Relating ion channel expression, bifurcation structure, and diverse firing patterns in a model of an identified motor neuron

Abstract

Neurons show diverse firing patterns. Even neurons belonging to a single chemical or morphological class, or the same identified neuron, can display different types of electrical activity. For example, motor neuron MN5, which innervates a flight muscle of adult Drosophila, can show distinct firing patterns under the same recording conditions. We developed a two-dimensional biophysical model and show that a core complement of just two voltage-gated channels is sufficient to generate firing pattern diversity. We propose Shab and DmNa v to be two candidate genes that could encode these core currents, and find that changes in Shab channel expression in the model can reproduce activity resembling the main firing patterns observed in MN5 recordings. We use bifurcation analysis to describe the different transitions between rest and spiking states that result from variations in Shab channel expression, exposing a connection between ion channel expression, bifurcation structure, and firing patterns in models of membrane potential dynamics.

Fig. 1

Different electrophysiological profiles recorded from Drosophila MN5. Intracellular recordings were made in whole-cell patch configuration from MN5 in wild-type adult Drosophila. At least four different firing behaviors can be observed in response to a 500 ms square-pulse current injection of 0.4 nA (black traces) or 0.8 nA (red traces). (A) Low-amplitude current stimulation elicits a single action potential (black trace), while increased stimulation elicits repetitive spiking that adapts in frequency (red trace). (B) Repetitive firing is elicited by low-amplitude stimulation, but commences only after a delay (black trace). Increasing the current amplitude eliminates the delay and increases the firing frequency (red trace). (C) Repetitive firing without a significant delay is induced even by low-amplitude current injection (black trace), while increasing the amplitude increases the firing frequency (red trace). (D) Low-amplitude stimulation produces a large initial spike, followed by triangular-shaped spikes which diminish in amplitude and broaden as stimulation continues (black trace). Larger stimulation amplitudes induce dampening oscillations that end in depolarization block (red trace)

Fig. 2

Steady-state curves resulting from variation in Shab channel expression. The steady-state current I_∞ (υ) was calculated according to Eq. (11) for different maximum amplitudes of the Shab current relative to the DmNa_υ current (a_K). The value of a_K was varied between 1 and 5, as indicated above the corresponding curve

Fig. 3

Bifurcation diagrams for different levels of Shab channel expression. Curves are comprised of the υ-values of the fixed points (υ_*) as a function of the current stimulation amplitude (Is). Note that the shapes of these curves are identical to the shapes of the I_∞(υ) curves (Fig. 2), since the curve of υ-values is obtained by calculating the zero crossings of I_s – I_∞(υ) using different values of I_s. Dashed line marks where I_s = 0, for ease of identifying the fixed points in the absence of stimulation. Shab channel expression (a_K) was varied between 1 and 5 (as indicated) in steps of 1.0 (A), or between 1 and 3 in steps of 0.2 (B). Open circles represent unstable fixed points, while filled circles represent asymptotically stable fixed points. Black, blue, and green circles represent, respectively, foci, nodes, and saddles. Parameters: υ_w = −1, η_w = 2, σ_w = 0.7, τ̄_w = 10 ms

Fig. 4

Transitions from rest to repetitive spiking for different levels of Shab channel expression. Membrane potential dynamics, nullclines, and trajectories in phase space for a_K = 3.0 (A1–A4), 2.0 (B1–B4), 1.4 (C1-C4)), or 1.2 (D1-D4). A1, B1, C1, D1: Responses to two 400 ms square pulses of current, the first at 1 pA below (black traces) and the second at I_cyc (gray traces). A2, B2, C2, D2: Nullclines I_cyc for υ (dashed) and w (solid) are plotted in phase space in the absence of stimulation (I_s = 0). Intersections of the nullclines are marked with circles to indicate stable (filled) or unstable (open) fixed points. Nodes are blue, saddle points are green, and foci are black. If there is more the one fixed point, the one located at the lowest membrane potential has the larger marker size. A3, B3, C3, D3: Nullclines and trajectories (solid black) of the system in phase space corresponding to the response to current stimulation 1 pA below I_cyc. Nullclines and fixed points are plotted as described above. A4, B4, C4, D4: Nullclines and trajectories of the system in phase space corresponding to the response to current stimulation at I_cyc. Parameters: σw = 0.7,. τ̄_w = 10 ms, with initial conditions (w₀, υ₀) = (0.025, −65). See Table 1 for all other parameters

Fig. 5

Firing behaviors produced by different combinations of Shab channel expression and stimulation amplitude. (A) Stimulating the membrane just below I_cyc when a_K = 2.0 causes a single spike followed by a depolarization that is sustained for the duration of the current pulse (black trace). Increasing the current amplitude by 100 pA elicits relatively high-frequency repetitive spiking (gray trace). Compare these responses to those of MN5 in Fig. 1(A). (B) Stimulating at I_cyc when a_K = 1.4 produces low-frequency repetitive firing with a long delay to first spike (black trace). Increasing the current by 100 pA abolishes the long delay and elicits higher-frequency repetitive firing (gray trace). Compare to MN5 recordings in Fig. 1(B). (C) Stimulating with an amplitude beyond I_cyc when a_K = 1.4 induces repetitive firing that commences shortly after stimulus onset. Increasing the current by 100 pA decreases the already short spike latency and increases the firing frequency. Compare to MN5 responses in Fig. 1(C). (D) Stimulating with a current well beyond I_cyc when a_K = 1.2 produces fast-onset repetitive spiking, but APs diminish slightly in amplitude after the first spike and have a triangular shape. Increasing the current by 100 pA produces an initial spike, but subsequent oscillations decrease in amplitude until full depolarization block is reached. Compare to MN5 responses in Fig. 1(D). Parameters same as in Fig. 4

Fig. 6

Elevated potentials and delayed firing profiles produced by variation in Shab channel activation and expression. Slower Shab channel activation results in the production of elevated potentials on which spikes ride for each a_K. Several different delayed firing profiles are also produced by varying a_K. Stimulating the membrane a little beyond I_cyc when a_K = 2.0 elicits repetitive spiking with a delay to first spike and a subsequent ISI of similar duration (black trace). Increasing the current amplitude by 100 pA decreases the spike latency and elicits higher-frequency spiking (gray trace). Spike shape is similar to MN5 spikes. (B) Stimulating beyond I_cyc when a_K = 1.4 produces a delay to first spike longer than the subsequent ISI (black trace). Increasing the current by 100 pA abolishes the long delay and elicits higher-frequency repetitive firing (gray trace). Note similarity to delayed firing profile of MN5 in Fig. 1(B). However, spikes in this case have different concavity to those recorded from MN5. (C) Stimulating the membrane just above I_cyc when a_K = 1.2 produces a much longer delay to first spike than the ISI (black trace). Increasing the current amplitude eliminates the long delay and increases the firing frequency (gray trace). Again, this profile is similar to that in Fig. 1(B). However, the spike shape is worse than that seen when aK = 1.4. Parameters: σ_w = 0.3, τ̄_w = 5 ms