Figure 1.
The Strength of the Na+ Window Conductance
The maximal value of m∞3h∞(V) with respect to V (Equation 6) is plotted as a function of θm.
Figure 2.
Responses of the Model Neuron to Noiseless Current Steps for Small Na+ Window Current (θm = −24 mV)
(A) The phase diagram of the model neuron in the Iapp-gd plane. The solid line represents the current threshold, Ith, as a function of gd. To the left of this line, the neuron is quiescent. To the right of the dashed line, the spike response is almost immediate after the current step onset. Stuttering emerges on the dotted line via a torus bifurcation.
(B) Top panel: the membrane potential V in response to a step of current for gd = 0.1 mS/cm2 and Iapp = 3.35 μA/cm2. The neuron exhibits non-delayed, high-frequency tonic firing. Bottom panel: the f-Iapp curve for gd = 0.1 mS/cm2 is discontinuous at the current threshold. The minimal frequency is 27.4 Hz.
(C) For gd = 0.39 mS/cm2, Iapp = 3.35 μA/cm2, the neuron exhibits delayed, high-frequency tonic firing.
(D) For gd = 1.8 mS/cm2 and Iapp = 4.2 μA/cm2, the neuron exhibits delayed stuttering.
Figure 3.
Responses of the Model Neuron to Noiseless Current Steps for Large Na+ Window Current (θm = −28 mV)
(A) Phase diagram of the model neuron in the Iapp-gd plane. Solid line: current threshold to action potential firing. Dashed line: to the right of the dashed line, the spike response is almost immediate after the current step onset. Period doubling bifurcations occur on the dotted lines, leading to doublet or complex firing (grey region).
(B) Voltage traces for gd = 0.39 mS/cm2. Top: for Iapp = 1.25 μA/cm2, the neuron displays low frequency delayed tonic firing. Bottom: for Iapp = 1.27 μA/cm2, the neuron fires doublets of action potentials.
(C) The steady-state f-Iapp curve of the neuron; gd is as in (B). The average firing frequency goes to zero at firing threshold. Between the two solid circles, the neuron fires doublets of spikes.
Figure 4.
Dependence of the Delay Duration tdelay on the Amplitude of the Current Step Iapp
(A) θm = −24 mV.
(B) θm = −28 mV. In the two panels, gd = 0.39 mS/cm2. Solid lines: noiseless input (D = 0). Dashed lines: noisy input with variance D = 0.01 μA2 × ms/cm4. The delay duration was averaged over 50 trials. Gray lines represent one standard deviation around the mean value of tdelay for noisy input.
Figure 5.
Bifurcation Diagrams and the Fast–Slow Analysis of the Model Neuron
(A–C) are for small and (D–F) for large Na+ window currents. Parameters in (A–C) are as in Figure 2C: θm = −24 mV, gd = 0.39 mS/cm2, Iapp = 3.35 μA/cm2. Parameters in (D–F) are as in the top panel in Figure 3B: θm = −28 mV, gd = 0.39 mS/cm2, Iapp = 1.25 μA/cm2.
(A,D) The bifurcation diagram of the fast subsystem in the V-b space.
(B,E) The frequency f of the limit cycle of the fast subsystem, plotted as a function of b (f-b curves).
(C,F) The functions b∞(VFP(b)) and F(b) (Equation 2) plotted as a function of b. Thin solid lines: stable fixed points; thin dotted lines: unstable fixed points; thick solid line: stable limit cycle (periodic state); thick dotted line: unstable limit cycle. Solid circles denote Hopf, saddle-node (SN), and saddle-node of periodics (SNP) bifurcation points. The value of b at rest (Iapp = 0) is brest, and b* is the value of b at steady state of the neuron. For bdelay, see text. In (C,F), the intersection of the curve b = F(b) with the diagonal dashed line determines the value of b* (open square point). Arrows represent the evolution of the neuron from rest to its steady state following a current step injection of amplitude Iapp.
Figure 6.
Fast–Slow Analysis of Neurons with Small Na+ Window Current Exhibiting a Depolarized Rest Potential or Stuttering
Fast–slow analysis is described for θm = −24 mV.
(A,B) Parameters are: gd = 0.39 mS/cm2, Iapp = 2.9 μA/cm2. (Except for Iapp, parameters are as in Figures 5A–5C and 2C. The neuron is quiescent at steady state.)
(C,D) The parameters gd = 1.8 mS/cm2 and Iapp = 4.2 μA/cm2 are as in Figure 2D, and the neuron stutters at steady state. For each case, we plot the bifurcation diagrams of the fast subsystem in the V-b space (A,C), and the functions b∞(VFP(b)) for fixed points and F(b) for limit cycles (Equation 2) as functions of b (B,D). Symbols are as in Figure 5.
Figure 7.
Voltage Traces and Firing Patterns of the Model Neuron in Response to a Noisy Current Step
(A) Delayed tonic firing for θm = −24 mV, gd = 0.39 mS/cm2, Iapp = 3.35 μA/cm2.
(B) Delayed stuttering for θm = −24 mV, gd = 1.8 mS/cm2, Iapp = 4.2 μA/cm2.
(C) Delayed tonic firing for θm = −28 mV, gd = 0.39 mS/cm2, Iapp = 1.25 μA/cm2. In all three panels, the variance of the noise is D = 0.01 μA2 × ms/cm4. The time course of the mean applied current, Iapp, is also plotted. The membrane potentials during the delay periods are magnified in the panels on the right. Note the presence of subthreshold oscillations in (A) and (B). The peaks of these oscillations are denoted by the arrows.
Figure 8.
Dependence of the Averaged Delay Duration on the Noise Variance, D
Black line: the delay duration, tdelay, computed and averaged over 50 trials of the noisy external current step is plotted as a function of log(D). Parameters: θm = −24 mV (small window Na+ current), gd = 0.39 mS/cm2, and Iapp = 3.08 μA/cm2. Gray lines: one standard deviation confidence limit of the averaged tdelay. Red line: fitting to the simulation results using the analytical prediction for small D (Equation 4). The arrow on the y-axis indicates the value of tdelay for a noiseless input (D = 0).
Figure 9.
Fourier Spectrum of the Membrane Potential Fluctuations During the Delay Period in the Model for Three Different Strengths of the Na+ Window Current
(A) θm = −22 mV, gd = 0.8 mS/cm2, Iapp = 4.9 μA/cm2; (B) θm = −24 mV, gd = 0.39 mS/cm2, Iapp = 3.23 μA/cm2; (C) θm = −28 mV, gd = 0.39 mS/cm2, Iapp = 1.25 μA/cm2.
Figure 10.
Noise-Induced Stuttering Activity for Small Na+ Window Current (θm = −24 mV)
(A,B) Voltage traces in response to a step current of amplitude Iapp = 3 μA/cm2.
(A) In the absence of noise in the input (D = 0), the membrane potential depolarizes during the current step but the neuron does not fire action potentials.
(B) Noise of variance D = 0.1 μA2 × ms/cm4 induces irregular bursts of action potentials (irregular stuttering).
(C,D) Voltage traces in response to a step current of amplitude Iapp = 3.2 μA/cm2.
(C) For D = 0, the neuron exhibits delayed tonic firing.
(D) For D = 0.1 μA2 × ms/cm4, the neuron stutters irregularly. For all panels, gd = 0.5 mS/cm2. Projection of the phase portraits on the V-h plane for constant values of b (0.046 in (A) and 0.22 in (C)) are shown below the traces in (A,C). These values of b are either the fixed-point value (in (A)) or the average value on the limit cycle (in (C)). Solid circles denote stable fixed points, solid lines denote stable limit cycles, and dotted lines denote unstable limit cycles.
Figure 11.
Experimental Results: Responses of FS Neurons to Just Suprathreshold Depolarizing Current Steps
(A) An example of a neuron displaying high rate delayed tonic firing after a delay. The time interval below the top horizontal bar is magnified in the inset on the right.
(B) A neuron exhibiting stuttering.
(C) A neuron that exhibits delayed irregular firing.
(D) A neuron that fires at low rates after a delay. The dotted line denotes −70 mV.
Figure 12.
Experimental Results: Fourier Spectrum of the Membrane Potential Fluctuations during Delay Periods in FS Neurons
Top: voltage time courses for three FS neurons exhibiting delayed firing.
Bottom: the Fourier spectra of the subthreshold membrane potentials during the delay before spiking for these three neurons.
(A) The minimal (just suprathreshold) firing frequency of this neuron is high (80 Hz). Pronounced subthreshold oscillations are observed during the delay period.
(B) The minimal firing frequency of this neuron is lower than in (A) (38 Hz). The subthreshold oscillations exhibited by this neuron are less pronounced than in (A).
(C) The minimal firing frequency of this neuron (4.3 Hz) is much lower than in (A) or (B). This neuron does not exhibit subthreshold oscillations during the delay period. Spectra were calculated over the time intervals denoted by the horizontal bars in the top panels.