Plasticity and learning Dayan and Abbot Chapter 8

Plasticity and learning Dayan and Abbot Chapter 8

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 CA 1 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 weights If 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,

Covariance rule • When Either rule produces Covariance rule is unstable

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 (nonlocality). We must add a constraint that weights are non-negative.

Multiplicative normalization • Oja rule (1982) • The rule implements the constraint dynamically:

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 • We analyze Eq. 8. 5

Single post-synaptic neuron • Solution in terms of eigenvalues of Q • Eigen values are positive, so solution explodes. • Asymptotically – e 1 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 e 1=n/sqrt(n). W in direction of e 1 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 e 1=n weight grows proportional to e 2=(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 e 1. 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 Spectral analysis also applicable to non-linear systems Dominant eigenvector uniform. Non-uniform receptive fields result from sub-dominant eigenvector Cortical receptive fields from LGN. ON-center (white) and OFF-center (black) cells excite cortical neuron.

Multiple postsynaptic neurons

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 u 1 is location on retina in coordinates – Feature u 2 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).

Supervised Hebbian learning • Weight decay: • Asymptotic solution is

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’