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Flexible Timing with Delay Networks – The Scalar Property and Neural Scaling

Joost de Jong (j.de.jong.53@student.rug.nl)1

Aaron R. Voelker (arvoelke@uwaterloo.ca)2

Hedderik van Rijn (d.h.van.rijn@rug.nl)1

Terrence C. Stewart (tcstewar@uwaterloo.ca)2

Chris Eliasmith (celiasmith@uwaterloo.ca)2

1Experimental Psychology, Grote Kruisstraat 2/1, Groningen, 9712 TS, the Netherlands

2Centre for Theoretical Neuroscience, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada

Abstract

We propose a spiking recurrent neural network model of ﬂex-

ible human timing behavior based on the delay network. The

well-known ‘scalar property’ of timing behavior arises from

the model in a natural way, and critically depends on how many

dimensions are used to represent the history of stimuli. The

model also produces heterogeneous ﬁring patterns that scale

with the timed interval, consistent with available neural data.

This suggests that the scalar property and neural scaling are

tightly linked. Further extensions of the model are discussed

that may capture additional behavior, such as continuative tim-

ing, temporal cognition, and learning how to time.

Keywords: Interval Timing; Scalar Property; Spiking Recur-

rent Neural Networks; Neural Engineering Framework; Delay

Network

Introduction

Time is a fundamental dimension against which our mental

lives play out: we remember the past, experience the present,

and anticipate the future. Humans are sensitive to a wide

range of temporal scales, from microseconds in sound local-

ization to tens of hours in circadian rhythms. It is somewhere

in between—on the order of hundreds of milliseconds to sev-

eral seconds—that we consciously perceive time and coordi-

nate actions within our environment (van Rijn, 2018). How

does our brain represent time as accurately as possible, and

how does it ﬂexibly deal with different temporal intervals?

Scalar Property

Given the centrality of time to our experience, it is no won-

der that timing and time perception have been the subject of

extensive empirical study over the past 150 years. Many per-

ceptual, cognitive, and neural mechanisms related to time per-

ception have been studied, and perhaps the most well-known

ﬁnding from the literature is the scalar property (Gibbon,

1977). The scalar property of variance states that the stan-

dard deviation of time estimates are linearly proportional to

the mean of the estimated time. The scalar property has been

conﬁrmed by a wide variety of experimental data (Wearden

& Lejeune, 2008). However, some research suggests that

the scalar property does not always hold. It was already

observed by Allan and Kristofferson (1974) that for well-

practiced subjects in interval discrimination tasks, the stan-

dard deviation was constant for a range of relatively short

intervals. Similar results were observed with pigeons, where

the standard deviation remained ﬂat for intervals up to around

500 ms (Fetterman & Killeen, 1992). Also, Grondin (2014)

notes that the scalar property of variance critically depends

on the range of intervals under consideration, and cites many

examples with increases in slope after intervals of about 1.3

seconds.

Most models of timing take the scalar property as a start-

ing point, or consider conformity to the scalar property as a

crucial test. This seriously undermines their ability to explain

violations of the scalar property. Here, we take the approach

of not assuming the scalar property a priori, but instead con-

struct a biologically plausible model that is trained to opti-

mally represent time. We then systematically explore ranges

of model parameters that lead the scalar property to be sat-

isﬁed or violated, and provide a theoretical framework that

aims to unify a variety of empirical observations.

Neural Scaling

Variance is not the only property of timing that scales with

the estimated time interval. The ﬁring patterns of individual

neurons also stretch or compress proportional to the timed

interval. In a recent study, Wang, Narain, Hosseini, and Jaza-

yeri (2018) show that neurons in striatum and medial pre-

frontal cortex (mPFC) scale in this manner. During the timed

interval, individual neurons display ramping, decaying, oscil-

lating, or more complex ﬁring patterns. In general, the spe-

ciﬁc shapes of temporal ﬁring patterns for a given neuron re-

main the same, but become stretched for longer intervals and

compressed for shorter intervals. Additionally, neurons in the

thalamus display a different kind of scaling: their mean level

of activity correlates with the timed interval. Both ﬁndings

have been explained using a recurrent neural network (RNN)

model (corresponding to neurons in striatum or mPFC) that

receives a tonic input (originating from the thalamus) to scale

the temporal dynamics of the network (Wang et al., 2018).

The units in the neural network exhibit neural ﬁring patterns

and scaling similar to those observed experimentally. The

model of timing we propose reproduces the same ﬁndings

as the RNN model described in Wang et al. (2018). These

ﬁndings suggest that, in order to perform timed actions as

accurately as possible, the brain is able to ﬂexibly scale its

temporal dynamics. This implies a tight connection between

the scalar property of variance and the temporal scaling of

individual neurons.

Neural Models of Timing

Many neurally inspired models of timing and time perception

have been proposed. Some models are based on ramping neu-

ral activity (Simen, Balci, deSouza, Cohen, & Holmes, 2011),

some decaying neural activity (Shankar & Howard, 2010) and

some on oscillating neural activity (Matell & Meck, 2004).

Interestingly, all these neural ﬁring patterns (and more com-

plex ones) have been observed by Wang et al. (2018) in

striatum and mPFC during a motor timing task. Therefore,

appealing to only one of these neural ﬁring patterns may

be insufﬁcient to fully explain timing performance. In line

with this observation, the recurrent neural network model by

Wang et al. (2018) exhibits a wide variety of ﬁring patterns.

However, their model does not show why this heterogene-

ity of ﬁring patterns is important for timing performance or

what the role is of ramping, decaying, or oscillating neurons

in timing performance. Randomly-connected recurrent neu-

ral networks—referred to as reservoir computers—produce a

wide variety of dynamics that can subsequently be extracted

by a read-out population (Buonomano & Maass, 2009). A

more structured approach to building a recurrent neural net-

work may highlight the functional relevance of different neu-

ral ﬁring patterns on timing performance.

One candidate for such a structured approach is the delay

network (Voelker & Eliasmith, 2018). The delay network is a

spiking recurrent neural network that approximates a rolling

window of its input history by compressing the history into

aq-dimensional state-vector. It has been observed that in-

dividual neurons in the delay network show responses simi-

lar to time-cells (MacDonald, Lepage, Eden, & Eichenbaum,

2011). Here, we use the delay network to explain both the

scalar property of timing and the scaling of individual neural

responses by comparing delay network data to empirical data

from Wang et al. (2018).

Methods

We ﬁrst discuss the mathematics behind the delay network.

Then, we show how to implement the delay network as a

spiking recurrent neural network using the Neural Engineer-

ing Framework (NEF; Eliasmith & Anderson, 2003). Lastly,

we discuss the details of our simulations that follow the ex-

perimental setup of Wang et al. (2018).

The Delay Network

The delay network is a dynamical system that maintains a

temporal memory of its input across a rolling window of

θseconds (Voelker & Eliasmith, 2018; Voelker, 2019). It

does so by optimally compressing its input history into a q-

dimensional state-vector. This vector continuously evolves

through time in a way that captures the sliding window of

history, while being amenable to representation by a popula-

tion of spiking neurons using the NEF (as explained in the

following subsection).

We consider the problem of computing the function y(t) =

u(t−θ), where u(t)is the input to the network, y(t)is the

output of the network, and θ>0 is the length of the window

in time to be stored in memory. In order to compute such a

function, the network must necessarily maintain a history of

input across all intermediate moments in time, u(t−θ0), for

θ0ranging from the start of the window (θ0=0), going back

in time to the end of the window (θ0=θ). This window must

then slide forwards in time once t>θ, thus always preserving

the input over an interval of length θ. Computing this func-

tion in continuous time is challenging, as one cannot merely

sample a ﬁnite number of time-points and shift them along;

the time-step of the system could be arbitrarily small, or there

may not even be an internal time-step as in the case of imple-

mentation on mixed-analog neuromorphic hardware (Neckar

et al., 2019).

The approach taken by Voelker and Eliasmith (2018) is

to convert this problem into a set of differential equations,

dx/dt =θ−1(Ax+Bu), where xis a q-dimensional state-

vector, and (A,B)are matrices governing the dynamics of

x. We use the (A,B)matrices from Voelker (2019; sec-

tion 6.1.3). This results in the approximate reconstruction:

u(t−θ0)≈P(θ0/θ)·x(t), where Pare the shifted Legendre

polynomials. Importantly, the dimensionality qdetermines

the quality of the approximation. This free parameter con-

trols the number of polynomials used to represent the win-

dow – analogous to a Taylor series expansion of the input

using polynomials up to degree q−1. Thus, qdetermines

how much of the input’s frequency spectrum, with respect to

the period 1/θ, should be maintained in memory. Another

notable property is that 1/θcorresponds to a gain factor on

the integration of x(t)that can be controlled in order to dy-

namically adjust the length of the window on-the-ﬂy.

The Neural Engineering Framework (NEF)

Given this mathematical formulation of the computations that

the neurons must perform in order to represent their past in-

put, we turn to the question of how to recurrently connect

neurons such that they perform this computation. For this,

we use the NEF (Eliasmith & Anderson, 2003).

In the NEF, the activity of a group of neurons forms a dis-

tributed representation of some underlying vector space x. In

particular, each neuron ihas some encoder (or preferred di-

rection vector) eisuch that this neuron will ﬁre most strongly

when xis similar to ei. To produce heterogeneity in the neu-

ral population, each neuron has a randomly assigned gain αi

and bias βi. Overall, the current entering each neuron would

ideally be αiei·x+βi. This input current determines the spik-

ing activity of the neuron, based on the neuron model. In

this work, we use the standard leaky integrate-and-ﬁre (LIF)

model. This results in a pattern of neural activity over time

ai(t)that encodes some continuous vector over time x(t).

If we have two groups of neurons, one representing x

and one representing y, and we want yto be some function

of x, then we can form connections from the ﬁrst popula-

tion to the second. In particular, we want to connect neu-

ron ito neuron jwith weights ωi j such that the total sum

from all the input connections will give the same result as

the ideal equation assumed above. In other words, we want

∑iai(t)ωi j =αjej·y(t)for all j(the bias βjcurrent is sup-

plied separately). The ideal ωi j are found using regularized

least-squares optimization.

Furthermore, this method for ﬁnding connection weights

can be extended to support recurrent connections (i.e., con-

nections from the neurons in one population back to itself).

These connections are solved for in the same manner, and, as

has been shown Eliasmith and Anderson (2003), the result-

ing network approximates a dynamical system of the form

dx/dt =f(x) + g(u), where xis the vector represented by

the group of neurons, uis the vector represented by the group

of neurons providing input to this group, and the functions f

and gdepend on both the functions used to ﬁnd the connec-

tion weights (as per the previous paragraph) and the tempo-

ral properties of the synapses involved (most importantly, the

postsynaptic time constant).

The result is that the NEF provides a method for generat-

ing a population of neurons (to represent the q-dimensional

state) and ﬁnding the ideal recurrent connections between

those neurons such that they compute the differential equa-

tions required by the delay network.

It should be noted that the resulting network is structured

exactly like a standard reservoir computer: a large number

of neurons are recurrently connected, an input is supplied to

that network, and we can decode information from the dy-

namics of the network by computing weighted sums of the

overall neural activity. However, rather than randomly gener-

ating the recurrent weights, we are using the NEF to ﬁnd the

optimal weights for storing information over a rolling win-

dow in time. This method has been shown to be far more

computationally efﬁcient and accurate than various forms of

reservoir computer for computing delays (Voelker, 2019).

An example of the resulting system is shown in Figure 1.

Here the network is optimized to represent the past θ=1 s of

its own input using q=6 dimensions. Part A shows the (one-

dimensional) input to the network over time. In this case, the

input is a Gaussian bump centred at t=0.5 seconds. The

resulting neural activity (for 50 randomly-chosen neurons) is

shown in Part B. Note that the neural activity at the begin-

ning (before the input bump) and at the end (after t>1.5 s) is

fairly constant. This is the stable background activity of the

network in the absence of any input. Since the network only

stores the last second, in the absence of any input it will settle

back to this state in ∼1 second.

Part C shows one example of decoding information out

of this network. In particular, we are decoding the function

y(t) = u(t−0.5)– that is, the output should be the same as the

input θ0=0.5 seconds ago. This output is found by comput-

ing the weighted sum of the spikes that best approximates this

value, again, using least-squares optimization to ﬁnd these

weights. That is, y(t) = ∑iai(t)di, where diis the decoding

weight for the ith neuron. We see that the network accurately

represents the overall shape, although the Gaussian bump has

become a bit wider, and the output dips to be slightly nega-

Figure 1: The Delay Network – Optimized to represent the

past 1second of input history using 6dimensions. (A): The

input to the network. (B): Neural activity of 50 randomly-

chosen neurons within the network. (C): Decoding informa-

tion from the network by taking the weighted sum of neu-

ral activity that best approximates the input from 0.5 seconds

ago. (D): Decoding all information from the past 1 second.

Each row is a different slice in time (from 0 to 1 second),

and uses a different weighted sum of the same neural activity.

The graph in part (C) is a slice through this image, indicated

by a dotted line. (E): The underlying low-dimensional state

information that represents the window.

tive before and after the bump. These are side-effects of the

neurons approximating the ideal math for the delay network,

and its compression of the input into 6 dimensions.

In Part D, we show the same process as in Part C, but for

all times in the past from right now (θ0=0 s) to the furthest

point back in time (θ0=1 s). This is to show that we can

decode all different points in time in the past, and the partic-

ular case shown in Part C is just one example (indicated with

a dotted line). Each of these different outputs uses the same

underlying neural activity, but different decoders diout of the

recurrent population.

Finally, Part E shows that we can also decode out the q-

dimensional state representation x(t)that the delay network

uses for its representation. These are the values that govern

the dynamics of the delay network, and they form a nonlin-

ear basis for all the possible functions that could be decoded

out from the neural activity. Indeed, each row in Part D can

also be interpreted as a different linear transformation of the

data shown in Part E. Voelker and Eliasmith (2018) derive

the closed-form mathematical expression that provides such a

transformation, thus relating all time-points within the rolling

window to this underlying state-vector.

These different views of the delay network can be seen

as a very clear example of David Marr’s Tri-Level Hypoth-

esis (Marr, 1982), which we use here to understand this sys-

tem at varying levels of abstraction. For instance, we may

consider only the implementational level, which consists of

leaky integrate-and-ﬁre neurons with recurrent connection

weights between them, a set of input weights from the stimu-

lus, and multiple sets of output weights. Or we may consider

the algorithmic level, where the system is representing a q-

dimensional state-vector xand changing that vector over time

according to the differential equations given in the previous

section. Or we may consider the computational level, where

the network is storing a (compressed) history of its own in-

put, and different slices of that input can be extracted from

that memory. All of these are correct characterizations of the

same system.

Simulation Experiment

In the original experiment by Wang et al. (2018), monkeys

were presented with a “cue” signal that indicated the inter-

val to be reproduced: red for a short interval (800 ms) and

blue for a long interval (1500 ms). Then, they were presented

with a “set” signal that marked the start of the interval. The

monkeys had to issue a response after the cued interval had

elapsed. We have attempted to match the relevant details of

their experimental setup as follows. The delay network (with

q=4) continually receives input from a control population

that scales θin order to produce intervals around 800 ms or

1500 ms. In effect, this gain population controls the length

of the window on-the-ﬂy. The effective value of θis 1, di-

vided by the value that the gain population represents. When

the value represented by the gain population is greater than 1,

it makes the length of the window shorter; when it is smaller

than 1, it makes the window longer. This enables us to choose

values for the gain population that will let the delay network

time intervals around 800 ms or 1500 ms. The delay network

receives input that is continually represented, along with the

history of this input. The input signal is a rectangular impulse

of 500 ms. The same read-out population decodes the delayed

input signal as θis varied.

Results

Scalar Property in the Delay Network

In order to quantify the scalar property in the spiking imple-

mentation of the delay network, we calculated the mean and

standard deviation of the decoded output at θseconds. We

performed this analysis for delay networks with a range of

values for θand qwhile keeping the number of neurons per

dimension ﬁxed at 500. We considered only positive values

around the peak of the decoded output. If the scalar prop-

erty holds, we should observe a linear relationship between θ

and the standard deviation of the impulse response. Our data

suggests that the scalar property critically depends on q(Fig-

ure 2). The relationship between the standard deviation, θ,

and qcan be described as follows. The standard deviation

remains constant for a range of short intervals and starts to

increase linearly after some value of θ. Both the location of

this transition and the slope of the linear increase depend on

q. This helps explain some previous differences in experi-

mental ﬁndings. For example, the ﬂat standard deviation for

≤500 ms observed by Fetterman and Killeen (1992) can be

explained by assuming that q=2 within our model.

Figure 2: Scalar Property. The standard deviation of the im-

pulse response plotted against θfor different values of q.

Neural Scaling in the Delay Network

Our simulations of the Wang et al. (2018) experiment pro-

duced results with a qualitative ﬁt to the empirical data (Fig-

ure 3). First, the standard deviation of the decoded output

increased with θ(also see previous section). Second, the neu-

ral responses were highly heterogeneous, with ramping, de-

caying, and oscillating neurons. These ﬁring proﬁles were

observed because they are linear combinations of the under-

lying state vector x(t)(see Figure 1E). Third, the responses

of individual neurons stretched or compressed with the length

of the timed response response, similar to the empirical data

from Wang et al. (2018).

Figure 3: Neural Scaling. A square input was provided to

the delay network, while varying the value of the gain input.

The peak and standard deviation of the decoded output scale

with the gain. The heterogeneous ﬁring patterns of individual

neurons also scale with gain. Here, neural ﬁring patterns of

three example neurons are shown that qualitatively ﬁt the data

from Wang et al. (2018). We focused on the ﬁrst period of the

neural response to the “set” stimulus. The top neuron shows

ramping activity, the middle neuron shows decaying activity,

and the bottom neuron shows oscillatory activity.

Discussion

The aim of the present study is to use the delay network to ex-

plain two ﬁndings in the timing literature: the scalar property

of variance and neural scaling. We did not assume the scalar

property a priori, but systematically explored the parameters

of the delay network that lead the scalar property to be satis-

ﬁed or violated. Our results suggest that the scalar property

critically depends on q. Notably, the time-cell data that was

analyzed in earlier work ﬁt best for q=6 (Voelker & Elia-

smith, 2018). The temporal range of conformity to the scalar

property and slope of the scalar property may be explained

by the number of dimensions the delay network uses (q): for

higher q, the range of short intervals with a constant standard

deviation increases, whereas the slope of the scalar property

decreases. We also found that scaling the dynamics of the de-

lay network produces scaling of neural ﬁring patterns, match-

ing empirical data (Wang et al., 2018). Our model suggests

that when the delay network represents its input history with

more dimensions, neural ﬁring patterns become more com-

plex, as additional linear combinations of higher-degree Leg-

endre polynomials are encoded by individual neurons. Fur-

thermore, these ﬁndings suggest that the scalar property and

the adaptive control of neural dynamics are tightly linked.

Previous Models

The delay network shares some features with previous neu-

ral models of timing, but there are also critical differences.

First, similar to previous RNN models, the delay network is

an RNN that uses population-level dynamics to time inter-

vals. However, previous RNN models use a random connec-

tivity approach to generate the necessary dynamics for ac-

curate timing, whereas the delay network explicitly deﬁnes

the required dynamics and optimizes neural connectivity to

implement those dynamics. Also, previous RNN models of

timing do not characterize how the input history is repre-

sented. Similar to memory models of timing (Shankar &

Howard, 2010), the delay network makes this connection ex-

plicit. Even though memory models and the delay network

both specify how input history is represented, the memory

models do not specify how to optimally scale the dynamics

of the network or compute arbitrary functions over the repre-

sented history. In contrast, the delay network is optimized to

recurrently represent time, and comes with a general frame-

work that links the input history, network representation, and

spiking neural activity. In sum, we believe that the delay net-

work is an improvement over previous models of timing by

both explicitly specifying how time is represented and imple-

menting that representation in a ﬂexible neural framework.

Extending the Delay Network

In this work, we have used the delay network to explain the

scalar property and neural scaling in a simple motor timing

task. However, the delay network may be used to explain a

wide variety of timing phenomena, including: continuative

timing, temporal cognition, and learning how to time.

Continuative Timing First, the delay network can be ex-

tended to account for time perception in a wide variety of re-

alistic situations. A classic dichotomy in the timing literature

is between prospective and retrospective timing. Prospective

timing is explicitly estimating an interval with knowledge be-

forehand that attention should be focused on time. On the

other hand, retrospective timing is estimating, in hindsight,

how long ago an event happened. However, this distinction

may be arbitrary, since in realistic situations, one often no-

tices the duration of an ongoing interval. For instance, you

may notice that a web page is taking too long to load but wait

an additional amount of time before checking your signal re-

ception. When this happens, one neither has earlier knowl-

edge that time should be attended to (prospective) nor the in-

struction to estimate how much time has passed since an event

(retrospective). Therefore, a more appropriate term for timing

in realistic situations would be continuative timing (van Rijn,

2018). The delay network, at any point in time, serves as a

rich source of information regarding the temporal structure

of ongoing events, including how long ago an event started

and stopped. This information can be used to infer how much

time has elapsed since a salient event and compared to the

typical temporal structure of an event in memory. Such com-

parisons could then facilitate decision-making, such as in de-

ciding whether to wait for an additional amount of time.

Temporal Cognition Second, time is a crucial factor in a

wide variety of cognitive processes. Timing models have

been successfully integrated in ACT-R (Taatgen, van Rijn,

& Anderson, 2007) and models of decision-making (Balc &

Simen, 2016). The delay network, built with the NEF, is

compatible with other cognitive models that have been devel-

oped in the same framework, or indeed any cognitive mod-

els that can incorporate neural networks. Therefore, a future

avenue of research will be to incorporate the delay network

into existing models of cognitive processes, such as action-

selection (Stewart, Bekolay, & Eliasmith, 2012) and working

memory (Singh & Eliasmith, 2006).

Learning to Time Third, the delay network may be used to

explain how timing is learned. In the experiment by Wang

et al. (2018), the monkeys trained extensively before they

could accurately perform the motor timing task. The mon-

keys received rewards according to the accuracy of their per-

formance. Another open question is how an optimal mapping

between cues and the gain population can be learned. There-

fore, future work will focus on modeling how timing is mas-

tered during reinforcement learning.

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