)
- utput
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input from olfactory bulb feeback interactions (association fibers intrinsic cnnection)
- utput
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"Hopfield networks", which have a recurrent structure and the development of which is inspired by statistical physics. They share the following features: Nonlinear computing units (or neurons) Symmetric synaptic connections (wij = wji) No connections of a neuron on itself (wii = 0) Abundant use of feedback (usually, all neurons are connected to all
- thers)
(Feedback means that a neurons sends a synapse to a neuron it also receives a signal from, so that there is a closed loop).
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state] previous its in remains j unit [if if 1
- if
1 { and *
v v v x x w v
i N 1 j i i i j ij i
= < > + = = ∑
=
j i if j i if * N 1
w x x w
ij j i ij
= = ≠ =
∑∑
= =
− =
N 1 i N 1 j j i ij
* * 2 1 E
v v w
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Initial condition Trajectory Minimum
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N1 N2 w12 w21 v1, v2, v3: inputs to neuron 1, 2 x1, x2, x3: outputs of neuron 1,2 w12, w21 etc: connection strength N3 w13 w31 w32 w23
Show that (1,1,1) is stable when all w = 1
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N1 N2 w12 w21 v1, v2, v3: inputs to neuron 1, 2 x1, x2, x3: outputs of neuron 1,2 w12, w21 etc: connection strength N3 w13 w31 w32 w23
Show that (-1,-1,-1) is stable when all w = 1
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Initial condition Trajectory Minimum
Explain attractor and basin of attraction
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E(1,1,1) = -3 E(-1,-1,-1) = -3 E(-1,1,-1) = 1 E(1,-1,1) = 1 E(-1,1,1)=1 (-1,-1,1) E(1,-1,-1) = 1 (1,1,,-1)
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Calculate example from notes
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Does it work well? Is it realistic?
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