Scale-Change Symmetry in the Rules Governing Neural Systems

Abstract

Similar universal phenomena can emerge in different complex systems when those systems share a common symmetry in their governing laws. In physical systems operating near a critical phase transition, the governing physical laws obey a fractal symmetry; they are the same whether considered at fine or coarse scales. This scale-change symmetry is responsible for universal critical phenomena found across diverse systems. Experiments suggest that the cerebral cortex can also operate near a critical phase transition. Thus we hypothesize that the laws governing cortical dynamics may obey scale-change symmetry. Here we develop a practical approach to test this hypothesis. We confirm, using two different computational models, that neural dynamical laws exhibit scale-change symmetry near a dynamical phase transition. Moreover, we show that as a mouse awakens from anesthesia, scale-change symmetry emerges. Scale-change symmetry of the rules governing cortical dynamics may explain observations of similar critical phenomena across diverse neural systems.

Keywords: Mathematical Biosciences; Statistical Mechanics; Systems Neuroscience.

Graphical abstract

Figure 1

Phase Transition in a Simple Neural Model (A) Each panel shows the two-dimensional lattice of nodes at a single time step. Each pixel represents one node (yellow, active; blue, inactive). A subset of the full lattice is shown for clarity. (B) As coupling strength C increases a sharp increase in time-averaged network activity occurs at a critical coupling strength C^* near C = 0.23. $\mathit{S}$ is averaged over 10⁴ time steps excluding a transient period of 10² time steps.

Figure 2

Scale-Invariance of Dynamical Rules Peaks at Criticality (A) Cartoon illustration of coarse-graining scheme. Each block of nodes at fine scale b is transformed probabilistically to one node at the coarse scale b + 1. (B) Examples of activity snapshots before and after coarse graining. (C) Upon coarse graining, the dynamical rules change the least ($\mathit{\zeta }$ is minimal) at criticality. Inset shows the coarse-graining transformation function with (k; x₀) = (76; 0.22). Block size was r = 8. (D) Shown are optimal coarse-graining functions for three example C values and six block sizes (legend in E specifies different values of r and τ). (E) Using the optimal coarse-graining function for each C resulted in the strongest scale-invariance of dynamical rules, i.e., lowest ${\mathit{\zeta }}_{\mathit{min}}$ around $\mathit{C}={\mathit{C}}^{\mathit{*}}$. This result held for multiple choices of block size and duration (see legend). (F) The valley in ${\mathit{\zeta }}_{\mathit{min}}$ as a function of coupling strength C became broader as p was increased. For A–E p = 0.001.

Figure 3

Scale-Invariance of Dynamical Rules Peaks at Phase Transition in a More Biologically Plausible Model (A) Each panel shows the two-dimensional lattice of neurons at a single time step. Each pixel represents one neuron (yellow, active; blue, inactive). The spatiotemporal dynamics was limited to small scales for strong inhibition (I = 2.0, bottom row), exhibited massive propagating waves and oscillations for weak inhibition (I = 0.01, top row), and had more complexity near the transition between these extremes (I = 0.65, middle row). (B) Time series of network activity reveals the prominent oscillatory activity of the weak inhibition regime (red). (C) As inhibition is increased, the boundary of the oscillatory regime near I = 0.65 (dashed line) is revealed by the drop in mean pairwise correlations. (D) Scale-invariance of dynamical rules peaked (${\mathit{\zeta }}_{\mathit{min}}$ is minimal) near the onset of the oscillatory regime. This held for blocks with different spatial sizes and durations (see legend).

Figure 4

Applying Our Approach to Continuous Synaptic Input (A) Mean pairwise correlations of binarized membrane potential for the realistic model. (B) Change in dynamical rules ${\mathit{\zeta }}_{\mathit{min}}$ governing the binarized membrane potential as a function of inhibition strength I for r = 8 (left), r = 16 (right), and different binarization thresholds (color). For all the cases shown τ = 1 and network size, L × L = 160 × 160.

Figure 5

Increase in Scale-Invariance of Cortical Dynamical Rules as Mouse Awakens (A) Genetically encoded voltage-sensitive fluorescence imaging was done to measure the spatiotemporal dynamics across one hemisphere of mouse cortex as it awoke from anesthesia. Each panel shows a snapshot of binarized activity (yellow, active; blue, inactive). The signal of each pixel arises from many neurons within a 33 × 33 μm² area. (B) Time series of binary network activity datasets. Under anesthesia (red), the dynamics exhibited relatively large-scale bursts, whereas the awake dynamics (blue) tended to be more diverse. (C) Mean pairwise correlation decreases as the mouse awakens. (D) Scale-invariance of dynamical rules increases (ζ_min decreases) as the mouse awakens. Results were qualitatively consistent for three different binarization thresholds (yellow, red, and blue) and two different coarse-graining block sizes (r = 8 and 16).

Figure 6

Scale-Invariance of Rules Versus Avalanche Size Distributions (A) Shown are avalanche size distributions obtained from the simple model with different values of coupling, C. The probability for large avalanches is prominent for strong coupling and dramatically lower for weak coupling. Distributions are shifted vertically for visual comparison. Black dashed line indicates a power law with exponent −1.5. (B) The parameter κ measures deviation between a measured avalanche size distribution and −1.5 power law. Near C = C *, we found minimal deviation from power law (κ = 1). (C) We found minimal change in rules ${\mathit{\zeta }}_{\mathit{min}}$ near κ = 1. (D) For the realistic model, avalanche size distributions exhibited high probability for large avalanches when inhibition was small (blue) and approximate power law distributions for stronger inhibition. (E) Near the onset of the oscillatory phase, we found the smallest deviation from power law (κ near 1). (F) Change in rules ${\mathit{\zeta }}_{\mathit{min}}$ was minimal near κ = 1.