Oscillations and collective excitability in a model of stochastic neurons under excitatory and inhibitory coupling

Italo'Ivo Lima Dias Pinto and Mauro Copelli

Phys. Rev. E 100, 062416 – Published 24 December 2019

Abstract

We study a model with excitable neurons modeled as stochastic units with three states, representing quiescence, firing, and refractoriness. The transition rates between quiescence and firing depend exponentially on the number of firing neighbors, whereas all other rates are kept constant. This model class was shown to exhibit collective oscillations (synchronization) if neurons are spiking autonomously, but not if neurons are in the excitable regime. In both cases, neurons were restricted to interact through excitatory coupling. Here we show that a plethora of collective phenomena appear if inhibitory coupling is added. Besides the usual transition between an absorbing and an active phase, the model with excitatory and inhibitory neurons can also undergo reentrant transitions to an oscillatory phase. In the mean-field description, oscillations can emerge through supercritical or subcritical Hopf bifurcations, as well as through infinite period bifurcations. The model has bistability between active and oscillating behavior, as well as collective excitability, a regime where the system can display a peak of global activity when subject to a sufficiently strong perturbation. We employ a variant of the Shinomoto-Kuramoto order parameter to characterize the phase transitions and their system-size dependence.

Received 12 June 2019
Revised 24 October 2019

DOI:https://doi.org/10.1103/PhysRevE.100.062416

Physics Subject Headings (PhySH)

Collective dynamics Continuous phase transition Neuronal dynamics

Statistical Physics & ThermodynamicsNetworksNonlinear Dynamics

Authors & Affiliations

Italo'Ivo Lima Dias Pinto and Mauro Copelli

Physics Department, Federal University of Pernambuco (UFPE), Recife, PE 50670-901, Brazil

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Vol. 100, Iss. 6 — December 2019

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Images

Figure 1
(a) Neuron represented as a three-state unit with unidirectional transitions. (b) Schematic representation of the coupling between populations, showing the coupling parameters for each kind of interaction.
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Figure 2
(a) Mean-field phase diagram for $a_{i e} = 12$ (see text for details). Filled symbols (square, circle, and triangle) are representative points of different phases, which we consider in detail in panel (c). The vertical line crosses a SNIC and a saddle-saddle bifurcation, as detailed in panel (b). (b) Mean-field bifurcation diagram showing the position and stability of $P_{1}$ for the range of $a_{e i}$ indicated by the vertical line of panel (a). Solid (dashed) lines represent stable (unstable) fixed points. (c) Time series (black line for excitatory neurons, gray for inhibitory neurons) of a simulation on a complete graph ( $N = 10^{4}$ and $a_{e e} = 15$ ) for different values of $a_{e i}$ as indicated by the symbols in panel (a). For $t < 200$ , the system is in the active phase with parameter $a_{e i} = 5$ . For $200 < t < 400$ the parameter is $a_{e i} = 10$ and the system displays global oscillations. For $t > 400$ , $a_{e i} = 13.5$ and the system exhibits collective excitability with irregular bursts of activity. (d) Projection on the $P_{1}^{(e)} \times P_{1}^{(i)}$ plane of an excursion in phase space in the collective oscillations phase $[200 < t < 400$ in panel (c)].
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Figure 3
(a) Zoom of the phase diagram in Fig. 2 with vertical lines at $a_{e e} = 14.5$ and $a_{e e} = 15$ showing paths in parameter space along which the order parameter $q$ is assessed in panels (b) and (c), respectively. (b) Order parameter $q$ for the mean-field approximation (solid line) and simulations (symbols) on complete graphs for $a_{e e} = 14.5$ and increasing system sizes. The inset presents the details of the hysteresis cycle at the coexistence region of the (mean-field) subcritical Hopf bifurcation [dashed line in (b)]. (c) Same as (b), but for $a_{e e} = 15$ . The inset shows the size dependence of $q$ as $a_{e i}$ crosses the mean-field supercritical Hopf critical line. From bottom to top, we show two sets of points in the nonoscillatory phase, with $a_{e i} = 6.8$ (crosses), $a_{e i} = 7$ (X's) and two sets in the oscillatory phase, with $a_{e i} = 7.5$ (empty squares) and $a_{e i} = 7.7$ (filled squares). The critical value obtained for this set of parameters was $a_{e i} = 7.19488$ , for which we obtain approximately $q \sim N^{- 0.22 \pm 0.0038}$ . (d) Time series for the coexistence region [inset of (b)] showing periods of oscillations interposed by periods with constant activity (black line for excitatory neurons, gray for inhibitory neurons). The time series was obtained for $a_{e e} = 14.5$ and $a_{e i} = 10.99$ .
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Figure 4
Order parameter for the mean-field approximation and simulations on random graphs (RGs) for (a) $a_{e e} = 14.5$ and (b) $a_{e e} = 15$ [compare with Figs. 3 and 3]. Both panels show simulations with different system sizes $N$ , with each neuron randomly connected to other $k = 10^{3}$ neurons. Results can be compared with a complete graph (CG) with $N = 10^{4}$ . The inset in (a) shows how the order parameter $q$ approaches the complete graph and mean-field values as we increase $k$ for a RG with $N = 10^{4}$ . Panel (c) presents a time series (black line for excitatory neurons, gray for inhibitory neurons) for a random graph with $a_{e e} = 15$ , $a_{i e} = 12$ , $N = 10^{4}$ , and $k = 10^{3}$ , showing the same regimes as in Fig. 2, with $a_{e i} = 5$ for $t < 200$ , $a_{e i} = 10$ for $200 < t < 400$ , and $a_{e i} = 16$ for $t > 400$ [compare with Fig. 2]. Panel (d) presents the density of active neurons around the absorbing-active phase transition, as predicted by the mean-field calculations [Eq. (10)] and compared with results obtained for a complete graph ( $N = 10^{4}$ and $10^{5}$ ) and random graph simulations ( $k = 20$ and $k = 10^{3}$ with $N = 10^{3}$ ). In this case we used the parameters $a_{e i} = a_{i e} = 1$ .
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