Figure 1.
Tonic spiking, bursting, and silence are mapped onto the (,
) bifurcation diagram.
(A) Tonic spiking, bursting, silence, and multistability of tonic spiking and silence are supported in the corresponding parameter regions labeled ,
,
, and
. Bursting is described by duty cycle, which is the ratio of burst duration to cycle period. Duty cycle is represented as a color map from 0 to 100%. The three empty red circles mark sample parameters sets with duty cycle 10%. The grey curve indicates the position of the saddle-node bifurcation for periodic orbits. The black curve indicates the position of the saddle-node bifurcation for equilibria. Examples of waveforms of bursting activity at four different parameter sets (
,
): (B) (−0.0105
, 0.0413564925
), (C) (−0.0075
, 0.041326
), (D) (−0.0075
, 0.038
), and (E) (−0.0105
, 0.038
). The point at (B) is near but not on the codimension-2 bifurcation point.
Table 1.
Temporal characteristics of bursting for different parameter values in region .
Figure 2.
The inverse-square-root laws in transient responses triggered by a pulse of current for parameter values that support silent (region ) and tonic spiking (region
) regimes.
(A–D) An individual burst was triggered by a hyperpolarizing pulse of injected current. Pulses of current were 0.03 s in duration and 0.1 in amplitude. The duration of individual bursts were (A) (
−0.0077
,
0.0415
) 10.327403 s, (B) (
−0.01043
,
0.0415
) 103.48097 s, and (C) (
−0.010496
,
0.0415
) 309.27622 s. (D) The log-log graph of burst duration of individual bursts plotted against
. The blue dots correspond to the burst duration measured at the respective values of
. The red curve is the graph of the curve fitted in the form
. Coefficients of the curve fit are
0.97239439,
−8.48809474, and
0.01050536. (E–H) Latency to spiking was shown by administering an individual pulse of injected current. Pulses were 0.03 s in duration and 0.2
in amplitude. The delays shown here are (E) (
−0.0107
,
0.04134
) 10.287 s, (F) (
−0.0107
,
0.041358041
) 103.378 s, and (G) (
−0.0107
,
0.0413580468
) 317.679 s. (H) latency to spiking for sampled parameter values (blue dots) and the graph of the curve fitted to
(red curve) were plotted in log-log scale against the sampled values of
. Coefficients of the fitted curve were
0.00709613 and
14.47354286. The parameter
was 0.04135804734566
.
Table 2.
Measure of burst duration and latency to spiking in response to inhibition from Figure 2 (A−C,E−G).
Figure 3.
Diagram of the cornerstone bifurcation (CS).
(A) The CS is located at the intersection of the saddle-node bifurcation for equilibria (; red curve) and the saddle-node bifurcation for periodic orbits (
; solid blue curve). The dashed blue curve is
, where a large amplitude stable orbit is born. The solid green curve is a period doubling bifurcation. A series of period doubling bifurcations occur between this curve and the dashed green curve, where the large amplitude regime terminates. For values of
larger than where this regime terminates, we consider four adjacent regions of the parameter space. In the region marked
, we observe only a small amplitude orbit, which corresponds to tonic spiking. In
, a large amplitude orbit co-exists with the small amplitude orbit. In
, a the large amplitude orbit becomes chaotic and vanishes in a period doubling cascade. In
, the tonic spiking regime co-exists with a stable equilibrium. In
, the small amplitude orbit and the stable equilibrium co-exist with a large amplitude orbit. In
, the large amplitude orbit becomes chaotic and vanishes in a period doubling cascade. (B–F) Representations of the dynamics of the system at different points in the parameter space. The orange curves represent trajectories, and the black arrows indicate the direction of motion of the phase point. The two sets of light green and blue curves represent the maximum and minimum of orbits on the slow motion manifolds for oscillations. The green and blue portions indicate the attracting and repelling segments of this manifold, respectively. The solid and dashed purple curves correspond to stable and unstable equilibria in the fast subsystem, respectively. Filled red dots represent stable equilibria, and unfilled red dots represent unstable equilibria. Solid and dashed vertical dark green lines represent stable and unstable simple periodic orbits, respectively. (B) The structure of the state space at the CS point. A saddle-node periodic orbit exists on the slow motion manifold for oscillations, and a saddle-node equilibrium exists on the slow motion manifold corresponding to the equilibria of the fast subsystem. (C) A stable periodic orbit and a saddle periodic orbit exist on the slow motion manifold for oscillations. (D) Periodic bursting is observed. The phase point moves as indicated by the black arrows in a clockwise fashion. (E) A stable equilibrium of the full system obstructs the stable segment of the equilibria of the fast subsystem. (F) Spiking co-exists with the silent regime.
Table 3.
Cycle period, duty cycle, and values for and
. Labeled parameter sets were used to produce activity found in Figure 4.
Figure 4.
Scaling of the metachronal-wave pattern in chains of coupled endogenously bursting neurons.
Neurons connected through inhibitory coupling with strongest connections to the immediate anterior neighbor. Metachronal Waves produced by three examples of the network. The parameter values for (,
) used to create these trajectories were as follows: (A) (−0.0069999
, 0.041319316864014
), (B) (−0.0040999
, 0.041268055725098
), and (C) (0.005905
, 0.04073603515625
).
Figure 5.
Temporal characteristics of metachronal wave pattern across a range of cycle periods.
The phase relations of a metachronal-wave pattern are determined by the duty cycle of each element in a chain of inhibitory coupled bursting neurons. The parameters are changed in a coordinated fashion to support different cycle periods while the duty cycle is kept constant. (A) The phases of oscillators are portrayed relative to the oscillator in the seventh segment. (B) The average number of spikes per burst varies linearly with cycle period as and
are changed. The black markers indicated average spike number with error bars. The grey line was the linear function for spike number fitted to our data:
.