Finite-size correlation behavior near a critical point: A simple metric for monitoring the state of a neural network

In this article, a correlation metric ${\kappa }_c$ is proposed for the inference of the dynamical state of neuronal networks. ${\kappa }_C$ is computed from the scaling of the correlation length with the size of the observation region, which shows qualitatively different behavior near and away from the critical point of a continuous phase transition. The implementation is first studied on a neuronal network model, where the results of this new metric coincide with those obtained from neuronal avalanche analysis, thus well characterizing the critical state of the network. The approach is further tested with brain optogenetic recordings in behaving mice from a publicly available database. Potential applications and limitations for its use with currently available optical imaging techniques are discussed.

Figure 1

System scheme. (a) A system of characteristic size $L$ is studied through boxes of side $W$. Only neurons inside the box are recorded. (b) Example of the time series of $A$, the total number of active neurons inside a window, as a function of time. An avalanche (filled with gray) is defined as the total activity above a threshold $c$, computed from the time at which $A$ becomes greater than $c$ to the next time that it is becomes lower than $c$. Panel (c) shows the cumulative avalanche size distribution function $F\left(s\right)$ as a function of avalanche size $s$ for three different situations: subcritical ($T=0.33$, open blue squares), supercritical ($T=0.31$, open red diamonds), and close to criticality ($T=0.318$ green filled circles). The dashed line represents the theoretical expectation for the avalanche size distribution expected at criticality, $F^{NA}\left(s\right)$. Curves were computed for $m=10$ values of $s$. (d) The connected correlation function of a window of size $W, C_W\left(r\right)$, for several values of $W$, computed at criticality. From left to right, $W=50$ (violet line), $W=150$ (cyan), $W=250$ (orange), $W=500$ (light green). The characteristic length $r_0$ for $W=500$ is marked with an arrow, as an example. (e) Characteristic length $r_0$ as a function of window size $W$ at the critical state. Results computed on a system of size $L=1000, k=24, \pi =0.01$, and $T=0.318$. In panels (b) and (c), a window size $W=500$ was used.

Figure 2

Avalanche size distribution computed on a window of size $W=500$, for different values of $T$ in panel (a) and for $T=0.3180\simeq T_C$ and several values of $W$ in panel (b). The dashed lines, in both panels, show a power law with exponent $-3/2$ as a guide to the eye. All parameters are the same as in Fig. 1.

Figure 3

Characteristic correlation length as a function of window size $W$ obtained at various control parameter values $T$ (indicated in the legend). The same results are plotted on a linear scale in panel (a) and on a linear-logarithmic scale in panel (b). All other parameters are the same as in Fig. 1.

Figure 4

Behavior of the different metrics (mean $\pm SD$) as a function of $T$ near the critical point of the neural model: ${\kappa }_S$ in panel (a) and ${\kappa }_C$ in panel (b) and $\mathit{AC}$(1) in panel (c). All other parameters are the same as in Figs. 2 and 3.

Figure 5

Numerical simulations demonstrating the behavior of the metrics in response to a slow change in the network excitability, here simulated by ramping up and down the model parameter $T$. Panel (a) shows the raster plot of a subset of 100 neurons as $T$ is varied. Panel (b) shows the evolution of $T$ as a function of time $t$. Panel (c) shows the estimated mean ($\pm$ SD) ${\kappa }_S$ and ${\kappa }_C$, computed over time segments of $n=2000$ steps. For ${\kappa }_C$, we used spatial windows $W\le 300$. For ${\kappa }_S$ an average of $\approx 95$ avalanches (range 14–280) were detected in each run and each temporal window. Panel (d) shows the first autocorrelation coefficient $AC\left(1\right)$ of the population rate fluctuations within the same windows. Panel (e) shows the errors of the estimators, computed as the average distance between the measured and the expected value, ${\kappa }_S^{*}, {\kappa }_C^{*}$, and $AC{\left(1\right)}^{*}$, as a function of the number $n$ of steps at $T=0.3180\simeq T_C$. Results in each panel are from four independent realizations of the numerical simulations. For avalanche analysis, since $n$ is variable, we considered $s_{\min }$ as 10 times the smaller avalanche size observed, and $s_{\max }$ as 0.1 of the largest observed avalanche size.

Figure 6

Experimental recordings. Analysis of neuronal spike data inferred from two-photon imaging from the Allen Institute database [35]. Panel (a) shows box-scaling results for the entire data with empty symbols and with lines for the segments of 1/8 of the time series. The inset shows the same data in log-linear axis. In panel (b) the symbols show ${\kappa }_C$ for different time windows as a function of the number of frames used, while the black lines show the mean $\pm$ standard error. Panel (c) shows the avalanche size distribution of the same data, for all time frames with filled symbols (17 155 avalanches), and for segments of 1/8 of the points [as in panel (a)] with lines. Panel (d) shows the ${\kappa }_S$ in the same format used in panel (b). For each time segment, ${\kappa }_C$ was computed from windows of size $100\mu \text{m}$ or larger, while ${\kappa }_S$ was computed using avalanche sizes ranging from twice the minimum observed avalanche size to half of the largest observed avalanche size.

Figure 7

Dynamical changes in response to visual stimuli exhibited by the experimental recordings according with the different metrics (same data as in Fig. 6). Panel (a) shows the raster plot, and panel (b) shows the average rate (number of spikes per frame) computed over time blocks related to different visual stimuli. Panels (c) through (e) show ${\kappa }_S, {\kappa }_C$, and $AC\left(1\right)$, respectively, for the same time blocks. These points are replotted in the right panels where panel (f) shows ${\kappa }_C$ as a function of $AC\left(1\right)$ and panels (g)–(h) show ${\kappa }_S$ as a function of $AC\left(1\right)$ and as a function of the rate, respectively ($r$ values correspond to linear regression coefficients). The visual stimuli, labeled in panel (a) and denoted by the vertical dashed lines, consisted of a sequence of 8 min. of static gratings (ST) followed by interstimulation period of gray screen, 8 min. of natural images (N), 5 min. of spontaneous activity (Sp), 8 min. of natural images, interstim gray screen, 8 min. of static gratings, interstim gray screen, 5 min. of natural movie (NM), 9 min. of natural images, and 9 min. of static gratings. The symbols in panels (f)–(h) correspond to the different stimuli kinds [colored as the labels on top of panel (a)]. Results computed from Ref. [35].

Figure 8

Changes in the level of response to natural images (N) and static gratings (ST) stimuli as a function of the network state, estimated by the three metrics. Panel (a) corresponds to ${\kappa }_S$, panel (b) to ${\kappa }_C$, and panel (c) to $AC\left(1\right)$. For static gratings, the response was computed on the 5 most responsive neurons on each orientation (total: 30 neurons). For natural images, the neurons for the eight natural images with most responsive neurons were considered (natural images 45 with 20 responsive neurons; 85, with 14 neurons; 41, 115, and 108, with 10 neurons; 69 and 36, with 8 neurons; 86 with 7 neurons, total: 87 neurons). All other parameters are the same as in Fig. 7.