Box scaling as a proxy of finite size correlations

Abstract

The scaling of correlations as a function of size provides important hints to understand critical phenomena on a variety of systems. Its study in biological structures offers two challenges: usually they are not of infinite size, and, in the majority of cases, dimensions can not be varied at will. Here we discuss how finite-size scaling can be approximated in an experimental system of fixed and relatively small extent, by computing correlations inside of a reduced field of view of various widths (we will refer to this procedure as “box-scaling”). A relation among the size of the field of view, and measured correlation length, is derived at, and away from, the critical regime. Numerical simulations of a neuronal network, as well as the ferromagnetic 2D Ising model, are used to verify such approximations. Numerical results support the validity of the heuristic approach, which should be useful to characterize relevant aspects of critical phenomena in biological systems.

Disentangling the critical signatures of neural activity

“…In particular, since the maximum system size (corresponding to N channels) both in our experimental data and in our model is fixed, we investigate how ξ changes with system sizes corresponding to different subsamples from the multi-array probe [64]. In fact, through simulations on control models that display a critical point [65], subsampling has been shown to be practically equivalent to consider systems of different sizes.…”

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

“…This measures were facilitated by the use of novel opto-genetic techniques [15], which allows for the recording of the individual activity of a relatively large number of neurons. In that work, a proxy of the standard finite size analysis, named Box-Scaling (B-S) was used, [16] in which the observation window, instead of the system size, is varied. An estimate of the correlation length ξ was found to grow linearly or logarithmically with window size depending if the system is near or far from the critical state, respectively.…”

“…We run simulations for several values of the control parameter T which previous results [16] indicate produces subcritical (for very high values of T ), supercritical (for very low values of T ) or critical dynamics. To accumulate enough statistics, we run 20 numerical simulations (lasting 10 5 time steps, discarding the initial 5000 time steps).…”

“…Metric based on finite-size scaling of correlations-Following previous work [16], we computed the connected correlation function on a window of size W , as the correlation of the fluctuations of the neuronal activity, with respect to it's instantaneous spatial average [12-14, 16, 21-25]: where δ(r − r ij ) is a smoothed Dirac δ function selecting pairs of neuron states at a distance r (in practice, we have computed C W (r) for integer values of r, averaging all points at distances (r − 0.5, r + 0.5]); r ij is the Euclidean distance from the site i to site j; δv i is the value of the signal v i of site i at time t, after subtracting the the instantaneous spatial average of signals…”

“…We measure C W (r) for several values of W , ranging from W min to W max . For equilibrium thermodynamic systems, the behavior of r 0 as a function of W , for fixed L, is known in the limiting cases: r 0 ∝ W for W L ξ at criticality, while r 0 ∝ ξ log(W/ξ) for ξ W min , where ξ is the standard correlation length, see [16,24].…”