PMC

The maximal value of m _∞ ³ h _∞(V) with respect to V () is plotted as a function of θ_m.

(A) θ _m = −22 mV, g _d = 0.8 mS/cm², I _app = 4.9 μA/cm²; (B) θ _m = −24 mV, g _d = 0.39 mS/cm², I _app = 3.23 μA/cm²; (C) θ _m = −28 mV, g _d = 0.39 mS/cm², I _app = 1.25 μA/cm².

(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.

(A) θ _m = −24 mV.
(B) θ _m = −28 mV. In the two panels, g _d = 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 t _delay for noisy input.

Fast–slow analysis is described for θ _m = −24 mV.
(A,B) Parameters are: g _d = 0.39 mS/cm², I _app = 2.9 μA/cm². (Except for I _app, parameters are as in A–C and 2C. The neuron is quiescent at steady state.)
(C,D) The parameters g _d = 1.8 mS/cm² and I _app = 4.2 μA/cm² are as in D, 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 _∞(V _FP(b)) for fixed points and F(b) for limit cycles () as functions of b (B,D). Symbols are as in .

(A) Delayed tonic firing for θ _m = −24 mV, g _d = 0.39 mS/cm², I _app = 3.35 μA/cm².
(B) Delayed stuttering for θ _m = −24 mV, g _d = 1.8 mS/cm², I _app = 4.2 μA/cm².
(C) Delayed tonic firing for θ _m = −28 mV, g _d = 0.39 mS/cm², I _app = 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, I _app, 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.

Black line: the delay duration, t _delay, computed and averaged over 50 trials of the noisy external current step is plotted as a function of log(D). Parameters: θ _m = −24 mV (small window Na⁺ current), g _d = 0.39 mS/cm², and I _app = 3.08 μA/cm². Gray lines: one standard deviation confidence limit of the averaged t _delay. Red line: fitting to the simulation results using the analytical prediction for small D (). The arrow on the y-axis indicates the value of t _delay for a noiseless input (D = 0).

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.

(A) The phase diagram of the model neuron in the I _app-g _d plane. The solid line represents the current threshold, I _th, as a function of g _d. 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 g _d = 0.1 mS/cm² and I _app = 3.35 μA/cm². The neuron exhibits non-delayed, high-frequency tonic firing. Bottom panel: the f-I _app curve for g _d = 0.1 mS/cm² is discontinuous at the current threshold. The minimal frequency is 27.4 Hz.
(C) For g _d = 0.39 mS/cm², I _app = 3.35 μA/cm², the neuron exhibits delayed, high-frequency tonic firing.
(D) For g _d = 1.8 mS/cm² and I _app = 4.2 μA/cm², the neuron exhibits delayed stuttering.

(A) Phase diagram of the model neuron in the I _app-g _d 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 g _d = 0.39 mS/cm² . Top: for I _app = 1.25 μA/cm², the neuron displays low frequency delayed tonic firing. Bottom: for I _app = 1.27 μA/cm², the neuron fires doublets of action potentials.
(C) The steady-state f-I _app curve of the neuron; g _d 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.

(A,B) Voltage traces in response to a step current of amplitude I _app = 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 I _app = 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, g _d = 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.

(A–C) are for small and (D–F) for large Na⁺ window currents. Parameters in (A–C) are as in C: θ _m = −24 mV, g _d = 0.39 mS/cm², I _app = 3.35 μA/cm². Parameters in (D–F) are as in the top panel in B: θ _m = −28 mV, g _d = 0.39 mS/cm², I _app = 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 _∞(V _FP(b)) and F(b) () 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 (I _app = 0) is b _rest, and b* is the value of b at steady state of the neuron. For b _delay, 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 I _app.