A firing rate model of Parkinsonian deficits in interval timing

Abstract

To account for deficits in interval timing observed in Parkinson's Disease (PD) patients, we develop a model based on the accumulating firing rate of a neural population with recurrent excitation. This model naturally produces the curvilinear accumulation of neural activity introduced to timing psychophysics by Miall (Models of Neural Timing, Elsevier Science, 1996), and implicated in Parkinsonian timing by Malapani and Rakitin (Functional and Neural Mechanisms of Interval Timing, CRC Press, 2003). The parameters essential for our model are the strength of the net neural feedback and the mean rate of inputs to the population from external brain areas. Systematic variations in these parameters reproduce the PD migration effect, in which estimates of long and short intervals drift towards each other, as well as uniform slowing of time estimates observed under other experimental conditions. For example, our model suggests that dopamine depletion in PD patients increases the neural feedback parameter and decreases the effective input parameter for populations involved in the production of time estimates. The model also explains why the migration effect will be associated with a violation of the scalar property, the linear increase in the standard deviation of time estimates with the duration of the target interval that is ubiquitous in healthy participants. We also show that the effect of systematically decreasing the input rate parameter in our model is equivalent to increasing thresholds, so that either of these changes may be associated with the Parkinsonian state.

Introduction

This paper is motivated by experiments that revealed specific deficits in the interval timing behavior of Parkinson's Disease (PD) patients (Malapani et al., 1998b, Malapani et al., 2002a, Malapani et al., 2002b, Malapani and Rakitin, 2003). In the most developed of these experiments, the timing abilities of PD patients were assessed using the Peak Interval (PI) timing task (Rakitin et al., 1998) immediately after learning a pair of target time intervals with the assistance of behavioral feedback and 24 h later without the benefit of such feedback (Malapani et al., 2002a). The results indicated two separable, dopamine-dependent error patterns associated with storage of time intervals (i.e., encoding) and retrieval from temporal memory (i.e., decoding) (Malapani et al., 2002a). These patterns cannot be immediately reconciled with established models of the timing process, especially Scalar Expectancy Theory, or SET (Gibbon et al., 1984, Church, 2003, Gibbon, 1992, Gibbon and Malapani, 2002). Our goal is to present a modification of SET based on simple models of the behavior of neural populations that can explain the Parkinsonian timing phenomena.

The PI task requires participants to produce behavioral responses (i.e., a burst of button presses) at one of two memorized time intervals. Originally developed for animal research (Catania, 1970, Church et al., 1994, Roberts, 1981), it was adapted for human use in healthy subjects (e.g., Rakitin et al., 1998, Rakitin et al., 2005, Hinton and Meck, 2004, Hinton and Rao, 2004, Lustig and Meck, 2001) and patients with neurological disease (e.g., Malapani et al., 1998a, Malapani et al., 1998b, Malapani et al., 2002a, Lustig and Meck, 2005). Given the high ratio of variance associated with timing vs. motor components (Rakitin, 2005) (as motor requirements are minimal and time ranges are of seconds in duration), the PI task is well suited for the assessment of timing functions in patient populations encountering severe motor deficits, as in PD. Indeed, PD patients experience specific problems when producing movements, such as increased reaction time (Bloxham et al., 1987, Evarts et al., 1981, Malapani et al., 1994), movement time (Benecke et al., 1986, Roy et al., 1993), and speech production time (Lieberman et al., 1992, Volkmann et al., 1992), as well as deficits in programming and synchronizing motor responses (Nakamura et al., Wing et al., 1984, Pastor et al., OBoyle, 1997). These motor deficits may add a constant (“motor”) variance in timing performance (Wing and Kristofferson), particularly in tasks with substantial motor requirements such as repetitive tapping (Ivry and Keele, Pastor et al., OBoyle, 1997, Wing et al., 1984).

Demonstrating separable properties of interval storage and retrieval in PD patients requires two distinct experimental sessions, the first performed in the presence of behavioral feedback and immediately following demonstration of a standard interval, and the second without feedback or demonstration. This was originally reported as the “encode–decode” task design (Malapani et al., 2002a). In this design, during the first session, the standard interval that subjects are to reproduce is first demonstrated. Then, “production” trials begin, in which subjects are asked to reproduce the standard intervals via timed button presses following a cue; on a fraction of these trials, the standard interval is again demonstrated to the subject. Additional behavioral feedback is delivered after other production trials in the form of a histogram indicating whether the response was too short or too long. Two separate blocks of such trials are performed, one with each of a longer (17 s) and a shorter (6 s) standard interval. In the second session, performed on the following day, subjects produce both intervals (in separate blocks) without further demonstration of the standard intervals and with no behavioral feedback. Therefore, participants learn and store the target time intervals only during the first session (henceforth referred to as the training session), but retrieve these intervals from temporal memory during both the training session and the second day's testing session.

PD patients' drug state (ON or OFF medication) on the two successive days varied according to their assignment to one of four experimental groups. The ON–ON group was provided with l-dopa during both training and testing sessions, and the OFF–OFF group was tested without l-dopa for both sessions. The ON–OFF group was provided with l-dopa during the training but not the testing session, and vice-versa for the OFF–ON group. Clinical measures for all four groups are given in Table 1. The motivation was the hope that by crossing patients' drug state with the availability of information about the accuracy of reproductions of the standard intervals, we could determine whether dopamine (DA) deficiency (associated with being in an OFF state) selectively affected memory storage (encoding), retrieval (decoding), or both.

Fig. 1 summarizes the resulting behavioral performance during the testing session. Correct estimates are obtained when both storage and retrieval occur ON medication (see panel A, left). However, panel B, left, shows that retrieving the trace of two different time intervals while OFF medication results in “migration,” a pattern of bi-directional errors such that reproductions of each interval drift in the direction of the midpoint of the intervals. This migration effect is seen for the OFF–OFF group (see panel D, left). However, when time intervals are stored OFF medication, but retrieved ON, panel C, left, shows that both intervals are overestimated. In addition to migration and overestimation of mean time estimates, retrieval of time estimates OFF medication results in a violation of the scalar property of timing variability (Figs. 1B, D, right panels). (The scalar property implies a linear proportionality between standard deviation and mean of time estimates, see below.) By contrast, for retrieval ON medication, the scalar property holds, regardless of whether intervals were encoded ON or OFF medication (A, C, right panels). Note that migration is always accompanied by a violation of the scalar property (specifically, by estimates for the shorter interval having a relatively broad distribution), a fact we will return to in the modeling below. While we show here only data from the testing session, similar migration and scalar timing effects occur for patients OFF l-dopa during the training sessions. This indicates that the availability of behavioral feedback is not a critical factor in producing the statistical trends just discussed.

These results forced a rethinking of some underlying assumptions of Scalar Expectancy Theory (SET), a framework for modeling timing originally developed by Gibbon (1977) in 1977 and still strongly influential in the field (Gibbon et al., 1984, Church, 2003, Gibbon, 1992, Allan, 1998, Gallistel and Gibbon, 2000, Wearden, 1999). SET is an information-processing model that describes the sources of timing errors that lead to the scalar property, the behavioral phenomenon in which distributions of time estimates for different length target intervals appear as scaled versions of a single fundamental distribution and which is robustly observed in both animals (Church, 2003, Gallistel and Gibbon, 2000) and humans (Rakitin et al., 1998, Gibbon et al., 1997b, Ivry and Hazeltine, 1995, Wearden and McShane, 1988). According to this model, pulses from an internal pacemaker are integrated by an accumulator when attention is directed to time (Lejeune, 1999, Meck and Church, 1983, Zakay and Block, 1997, Rakitin, 2005). The value of the accumulator increases linearly with time, and is stored in memory upon the occurrence of reinforcement or feedback (Gibbon et al., 1984, Jones and Wearden, 2003). A ratio-based decision process then compares values of the accumulator to remembered values in order to determine responses on future trials (Allan and Gibbon, 1991, Allan and Gerhardt, 2001). The SET model parameters can fluctuate from trial to trial, but their overall statistics are independent of the target time interval. This produces the scalar property (Gibbon et al., 1984, Church, 2003), and explains the fact that most experimental manipulations produce monotonic changes across a range of target intervals (Malapani and Rakitin, 2003). Indeed, SET could accommodate the unidirectional, proportional rightward shift seen in the OFF–ON groups by relative slowing of the pacemaker rate, which sets the temporal slope of the accumulator (see Gibbon and Malapani, 2002 for a more detailed discussion). The migration effect, however, cannot be reconciled with SET or other similar timing theories (e.g., Zakay and Block, 1997). The same is true of the violation of the scalar property for times produced OFF l-dopa. As a result, new modeling approaches are necessary.

Malapani and Rakitin (2003) produced a theoretical account of both the migration effect and uniform overestimation trends in PD interval timing data by modifying the accumulator so that its value increased with a curvilinear dependence on time. The critical notion was that if l-dopa altered the curvature of the accumulators, then functions associated with timing ON and OFF l-dopa could be made to cross, and duration-dependent errors like migration could emerge. The underlying computational model was a stochastic, neural network-type architecture comprised of binary-valued neural units proposed by Miall (1996), in which the mean value of network activity exponentially approaches an equilibrium value from its initial state. In this paper, we establish a different, idealized model that also produces curvilinear accumulation of network activity, but which is immediately tractable analytically and which may be more directly related to the firing rate dynamics of neural populations.

Specifically, we consider a related class of ‘leaky integrator’ models derived from idealized equations for the firing rate of a recurrently connected neural population receiving input from another, separate neural population. Models of this type have a long history in neural modeling (cf. Seung, 1996, Seung et al., 2000, Wilson and Cowan, 1972, Usher and McClelland, 2001, Hopfield, 1984, Abbott, 1991, McClelland, 1979, Grossberg, 1988). The input to the recurrent population and its firing rate, respectively, correspond to the pacemaker and accumulator components of SET. In the variant that we use here, the time-dependent neural activation (firing rate) can display three qualitatively different trajectories, either approaching a steady state (as for the model of Miall, 1996), increasing linearly in time, or increasing at an accelerating rate. Therefore, this model can be used to further study the effects of curvilinear accumulation in interval timing.

In particular, this variety of temporal dynamics allows us to address an additional feature of the behavioral data not treated in (Malapani and Rakitin, 2003)—that of the scalar property of variability in time estimates. In particular, we show that model parameter values corresponding to the ON-drug condition produce timing distributions with a fixed ratio of mean to standard deviation (the scalar property). This results from a linear accumulation of firing rates in time, as in classical models (Gibbon, 1992). In contrast, parameter values modeling the OFF-drug condition result in violation of the scalar property, with proportionally broader distributions at shorter intervals (in agreement with experimental data).

The present paper extends the work of Malapani and Rakitin (2003) in three additional ways. First, we use explicit solutions to our simple firing rate-based model to show the equivalence of timing models that use two distinct mechanisms to time different intervals. We also group parameters into sets that have identical effects, identifying only two combined parameters that determine the model's dynamics. Finally, we note that our model, while substantially removed from the underlying physiology, is parameterized by values that may nevertheless be related to averaged or ‘mean field’ descriptions of asynchronously firing neural populations (e.g., Wilson and Cowan, 1972, Fourcaud and Brunel, 2002, Nykamp and Tranchina, 2000). Therefore, we are able to draw preliminary conclusions about the changes in effective parameters in neural timing circuits that may result from transitions from ON- to OFF-drug states.

The balance of the paper proceeds as follows. In the Experimental procedures section, we introduce the firing rate model, discuss the form of its solutions, and identify simplifying parameter combinations. Next, we give the results of the model for both trends in mean values of time estimates across the various experimental conditions, and adherence to or violation of the scalar property. Here, we emphasize how parameters must vary across experimental conditions in order to reproduce the trends observed in the data of Malapani and Rakitin (2003), and develop several predictions that follow from these necessary variations in parameters. We then show that these conclusions hold regardless of whether thresholds or accumulator slopes are changed to time different intervals, and identify an equivalence between increasing external input to the recurrent network and decreasing thresholds. Finally, we discuss our findings in the context of physiological mechanisms, covering both the model's shortcomings and its predictions for future experiments.