Fig 1.
Stochasticity in AP generation under constant stimulation conditions.
If 10 pulses are applied at threshold intensity 5 APs are expected but a repetition of the experiment may cause 4 or 7 APs. Stronger stimulation shows a precise response of one AP per pulse (bottom) but still some variation in the spike shapes. In an original model of the HH type (red examples) a noise term is added to include a stochastic behavior (blue). Recorded membrane voltages of retinal ganglion cells were redrawn from [13], simulations were calculated with the CRRSS model.
Fig 2.
Compartment model of a myelinated axon.
A: Geometry. The active compartments (nodes of Ranvier) are marked in red, the passive ones (internodes) in blue. B: Corresponding electrical network model. Independent noise currents Inoise,n are added to the active compartments. The stimulating current from the electrode defines the extracellular potential (Ve) which causes a change of intracellular potential (Vi) depending on capacitance Cm and conductance Gm of the membrane and the intracellular resistance R. All influences from outside are marked via ellipses in magenta. C: Top view of the electrode and axon with equipotentials (left) and voltage profiles of active (red) and passive (blue) compartments (right). The excitation is initiated at the NoR closest to the electrode (thick red line). The uppermost 30 (of 101) compartments are not shown; simulation without noise.
Table 1.
Model parameters and formulas for the opening and closing rates of the ion channels.
Fig 3.
Relationship between firing efficiency, relative spread (RS) and dynamic range (cyan).
The spread is plotted in magenta. The black curve is the Gaussian fit for five spiking probabilities (marked with x) with intensities 600, 650, 700, 750 and 800 μA. 700 μA represents the threshold value, i.e. the one in which a spike occurs in 50% of the stimulus trials. The dynamic range (normalized to threshold) corresponds to 2.56 times RS.
Fig 4.
RS increases linearly with knoise.
A: Spiking probability functions for three knoise values. Numerical evaluation was performed at 100 intensities, from 0.6 threshold to 1.6 ·threshold in 0.01 steps. For each point 500 runs were performed to estimate the spiking probability, then a cumulative Gaussian distribution was fitted. Electrode distance 500μm, myelinated axon d = 1 μm. B: RS values from A as functions of knoise underlines the assumption that for small knoise values RS is proportional to knoise. Linear fit: RS = 14458*knoise– 0.588, R2 = 0.956.
Fig 5.
Comparison of HH10, HH1, and CRRSS at adjusted knoise with Verveen’s formula (black). The CRRSS model of the myelinated fiber with knoise = 0.0038 (green) shows exactly the same slope of the straight line as Verveen, but slightly underestimates the vertical displacement. The HH10 model (knoise = 0.00042) of the myelinated fiber (blue) and the HH1 model (knoise = 0.0038) of the unmyelinated fiber (red) show slight deviations in slope and vertical displacement. Myelinated axon models were calculated with idealized internodes assuming Gm = 0 and Cm = 0 in Fig 2B.
Table 2.
Change of RS by doubling noise transmission time Dt.
Fig 6.
RS as function of axon diameter for different electrode distances z from a myelinated fiber.
NoR length 2.1 μm, internode length 100*diameter, HH10.
Table 3.
Thresholds and RS for the HH1, HH10 and CRRSS model.