INTRODUCTION

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How do we calculate time intervals? No sensory input is required to keep track of the elapsing time between two arbitrary events, and the accuracy of this calculation depends on a number of factors such as age (Block et al. 1998; Craik and Hay 1999), drugs use (Frankenhaeuser 1959), body temperature (Hancock 1993), degenerative brain diseases (Nichelli et al. 1993), and concurrent cognitive loads (Marmaras et al. 1995; Zakay 1993). Simply counting spikes could appear, at first, as the most natural choice for an internal time-keeping system. However, simple pacemaker-based systems (Treisman 1963) cannot be reconciled with distinctive features of the variability in the subjects responses, which is large, it is essentially independent from the interval length and it decreases with training. To take into account these effects, memory stages and decision processes (Gibbon et al. 1984) or fallible stochastic counters (Killeen and Taylor 2000) have been included in pacemaker-based systems, and alternative theories using memory dynamics (Staddon and Higa 1999), neural networks (Miall 1996), or connectionist models (Church 1989; Church and Broadbent 1991) have been proposed. Both kinds of implementations, reviewed by Ivry (1996) and Gibbon et al. (1997), have features based on experimental observations but do not give insights into the biophysical processes that could be involved at the single neuron level. Although it has been found that the basal ganglia are activated during time intervals encoding (Rao et al. 2001), there is still no direct experimental evidence on what kind of neural circuits we use. Since linking behavioral data to biophysical mechanisms is an important step to elucidate the neural substrates of time interval perception, we used a different approach with a biophysical model. Experiments suggested that the calculation of a time interval requires attentional resources (Zakay and Block 1996) and that excitation and inhibition are the driving forces of selective attention (Ghatan et al. 1998; Kastner et al. 1998). Thus we have implemented a simple time-keeping system with a counter and a neuron, which was activated by a background synaptic input representing attentional effects. We hypothesized that the properties of the excitatory and inhibitory components of this background activity could quantitatively represent the characteristics of the attentional state and determine the subjective calculation of the elapsing time. We found that the strength of the inhibitory component, and the variability of the excitatory one, were the only parameters required to quantitatively reproduce the subjective production and estimation of time intervals.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Experiments

To test our model, among the many experiments investigating interval timing in humans, we selected three representative examples (from different laboratories). The selected experiments were designed to investigate using a prospective paradigm different questions, such as the dependence on interval length under control conditions (Predebon 1995), the effects of different cognitive loads (Marmaras et al. 1995), and the difference between production and estimation of the same interval under the same cognitive load (Zakay 1993). To simplify the interpretation of the results, we did not take into account experiments based on retrospective paradigms or temporal discriminations, where interplay of time-keeping activities with additional processes, such as the internal replay of memorized events, could be expected. Thus, in all cases, subjects were informed that an interval production or estimation would be requested. Complete details about the different experimental setup can be found in the original papers. Here we briefly discuss only those relevant to our work. Predebon (1995) used intervals of 10, 18, 26, 34, 42, and 50 s estimated under control conditions ("empty" time). Intervals were estimated by independent groups of 18 subjects. Zakay (1993) used a single control interval (12 s) to show how interval production or estimation depended on concurrent cognitive loads, of increasing complexity, based on the Stroop color-word test. Each interval under each cognitive load (including empty time) was produced or estimated by an independent group of 20 subjects. Marmaras et al. (1995) reported intervals of 15, 30, and 60 s produced under different cognitive loads of increasing complexity. Cognitive loads were simple tasks, such as watching a ball in motion on a screen (condition C2), listening to a news radio program (C3), counting the times the ball touched the edges of the screen (C4), or more demanding tasks, such as solving easy arithmetic problems (C5) and answering to questions requiring a memory search (C6). In contrast to the previous experiments, in this case all subjects (n = 92) were involved in all tests. In addition, the raw data were made available (courtesy of G. Dounias, University of Aegean, Greece).

Model

COMPUTATIONAL DETAILS. The network connectivity and the passive and active properties of the neurons involved in interval timing are not known. A simple implementation of a timekeeping system requires a neuron (representing a probable network and not necessarily with pacemaker properties), a suprathreshold synaptic input, a basic reference interval at which the neuron is expected to fire action potentials (INT_EXP), and a spike counting mechanism.

View larger version (41K): [in this window] [in a new window]   Fig. 1. Schematic representation and basic results of the model. A: random (Poisson) excitatory and inhibitory synaptic activity, modeling attentional processes, elicited somatic action potentials that were used to produce or estimate time intervals. Inset: 8 s of somatic membrane potential from a typical simulation (isyn = 0.15 nS). B: production () and estimation () of a 10-s control interval using different levels of background inhibitory activity. C: coefficient of variation (CV) of production () and estimation () of control intervals as a function of the excitatory mean interstimulus interval (ISI) variability (left, 10 s, isyn = 0.15 nS) or the interval length (right).

INTERVALS CALCULATION AND FITTING PARAMETERS. Following experimental suggestions (Ivry and Hazeltine 1995), the same mechanism (counting the spikes generated by the model neuron, in our case) was used to estimate or produce an interval. However, the production and the estimation of a time interval are two rather different operations. In a time-estimation protocol, the experimenter himself/herself starts and stops the interval. When an interval production is requested, the subject himself/herself controls the start and stop signals. In both cases, we assume that a subject produces or estimates intervals by using the (internal) expected frequency, 1/INT_EXP, and the number of output spikes, N_S. Thus, in our simulations, the estimate (T_E) of a control interval (T_C) was calculated from the number of spikes generated by simulations T_C-s long as T_E = N_S * INT_EXP. To produce a control interval, a simulation lasted until the neuron generated the expected number of spikes, N_S = T_C/INT_EXP. The total simulation length was the model production of a T_C = N_S * INT_EXP-s interval. An 8-s window of the somatic membrane potential from a typical simulation is shown in Fig. 1A.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Experimentally, produced intervals showed a positive relationship with increasing cognitive loads, whereas estimated intervals were negatively related (Zakay 1993). In Fig. 1B, we show a series of simulations of production and estimation of a 10-s control interval as a function of the inhibitory synaptic strength (peak conductance, _isyn). The average values for the interval production increased (Fig. 1B, ) and estimation decreased (Fig. 1B, ) with the strength of the inhibitory conductance, reproducing the experimental findings and suggesting that the inhibitory input could be a useful indicator of the concurrently attended activities unrelated with timing. Another distinctive feature of psychological experiments on interval timing is the large range of values observed for the CV of the subjects responses (Gibbon et al. 1997; Marmaras et al. 1995; Predebon 1995; Zakay 1993), and its independence from experimental conditions (such as interval length or strength of cognitive load). In Fig. 1C (left) we present simulations findings for production () and estimation () of a 10-s interval as a function of SD, the variability of the excitatory mean ISI. The CV strongly increased with SD for both production and estimation, spanning the entire range of values observed experimentally (Gibbon et al. 1997). A small change was found with the interval length, as shown in Fig. 1C (right) for intervals produced () of estimated () using different values of SD. In agreement with a general experimental finding (Block et al. 1998), the CV for intervals estimation was greater than for production, especially for larger SD values.

To test the model on more quantitative grounds, we fitted the experimental results on production and estimation of time intervals obtained from human subjects under different cognitive loads from the above-mentioned authors (Marmaras et al. 1995; Predebon 1995; Zakay 1993). A comparison between experiments and model findings is shown in Fig. 2. In Fig. 2A, intervals of different lengths were estimated under control conditions, i.e., no additional activities (Predebon 1995). Average values and SDs obtained from the model were in quantitative agreement with experiments (Wilcoxon test, P > 0.1). In the next example, Fig. 2B, model findings are compared with experiments on production and estimation of a 12-s control interval under increasing cognitive loads based on the Stroop color-word test (Zakay 1993). Quantitative agreement with experiments (P > 0.9) was obtained in both cases. Finally, Fig. 2C, intervals of 15, 30, and 60 s were produced under increasing cognitive loads (indicated as C1, ... , C6). It should be stressed that conditions C1, ... , C4 refer to basic information processing, whereas conditions C5 and C6 involved higher brain function (see METHODS). Quantitative agreement with experiments (P > 0.3) was also obtained for these cases.

View larger version (31K): [in this window] [in a new window]   Fig. 2. Quantitative modeling of experiments, using time intervals produced or estimated under different cognitive loads. A: experimental () and model findings () for intervals produced under control conditions (no concurrent activities), plotted as a function of interval length. B: comparison between experimental production () or estimation (), and simulation findings () for a 12-s control interval under different cognitive loads; ET was the control condition. C: for each experimental condition (C1, ... , C6), results from experiments () and model () are plotted for intervals of 15, 30 and 60 s; C1 was the control condition. Experimental data for A-C are from Predebon (1995), Zakay (1993), and Marmaras et al. (1995), respectively. In all cases, error bars are SDs.

Often, experimental distributions of intervals are not symmetric but show some skewness, suggesting an asymmetrical source of variance in the time-keeping systems. In rats trained for a temporal generalization task (Church et al. 1991), it resulted largely from responses not controlled by timing. Its origin and the basic biophysical mechanisms for perception or production of intervals in humans are unknown. Our model suggests a simple and physiologically plausible explanation. In fact, from trial to trial, the excitatory mean ISI was changed, from its average value, according to a normally distributed random variable with a standard deviation of SD, one of our fitting parameters. High SD values increase the probability to have very long or very short ISIs. However, whereas long ISIs result in the production of long intervals, very short ISIs cannot produce very short intervals, since several synaptic pulses will be missed because of the membrane properties and the refractory period of the neuron. As illustrated in Fig. 3, this increases the asymmetry in the distributions. The model was able to quantitatively predict both shape and location of interval distributions, as illustrated by the typical examples shown in Fig. 4. Experimental distributions from one laboratory (Marmaras et al. 1995) are compared with simulation findings for 15-, 30-, and 60-s control intervals under three different cognitive loads (C1, C3, and C6). In all cases (including those not shown in Fig. 4), the Kolmogorv-Smirnov two-sample test confirmed that the distributions obtained from experiments and simulations were the same (Fig. 4, P values are shown in the inset of each panel).

View larger version (32K): [in this window] [in a new window]   Fig. 3. The asymmetry in the distributions depends on the variability of the excitatory synaptic input. Distributions of estimates of a 10-s control interval, normalized with respect to their peak value, for SD = 0 (), SD = 10 (), and SD = 20 (). In all cases, isyn = 0.15 nS, and a bin size of 2 s was used.

View larger version (38K): [in this window] [in a new window]   Fig. 4. Typical distributions of intervals produced by the model () and in the experiments by Marmaras et al. (1995) (). Results for 3 cognitive load conditions (C1, C3, and C6) are shown. Bin sizes were 2, 5, and 10 s for 15, 30, and 60 s, respectively, and P values from a Kolmogorov-Smirnov 2-sample test are shown in each case.

The model parameters fitting the experimental data are summarized in Fig. 5. The inhibitory component increased with the cognitive load for both production and estimation (Fig. 5A, left), supporting our assumption that it could be a useful indicator of the concurrent level of nontemporal information processing. Interval estimation systematically involved additional processing (i.e., higher inhibition), with respect to production under the same cognitive load (Fig. 5A, left, compare  and ). Concurrent processing of simple cognitive loads was essentially independent from the length of the interval to produce (Fig. 5A, right, C1, ... , C4). Complex tasks (Fig. 5A, right, C5 and C6), however, required higher values for the inhibitory conductance, suggesting that higher brain functions use additional attentional resources that progressively inhibit the time-keeping system with the length of the interval to produce. Because cognitive load and interval length did not influence the variability of the excitatory mean ISI (Fig. 5B), our model supports the view that additional processes not related to timing, such as the actual attentional context, possibly influenced by personal experiences or training, affect the spread of the produced/estimated intervals. However, under the same cognitive load, time estimation systematically resulted in a lower variability for the excitatory frequency, with respect to production of the same interval (Fig. 5B, left, compare  and ).

View larger version (28K): [in this window] [in a new window]   Fig. 5. Model parameters fitting experimental findings. A: peak inhibitory conductance, isyn, as a function of experimental conditions (left), or interval length (right). B: SD of excitatory mean ISI as a function of experimental conditions (left), and interval length (right). Left: the findings for production/estimation of 12-s intervals [ and , control condition (ET), and the Stroop color-word tests W, CW, and CWA] are shifted to the right for clarity.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Models of interval timing (reviewed by Gibbon et al. 1997; Ivry 1996) have been traditionally implemented along two main lines using systems based on a pacemaker and a spike counter or representing different intervals with distinct elements corresponding to specific durations. Both kinds of implementations have features based on experimental observations, but none of them uses a realistic (biophysical) implementation and reaches the kind of quantitative agreement with experiments that we have shown here. Thus, a direct comparison with our model was not possible. We have indicated a plausible link between biophysical mechanisms and behavioral data on interval timing, using a basic framework that, in principle, could be used as a "canonical model" (Shepherd 1992) to study other psychophysical measurements that obey to the same laws. The results suggest to investigate experimentally if the same brain regions that are activated during production/estimation of time intervals are also active during production/estimation of lengths or weights.

All the main characteristics of intervals produced or estimated under different cognitive loads were quantitatively reproduced without the use of special purpose circuits or networks. According to our model, an internal time-keeping system could be composed by a very simple network (or even single neurons), firing APs elicited by a synaptic input modulated by attention, and fed into a spike counter. It should be noted, however, that its in vivo implementation most likely requires a network rather than a single neuron. This would take into account problems such as the output robustness with input synchronization, which we have minimized using simple synaptic inputs to model the attention-based synchronized activation of different population of synapses. In fact, both the excitatory and inhibitory inputs could be generated by afferent fibers activated at different stages of the cognitive loads. Thus they may not be synchronized, although they may fire at approximately the same rate. In this case, a network may be the most appropriate solution to detect and elaborate this kind of features (Hopfield and Brody 2001). Although a biophysical implementation of a spike count detector was outside the scope of the present work, we would like to note that a spike count network could be arranged using the intrinsic temporal integration properties of neurons. However, the relatively slow time constants that would be involved with intervals in the minutes range make this implementation unlikely. A time-dependent associative neuronal network, with an output pattern coding the number of input spikes and complemented by an appropriate learning process, may be a more suitable solution.

Only two fitting parameters were used: the level of inhibition and the variability of the excitation. Both are closely related to the physiological mechanisms that drive a neuron to fire action potentials. Because they indirectly control the mean firing rate and variability, the quantitative reproduction of the experimental average values and SDs may be somewhat expected. However, in contrast with the existing models, we were able to predict, in terms of the background synaptic activity, the experimental distributions of the produced/estimated intervals, which were not used in the fit procedure.

One of the unique features of this model is that it does not explicitly uses memory stages and dynamics (Staddon and Higa 1999), although our assumption of the existence of an expected firing frequency was an implicit use of a memory store. As we have shown here, interaction of the timing processes with memory was not necessary to quantitatively account for the intervals (and their variability) produced or estimated by humans under different conditions. This, however, may be a limitation of the model, because it may not take into account with the same accuracy experiments using retrospective paradigms or temporal discriminations, which explicitly need memory of past events to calculate an interval.

It could also be argued that some constrains should be imposed on the two parameters. For example, because we assumed that they depend on attention, for a given cognitive load their values should be fixed for any interval length or experimental protocol (production or estimation). However, using these constrains it was not possible to obtain a quantitative fit in all cases. Rather than consider this a weakness of the model, we interpret this finding as a hint on the processes underlying attention for specific tasks. For example, the model suggested that cognitively more demanding tasks (such as C5 and C6) increase their effects on time production with interval length (Fig. 5A, right). Furthermore, a given cognitive load could result in different information processing activities during estimation or production. In particular, in the experiments using a Stroop color-word test (Zakay 1993) and in the model (Fig. 5A), a lower variability was found for estimation with respect to production. This is surprising since, in general, experimental findings (Block et al. 1998) and the model (Fig. 1C), predict a greater variability for estimation when an interval calculation is hindered by a given task. The model thus suggests that, in this particular experiment, subjects used additional processes that resulted in estimations more accurate than productions. The effect of these additional processes was reflected in the higher level of inhibition required by time estimation (Fig. 5A), suggesting that they were not directly related to time calculation. The model can thus point out specific aspects of the experimental conditions.

One of the fundamental characteristics, that any computational model of interval timing must be able to reproduce, is the variability of the experimentally produced/estimated intervals, which does not show a clear trend among tasks, interval lengths, and species (see Fig. 3 in Gibbon et al. 1997). In models using distributed intervals, interpretation of experimental data assumes that the underlying timing mechanisms conform to the (empirical) Weber's law, according to which the observed standard deviation is a constant proportion of the produced/estimated interval. Clock-based systems use the Scalar Expectancy Theory (Gibbon 1977, 1992), in which memory effects account for the Weber fraction of the observed variability. In both cases, there are no detailed indications of the possible mechanisms involved at the single neuron level. According to our model, the firing variability of the neuron (or network) involved in the interval calculation depends on the synaptic background activity. We propose that it is related to the instantaneous attentional context as suggested by studies showing that ongoing synaptic activity of neuronal populations is responsible for the large variability in the evoked cortical responses to the same stimulus (Steinmetz et al. 2000) and that the attentional state increases neuronal firing synchronization (Arieli et al. 1996).

For the excitatory background, we used a mean ISI that was sampled from a normal distribution from trial to trial (see METHODS). As already noted by other authors (Gibbon et al. 1984), this approach results in a duration-independent variability and corresponds to a mode of operation in which the attention given to the elapsing time (modeled by the mean excitatory frequency) is engaged once the subject starts the trial. However, essentially the same results would have been obtained by changing (at random times) the mean excitatory frequency during a simulation, modeling fluctuation of attention during a given trial. This suggests an experimentally testable prediction. In fact, which mode is effectively used by humans could be tested by comparing the variability of the same interval produced or estimated many times during a single long trial or during multiple trials (with no feedback in both cases, to prevent learning effects). Our model predicts that if the subject's attention is engaged at the beginning of each trial, the variability should be lower for intervals obtained during a long trial.

In the experiments, asymmetric distributions were essentially caused by the occasional production/estimation of intervals much longer (~3-4 times) than the average value, especially under strong cognitive loads (see, for example, the distributions for condition C6 in Fig. 4). With the exception of the work by Bugmann (1998), who used a neural network with probabilistic internal feedback to model experiments on short (<1 s) durations discrimination, none of the current models is able to explain the asymmetry at the single neuron level. Our model predicts that it could be caused by the intrinsic membrane properties and by the refractory period of the neuron, which prevent very short ISIs to drive the involved neurons at high frequency. Furthermore, an additional asymmetrical source of variance is expected from the way in which, according to our model, humans produce or estimate intervals (see METHODS) and is caused by the inverse relationship that holds between the estimated interval (T_E) and the average interval between two spikes (INT_E) elicited by the synaptic activity, T_E ~ (T_C/INT_E)*INT_EXP. This may explain why experiments on verbal estimations usually result in CVs higher than production (Block et al. 1998) (see also Fig. 1C).

Concluding remarks

The overall picture emerging from this work suggests that all the main characteristics of estimated or produced time intervals could be quantitatively explained, at the single neuron level, in terms of attentional effects. More generally, the model further supports the view that anytime attention is focused on a given task two essentially independent processes are activated: an excitatory background on all the neural systems involved with the task execution, at a mean frequency that fluctuates according to reasons that are not related to the specific task, and an inhibitory activity from those brain regions involved with any concurrently attended, but unrelated, activity (such as additional cognitive loads, environmental changes, diseases, etc). Within this framework, for example, the shorter duration estimated for an auditory stimulus with respect to a visual one (Wearden et al. 1998) could be interpreted in terms of the additional attentional resources required to elaborate a visual input with respect to an auditory one. Changes in the attention-based processes could also explain why a number of factors such as age (Block et al. 1998; Craik and Hay 1999), drugs use (Frankenhaeuser 1959), body temperature (Hancock 1993), degenerative brain diseases (Nichelli et al. 1993), and concurrent cognitive loads (Marmaras et al. 1995; Zakay 1993), interfere with the correct production or perception of a time interval.