From Quasiperiodic Partial Synchronization to Collective Chaos in Populations of Inhibitory Neurons with Delay

Diego Pazó and Ernest Montbrió

Phys. Rev. Lett. 116, 238101 – Published 7 June 2016

Abstract

Collective chaos is shown to emerge, via a period-doubling cascade, from quasiperiodic partial synchronization in a population of identical inhibitory neurons with delayed global coupling. This system is thoroughly investigated by means of an exact model of the macroscopic dynamics, valid in the thermodynamic limit. The collective chaotic state is reproduced numerically with a finite population, and persists in the presence of weak heterogeneities. Finally, the relationship of the model’s dynamics with fast neuronal oscillations is discussed.

Received 11 November 2015

DOI:https://doi.org/10.1103/PhysRevLett.116.238101

Physics Subject Headings (PhySH)

Coupled oscillators

Biological neural networks Nonlinear time-delay systems

Networks

Authors & Affiliations

Diego Pazó¹ and Ernest Montbrió²

¹Instituto de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, 39005 Santander, Spain
²Center for Brain and Cognition, Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08018 Barcelona, Spain

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Vol. 116, Iss. 23 — 10 June 2016

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Images

Figure 1
Quasiperiodic partial synchronization (a)–(d) and collective chaos (e),(f) in an inhibitory network of $N = 1000$ identical QIF neurons with $η_{j} = \bar{η} = τ = 1$ , $τ_{s} = 10^{- 3}$ , and: (a),(b) $D = 2.5$ , $J = - 1.65$ , (c),(d) $D = 2.5$ , $J = - 1.85$ , and (e),(f) $D = 3$ , $J = - 3.8$ . (a),(c),(e) Raster plots of 200 randomly selected neurons. Dots correspond to firing events, and neurons are indexed according to their firing order. (b),(d),(f) Return plots for $10^{4}$ interspike intervals ${ISI}_{j} (k) = t_{j}^{k + 1} - t_{j}^{k}$ of an arbitrary neuron $j$ .
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Figure 2
Macroscopic dynamics of quasiperiodic partial synchronization (a)–(d), and collective chaos (e),(f); same parameters as in Fig. 1 are used. Black curves are obtained integrating the FREs (4), with $Δ = 0$ . Red curves in (a),(c),(e) are the firing rates obtained from numerical simulation of the population, Eqs. (1)–(3). Right panels (b),(d),(f): Phase portraits of the FREs. The unstable fixed point (brown circle) and the unstable orbit (brown, dashed line) correspond to incoherence and full synchronization, respectively. The three largest Lyapunov exponents of the chaotic attractor in panel (f) are ${0.055, 0, - 0.232}$ .
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Figure 3
Stability regions of the incoherent and fully synchronized states for $Δ = 0$ . Shaded region: Incoherence is stable. Hatched region: Full synchrony is unstable. Dark gray (blue) and the light gray (red) lines are the loci of subcritical and supercritical Hopf bifurcations of incoherence, respectively. The approximate periodicity of the phase diagram with $D$ stems from the $ISI = π$ of the uncoupled neurons. Inset: Three largest (collective) Lyapunov exponents in the range $- 6 < J < - 1.7$ for $D = 3$ , computed numerically from Eq. (4) using the method in [45]. Note the supercritical Hopf bifurcation at $J_{H}^{(1)} = - 2.116 \dots$ and the accumulation of period-doubling bifurcations at $J \approx 3.5$ .
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Figure 4
(a) Sketch of the transition between full synchronization and QPS as $D$ is changed ( $J < - 2.54$ , $Δ = 0$ ); the $y$ axis is an arbitrary coordinate like, e.g., the minimal value attained by the firing rate. Full synchrony undergoes two transcritical bifurcations. The first one, at $D_{c}^{(1)}$ , gives rise to QPS which may, eventually, undergo secondary instabilities (not depicted) leading to chaos. At $D = π$ , full synchrony recovers its stability abruptly. (b) Same sketch for $Δ \neq 0$ . Left transcritical bifurcation evaporates while the right one becomes a saddle-node bifurcation.
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Figure 5
(a) Raster plot and (b) Time series of the mean firing rate of a population of $N = 1000$ heterogeneous ( $Δ = 0.025$ ) QIF neurons, with inhibitory coupling ( $J = - 3.8$ ) and delay ( $D = 3.5$ ). (c) Time-averaged ISIs of individual neurons vs neuron index (red points)—labels are associated to values of $η_{j}$ . As a double check, the FREs (4) are simulated for the same parameters and used to drive individual neurons. In this way, we obtain the $⟨ {ISI}_{j} ⟩$ depicted by black circles. Note the horizontal segments corresponding to clusters of neurons with identical $⟨ ISI ⟩$ . (d) Lyapunov exponents $λ_{j}$ of individual neurons driven by FREs (4). The $λ_{j}$ ’s are all negative, while $λ_{j} = 0$ for the $Δ = 0$ case in Fig. 1.
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