Article

A Neural Mechanism for S

ensing and Reproducing aTime Interval

Highlights

d Non-human primates use a Bayesian strategy to reproduce

time intervals

d Parietal cortex conveys distinct timing signals for

measurement and production

d Parietal neurons represent an estimate of elapsed time on a

trial-by-trial basis

d The brain encodes time prospectively during time interval

reproduction

Jazayeri & Shadlen, 2015, Current Biology 25, 2599–2609October 19, 2015 ª2015 Elsevier Ltd All rights reservedhttp://dx.doi.org/10.1016/j.cub.2015.08.038

Authors

Mehrdad Jazayeri, Michael N. Shadlen

Correspondencemjaz@mit.edu

In Brief

Sensorimotor function relies on an

ongoing temporal coordination of

sensory events with motor plans. Using

an interval reproduction paradigm,

Jazayeri and Shadlen find that neurons in

sensorimotor cortex encode time

prospectively in terms of upcomingmotor

plans. Results highlight a simple

preplanning model for sensorimotor

coordination.

Current Biology

Article

A Neural Mechanism for Sensingand Reproducing a Time IntervalMehrdad Jazayeri1,* and Michael N. Shadlen21Department of Brain and Cognitive Sciences and McGovern Institute for Brain Research, Massachusetts Institute of Technology,

Cambridge, MA 02139, USA2Department of Neuroscience, Zuckerman Mind Brain Behavior Institute, Kavli Institute of Brain Science, and Howard Hughes MedicalInstitute, Columbia University, New York, NY 10032, USA

*Correspondence: mjaz@mit.edu

http://dx.doi.org/10.1016/j.cub.2015.08.038

SUMMARY

Timing plays a crucial role in sensorimotor function.However, the neural mechanisms that enable thebrain to flexibly measure and reproduce time inter-vals are not known.We recorded neural activity in pa-rietal cortex of monkeys in a time reproduction task.Monkeys were trained to measure and immediatelyafterward reproduce different sample intervals.While measuring an interval, neural responses hada nonlinear profile that increased with the durationof the sample interval. Activity was reset during thetransition from measurement to production andwas followed by a ramping activity whose slope en-coded the previously measured sample interval. Wefound that firing rates at the end of the measurementepoch were correlated with both the slope of theramp and the monkey’s corresponding productioninterval on a trial-by-trial basis. Analysis of responsedynamics further linked the rate of change of firingrates in the measurement epoch to the slope of theramp in the production epoch. These observationssuggest that, during time reproduction, an intervalis measured prospectively in relation to the desiredmotor plan to reproduce that interval.

INTRODUCTION

Timing is essential for many different aspects of brain function,

from classical and instrumental conditioning to complex cogni-

tive faculties such as coordinating thoughts and actions in hu-

mans [1–3]. A basic understanding of the neural basis of interval

timing could shed light on how the brain integrates information

about the recent past with plans for the near future, a process

that is at the core of central information processing.

To incorporate knowledge about elapsed time into behavior,

neural circuits must be able to measure and produce time inter-

vals. These capacities are typically discussed under the rubrics

of ‘‘sensory timing’’ and ‘‘motor timing’’ [4]. Animal models of in-

terval timing in the sub-second to seconds range have focused

on relatively simple tasks. Sensory timing tasks are typically con-

cernedwith the ability to anticipate a sensory event [5] or to cate-

Current Biology 25, 2599–2

gorize time intervals [6, 7]. Motor timing tasks, on the other hand,

focus on the ability to produce fixed time intervals in ongoing

movements [8] or in actions learned through trace [9, 10] and op-

erant conditioning [11–14]. However, in many natural settings,

sensory and motor aspects of timing are heavily intertwined.

For example, in sports, music, and imitation, humans continu-

ously measure time intervals and use those measurements to

control the timing of their actions. To investigate themechanisms

that flexibly link sensory and motor timing capacities, we devel-

oped a time reproduction task for rhesus monkeys in which the

animals measured an interval demarcated by two time markers

and reproduced it by a proactive saccade.

Neural mechanisms of interval timing engage multiple brain

areas and are thought to depend on timescale [3, 4]. In the

sub-second to seconds range, where temporal processing is

crucial for anticipation, prediction and planning in sensorimotor

function, correlates of interval timing have been reported in

higher cortical areas, the basal ganglia, and the cerebellum [4].

We focused our work on the lateral intraparietal cortex (LIP),

which is thought to play a central role in sensorimotor function

[15–20] and where neurons represent elapsed time during both

sensory [5, 6] and motor timing tasks [13, 21].

We found that single neurons in LIP conveyed information

about the animal’s internal estimate of elapsed time during

both measurement and reproduction of time intervals. The

response profiles associated with themeasurement and produc-

tion of time intervals were remarkably different yet linked such

that modulation of activity during the measurement predicted

the response dynamics during the ensuing production.

RESULTS

BehaviorWe trained monkeys on an interval reproduction task [22], which

we refer to as the ‘‘Ready, Set, Go’’ (RSG) task. The task

(Figure 1A) consisted of two successive epochs. In the first

‘‘measurement epoch,’’ monkeys measured a sample interval,

demarcated by two peripheral flashes, a Ready cue (‘‘Ready’’

from here on) followed by a Set cue (‘‘Set’’ from here on). In

the immediately ensuing ‘‘production epoch,’’ monkeys had to

reproduce the sample interval by making a self-initiated

saccadic eye movement to a visual target (i.e., no explicit ‘‘Go’’

cue was presented). On each trial, the sample interval was drawn

randomly from a discrete uniform distribution ranging between

529 and 1,059 ms (Figure 1B). We refer to this distribution as

609, October 19, 2015 ª2015 Elsevier Ltd All rights reserved 2599

A C

B

D E F

Figure 1. The RSG Task and Behavior

(A) Sequence of events during a trial. The monkey fixated a central spot. A saccade target was then presented in the response field (RF) of the neuron (gray zone)

followed in succession by two transiently flashed peripheral cues (Ready followed by Set). The monkey then made an eye movement to the saccade target. To

receive maximum reward, monkeys were required to time their saccade to the target so as to match the time interval between Set and saccade (production

interval) to the interval between Ready and Set (sample interval). The target changed color to green on successful trials.

(B) Sample intervals were drawn from a discrete uniform distribution with 10 values ranging between 529 and 1,059 ms.

(C) Reward schedule. The width of the window for which animals received liquid reward (green area) scaled with the sample interval (see Supplemental

Experimental Procedures). The amount of reward within this window increased linearly with accuracy, as shown schematically by the size of liquid drops to the

right. No feedback or reward was provided for production intervals outside the green region.

(D) Production interval as a function of sample interval. Mean production intervals (open and filled circles) for the twomonkeys (Yo andWi) are plotted as a function

of sample interval (SEMs are smaller than the circles). Gray histograms show the distribution of production intervals for each sample interval for bothmonkeys. On

average, production intervals deviated from the line of equality (diagonal dashed line) and toward the mean of the sample interval distribution (horizontal dashed

line). The inset shows the magnitude of the bias for the longest sample interval (ordinate) versus the bias for the shortest sample interval (abscissa) across the 58

(legend continued on next page)

2600 Current Biology 25, 2599–2609, October 19, 2015 ª2015 Elsevier Ltd All rights reserved

the ‘‘prior.’’ The production interval was measured as the interval

between Set and when monkeys acquired the saccade target. In

timing tasks, the variability of responses usually scales with the

duration of the interval [1]. To compensate for this so-called

‘‘scalar variability,’’ the width of the window in which monkeys

received reward scaled with the sample interval (Figure 1C).

This manipulation made the task difficulty more or less the

same for all the sample intervals. To encourage animals to be

as accurate as possible, we scaled the reward size as a function

of accuracy (see Supplemental Experimental Procedures).

Production intervals increased monotonically with the sample

interval (Figure 1D) and were more variable for longer sample in-

tervals, which is consistent with the scalar variability of timing.

However, production intervals exhibited systematic biases to-

ward the mean of the prior distribution (horizontal line). The

magnitude of the bias was larger for sample intervals at the

long end of the prior distribution (Figure 1D, inset) where mea-

surements are more variable (due to scalar variability). Following

earlier work in humans [22], we found that this trade-off between

bias and variance was accurately captured by a Bayesian model

that optimizes performance by exploiting knowledge about the

prior distribution to reduce the variability associated with uncer-

tain measurements (Figures 1E and 1F).

PhysiologyWe placed the saccade target in the response field (RF) of indi-

vidual LIP neurons and recorded their spiking activity as the

animals performed the RSG task. Neural responses were modu-

lated during the measurement epoch between Ready and Set,

underwent a reset after Set, and increased before saccade initi-

ation (Figure 2B). These observations were consistent across LIP

and were readily evident in the population-average response

profile (Figures 2C–2E). After the appearance of Ready, which

was well outside the RF, LIP responses initially declined and

stayed low for approximately 500 ms and then increased mono-

tonically until Set was presented (Figure 2C). The appearance of

Set, whichwas also flashedwell outside the RF, triggered a dip in

the activity followed by a monotonic rise (Figure 2D). Activity

associatedwith the dip (100–250ms after Set) did not vary signif-

icantly with the sample interval (regression; beta = 0.44, confi-

dence interval [CI] = [9.33 10.21], p = 0.46). In the production

epoch, LIP responses increased monotonically and reached a

plateau shortly before saccade initiation (Figure 2E). At the

plateau phase (100–250 ms before saccade initiation), re-

sponses did not vary significantly with the sample interval

(regression; beta = 1.49, CI = [12.82 9.83], p = 0.40).

Our initial analysis focused on two timewindows, (1) an interval

near the time of Set and (2) an interval between Set and saccade.

The average firing rate near the time of Set (from 50 ms before to

recording sessions for the two animals (open and filled circles). Themagnitude of t

t test, p < 1e7).

(E) The fit (solid line) of a Bayesian model to the behavior of the twomonkeys. Data

with measurement and production noise derived from the fit of the Bayesian mo

(F) Comparison of the variability and bias of production intervals between the an

Bias’’) wasmeasured as the root-mean square of the production interval bias acro

as the root-mean square of the SD across sample intervals. The data show that th

model fits (ordinate) accurately captures the corresponding values derived from th

all behavioral sessions, showing that monkeys’ behavior was stable across sess

Current Biology 25, 2599–2

50 ms after Set) increased monotonically with sample intervals

(Figure 3A), both across the population (regression: beta =

17.90 sp/s/s, CI = [10.92 24.87], p < 1e6) and for 22 out of

58 individual neurons (t test, p < 0.05). In the interval between

Set and saccade, the buildup rate (estimated 200–500ms before

saccade time) was progressively shallower for longer sample in-

tervals (Figure 3B), both across the population (regression:

beta = 144.62 sp/s/s/s; CI = [164.15 125.09], p < 1e10)

and for 40 out of 58 of individual neurons (t test, p < 0.05). These

analyses indicate that both the activity near the time of Set and

the following ramping activity (i.e., linear increase of firing rates)

during the production epoch provide a correlate of the sample

interval.

Linking Neural Activity to BehaviorTo assess the link between the neural activity and behavior, we

first focused on the production epoch. Computing a reliable es-

timate of the buildup rate from spike times of individual neurons

in single trials is challenging. To tackle this problem, wemodeled

the ramping activity by a non-homogeneous Poisson process

with a linear rate function and used the spike times 200–

500 ms before saccade time to estimate the linear rate function,

which corresponds to the slope of the buildup rate. Using this es-

timate, we found a significant negative correlation between the

buildup rate and the production interval for every sample interval

(correlation coefficient =0.26 ± 0.04 [mean ± SE], p < 0.005 for

all 10 intervals). Importantly, the correlation analysis was per-

formed for each sample interval independently so that the pair-

wise relationships could not be attributed to an indirect effect

of the sample interval on the firing rates and the behavior (see

Figure S1 for an alternative analysis combining values across

all trials). This significant negative correlation indicates that,

for every sample interval, trials with a steeper buildup rate

were followed by shorter production intervals, consistent with

previous electrophysiological recordings in motor timing tasks

[12, 21, 23–26].

We then asked whether the activity in the measurement epoch

predicts the buildup rate in the production epoch. We measured

the trial-by-trial correlation between the buildup rate and the

activity at the time of Set and found that the quantities were

negatively correlated for every sample interval (correlation coef-

ficient = 0.23 ± 0.02 (mean ± SE), p < 1e6 for all 10 intervals).

Again, by performing the correlation analysis for each sample in-

terval independently, we ensured that the pairwise correlation

was not due to an indirect effect of the sample interval (see Fig-

ure S1 for an alternative analysis combining values across all

trials). Moreover, this correlation was not explained by an auto-

correlation in LIP firing rates within a trial because there was

no significant correlation between buildup rate and LIP activity

he bias was significantly larger for the longest sample interval (37.08 ± 5.55ms;

points are replicated from (A). The inset shows theWeber fractions associated

del to each behavioral session (see Supplemental Experimental Procedures).

imals’ behavior and the Bayesian model across sessions. The bias (‘‘Average

ss all sample intervals. The variability (‘‘Average Var1/2’’) was similarly measured

e Average Bias (black) and the Average Var1/2 (red) predicted from the Bayesian

e animal’s behavior. The inset is a plot of Average Var1/2 versus Average Bias for

ions.

609, October 19, 2015 ª2015 Elsevier Ltd All rights reserved 2601

−2000 −1000 0 1000

Time (ms)

SetTarget Ready Saccade

yo080515a

A B

SaccadeReady

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SetEC D

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Sample interval (ms)

−500 0 −2500 529 1059

Time (ms)

−200 0 200 400

ips

lu

sts

Figure 2. LIP Electrophysiology in the RSG Task

(A) Recording site. Top: a macaque brain is shown schematically along with a coronal plane through the intraparietal sulcus (ips). Bottom: the structural MRI of a

coronal section of one of the monkeys (Yo) after the placement of the recording cylinder. All recorded neurons were within the ventral portion of area LIP along the

lateral bank of the ips (red).

(B) Raster plot of the spiking activity of a single neuron from before the onset of the saccade target through completion of the saccadic eyemovement. Each trial is

displayed as a row of spikes (black tics) aligned to the time of Set. Colored symbols mark the times of target onset (blue), Ready (red), Set (orange), and when the

saccade reaches the target (green). Trials are sorted by duration of the sample interval; the order was random in the experiment.

(C) Response averages (N = 58) aligned to the onset of Ready. Sample intervals are indicated by colors (see legend for E). Each trace terminates at the time of the

corresponding Set, indicated by a filled circle. After an initial decline, activity increased monotonically with elapsed time.

(D) Response averages aligned to the time of Set. After Set, responses underwent a stereotyped dip and then diverged according to the sample interval. Filled

colored circles are identical to those in (A).

(E) Response averages aligned to the time of saccade. Firing rate increased toward a common plateau approximately 200 ms before saccade initiation. The

buildup of firing rate was shallower for longer sample intervals.

In (C)–(E), error bars indicate SEM.

early in the measurement epoch or during the dip after the Set

(p > 0.05).

The results from the simple pairwise correlation analyses is

consistent with a model in which the sample interval controls

the activity at the time of Set (Figures 2D and 3A), which in turn

sets the buildup rate (Figures 2E and 3B), which in turn forecasts

the saccade initiation time. However, with four interrelated vari-

2602 Current Biology 25, 2599–2609, October 19, 2015 ª2015 Elsevi

ables (sample interval, Set activity, buildup rate, and production

interval), it is important to ascertain that pairwise correlation be-

tween any two variables is not explained by their association with

the other two variables. Thus, we performed a partial correlation

analysis looking at the strength of association between each pair

of variables while the variation in the other variables are ac-

counted for—for example, the correlation between buildup rate

er Ltd All rights reserved

Sample interval (ms)529 1059

30

35

Sample interval (ms)529 1059

A B

−400 −300 −200 −100 0 100

0

5

10

Cou

nt

Slope (sp/s/s/s)

20

60

100

Set

act

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(sp

/s)

Bui

ld-u

p ra

te (

sp/s

/s)

−10 2 14 26 38 50

0

5

10

Cou

nt

Slope (sp/s/s)

Figure 3. Representation of the Sample In-

terval in the RSG Task in LIP Activity

(A) Average firing rate from 100 ms before to 50 ms

after Set (Set activity), as a function of sample in-

terval. Firing rate near the time of Set increased

monotonically with elapsed time. Error bars are

SEM (N = 58 neurons). For each neuron, we

computed the slope of the regression line relating

Set activity to sample interval. Inset histogram

shows the distribution of slopes across 58 neu-

rons. Dashed line indicates the mean regression

slope; filled bars indicate slopes significantly

different from zero (regression; t test, p < 0.05 for

22 out of 58 cells).

(B) Average buildup rate of the neural response

from 500 to 200 ms before the saccade, as a

function of sample interval. The buildup was more

gradual with longer sample intervals. Error bars are SEM (N = 58 neurons). Inset histogram shows the distribution of regression coefficient in a linear regression of

the buildup rate against sample interval for 58 neurons. Dashed line indicates mean; filled bars indicate slopes significantly different from zero (regression; t test,

p < 0.05 for 40 out of 58 cells).

and the production interval while accounting for the sample inter-

val and Set activity. Table 1 shows the magnitude of the simple

pairwise correlations (top right) and pairwise partial correlations

(bottom left). The results of the partial correlation analysis upheld

the original results about the linkage between (1) the sample in-

terval and Set activity, (2) the Set activity and the buildup rate,

and (3) the buildup rate and the production interval. This analysis

also led to two additional findings. First, the association between

sample interval and buildup rate was greatly diminished when

the variations in the Set activity were accounted for. Second,

the correlation between Set activity and production interval

was no longer statistically significant after accounting for buildup

rate and sample interval. Both of these observations strengthen

the interpretation of the pairwise correlations in terms of an asso-

ciation between the buildup rate with the production interval and

an association between the Set activity with the sample interval.

Response Dynamics in LIPLIP responses preceding saccade initiation increased approxi-

mately linearly (Figure 2E). This ramping activity is consistent

with an increase in the salience of the saccade target [27, 28]

mediated by an evolving motor plan [21] or an expectation of

reward [18]. In contrast, the response dynamics in the measure-

ment epoch (Figure 2C) is unexpected. In this epoch, neither the

motor plan nor the reward times were known before the appear-

ance of Set.More specifically, it is unclear why the responses un-

derwent a slow and prolonged suppression until approximately

500 ms after Ready and why they increased with a decelerating

nonlinearity until the time of Set.

One possibility is that LIP response dynamics are explained by

modulations of attention, induced by the sensory and motor

events in the task, irrespective of computations needed for inter-

val reproduction. For example, visual flashes could modulate

exogenous attention. Similarly, the firing rate dynamics after

Ready could be due to a slow shift of endogenous attention to

the saccadic target. To examine these possibilities, we designed

a control task, which we refer to as the Ready-Go (RG) task (Fig-

ure 4). The sequence of events in the RG task were identical to

the RSG task: fixation followed by the presentation of Ready

and Set (demarcating an interval sampled from the prior), fol-

Current Biology 25, 2599–2

lowed by a proactive saccade to a visual target, followed by

reward for accurately timed saccades. However, in the RG

task, monkeys had to produce a fixed 1,588-ms interval from

the time of Ready, irrespective of when Set was presented (Fig-

ure 4A). Importantly, the RG task maintained the spatial and

temporal structure of sensory and motor events but inverted

the relationship between the duration of two epochs such that

longer Ready to Set intervals were followed by shorter Set to

saccade intervals. Accordingly, if LIP responses were explained

bymodulations of attention due to sensory andmotor events, we

would expect to see the same response dynamics in the two

tasks.

Both monkeys learned the RG task, as evidenced by the in-

verted relationship between the duration of the two epochs:

the interval between Set and saccade was progressively shorter

for longer Ready-Set intervals (Figure 4B). There were, however,

systematic biases away from the target 1,588 ms. In particular,

saccades were initiated earlier when Set occurred shortly after

Ready and later when Set occurred at a longer interval. This

counterintuitive observation is predicted by a Bayesian model

similar to the RSG task in which knowledge about the prior dis-

tribution of the time of Set helps to reduce variability and improve

performance (Supplemental Experimental Procedures).

In the RG task, LIP activity began to rise shortly after Ready

(Figure 4C), underwent a transient dip after Set (Figure 4D),

and continued its climb to a maximum firing rate shortly before

saccade initiation (Figure 4E). Notably, responses increased

with an approximately constant buildup rate in both Ready-Set

and Set-saccade epochs. The response dynamics in the RSG

and RG tasks differed in two important ways (Figure 5). First,

the responses immediately after Ready were remarkably

different. Unlike the RSG task where responses were sup-

pressed for nearly 500 ms, in the RG task, responses underwent

a rapid dip and recovery, whichwas similar to how LIP responses

changed after Set in the RSG task. Second, unlike in the RSG

task, where responses increased nonlinearly 500 ms after

Ready, in the RG task, responses exhibited a ramp-like activity

with firing rates increasing linearly until the time of Set. The linear

rise of activity before Set in the RG task was similar to the ramp-

ing activity of the RSG task before saccade initiation.

609, October 19, 2015 ª2015 Elsevier Ltd All rights reserved 2603

Table 1. Simple and Partial Pairwise Correlations between the

Sample Interval, the Activity near the Time of Set, i.e., the Set

Activity, the Buildup Rate Prior to Saccade Initiation, and the

Production Interval

Sample

Interval

Set

Activity

Buildup

Rate

Production

Interval

Sample interval – 0.13* 0.27* 0.62*

Set activity 0.07a,* – 0.26* 0.11*

Buildup rate 0.04a,* 0.23a,* – 0.37*

Production interval 0.58a,* 0.03a 0.26a,* –

The pairwise correlations are shown in the upper right triangle of the table

and the partial correlations in the bottom left triangle (see footnote below).

The partial correlation between each pair of variables is measured after

accounting for the effect of the other two variables. The asterisks (*) corre-

spond to correlation values that are significantly different from zero

(p < 0.01). The diagonal values (dashes) correspond to correlation of a

variable with itself.aPartial correlation

Importantly, the differences between the response dynamics

in the two tasks are not explained by a difference in the

baseline activity or response gain of the recorded neurons,

as evidenced by the similar average firing rate in both

tasks early in the measurement epoch (Figure 5). Since the

RG and RSG tasks had identical sensory and motor compo-

nents, the striking differences in firing rate modulations

associated with the two tasks rule out an interpretation of

the prolonged suppression in the RSG task in terms of

low-level sensory interactions or gradual shifts of attention

from Ready to the RF in anticipation of the saccade. Instead,

the differences in firing rate dynamics must be related to

the differences in temporal demands of the two tasks: in the

RG task, the desired production interval (1,588 ms) was

known through the trial, whereas in the RSG task, the desired

interval changed depending on the time between Ready

and Set.

Computational Models of Response Dynamics in theMeasurement EpochWe considered several computational models to explain the

response profile in the measurement epoch of the RSG task.

Each model sought to predict the observed dynamics based

on the dynamics of a candidate variable such as the anticipation

of Set or the anticipation of reward. In all models, we allowed our-

selves the freedom to scale and offset the predicted dynamics to

fit the neural data. This procedure ensures that the nonlinearities

in the neural data could only be explained by the variable of inter-

est and not the model parameters (see Supplemental Experi-

mental Procedures).

Anticipation of Set

The earliest time when Set was presented was 529 ms, which

matches the observed 500-ms delay in the rise of LIP responses

after Ready. Based on this observation and motivated by previ-

ous work [5, 29], we asked whether LIP responses represent a

hazard function of the time of Set (the probability that Set will

occur now, given that it has not yet occurred). As evident from

Figure 6A, the best fit of this model is inadequate since the

hazard function associated with a uniform prior distribution in-

2604 Current Biology 25, 2599–2609, October 19, 2015 ª2015 Elsevi

creases expansively, whereas LIP firing rates exhibited a

compressive nonlinearity.

Anticipation of Reward

Since reward expectation modulates LIP activity [18], we asked

whether the observed response dynamics represent the proba-

bility of expected reward. Since the animal never received

reward before the time of Set, we can reject the strongest

form of this hypothesis—that firing rates represent the instanta-

neous probability of reward. However, we considered a more

nuanced version of this hypothesis in which the firing rate

before Set represents the probability that the animal will receive

reward at the end of that trial (see Supplemental Experimental

Procedures), which we estimated directly from the average

reward the animal received. The response dynamics predicted

by this model also do not match the observed LIP dynamics

(Figure 6B).

Anticipation of the Expected Time of Reward

This is a timing model, but one in which firing rates adopt a slope

that is predictive of the expected time (as opposed to the prob-

ability) of reward. This schemewasmotivated by the observation

of response dynamics in the production epoch of the RSG task

where the buildup rate was adjusted such that the ramping activ-

ity reached a terminal firing rate shortly before the expected time

of reward. We formulated this model by a first order linear differ-

ential equation that adjusts the instantaneous slope of firing rate

according to the expected time of reward (see Supplemental

Experimental Procedures). As shown in Figure 6C, this model

predicts a decelerating response dynamics after 529 ms, which

is consistent with our data. However, it additionally predicts a lin-

early rising activity in the early part of the measurement epoch,

which is similar to what we observed in the RG task (Figure 4C),

but not in the RSG task.

Bayesian Estimate of the Sample Interval

Our analysis of behavior showed that the production interval was

based on a Bayesian estimate of the sample interval (Figures 1E

and 1F). This finding motivates a model in which the activity dur-

ing themeasurement epoch reflects the Bayesian estimate of the

sample interval. This idea is appealing because the Bayesian es-

timate is bounded by the range of the prior distribution, which

could explain the 500-ms delay in the rise of responses after

Ready. Moreover, the Bayesian regression toward the mean of

the prior might explain the nonlinear response dynamics begin-

ning 500 ms after Ready. As evident in Figure 6D, this scheme

is broadly consistent with a rise of firing rates after approximately

500 ms and the decelerating nonlinearity, albeit somewhat

different from the LIP response profile. This difference, however,

might be due to our particular implementation of the Bayesian

model (see Discussion).

Preplanning the Production Dynamics

A simple observation from LIP response dynamics is that both

the instantaneous slope of the activity as a function of sample in-

terval before Set and the slope of the ramping activity after Set

decrease progressively with longer sample intervals. We there-

fore considered a ‘‘preplanning’’ hypothesis in which the slope

of activity in the measurement interval is predictive of the slope

of activity in the production epoch. This is an attractive

computational strategy as it obviates the need to rapidly

compute the buildup rate during the transition between the mea-

surement and production epochs. Based on this scheme, the

er Ltd All rights reserved

A

Saccade

Set

ReadyTim

e

1588 ms

529

588

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Sample interval

−500 0 −250

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0 529 1059

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Time (ms)

Firi

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n = 14002==== 44444 00000

200Count

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Figure 4. The RG Control Experiment

(A) Task design. The animal had to make a saccade to visual target 1,588ms after Ready. Tomake the sensory events of the RG and RSG tasks identical, we also

presented a Set cue in the RG task. The interval between Ready and Set was drawn randomly from the same distribution used in the RSG task (Figure 1B).

(B) Animals’ behavior in the RG task. The behavior is plotted in the same format as in the RSG task (Figure 1D); i.e., the Set-saccade intervals as a function of the

Ready-Set interval. In this task, maximum reward was given when the Ready-saccade interval (i.e., the sum of the Ready-Set and Set-saccade intervals) was

1,588 ms (anti-diagonal). As expected, the Set-saccade intervals were on average shorter for longer Ready-Set intervals. However, saccade times exhibited

systematic biases away from the anti-diagonal.

(C–E) The time course of average population activity (N = 39) plotted in the same format as in Figures 2C–2E.

instantaneous slope of activity at the time of Set should provide

an estimate of the buildup rate after Set. To test the hypothesis,

we constructed a model in which the instantaneous slope of

activity at the time of Set was linearly related to the buildup

rate after Set (Experimental Procedures). This model captures

the response dynamics in the measurement epoch better than

the other models we considered (Figure 6E).

The preplanning model has the additional virtue that it pro-

vides a common framework to explain responses in both the

RSG and RG tasks. To demonstrate this point, we developed a

data-driven analog of the preplanning model in which we aimed

to fit the firing rate dynamics prior to Set based on the observed

slope of the ramping activity after Set. To do so, we constructed

a piecewise linear function in which the slope of each line

segment was derived directly from the corresponding slope after

Current Biology 25, 2599–2

Set. As shown by the red traces in Figure 4, the piecewise linear

functions constructed in this way matched the observed

response dynamics for both tasks remarkably well (R2 = 0.98

and 0.95 for RSG and RG tasks, respectively). This result indi-

cates that the response profiles before Set in both tasks are

consistent with the preplanning hypothesis.

DISCUSSION

Previous animal studies of interval timing havemainly focused on

either sensory or motor aspects of timing [4]. To understand the

mechanisms by which sensory and motor aspects of timing are

coordinated, we recorded neural activity in area LIP of monkeys

performing a time interval reproduction task. Monkeys repro-

duced time intervals by making saccades toward a visual target

609, October 19, 2015 ª2015 Elsevier Ltd All rights reserved 2605

529 706 882 1059

25

35

45

Time after Ready (ms)

Firi

ng r

ate

(sp/

s)

RSG task

RG task

Prediction fromactivity after Set [a.u.]

Figure 5. Comparison of LIP Responses in the RSG and RG Tasks

The black and gray traces show the average LIP activity between Ready and

Set in the RSG and RG tasks, respectively. The spike train from each trial was

convolved with a 50 ms boxcar filter, and the smoothed firing rates were used

to compute a running mean across neurons and trials that combined all spikes

elicited after Ready and before 50 ms after Set. The superimposed red curves

are fits based on the preplanning model. Each red trace is a piecewise linear

functionwith ten line segments (for ten sample intervals). The slope of each line

segment was derived directly from the corresponding ramping activity in the

production epoch. Specifically, the slope of the line segment between each

pair of consecutive sample intervals, ti and ti + 1, was equal to the slope of the

ramping activity associated with reproducing ti. We then used linear regression

(i.e., two parameters: offset and scaling) to fit this piecewise linear function to

LIP firing rates.

inside the RF of individual neurons in area LIP. As many previous

studies have shown, this experimental design exploits the

spatial tuning of LIP neurons to investigate the computational

mechanisms of various cognitive functions such as anticipation,

planning, and decision making [13, 18, 28, 30–33]. Here, we

adapted this scheme to study the neural computations associ-

ated with time interval reproduction.

LIP firing rates in both measurement and production epochs

were modulated by the sample interval, and the trial-by-trial

fluctuations of neural activity in both epochs were predictive of

the fluctuations of animals’ production intervals. Our partial

correlation analyses (Table 1) support the presence of associa-

tions between (1) the sample intervals and firing rates in themea-

surement epoch, (2) the firing rates in the measurement epoch

and the ramping activity in the production epoch, and (3) the

ramping activity in the production epoch and the production

intervals.

Response DynamicsIn the production epoch, firing rates leading up to the saccade

increased linearly and reached a fixed plateau regardless of

the duration of the sample interval. This observation was rein-

forced by the RG control experiment. There, the ramping activity

began shortly after Ready, which is consistent with the fact

that the desired interval was specified with respect to Ready.

These observations are not surprising as ramping activity

in anticipation of a potentially rewarding motor response has

2606 Current Biology 25, 2599–2609, October 19, 2015 ª2015 Elsevi

been observed in many previous reaction time and self-timed

tasks [12, 21, 23–26, 34–36]. This pattern of activity either might

be related to the anticipation of an imminent reward or might

reflect an ongoing saccadic motor plan.

The second, more surprising observation was the discovery of

an evolving signal during the measurement epoch of the RSG

task. In this epoch, responses began to rise approximately

500 ms after Ready and increased monotonically until the time

of Set. Our first concern was that these features might be ex-

plained by low-level sensory responses or by attentional modu-

lations during the trial. To evaluate those possibilities, we

designed a control (RG) task that comprised the same exact

sensory and motor elements and compared responses in the

Ready-Set interval. We found striking differences between the

response in the RSG and RG tasks that were not explained by

differences in baseline activity or gain differences (Figure 5).

More generally, no linear transformation of average firing rates

could explain the difference between the profiles of the firing

rate dynamics in the two tasks. This observation suggests that

the activity during the Ready-Set interval of the RSG task is un-

related to low-level stimulus-driven interactions and cannot be

explained by a gradual shift of attention to the location of Set

or the saccade target.

We then formulated a series of models that sought to explain

the response dynamics in the measurement epoch in terms of

anticipation of Set or reward. Anticipation of Set is captured by

a hazard-like function of Set time [5], which predicts an acceler-

ating nonlinearity—opposite to the decelerating nonlinearity

observed in the data (Figure 6A). A model based on experienced

reward [17, 18] also failed because the probability of reward as a

functionof timedidnotmatch theLIP responseprofile (Figure6B).

A thirdmodel that tested the responsedynamics against anantic-

ipation of the time of reward (or the time of saccade, which

occurred at the same time in our study) could predict the decel-

erating nonlinearity in the measurement epoch of the RSG task

(Figure 6C). However, it additionally predicts that LIP responses

should exhibit identical ramping activity in the first 500 ms of

the RG and RSG tasks, which is not supported by data. In sum,

although LIP responses are likely to be influenced by sensory,

motor, or reward anticipations, our modeling results render it

highly unlikely that these functions explain LIP response dy-

namics during the measurement epoch of the RSG task.

Two Novel Hypotheses: Bayesian Estimation andPreplanningLIP responses in the measurement epoch of the RSG task

cannot be interpreted as a direct measure of elapsed time

because firing rates do not increasemonotonically with time until

approximately 500 ms after Ready. According to our analysis of

behavior, production intervals were based on the Bayesian esti-

mate of the sample interval. We therefore developed a model

that tested whether LIP responses in the measurement epoch

were consistent with a Bayesian estimate of the sample interval.

This model, like themodel associated with the hazard rate of Set,

explains the 500-ms delay in the rise of response after Ready,

and it additionally predicts a decelerating nonlinearity thereafter

(Figure 6D). Although the exact predicted profile was somewhat

different from LIP activity, we think that this model deserves

further investigation. In our implementation of the Bayesian

er Ltd All rights reserved

300 650 1000

26

32

38

Time after Ready (ms)

Firin

g ra

te (s

p/s)

r2=0.86 r2=0.81

Set hazard Expected reward

A B

r2=0.93 r2=0.92 r2=0.97

Reward time Bayesian estimate Preplanning

EC D

Figure 6. Model Comparison

(A–E) Comparison of the activity profile in the RSG task (black traces, same in all panels) from 300 ms after Ready to the time of the latest possible Set with

predictions of different models (red). Themodels are constructed as a linear function of the subjective hazard of the Set (A), the expected reward (B), the expected

time of reward (C), the Bayesian estimate of the sample interval (D), and the preplanning model in which the instantaneous slope predicts the slope of the

corresponding ramping activity in the production epoch (E).

estimator, we assumed that the animal has accurate knowledge

about the prior distribution of sample intervals and uses a least

square cost function. It may be that an alternative implementa-

tion with more realistic assumptions about the prior and cost

function could capture the observed dynamics more accurately.

Future experiments involving multiple prior distributions will pro-

vide a critical test for this possibility.

The last model we considered was motivated by the observa-

tion that both the instantaneous slope of responses before Set

and the slope of the ramping activity after Set decreased with

longer sample intervals. We therefore developed a preplanning

model in which the instantaneous slope of the response during

the measurement epoch anticipates—or preplans—the buildup

rate of the ensuing ramp after the Set and dip in activity. In this

model, it is expected that LIP activity begins to rise with an initial

slope when the earliest Set is anticipated (529ms), and the slope

is reduced dynamically to adjust for the attenuated buildup rate

needed for producing longer production intervals. For this

scheme to work, the brain must possess implicit knowledge of

(1) the fixed delay between Set and the beginning of the ramping

activity, (2) the difference in firing rate at the beginning of the

ramping phase and some threshold level, and (3) the fixed delay

between when responses reach a terminal point and saccade

initiation. When we estimated these values from the average

LIP activity, we found that the preplanning model was able to

capture the observed LIP response profile quite well (Figure 6E).

Moreover, this model readily explains the response dynamics in

both RSG and RG tasks, as evidenced by the quality of fits

derived directly from slopes of ramping activity after the Set (Fig-

ure 5, red). To test this model more definitively, we must record

from multiple neurons simultaneously to have a more accurate

estimate of the instantaneous slope on single trials.

Unresolved QuestionsOne of the notable features of firing rate dynamics is the pres-

ence of a strong and transient suppression after Set in the

RSG task. The function of this so-called dip, which has been

observed in different oculomotor areas [24, 37–39], is unclear.

In our work, the dip transiently masks the information about

the interval, and it is not clear how this information reappears

after the dip. The suppression during the dip may be due to a

Current Biology 25, 2599–2

temporary shift of exogenous attention to Set [40]. However,

regardless of the source of this suppressive effect, the recovery

of signal after the dip implies that either the recurrent networks in

LIP can maintain an estimate of the buildup rate through the

externally triggered suppression or the signals associated with

the buildup rate are available in other brain areas that do not un-

dergo such strong suppression, or the maintenance of the infor-

mation in mediated by an intrinsic cellular mechanism [41].

An important issue is whether LIP plays a causal role in interval

timing or whether it is modulated indirectly through anatomical

connections that support functions other than timing [5, 6, 13,

15, 18–21, 28, 42–48]. Without a careful causal manipulation,

we cannot distinguish between these two possibilities, nor can

we rule out a third possibility that LIP neurons multiplex behav-

iorally relevant variables [49] including time [5, 6]. Regardless,

however, our analysis of how LIP signals in the measurement

epoch seem to preplan the buildup rate in the production epoch

raises the speculative but intriguing hypothesis that the sense of

time could be established through an embodied scheme that

models the parameters of a motor plan.

EXPERIMENTAL PROCEDURES

General Procedures

Behavioral protocols, animal care, and surgical procedures were all in accor-

dance with the US National Institutes of Health Guide for the Care and Use of

Laboratory Animals and were approved by the University of Washington Ani-

mal Care Committee. We recorded from 97 well-isolated LIP neurons in the

ventral portion of area LIP (LIPv) in the right hemisphere of two monkeys (Yo

and Wi), 58 in the RSG task (35 in Yo and 23 in Wi) and 39 in the RG task (26

in Yo and 13 in Wi). We analyzed the behavior using a Bayesian model

following earlier work [22]. Population, single-cell, and single-trial analyses

were based on either the average firing rates (e.g., at the time of Set) or the

change of firing rate per unit time (e.g., buildup rate before saccade initiation).

General experimental procedures, behavioral tasks, electrophysiological

recording technique, data analysis, and mathematical models are described

in detail in the Supplemental Experimental Procedures.

SUPPLEMENTAL INFORMATION

Supplemental Information includes Supplemental Experimental Procedures

and one figure and can be found with this article online at http://dx.doi.org/

10.1016/j.cub.2015.08.038.

609, October 19, 2015 ª2015 Elsevier Ltd All rights reserved 2607

AUTHOR CONTRIBUTIONS

M.J. and M.N.S. designed the experiment, analyzed the data, and wrote the

manuscript. M.J. collected the data.

ACKNOWLEDGMENTS

We are grateful to A.D. Boulet and K. Ahl for technical assistance and to Ros-

sana Chung and Leslie Gentleman for proofreading. This work was supported

by a fellowship from Helen Hay Whitney Foundation, HHMI, and research

grants EY11378, EY01730, and RR000166.

Received: July 15, 2015

Revised: August 15, 2015

Accepted: August 17, 2015

Published: October 8, 2015

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Current Biology

Supplemental Information

A Neural Mechanism for Sensing

and Reproducing a Time Interval

Mehrdad Jazayeri and Michael N. Shadlen

Fig S1. Related to Figure 3; Trial­by­trial correlations. The left panel shows

production intervals as a function of buildup rates, and the right panel shows buildup

rates as a function of Set activity for all trials for which the buildup rate could be

estimated. To combine values across neurons, variables were first normalized (zscore)

for each neuron independently. The marginals of the the variables on the ordinate and

abscissa are shown on the left and below each panel. The two ellipses (red) represent a

2D gaussian fit to the data points at 1 and 2 standard deviations from the mean.

Supplemental Experimental Procedures

Monkeys were surgically implanted with a titanium head­holding device attached to the

skull. During the experimental sessions, monkeys were comfortably seated in a dark

quiet room with their heads fixed, and viewed stimuli binocularly from a distance of 57

cm. All stimuli were presented on a fronto­parallel 15­inch Hewlett Packard HP71 CRT

monitor at a resolution of 1024x768 at a refresh rate of 85 Hz. We used the software

Expo (http://corevision.cns.nyu.edu/) to deliver stimuli and control behavioral

contingencies. Eye positions were sampled at 1kHz from signals recorded with an

infrared camera (Eyelink 1000; SR Research Ltd.; Ontario, Canada). We recorded

action potential extracellularly through a craniotomy over the right intraparietal sulcus

with either glass­coated tungsten electrodes (Alpha Omega Co.; Alpharetta, GA, USA)

or quartz­platinum/tungsten microelectrodes (Thomas RECORDING GmbH; Giessen,

Germany). Eye movement and extracellular signals were stored using a standard data

acquisition system (Plexon Inc.; Dallas, Texas, USA).

Behavioral task

In the main RSG task, each trial began with the appearance of the central fixation point

(FP) that monkeys were required to fixate. While fixating and after a 500 ms delay, the

saccade target was presented. After a variable foreperiod (0.2­2.0 sec, truncated

exponential distribution), two 105­ms flashes (Ready and Set) were presented.

Typically, Ready was presented diametrically opposite to the saccade target, and Set

was presented 4 to 10 deg away from the FP, equidistant from Ready and the saccade

target. Monkeys were trained to measure the sample interval (ts) between Ready and

Set, and reproduce that interval by a proactive saccade to the target. Production interval

(tp) was measured from midpoint during the flashed Set (i.e. ~ 50 ms after its onset) to

when the monkey acquired the target.

When the difference between ts and tm was smaller than an experimentally specified

window, the color of the saccade target changed to green and monkeys received an

immediate liquid reward. To compensate for the increased variability associated with

longer ts [S1,S2] the width of the rewarded window was scaled with the sample interval.

The scaling constant was adjusted so that approximately 60% of trials were rewarded.

The rewards were graded linearly such that the volume was largest for accurate

reproduction and declined to 0 at the boundary of the acceptance window. Monkeys’

behavior was stable and consistent in all recorded sessions. In some sessions, we

increased reward rate near the end of the session by increasing the width of reward

window by nearly 10% to encourage the monkey to complete more trials. These

manipulations did not have a significant effect on the average root­mean­squared­error

(RMSE) of production intervals.

The RG task we used as a control experiment was identical to the RSG task in sensory

and motor events but required monkeys to make a saccade to the target 1588 ms after

the Ready, irrespective of when Set was presented. The duration of Ready­Set was

drawn from the same discrete uniform distribution that was used in the RSG task. The

rewarded window in the RG task was also scaled with the production interval.

Bayesian model for the RSG task

Our Bayesian model follows directly from our work on humans performing the RSG task

[S3]. Briefly, a Bayesian observer makes a noisy measurement of the ts. We modeled

the measurement noise as a random variable with a zero­mean Gaussian distribution

whose standard deviation scales with the ts (i.e., scalar variability). Accordingly, the

relationship between the measured interval, tm, and sample interval, ts, can be written

as:

(1)

where p(tm|ts) denotes the conditional probability of the measured interval (tm) as a

function of ts, and wm is the Weber fraction associated with scalar variability in the

measurement process.

The Bayesian observer combines the probabilistic information conveyed by tm with

knowledge about the distribution of sample intervals, p(ts), which is a uniform distribution

ranging between 529 and 1059 ms, to compute the posterior distribution over ts:

(2)

Following previous work on humans [S3], we assumed that the observer uses the

posterior to infer an optimal estimate, te, that minimize the squared difference between

the sample and estimate. In other words, we considered a quadratic cost function (te ­

ts)2, which is associated with a Bayes Least Squares (BLS) estimator. The BLS

estimator uses the mean of the posterior as its optimal estimate:

(3)

The Bayesian observer then attempts to reproduce the estimate, te. We assumed that

the production process is also noisy, and modeled the production noise as a random

variable with a zero­mean Gaussian distribution whose standard deviation scales with

the estimate (i.e., scalar variability):

(4)

where p(tp|te) denotes the conditional probability of the production interval, tp, as a

function of te, and wp is the Weber fraction associated with production process that may

be different from wm.

We fitted this model to the monkeys’ behavior by maximizing the likelihood of the model

parameters, wm and wp, across all ts and tp values.

We quantified the monkeys’ behavior and the corresponding behavior of the Bayesian

model using two parameters, the average bias and the average variance. These two

metrics, which we denote them by BIAS and VAR, respectively, are directed related to

the overall mean­squared error (MSE) of the production intervals as follows:

(5)

where index specifies the value of ts , and refers to the production time in

the repetition of . As indicated by this equation, MSE can be written as the

sum of two terms: the sum of squares of bias (i.e., BIAS2) and the sum of variances

(i.e., VAR).

Bayesian model for the RG task

In the RG task, the animals’ behavior was influenced by the time of the Set cue. This

influence seems perplexing because the monkey is rewarded for reproducing a 1588

ms interval on all trials, regardless of the intervening Set cue. There seem to be two

general possibilities for explaining this influence: Set presents itself as a distractor, or

alternatively, the monkey actively uses the Ready­Set interval as a cue to improve

performance. We can eliminate the first possibility by examining the variability of the

production intervals in the RG task. If Set were to act as a distractor, it would increase

the variability, but it appears to do the opposite. We base this conclusion on the

following analysis.

We computed the predicted standard deviation of the tp in the RG task, which is equal to

the mean tp multiplied by wp. Our estimate of wp from the fits of the Bayesian model to

the behavior in the RSG task (wp = 0.14) predicted the standard deviation of tp in the RG

task to be ~224 ms. In comparison, the standard deviation we observed in the RG task

was ~150 ms, which is approximately 33% below expectation. It is therefore, not

plausible to assume that the Set was a simple distraction. Instead, monkeys seemed to

have used the Ready­Set interval to reduce variability in their behavior.

It may seem at first unintuitive that using the time of the Set cue would improve

performance, but in fact it could. To understand why, it is useful to consider a

hypothetical experiment in which the Set occurs at the same time on every trial. In that

case, the superior strategy is to rely on Set because it would allow the monkey to

produce the shorter time from Set to the target 1588 ms, which will be less variable, as

per Weber’s law. For example, if ts were fixed to half the 1588 ms (i.e., 794 ms), the

subject could learn to wait until the Set, and then produce a fixed 794 ms, which would

make production times twice as accurate as timing 1588 ms from the Ready cue.

What about the case in which the Ready­Set is drawn from a distribution? In that case,

an optimal Bayesian observer would still not ignore the Set altogether; instead it would

used it probabilistically and estimate the Ready­Set interval by integrating the observed

interval with the prior distribution from which it is drawn. The benefit is of course

attenuated by the additional uncertainty of the prior distribution but it is not eliminated

altogether, so long as the prior distribution can be exploited. In other words, the same

observer model we used to explain the behavior in the RSG task could also explain the

effect of Set on the behavior of the two monkeys in the RG task. The only difference is

that, in the RG task, the observer reproduces 1588ms – te.

Electrophysiology

A 9 Kg male (Yo) and a 6.8 Kg female (Wi) rhesus monkey (Macaca mulatta), both 7

years old, participated in this experiment. We recorded from 97 well­isolated LIP

neurons in the ventral portion of area LIP (LIPv) in the right hemisphere of two monkeys

(Yo and Wi), 58 in the RSG task (35 in Yo, and 23 in Wi), and 39 in the RG task (26 in

Yo, and 13 in Wi). LIPv was identified based on (1) anatomical landmarks along the IPS

[S4] and in registration with structural MRI scans distributed with the CARET software

package [5] and (2) transition of white and gray matter during recording.

Most LIP neurons have oculocentric response fields (RF) [S6] and their firing rate

increases when monkeys plan a saccadic eye movement to the RF [S7]. Many LIP

neurons also exhibit persistent activity in the memory period of the memory saccade

task [8,9]. We reasoned that neurons with persistent activity would be a good candidate

for studying temporal anticipation and planning, as they are capable of reflecting the

animal’s internal state in the absence of direct visual input [S10,11]. We thus recorded

from cells that had spatially selective sustained activity in the memory period of a

memory­saccade task [S8]. The cell’s response during the memory period of the

memory saccade task had to be at least 50% larger than firing rate for the memory

period of saccades made to a target opposite to cells’ response field (RF). We recorded

from neurons with RF located at eccentricities ranging between 3.8 to 20 deg. For RFs

located at eccentricities over 14 deg, we moved the fixation point to ensure that the

target was not too close to the edge of the monitor. During the RSG and RG tasks, the

saccade target was placed in the central part of the neuron’s RF (ranging between 4

and 20 deg across sessions), and spiking activity was recorded. Ready and Set were

displayed outside the RF. For many cells, we verified the stability of recordings by

monitoring the spatial selectivity and persistent activity in the memory­saccade task

intermittently throughout the session.

Data Analysis

We estimated response strength at the time of Set by averaging firing rates within a

window of 50 ms before to 50 ms after Set. We estimated the slope of the ramping

activity in the production epoch from firing rates within a window of 500 to 200 ms

before saccade initiation. Results were qualitatively the same for moderate variations of

the analysis windows (tested up to 50 ms). For each trial, we fitted a first­order rate

function of an inhomogeneous Poisson process to each trial’s spike times

(maximum­likelihood procedure), and used the linear term of the fitted rate function as

our estimate of the slope for that trial.

We used linear regression to quantify the correspondence between LIP activity and the

sample interval. To estimate the slope of the regression line, we used ordinary least

squares for single cell analyses and weighted least squares for the population analysis.

For the weighted least squares regression analysis, the error term associated with each

cell was weighted by the reciprocal of the standard error of that cell’s mean firing rate.

Confidence intervals and p values for the fits were derived from the standard error of the

slope of the regression line [S12].

Models

We developed a series of models to understand which of several candidate quantities

would best capture the observed nonlinear response dynamics (firing rate vs. time)

during the measurement epoch of the RSG task. In each case, we computed the

quantity of interest as a function of time, f(t), and developed a model in which the firing

rate, y(t), was a linear function of f(t); i.e. y(t) = Af(t)+B. Importantly, the linearity of this

transformation ensures that the response nonlinearlities can only be explained by the

quantity of interest and not the parametric form of the model. Furthermore, it ensues

that all models have the same number of parameters (A and B) that simply control the

scaling and offset of the fit to the data.

1. Anticipation of Set. To test this model, we computed the probability distribution of

measured intervals, p(tm), from the prior probability, p(ts), and p(tm|ts), as described in the

Bayesian fit to the behavior (Equation 1), and used the hazard of p(tm) as a subjective

hazard for Set, hSet(t).

(6)

(t ) (t |t )p(t )dtp m = ∫

p m s s s

(7)

(t )hSet =p(t )m

1−P(t )m

where p(tm) and P(tm) refer to the probability density and distribution function,

respectively. We then modeled the firing rate, y(t), as a linear function of hSet(t), and

computed the best fit of this model to activity from 300 ms after Ready to the time of

Set. Our conclusions were the same for a variant of this model in which firing rate are

modeled as a linear function of the objective hazard (computed from p(ts) instead of

p(tm)).

(8)

(t) Ah (t) y = Set + B

2. Anticipation of Reward. To test this model, we first computed the expected reward as

a function of time in terms of the hazard of Set; i.e., the probability of receiving reward,

given that Set has not yet occurred. This probability provides an empirical measure of

the probability of reward as a function of time during the measurement epoch. At the

beginning of the measurement epoch and until the time of the earliest possible Set, all

sample intervals are equally probable, and therefore, the probability of reward is equal

to its grand average across all trials. As time elapses beyond the earliest possible Set

time, the shortest sample interval is no longer possible, and the probability of reward

gets updated to the average of all trials except those associated with the shortest

sample intervals. This updating process continues after each possible sample interval

so that the probability of reward dynamically reflects the animals’ expectation based on

the remaining probable sample intervals. If we denote the proportion of rewarded for

trials as a function of the sample interval by gRew(ts), this hazard­like function for reward,

hRew(t), can be written mathematically as follows:

(9)

(t )hRew = Σ p(t )t >ts s

Σ g (t )p(t )t >ts Rew s s

This function has a piecewise linear form as it gets updated only at discrete times

associated with each sample interval. We used a linear interpolation scheme to smooth

the function, and then modeled firing rate, y(t), as a linear function of this interpolated

function. We computed the best fit of this model to activity from 300 ms after Ready to

the time of Set. Our conclusions were the same for a variant of this model in which we

replaced the proportion of rewarded trials by the expected reward magnitude computed

as the reward magnitude multiplied by the probability (recall that, when the animal was

rewarded, the magnitude of reward was dependent on accuracy).

(10)

(t) Ah (t) y = Rew + B

3. Anticipation of the expected time of reward. In the production epoch of the RSG task,

and throughout the RG task, activity increased linearly with a slope that allowed firing

rate across trials to reach a terminal value shortly before the expected time of reward

(and saccade). Motivated by this observation, we developed a model of LIP activity in

the measurement epoch in which the instantaneous slope was adjusted to create a

ramp that would reach the terminal firing rate at the expected time of reward. At the

beginning of the measurement epoch and until the time of the earliest Set, all sample

intervals are equally probable, and therefore, the expected time of reward is the twice

as long as the mean sample interval. As time elapses, some of the sample intervals are

no longer probable, and therefore, the expected time of reward gets progressively

delayed to reflect the expected time of reward for the remaining possible sample

intervals. Mathematically, this model can be written as:

(11)

dtdr = R −rmax

E[t |t]−Δt −tRew Sacc

in which dr/dt is the instantaneous slope, Rmax is an arbitrary terminal (maximum) firing

rate, E[tRew|t] is the expected time of reward given time t, and ∆tSacc is the fixed delay

between when firing rate reached Rmax and when reward is delivered. We inferred the

expected time of reward empirically from the animals’ production intervals, as follows:

(12)

[t |t]E Rew = Σ p(t )t >ts s

Σ E[t |t ]p(t )t >ts p s s

where E[tp|ts] is the expected (average) production interval for sample interval, ts. To

estimate ∆tSacc, we noted that when average firing rates reached their common terminal

firing, the response variance was at its minimum. Therefore, we computed the earliest

time when response variance was within 2 standard deviation of this minimum value.

To compute response variance, we first constructed peri­event time histograms (PETH)

for each neuron’s spike times within a window of 400 ms before saccade initiation using

a 100 ms boxcar filter. We then subtracted from PETH, the average firing rate for each

time point across trials to compute residual (i.e., fluctuations) of firing rates around

mean (PETHres). To estimate the minimum variance, we combined PETHres across

neurons, sorted trials into 20 bins based on the production intervals, measured average

PETHres within each sorted bin, and computed the variance across the bins. To compute

the standard deviation of the variance, we used 100 bootstrap estimates of the variance

by resampling from the combined PETHres across cells and trials. Finally, we determined

the earliest point before saccade initiation when the average variance was within 2

standard deviations of the minimum variance (i.e., the time point at which firing rates

had reached their terminal value).

We then solved Equation 11 for r(t), and modeled the observed firing rate, y(t), as a

linear function of r(t), and computed the best fit of this model to activity from 300 ms

after Ready to the time of Set.

(13)

(t) Ar(t) y = + B

Our conclusions were insensitive to reasonable variations of analysis parameters

including 80 to 200 bootstraps, 50 to 150 ms boxcar filtering of the PETHs, and different

measures of significance for change of variance (1 to 3 standard deviations away from

the average variance).

4. Bayesian estimate of the sample interval. For this scheme, we modeled the firing rate

as a linear function of the BLS estimate (Equation 3) and computed the best fit of this

model to activity from 300 ms after Ready to the time of Set.

(14)

(t) Af (t) y = BLS + B

5. Preplanning the production dynamics. According to this model, response dynamics in

the measurement epoch provide a moment­to­moment (if Set were to occur now)

prediction of the build­up rate in the ensuing production epoch. To formalize this model,

we assumed that the instantaneous slope of the activity in the measurement epoch is

linearly related to the slope of the build­up rate in the production epoch, which is

determined by the ratio of a fixed firing rate excursion, ∆r, to the duration of the build­up,

which begins ∆tSet after Set, and continues until ∆tSacc before saccade initiation. We can

write this model mathematically as follows:

(15)

dtdr = Δr

t −Δt −Δtp Sacc Set

We estimated ∆tSacc as described above for model 3 (Anticipation of the expected time

of reward). For ∆tSet, we used the same exact strategy: we first determine that

responses reached their lowest variance approximately 150 ms after Set, and

proceeded by looking for the latest point in time after Set that response variance was

still within two standard deviation of the minimum value.

We then solved Equation 15 analytically for r(t), and modeled the observed firing rate,

y(t), as a linear function of r(t), and computed the best fit of this model to activity from

300 ms after Ready to the time of Set.

(16)

(t) Ar(t) y = + B

Our conclusions were insensitive to reasonable variations of analysis parameters

including 80 to 200 bootstraps, 50 to 150 ms boxcar filtering of the PETHs, and different

measures of significance for change of variance (1 to 3 standard deviations away from

the average variance).

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