Testing the memory
To test the memory you have to start it in some initial states. As an initial
state you can either:
-
use one of the stored patterns. For example the last one which you have
imposed and which is still visible on the screen.
-
use a new random pattern by pressing "Ramdomize".
-
construct a pattern by clicking (press "Clear Display" before).
-
use a stored pattern and change a few pixels by clicking (use either "First
Pattern or "Last Pattern").
Test the retrieval properties by pressing "Test". Make sure that you do
NOT MEMORIZE the test pattern.
Pattern
A pattern u is a specific pixel set Xui
= +1 defined for all N neurons (pixels), { Xui
| 1 <= i <= N }. The patterns are labeled by the index
u with 1 <= u <= p.
States
A network state is the set { Si(t) | 1 <= i
<= N }, where Si is the neuronal activity with
Si(t) = +1 for a neuron which is active (red pixel)
at time t and Si(t) = -1 for an inactive
neuron (blue pixel).
Overlap
The overlap is a measure of similarity between the momentary network state
and one of the patterns. For example, the overlap with pattern u
is defined as mu = (1/N) 
i
(XuiSi). The overlap
is one if the momentary state of the network is identical with one of the
patterns.
Theoretical capacity
In the limit of a large network (N growing to infinity) a network
with N neurons can store p = a.N patterns. For the
standard Hebb-Hopfield learning rule and random patterns a = 0.14.
Optimized learning rules yield higher capacities. The capacity is also
large if the patterns are less correlated.
Correlation between patterns
The correlation between patterns u and v can be defined as
Cuv = (1/N) i
Xui Xvi.
Orthogonal patterns have Cuv = duv .