Characterizing spreading dynamics of subsampled systems with nonstationary external input

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Characterizing spreading dynamics of subsampled systems with nonstationary external input

Jorge de Heuvel, Jens Wilting, Moritz Becker, Viola Priesemann, and Johannes Zierenberg

Phys. Rev. E 102, 040301(R) – Published 23 October 2020

Abstract

Many systems with propagation dynamics, such as spike propagation in neural networks and spreading of infectious diseases, can be approximated by autoregressive models. The estimation of model parameters can be complicated by the experimental limitation that one observes only a fraction of the system (subsampling) and potentially time-dependent parameters, leading to incorrect estimates. We show analytically how to overcome the subsampling bias when estimating the propagation rate for systems with certain nonstationary external input. This approach is readily applicable to trial-based experimental setups and seasonal fluctuations as demonstrated on spike recordings from monkey prefrontal cortex and spreading of norovirus and measles.

Received 11 October 2019
Revised 24 April 2020
Accepted 21 September 2020

DOI:https://doi.org/10.1103/PhysRevE.102.040301

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Neuroscience, neural computation & artificial intelligence Spreading Stochastic processes

Spiking neurons

Epidemic spreading

Interdisciplinary PhysicsPhysics of Living SystemsStatistical Physics & Thermodynamics

Authors & Affiliations

Jorge de Heuvel ¹, Jens Wilting¹, Moritz Becker^1,2, Viola Priesemann ^1,*, and Johannes Zierenberg ^1,*

¹Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
²Department of Computational Neuroscience, Third Institute of Physics–Biophysics, Georg-August-University, 37077 Göttingen, Germany

^*These authors contributed equally to this work.

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Issue

Vol. 102, Iss. 4 — October 2020

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Images

Figure 1
Unbiased estimation of internal autocorrelation time $τ$ for a subsampled system with time-dependent input rate can be achieved after subtracting trial-ensemble average from activity. (a) Step function as an example for a time-dependent input rate. (b) Subsampled activity $a_{t}^{i}$ of a branching process with constant internal autocorrelation time $τ$ and nonstationary input rate shows nonstationary behavior in regimes II and III. The colored lines are individual trials, and the black solid line is the trial-ensemble average ${\bar{a}}_{t}$ (over 200 trials). (c) Linear regression slope estimates $r_{k}$ (blue dots) of time lag $k$ for the process in (b) do not decay exponentially as expected from the process' internal autocorrelation (red line), which makes an unbiased estimation of $τ$ impossible. (d) Time series (b) corrected by subtracting trial-ensemble average: ${\tilde{a}}_{t}^{i} = a_{t}^{i} - {\bar{a}}_{t}$ . (e) For corrected time series (d), the $r_{k}$ decay exponentially, such that $τ$ can now be inferred without bias. Simulation parameters: Trial length $T = 10 000$ steps, internal autocorrelation time $τ = 20$ steps, number of trials $N = 200$ , mean (fully sampled) baseline activity $〈 A_{0} 〉 = 1000$ , subsampling fraction $α = 0.05$ , relative step height $〈 h_{up} 〉 / 〈 h_{down} 〉 = 2.6$ , and step duration $c = 200$ steps.
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Figure 2
Robustness of our estimate to variability in a nonstationary input (step function with step-duration $c = 10 τ$ ). (a) Variability in the onset time $t_{step}$ with standard deviation $σ_{t}$ . (b) Variability in the step height $Δ 〈 h 〉$ with standard deviation $σ_{h}$ . The color in the matrix indicates the relative error between the estimated autocorrelation time $\hat{τ}$ and the internal autocorrelation time $τ$ of the branching process. The 5% error bound was fitted (black lines) and scales as $σ \propto τ^{γ}$ . Simulation parameters: $T = 1000 τ$ steps, $N = 300, 〈 A_{0} 〉 = 5000, α = 0.01$ .
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Figure 3
Application of our new approach to experimental data. Top (a)–(c) The intrinsic timescales $τ$ in the macaque prefrontal cortex have been inferred with our new approach from spike recordings during a trial-based visual short-term memory task [35]. (a) Example trial-ensemble average of stimulus-evoked nonstationary neural responses. (b) Autocorrelation functions $r_{k}$ of (a) before (orange) and after the correction (blue) hardly differ. (c) Intrinsic timescales inferred from uncorrected data are systematically but not very strongly overestimated (less than 10%). (d) Numerical robustness validation for typical experimental recordings, resembling a typical evoked potential: Nonstationary external input (blue) with twofold increase during stimulus presentation (total duration of $c = 375 ms$ ) for $N = 300$ trials of length $T = 5 s$ with sampling frequency $f = 1 kHz$ and subsampling fraction $α = 0.01$ . An example trial realization $a_{t}^{i}$ is shown as a black line. The impact on estimating $τ$ under variance of the stimulus onset ( $σ_{t}$ ) is evaluated for various intrinsic autocorrelation times $τ$ as in Fig. 2. Bottom (e)–(g) The infectious spreading dynamics of norovirus and measles have been inferred with case report data from the Robert Koch Institute [36]. (e) Reported infection numbers (blue lines) and the time-dependent trial-ensemble average (black line) for norovirus reveal seasonal nonstationarities. (f) With our method, the seasonality was mostly removed from the autocorrelation function $r_{k}$ of (d). (g) In contrast to the neural recordings, the infectious spreading dynamics inferred from the uncorrected disease data are systematically underestimated.
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