Excitatory-inhibitory branching process: A parsimonious view of cortical asynchronous states, excitability, and criticality

Letter
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Excitatory-inhibitory branching process: A parsimonious view of cortical asynchronous states, excitability, and criticality

Roberto Corral López, Víctor Buendía, and Miguel A. Muñoz

Phys. Rev. Research 4, L042027 – Published 14 November 2022

Abstract

The branching process is the minimal model for propagation dynamics, avalanches, and criticality, broadly used in neuroscience. A simple extension of it, adding inhibitory nodes, induces a much-richer phenomenology, including an intermediate phase, between quiescence and saturation, that exhibits the key features of “asynchronous states” in cortical networks. Remarkably, in the inhibition-dominated case, it exhibits an extremely rich phase diagram that captures a wealth of nontrivial features of spontaneous brain activity, such as collective excitability, hysteresis, tilted avalanche shapes, and partial synchronization, allowing us to rationalize striking empirical findings within a common and parsimonious framework.

Received 30 March 2022
Revised 6 July 2022
Accepted 13 October 2022

DOI:https://doi.org/10.1103/PhysRevResearch.4.L042027

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.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Biological neural networks Complex systems Fluctuations & noise

Critical phenomena

Interdisciplinary PhysicsStatistical Physics & ThermodynamicsPhysics of Living Systems

Authors & Affiliations

Roberto Corral López ¹, Víctor Buendía ^2,3, and Miguel A. Muñoz ¹

¹Departamento de Electromagnetismo y Física de la Materia and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, E-18071 Granada, Spain
²Department of Computer Science, University of Tübingen, 72076 Tübingen, Germany,
³Max Planck Institute for Biological Cybernetics, 72076 Tübingen, Germany

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Issue

Vol. 4, Iss. 4 — November - December 2022

Subject Areas

Complex Systems

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Images

Figure 1
(a) Sketch of an excitatory-inhibitory branching process on a tree; active inhibitory units (red squares) reduce the probability of propagation from active excitatory units (blue circles). Empty symbols stand for inactive units and full/dashed lines for fulfilled/unfulfilled processes. (b) Transition rates for the “excitatory-inhibitory contact process” (EI-CP). (c) Illustration of a two-dimensional (2D) lattice with a central cluster of active nodes.
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Figure 2
Results for the excitatory-inhibitory contact process (EI-CP) on fully connected networks as analytically obtained from Eq. (1) for $α = 1 / 2, r_{i} = 0$ , and excitation-dominated initial conditions (note that inhibition-dominated conditions always lead to the quiescent state). (a) Phase portrait in the $r − λ$ plane: Active phase (blue), quiescent phase (red), excitable quiescent phase (purple), and bistable regimes (green). The full red line $λ_{1} (r) [λ_{3} (r)$ in blue] marks continuous (discontinuous) transitions between quiescent and active states. These two lines come together at a tricritical point (yellow star). The line $λ_{2} (r)$ marks a Hopf bifurcation, separating the standard quiescent phase from an excitable quiescent one, where the quiescent state is locally unstable, but globally stable. (b)–(d) Overall stationary activity $ρ = ρ_{e} + ρ_{i}$ as a function of $λ$ for three different values of $r$ as marked and color coded in (a): Continuous transition (b), discontinuous transition with a regime of bistability between an active state and a quiescent state (c), or between an active and an excitable quiescent state (d). (e)–(g) Flow diagrams in the $ρ_{e}, ρ_{i}$ plane for the three points marked in panel (a); the background color stands for the phase and its color intensity is proportional to the vector-field module, the colored lines are the nullclines ${\dot{ρ}}_{e} = 0$ (green) and ${\dot{ρ}}_{i} = 0$ (purple), respectively, and the black line ( $ρ_{i} = ρ_{e} / r$ ) separates zone 1 (inhibition dominated) from zone 2 (excitation dominated). Characteristic trajectories are depicted as arrowed orange lines, while colored points stand for stable steady states.
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Figure 3
Avalanche shapes rescaled with their duration $T_{r}$ at different points of the mean-field phase diagram (characterized by $r$ ). Simulations are performed for the noisy version of Eq. (1) with small excitatory initial conditions (see SM-IV.B [55]). (i) At the inhibition-free critical point (blue curves; $r = 0$ and $λ_{c} = 2$ ) avalanches for different durations, $T_{r = 0}$ , are scale free as their rescaled curves collapse onto a universal inverted-parabola shape using the BP exponent $γ = 2$ [13, 15]. (ii) In the presence of inhibition, the curves at the critical point (orange curves; $r = 0.22$ and $λ_{c} = 3$ ) are scale invariant with BP exponents and they collapse onto a slightly “tilted” nonparabolic curve (see also [61]). (iii) Within the excitable phase (red curves; $r = 0.5$ and $λ = 10$ ), i.e., away from bifurcations, one observes duration-dependent (non-scale-invariant) skewed nonparabolic shapes.
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Figure 4
Features of the asynchronous phase. (a),(b) Analytical results for sparse (annealed) random networks. (a) Stationary activity where the dashed line indicates the end of the quiescent phase and stripes signal the bistability region between the asynchronous and standard-active phase. (b) Henrici index gauging the level of non-normality in the $r, λ$ diagram (see SM-IV.C [55]). (c)–(l) Results for a 2D lattice (which helps visualization) with $N = 10^{4}$ . (c) Section of the phase diagram ( $r = 0.7$ ), illustrating the discontinuous transition with bistability and (d) coefficient of variation ( $C V$ ) for different values of $λ$ . (e) Lagged cross correlations ( $C C$ ) between excitatory and inhibitory time series; inhibition follows closely excitation with a delay $τ$ . Total excitatory and inhibitory activity as a function of time for $r = 0.7$ is plotted in (f) for the AS phase ( $λ = 200$ ) and in (h) for the standard active phase ( $λ = 1000$ ); (g)–(i) same as in (f) and (h), respectively, plotted in the standardized ${\tilde{ρ}}_{e}, {\tilde{ρ}}_{i}$ plane (see SM-IV.C [55]), as an illustration of the diverse nature of cross correlations in both phases. (j) Stimulation experiment where a fraction $Δ_{e}$ of excitatory nodes (on a 2D lattice) is transiently activated; in the bistability region ( $r = 0.7, λ = 750$ ), this can potentially drive the system from the AS phase to the standard active phase, much as in experimental setups [72]. (k)/(l) Snapshots of the system before (k) and after (l) the perturbation ( $N = 20^{2}$ ). See SM [55] for further details.
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