Dragon-king-like extreme events in coupled bursting neurons

We present evidence of extreme events in two Hindmarsh-Rose (HR) bursting neurons mutually interacting via two different coupling configurations: chemical synaptic- and gap junctional-type diffusive coupling. A dragon-king-like probability distribution of the extreme events is seen for combinations of synaptic coupling where small- to medium-size events obey a power law and the larger events that cross an extreme limit are outliers. The extreme events originate due to instability in antiphase synchronization of the coupled systems via two different routes, intermittency and quasiperiodicity routes to complex dynamics for purely excitatory and inhibitory chemical synaptic coupling, respectively. For a mixed type of inhibitory and excitatory chemical synaptic interactions, the intermittency route to extreme events is only seen. Extreme events with our suggested distribution is also seen for gap junctional-type diffusive, but repulsive, coupling where the intermittency route to complexity is found. A simple electronic experiment using two diffusively coupled analog circuits of the HR neuron model, but interacting in a repulsive way, confirms occurrence of the dragon-king-like extreme events.

Figure 1

Two HR systems coupled via inhibitory chemical synapses. (a) Temporal dynamics of $x_{\parallel }$ for $k_{1,2}=-0.17$. Horizontal dashed (red) line depicts $H_s=2.44$. (b) Plot of $x_1$ vs. $x_2$, a projection of the APS manifold. The error dynamics of the coupled system is confined to APS manifold in a dense black (blue) region, but occasionally moves out along the transverse direction (in-phase synchrony).

Figure 2

Numerical PDF of events for $k_{1,2}=-0.17$. It is obtained for truncated events' size $x_{\parallel }\ge 1.2$. A power law fits into a range of event size (1.2–2.5) [dashed (red) line] in the log-log scale with an exponent $-3.0$. Events above 2.5 are outliers and create the humpy dragon-king.

Figure 3

Extreme events in $k_1-k_2$ plane. Colored region denotes extreme events that cross the $H_S$ line. A range of colors indicates a count of events counted in a color bar in log scale. IRC in first quadrant and QRC in third quadrant prevail. Inset shows an enlarged view of the marked box.

Figure 4

Evolution of temporal dynamics of coupled neurons with increasing chemical synaptic interactions. (a–c) IRC: periodic bursting, intermittency, and chaos for $k_{1,2}=0.04,0.0532,0.07$, respectively. (d–f) QRC: periodic bursting, quasiperiodicity, and chaos for $k_{1,2}=-0.04,-0.06,-0.07$, respectively. Horizontal dashed lines denote the $H_S$ line.

Figure 5

Temporal dynamics of two mutually interacting HR systems via chemical synapses. (a) Two bursting variables $x_1$ and $x_2$ in black (red) and gray (blue), respectively, appear (upper panel) out-of-phase in time for $k_{1,2}=-0.07$, except three pairs of spikes firing in unison irregularly and forming three events in a $x_{\parallel }$ plot (lower panel). (b) Each spike in two bursts are out-of-phase most of the time for $k_{1,2}=0.07$, except two pairs of spikes that overlaps perfectly in-phase synchrony (upper panel) and reflected as extreme events in the lower plot of $x_{\parallel }$. Dashed horizontal lines indicate $H_S$.

Figure 6

Extreme events in two repulsively coupled HR systems. Oscilloscope pictures of voltage analog of $x_{\parallel }$ in time from electronic experiment in (a). Two bursting voltage $x_1$ in gray (cyan) and $x_2$ in white (yellow) in upper traces in (b); $x_1$ and $x_2$ (upper traces) are in antiphase burst synchronization except once two spikes in white (yellow) and gray (cyan) coincide in-phase (upper traces) when one large event in gray (red) is seen in the voltage analog of $x_{\parallel }$ (lower trace).

Figure 7

Experimental PDF of events in log-log scale. Event sizes $1.1<|x_{\left|\right|}{|}_n<2.4$ follow a power law that fits a dashed line (red) of slope $\approx -3.25$. Larger events are outliers giving a humpy dragon-king-like distribution.

Figure 8

Temporal dynamics of $x_{\parallel }$ for purely inhibitory coupling. Varying $k_1$ =$k_2$=$k$. Upper row left and right panels for $k=-0.07,-0.08$, respectively. Lower row left and right panels are for $k=-0.2,-0.35$, respectively.

Figure 9

Temporal dynamics of $x_{\parallel }$ is given for purely attractive coupling. Left and right panels are for $k_1=k_2=0.18$ and 0.35, respectively. The large events occur more frequently giving a rise of the threshold level $H_S$ (red dashed line) above all the events.

Figure 10

PDF of events for $k_1=k_2=0.35$. The dashed vertical line (red line) is the threshold of extreme events showing zero probability of any event.

Figure 11

Two HR systems coupled via repulsive diffusive coupling. (a) Temporal dynamics of $x_{\parallel }$ for $k_{1,2}=-0.018$. Horizontal dashed (red) line depicts $H_S=2.53$. (b) $x_1$ vs. $x_2$ plot, a projection of the APS manifold. The error dynamics of the coupled system is confined to APS manifold in a dense black (blue) region, but occasionally moves out along the transverse direction (in-phase synchrony).

Figure 12

Schematic circuit diagram of a single HR system.