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 two-dimensional 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.

Figure 1

Average autocorrelation coefficient as a function of the control parameter for Chialvo's neuron map in the excitable [panel (a)] and chaotic [panel (b))] regimes, for 100 samples. The inset in panel (a) depicts the typical hysteresis for the excitable regime of this model. Panel (c) shows the same quantities for the FHN model. In all the panels, the evolution of AC is shown for increasing (black crosses) and decreasing (red circles) control parameter values.

Figure 2

Adaptive control of the Chialvo's neuron map in the excitable regime. All of the results were averaged over 100 samples. Data correspond to numerical solutions of the model starting from ten different initial conditions of the parameter $k$ from 0.024 (black) to 0.033 (magenta). Different colors denote the evolution of the variables toward the bifurcation point for each initial condition. Panels (a) and (c) show the evolution of $\text{AC}\left(1\right)$ and control parameter $k$, respectively. Panel (b) shows $\text{AC}\left(1\right)$ as a function of the control parameter $k$. Parameters $(a,b,c)=(0.89,0.60,0.28)$.

Figure 3

Adaptive control of the Chialvo neuron map in the excitable regime. Panel (a) shows the evolution of the two variables (blue dotted lines) and the nullclines of Eq. (2) (black solid lines) without control (fixed $k=0.031$) corresponding to the data in panels (d) and (g). Panels (b) and (c) show the evolution and the nullclines after the control is switched on and after two perturbations [panels (e) and (h), and panels (f) and (i), respectively].

Figure 4

Adaptive control of Chialvo's neuron map in the chaotic regime. Same format as in Fig. 2. Numerical solutions of the model starting from ten different initial conditions of the parameter $k$ from 0.015 (black) to 0.025 (orange), for 100 samples.

Figure 5

Example of the control applied to the variable $k$ in the Chialvo neuron map in the chaotic regime, with parameters $(a,b,c)=(0.89,0.18,0.28)$. In panel (a) the nullclines of Eq. (2) are displayed by black lines while the blue dotted lines show the trajectory for $k=0.025$. Trajectories in blue are the same as displayed in the first half of panels (a) and (d). Panel (b) shows the nullclines (dashed lines) and the symbols are the trajectory of the system after small perturbations, $p_1,p_2,p_3,$ and $p_4$, are applied to the system now under control, which can be seen in the second half of panels (c) and (d).

Figure 6

Adaptive control of the FitzHugh-Nagumo model. Numerical solutions of the model applying the control starting from 14 different initial conditions of the control parameter $I$ (different colors), from 0.105 (black) to 0.118 (turquoise) for 100 samples.