Lecture 9 - Edward Rolls

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Biologically Plausible Neural Networks

Terms

r'j - input neuron firing rate

ri - outpur neuron firing weight

hi - Activation (used because of the physics concept of a field)

wij - weight of the synaptic connection.

Brain Mechanisms of Emotion

Emotion - states ellicited by rewards (something you'll work for) and punishment.(something you'll work to avoid). Naturally there are primary reinforcers and secondary rewards and/or punishments. Secondary reinforcers get paired with primary reinforcres in a kind of pattern matching to ultimately produce the output on their own, much like the case of classical conditioning. For the case of emotions the outputs go to the autonomic and endocrine systems, the implicit action system (eg. basal ganglia) and the explicit actions systems (long range planning).

Activation Functions

Modelsers can make use of many activationfunctions, some more plausible than others. Linear, threshold linear, sigmoid, and binary threshold. The advantage of the sigmoid is that it saptures some of the properties of both kinds of activation functions.
     All of these have important aplications for learning. First, semm to only get changes in the case of very strong activations. If the post synaptic neuron is also strongly activated than LTP will result. Synapses which are not active will weaken (heterosynaptic LTD). Likewise, if the post-synaptic neuron is not activated, then the strong presynaptic activation will produce homsynaptic LTD.

Pattern Association Memory

Memory is dependent on the strength o the local synaptic weights, as is learning. In learning, the change in weight is proportional to ri (postsynaptic firing) and r"j (presynaptic firing) Retrieval now depends on the new weights. And can every hapen in as little as 15ms. This is very handy in a neuron with 20,000 or so inputs to any given neuron.
     We can see the effect of this, by considering the abilities of such a network to learn different patterns. Not too surprisingly, given what we learned in the first week, such a pattern associator learns orthogonal sets very well (although such is dependent on a non-linearity) and will even generalize to similiar patterns (in particualar the prototype patterns) showing a surprisable amount of graceful degradation (or fault tolerance, for the computer engineers). Likewise, when non-independant patterns are presented a sizable amount of interference results, but this isn't neccessarily bad.
     Thus, such simple, biologically plausible, pattern associators show three important properties.

• Generalisation - mathematically it can be shown that such a network is actually calculatiing, through the dot or inner product, the Pearson's Product moment coorelation. In effect, each neuron calculates the similarity of any one pattern with the patterns previous ly presented it.
• Graceful Degradation
• Prototype extraction.
• Speed - recall is very fast and learning as well (10-20ms)
• Local Learning Rule

Pattern Associator Capacity

So, how many patterns can such a network store? Well, in the case of localised representations the number of patterens is just the number of inputs per neuron. However, in distributed networks thecapacity is considerably less, unless one uses a modified Hebb rule which captures the essences of heterosynaptic LTD. Moreover, if add non-linear neurons, then the modified Hebbian rule plus a sparse representation (say only 5% active at any one time) now produces much more than N, say three times as much (approximately 50,000).
   Moreover, such simple pattern associators or "perceptrons" can escape the XOR problem through the use of expansion recoding. That is, if have two inputs coming into the system, the linear seperability problem can be efectively circumvented by having the next four neurons, represent the four possible states of the network. (Incidentally, granule cells in the brain seem to be doing exactly this within the brain.

Linear Algebra Review

Linearly Dependent - set of vectors in which at least one can be written as a linear combination of two or more others.

Linear Independece - set of vectors which cannot be described by the linear combination of two or more other vectors in the set.

Linear Seperability - at set of vectors in which a subset can be seperated via a hyperplane. It turns out the nearly all the inputs to the brain are both Linearly independent, and linearly seperable.

Networks and Emotion Recap

Recently, has been demonstrated that the link between emotions and secondary reinforcers is purely through pattern association to the original unconditioned stimulus.