Fluctuations and information filtering in coupled populations of spiking neurons with adaptation

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Fluctuations and information filtering in coupled populations of spiking neurons with adaptation

Moritz Deger, Tilo Schwalger, Richard Naud, and Wulfram Gerstner

Phys. Rev. E 90, 062704 – Published 1 December 2014

Abstract

Finite-sized populations of spiking elements are fundamental to brain function but also are used in many areas of physics. Here we present a theory of the dynamics of finite-sized populations of spiking units, based on a quasirenewal description of neurons with adaptation. We derive an integral equation with colored noise that governs the stochastic dynamics of the population activity in response to time-dependent stimulation and calculate the spectral density in the asynchronous state. We show that systems of coupled populations with adaptation can generate a frequency band in which sensory information is preferentially encoded. The theory is applicable to fully as well as randomly connected networks and to leaky integrate-and-fire as well as to generalized spiking neurons with adaptation on multiple time scales.

Received 19 November 2013
Revised 11 September 2014

DOI:https://doi.org/10.1103/PhysRevE.90.062704

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

Moritz Deger^1,*, Tilo Schwalger^1,†, Richard Naud², and Wulfram Gerstner¹

¹School of Computer and Communication Sciences and School of Life Sciences, Brain Mind Institute, École polytechnique fédérale de Lausanne, Station 15, 1015 Lausanne EPFL, Switzerland
²Department of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, Ontario, K1N 6N5 Canada

^*Corresponding author: moritz.deger@epfl.ch
^†Corresponding author: tilo.schwalger@epfl.ch

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Vol. 90, Iss. 6 — December 2014

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Figure 1
Schematic of the spike response model (a GLM for spike generation) with exponential escape noise and the derived quasirenewal population model.
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Figure 2
Quasirenewal approximation of the population dynamics. Coarse-grained population activity $\bar{A} (t)$ [ $\bar{A} (t) = n_{0} (t) / (N Δ t)$ , where $n_{0} (t)$ is the spike count in $[t, t + Δ t)$ , with $Δ t = 2$ ms, black] resulting from 500 randomly connected adapting neurons (4) receiving a common input current [ $I (t)$ , blue]. The theoretical expectation $a (t)$ [green, Eq. (1)] based on QR approximation (7), and expected fluctuations [red, one standard deviation (s.d.), $a (t) \pm {(N Δ t)}^{- \frac{1}{2}} a {(t)}^{\frac{1}{2}}$ ]. At time $t$ , $a (t)$ depends on the actual history $\bar{A} (t^{'})$ (black) for $t^{'} < t$ . Standard parameters (see Appendix pp1), except for $I (t)$ as shown and $c = 5 s^{- 1}$ . Inset shows mean (green) and s.d. (red) of deviations $δ A (t) = \bar{A} (t) - a (t)$ as a function of $\sqrt{a}$ , averaged over 25 repetitions of the displayed $I (t)$ (dots) vs theory (lines).
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Figure 3
Spectral density of the population activity in random networks. Simulation (light colored lines) vs theory (10) (solid lines). (a) Single populations ( $K = 1$ ); [(b)–(d)] coupled exc.-inh. network ( $K = 2$ ) [in (b) green lines show the amplitude of the cross spectrum]. [(c)–(d)] The limit $ω \to 0$ in dependence on parameters for the $K = 2$ case shown in (b); symbols mark simulation results for different connection probabilities ( $▽, △, \circ$ : $p = 0.2, 0.5, 1$ ) while keeping $J_{s} = N \bar{w}$ fixed; solid lines show theory (10). Symbols mostly fall on top of each other indicating independence of results with respect to $p$ . In (c) colors are as in (b), in (d1)–(d3) only the exc. population is shown, colors denote number of neurons (blue, green, red, and cyan: $N = 50, 250, 500, 1000)$ . Standard parameters [marked by arrows in (c) and (d), cf. Appendix pp1] were used except as indicated.
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Figure 4
Signal processing and coherence shaping in a feed-forward chain of recurrently connected neural populations, simulation vs theory. Left: Schematic of the network. Right: Spectral coherences $Γ_{i 1}$ (13) of $I_{1}$ and $A_{i}^{exc .}$ , theory (13) (lines) vs simulation result (light colored lines). Colors indicate coherences of signals $I_{1}$ and population activity $A_{i}$ . The mutual information rate is closely related to the spectral coherence, see text. Standard parameters, except for increased coupling $J_{s} = 10$ . The input current $I_{1} (t)$ is a Gaussian white noise with spectral density ${\tilde{C}}_{I, 1} (ω) = 3 s^{- 1}$ .
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Figure 5
Spectral density of the population activity in random networks [as in Fig. 3–3] with comparison to earlier theories (special cases). Simulation (light colored lines) vs theory (10) (solid lines). For comparison, uncoupled renewal processes ( $\tilde{B} = 1$ ) (dotted lines); coupled renewal processes ( ${\tilde{B}}^{- 1} = 1 - {\tilde{R}}_{0} J$ ) (dash-dotted lines); Hawkes process ( ${\tilde{B}}^{- 1} = 1 - {\tilde{R}}_{0} J$ , ${\tilde{C}}_{0} = A_{0}$ ), (dashed lines). (a) Single populations ( $K = 1$ ); (b) coupled exc.-inh. network ( $K = 2$ ). Blue dash-dotted and solid lines coincide because inhibitory neurons here have no adaptation ( $J_{a} = 0$ ). Standard parameters were used except as indicated.
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Figure 6
Illustration of negative correlations between $δ m_{k} (t)$ and $δ m_{k + 1} (t + Δ t)$ . (a) Expected number of spikes from group $k$ in bins $[t, t + Δ t]$ and $[t + Δ t, t + 2 Δ t]$ . (b) A large fluctuation of $δ m_{k} (t)$ leads to reduction of the number of available spikes $m_{k + 1} (t + Δ t)$ . As a consequence, $δ m_{k + 1} (t + 1)$ tends to be small.
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