Self-organized bistability and its possible relevance for brain dynamics

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Self-organized bistability and its possible relevance for brain dynamics

Victor Buendía, Serena di Santo, Pablo Villegas, Raffaella Burioni, and Miguel A. Muñoz

Phys. Rev. Research 2, 013318 – Published 16 March 2020

Abstract

Self-organized bistability (SOB) is the counterpart of “self-organized criticality” (SOC), for systems tuning themselves to the edge of bistability of a discontinuous phase transition, rather than to the critical point of a continuous one. The equations defining the mathematical theory of SOB turn out to bear a strong resemblance to a (Landau-Ginzburg) theory recently proposed to analyze the dynamics of the cerebral cortex. This theory describes the neuronal activity of coupled mesoscopic patches of cortex, homeostatically regulated by short-term synaptic plasticity. The theory for cortex dynamics entails, however, some significant differences with respect to SOB, including the lack of a (bulk) conservation law, the absence of a perfect separation of timescales and, the fact that in the former, but not in the second, there is a parameter that controls the overall system state (in blatant contrast with the very idea of self-organization). Here, we scrutinize—by employing a combination of analytical and computational tools—the analogies and differences between both theories and explore whether in some limit SOB can play an important role to explain the emergence of scale-invariant neuronal avalanches observed empirically in the cortex. We conclude that, actually, in the limit of infinitely slow synaptic dynamics, the two theories become identical but the timescales required for the self-organization mechanism to be effective do not seem to be biologically plausible. We discuss the key differences between self-organization mechanisms with/without conservation and with/without infinitely separated timescales. In particular, we introduce the concept of “self-organized collective oscillations” and scrutinize the implications of our findings in neuroscience, shedding new light into the problems of scale invariance and oscillations in cortical dynamics.

Received 13 November 2019
Accepted 21 February 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.013318

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)

Critical phenomena Neuroscience, neural computation & artificial intelligence Self-organized criticality

Interdisciplinary PhysicsStatistical Physics & ThermodynamicsPhysics of Living Systems

Authors & Affiliations

Victor Buendía ^1,2,3, Serena di Santo⁴, Pablo Villegas ⁵, Raffaella Burioni^2,3, and Miguel A. Muñoz ^1,2

¹Departamento de Electromagnetismo y Física de la Materia e Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, E-18071 Granada, Spain
²Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, via G.P. Usberti, 7/A - 43124, Parma, Italy
³INFN, Gruppo Collegato di Parma, via G.P. Usberti, 7/A - 43124, Parma, Italy
⁴Morton B. Zuckerman Mind Brain Behavior Institute Columbia University, 10027 New York, USA
⁵Istituto dei Sistemi Complessi, CNR, via dei Taurini 19, 00185 Rome, Italy

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Issue

Vol. 2, Iss. 1 — March - May 2020

Subject Areas

Interdisciplinary Physics

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Images

Figure 1
Phase portraits and nullclines for the deterministic dynamics at an individual site of the Landau-Ginzburg theory, Eq. (2). Insets: the nullcline for $\dot{ρ} = 0$ is colored in black, while nullclines for $\dot{R} = 0$ —for three different values of the baseline level of synaptic resources $ξ$ – are plotted in orange, purple, and green color, respectively. The small grey arrows represent the vector field for $(\dot{ρ}, \dot{R})$ and the light purple curve represents a limit-cycle trajectory. [Inset (a)] For $τ_{D} / τ_{R} = 0.1$ and $τ_{R} = 10^{3}$ , the system may display a down state (orange nullcline, $ξ = 0.2$ ), a limit cycle (purple nullcline, $ξ = 1.5$ ) or an up state (green nullcline, $ξ = 3.5$ ). [Inset (b)] For $τ_{D} / τ_{R} = 0.001$ and $τ_{R} = 10^{7}$ the system approaches the SOB behavior and the dependence on the control parameter $ξ$ becomes very weak, meaning that for a wide range of values of the control parameter above the down state ( $ξ = 1, 5, 10$ for orange, purple, and green nullclines, respectively) the system sets in the oscillatory phase. [Inset (c)] When the separation of timescales is very low, the system only displays bistability between up and down states with no oscillations whatsoever, no matter the value of $ξ$ . (Main) The global behavior of the system depending on the timescale separation is coded in the main figure with different background colors: purple when the limit cycle appears, and green when only bistability is possible. Parameters are set to $I = 10^{- 3}, a = 0.6, b = 1.3$ , in both cases.
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Figure 2
Avalanches in the limit of a large separation of timescales, for the spatially extended (two dimensional) version of the Landau-Ginzburg theory for cortical dynamics, Eq. (2). Probability distribution for avalanche durations $T$ (a), avalanche sizes $S$ (b) and average avalanche size as a function of duration (c) in double logarithmic scale, for square-lattice systems of sizes: $N = 2^{12}, 2^{14}$ , and $2^{16}$ . The dashed lines are plotted as a guide to the eye, and have slopes corresponding to the expectations for an unbiased branching process ( $α = 2, τ = 3 / 2$ , and $γ = 2$ , respectively). The “bumps” correspond to anomalously large events, i.e., synchronized spiking events. The cut-offs/bumps change with $N$ obeying finite-size scaling as in the theory of self-organized bistability [25]. Parameters: $b = 0.5 a = 1, I = 10^{- 7}, τ_{D} = 10^{4}, τ_{R} = 10^{6}$ , and $D = 1$ .
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Figure 3
Weak dependence on the control parameter in the limit of large separation of timescales, for the spatially extended (two dimensional) version of the Landau-Ginzburg theory for cortical dynamics, Eq. (2). Different colors represent different values of $ξ$ . There is essentially no visible change in the probability distributions for avalanche durations $T$ (a), avalanche sizes $S$ (b), and average avalanche size as a function of duration (c), for different values of $ξ$ , in particular, $ξ = 1, 3, 5, 7, and 10$ . The dashed lines are plotted as a guide to the eye and their slopes correspond to unbiased branching process exponents. Parameters are fixed as in Fig. 2 with $N = 2^{14}$ .
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Figure 4
Two forms of self-organization. Diagram showing the steady-state solutions $ρ^{*}$ of Eq. (A2) as a function of $E$ . The solid line indicates stable fixed points, while the dashed line indicates unstable ones. Taking $E$ as a control parameter there are three different regions: a down state, a bistability region, and an up state. When a dynamics for $E$ is added, the system may display oscillations. For the sake of comparison, nullclines are displayed for both the SOB, Eq. (A3) (horizontal dotted line) and the Landau-Ginzburg for the cortex, Eq. (B1) (curved dotted line) models. Note that the nullclines in these two cases tend to cut tangentially at the bifurcation point, $ξ = a$ —becoming locally indistinguishable—as the large timescale separation limit is approached. Gray arrows represent the corresponding vector field for the case of the Landau-Ginzburg model. Parameters have been set to $a = 0.6, b = 1.3, h = 0.005, ɛ = 0.1, and ξ = 1.0$ .
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Figure 5
Invariance on $ξ$ . Behaviour of the Landau-Ginzburg system as a function of the control parameter $ξ$ , as the limit $Δ \to 0$ is approached. Each color represents a different phase (see legend). Note that as the separation of timescales increases (i.e., as $Δ \to 0$ , in the SOB limit) the oscillatory phase takes more space until it comprises the whole region $ξ > a$ . Parameter values: $a = 0.6, b = 1.3$ .
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Figure 6
Width of the region in which collective oscillations emerge in the model for adaptation model of Levenstein et al. [60]. It shows that either increasing $k$ or, alternatively, decreasing the ratio $ω / b$ enhances the region of collective oscillation, as in Fig. 5. Taking, either of these combinations of parameters to their limit $k \to \infty$ or $ω / b \to 0$ one observes a huge range for self-organized collective oscillations.
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