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A Neurocomputational Account of Ecologically Plausible, Flexible Timing with Legendre Memory

Authors:

Abstract

Most computational models of timing rely on well-defined start- and stop-signals, however, these are quite rare in our natural environments. Moreover, theories typically propose different mechanisms to account for retrospective and prospective timing, an assumption that is difficult to align with naturalistic, continuative types of timing. Here we propose a spiking recurrent neural network model of flexible human timing behavior: Legendre Memory Timing (LMT). Our model continually and optimally represents the history of its input by compressing the input into a q-dimensional state, consisting of Legendre polynomials. At any point in time, the network represents a rolling window of its input history that spans from the current time to θ seconds in the past and uses this window to assess time (Voelker & Eliasmith, 2018). Where previous models require constrained ramping, decaying, or oscillating neural activity, our model is not restricted to a single firing pattern, but - consistent with available experimental data from monkeys (Wang, Narain, Hosseini, and Jazayeri, 2018) - utilizes heterogeneous firing patterns in individual spiking neurons that temporally scale with the timed interval. Without an explicit clock, or any specific clock-focussed assumptions, this model accounts for a number of key timing phenomena. For example, the scalar property naturally arises from our model and is functionally linked to the numbers of dimensions that are used to represent input history. Moreover, the model explains why constant standard deviation may be observed for well-trained subjects (e.g., Fetterman & Killeen, 1992). The model suggests that the scalar property and neural scaling are tightly linked: inter-trial variability in timing responses are correlated with inter-trial variability in neural scaling. In doing so, the model explains how accurate timing performance, both prospective and retrospective, may be accomplished in more ecological settings without relying on clear start- and stop-signals (van Rijn, 2018). References Fetterman, J. G., & Killeen, P. R. (1992). Time discrimination in Columba livia and Homo sapiens. Journal of Experimental Psychology: Animal Behavior Processes, 18(1), 80–94. DOI: 10.1037/0097-7403.18.1.80 van Rijn, H. (2018). Towards Ecologically Valid Interval Timing. Trends in Cognitive Sciences, 22(10), 850–852. DOI: 10.1016/j.tics.2018.07.008 Voelker, A. R., & Eliasmith, C. (2018). Improving Spiking Dynamical Networks: Accurate Delays, Higher-Order Synapses, and Time Cells. Neural Computation, 30(3), 569–609. DOI: 10.1162/neco_a_01046 Wang, J., Narain, D., Hosseini, E. A., & Jazayeri, M. (2018). Flexible timing by temporal scaling of cortical responses. Nature Neuroscience, 21(1), 102–110. DOI: 10.1038/s41593-017-0028-6
Spiking Neural Network Model of
Prospective and Retrospective Timing
Explains Violation of Scalar Property and
Temporal Scaling of Neural Responses
Joost de Jong*1,2
Aaron R. Voelker3
Hedderik van Rijn1,2
Terrence C. Stewart4
Chris Eliasmith3
*joost.de.jong@rug.nl
1. University of Groningen,
Department of Experimental
Psychology
2. University of Groningen,
Behavioral and Cognitive
Neurosciences
3. University of Waterloo,
Centre for Theoretical
Neuroscience
4. National Research Council of
Canada
Model
(2) C. Eliasmith, C. H. Anderson, Neural engineering:
computation, representation, and dynamics in
neurobiological systems (2003)
(3) J. G. Fetterman, P. R. Killeen, Journal
of Experimental Psychology: Animal
Behavior Processes. (1992).
Implementation
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dim
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Temporal Basis Functions
Results
−0.34
−0.58
0.98
0.66
−0.60
−0.02
Coefficients
Linear combination
Violation of Scalar Property3
Temporal Scaling of Neural Responses4
(1). A. R. Voelker, C. Eliasmith,
Neural Computation (2018).
(4) J. Wang, D. Narain, E. A.
Hosseini, M. Jazayeri, Nature
Neuroscience (2018).
Part of this project was supported by the research
programme “Interval Timing in the Real World: A functional,
computational and neuroscience approach”, project number
435-16-005,financed by the Netherlands Organisation for
Scientific Research (NWO)
Prospective
The network1computes coefficients on temporal
basis functions to represent atemporal window. It is
implemented as aRecurrent Spiking Neural
Network with the Neural Engineering Framework2
Retrospective
We can flexibly control
the size of the temporal
window (θ).
In prospective timing
we approximately know
the length of intervals
beforehand, so we can
match the window size
to the target interval,
resulting in
optimal timing.
On the other hand, in
retrospective timing,
we don’t know how
long the intervals will
be,so the window size
remains fixed,resulting
in less accurate timing.
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time (s)
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Temporal Window
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time (s)
Represented
Real
Decoding at t=0
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time (s)
ABCD
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ABCD
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time (s)
Temporal Window
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time (s)
0
0.0 2.5 5.0 7.5 10.0
time (s)
Window Size
2.5
5.0
7.5
10.0
Adjusting window size
(θ)and reading out at
the end of the window
allows for optimal
interval production
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