Download Document

3.2. Neurons and their networks
3.2.1 Biological neurons
Tasks such as navigation, but also
cognition, memory etc. happen in the
nervous system (more specifically the
brain).
The nervous system is made up of
several different types of cells:
- Neurons
- Astrocytes
- Microglia
- Schwann cells
Neurons do the computing, the rest is
infrastructure
Astrocytes
Star-shaped, abundant,
and versatile
Guide the migration of
developing neurons
Act as K+ and NT buffers
Involved in the formation
of the blood brain barrier
Function in nutrient
transfer
Microglia
Specialized immune
cells that act as the
macrophages of the
central nervous
system
Schwann cells and Oligodendrocytes
Produce the
myelin sheath
which provides
the electrical
insulation for
neurons and
nerve fibers
Important in
neuronal
regeneration
Myelination – electrically insulates the
axon, which increases the transport
speed of the action potential
Types of neurons
Brain
Sensory
Neuron
Lots of
interneurons
Motor
Neuron
What they look like
...or schematically
In fact, things are a bit more crowded
Neurons
communicate with
each other, we will
see later how this
works. This will be
the "neural network"
Thus, neurons need to be able to conduct
information in 2 ways:
1.From one end of a neuron to the other
end.This is accomplished electrically via
action potentials
2.Across the minute space separating one
neuron from another. This is accomplished
chemically via neurotransmitters.
Resting potential of neurons
K+
Cl-
Na+
Outside of Cell
Cell Membrane at rest
Na+
K+
A-
- 70 mV
ClInside of Cell
Potassium (K+) can
pass through to
equalize its
concentration
Sodium and
Chlorine
cannot pass
through
Result - inside is
negative relative to
outside
Now lets open a Na channel in the
membrane...
If the initial amplitude of the GP is sufficient, it will
spread all the way to the axon hillock where V-gated
channels reside. At this point an action potential can
be excited if the voltage is high enough.
N.B. The gating properties of ion channels were
determined long before it was known they
existed from electrical measurements
(conductivity of squid axons to Na and K)
Similar for the transport
of K – the different
coefficients imply the
number of opening and
gating bits...
With modern crystalography, these effects have
been observed...
Transport of the action potential, like
a row of dominos falling...
This goes a lot faster with myelinated
axons – saltating transport...
Once at the syapse,
the signal is
transmitted
chamically via
neurotransmitters
(e.g. Acetylcholin)
These are then
used to excite a
new graded
potential in the
next neuron
This graded
potential can
be both
positive and
negative,
depending
on the
environment
The intensity of the signal is given by
the firing frequency
These properties are caricatured in
the McCulloch-Pitts neuron
Learning happens when the weights wij
are changed in response to the
environment – this needs an updating rule
Common in informatics is the iterative
learning, which needs a teacher. I.e.
The weights are adjusted so that in
every learning step, the distance to
the correct answer is obtained.
This is known as the perceptron
Input Signals
Out put Signals
With the use of hidden layers, not
linearly separable variable can be
learnt...
Input
layer
First
hidden
layer
Second
hidden
layer
Output
layer
An example: letter recognition
The problems that can be solved depend on
the structure of the network
3.2.2 Hebbian learning
This means that a synapse gets stronger as
neighbouring cells are more correlated
Hebb’s Law can be represented in the form of
two rules:
1. If two neurons on either side of a connection
are activated synchronously, then the weight of
that connection is increased.
2. If two neurons on either side of a connection
are activated asynchronously, then the weight of
that connection is decreased.
Hebb’s Law provides the basis for learning
without a teacher. Learning here is a local
phenomenon occurring without feedback
from the environment.
i
j
wij ( p)   y j ( p) xi ( p)
Output Signals
Input Signals
Hebbian learning in a neural network
A Hebbian Cell Assembly
By means of the Hebbian Learning Rule, a
circuit of continuously firing neurons could be
learned by the network.
The continuing activation in this cell
assembly does not require external input.
The activation of the neurons in this circuit
would correspond to the perception of a
concept.
A Cell Assembly
Input from the environment
A Cell Assembly
Input from the environment
A Cell Assembly
Input from the environment
A Cell Assembly
Input from the environment
A Cell Assembly
Note that the input from the
environment is gone...
A Cell Assembly
Hebbian learning implies that weights can only
increase. To resolve this problem, we might
impose a limit on the growth of synaptic
weights. It can be done by introducing a nonlinear forgetting factor into Hebb’s Law:
wij ( p)   y j ( p) xi ( p)   y j ( p) wij ( p)
where  is the forgetting factor.
The forgetting factor usually falls in the
interval between 0 and 1, typically between
0.01 and 0.1, to allow only a little “forgetting”
while limiting the weight growth.
First simulation of Hebbian learning
• Rochester et al. attempted to simulate the
emergence of cell assemblies in a small
network of 69 neurons. They found that
everything became active in their
network.
• They decided that they needed to include
inhibitory synapses. This worked and cell
assemblies did, indeed, form.
• This was later confirmed in real brain
circuitry.
In fact, these inhibitory connections
are distance dependent and as such
give rise to structure
Exciation happens within columns
and inhibition further away
Long range inhibition and short range
activation gives rise to patterns
1
Connection
strength
Excitatory
effect
0
Inhibitory
effect
Distance
Inhibitory
effect
See also the excursion into pattern formation in Sec 3.6
Feature mapping Kohonen model
Kohonen layer
Kohonen layer
Input layer
1
0
Input layer
0
1
(b)
(a)
 ( xi  wij ), if neuron j wins the competitio n
wij  
0,
if neuron j loses the competitio n

Competitive learning
Set initial synaptic weights to small random values,
say in an interval [0, 1], and assign a small positive
value to the learning rate parameter .
1/ 2


2
jX ( p)  min X  W j ( p)  [ xi  wij ( p)] 
j
 i 1

Update weights:
n
,
wij ( p  1)  wij ( p)  wij ( p)
j(p) is the neighbourhood function centred around jX
Iterate...
To illustrate competitive learning, consider the
Kohonen network with 100 neurons arranged in
the form of a two-dimensional lattice with 10
rows and 10 columns. The network is required
to classify two-dimensional input vectors  each
neuron in the network should respond only to
the input vectors occurring in its region.
The network is trained with 1000 twodimensional input vectors generated randomly
in a square region in the interval between –1
and +1. The learning rate parameter  is equal
to 0.1.
Initial random network
1
0.8
0.6
0.4
W(2,j)
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1
-0.8
-0.6
-0.4
-0.2
0
W(1,j)
0.2
0.4
0.6
0.8
1
After 100 steps
1
0.8
0.6
0.4
W(2,j)
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1
-0.8
-0.6
-0.4
-0.2
0
W(1,j)
0.2
0.4
0.6
0.8
1
After 1000 steps
1
0.8
0.6
0.4
W(2,j)
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1
-0.8
-0.6
-0.4
-0.2
0
W(1,j)
0.2
0.4
0.6
0.8
1
After 10000 steps
1
0.8
0.6
0.4
W(2,j)
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1
-0.8
-0.6
-0.4
-0.2
0
W(1,j)
0.2
0.4
0.6
0.8
1
Or for letter recognition
In the cortex, this gives rise to the
homunculus, the spatial distribution of
nerve cells responsible for senses
Similar for other features in the cortex
3.2.3 Associative networks
x1
1
y1
x2
2
y2
xi
i
yi
xn
n
yn
Output Signals
Input Signals
In a Hopfield Network, every neuron
is connected to every other neuron
Topological state analysis for a
three neuron Hopfield network
y2
(1, 1, 1)
(1, 1, 1)
(1, 1, 1)
(1, 1, 1)
y1
0
(1, 1, 1)
(1, 1, 1)
(1, 1, 1)
y3
(1, 1, 1)
W
M
T
Y
Y
 m m M I
m1
The stable state-vertex is determined
by the weight matrix W, the current
input vector X, and the threshold
matrix . If the input vector is
partially incorrect or incomplete, the
initial state will converge into the
stable state-vertex after a few
iterations.
Si (t  1)  sgn(  j wij S j (t )   i )
Energy function of Hopfield net:
multidimensional landscape
1
H    wi , j Si S j
2
wij  v v
p p
i j
Example: Restoring corrupted memory
patterns
Original T
Half is
corrupted
20% of T
corrupted
Recap Sec. 3.2
The brain is a network of neurons, whose
properties are important in how we learn
Within neurons, signals are transported
electrically, between chemically
This can be abstracted in a McCulloch Pitts
neuron
Hebbian learning makes strong connections
stronger (leads to pattern formation)
This is taken further in Kohonen networks and
competitive learning