Fig 1.
Response of the model to a step current of 2 μA/cm2 input as shown in the top.
A. Voltage trace (V) with reversal potentials for sodium (Orange) and potassium (Blue).B. Firing rate (Fr) of neuron model. C. Intracellular sodium (Orange) and extracellular potassium (Blue) concentration (C) dynamics. D. Spike amplitude and reversal potential for sodium (Orange) for the trace shown in A. E. First and last spike (peaks aligned).
Fig 2.
Response of the model to a step current with colored noise filtered at 500Hz (mean input is 1μA/cm2 and standard deviation 1.05μA/cm2, shown above).
A. Voltage trace (membrane potential) of the model responding to a noisy step current (Top of panel). B. Zoom of the voltage trace from panel A at the beginning and towards the end of stimulation showing the evolution of the reversal potentials for sodium (orange) and potassium (blue), as in Fig 1. C. Phase portraits of the steady state of the fast spike generating sub-system, when imposing the average reversal potentials for sodium (orange) and potassium (blue) of panel B as parameters. Vertical axes show the voltage and the horizontal axes show the potassium current gating-variable (nK). Empty dots are the unstable nodes, filled dots the stable nodes, and the orbits are stable limit cycles. D. Evolution of the maximum voltage of the system attractors. Empty dots represent the unstable nodes, filled dots the stable nodes, and the black line denotes the maximum voltage of the stable limit cycles (action potential peak).
Fig 3.
Characteristic phase portraits in the extracellular potassium / applied current space.
Different combinations of extracellular potassium and input current yield different phase portraits of the fast spike generating sub-system. A. The background color represents the characteristic response of that area; Subthreshold or depolarization block: White; Bistable: Dashed; Spiking: Gray. The different regions are separated by the disappearance of the stable node (gray line) and the limit cycle onset (black dots). Examples of phase portraits in each region of the extracellular potassium—input current plane are portrayed; B. subthreshold, C. bistable, and D. spiking state.
Fig 4.
Rodent cortical neurons exposed to high extracellular potassium show intermittently interrupted firing.
Example cell; Response of a neuron to a just suprathreshold stimulus in low (3 mM; Top) and high (10 mM; Bottom) extracellular potassium conditions. The suprathreshold current is taken as the current needed to elicit the first spike when injecting a ramp with a shallow slope. The small spikelets visible in the current trace are artifacts resulting from limited capacitive coupling of two channels at the digitizer (i.e., of the action potentials present in the voltage trace), and are not reflective of current injected into the neuron.
Fig 5.
Activity-dependent decrement in action potential amplitudes.
A. Membrane potential of a rodent cortical neuron (lower trace), subjected to prolonged DC current punctuated by brief hyperpolarizing pulses with different durations (top trace). B. Plot of peak voltage for each action potential shown in A. Note that action potential peaks fail to recover after even 1 second long breaks in current stimulation. C. Overlay of the first and last action potentials shown in A (aligned at 5 ms).
Fig 6.
Extracellular-potassium- and intracellular-sodium-dependent bistable area.
Same bifurcation diagram portrayed in Fig 3 for different intracellular sodium concentrations [Na+]i. [Na+]i controls the input current required to transition from resting to spiking regimes. [Na+]i accumulation shifts the bistable region to higher current values.
Fig 7.
Consequences of simultaneous [Na+]i and [K+]o slow dynamics on the fast spike generating dynamics.
A. Characteristic response of the reduced model (receiving a constant input current stimulus of 1μA/cm2) along the reversal potential plane. The characteristic response can be split in three categories; stable-resting state (purple areas in the lower left corner and top of the graph, representing the subthreshold regime and depolarization block, respectively), the spiking state (green), and the bistable state (yellow). Three example trajectories of the complete system (Including the slow activity-dependent concentration dynamics) simulated during 10 seconds with an irregular input (mean of 1μA/cm2 and standard deviation of 1μA/cm2). Initial conditions represented by a circle, and the state of the system 10 seconds later with a triangle. B. Membrane potential trace of the trajectory with initial conditions EK = −55.1mV and ENa = 66.7mV displayed in A. Left panel shows the first 400ms of simulation (marked with a circle), and the right panel shows the last 400ms out of the 10 seconds simulation (marked with a triangle). C. Membrane potential trace of the trajectory with initial conditions EK = −63.9mV and ENa = 75.7mV displayed in A. B. Membrane potential trace of the trajectory with initial conditions EK = −74.2mV and ENa = 69.3mV displayed in A.
Fig 8.
Summary of the dynamics represented in the model.
The figure illustrates the flux of ions through the fast sodium (INa), the delayed rectifier IK spike generating currents, the Na-K-ATPase pump, and through the neuronal membrane (IL). The mathematical representation of the currents flowing through proteins can be found in Eqs (2), (3), (8) and (4). The change of the membrane potential due to those currents is represented by Eq (1).