## Pros and Cons of the Accumulator Model of PD Timing Effects

As discussed earlier, SET and related models impose timescale independence as a central feature of the stochastic modeling of timing. This scale independence limits SET's capacity to explain duration-dependent timing errors. This simple pacemaker-accumulator model sheds light on the specific problematic features of SET. SET assumes that memory function in the accumulator as well as in short-term memory is linear in the sense that the function relating objective and subjective time is a straight line with an intercept of zero (barring the effects of attention on the latency to start and stop timing). The current model instead produces a curvilinear relationship between subjective and objective time, controlled by two parameters. If both parameters are under experimental control, such curvilinear functions can give rise to migration or other duration-dependent phenomena. If we divorce this mathematical fact from the implementation at hand, then we can modify other aspects of the SET information-processing model to serve as the source of migration. For example, the current working explanation of the cognitive mechanisms of migration assigns the effect to the short-term memory store, typically governed by the K* parameter in SET. The current findings suggest that SET can be modified to account for migration by bifurcating K* into two parameters, an exponent and a horizontal limit of a bounded-exponential function.

While the most important outcome from these simulations is the light it sheds on the manner in which limitations of SET can be overcome to account for migration, this model has certain features that interest us with respect to modeling timing in the basal ganglia. An important aspect of basal ganglia anatomy is the dichotomy between the direct and indirect pathways from the striatum to the globus pallidus. Both of these pathways originate in the striatum (Bejjani et al., 1997; Benabid et al., 1991, 1998; Limousin et al., 1995a, 1995b, 1998; Mitchell et al., 1986, 1995; Molinuevo et al., 2000; Pollak et al., 2002; Yelnik et al., 2000), rely on dopaminergic function (Young and Penney, 1984), and terminate at the globus pallidus (Albin et al., 1989; Crossman, 1989; DeLong, 1990; Filion et al., 1991; Miller and DeLong, 1987; Robertson et al., 1990, 1991; Tronnier et al., 1997). The net influence of activity in these pathways is opposite in that the direct pathway upregulates pallidal activity, while the indirect pathway downregulates pallidal activity. Thus, we suggest that this pacemaker-accumulator model that relies on competition between excitation and inhibition to produce the curvilinear psychophysics necessary for modeling the migration effect captures a crucial aspect of the brain anatomy thought to underlie timing in general. This analogy is consistent with Meck's assertion that the striatum implements the SET accumulator (Gibbon and Church, 1984; Gibbon et al., 1984)

if one assumes that the striatum is the input node to an accumulator implemented by the entire striatal-pallidal system.

Although oversimplified in terms of the model's relation to brain anatomy, such implementations are still useful if they generate testable hypotheses. The most salient testable hypothesis derived from this model is the following: if the form of the psychophysical functions generated by the model is independent of the intervals being timed, then there will exist some set of two time values that do not migrate, but rather show monotonic errors. The direction of these errors will depend on their absolute values relative to the crossover point of the two psychophysical functions underlying encoding and decoding.

There are two major shortcomings of this model. First, the model does not currently predict the magnitude of the PD timing effects, only their directions. In the future we will use nonlinear searches of wider parameter spaces to determine the relative change necessary in the parameter to simultaneously account for both PD effects. Second, we have as yet made no effort to determine whether this new model and the proposed changes to SET negatively affect other critical psychophys-ical properties, especially the scalar property of timing variability. In general, alterations to the pacemaker-accumulator should not have a major impact on determining overall timing variability because the pacemaker-accumulator system contributes little variability itself and operates prior to the working memory and decision processing stages that contribute the majority of scalar variability (Gibbon and Church, 1984; Gibbon et al., 1984). However, we are offering a conceptual change in the relationship between subjective and objective time that entails an inequality in the ratio between two target times and the ratio between their corresponding internal representations. This could have profound effects on variability scaling, depending on the curvature of the functions relating subjective and objective time.

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