Ch. 19 Synaptic plasticity and learning

Neural Computation UnitBiological physics theory Unit

Ph.D candidateHiroaki Hamada

Synaptic plasticity and learning Learning and neural mechanism

Synaptic structure

Potentiation of synaptic efficacy (long-term potentiation, LTP)

Reduction of synaptic efficacy (long-term depression, LTD)

Learning rule (Hebbian rule): Pre- and Post-synaptic neurons’ activity

Contents

19.1 Hebb rule and experiments19.2 Models of Hebbian learning

- Oja’s rule

19.1 Hebb rule and experiments

Hebbian Learning (Hebb, 1949) Synaptic plasticity (1970’s studies) ‘When an axon of cell A is near enouth to excite

cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cell such that A’s efficiency, as one of the cells firing B, is increased.’

“fire together, wire together ”(Shatz, 1992)

Stabilization of neural dynamics Hopfield network

Unsupervised learning No notion about ‘good’ or ‘bad’

Donald O. Hebb (1904-85)from wikipedia

19.1.1 Long-term potentiation (LTP)

LTP is a persistent strengthening of synapses based on pre-and-post synaptic activity.

A. presynaptic stimulation and postsynaptic recording- means of a second electrode- Test stimulation

B. A sequence of high-frequency stimulation

C. presynaptic pulse stimulation and see response.

- check the response to pulse stimulation- Compare to the first stimulation

Whether increase of relative amplitude

Example: Voltage dependence of LTP

LTP and LTD are dependent on amplitude of the membrane potential of the post-synaptic neuron.

19.1.2 Spike-timing-dependent plasticity (STDP)

Paring window is spike-timing dependent

Bi and Poo(2001) for the review

A. intracellular electrodes in both pre-and-post synaptic neurons (j and i). Test stimulation to the pre-

synaptic neuron B. both neurons are

repeatedly stimulated at a very precise timing.

C. test stimulation and observe response.

Pairing timing decides increase or decrease of synaptic plasticity, and its amplitude.

ij

tjf: timing of firing in the j-th neuron

tif: timing of firing in the i-th neuron

Short Abridgement

Two main factors for synaptic plasticity

- Neural activity of both pre-and-post synaptic neurons- Timing of pairing (STDP rule)

19.2 Models of Hebbian learning

Firing rate models are used for the artificial networks.

Two aspects of weight change based on the Hebbian Rule

Locality and Joint activity

Bilinear term or higher order terms are necessary for the Hebbian Rule such as

19.1

Undetermined function F()

Apply Taylor expansion w.r.t. vi = vj = 0

Joint activity

In the case of c11corr is constant….

If c11corr > 0, called the Hebbian Rule.

If c11corr < 0, called the anti-Hebbian Rule.

But what if c11corr is dependent on wij….?

Β: usually 1 but can be higher orderIf γ2 is constant, called hard-bound rule.If γ2 is not constant, called soft-bound rule.

Oja’s rule (Oja, 1982)

An upper-bound model

Oja’s rule is mathematical proven to be principal component analyzer (Oja, 1982; see exercises).

Oja’s ruleHebbian archetype

19.2.2 Pair-based models of STDP

Under spike models Updates (19.10)

where

: the dependence of the update on the current weight of the synapse

19.2.2 Pair-based models of STDP

Under spike models Updates (19.10)

where

: the dependence of the update on the current weight of the synapse

the standard form (19.11)

whereThe time course of the learning window (exponential function)

Non-Hebbian contribution (like c1

corr)

19.2.3 Generalized STDP models Two problems with pair-based STDP models

1. Increase of Repetition frequency -> depressing interaction increased (model prediction)- Experiments showed it did not happen (Sjöström et al.,

2001) meaning increase of net potentiation.2. Other experiments were done with different stimulation protocols such as repeated symmetrical triplets’ stimulation (pre-post-pre and post-pre-post)- A model indicates both responses should be same since each

includes the other in case of repeated protocol.

Triplet model (Pfister and Gerstner (2006))

Different decays of traces for LTP and LTDfaster decay

slower decay

A trace contributed by j-th neuron

LTD

LTP

Good fitting to experimental results (H.-X. Wang et al., 2005).

Short Abridgement 2

Different Hebbian forms (Oja’s rule) Triplet model is currently a biologically

plausible model.

Show that Oja rule converges to the state |w2|=1

The Oja rule in the matrix form:

Show that Oja rule converges to the state |w2|=1

The Oja rule in the matrix form:

Ch19 (b) Show that only the eigenvector e1 with the largest eigenvalue is stable

Assume that a weight vector w = e1 + ε ek has a small perturbation ε << 1 in the principal direction.

therefore

λ1 is stable when λ1 > λk for every k but otherwise ε grows