Fig 1.
Power-law dynamics of individual Hodgkin-Huxley gating variables.
(A) The response of the individual gating variables (n, m, and h) using the analytical (solid) and numerical (dotted) solutions to an identical voltage command (30 mV for n, -55 mV for m, -70 mV for h) with different values of the fraction order derivative (η). (B) The long term response of the individual gates (, x = m,n or h) for all the voltages and values of η for the analytical and numerical solutions. The arrow points to the inflection point of the sigmoidal curve. For
some numerical solutions were unstable. (C) For the n and h gates we fitted a dual exponential process to the temporal response to voltage commands for all values of η resulting in a fast and slow time constant (
, x = m, n, or h). We fitted a single time constant to the m gate.
Fig 2.
The effect of power-law behaving Hodgkin-Huxley gating variables on the shape and properties of the action potential.
(A) Action potential shapes generated with the minimum input current as a function of the order of the fractional derivative (η) for the respective gate. (B) The action potential current threshold as a function of input current and η. (C) Phase plot of the action potentials generated at minimum input current as function of η. The red squares indicate the crossing of threshold detection (dv/dt > 20 mV/ms). (D) The voltage threshold calculated from the phase plots in C.
Fig 3.
Action potential spiking patterns due to power-law conductances in response to constant current input.
(A-D) Spiking patterns generated with power-law behaving n gate. (E-G) Spiking patterns generated with power-law behaving h gate. Each set of simulations done with identical input current and varying the order of the fractional derivative (η).
Fig 4.
Action potential patterns generated by a Hodgkin-Huxley model modified with power-law behaving n (A-C) and h (D-F) gates.
Each panel has the information of the current input (I) and value used for the respective fractional order derivative (η).
Fig 5.
Phase transition diagrams of the spiking patterns generated by the Hodgkin-Huxley model with power-law behaving conductances.
The power-law dynamics was implemented with a fractional order derivative of order η for the respective gating variables. (A) Potassium conductance activation n gate. (B) Sodium conductance activation m gate. (C) Sodium conductance inactivation h gate. RS, resting state; PS, phasic spiking; MMO, mixed-mode oscillations; TS, tonic spiking; SWB, square-wave bursting; and PPB, pseudo-plateau bursting. Spiking responses and boundaries were manually classified based on the first 1,500 ms of simulation.
Fig 6.
The contribution of the memory trace to Mixed Mode Oscillations in the Hodgkin-Huxley model with power-law n gate.
The power-law dynamics was implemented with fractional derivative of order η = 0.7 and constant input current I = 23 nA. (A-E) Examples over a long (left) and short (right) time window of the voltage, memory trace, and gate values. The gray line is the identical simulation with η = 1.0. (F-H) Phase plane analysis of the same responses. (F) Phase plot of the sodium (INa) vs Iw = potassium + leak + injected currents. The red line indicates the balance current and the red square indicates the presence of an attractor. (G) Zoom in the attractor in F. (H) Same data as in G but plotting the imbalance current (INa+Iw) vs Iw. The * indicates where Iw starts compensating for INa.
Fig 7.
Phase plane analysis of the Transient Spiking pattern (see Fig 4B).
(A) Comparison of the current trajectories of the power-law (black) and classic (gray) Hodgkin-Huxley model. The power-law model had a fractional order derivative of η = 0.4 and input current I = 8 nA. The red box indicates the area of the attractor. (B) Phase plane of the attractor in A. S1 to S4 indicate spikes and RS is the resting state.
Fig 8.
The contribution of the memory trace to square wave bursting and pseudo plateau potential spiking patterns in the Hodgkin-Huxley model with power-law behaving h gate.
(A) Voltage traces for square wave bursting and two types of pseudo plateau potential (pituitary and cardiac types). The gray plot corresponds to the classic Hodgkin-Huxley model. (B-E) the temporal behavior of the h memory trace, and gating variables. The square wave bursting was generated with a fractional order derivative of η = 0.4 and the input current I = 10 nA; the Pituitary type was generated with η = 0.2 and I = 20 nA; and the cardiac type was generated with η = 0.2 and I = 9 nA.
Fig 9.
Phase plane plots of the square wave bursting and the two types of pseudo plateau potential spiking patterns in the Hodgkin-Huxley model with power-law behaving h gate.
Same settings as in Fig 8. (A) Phase plot of the sodium (Ina) vs Iw = potassium + leak + injected currents, the red square shows the place of the attractor. (B) The attractors from A plotting balanced current (Ina+Iw) vs Iw.