## An Accumulator Model that Produces Migration to the Mean

Miall (1989) developed two simple models capable of representing the passage of time within the activity of a simple neural network model. The beat frequency model supposes a bank of pulse generators with varying frequencies that serves as the internal time basis, and a register of neural nodes that responds to coincident activity, or "beats," among the pulse generators. This model has been employed by other researchers to model the interactions between cortical and basal ganglia structures involved in timing (Miall, 1996; Matell et al., this volume). Here we examine the properties of Miall's pacemaker-accumulator model. This model's architecture is similar to that of the structures of the same name found in the information-processing model of SET and other theories. This fact was the principle reason we chose the pacemaker-accumulator model over the beat frequency model for investigation.

Another reason was that unlike the beat frequency model, the pacemaker-accumulator model represents time continuously. That is, the beat frequency model is a time detector in that it signals the occurrence of a unique state of the time basis — namely, the co-occurrence of pulses from a fixed subset of a larger set of variable frequency oscillators. In contrast, the pacemaker-accumulator maps real time onto

FIGURE 20.2 Architecture of Miall's pacemaker-accumulator artificial neural network. A Poisson pacemaker (indicated by the diamond) feeds simultaneously to an accumulator (indicated by circles) and is ultimately summed by a third process (marked as a square). The model discussed in the text included 1000 accumulator neurons. Accumulator nodes can transition from the off state (light circles) to the on state (dark circles) with each pulse of the pacemaker with a probability defined by the gain parameter, but they can also transition from the on state to the off state at any time with a probability defined by the decay parameter. (Adapted from Miall, R.C., Time, Internal Clocks and Movement, Vol. 115, Pastor, M.A. and Artieda, J., Eds., Elsevier/North-Holland, Amsterdam, 1996, pp. 69-94.)

FIGURE 20.2 Architecture of Miall's pacemaker-accumulator artificial neural network. A Poisson pacemaker (indicated by the diamond) feeds simultaneously to an accumulator (indicated by circles) and is ultimately summed by a third process (marked as a square). The model discussed in the text included 1000 accumulator neurons. Accumulator nodes can transition from the off state (light circles) to the on state (dark circles) with each pulse of the pacemaker with a probability defined by the gain parameter, but they can also transition from the on state to the off state at any time with a probability defined by the decay parameter. (Adapted from Miall, R.C., Time, Internal Clocks and Movement, Vol. 115, Pastor, M.A. and Artieda, J., Eds., Elsevier/North-Holland, Amsterdam, 1996, pp. 69-94.)

subjective time upon each occurrence of a pacemaker pulse. The specific form of this mapping is of particular interest because it allows for both standard timing effects, such as the ability to subdivide intervals, and the more unique aspects of timing that are of interest here.

The conceptual design of the pacemaker-accumulator model is presented in Figure 20.2. The pacemaker consists of a pacemaker that emits discrete pulses. The pacemaker is a Poisson process in that the number of pulses emitted per unit time has both a central tendency and variability, and successive pulses are independent with respect to time. The accumulator consists of a bank of neural nodes that can be in either an active ON state or an inactive OFF state. Each pulse of the pacemaker has a certain probability of turning on a node that is currently off. This probability is referred to as the gain parameter because it determines the basic rate of accumulation. The corresponding probability of a node transitioning from the on state to the off state is referred to as the decay parameter. Transitions to the off state can occur between pacemaker pulses. These two parameters together determine the dynamical evolution of the total number of nodes that are in the on state. A summation process computes this total, which is the model's current estimate of real time.

This model was implemented in MATLAB code, listed in Appendix A. The function defined in that code returns the total number of active nodes per pacemaker pulse for an arbitrary number of simulations, nruns. The variable nactive stores and returns the results. The variable steps store the Poisson deviates (with mean itsper-step) that determine the number of main program loop iterations for each of the nsteps pacemaker pulses. The Poisson deviates are generated using a published algorithm implemented in the code listed in Appendix B. The main program consists of nested loops. The outer loop with counter variable j is iterated once for each simulation. Every iteration of the outer loop clears the storage variable units, used to represent the bank of nodes constituting the accumulator, and initiates a second, inner loop. The inner loop indexed with variable i iterates once for each pulse of the hypothetical pacemaker. Within this loop variable ponmat and poffmat are used to determine whether nodes change state using the gain (pon) and decay (poff) parameters, respectively. This is done by comparing a random array of uniform deviates on the interval 0 to 1 to one of the parameters, and then combining the results with the node bank's current state using Boolean operators. Finally, the number of active nodes in units is summed and then stored in the appropriate location of the output storage matrix.

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