Table 1.
Typical values for the Morris-Lecar model, Equations (7)–(8).
Fig 1.
Action potential within Morris-Lecar model.
A stimulus is applied to the resting cell at t = 400ms. Corresponding behavior of recovery variable W(t) is shown in the inset.
Fig 2.
Time series and phase space diagrams for modified Morris-Lecar model.
Upper left: Single neuron (α = 0.7, V0 = 6.2mV) firing once after stimulus is applied at t = 0ms and approaching a stable equilibrium. Upper right: Single neuron (α = 1 and V0 = 6.2mV) entering a stable limit cycle after initial stimulus. Phase space trajectories for each case are shown in the panel below the corresponding time-series plot.
Fig 3.
Map in (V0, α) space of resonant frequencies using typical parameters for Morris Lecar model.
Regions colored in white correspond to points where no stable limit cycles exist in the absence of a stimulation current.
Fig 4.
Bifurcation diagrams in extended parameter space.
These diagrams, created using Xppaut, depict the occurrence of stable, equilibrium points (solid black line), unstable equilibrium points (dashed black line), stable limit cycle (thick green line) and unstable limit cycle (thin, red line) as α (left) and V0 (right) are varied. From left to right in the left panel, a stable equilibrium branches into an unstable limit cycle through a subcritical Hopf bifurcation point near α ≈ 1. As α is increased, a stable limit cycle emerges which collapses into a stable equilibrium at the supercritical Hopf bifurcation point near α ≈ 1.5.
Fig 5.
Limit cycle and quiescent state achieved by stimulation.
Left: Auto-generation of pulses in modified Morris-Lecar system with α = 1, V0 = 6.2mV. Right: An initial stimulus of 80 μA/cm2 applied for 5ms is sufficient to pull the system off the limit cycle to a stable equilibrium.
Fig 6.
Vulnerable window in the modified Morris Lecar model depicted in phase space.
The system’s limit-cycle trajectory for α = 1, V0 = 6.2mV is shown (thin black line) with the vulnerable region indicated by a thick red line for varying values of V0 with α = 1. Within the vulnerable window, a short stimulation takes the system from its stable limit cycle to a stable equilibrium point in phase space. The size of the vulnerable window increases as (V0, α) approaches the Hopf bifurcation point.
Fig 7.
Long-time steady states for cables of varying lengths.
Fixing all parameters except cable length generically yields a region of vulnerability in extended cables. For small (left) and large (right) cables, an initial stimulation sufficient to quiesce a single cell causes a transient quiescent region which results in full synchronization. For a range of cable lengths (center) the stimulation results in the entire cable approaching an equilibrium state.
Fig 8.
Variation of diffusion constant for fixed cable size.
The similarity of these results to those in Fig. 7 shows results consistent with the continuum predictions.
Fig 9.
Larger stimulation current (i0 = 800μA/cm2) is used for a shorter time (T0 = 50Δt) in the middle of the cable.
The modified stimulation protocol produces qualitatively similar results to those shown in Fig. 7
Fig 10.
Vulnerable window of the one-dimensional cable shown in phase space.
The system’s limit-cycle trajectory for α = 1, V0 = 6.4mV is shown (thin black line) with the vulnerable region indicated by a thick red line. Localized stimulations applied within this window result in a long-time quiescent state for the entire system. Regions of instability are shown in dashed blue line. For all other points (thin black line), localized stimulations only give rise to transient effects, and the entire cable eventually returns to synchronized oscillations.
Fig 11.
Complex spatiotemporal pattern generated with increased stimulation time near the crossover between quiescent and synchronized steady-states.
A standard, second-order stencil was used for evaluation of the spatial derivative.
Fig 12.
Emergent bursting in the center cell.
Time series for the membrane potential V at the center cell in the one-dimensional cable shown in Fig. 11.
Fig 13.
Complex spatiotemporal pattern generated with increased stimulation time near the crossover between quiescent and synchronized steady-states.
A fourth-order, five-point stencil was used for evaluation of the spatial derivative. Compare to Fig. 11.