This presentation was meant to explain the contents of Ch. 19 of the book, Neuronal Dynamics http://neuronaldynamics.epfl.ch

1. Ch. 19 Synaptic plasticity and learning
Neural Computation Unit
Biological physics theory Unit
Ph.D candidate
Hiroaki Hamada

2. 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

3. Contents
19.1 Hebb rule and experiments
19.2 Models of Hebbian learning
- Oja’s rule

4. 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

5. 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

6. Example: Voltage dependence of LTP
LTP and LTD are dependent on
amplitude of the membrane potential
of the post-synaptic neuron.

7. 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
tj
f: timing of firing in the j-th neuron
ti
f: timing of firing in the i-th neuron

8. Short Abridgement
 Two main factors for synaptic
plasticity
- Neural activity of both pre-and-post
synaptic neurons
- Timing of pairing (STDP rule)

9. 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 c11
corr is constant….
If c11
corr > 0, called the Hebbian Rule.
If c11
corr < 0, called the anti-Hebbian Rule.

10. But what if c11
corr is dependent on wij….?
Β: usually 1 but can be higher order
If 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

11. 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

12. 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)
wher
e The time course of the
learning window
(exponential function)
Non-Hebbian
contribution (like c1
corr)

13. 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 LTD
faster decay
slower decay
A trace contributed by j-th neuron
LTD
LTP
Good fitting to experimental results
(H.-X. Wang et al., 2005).

14. Short Abridgement 2
 Different Hebbian forms (Oja’s rule)
 Triplet model is currently a biologically
plausible model.

15. Show that Oja rule converges to the state |w2|=1
The Oja rule in the matrix form:

16. Show that Oja rule converges to the state |w2|=1
The Oja rule in the matrix form:

17. 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