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Fig. 1. Structural–kinetic (S–K) phase diagram of excitability status in the Hodgkin–Huxley model. (Left) Ten thousand realizations of a full Hodgkin–Huxley model, following the procedure described in Ori et al. (4). ĜX is the maximal membrane conductance to ion X relative to the Hodgkin–Huxley standard model values G¯X, where G¯Na=120 mS/cm2 and G¯K=36 mS/cm2. k^ is a linear scalar of ionic channel transition rate function. The subscripts of k^res and k^ex depict transition rate functions contributing to restoring and exciting forces: αn(V) and βm(V) for the former and αm(V) and βn(V) for the latter. Parameters (maximal sodium and potassium conductance, leak conductance, membrane capacitance, and the six rate equation functions) vary randomly and independently over a ±0.25 range compared to their values in the original Hodgkin–Huxley model. The resulting membrane responses to above-threshold stimuli are classified (different colors) to three excitability statuses: excitable (2,225; blue), not excitable (4,884; orange), and oscillatory (2,891; green). (Right) Dataset of Left replotted with colors depicting response peak amplitude clusters, classified to four bins indicated in the horizontal color bar.
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Fig. 2. Dynamic clamp constructed phase diagram of excitability in naive (i.e., no channel mRNA injected) Xenopus oocyte. (A) Equivalent electrical circuit of the basic system configuration (Top and Middle). An oocyte is impaled by two sharp electrodes. Signals from the voltage measuring electrode are read by a real-time processor that calculates Hodgkin–Huxley sodium and potassium currents, feeding the sum of these currents back to the oocyte through the current injecting electrode. The oocyte contributes membrane capacitance and leak conductance; the dynamic clamp algorithm contributes sodium and potassium voltage-dependent conductance. Excitability may be induced (Bottom) upon activation of the dynamic clamp algorithm. The nature of the system’s response to stimuli (in this case, 0.5 ms, 10 μA; Methods) depends on the scaling of the Hodgkin–Huxley parameters implemented in the dynamic clamp algorithm (Bottom; ĜNa= [0,10], α^m(V)= [0.75,1.25]; see Results and Discussion for scaling notation). (B) S–K phase diagrams. The ranges of S and K accessible for scanning by dynamic clamp differ between experiments; they are dictated, mainly, by leak and capacitance contributed by the oocyte and affected by the quality of electrode–membrane interactions. In both experiments, soft but relatively well-defined borders separate nonexcitable from excitable phases (colors depict peak response amplitude clusters, classified to four bins indicated in horizontal color bars). The diagrams were constructed under different conditions: in Top, the parameters of sodium and potassium conductance were taken from the Hodgkin–Huxley canonical model. In Bottom, the potassium conductance parameters are those of the Kv1.3 channel (22). Both experiments were conducted in the same oocyte.
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Fig. 3. Dynamic clamp and excitability in Kv1.3 mRNA-injected Xenopus oocyte. (A) Taking the experimental system described in Fig. 2 a step further by relocating the voltage-dependent potassium conductance from the dynamic clamp algorithm into the biological domain (Top). This is achieved by injection of mRNA that codes the voltage-dependent Kv1.3 potassium channel (Middle). Within a couple of days, the channels are extensively expressed in the oocyte membrane. Upon activation of the dynamic clamp, excitability emerges with biological capacitance, leak, and potassium conductance, while the sodium conductance and its related kinetics are computationally expressed. The nature of the system’s response depends on the scaling of the Hodgkin–Huxley parameters implemented in the dynamic clamp algorithm (Bottom; ĜNa= [0,16], α^m(V)= [0.5,1.6]; see Results and Discussion for scaling notation), as well as on the kinetics and density of the expressed Kv1.3 channels. (B) Maximal potassium conductance, potassium Nernst potential, and membrane capacitance are estimated by standard voltage-clamp procedures. (Top) Cell capacitance is estimated from the capacitive current step upon instantaneous switch from −2.66 V/s to +2.66 V/s (in a range between −70 and −110 mV). (Bottom) Tail current protocol (Inset) to establish potassium Nernst potential based on current reversal; depolarizations to +40 mV followed by hyperpolarization to different values. G¯K was estimated from maximal current and Nernst potential and validated using tail current data.
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Fig. 4. Dynamic clamp constructed phase diagram of excitability in Kv1.3-injected oocytes. Top Right, Top Left, and Bottom Left demonstrate phase diagrams of three different experiments, where biological capacitance, leak, and potassium conductance are contributed by the oocyte, while the sodium conductance and its related kinetics are computationally expressed in the dynamic clamp algorithm. Note S–K planes with relatively well-defined borders that separate nonexcitable from excitable phases. (Bottom Right) Summary of 12 experiments, showing that a positive slope diagonal separating nonexcitable from excitable phases (demonstrated in the three other panels) is consistent across experiments. To this end, a histogram of response amplitudes was generated for each experiment. Of the 9,889 responses in all 12 experiments, 529 responses were identified within an intermediate range [0.2, 0.8] of response amplitudes (the amplitude of the passive response to the stimulus was defined as zero). The S–K coordinates of these 529 responses are plotted in Bottom Right, together with a fitted straight line (S=0.7+0.4K; 99% confidence bands for mean predictions). Numerical symbols depict the different experiments. The slopes of the individual lines fitted separately for each of the 12 experiments are 0.66, 0.62, 0.62, 0.75, 0.58, 1.01, 0.49, 1.46, 0.67, 1.25, 2.2, and 0.73. Note that these slopes are significantly less steep compared to the slope of the Hodgkin–Huxley model reported by Ori et al. (4).
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Fig. 5. The adequacy of the S–K parameterization estimated by observing multiple instantiations of the same S–K coordinate. A Kv1.3 mRNA-injected oocyte (Cm oocyte= 0.33 μF, GLeak oocyte=0.035 mS, G¯Kv1.3 oocyte=0.4 mS) is coupled to a dynamic clamp algorithm where the Hodgkin–Huxley sodium, potassium, and leak conductance (G¯Na injected, G¯K injected, and GLeak injected, respectively) and their related kinetics are computationally expressed. A set of 3,000 instantiations was prepared, within the following ranges: G¯Na injected= [2.46, 4.95], G¯K injected= [0.40, 0.73] or set to zero (see below), GLeak injected= [0.06, 0.10], { α^m(V), β^m(V), α^h(V), β^h(V), α^n(V), β^n(V) } = [0.75, 1.25]. The membrane response to a 0.5-ms stimulus (14.5 μA) was recorded. Different instantiations were delivered at a rate of 5/s. Assuming α^Kv1.3(V)=β^Kv1.3(V)=1 and considering both potassium conductance (G¯Kv1.3 oocyte and G¯K injected), the values of S and K for each instantiation were calculated and binned to 0.02 resolution. For clarity, the total of 3,000 traces was down-sampled by a factor of 3, and each of the sampled responses is plotted in its corresponding S–K coordinate bin in Top Left. The brown colored traces are those corresponding to cases where G¯K injected=0; the red colored traces are those corresponding to cases where G¯K injected= [0.40, 0.73]. (Top Right) Contour plot of mean peak amplitudes for all binned S–K coordinates (3,000 spikes). (Bottom) Examples of several responses in four of the S–K coordinates (arrows depict stimulus time; traces were smoothed by moving average of 120 μs). The responses are fairly resilient to the actual set of parameters used to determine a given coordinate. Even delicate response features (e.g., the poststimulus subthreshold depolarization in the rightmost traces), not accounted for in the construction of the theory, are nicely caught by the S–K coordinates. Note that ringing in several of the fast–high-amplitude spikes (depicted by gray oval in Bottom) is due to limits imposed by the rate of the real-time processor and/or the current injection device (Methods).
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Fig. 6. Activity dependence and hysteretic dynamics due to the impacts of Kv1.3 slow inactivation kinetics. (Left) Membrane responses of a Kv1.3-expressing oocyte as a function of S–K coordinates in a directed walk within the diagram. A gradual (up and down ramp; total trajectory ca. 450 s) change is implemented in the dynamic clamp parameters of Hodgkin–Huxley sodium conductance (G¯Na and α^m(V)). The spikes are plotted over an S–K diagram constructed for that same oocyte. Each trace depicts membrane response to a short above-threshold current stimulation. As the system is moved toward the excitable phase (top left corner of the diagram), the oocyte responds to the stimulus with a self-propelled depolarization that becomes a fully blown action potential. At some point, as might be expected, the structural exciting force (S) is so high and the kinetic restoring force (K) is so low ({0.95, 0.42}, depicted by arrow in Right) that the membrane cannot hyperpolarize back to resting potential and remains stuck in some depolarized not excitable value. Upon return, a clear hysteresis is revealed, reflecting recovery of the Kv1.3 from long lasting inactivation. (Right, Insets) Two repetitions of the same ramp protocol in another oocyte, demonstrating reversibility of the hysteresis phenomenon.
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