Fig 1.
Properties of the fractional-order passive membrane.
(A) Strength-duration curves, derived from the fractional passive membrane, are shown as a function of fractional-order α (Eq 9). (B) The magnitude (top) and phase
(bottom) of the complex impedance of the fractional-order passive membrane are shown as a function of the normalized frequency ωτ, for different values of α. (C) The normalized membrane potential Vm/(Im Rm) response following a current step is shown as a function of normalized time t/τ on a linear (top) and logarithmic (bottom scale), for different values of α.
Fig 2.
Properties of the fractional-order Hodgkin-Huxley spike.
(A) The membrane potential Vm, sodium current INa, potassium current IK, and voltage memory trace vmem are shown as a function of time, for different values of fractional-order α. (B). Vm maximum and minimum (left), INa and IK peak current magnitude, and hpeak (the sodium inactivation gating variable at the time of peak INa current) are shown as a function of α. Spikes are elicited by a brief 0.1-ms duration, 1.5x threshold stimulus.
Fig 3.
Refractoriness in the fractional-order Hodgkin-Huxley model.
The minimum time period between stimuli (top), the time to the spike peak (middle), and their difference (bottom) are shown as a function of fractional-order α. Spikes are elicited by a brief 0.1-ms duration, 1.5x threshold stimulus.
Fig 4.
Repetitive firing in the fractional-order Hodgkin-Huxley model.
(A) The membrane potential Vm, sodium current INa, potassium current IK, and voltage memory trace vmem are shown as a function of time in response to a constant applied current, Iapp = 20 (left), 100 (middle), and 140 (right) μA/cm2, for different values of fractional-order α. (B) The instantaneous spike frequency is shown as a function of the interspike interval (ISI) number for different values of α.
Fig 5.
Repetitive firing in the first-order Hodgkin-Huxley model with scaled ionic current conductances.
The membrane potential Vm is shown as a function of time in response to a constant applied current, Iapp = 20 (left), 100 (middle), and 140 (right) μA/cm2, for different values of fractional-order α. Sodium, potassium, and leak conductances, gNa, gK, and gL, respectively, are scaled, individually (A-C) and combined (D, top), such that peak current measurements are equivalent to values for particular value of α, as described in the text. (D, bottom) The instantaneous spike frequency is shown as a function of the interspike interval (ISI) number for different values of α.
Fig 6.
Spike properties in the fractional-order Hodgkin-Huxley model.
(A) Bifurcation diagram of Vm, sodium current INa, and potassium current IK, showing steady-state values and limit cycle maximum and minimum, as a function of the applied current Iapp, for different values of α. (B) The critical values denoting Iapp lower and upper limits for spiking (Hopf bifurcations), I1 and I2, respectively, are indicated (fractional-order Hodgkin-Huxley model (fHH), solid lines). (C) The spike frequency (top) and amplitude (bottom) are shown as a function of Iapp and α. In B and C, values for I1, I2, and spike frequency and amplitude are shown for the first-order model with scaled conductances for comparison (dashed lines, see Fig 5 and main text for more details). In the bottom panel of C, the solid and dashed lines are nearly identical.
Fig 7.
Sub-threshold impulse and voltage response in the passive fractional-order cable equation.
(A) The impulse response function G(x/λ, t/τ) is shown as a function of space x, normalized by space constant λ, at times t = 0.05τ and t = τ, where τ is the time constant, on a linear (top) and logarithmic (bottom) scale, for different values of fractional-order α. (B) G(x/λ, t/τ) is shown as a function of normalized time t/τ at location x = 0 (top) and x = λ (bottom). (C) The normalized voltage response to a current step input at the origin x = 0 is shown as a function of normalized time t/τ at locations x = 0 and x = λ. The voltage response in the membrane patch is shown for comparison (dashed lines, Fig 1C). (D) The normalized position of stimulus propagation x/λ is shown as a function of normalized time t/τ (the time at which the normalized voltage response is 0.5) (E) The pseudo-velocity, given by the slope of the stimulus propagation, in units of λ/τ, is shown as a function of α.
Fig 8.
Spike propagation in the fractional-order Hodgkin-Huxley nerve axon following a brief stimulus pulse.
(A) A space-time plot of the membrane potential Vm(x, t) is shown for different values of fractional-order α. (B) The peak sodium current INa, potassium current IK, leak current IL, and voltage memory trace vmem magnitude are shown as a function of position along the cable x, for different values of α. (C) Spike propagation velocity is shown in the fractional-order Hodgkin-Huxley (fHH) nerve axon, as a function of α, for different values of longitudinal conductance g (solid lines). Velocity measurements are also shown (dashed lines) for simulations in which the sodium gNa and potassium gK conductances are scaled such that peak current measurements are equivalent to values for particular value of α. See text for more details. In A and B, axon conductance g = 7.06 μS. Propagating spikes are elicited by a brief 0.1-ms duration, 500-μA/cm2 stimulus at x = 0.
Fig 9.
Spike propagation in the fractional-order Hodgkin-Huxley nerve axon during a constant stimulus.
(A) Spike propagation velocity (top) and the change in velocity, as a percentage of the final velocity (bottom), are shown as functions of spike number, for different values of fractional-order α and applied current amplitude Iapp. (B) Velocity measurements are shown for simulations in which the sodium, potassium, and leak conductances, gNa, gK, and gL, respectively, are scaled such that peak current measurements are equivalent to values for particular value of α and location x, as described in the text. Axon conductance g = 7.06 μS. Propagating spikes are elicited by a constant stimulus at x = 0.
Fig 10.
Electrical activity in a fractional-order Hodgkin-Huxley neural network.
(A) Rastergram of spikes in the neural network for different values of fractional-order α. Synaptic connections and network architecture were identical in all simulations. (B) The pseudo-electroencephalogram (pEEG, Eq 25) and (C) firing rate are shown as functions of time, for the simulations in A. Firing rate is measured in a sliding 50-ms window, with 10-ms steps. (D) Interspike interval (ISI) histograms are shown for each simulation. The gray bar denotes the 50-ms applied stimulus, during which a 40-μA/cm2 current was applied to 13 randomly chosen neurons. (E) The mean network activity duration ± standard error of the mean and (F) the fraction of sustained network activity are shown in the fractional-order Hodgkin-Huxley (fHH) neural network, as a function of α (solid lines). Network simulations in which the sodium, potassium, and leak conductances, gNa, gK, and gL, respectively, are scaled such that peak current measurements are equivalent to values for particular value of α, as described in the text, are shown for comparison (dashed lines). Values in E and F are calculated for 12 network architectures.
Fig 11.
Synaptic activity in a fractional-order Hodgkin-Huxley neural network.
(A) The excitatory and inhibitory synaptic currents, IsynE and IsynI, respectively, averaged over all network neurons, are shown as a function of time, for different values of fractional-order α, for the same network as shown in Fig 10A–10D. The gray bar denotes the 50-ms applied stimulus, during which a 40-μA/cm2 current was applied to 13 randomly chosen neurons. (B) The mean of excitatory and inhibitory current charge magnitude, QsynE and QsynI, respectively, ± standard error of the mean, are shown as a function of α, calculated for 12 network architectures (solid lines). Network simulations in which the sodium, potassium, and leak conductances, gNa, gK, and gL, respectively, are scaled such that peak current measurements are equivalent to values for particular value of α, as described in the text, are shown for comparison (dashed lines).