Fig 1.
Schematic drawing of the neuron model in the extracellular environment.
The concentration of extracellular potassium changes with the activity of voltage-gated potassium channels (red) and the Na+/K+-ATPase pump (green). The model equations governing the spike-generating inward sodium (blue) and outward potassium (red) currents correspond to those in the original Wang-Buzsáki model, retaining their fast dynamics. A slower timescale arises from the dynamics of the extracellular potassium concentration.
Fig 2.
(A) Each bursting cycle consists of two phases: a spiking phase and a quiescent phase, visible in the voltage trace. The concentration waxes with each spike and wanes during the quiescent phase. (Iapp = 0.5 μA/cm2, Imax = 1 μA/cm2). (B) The spiking phase commences at a saddle-node (SN) bifurcation of the fast subsystem (red square). During the spiking phase, the average outward potassium flux through potassium channels surpasses the average inward potassium flux through the Na+/K+-ATPase leading to an accumulation of extracellular potassium. This phase terminates at a homoclinic (HOM) bifurcation of the fast subsystem (blue circle). From this bifurcation, the quiescent phase initiates, accompanied by a continuous decrease in extracellular potassium concentration due to the Na+/K+-ATPase current superseding the potassium current through the voltage-gated channels. This eventually results in a new spiking phase. (C) Minimal and maximal values of [K+]out during an oscillation as a function of pump capacity (Imax) for different Iapp. For each curve from right to left (decrease in pump capacity Imax): onset of bursting (inter-spike interval transitions from unimodal to bimodal), widening of the range of [K+]out covered by the oscillations, finally burst termination (via depolarization block for Iapp = 0.3 and 0.5 μA/cm2, or initially double period and tonic spiking before entering depolarization block for Iapp = 0.7 and 0.9 μA/cm2).
Fig 3.
Bifurcations of the fast subsystem and bursting dynamics.
The bistability of the fast subsystem is a requirement for the emergence of bursting dynamics in our system. (A) Two-parameter (extracellular potassium, [K+]out, and applied current, Iapp) bifurcation diagram of the fast subsystem. The black vertical line indicates the range of [K+]out values that the complete system traverses during the burst cycle shown in Fig 2A. The burst trajectory passes through the bistable region, bounded by the homoclinic (HOM) bifurcation at higher [K+]out and the saddle-node (SN) bifurcation at lower [K+]out. During the spiking phase of the burst, [K+]out builds up, whereas it depletes during the resting part. The black line extends into the tonic spiking domain of the fast subsystem due to the time spent near the ghost of the SN bifurcation. (B) Phase portrait of the complete burst (in voltage and [K+]out) onto the one parameter ([K+]out) bifurcation diagram of the fast subsystem. The black solid line demonstrates the same bursting dynamics as in Fig 2A.
Fig 4.
Hysteresis loop of the slow subsystem overlaid with the bifurcations of the fast subsystem.
The extracellular potassium ([K+]out) dynamics of the reduced, slow subsystem entails a hysteresis loop oscillator. The slow oscillation organises around the bistable region of the fast subsystem, as also shown in Fig 3. The slow subsystem trajectory for the resting branch is calculated by inserting steady state fast variables as a function of the slow concentration into the [K+]out time derivative (see Methods, Eq 9), while the spiking branch requires averaging (see Methods, Slow-fast method, Eq 14). The red and blue lines represent the location of the saddle-node (SN) and homoclinic (HOM) bifurcations of the fast subsystem according to ([K+]out), respectively. Note that, after going through the SN bifurcation, the system remains near the SN ghost for a while before spiking is resumed (brown dotted line). Parameters as in Fig 2A.
Fig 5.
Shear transformation of the saddle-node loop (SNL) unfolding induced by the Na+/K+-ATPase.
Including a Na+/K+-ATPase into the neuron model affects the fast subsystem bifurcation diagram and thereby enables bursting dynamics in the complete system. (A) and (B): Voltage traces from the models with an electroneutral (akin to [15]) and with an electrogenic Na+/K+-ATPase for different values of the applied current Iapp (0.1, 0.25, 0.5, 0.9 μA/cm2, top to bottom). Tonic spiking and bursting dynamics occur in the paradigm with pump. (C) Two-dimensional bifurcation diagram of the fast subsystem with the electroneutral pump. The generic unfolding of the SNL bifurcation (green circle) includes a bistable area between the homoclinic (HOM) and saddle-node (SN) bifurcation lines. (D) In the model with electrogenic pump, the area of bistability is sheared. This opens up the possibility for a [K+]out hysteresis loop (yellow region) for values of Iapp between the SNL (green circle) and the HOM inflection point (purple square). Imax = 1 μA/cm2 in all panels.
Fig 6.
Bifurcations of the fast subsystem with an overlaid phase portrait around the bursting region.
The bifurcations of the fast subsystem together with the flow of dynamics is shown in the vicinity of bursting area. (A) Zoom-out of the two-parameter bifurcation diagram of the fast subsystem depicted in Fig 3A. Depending on the applied current (Iapp) and extracellular potassium concentration ([K+]out), the fast subsystem exhibits mono-, bi-, or even multistability. The resting region in the top right corner, above the fold of limit cycles (FLC) is characterised by bistability between two fixed points. The region between the Hopf and FLC bifurcations is multistable between two fixed points and one limit cycle. The bistable region enclosed between the homoclinic (HOM), saddle-node (SN), and Hopf bifurcations enables bursting. Here, the bistability occurs between one limit cycle and one fixed point. The burst is possible for values of Iapp between the ones corresponding to the saddle-node-loop (SNL) and HOM inflection point. The area marked as D highlights where the bursting shown in Fig 2A occurs. The vertical arrows indicate the flow field direction of the complete system. For the higher [K+]out than the Hopf line, the one branch of the complete system flow field direction which is going to higher [K+]out and indicating the depolarization block branch is not shown (see panel B, rest branch with positive time derivative of [K+]out). Two specific cases are selected from the region of depolarization block (B) and tonic spiking (C) of the complete system dynamics for further study. (B) Reduced slow subsystem dynamics overlaid on the bifurcation of the fast subsystem in the case marked as B in panel A (Iapp = 0.9 μA/cm2). This case corresponds to a transition to the depolarization block in the complete system. The arrangement of the resting and spiking branches indicates that, regardless of where the system starts on this diagram, it will eventually reach the resting branch with the higher time derivative of [K+]out and follow that line up to the depolarization block. (C) Reduced slow subsystem dynamics overlaying on the bifurcation map of the fast subsystem in the case marked as C in panel A (Iapp = 0.25 μA/cm2). This case corresponds to tonic firing in the complete system. The black straight vertical line represents d[K+]out/dt = 0, and the arrow indicates the spiking branch of the slow subsystem crossing this line and having a stable fixed point. Accordingly, the complete system remains tonically firing. Imax = 1μA/cm2 for all the panels.
Fig 7.
Dynamics of the complete system for varying pump density.
Depending on the maximal pump current (Imax) and applied current (Iapp), the complete system can show different dynamics (rest, tonic spiking, bursting, and depolarization block). This figure focuses on a low potassium concentration. (A) Bifurcations in the complete system as well as the corresponding fast subsystem saddle-node-loop (SNL) and HOM inflection point. The points labelled B, C, and D in this figure correspond to the same parameter sets as shown in Fig 6 with the same labelling. The area between the saddle-node-loop (SNL) and HOM inflection point lines of the fast subsystem represents the parameter space where bursting can potentially occur, according to the fast subsystem analysis. Within this region, the complete system displays bursting behaviour only in the yellow zone. Numerically, the boundary of the bursting region is verified by detecting the transition of the interspike interval distribution from unimodal (tonic spiking) to bimodal (bursting). This boundary aligns with the period-doubling (PD) bifurcation line of the complete system. (B) Two-parameter (extracellular potassium [K+]out, and Iapp) fast subsystem bifurcation diagram for the case indicated by E in panel A (Imax = 2 μA/cm2, Iapp = 1 μA/cm2). The necessary condition for bursting, i.e., a bistable region bounded by homoclinic (HOM) and saddle-node (SN) bifurcations from above and below, respectively, is satisfied. (C) Reduced slow subsystem dynamics overlaid onto the bifurcation map of the fast subsystem corresponding to letter E in panels A and B. The spiking branch of the reduced slow subsystem intersects the line d[K+]out/dt = 0, indicated by the black arrow. This intersection signifies the presence, on the spiking branch, of a stable fixed point of the reduced slow subsystem, within the bistable region of the fast subsystem. Consequently, the complete system exhibits tonic spiking, despite the fast subsystem meeting the conditions for bursting (D) One-parameter bifurcation diagram ([K+]out, Imax) of the complete system; Iapp = 0.5 μA/cm2. Insets: zoom in near the borders of the bursting region, where the complete system undergoes cascades of supercritical period doubling bifurcations. Note that this bifurcation diagram is not exhaustive. (E) Consecutive samples along the period-doubling cascade shown in the inset on the right-hand side of panel D. E_5 to E_2: stable limit cycles before the first PD (case 5: I_app = 1.5897 μA/cm2), between the first and second PD (cases 4 and 3: I_app = 1.5767, and 1.5037 μA/cm2, respectively) and between the second and third PD (case 4: I_app = 1.4903 μA/cm2). E_1: chaotic trajectory further down the cascade (I_app = 1.45 μA/cm2), obtained by direct simulation for 90 seconds.
Table 1.
Nomenclature.