Figure 1.
Schematic of a mechanically-injured node of Ranvier
depicted with a mix of intact-looking and severely-blebbed axolemma (as labelled) such as seen in transmission electromicrographs of stretch-injured optic nerve nodes [2]. In pipette aspiration bleb injury, the cortical actomyosin-spectrin skeleton progressively detaches [6]. Our model considers a node as one equipotential compartment in which actual spatial arrangements of pumps and channels are irrelevant. However, the fraction of Nav channels in the injured portion of the membrane, along with the severity of their gating abnormality, are model parameters.
Table 1.
Parameters for node of Ranvier with pump.
Figure 2.
Nav-CLS induced spontaneous activities of injured node: steady state with fixed EIon, tonic spiking, tonic subthreshold oscillations (STOs).
By setting Imaxpump, gNaleak, and gKleak to zero and eliminating Eqs (11) and (12), the EIon are artificially maintained at fixed values. Three sets of values are considered: left column, ENa = 50 mV, EK = −77 mV; middle column: ENa = 42 mV, EK = −77 mV; right column: ENa = 42 mV, EK = −71 mV. For the first 3 rows, Nav-CLS have Gaussian distributions (mean±SD): (A,B,C) 1.3±0.4 mV; (D,E,F) 8±1 mV; (G,H,I) 15±1 mV. For the last rows,(J,K,L) bifurcation diagrams (solution of Vm in terms of LS) are plotted, and there, for computational tractability, single gNa populations (i.e. f = 1, no Gaussian “smears”) are used. As labeled, the solid line, dashed line, filled dots, open dots and “HB” respectively denote: stable fixed point (i.e. resting potential (RP)), unstable fixed point, stable limit cycle (SLC, i.e., tonically firing APs), unstable limit cycle (ULC) and Hopf bifurcation point (HB). When ENa alone changes (J to K), the bifurcation structure shows only slight changes in the amplitude of SLC and the locations of subcritical HB on both ends. When EK changes (K to L), the HBs on both ends shift to the left (i.e. towards relatively smaller ENa) and the previously subcritical HB (right side) becomes supercritical. Across the 3 columns, note that if the system did have pumps, interactions between the EIon and Ipump would continually and slowly change EIon thereby repeatedly evoking these activity patterns.
Figure 3.
Ectopic steady-state Vm excursions are tuned by Ipump.
(A) From the spontaneous tonic firing regime of Figure 9 in [11], a random point (LSi = [0,15]mV, fi = [0.5,0.5]) is selected; as illustrated, Vm oscillation amplitude varies as Imaxpump is set at 30 and 95 µA/cm2. (B) Ipump changes from a constant 10 µA/cm2 (associated with a small periodic Vm fluctuation) to a fluctuating 23±0.5 µA/cm2 (associated with a train of APs). (C) Corresponding total INa. (D) With growing Imaxpump, voltage excursions (blue) increase and their oscillation frequencies (red) decrease.
Figure 4.
Transitions between STO and burst behaviors.
(A) Upper: Vm (black solid line) at an injured node and the varying EIon (ENa and EK; blue dotted lines). Three gNa populations are used: LSi = [0,2,26.5]mV and fi = [0.72,0.08,0.2] with Voli = Volo = 10−15 m3. Initiation and termination times of a burst of spikes (pink star, green dot, respectively) are used in Figure 5. Lower: corresponding Na+ currents, as labelled. (B) Equilibrium values of Nav channel activation and inactivation variables, mi and hi. (C) The steady-state open probabilities, : with a mild injury gwindow(V) magnitude is slightly less at 0 mV (vertical line and circles) but much enlarged at voltages near the normal RP (−65.5 mV for fixed [ion] condition). (D) Another example: LSi = [0,2,20]mV and fi = [0.72,0.08,0.2] and Voli = Volo = 3×10−15 m3, with expanded detail showing STOs. Note the difference of time scales in (A) and (D), reflecting the fact that a 3-fold lower axonal surface-to-volume ratio in (D) slows the rate of [ion] changes.
Figure 5.
Burst dynamics explained with three-dimensional Vm trajectories and bifurcation diagrams
(for bursts in Figure 4A with Imaxpump = 95 µA/cm2). Note: x-axis scales are different is each plot. (A) The first burst of spikes in Figure 4A plotted as a function of time are now plotted in 3-D as a function of ENa and EK (blue arrow: direction of Vm trajectory). (B) Bifurcation diagram for fixed EK and Ipump as per Figure 4A pink star; ENa is the slowly varying parameter. Lines and points are labeled (see also abbreviation list) and have the same meanings in (C). For ENa at its Figure 4A pink star value (40.2 mV), the only stable solution is a periodic orbit (pink oval), with each cycle corresponding to a spike (in a burst). The oval-loop symbolizes one cycle of Vm oscillation at fixed ENa. During a burst the ENa decline shifts orbits leftward into the bistability regime (pink area) where there exist two stable solutions: a limit cycle and a stable fixed point. (C) Bifurcation diagram for EK and Ipump fixed as per Figure 4A green dot. For ENa at its Figure 4A green dot value (37.27 mV), the system (large green dot) is within the bistability regime (green area). Vm, attracted by this fixed point, has STOs (drawn as the green loop) until ENa increases and superthreshold-oscillations (spikes) return. The PD region corresponds to period-doubling bifurcations. (D) Two-parameter phase diagram for EK and ENa. Pink solid and dashed curves represent LP (saddle-node bifurcation) and HB, respectively, when Ipump is fixed as in B. The green solid, dashed, and dash-dotted curves represent LP, HB and PD, respectively, when Ipump is fixed as in C. With varying Ipump the bistability regime shifts from the pink to the green area (the zone with both colors is the overlap of these two areas). The gray area between two green dash-dotted curves is a zone with PD bifurcations. The black loop shows EK and ENa orbits during a burst.
Figure 6.
Classes of Nav-CLS-induced activities with two injured and one intact gNa population.
(A) Membrane excitability map of injured node with LSi = [26.5,2,0]mV. White, pink, green and blue regions represent different stable state activities: quiescence, bursts, tonic firing, and bistability between quiescence and tonic firing. Here the fraction of intact gNa is f3 = 1−f1−f2. (B) and (C), typical Vm trajectories when the system is in bursting and bistability regimes, respectively.
Figure 7.
Noise induces or modifies bursting.
(A) Without noise, bursts are as shown in Figure 6B but here, under the influence of noise (σ = 0.1 µA/cm2), the bursting periods are random. Right (here and in B): an expanded section showing STOs induced by noise via coherence resonance. (B) Without noise the system is in the bistability regime as shown in Figure 6C, but here, noise (σ = 0.4 µA/cm2) causes Vm to randomly visit quiescent or repetitive firing states, hence producing noisy bursts.
Figure 8.
Classes of Nav-CLS-induced activities.
(A) Nav-CLS of Gaussian distributions with mean±SD(LS) = [0.5±0.2, 1±0.4, 2.5±1.0, 5±5, 10±5, 20±5, 27±5, 30±5, 20±10]mV. (B)–(I) various activities of a node are demonstrated for increased injury (i.e. > mean(LS)): quiescence (B), burst (C,D), tonic firing (E,F,G), burst (H) and quiescence (I). Insets in (B) and (I) show the effect of injecting a 12 µA/cm2 stimulating current. (J) Doubling the SD(LS) in G from 5 to 10 mV slightly increases the tonic firing rate. (K) Instantaneous firing rate (inverse of the interval between two spikes) for tonic or within-burst spikes, versus mean(LS), corresponding to firing patterns C–H (see color coding). (L) Spike (tonic or intraburst) voltage excursions begin to decrease for mean(LS) >10 mV.
Figure 9.
Increased Imaxpump shortens bursting periods (BP) and burst duration (BD).
(A)–(E) Five values of Imaxpump as labeled elicited activities as shown with LS: Gaussian distribution, 2.5±1 mV (steady-state values in the absence of CLS injury are listed in Table 2). Note the different time scales in each panel. For a bursting node, Ipump is periodic with a period equal to BP. Inset in (A) shows the effect of injecting a 12 µA/cm2 stimulating current. (F) For direct comparison of BPs and burst durations, Ipump traces from panels A–E (color coding preserved) are plotted together.
Table 2.
Steady-states for intact nodes (Vm = −59.9 mV) as Imaxpump varies (units as in Table 1).
Figure 10.
Partial Nav-CLS in a population and apparently first order activation kinetics.
Equilibrium activation values (m3, and m ) of co-existing intact Nav and CLS Nav channels plotted as a function of Vm. Black solid and black dashed line represent m3 for intact and 20 mV CLS channels with f = 1. For a membrane with large Nav-CLS injury given by: LSi = [0,20]mV and fi = [0.3,0.7], the gray bold line denotes
. Red line: m of the intact channels.