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Introduction
• Learning occurs through synaptic plasticity
• Hebb (1949): If neuron A often contributes to the firing of neuron B, then the synapse from A to B should be strengthened– Stimulus response (Pavlov)
– Converse: If neuron A does not contribute to the firing of B, the synapse is weakened
– Hippocampus, neocortex, cerebellum
LTP and LTD at Shaffer collateral inputs to CA1 region of rat hippocampal slice. High stimulation yields LTP. Low stimulation yields LTD. NB: no stimulation yields no LTD
Function of learning
• Unsupervised learning (ch 10)– Feature selection, receptive fields, density estimation
• Supervised learning (ch 7)– Input-output mapping, feedback as teacher signal
• Reinforcement learning (ch 9)– Feedback in terms of reward, similar to control theory
• Hebbian learning (ch 8)– Biologically plausible + normalization
– Covariance rule for (un)supervised learning
– Occular dominance, maps
Rate model with fast time scale
• Neural activity as continuous rate, not spike train
V is output neuron, u is vector input neurons, w is vector of weightsIf tau_r small wrt learning time:
Basic Hebb rule
• V and u are functions of time. Makes dynamics hard to solve. Alternative is to assume v,u from distribution p(v,u) and assume p time independent.
Using v=w. u we get
Basis Hebb rule
• Hebb rule is unstable, because norm always increases
• Continuous differential equation can be simulated using Euler scheme
Covariance rule
• Basic Hebb rule describes only LTP since u, v positive
• LTD occurs when pre-synaptic activity co-occurs with low post synaptic activity
• Alternatively,
BCM rule
• Bienenstock, Munro, Cooper (1982):
Requires both pre and post synaptic activity for learning
For fixed threshold, the BMC rule is also unstable
BMC rule
• BMC rule can be made stable by threshold dynamics
• Tau_theta is smaller than tau_w
• BMC rule implements competition between synapses– Strenghtening one synapse, increases the threshold, makes
strengthening of other synapses more difficult
• Such competition can also be implemented by normalization
Synaptic normalization
• Limit the sum of weights or sum of squared weights
• Impose this constraint rigidly, or dynamically
• Two examples:– Rigid scheme for sum of weights constraint (subtractive norm.)
– Dynamic scheme for sum of squared weights (multipl. Norm.)
Subtractive normalization
• Subtractive normalization ensures that sum w does not change
Not clear how to implement this rule biophysically (non-locality).
We must add a constraint that weights are non-negative.
Unsupervised learning
• Adapting the network for a set of tasks – Neural selectivity, receptive field
– Cortical map
• Process depends partly on neural activity and partly not (axon growth)
• Ocular dominance– Adult neurons favor one eye over the other (layer 4 input from
LGN)
– Neurons are clustered in bands or stripes
Single post-synaptic neuron
• Solution in terms of eigenvalues of Q
• Eigen values are positive, so solution explodes.
• Asymptotically
– e1 is the principle eigen direction
– Neuron projects input onto this direction:
Single post-synaptic neuron
• Example with two weights. • Weights grow indefinite, one positive one negative. Choice
depends on initial conditions. • Limit to [0, 1] yields different solutions depending in init value
Single post-synaptic neuron
• Subtractive normalization. Averaging over inputs:
• Analysis in terms of eigenvectors:– In ocular dominance e1=n/sqrt(n). W in direction of e1 has rhs
equal to zero. Ie this component of w is unaltered
– In other directions normalizing term is zero
– W asymptotically dominated by second eigenvector
Hebbian development of ocular dominance
Subtractive normalization may solve this, since e1=n weight grows proportional to e2=(1,-1)
Single post-synaptic neuron
• Using the Oja rule
• Show that each eigenvector of Q is solution.
• One can show that only principal eigenvector is stable.
Single post-synaptic neuron
• A: Behavior of– Unnormalized Hebbian learning
– Multiplicative normalization (Oja rule)
gives w propto e1. This is similar to PCA.
• B: Shifting mean of u may yield different solution
• C: Covariance based learning corrects for mean
• Saturation constraints may alter this conclusion
Hebbian development of ocular dominance
• Model layer 4 cell with input from two LGN cells, each associated with different eye.
Hebbian development of orientation selectivity
Cortical receptive fields from LGN. ON-center (white) and OFF-center (black) cells excite cortical neuron.
Spectral analysis also applicable to non-linear systemsDominant eigenvector uniform.Non-uniform receptive fields result from sub-dominanteigenvector
Hebbian development of ocular dominance stripes
• A: model with right and left eye inputs drive array of cortical neurons
• B: ocular dominance maps. Top: light and dark areas in top and bottom cortical layer show ocular dominance in cat primary cortex. Bottom: model of 512 neurons with Hebbian learning
Hebbian development of ocular dominance stripes
• Use 8.31 with W=(w+,w-) the n*2 matrix. See book.
• Subtractive normalization dw+/dt=0
• Ocular dominance pattern given by largest eig. vector of K
Hebbian development of ocular dominance stripes
• Suppose K translation invariant – Periodic boundary conditions simulating a patch of cortex ignoring
boundary effects
– Eigenvectors are
– Eigenvalues are Fourrier components
– Solution of learning is spatially periodic (viz. fig 8.7)
Feature based models
• Multi dimensional input (retinal location, ocular dominance, orientation preference, ....)
• Replace input neurons by input features, W_ab is selectivity of neuron a to feature b– Feature u1 is location on retina in coordinates
– Feature u2 is ocularity (how much is the stimulus prefering left over right eye), a single number
• The coupling to neuron a describe the preferred stimulus
• Activity of output a is
Feature based models
• Output is soft-max
• Combined with lateral averaging– Self-organizing map (SOM)
– Elastic net
Feature based models
optical imaging shows ocularity and orientation selectivity in macaque primary visual cortex. Dark lines are ocular dominance boundaries, light lines are iso-orientation contours. Pin wheel singularities, linear zones
Feature based models
• Elastic net output• SOM, competitive Hebbian rules can produce similar output
Anti-hebbian modification
• Another way to make different outputs specialize is by adaptive anti-Hebbian modification• Consider Oja rule:
• Each output a will be identical • Anti-Hebbian modification is shown at synapses from parallel fibers to Purkinje cells in
cerebellum.
• Combination yields different eigenvectors as outputs
Timing based rules
• Left: in vitro cortical slice. Right: in vivo xenopus tadpoles
• LTP when pre-synaptic spike precedes post-synaptic spike
• LTD when pre-synaptic spike follows post-synaptic spike
Timing based rules
• Simulating spike-time plasticity requires spiking neurons
• Approximate description with firing rates
• H(t) positive/negative for t positive/negative
Timing based plasticity and prediction
• Consider array of neurons labeled by a with receptive fields f_a(s) (dashed and solid curves)
• Timing based learning
rule. Stimulus s moves
from left to right.
Timing based plasticity and prediction
• If a left of b, then link a to b is strengthened and link b to a is weakened. Receptive field of neuron a is asymmetrically deformed (A solid bold line)
• Prediction: next presentation of s(t) will activate a earlier.
• In agreement with shift of place field mean when rats run around track (B).
Classification and the Perceptron
• If output values are +/- 1 the model implements a classifier, called the Perceptron:
– The weight vector defines a separating hyper plane: = gamma.
– The perceptron can solve problems that are ‘linearly separable’