The discussion above and the demonstrations we have done with CSIM suggest the possibility of a compiler which would take a statement in a simple parallel programming language such as the one given above and would write a script in for example the CSIM language which would construct a spiking neural network which would enact the computational meaning of the statement. Instead of train- ing spiking neural nets to perform given tasks, we are constructing them, as is also done in Maass (1996). There are already examples of such compilers from Occam-like languages into sigmoid neural networks. See Neto et al (2003). The implementation of addition, subtraction and multiplication which we give, for purposes of exposition, is very inefficient computationally. However, using the same basic ideas it should be clear that we can implement other, more efficient, algorithms for arithmetic. For example, we could implement a variable as an array or pairs of oscillators in such a way that when one of the pairs has been advanced as far as possible it returns to zero and the next pair is advanced by one step, as in the usual decimal notation. We could then build in an addition and multiplication table, and proceed, as we did, with effort, in school.
Chapter 5
Analysis of STDP Learning
properties
5.1
Introduction
The work presented in this chapter is focussed on analysis, at a fundamental level, of a form of STDP learning within highly recurrently connected networks of spiking LIF neurons. In addition to STDP learning, the synaptic weights are also modified by a normalisation process. This normalisation allows for the norm of the weight vector to each network neuron to be maintained, and allows for the synaptic weights to assume any value between a minimum and maximum weight value. The reason for using such a normalising procedure is so that successive ap- plications of STDP weight changes does not create a bimodal weight distribution in which the synaptic weights will eventually attain either the maximal or min- imal possible value. The bimodal nature of such un-normalised STDP learning was shown in Song et al (2000). Using the normalisation means that the synaptic weights are able to assume a range of intermediate values between the minimum and maximum, making the learning meaningful and stable at these intermediate values in the case of long term application of STDP learning.
Introduction 119 We show here that, the application of a form of STDP within a highly recur- rent spiking neural net based upon the Liquid State Machine (LSM) leads to an approximate convergence of the synaptic weights. Convergence is a desirable property as it signifies a degree of stability within the network. This approximate convergence is not formally proved, but its presence is indicated experimentally. An activity link L is defined which describes the link between the spiking ac- tivity on a connection and the weight change of the associated synapse. This activity link L also enables perfect convergence to be defined as the point as which the input weight vector, W , of each neuron in a network are aligned with their respective activity link vectors. It is shown that under specific conditions Hebbian and Anti-Hebbian learning can be considered approximately equivalent processes. Also, it is shown that such a network habituates to a given stimulus and is capable of detecting subtle variations in the structure of the stimuli itself. It is also shown that it is possible to extend the firing duration of highly recurrent neural networks in a pre-determined manner. A method for storing precise spike trains within recurrent networks is also suggested.
There are forms of STDP learning other than the most basic Hebbian STDP learning. Two major disadvantages of basic Hebbian learning mean that it is unstable, see Bienenstock et al (1982): Firstly, the tendency is that weights increase or decrease exponentially with the end result being that synaptic weight values will lie only at the extremes of their operational range after sufficient exposure to learning; Secondly, basic Hebbian learning has no mechanism to allow the synaptic weights of unused connections to decrease in magnitude. In order to address these shortcomings of basic Hebbian learning, Bienenstock et al (1982) introduced the Bienenstock, Cooper & Munro (BCM) model of synaptic modification learning.
BCM theory incorporates modifications that make it a more favourable form of learning, over basic Hebbian learning. The BCM model includes a time constant that allows synaptic weights of unused connections to decay to some base weight value. Full details can be seen in Bienenstock et al (1982).
ST DP + N learning 120 with normalisation (ST DP + N ) learning regime to be used on the synapses of the recurrently connected networks. The activity link vector L is then defined. Having defined the learning rule, it is then applied during a series of experiments. The initial result is that when a recurrently connected spiking neural network is stimulated by an external spiking source and the synaptic weights are modified using the ST DP + N learning that, the weights converge, as shown by a plot of the average synaptic weight change for the whole network w.r.t. the number of iterations. An explanation is then given, which describes how, under certain conditions, Anti-Hebbian learning is approximately equivalent to Hebbian learn- ing. This approximate equivalence is then shown experimentally. Finally, it is shown experimentally that such a network with ST DP + N learning can become habituated to an input stimuli and that, even once this converged state of habit- uation is attained, the network is capable to respond to a change in the nature of the input stimuli by undergoing another sharp change in synaptic weights be- fore once again converging and becoming Habituated to this new form of input stimuli. The reason for these basic experiments are to establish firstly that, the learning regime is stable, but more importantly, to investigate aspects of Hebbian and Anti-Hebbian learning, such as their approximate equivalence under certain conditions, that have not been considered in previous research.