Self Tuned Criticality: Controlling a neuron near its bifurcation point via temporal correlations

Previous work showed that the collective activity of large neuronal networks can be tamed to remain near its critical point by a feedback control that maximizes the temporal correlations of the mean-field fluctuations. Since such correlations behave similarly near instabilities across nonlinear dynamical systems, it is expected that the principle should control also low dimensional dynamical systems exhibiting continuous or discontinuous bifurcations from fixed points to limit cycles. Here we present numerical evidence that the dynamics of a single neuron can be controlled in the vicinity of its bifurcation point. The approach is tested in two models: a 2D generic excitable map and the paradigmatic FitzHugh-Nagumo neuron model. The results show that in both cases, the system can be self-tuned to its bifurcation point by modifying the control parameter according to the first coefficient of the autocorrelation function

Scale-free correlations in the dynamics of a small (N ~ 10000) cortical network

The advent of novel opto-genetics technology allows the recording of brain activity with a resolution never seen before. The characterisation of these very large data sets offers new challenges as well as unique theory-testing opportunities. Here we discuss whether the spatial and temporal correlation of the collective activity of thousands of neurons are tangled as predicted by the theory of critical phenomena. The analysis shows that both, the correlation length $\xi$ and the correlation time $\tau$ scale as predicted as a function of the system size. With some peculiarities that we discuss, the analysis uncovers new evidence consistent with the view that the large scale brain cortical dynamics corresponds to critical phenomena.Comment: 8 pages, 6 figure

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

In this note, a correlation metric $\kappa_c$ is proposed which is based on the universal behavior of the linear/logarithmic growth of the correlation length near/far the critical point of a continuous phase transition. The problem is studied on a previously described neuronal network model for which is known the scaling of the correlation length with the size of the observation region. It is verified that the $\kappa_c$ metric is maximized for the conditions at which a power law distribution of neuronal avalanches sizes is observed, thus characterizing well the critical state of the network. Potential applications and limitations for its use with currently available optical imaging techniques are discussed.Comment: 5 pages, 5 figure