A Parallel Race Network

This paper presents a generalization of the race models involving parallel channels. Previous versions of this class of models involved only serial channels, either dependent or independent. Further, concrete applications of these models did not involve variability in the accumulator size (Logan, 1988) or were based on a specific distribution (Bundesen, 1990). We show that the distributions of response times predicted by the parallel race model can be solved analytically to yield a Weibull distribution using asymptotic (large n) theories. The model can be manipulated to predict the effects of reward on ROC curves and speed-accuracy decomposition. However, the major contribution of this paper is the implementation of a learning rule that enables networks based on a parallel race model to learn stimulus-response associations. We call this model a parallel race network. Surprisingly, with this architecture, a parallel race network learns a XOR without the benefits of hidden units. The model described here can be seen as a reduction of information system (Haider and Frensch, 1996) and is compatible with a priority learning system. An emergent property of this model is seriality: in some conditions, responses are performed with a fixed order although the system is parallel. Finally, the mere existence of this supervised network demonstrates that networks can perform cognitive processes without the weighted sum metric that characterizes connectionist models. A parallel race network 3 Introduction The aim of this text is to expand our knowledge of the race models by showing that there exist similarities between the race models and the connectionist networks. The major objective of this paper is to introduce a learning rule for race models. With it, race models and connectionist models can be compared on similar grounds. We will not undertake this comparison in this paper, its purpose being just to describe the parallel race network. To this end, we will first cast the race model into the form of a network of connections. A second objective of this paper is to describe the response times (RT) predicted by race models. Many predictions will be done in terms of the distribution of response times (averages can easily be derived from them). Finally, networks differ on what is being transported by the connections (sometimes called the channels hereafter). We show that race models don’t transport strength of activation. One important feature that we will address is redundancy of the channels. Redundancy turned out to be necessary to simplify the mathematics of the parallel race model. Overview of the race models Accumulator models and random walk models are generalization of the Signal Detection Theory (SDT) because they share the same coding in the form of samples coming from the senses. However, instead of taking a single sample from the stimuli, these two classes of models can take an arbitrary number of samples, this number changing from trial to trial. In these models, the samples can be evidence for one response or an alternative response. In some variants, the samples can also be evidence for both responses, or neither. Following Smith and Vickers (1988), the distinction between random walks and race models pertains on the evidence A parallel race network 4 accumulation process. For the former, the accumulation process is not independent because an evidence for one response implies a reduction of the amount of evidence for the alternative response. For the latter, the evidence accumulations are done in independent accumulators. However, the first accumulator filled triggers a response. Thus, the name of race model. The class of accumulator models can be further broke down by whether the evidence collected are discrete, called simple accumulator models by Luce (1986) or continuous, called strength accumulator model. Examples of simple accumulator models are given by Audley and Pike (1965) where the sampling process takes a fixed amount of time, and by Pike (1973), where the time between two samples is continuous. An example of strength accumulator is given by Smith and Vickers (1988) where the time between two samples is fixed. Another important distinction between the race models is whether the channels bringing evidence to the accumulator are dependent or independent (Meijers and Eijkman, 1977). We describe one accumulator model, the Poisson race model because it recapitulates in a nutshell the following sections of this paper. The Serial Poisson Race Model One of the first quantitative description of the Poisson race models was described by Pike (1973, 1965; also see Van Zandt, Colonius and Proctor, 2000). Architecture: This model is essentially composed of a single channel that encodes information (under the form of spikes of activation) that is brought to a unit accumulating the spikes (an accumulator) containing K slots (see Figure 1, panel a). As soon as the accumulator is filled, a response is emitted. This model is serial (a sequential sampling model) because the spikes travel one at a time. It is also based on a race model since a decision criterion can be formulated by a A parallel race network 5 rule akin to “As soon as you have K evidences, respond”. In other word, the accumulator is a hard threshold unit. Insert Figure 1 about here Response times: From this, it follows that the response times of the output o is given by: