The asymmetry of a defect can affect other neural transport properties such as the signal propagation speed. This effect can be readily detected by measuring the speed of an AP running on a repeated asymmetric defect like a ratchet, as the triangular example of period L and ratio γ in the inset of figure 5. The main plot of figure 5 shows that this speed will rise slightly above the standard AP speed on a defect-free neuron (green line) for the K ratchet and decline considerably below that for the Na ratchet. For L larger than the typical AP width, around 2 × 104µm, of the squid giant axon, the AP stays mostly on a rising or falling ramp during its running time. When L exceeds the values marked by the crosses, the AP will be extinguished because the channel densities there are too dilute to support signal propagation. The crosses appear only for those channel density defects with a large γNa close to 1. It reflects again the fact of abundantly expressed ion channels, as the large γ for observing phases (II) and (III) in figure3(a) indicates. A clear trend illustrated in figure5is that a leftward-moving signal (blue) always propagates faster than its rightward-moving (red) counter signal on an Na ratchet. The speed difference can reach 1 m s−1 for γNa= 1. It is comparable with the conduction velocity difference 12–18 m s−1 under a temperature variation of 6.3–18.5◦C [33]. For infinitely dense ratchet zigzags at L → 0, the ramp period shrinks to zero. The traveling AP will behave like a shape-invariant soliton running on a defect-free neuron. Nevertheless, its speed will differ from the defect-free speed (green line), since the total channel amount at L → 0 is less than that of the defect-free case if γ > 0. How the speed approaches its limiting value at L = 0 cannot
0 0.5 1 1.5 2
Figure 5. In the inset, a rightward-moving signal (red) and a leftward-moving signal (blue) travel on a ratchet defect of ratio γ and period L. These traveling signals are externally generated and not induced by the ratchet defects. The main plot depicts how the rightward-moving (red) and leftward-moving (blue) speeds vary withγ and L. Therein, γNaandγK marked on the curves denote the defect ratios of the corresponding Na and K ratchets, respectively, and the green straight line represents the defect-free speed.
be solved numerically, owing to the increasing non-smoothness of the differential equation at L → 0. Consequently, the curve information is missing at small L in figure5.
3. Discussion
A neuron is an intriguing biological structure, which is frequently integrated into and/or manipulated using manmade solid devices for applications in bionanotechnology and neural engineering [34]. For instance, nanowire arrays have been put on neurons for non-invasive and highly sensitive detection, stimulation and inhibition of neuronal signal propagation [35].
Carbon nanotubes have been integrated with neurons to promote neuronal electrical activity [36]. Furthermore, reconstitution methods have been developed for inserting voltage-gated ion channels into cell-sized giant unilamellar vesicles, allowing for the study of the electric dynamics of different ion channel distributions [37]. Clearly, the effects of inhomogeneous channel density demonstrated above may further enrich the manipulation versatility in these newly emerging fields. Experimentally, it should not be difficult to observe the neural dynamics predicted in this study, since the defects considered here are about a centimeter wide and located on a squid giant neuron, which is typically several centimeters long. A defect of this scale can be precisely positioned and controlled, for instance, by micropipettes filled with solutions of channel blockers or filament depolymerization agents [10]. The injected solutions will diffuse
and most likely generate a density defect close to a Gaussian form as depicted in figures 1 and 2. Such an experiment should also be feasible for other thinner neurons, as long as the channel density perturbation can be controlled locally.
In addition to applications in engineering, the present results might provide information on some neuronal diseases. It is well known that improper channel localization can cause communication problems in neuronal networks [18], and channel dysfunction can lead to diseases in many tissues [38]. However, most reports on such diseases are limited to macroscopic descriptions and seldom discuss their origin at the subcellular level. In fact, the channels in some diseases could be non-uniformly crippled, blocked or transported during the channel trafficking process. Thus, some abnormal behaviors might be a combined effect of the several basic dynamical patterns shown in figure 2. For instance, recent studies have demonstrated that a reduced expression of A-type K channels in primary sensory neurons led to mechanical hypersensitivity [39] and that genetic elimination of the K channel Kv4.2 in the dorsal horn neurons enhanced sensitivity to tactile and thermal stimuli [40]. Unless the channels on neurons are uniformly eliminated, this reduction might cause hypersensitivity phenomena similar to the irregular firing occurring after the K channel density defect demonstrated in figure 1. To compare other patterns in figure 2 with possible channel density-related neural diseases, subcellular measurements of the distributions of channels, or their anchoring proteins [41], are crucial. Upon comparison, we still need to note that the patterns in figure2 are for unmyelinated squid giant axons. If a channel density disease occurs on a myelinated axon, its dynamics will be more complex, since the system contains more degrees of freedom.
Nevertheless, some complex dynamics might share common properties with and could be more easily understood through the simpler patterns in figure2.
Another intriguing comparison is between the current findings and those in other excitable systems. It is widely known that heterogeneity is a big issue of concern in excitable systems such as cardiac tissues. Therein, heterogeneous cellular types in a multicellular system can cause active activities. For instance, modeling studies have shown that passive defects, such as non-excitable fibroblasts, coupled to excitable ventricular myocytes can cause spontaneous oscillations [42–44], which has been demonstrated in experiments as ectopic excitations [46,52]. Normal ventricular myocytes coupled to ischemic myocytes cause spontaneous oscillations [47, 48] due to elevated resting potential of the ischemic cells. In the present study, reducing the local density of K channel in the HH model also elevates the local resting potential. Thus, the pacemaker patterns of neurons in figures1and2(a) can be regarded as a subcellular version of the oscillatory behaviors of heterogeneous tissues. However, to our knowledge, other patterns in figure 2have not been reported in tissues. It might be ascribed to two differences between the current neuron system and usual tissue models. Firstly, whereas the voltage dynamics in figure 2 evolves in a continuous space along the axon, that in usual tissue models lives in discrete networks. Secondly, usual tissue models do not contain cell types playing the role of an excitability suppressor such as the Na channel defect in neurons. From this aspect, the dynamics of excitable neurons seem to be more versatile than those of excitable tissues.
4. Conclusion
In summary, the present study systematically demonstrates how the physical properties of a biological mesoscopic system will be changed by a local defect. Generally, a local defect is
an inhomogeneous position deviating from a uniform phase. This slight deviation usually has a dramatic impact on the transport properties and wave propagation of a system, especially a quasi-one-dimensional system. Intuitively, a defect would be expected to suppress the mass and energy flow and to monotonically reduce the conductance or information transfer through a system; this is indeed true for normal solid materials, which are usually non-excitable [49–53].
However, how a defect will change the behavior of an excitable one-dimensional cable, such as a neuron, is highly non-trivial and hard to predict. To investigate this, we focused on a family of basic local ion channel density perturbations in this study. A systematic survey of the defect parameter spaces revealed several fundamental short- and long-term defect dynamics in the HH model and yielded the critical defect size for qualitatively different neural dynamics. Our results may (i) serve as a simple example illustrating the typical defect dynamics in general excitable cables, (ii) give hints on how to utilize defects for manipulating neuronal signals for biotechnological applications, (iii) quantitatively elucidate disease-related irregular firings at the subcellular level and (iv) enrich our understanding of the challenging issue of channel density heterogeneity-induced neural behaviors [18]. We expect that the present study will stimulate further theoretical and experimental works in these areas.
Acknowledgments
We thank Cheng-Chang Lien and Chung-Chuan Lo for helpful discussions and acknowledge support from the National Science Council in Taiwan through grant no. NSC 100-2112-M-009-003.
Appendix
Here we briefly introduce the Hodgkin–Huxley model of heterogeneous ion channel densities used in our calculations. Generally, neurons contain numerous types of voltage-dependent ion channels; in contrast, the squid giant axon has only Na and K ion channels. Let d, Ri and Cm
denote the axon diameter, specific resistance of the axoplasm and transmembrane capacitance of a squid giant axon, respectively. The dynamics of the transmembrane voltage Vm(x) at position x along the axon can be described by a resistor–capacitor circuit equation furnished with a diffusion term caused by current leakage [54],
∂Vm
∂t = 1 Cm
d 4Ri
∂2Vm
∂x2 − (INa+ IK+ IL)
, (A.1)
where the Na, K and leakage (L) transmembrane currents INa, IKand IL follow the Ohm’s law IS = gS(Vm− VS) with the equilibrium potentials VS and the conductances gS, where S stands for Na, K and L. These conductances are voltage dependent and given by
gK(Vm, t) = ¯gKn4(Vm, t), (A.2)
gNa(Vm, t) = ¯gNam3(Vm, t) h(Vm, t), (A.3)
gL(Vm, t) = ¯gL, (A.4)
where the constants ¯gNa and ¯gK denote the maximum conductances when the Na and K channels, respectively, are completely open and ¯gL represents the voltage-independent leakage conductance. The gating probabilities m, n and h describe how the fast and slow gates of the
Na ion channel, as well as the gate of the K ion channel, change with Vm, respectively. They follow the first-order kinetic
dX
dt = αX(1 − X) − βXX (X = m, n, h) (A.5)
with the empirical Vm-dependent rate constants αm= −0.1(Vm+ 40)
e−0.1(Vm+40)−1 , βm= 4e−(Vm+65)/18, αh = 0.07e−0.05(Vm+65), βh= 1
1 + e−0.1(Vm+35), αn = −0.01(Vm+ 55)
e−0.1(Vm+55)− 1 , βn= 0.125 e−0.0125(Vm+65).
Equations (A.1)–(A.4) form the original four-dimensional HH equations on the(Vm, m, n, h) space. Since the system described by this equation is homogeneous in space, how an AP evolves is independent of where it is initiated. This translational symmetry will be broken by a local perturbation like a channel density defect. To describe such inhomogeneous channel distributions, the constants ¯gSin (A.2)–(A.4) are multiplied by some position-dependent weights wS(x),
gK(Vm, t, x) = wK(x) ¯gKn4(Vm, t), (A.6) gNa(Vm, t, x) = wNa(x) ¯gNam3(Vm, t) h(Vm, t), (A.7)
gL(Vm, t, x) = wL(x) ¯gL. (A.8)
These weight functions lying between 0 and 1 characterize to what extent the conductance is reduced at x. For wS(x) ≡ 1 for all x, the gS in (A.6)–(A.8) return to those conductances in (A.2)–(A.4) in the original HH equations for homogeneous axons. If the conductance is linearly proportional to the density of the functioning channels, wS(x) represents the percentage of the density of these channels at x over the uniform channel density in the original HH model. In the calculation, the following standard parameter values were used:
Ri = 35.4 cm−1, d = 476 µm, ( ¯gNa, ¯gK, ¯gL) = (120, 36, 0) mS cm−2, Cm= 1 µF cm−2 and (VNa, VK, VL) = (50, −77, 0) mV.
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