Spike-Conducting Integrate-and-Fire Model

Abstract

Modeling is a useful tool for investigating various biophysical characteristics of neurons. Recent simulation studies of propagating action potentials (spike conduction) along axons include the investigation of neuronal activity evoked by electrical stimulation from implantable prosthetic devices. In contrast to point-neuron simulations, where a large variety of models are readily available, Hodgkin-Huxley-type conductance-based models have been almost the only option for simulating axonal spike conduction, as simpler models cannot faithfully replicate the waveforms of propagating spikes. Since the amount of available physiological data, especially in humans, is usually limited, calibration, and justification of the large number of parameters of a complex model is generally difficult. In addition, not all simulation studies of axons require detailed descriptions of nonlinear ionic dynamics. In this study, we construct a simple model of spike generation and conduction based on the exponential integrate-and-fire model, which can simulate the rapid growth of the membrane potential at spike initiation. In terms of the number of parameters and equations, this model is much more compact than conventional models, but can still reliably simulate spike conduction along myelinated and unmyelinated axons that are stimulated intracellularly or extracellularly. Our simulations of auditory nerve fibers with this new model suggest that, because of the difference in intrinsic membrane properties, the axonal spike conduction of high-frequency nerve fibers is faster than that of low-frequency fibers. The simple model developed in this study can serve as a computationally efficient alternative to more complex models for future studies, including simulations of neuroprosthetic devices.

Keywords: action potential propagation; auditory nerve; computational model; electrical stimulation; spike conduction.

Figure 1.

Response properties of single-compartment models. A, Spike responses of the sEIF (green), bEIF (blue), and WB (gray) models driven by a step current input. B, f-I curves of the models. Step currents of varied amplitudes were injected and the numbers of spikes in 1000 ms were calculated. C, Voltage dependence of the exponential growth factors of the sEIF and bEIF models. D, Depolarizing and repolarizing spike currents of the bEIF model. The horizontal corresponds to the expanded time in A. E, Spike shapes of the bEIF and WB models. F, Membrane currents of the bEIF and WB models. In E, F, traces are aligned such that time 0 corresponds to the peak timings of the action potentials shown in A.

Figure 2.

Response properties of myelinated axon models. A, Schematic drawing (top) and modeled electric circuit (bottom) of a myelinated axon with a diameter D, nodal length L_n and internodal length L_i. Each nodal compartment has a capacitance c_m and (voltage-dependent) resistance r_m and is interconnected with neighboring nodes with an axial resistance r_a. See Table 4 for the default parameter values. In panels B, C, G, H, voltage responses at every 10th node (i.e., each 2 mm apart) are shown. Currents were injected intracellularly into the node #20 (gray traces). B, Spike conduction along the modeled myelinated axon simulated with the WB model. C, Spike conduction along the modeled myelinated axon simulated with the bEIF model. D, Shapes of conducted spikes of the bEIF and WB models. The peaks of both traces are aligned at time 0. E, Dependence of simulated conduction velocity u (m/s) on the axon diameter D (μm). The dotted curve shows a square root fit by $u=4.1\sqrt{D}$. F, Dependence of conduction velocity u (m/s) on the internodal length L_i (μm). The dotted curve shows a square root fit by $u=0.395\sqrt{L_i}$. G, Spike conduction along the modeled myelinated axon simulated with the sEIF model. Inset, Expanded traces around the spike generation. In panel G, we used a time step of 0.2 μs to faithfully simulate the rapidly increasing membrane potentials. H, Spike conduction along the modeled myelinated axon simulated with the bEIF model, with an instantaneous potential reset instead of the repolarizing current. I, Dependence of conduction velocity u on the ceiling value A_T of the bEIF model (blue). Conduction velocity of the WB model (5.7 m/s) is also shown as a reference (thicker gray line). When necessary (typically for large and small values of A_T), the starting voltage for repolarization current V_rep was readjusted (down to -20 mV from the default value of +15 mV using a step of 5 mV) to make sure the membrane potential returned to rest after spiking.

Figure 3.

Response properties of unmyelinated axon models. A, Schematic drawing (top) and modeled electric circuit (bottom) of an unmyelinated axon with a diameter d. Each nodal compartment of length l has a capacitance c_m and (voltage-dependent) resistance r_m and is interconnected with neighboring nodes with an axial resistance r_a. See Table 5 for the default parameter values. In panels B, C, voltage responses at every 25th node (i.e., each 0.5 mm apart) are shown. Currents were injected intracellularly into the node #50 (gray traces). D, Dependence of simulated conduction velocity u (m/s) on the axon diameter D (μm). The dotted curve shows a square root fit by $u=0.42\sqrt{D}$.

Figure 4.

Responses of myelinated axon models to extracellular stimulation. A, Schematic drawing of a myelinated axon stimulated with extracellular current injection. The extracellular voltage at each node is determined by the distance between the node and the stimulus electrode (see Materials and Methods for the equations). B, Spike conduction along the modeled myelinated axon simulated with the WB model. C, Spike conduction along the modeled myelinated axon simulated with the bEIF model. In panels B, C, voltage responses at every 10th node (i.e., each 2 mm apart) are shown. Gray arrows indicate the intracellular responses caused by the extracellular negative current injection. D, Location-dependent voltage responses to extracellular stimulation. Simulated membrane potentials at the offset of extracellular current stimulation (-1 mA, 0.1 ms) are plotted as a function of the distance from the node #20 (gray traces in B, C), which is the closest node to the stimulus electrode, with a separation of 1 mm (see inset for a schematic drawing).

Figure 5.

Response properties of AN axon models. See Table 7 for the default parameter values. A, Spike responses of the low-frequency (red) and high-frequency (blue) single-compartment AN models driven by step current inputs. B, Same traces as in A but with a rescaled time axis. C, Spike conduction along the modeled low-frequency myelinated AN axon. D, Spike conduction along the modeled high-frequency myelinated AN axon. In panels C, D, voltage responses at every five nodes are shown.