Reverse-engineered models reveal differential membrane properties of autonomic and cutaneous unmyelinated fibers

Abstract

Unmyelinated C-fibers constitute the vast majority of axons in peripheral nerves and play key roles in homeostasis and signaling pain. However, little is known about their ion channel expression, which controls their firing properties. Also, because of their small diameters (~ 1 μm), it has not been possible to characterize their membrane properties using voltage clamp. We developed a novel library of isoform-specific ion channel models to serve as the basis functions of our C-fiber models. We then developed a particle swarm optimization (PSO) framework that used the isoform-specific ion channel models to reverse engineer C-fiber membrane properties from measured autonomic and cutaneous C-fiber conduction responses. Our C-fiber models reproduced experimental conduction velocity, chronaxie, action potential duration, intracellular threshold, and paired pulse recovery cycle. The models also matched experimental activity-dependent slowing, a property not included in model optimization. We found that simple conduction responses, characterizing the action potential, were controlled by similar membrane properties in both the autonomic and cutaneous C-fiber models, but complicated conduction response, characterizing the afterpotenials, were controlled by differential membrane properties. The unmyelinated C-fiber models constitute important tools to study autonomic signaling, assess the mechanisms of pain, and design bioelectronic devices. Additionally, the novel reverse engineering approach can be applied to generate models of other neurons where voltage clamp data are not available.

Fig 1

Comparison of our isoform-specific model of Na_V1.8 (solid line) and the Schild and Kunze Na_S model (dashed line) [14] to experimental data (markers and grey shading) [–22]: a) steady-state gating parameter values, b) activation time constant, d) fast-inactivation time constant, e) slow-inactivation time constant, c) novel Na_V1.8 model current responses to voltage clamps, and f) Schild Na_S model current responses to voltage clamps.

Fig 2. Particle swarm optimization (PSO) algorithm to identify parameter values of model C-fiber axons and meet performance criteria.

The evaluation, election, and update steps are repeated until termination criteria are met.

Fig 3. Conduction responses in optimized C-fiber models (PSO Autonomic and PSO Cutaneous) and published C-fiber models compared to experimental recordings (Table 1).

a) Conduction velocity (left: autonomic, right: cutaneous), b) action potential duration (left: autonomic, right: cutaneous), c) intracellular threshold (cutaneous), d) strength-duration (autonomic), e) strength-duration (cutaneous), f) threshold recovery cycle (autonomic), and g) threshold recovery cycle (cutaneous). In panels a to c, experimental ranges are shown as grey shaded regions and the error bars and data points represent the conduction responses for three simulated fiber diameters (center = 1 μm, range = [0.5 μm, 1.5 μm]). In panels d to g, the data points denote experimental data, and the shaded areas correspond to the range of conduction responses from the three simulated diameters; the lines for the previously published models correspond to the conduction responses for 1 μm fibers.

Fig 4

Activity-dependent slowing of action potential conduction in optimized C-fiber models with 0.5, 1, 2, and 4 Hz stimulation compared to experimental data with 2 Hz stimulation for a) autonomic [35] and b) cutaneous [4] C-fibers. Lines show data for 1 μm C-fiber models; shaded regions show range for 0.5 to 1.5 μm C-fiber models. Note: the % change in conduction velocity for a) autonomic and b) cutaneous C-fiber models differ in scale.

Fig 5

Transmembrane potential (a and c) and proportions of total ionic current (b and d) during the action potential and afterpotentials for the autonomic (left) and cutaneous (right) C-fiber models. In b and d, inward currents are shown as a negative proportion of the total current and outward currents are shown as a positive proportion of total current. The three vertical dashed lines correspond to the action potential initiation, action potential peak, and action potential shoulder. Only the dominant ion currents are visible, but all ion channels were included in the simulated models.

Fig 6. Sensitivity of autonomic C-fiber conduction responses to model parameters.

Sensitivity of a) conduction velocity, b) chronaxie of the strength-duration curve, c) action potential duration, and d) recovery cycle root mean square (RMS) error compared to experimental data (see Methods) are shown for 50% changes in parameter values (max conductance, max pump current, or Ra) for 1 μm autonomic C-fiber models. Bolded ion channels and pumps are the minimum set of ion channels and pumps needed to reproduce the experimental conduction response. When the ion channels underlined in red were removed, the model no longer matched the experimental data.

Fig 7. Sensitivity of cutaneous C-fiber conduction responses to model parameters.

Sensitivity of a) conduction velocity, b) chronaxie of the strength-duration curve, c) action potential duration, d) recovery cycle root mean square (RMS) error compared to experimental data (see Methods), and e) intracellular activation threshold are shown for 50% changes in parameter values (max conductance, max pump current, or Ra) for 1 μm cutaneous C-fiber models. Bolded ion channels and pumps are the minimum set of ion channels and pumps needed to reproduce the experimental conduction response. When the underlined ion channels were removed, the model no longer matched the experimental data.

Fig 8. Geometry and electrical properties of C-fiber models.

a) Fiber geometry with 8.33 μm-long segments and 30 nm periaxonal space. b) Cable model representation of a single segment, showing ion channels and pumps present in the C-fiber membrane.