If both Na and K channels are deficient around the same site, perhaps owing to local cytoskeletal disruption, a blended defect is formed. This defect may become a stimulated pulse generator, which will generically elicit APs on the same side of the stimulation (figure 2(c)). However, some defects can elicit APs on the opposite side (figure 2(d)) or with different frequencies on different sides (figure 2(e)). Under incommensurate frequencies, the latter dynamics will
become a quasi-periodic motion distinct from the three phases in figure3(a). Some exceptional defects cease firing after a particular number of pulses and behave like a bursting generator (figure2(f)). Moreover, some defects will trap an approaching AP and freeze the depolarization at the defect site (figure 2(g)). This dynamical system has at least two fixed-point attractors, corresponding to two different resting potentials. Some defects resemble an elastic scatter (right in figure 2(h)), where the time lag during the collision is reminiscent of a soft wall collision.
Some defects resemble a signal rectifier, allowing a leftward-moving AP to pass through, but blocking a rightward-moving AP (left in figure 2(h)). Some defects resemble a switch, with a low Vm hump state (ON) that can be turned to the high Vm hump state (OFF) by a signal. The ON state allows an AP to pass, whereas the OFF state blocks all APs (left in figure 2(i)). That is, such defects allow at most only one signal to pass through.
In these patterns, one has to distinguish (A) the pulses induced by the defects from (B) the pulses generated externally to stimulate and probe the defects. In figures1and2(a), the pulses belong to (A) and any initial neural state will evolve to such a kind of cyclic motion, irrespective of whether an external pulse is launched to stimulate the defect or not. In figures 2(b)–(f), the pulses before and after around 10 ms belong to (B) and (A), respectively. If the defects in figures 2(b)–(f) are absent, an external pulse is not able to trigger any ceaseless firing and any initial neural state will converge to the resting state. If they are present, an initial state will converge either to the resting state or cyclic pulse emission, depending on the basins where the initial state is located. The initial states with incoming pulses (those before about 10 ms) in figures2(b)–(f) are examples that evolve to cyclic pulse emissions. In figures2(g)–(i), the defects are not pulse generators. After the external pulses of type (B) pass through those defects, the systems converge to different resting states. Note that an initial state here can have two different interpretations. Firstly, it can be a state around the defect area and the externally generated pulse is not a part of the state. Secondly, it can be a state of the whole neuron and different external pulses belong to different inertial states. The first interpretation has a vivid physical picture, while the second one is mathematically more rigorous.
The contour patterns demonstrated in figure2reveal several basic types of defect dynamics generated by combined Na and K channel density heterogeneities. Some of these patterns are easy to understand. For example, our intuition might tell us that the one-sided pacemaker in figure 2(c) is created by a two-sided pulse generator with a pulse suppressor on its right-hand side to extinguish rightward pulses. Indeed, an Na defect is closely adjacent to a K defect, as shown at the bottom of figure2(c). However, a slight modification on each of these defects could largely change the pattern. It is generally hard to predict beforehand which pattern will come out of a given defect combination. Some of the basic defect dynamics in figure2are reminiscent of the fundamental neurocomputational properties generated, for instance, by Izhikevic’s simple model [32]. Examples include the typical bursting and pacemaker-like dynamics, which can be found in both the simple model and our systems with channel density defects. Nevertheless, while the former is controlled by the input current, the latter is tuned by the defect shape. Note that the neuron lengths considered in figure2are sufficiently long, so that most patterns in the panels are robust against small variations of the parameters γNa, σNa, γK, σKand d. Exceptions are the defect in (f) and the reflectors in (h) and (i), whose parameters still slightly depend on the selected neuron lengths.
The blended defects in figures 2(c)–(i) can be characterized by the widths σNa and σK, the ratios γNa andγK of the Na- and K-deficient regions and the position d of the K depletion center relative to the Na depletion center. ForγNa= γKandσNa= σK, this five-parameter space
Figure 4. Phase diagram for blended defects of Gaussian (solid) and square (dash) weights. The four red curves denote the Hopf bifurcations at d = 10, 9, 8 and 7.5 × 104µm (left to right). The seven blue curves represent the transit–extinction transition for a leftward-moving AP at d = 10, 7.5, 5, 3, 2, 1.1 and 0 × 104µm, whereas the green curve represents the same transition for a rightward-moving AP at d = 5 × 104µm. The four black dashed curves indicate the Hopf bifurcations at d = 5, 4.5, 4 and 3.5 × 104µm (left to right) for square defects. The outer cyan extinction funnel and the inner orange Hopf bifurcation funnel are formed by lifting the blue and red curves, respectively, to the corresponding d’s.
is projected onto a three-dimensional diagram in figure 4. The red curves on the(γ, σ ) plane correspond to the Hopf bifurcations at different d’s. The (γ, σ ) values to the left of each red curve have a resting potential solution, whereas those to the right indicate the spontaneous emission of APs. The blue curves denote the transit–extinction phase boundary in analogy to that in figure 3(a). The (γ, σ) values to the left of each blue (green) curve allow a leftward-(rightward-)moving signal to pass, whereas those to the right block transmission. The existence of the non-zero narrow area enclosed by the blue and green curves at the same d, say d = 5 × 104µm, allows us to find the rare rectifier defect (left in figure2(h)).
Apparently, the spontaneous pulse generator regime denoted by phase (II) in figure 3(a) shrinks to a smaller area in figure 4and the stimulated pulse generator regime represented by phase (III) in figure 3(a) even disappears in figure 4. This excitability suppression is due to the involvement of the Na channel deficiency in the blended defect. The suppression extent is significant for Gaussian weights with long tails (red solid curves in figure 4), but weak for square weights with sharp boundaries (black dashed curves in figure 4). Lifting the blue and red curves to the corresponding d’s results in the formation of an outer cyan extinction funnel and an inner orange Hopf bifurcation funnel plotted in the three-dimensional space (figure4).
While the former converges to the rightmost blue curve on the(γ, σ ) plane at d → 0, the latter disappears in the same limit. The disappearance at d ≈ 0 is because the Na- and K-deficient regions almost overlap each other and the depolarization effect of the K deficiency is nearly completely compensated by the hyperpolarization effect of the Na deficiency. In this limit, the blended defect cannot become a pulse generator and the system looks like it is defect-free.
But it differs from the defect-free case in that this defect can extinguish an approaching signal.
The size of the extinction funnel indicates the suppression ability of the blended defect or, more explicitly, how likely a defect chosen from figure 4 can extinguish an oncoming signal.
Moreover, the size of the Hopf funnel represents the defect excitability or how likely it is that a chosen blended defect will be a pulse generator.
If the restriction σNa= σK is lifted, the shapes of the two funnels in figure 4 will vary with σNa and σK independently. ForσK< σNa, the blended defect becomes less excitable and the whole Hopf funnel under a given σNa will shrink to zero at σK→ 0. This limiting case corresponds to a pure Na channel density defect, which will never generate any pulses. In contrast, for σNa< σK, the defect becomes more excitable and the Hopf funnel under a given σKwill deform to a d-independent perpendicular funnel atσNa→ 0. In that case, only a pure K channel density defect is present. The behavior caused by this defect is obviously independent of its distance d to any virtual Na density defect of zero width. The projected curve of that d-independent funnel on the(γK, σK) space is exactly the red solid boundary of phase (II) in figure3(a). Interestingly, the stimulated emission denoted by phase (III) in figure3(a) does not exist in the phase diagram of blended defect in figure 4, but appears and grows up gradually in the limitσNa→ 0 when the suppression effect of the Na density defect diminishes. Finally, the funnel deformation trends under different defect relative depths,γNa6= γK, are similar to those underσNa6= σK.