575. 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).

576. 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

577. 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

578. Microglia
Specialized immune cells that act as the macrophages of the central nervous system

579. Schwann cells and Oligodendrocytes
Produce the myelin sheath which provides the electrical insulation for neurons and nerve fibers
Important in neuronal regeneration

580. Myelination – electrically insulates the axon, which increases the transport speed of the action potential

581. Types of neurons Brain Sensory Neuron Lots of interneurons Motor

582. What they look like

584. ...or schematically

585. In fact, things are a bit more crowded

586. Neurons communicate with each other, we will see later how this works
Neurons communicate with each other, we will see later how this works. This will be the "neural network"

587. Thus, neurons need to be able to conduct information in 2 ways:
From one end of a neuron to the other end.This is accomplished electrically via action potentials
Across the minute space separating one neuron from another. This is accomplished chemically via neurotransmitters.

588. Resting potential of neuronsK+
Cl-
Na+
Outside of Cell
Cell Membrane at rest
Na+
- 70 mV K+ A-
Cl-
Inside of Cell
Potassium (K+) can pass through to equalize its concentration
Sodium and Chlorine cannot pass through
Result - inside is negative relative to outside

589. 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.

592. 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...

593. With modern crystalography, these effects have been observed...

594. Transport of the action potential, like a row of dominos falling...

595. This goes a lot faster with myelinated axons – saltating transport...

597. 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

598. This graded potential can be both positive and negative, depending on the environment

600. The intensity of the signal is given by the firing frequency

601. 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

602. Common in informatics is the iterative learning, which needs a teacher
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

603. With the use of hidden layers, not linearly separable variable can be learnt...

604. An example: letter recognition

605. The problems that can be solved depend on the structure of the network

606. 3.2.2 Hebbian learning
This means that a synapse gets stronger as neighbouring cells are more correlated

607. 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.

608. Hebbian learning in a neural network

609. 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.

610. A Cell Assembly
Input from the environment

611. A Cell Assembly
Input from the environment

612. A Cell Assembly
Input from the environment

613. A Cell Assembly
Input from the environment

614. A Cell Assembly
Note that the input from the
environment is gone...

615. A Cell Assembly

616. Hebbian learning implies that weights can only increase
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 non-linear forgetting factor into Hebb’s Law:
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.

617. 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.

618. In fact, these inhibitory connections are distance dependent and as such give rise to structure

619. Exciation happens within columns and inhibition further away

620. See also the excursion into pattern formation in Sec 3.6
Long range inhibition and short range activation gives rise to patterns
See also the excursion into pattern formation in Sec 3.6

621. Feature mapping Kohonen model

622. Iterate... 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 .
Update weights:
j(p) is the neighbourhood function centred around jX
Iterate...

623. 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 two-dimensional input vectors generated randomly in a square region in the interval between –1 and +1. The learning rate parameter  is equal to 0.1.

624. Initial random network

625. After 100 steps

626. After 1000 steps

627. After steps

628. Or for letter recognition

629. In the cortex, this gives rise to the homunculus, the spatial distribution of nerve cells responsible for senses

630. Similar for other features in the cortex

631. In a Hopfield Network, every neuron is connected to every other neuron
3.2.3 Associative networks
In a Hopfield Network, every neuron is connected to every other neuron

632. Topological state analysis for a three neuron Hopfield network

633. 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.

634. Energy function of Hopfield net: multidimensional landscape

635. Example: Restoring corrupted memory patterns
20% of T corrupted
Half is corrupted
Original T

637. 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