Critical Drift in a Neuro-Inspired Adaptive Network

Silja Sormunen, Thilo Gross, and Jari Saramäki

Phys. Rev. Lett. 130, 188401 – Published 2 May 2023

Abstract

It has been postulated that the brain operates in a self-organized critical state that brings multiple benefits, such as optimal sensitivity to input. Thus far, self-organized criticality has typically been depicted as a one-dimensional process, where one parameter is tuned to a critical value. However, the number of adjustable parameters in the brain is vast, and hence critical states can be expected to occupy a high-dimensional manifold inside a high-dimensional parameter space. Here, we show that adaptation rules inspired by homeostatic plasticity drive a neuro-inspired network to drift on a critical manifold, where the system is poised between inactivity and persistent activity. During the drift, global network parameters continue to change while the system remains at criticality.

Received 17 June 2022
Revised 4 February 2023
Accepted 3 April 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.188401

Physics Subject Headings (PhySH)

Network evolution Neuronal dynamics Self-organized criticality

Stochastic dynamical systems

Neuronal network models

Interdisciplinary PhysicsNonlinear Dynamics

Authors & Affiliations

Silja Sormunen ^1,*, Thilo Gross ^2,3,4, and Jari Saramäki ¹

¹Department of Computer Science, Aalto University, 00076 Espoo, Finland
²Helmholtz Institute for Functional Marine Biodiversity at the University of Oldenburg (HIFMB), Oldenburg 26129, Germany
³Alfred-Wegener Institute, Helmholtz Centre for Marine and Polar Research, Bremerhaven 27570, Germany
⁴Institute for Chemistry and Biology of the Marine Environment (ICBM), Carl-von-Ossietzky University, Oldenburg 26129, Germany

^*Corresponding author. silja.sormunen@aalto.fi

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Issue

Vol. 130, Iss. 18 — 5 May 2023

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Images

Figure 1
The onset of activity in static ER networks of different sizes $N$ . (a) The average fraction of firing nodes $⟨ [F] ⟩$ (average taken over time) before $t_{\max} = 5000$ , with 5% of the nodes initialized as firing. The inset shows the time series of the fraction of firing nodes for networks of size $N = 10^{5}$ with different mean degrees. (b),(c) The maximum size and duration of finite avalanches (lasting less than $t_{\max}$ ) in 1000 successive runs. Both quantities display a sharp peak at a critical value $⟨ k ⟩_{N, static}^{*}$ , which moves closer to the theoretical estimate as $N$ increases. All results are averaged over 30 network realizations for each mean degree. In these and subsequent figures, we set $β = 0.7$ , $δ = 0.95$ , and $γ = 0.4$ .
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Figure 2
Critical drift in evolved networks with different initial mean degrees. (a)–(d) Time evolution of the leading eigenvalue $λ_{1}$ , mean degree $⟨ k ⟩$ , mean excess degree $⟨ q ⟩$ , and Pearson correlation coefficient $ρ$ of the nodes’ in- and out-degrees. The simulation is initialized with 5% of the nodes in the firing state. The parameter values are $N = 10^{5}$ , $l = 10^{- 3}$ , $g = 10^{- 6}$ , and $η = 1 / (100 N)$ . For analysis on their effect on the drift, see SM, II and III [25]. The yellow dashed line in (a) shows $⟨ k ⟩_{N = 10^{5}, static}^{*}$ (determined in SM, I [25]). (e) We freeze the networks at different time points shown in panel (b). These are chosen so that for each network, point A lies before the start of the drift while points B–D correspond to the drift phase. For each time point, the dashed vertical line marks the mean degree in the evolved network at that point, while the curve in the same color shows results for the networks obtained by manipulating the mean degree. The top row shows average activity $⟨ [F] ⟩$ before $t_{\max} = 5000$ with 5% of the nodes initialized as firing. The other two rows show the maximum size and duration of finite avalanches in 1000 successive runs. Results are averaged over 30 network realizations for all networks obtained by manipulating $⟨ k ⟩$ . We observe that during the initial phase (point A), the mean degree lies clearly under or above the onset of activity, while during the drift phase (points B–D), the networks reside at the onset of activity. As the network evolves, the onset of activity happens at higher values of $⟨ k ⟩$ , which confirms the existence of the manifold. Note that the divergence peaks of avalanche sizes and durations are not exactly at the point where $⟨ [F] ⟩$ becomes nonzero (top row); however, these three measures converge to the critical value (or slightly above it due to finite values of $g$ and $l$ ) as $N$ is increased (see SM, IV [25]).
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