Modelling neurons in an extracellular environment

In order to study the emergence of slow rhythmic activity via the interactions of fast spiking and slow concentration dynamics (i.e., fast and slow subsystems, respectively. see next sections), a neuron model equipped with only basic fast spike-generating ion channels is placed in an external environment of fixed volume that contains extracellular potassium ions, whose concentration, [K⁺]_out, can change (Fig 1). The interplay between concentration dynamics and neuronal voltage dynamics is modulated by the Na⁺/K⁺-ATPase in two-fold manner: First, the pump current directly affects the voltage dynamics. Second, the pump changes concentration gradients which, in turn, alter ionic reversal potentials. It will be shown in the subsequent section that this parsimonious setup suffices to create slow rhythms.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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.

https://doi.org/10.1371/journal.pcbi.1011751.g001

The neuron model, inspired by the Wang-Buzsáki model, comprises four currents: The depolarising Na⁺-current and repolarising K⁺-current are part of the spike-generating mechanism. A leak current balances the membrane potential in the rest state. For completing our model, we add a fourth current as a hyperpolarising pump current (I_p). This pump extrudes three Na⁺ ions out of the cell while two K⁺ ions are pumped in. The pump maintains the ionic gradient of the cell, which drives spiking activity. The activity of the pump is regulated by the extracellular potassium concentration, with higher [K⁺]_out resulting in increased pump strain (Eq 7). However, the pump’s activity is limited by its maximum current capacity (I_max), which is influenced by both the electrophysiology characteristic of the pump and its density on the neuron’s membrane. For the mathematical details, see Methods.

With each action potential, K⁺ ions flow out of the cell through the voltage-dependent K⁺ channels and accumulate in the extracellular space. The elevated extracellular potassium concentration ([K⁺]_out), in turn, not only affects the pump current but also influences the neuron’s excitability state via its effect on the reversal potential. Therefore, the dynamics undergoes a sequence of transitions, which form the basis for the rhythmic bursting behaviour and are explained and analysed in more detail in the following sections.

Pump- and extracellular potassium-mediated slow bursting

In this section, we first provide a brief illustrative description of the generic burst mechanism, leaving a detailed analysis of the conditions for its existence to the following sections.

An example trace of voltage and potassium concentration, respectively, for the slow bursting dynamics in a dedicated extracellular volume is depicted in Fig 2A. Each burst cycle contains two alternating states: quiescence and spiking. The rhythmicity of the voltage shows an inter-burst period of about one second. This period is about 30 times longer than the dynamics of one complete action potential in the spiking phase. The number of spikes per burst is fixed (i.e., bursting is deterministic).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Bursting dynamics overview.

(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. (I_app = 0.5 μA/cm², I_max = 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 (I_max) for different I_app. For each curve from right to left (decrease in pump capacity I_max): 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 I_app = 0.3 and 0.5 μA/cm², or initially double period and tonic spiking before entering depolarization block for I_app = 0.7 and 0.9 μA/cm²).

https://doi.org/10.1371/journal.pcbi.1011751.g002

The basic cycle undergone during a burst is summarised in Fig 2B. The burst relies on the bistable nature of the neuron’s dynamics between spiking and quiescence. During the spiking period, the outward flow of potassium ions is considerable. The inter-spike interval does not provide sufficient time for the Na⁺/K⁺-ATPase to pump back all K⁺ ions that left the cell during the spike. Consequently, K⁺ ions accumulate in the extracellular space, resulting in a saw-tooth-like build-up of [K⁺]_out (Fig 2B, spiking phase). Furthermore, during the spiking phase of each burst, the inter-spike interval progressively prolongs as a result of K⁺ accumulation in the extracellular space. Eventually, the rise of [K⁺]_out reaches a level where spiking is not possible anymore (the quiescent phase). Spiking is terminated by a homoclinic bifurcation (HOM) in the fast subsystem, indicated by the blue circle in Fig 2B. In the quiescent phase, the pump activity now suffices to reduce extracellular potassium, as the outward current via the potassium channels is significantly reduced in the absence of spike generation. Ultimately, the decrease of [K⁺]_out, during quiescence results in another bifurcation of neuronal dynamics, corresponding to saddle-node bifurcation (SN) of the fast subsystem, see red square in Fig 2B. In passing this bifurcation, neuronal dynamics are back in the spiking region and the cycle starts all over again.

In Fig 2C, the amplitude of potassium concentration change is plotted as a function of the pump current density for different applied currents. At lower input currents, a decrease in pump density causes the potassium oscillation amplitude to grow. At high input currents, there is an additional jump-like increase in potassium oscillation amplitude that is caused by the chaotic dynamics of period doubling in the complete system bifurcation which is discussed in more detail below.

In summary, the spiking dynamics of the neuron impacts [K⁺]_out, the latter of which, in turn, influences the K⁺ reversal potential and hence the neuron’s voltage dynamics. In the next sections, it will be shown that the identified mechanism is, indeed, a hysteresis loop where one branch of the bistable system corresponds to the spiking state and the other branches constitutes the quiescent state. The properties of the hysteresis loop are then further analysed in the slow-fast analysis section. The remainder of the article focuses on establishing how generic this mechanism is.

Slow-fast analysis

Neuronal bursting consists of an oscillation governed by two time scales: one corresponding to the fast spiking dynamics and one corresponding to the slower dynamics related to concentration changes. Typically, these two different time scales result from either the presence of fast and slow currents [26], stochastic escape [27], external drive, synaptic delays [6], or as here, the slow variation of ions. All these cases benefit from a slow-fast analysis [26].

In the mechanism for the generation of slow rhythms proposed here, the fast subsystem is governed by the millisecond kinetics of the ion channels that shape the neuron’s action potentials. In contrast, the slow subsystem is governed by the timescale of slow changes in [K⁺]_out. The bursting rhythmicity of the system is thus regulated by an interplay between these two subsystems, which crucially depends on the activity of the Na⁺/K⁺-ATPase.

In slow-fast analysis, it is assumed that the timescale difference between the two subsystems is sufficiently large for them to be investigated separately. We will first subject the fast subsystem to bifurcation analysis while assuming that the slow subsystem undergoes only negligible variations on the timescales of the fast subsystem. Hence, [K⁺]_out can be treated as constant and the bifurcation parameter in this scenario. Subsequently, we will explore the dynamics of the slow subsystem assuming time-averaged values of the fast subsystem dynamics. More details on the method are to be found in the Methods section.

[K⁺]_out is a variable of interest featured in all figures related to slow-fast analysis. For consistency, it is depicted on the y-axis in all the figures addressing the dynamics of the system. To facilitate the comparison with other studies, the more conventional representation of the fast bifurcation diagram and slow dynamics (presenting bifurcation parameters on the x-axis) is given in S1 Fig.

Fast subsystem bifurcation analysis

For analysis of the fast, spike-generating subsystem, extracellular potassium concentration is approximated as a constant parameter. The bifurcation diagram in Fig 3 summarises the transitions in dynamics induced by changes in [K⁺]_out.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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, I_app) 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.

https://doi.org/10.1371/journal.pcbi.1011751.g003

Fig 3A shows a two-parameter bifurcation diagram of the fast subsystem spanned by the parameters extracellular potassium ([K⁺]_out) and applied current (I_app) in the vicinity of the region used for the simulated traces in Fig 2A. Overlaid on the bifurcation diagram of Fig 3A are the dynamic changes of the complete system during a burst cycle with the parameters of Fig 2A shown as the black vertical line. If [K⁺]_out is low, the only stable state for the fast subsystem is a stable limit cycle (the dark grey area in both panels of Fig 3). In this region, the neuron spikes tonically. If [K⁺]_out is increased, the dynamics of the fast subsystem traverse the saddle-node (SN) bifurcation (the red line in Fig 3A). From the SN bifurcation, a saddle and a stable node emerge. When [K⁺]_out exceeds the SN bifurcation value, the neuron enters the bistable region (the white region in both panels of Fig 3). In the bistable region, the fast subsystem of the neuron model has two stable states, a stable limit cycle and a stable fixed point, separated by a separatrix originating from the stable manifold of the saddle. By raising [K⁺]_out even more, the blue line of a homoclinic orbit bifurcation (HOM) is crossed, at which the limit cycle of the action potential collides with the saddle and is annihilated. If [K⁺]_out passes this line, the fast system dynamics end up in the monostable zone, because at the HOM bifurcation, the stable limit cycle disappears. Hence, in the light grey part of Fig 3A, the neuron is in a quiescent state.

To better understand the bursting dynamics, the voltage trajectory of one burst cycle in the complete system (black solid line) is superimposed on the codimension-one ([K⁺]_out) bifurcation diagram of the corresponding fast subsystem (coloured lines) in Fig 3B. The black trajectory depicts the burst dynamics presented in Fig 2A, however, instead of time the corresponding levels of [K⁺]_out are shown. The resulting phase portrait encompasses the complete system dynamics of the burst: In the dark grey region of Fig 3B the fast subsystem is in the tonic spiking mode. Here, the complete system (black trace) is going through a spiral path. Each rotation corresponds to one action potential. With each action potential, more K⁺ accumulates in the extracellular space (see also Fig 2). By increasing [K⁺]_out, the system enters the fast subsystem’s bistable region (white area) and continues on the spiking branch. Note that if the neuron model was subjected to strong perturbations, jumps between the two attractors, a stable fixed point and a stable limit cycle, would occur. The effective increase in [K⁺]_out per spike diminishes as spiking progresses because the pump becomes more and more active due to the increase of extracellular potassium (Eq 7). Furthermore, the inter-spike interval increases because of the changes in the spiking dynamics caused by changes in the potassium reversal potential due to the accumulation of extracellular potassium.

Spiking continues until, eventually, [K⁺]_out reaches the HOM bifurcation (blue circle) and the stable limit cycle of the fast subsystem is annihilated. Here, the system enters the quiescent phase of the fast subsystem (light grey zone). In this phase, the only stable state is the rest state. Hence, the system trajectory leaves the spiking spiral and converges to the stable fixed point (orange line). In the rest state, [K⁺]_out is efficiently pumped back into the cell by the Na⁺/K⁺-ATPase (see also Fig 2). The pump activity increases the resting state voltage until an SN bifurcation is reached (red square). The saddle and the stable node collide and annihilate at this point. Thereby, the resting state loses stability and the system converges back onto the tonic spiking dynamics (dark grey), where the next burst cycle is initiated; for a zoom out of the bifurcation diagram in Fig 3B see S1 Fig.

Notable in both panels of Fig 3, there is a gap between the SN bifurcation and the onset of the first spike in a burst cycle. In this region, the voltage dynamics cannot, after passing the SN bifurcation, immediately jump to the spiking branch. The dynamics is slowed down by the proximity of the saddle-node bifurcation in the eigendirection associated with the eigenvalue that is still close to zero. This phenomenon is often referred to as ghost of the SN bifurcation (or, for brevity, the SN ghost in the rest of the article). Thus, the decrease of the [K⁺]_out does not immediately stop when the system reaches the SN bifurcation. [K⁺]_out is further reduced when moving away from the SN ghost until it can finally enter the spiking phase.

Slow subsystem analysis

The previous section analysed the spiking behaviour with the slow variable, specifically, the extracellular potassium concentration ([K⁺]_out) treated as a parameter. The fast subsystem bifurcations that initiate and terminate spiking were identified. In this section, the dynamics of the slow subsystem is explored. The time scale of the slow subsystem is determined by the effect of spiking on [K⁺]_out in relation to the balancing effect of the Na⁺/K⁺-ATPase. Note that the effective pump rate depends on both the extracellular potassium concentration as well as the density of ATPase proteins (indirectly defined by I_max, see the Eq 7). Relevant quantities for the slow subsystem are thus the amount of K⁺ leaving the cell per spike and the effective pump rates during the spiking and quiescent phases.

The slow concentration dynamics can be isolated by applying the method of averaging to the fast dynamics. It exploits the effect that the slow subsystem cannot track the rapid changes in the fast subsystem; effectively, it only senses the average impact of one complete spike. To calculate this average impact on the change of [K⁺]_out, the pump current and the current carried by the fast spiking potassium channels are integrated over one spike period. The averaged quantity of the fast subsystem currents determines the time derivative of the slow subsystem and, consequently, its dynamics (see Eq 14). Note that when the fast subsystem is quiescent (i.e., no spiking), averaged and non-averaged fast subsystem dynamics are equal. With this averaging method, we can describe the dynamics of the slow subsystem as a reduced model that can capture the complete system dynamics. See Methods for more details on the averaging method and [26] for an introduction.

Fig 4 shows the phase portrait of the reduced slow subsystem, in terms of [K⁺]_out and its temporal derivative, for the specific parameters used in Fig 2. In analogy to Fig 3, in Fig 4 areas of tonic spiking (dark grey), bistability (white), and quiescence (light grey) of the fast subsystem are marked. The HOM and SN bifurcations of the fast subsystem mark the borders of these zones, (blue and red horizontal lines). Trajectories of the reduced slow subsystem corresponding to spiking and quiescence of the fast subsystem are indicated by the labels ‘spiking’ and ‘rest’, respectively.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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.

https://doi.org/10.1371/journal.pcbi.1011751.g004

Fig 4 reveals the hysteresis nature of the slow subsystem (reduced system), which revolves around the bistability of the fast subsystem. The bistability of the fast subsystem serves as the prerequisite for the existence of the hysteresis loop in the reduced slow subsystem. For clarification of the mechanism, we follow the [K⁺]_out hysteresis loop in Fig 4 through the same sequence of bifurcations as in Fig 3B. If our fast subsystem is spiking (equivalent to the spiral path in Fig 3B), this corresponds to the spiking branch of the hysteresis loop in Fig 4 and the time derivative of [K⁺]_out is positive. This means that in each cycle of an action potential, the mean K⁺ flux into the extracellular space is larger than the mean of the K⁺ flux via the pump, which reabsorbs potassium ions into the neuron. In other words, extracellular K⁺ accumulates continuously in extracellular space. However, while the slow subsystem slides up on this branch (from right to left) through the bistable region, the potassium increment reduces with every additional spike. With the parameters used here, the spiking branch of the slow subsystem terminates at the HOM bifurcation before the derivative of [K⁺]_out turns negative. This observation depends on the system’s parameters, as it is further explored below. After the HOM bifurcation of the fast subsystem, the slow subsystem falls back to the stable resting branch.

In the quiescent state of the fast subsystem (light grey region), the K⁺ flux via the Na⁺/K⁺-ATPase dominates, as there is no spiking and, consequently, no major flux of potassium out of the neuron. Therefore, the time derivative of [K⁺]_out is negative (Fig 4, rest branch). [K⁺]_out decreases and potassium is pumped back into the neuron. The slow subsystem follows the quiescence branch from higher to lower [K⁺]_out through the bistable zone (equivalent to sliding on the orange line in Fig 3B). At the SN bifurcation (red line), the stable node disappears from the fast subsystem’s dynamics. As a consequence, the silent branch comes to an end at a lower [K⁺]_out. Subsequently, the reduced slow subsystem undergoes a transition to the spiking branch after passing through the ghost of the SN bifurcation. This completes the hysteresis loop of the reduced slow subsystem.

In summary, the bursting process is enabled by the bistability of the fast subsystem and corresponds to a hysteresis loop. The bistability allows the system to alternate between quiescence (where the activity of the Na⁺/K⁺-ATPase suffices to extrude potassium from the extracellular space) and spiking (marked by accumulation of potassium in the extracellular space).

Generic occurrence of the burst mechanism

The previous sections described how, for specific parameter settings, class I conductance-based neuron models can exhibit slow bursting when the dynamics include the activity-dependent concentration changes in an extracellular space of fixed volume. The following section investigates the robustness and prevalence of the described burst mechanism and its dependence on model parameters.

The pump’s essential contribution to the bursting mechanism.

To investigate the influence of the electrogenic Na⁺/K⁺-ATPase, we switched off the pump current in the model’s voltage equation, yielding a system similar to [15] (see Fig 5A, and compare it to the full model with an electrogenic pump current Fig 5B). At higher input currents, the Na⁺/K⁺-ATPase switches the system from turning towards the depolarization block in all cases to a system that can transition from rest via tonic firing to proper bursting and only then depolarization block. The electrogenic pump activity shifts the depolarization block to higher inputs.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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 I_app (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 I_app between the SNL (green circle) and the HOM inflection point (purple square). I_max = 1 μA/cm² in all panels.

https://doi.org/10.1371/journal.pcbi.1011751.g005

As the mechanism of bursting relies on a region of bistability in the fast subsystem, one may note that the generic unfolding of a codimension-two saddle-node loop (SNL) bifurcation [28] from the neuron model with an electroneutral pump includes a region of bistability (Fig 5C) [29]. For the burst mechanism to work, however, the fast dynamics along the [K⁺]_out direction (with constant I_app in Fig 5C) needs to be bounded by HOM and SN branches for higher and lower values of [K⁺]_out, respectively (compare Figs 5C and 3A). This is possible, if the bifurcation diagram is sheared such that the HOM and SN branches bend to the right. Interestingly, such shearing occurs as a result of adding a K⁺-dependent electrogenic pump (Eq 7) to the neuron model with an SNL bifurcation (see Fig 5D). The pump current acts as an applied current that depends on [K⁺]_out; in our model it is a sigmoidal function with half activation at 11 mM. The pump induces a shear transformation of the canonical unfolding of the SNL bifurcation (Fig 5D). The larger the maximum pump current (I_max), the larger the shear. The Na⁺/K⁺-ATPase density in the membrane (directly related to I_max), hence is a crucial parameter for the emergence of bursting by the minimalistic mechanism proposed here.

The region of burst occurrence in the fast subsystem (yellow area in Fig 5D) is bounded by the SNL point (green circle) and the inflection on the HOM branch (purple square). For I_app beyond the inflection point, spiking continues until [K⁺]_out reaches a fold of limit cycles (FLC), which is analysed in the next section.

In summary, the burst mechanism relies on two essential ingredients from the neuron’s fast subsystem, which enable the hysteresis loop:

1. For low [K⁺]_out, the onset of spiking in the fast subsystem occurs via a saddle-node on invariant cycle (SNIC) bifurcation (typically present in neurons with class I excitability). This implies that the neuron model also undergoes an SNL bifurcation; at higher [K⁺]_out [22,25], opening up the bistable region between a homoclinic and a SNIC branch (Fig 5C).
2. The electrogenic Na⁺/K⁺-ATPase counteracts the K⁺ efflux and mediates the coupling between [K⁺]_out and neuronal voltage dynamics. The pump’s activity shears the bistable region such that it can be traversed by changes in [K⁺]_out while being bounded by a fixed-point region (quiescence) from above and a limit-cycle region (tonic spiking) from below (Fig 5D).

Dynamics in the proximity of the bursting region.

To comprehensively understand the model’s dynamics, we need to take a detailed look at the dynamics occurring in the vicinity of the bursting in the full system to shed light on factors that constrain bursts, the dependence on parameters, and the robustness of the described mechanism.

Zooming further out of the K_out-I_app plane depicted in Fig 3A reveals additional structures in which the region of bistability of the fast subsystem is embedded (Fig 6A). The vertical trajectories overlaid on the fast subsystem bifurcation show the direction of the slow subsystem (i.e., complete system) dynamics. According to Fig 6A, variation of the applied current resulted in different dynamics of the complete system. The slow bursting is possible in the range of I_app between SNL (green circle) and the HOM inflection point (purple square), as is the case in the yellow area in Fig 5D. The specific bursting dynamics of Fig 2A is found in the parameter region D in Fig 6A.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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 (I_app) 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 I_app 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 (I_app = 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 (I_app = 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. I_max = 1μA/cm² for all the panels.

https://doi.org/10.1371/journal.pcbi.1011751.g006

For I_app higher than the HOM inflection point (purple square in Fig 6A), the complete system dynamics ends up in depolarization block. After passing through a Hopf bifurcation, the neuron loses the ability to spike in a fold of limit cycles (FLC) bifurcation. This can be referred to as a depolarization block. The letter B in Fig 6A corresponds to one example that converges towards depolarization block. If the system starts in low [K⁺]_out at position B, i.e., the tonic spiking region of the fast subsystem (dark grey area), with each spike, [K⁺]_out increases, moving the system upwards along the flow field direction of the complete system (filled black arrows in Fig 6A). The system passes the bistable zone but does not encounter the HOM bifurcation. The limit cycle terminates in an FLC bifurcation instead. The same can also be seen in the dynamics of the reduced slow subsystem for the same modelling parameters (Fig 6B). Here, if we start from the spiking branch (solid line), the time derivative of [K⁺]_out is positive. Hence, the system slides on the spiking branch from lower to higher [K⁺]_out, until it reaches the FLC bifurcation. At this point, the spiking branch terminates, and the system jumps to the resting branch with a positive time derivative of [K⁺]_out instead of the normal resting state. The time derivative of [K⁺]_out on this branch is positive because this state corresponds to an upstate voltage at which more voltage-gated channels are in a conducting state. With the given I_max, the pump is insufficient to compensate and [K⁺]_out continues to increase until the silent branch eventually approaches zero in a very large [K⁺]_out. If the system starts at [K⁺]_out higher than the SN bifurcation of the fast subsystem and is on the other resting branch of the slow subsystem (left side of the Fig 6B), however, the negative time derivative moves the system along this branch in the direction of decreasing [K⁺]_out. As it progresses, the system reaches the SN bifurcation, transitioning into a spiking state, and eventually leading to the depolarization block.

At I_app below the SNL bifurcation in Fig 6A, tonic spiking is possible. An example from the tonic spiking area is indicated by the letter C in Fig 6A. The bistable region, which is a requirement for the bursting mechanism under investigation, is absent in the fast subsystem bifurcation for this case. As shown in Fig 6A, in the vicinity of the position marked C, two dynamical regimes for the fast subsystem exist: tonic spiking and quiescence. Why has the complete system a stable state at tonic spiking? The answer can be read of Fig 6C, which depicts the time derivative of the reduced slow subsystem. The spiking branch of the reduced slow subsystem crosses zero with a negative slope (see arrow in Fig 6C); the reduced system has a stable fixed point. This fixed point is in the region where the only attractor in the fast subsystem is a stable limit cycle. The silent branch in Fig 6C is lower than zero. Consequently, if we start on this branch, the reduced slow subsystem eventually jumps to the spiking branch and reaches the stable fixed point. Thus, the complete system in the region around C ends up in the spiking regime.

Note that, in Fig 6A, for I_app lower than the HOM inflection point of the fast subsystem and [K⁺]_out higher than HOM bifurcation in the bistable region of the fast subsystem, the complete system dynamics are similar to the dynamics of Fig 6B, where [K⁺]_out is higher than the fast subsystem SN bifurcation. Therefore, two directions for the complete system dynamics exist. The first one increases [K⁺]_out, resulting in depolarization block. The second one decreases [K⁺]_out, in contrast to the scenario in Fig 6B, resulting in either tonic spiking or bursting, in dependence of the size of I_app. Hence, the dynamics of the complete system exhibits bistability (depolarization block versus either bursting or tonic spiking).

In this section, we demonstrated that our basic neuron model possesses the ability to not only generate slow rhythmic bursting activity but also to exhibit other significant dynamics such as tonic spiking and depolarization block. Each of these dynamics plays a crucial role in both physiological and pathological scenarios.

Dependence of the bursting mechanism and other dynamics on the pump density.

We saw that Na⁺/K⁺-ATPases are crucial for the feedback from the fast to the slow dynamics. The pump density, denoted in our model by I_max, is thus a natural parameter to investigate the robustness of the described bursting mechanism, regarding both the existence and extent of a bursting region. In Fig 7A, the complete system dynamics is explored in the I_max-I_app plane. Four regions are identified, each associated with a distinct regime of the complete system: rest, tonic spiking, bursting, and depolarization block. As a reminder: In the previous sections we examined representative cases from the bursting (Figs 2–4 and label D in Fig 6A), depolarization block (label B in Fig 6A and 6B), and tonic spiking (label C in Fig 6A and 6C) regions, all of which shared the same I_max value (1 μA/cm2). In contrast, here I_max is varied.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. Dynamics of the complete system for varying pump density.

Depending on the maximal pump current (I_max) and applied current (I_app), 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 I_app) fast subsystem bifurcation diagram for the case indicated by E in panel A (I_max = 2 μA/cm², I_app = 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.

https://doi.org/10.1371/journal.pcbi.1011751.g007

First, we can see in Fig 7A that the larger the pump density, the larger the applied current needs to be to achieve depolarization block. For completeness, note that, as mentioned in the analysis of Fig 6, in the regions labelled as rest, tonic spiking and bursting, the complete system can also directly converge to depolarization block when the initial potassium concentration is large, depending on the other initial conditions. In contrast, in the region labelled as depolarization block, i.e., for I_app beyond the HOM inflection point of the fast subsystem (purple dotted curve), the depolarization block will inevitably be reached, even starting from low [K⁺]_out. Numerical continuation of the complete system shows that the limit cycle corresponding to tonic firing disappears in a fold of limit cycle bifurcation (pink curve in Fig 7A; pink star in Fig 7D). Beyond this bifurcation, only the large-[K⁺]_out stable branch of fixed points remains (orange solid curve in Fig 7D), leaving depolarization block as the only possible outcome.

We now examine the bursting region. In the preceding sections, we established the essential role of the pump current in opening up a range of applied current values for which bursting can potentially take place: for a constant I_app between the fast subsystem SNL bifurcation and HOM inflection point, the fast subsystem possesses a region of bistability, bounded by the homoclinic and saddle-node bifurcations (Fig 5D). This configuration is a necessary condition for the formation of the hysteresis loop that drives the bursting dynamics. The bursting region (yellow area in Fig 7A) is therefore constrained by the SNL bifurcation and HOM inflection lines of the fast subsystem (depicted in dotted green and purple lines, respectively). The bursting region initially expands with Imax as these two lines diverge, until at some point, although the requirement for bursting is still met, it shrinks and terminates. In the remaining area between the two lines, the complete system exhibits tonic spiking instead of bursting.

We conducted a case study in this particular zone, at the point in parameter space position marked E in Fig 7A. The corresponding slow-fast analysis diagrams are shown in Fig 7B and 7C. The fast subsystem does indeed fulfil the essential prerequisites for bursting: bistability between quiescence and tonic spiking for potassium concentrations comprised between the homoclinic and saddle-node bifurcations (Fig 7B). However, the reduced slow subsystem features a stable fixed point inside this region of bistability, marked with an arrow in Fig 7C. It corresponds to a stable limit cycle of the complete system, during which the contributions of the pump and voltage-gated channels to potassium dynamics cancel each other out on average. In this situation, in contrast to the case studied in Fig 3–4, the hysteresis loop fails to form and the complete system stabilizes in a tonic firing state (position E in Fig 7B) instead of bursting.

When approaching the bursting region from the tonic spiking region, the stable fixed point of the reduced slow subsystem moves progressively closer to the homoclinic bifurcation of the fast subsystem. Near the bifurcation points of the fast subsystem, the slow-fast method and analysis of the reduced slow subsystem fails because the timescale of the fast subsystem increases to the point the separation of the fast and slow timescale is not possible anymore. To better understand the transition between tonic spiking and bursting, we performed a numerical continuation of the complete system. We find that the limit cycle corresponding to tonic spiking destabilizes in a supercritical period doubling cascade to become the bursting solution (light blue curve in Fig 7A, light blue dots in Fig 7D). An in-depth investigation of the transfer to the bursting state in this region as well as of mechanisms that add extra spikes during bursting (also termed spike-adding mechanisms [30]) is beyond the scope of this work. We can, however, predict that it involves both chaos-induced (see period-doubling route to chaos in Fig 7D and 7E) and canard-induced (see jump-on canard solution in S2 Fig) mechanisms, similarly to what was reported for the phenomenological Hindmarsh-Rose neuron model, for example in [31]. Such phenomena are commonly found in Fold/HOM bursters and have been thoroughly analysed in several models of excitable cells [32–37]. Period doubling has previously been shown to be inducible by changes in the extracellular bath concentration [16].

From this section, it becomes clear that the Na⁺/K⁺-ATPase plays a pivotal role in various dynamics possible in models with class I spike generating dynamics, when extracellular potassium dynamics are taken into account. A higher pump density contributes to postponing the depolarization block. Moreover, while the pump can establish the necessary conditions for bursting, a pump current that is too strong can also suppress bursting incidents.

Finally, we briefly note that the mechanism was also tested in the presence of dynamic Na⁺ concentrations. To this end, we modified the model by incorporating intracellular sodium concentration ([Na⁺]_in) dynamics through Eqs 11 and 12 and by adjusting the Na⁺/K⁺-ATPase equation to also account for changes in [Na⁺]_in (see Eq 13). The result is shown in S3 Fig. The desired bursting mechanism is present in the modified model, too. However, including [Na⁺]_in dynamics in the model can introduce more intricate behaviours. For example, in of S3A Fig, sodium slightly accumulates over time and can result in alterations of spike amplitude and spiking threshold [38]. These dynamics, however, are not the focus of this paper.

The relevance of the Na⁺/K⁺-ATPase

Our findings highlight the large influence of Na⁺/K⁺-ATPases [39,40] on the dynamics of neuronal spiking. Different isoforms of the pump are known as generic players in the homeostasis of extracellular potassium, the resting potential, as well as cell volume. Previous experimental evidence already indicated that neuronal Na⁺/K⁺-ATPase function extends beyond the stabilization of extracellular potassium. Along these lines, it has been argued that if most of the potassium secreted by neurons was siphoned off by astrocytes, neuronal activity could not continue for tens of seconds because of imbalances in potassium concentration [41]. Our model indicates yet another function of the Na⁺/K⁺-ATPase: as an enabler of rhythmic activity when extracellular potassium concentration is elevated. While previous studies assigned a mechanistic role to the Na⁺/K⁺-ATPase for rhythmic bursting in pattern generators [42,43], it was the interaction between h-currents and the pump for control of Na⁺ that was emphasized; the burst mechanism proposed in this paper was not investigated. The involvement of the Na⁺/K⁺-ATPase in the induction of rhythmic activity is consistent with a previous proposal [44]. Kager and colleagues indicated that [K⁺]_out or voltage-dependent ion channels that elicit inward currents can generate seizure-like rhythmic activity, provided that the inward current secondarily triggers the release of K⁺ into a confined extracellular space–a set of conditions met by the Na⁺/K⁺-ATPase.

Our model predictions match a number of observations in pathological states. For example, our results that a higher pump density shifts the initiation of the bursting mode and depolarization block towards higher input currents agree with the experimental observation that the Na⁺/K⁺-ATPase helps to inhibit seizures and SD. Specifically, inactivation of the neuron-specific alpha3-isoform of Na⁺/K⁺-ATPase has been shown to induce seizures in mice [45]. Further, the loss-of-function mutation that causes chorea-acanthocytosis is accompanied by an impaired capacity of the Na⁺/K⁺-ATPase in neurons, and epilepsy is among the symptoms of this neurodegenerative disorder [46], potentially fostered by the bursting mechanism at hand. Even partial inactivation of the Na⁺/K⁺-ATPase with ouabain has been shown to cause SD-like depolarization: An epileptic population spike is followed by an ouabain-induced SD [47], which is consistent with the cellular dynamics observed in the model presented here. Moreover, a shortage of oxygen or ATP as it may, for example, occur during ischemia, renders the pump less effective. We demonstrate that an insufficient pump current can first induce bursting and then facilitate depolarization block. This agrees with depolarization events occurring in hypoxic and ischemic conditions [47,48] as well as previous suggestions that hypoxia or an impaired pump can move neurons into seizure states [44,49]. In addition, an experimental study has revealed a correlation between seizures and lowered oxygen pressure levels. Decreases in oxygen pressure can first induce seizures but also stop them at even more oxygen-deprived levels [50]. Our model, in alignment with these experimental findings, indicates that–although an insufficient pump current can trigger seizures (i.e., bursting)–a too strong reduction in pump activity can again terminate bursting (and thus potentially seizures) because the pump’s activity positively correlates with oxygen levels [49].

The Na⁺/K⁺-ATPase density sets four dynamical regimes

The present biophysical model also demonstrates the ability to exhibit both bursting and depolarization block, which aligns with previous observations indicating that seizures and spreading depolarizations (SD) can originate from the same neuronal population [51,52]. Depending on Na⁺/K⁺-ATPase density and input strength, the complete concentration-dependent system exhibits four different regimes: rest, tonic spiking, bursting, and depolarisation block, which have been associated with a range of healthy and/or pathological states. For example, seizures can be triggered by a handful of neurons going through bursts of activity [53]. In SD, the phase of neuronal hyperactivity preceding the depression can arise from synchronized activity of bursting neurons, as observed by population spikes in electrocorticography (ECoG) traces [54]. In the wave of death (i.e. the massive and simultaneous depolarization of neurons [55]) or when the SD wave changes from hyperactivity to silence, depolarization-block-like dynamics could be observed [51]. The changes in extracellular potassium concentration during bursting and depolarization block in our model are consistent with experimental observations during seizures and SD, as well as during sleep rhythms [56–58].

The SNL bifurcation as an organisational center for bursting

At the core of the present burst mechanism is the SNL bifurcation and the bistable region initiating from it in the fast subsystem. The latter is bounded by the HOM and SN branches, i.e., the two codimension-one bifurcation branches that appear from the codimension-two SNL bifurcation. Interestingly, the majority of seizures observed in patients and experimental models are assumed to rely on an SN/HOM bifurcation [52]. The minimalistic model by Depannemaecker et al. discovered this kind of onset-offset combination (SN/HOM) in their oscillations. Unlike the present model with a one-dimensional slow subsystem, their model features a two-dimensional slow subsystem. Other mechanisms for bursting do not involve SN or HOM bifurcations at the onset and offset of the burst (e.g., [15,16,51,59]).

The SNL bifurcation itself originates from the unfolding of the codimension-3 degenerate Takens–Bogdanov singularity which is present in a large class of conductance-based models [19]. We demonstrate that the minimal bursting mechanism analysed here requires the involvement of the Na⁺/K⁺-ATPase and ordinary K⁺ and Na⁺ channels, which are omnipresent throughout the brain.

Proximity of an SNL bifurcation can be found in a generic class of neuron models, i.e. all models starting with class I excitability [21,22], which encompass a significant proportion of neuron models, ranging from isolated gastropod somata to mammalian hippocampal neurons [25]. Note that in the bistable region between the HOM and SN bifurcations, the presence of noise in the system or its input can result in stochastic bursting that emerges from switching between the stable limit cycle attractor and the fixed-point dynamics (see, for example [38]).

Here, in contrast, the bursting mechanism is deterministic and stems from the feedback mediated by the Na⁺/K⁺-ATPase. Specifically, the electrogenic pump induces a distortion of the bifurcation diagram of the fast subsystem, which is crucial for the actual occurrence of deterministic bursting. If the pump, however, is not included in the voltage dynamics (as, for example, in [15,60,61]), a shear-induced hysteresis loop is not observed.

The relevance of potassium homeostasis

The dynamics described in this study does not explicitly consider other ionic transporters, or external potassium regulators, such as diffusion, or glial cells [16,17,62–66]. The model’s extracellular space can, however, be interpreted as an effective compartment that also subsumes possible additional potassium homeostasis mechanisms. Hence, the pathological dynamics introduced in our mechanism could arise from anomalies in these kinds of regulators. This hypothesis is consistent with the experimental observation that dysfunctional astrocytes are crucial players in epilepsy, assuming that extracellular potassium homeostasis is impaired as a function of the astrocytic impairment [67]. Other experiments have implicated dysfunctional astrocytes in SD [68]. We note that although our model produces intriguing dynamics at the level of a single cell, it cannot generate some of the more complex behaviours, such as recovery from depolarization block. This recovery requires the explicit involvement of other potassium regulators [17], exceeding the scope of this paper.

Relation to previous findings

Our study demonstrates that the interplay between potassium dynamics and neuronal excitability in class I neuron models with SNIC dynamics can give rise to slow bursting under minimal conditions. The general idea of generating diverse spiking dynamics via feedback from increased extracellular potassium has been previously reported and is supported by experimental evidence. For example, fluctuations in extracellular potassium levels have been observed to co-vary with the electroencephalogram (EEG) and field-potential waves during sleep or seizures [56,57,69]. Increases in extracellular potassium have been reported during pathological conditions like seizures and spreading depression [70–72]. Moreover, prior studies have shown that accumulated extracellular potassium does not rapidly diffuse away from neurons, thereby providing a feedback signal that can influence neuronal dynamics [73–77].

On the modelling side, a number of studies have explored the generation of slow deterministic bursting involving changes in ion concentrations. Sætra et al. [78] employed a Pinsky-Rinzel model to highlight the significance of considering ionic concentration dynamics, diffusion, and electrical drift. Bazhenov et al. [12] showed that increasing extracellular potassium levels, alongside various currents triggers rhythmic activity in both individual cells and networks. Also, Kager et al. [44], reported a slow bursting activity (around 0.06 Hz), arising from the interplay of potassium and sodium concentration dynamics in a complex multi-compartmental model with dendrites and glia. In contrast, Øyehaug et al. [59] utilized a similar model but attributed the slow oscillatory behaviour to the slow dynamics of glial cells.

In two papers, Hübel et al. [16,17] demonstrated that the Na⁺/K⁺-ATPase pump alone is insufficient to recover the system from depolarization block (referred to as free energy-starvation). Through complete system bifurcation analysis, they revealed that models augmented solely with the pump as a homeostatic regulator and within certain ranges of pump strengths, exhibit bistability between physiological states and depolarization block. They further showed that adding another ionic regulatory mechanism, such as a bath or glia, to the model can restore the system from depolarization block back to physiological conditions after a significant phase space excursion in ionic variables. The bistability between physiological cell state and a state of free energy starvation, i.e., depolarisation block, studied in their work, also exists in our model. It is important to note, firstly, that this bistability is different from the bistability in the fast subsystem that the burst mechanism relies on, and secondly, that the free energy depleted depolarisation block, a property of the whole system, is not the same as the Hopf curve in our fast bifurcation diagram, which also terminates spiking. Barreto et al. [15] developed a model incorporating sodium and potassium concentrations as slow subsystems but no pump current in the voltage equation. Their model is capable of producing slow-wave bursting particularly with very high variations in concentration, notably extracellular potassium. Prior to this work, Cressman et al. [60] examined a similar model. By sketching the bifurcation diagram of their slow subsystem, they investigated the effect of different potassium homeostasis (specially bath diffusion and glia) mechanisms on their model. Building upon a similar framework, Wei et al. [51] aimed to enhance the realism of neuron modelling by introducing additional elements such as more realistic glial ionic currents, oxygen and volume dynamics, and the Goldman–Hodgkin–Katz formalism. As a result, their model exhibited a diverse spectrum of dynamics including steady-state, seizure, spreading depression, tonic firing, and wave of death. Depannemaecker et al. [52], in turn, presented a more minimalistic model in which the potassium concentration in the bath compartment is regulated by diffusion. Dependent on the bath’s potassium concentration, this model can generate diverse rhythms. The majority of previous modelling work, however, has predominantly focused on ion-channel-based mechanisms with channel dynamics slower than those typically required for spike generation. Alternatively, concentration dynamics paired with fast spike generating channels were augmented by additional features like multiple compartments or homeostasis mechanisms. In contrast to some earlier studies on bursting induced by ionic concentration [15,60], which required multiple slow variables (termed slow-wave burst [9,19]), our model demonstrates that a single slow concentration ([K⁺]_out) suffices for burst formation. This type of burst is categorised as a hysteresis loop burst [9,19]. In our model, two conditions need to be met for the hysteresis loop to form: First, a bistable region must be part of the bifurcation structure, which is typical for the SNL unfolding in class I neurons. Second, the bifurcation structure must be arranged such that the slow variable can traverse the saddle-node and HOM branches that delimit the bistable region, to respectively initiate and terminate spiking. In our model, this tuning is achieved through the electrogenic action of the ubiquitous Na⁺/K⁺-ATPase. Due to the universality of the identified mechanism, which applies to any neuron with class I excitability, our results demonstrate the sufficiency of the Na⁺/K⁺-ATPase for the generation of slow rhythmic bursting across a large set of neuronal cell types.

Summary

In summary, we present a minimal mechanism for deterministic bursting activity that arises from the interplay of very common spiking dynamics with changes in the extracellular potassium concentration mediated by a ubiquitous Na⁺/K⁺-ATPase. While other, more complex models, exhibit deterministic bursting too, the mechanism described in this study relies only on a minimal set of physiological assumptions in neurons and its dynamics agree with experimental observations in pathological states like epilepsy or spreading depolarization. The underlying mechanism strengthens the role of Na⁺/K⁺-ATPases in the generation of slow rhythms and their relevance as a therapeutic target in pathology. While more complex models may capture bursting dynamics faithfully, the minimal model enabled us to deduce the exact mechanism involved in the bursting sequence. We anticipate that this detailed understanding, as well as the efficiency of the model in numerical simulations, will facilitate analyses aimed at mechanistically linking the biophysics of neurons to the behaviour of the networks that embed them.

Model description

In the following, we describe the parsimonious biophysical model, composed only of two fast spike-generating currents and a Na⁺/K⁺-ATPase pump as well as extracellular potassium concentration ([K⁺]_out) dynamics.

Our model is based on the Wang-Buzsáki model [79]. As it is shown in Fig 1, the model consists of two compartments: the neuron and its extracellular space. The outward potassium current through the neuron’s voltage-dependent channels will increase the concentration of [K⁺]_out. Conversely, [K⁺]_out is decreased by the Na⁺/K⁺-ATPase pump activity. We calculated these changes by the mass conservation equation of [K⁺]_out. Variation in [K⁺]_out affects the reversal potential of the potassium, which is calculated by the Nernst equation.

The Wang-Buzsáki model.

The Wang-Buzsáki (WB) model is a modified version of the famous Hodgkin-Huxley (HH) model. However, contrary to the HH model for invertebrate motoneurons, the WB model is categorized as a class I neuron model, mimicking fast-spiking vertebrate interneurons dynamics. The model has two fast spike-generating sodium and potassium voltage-dependent channels [79]. The potassium and sodium currents are described below: (1) (2) where the sodium channel inactivation variable h and potassium channel activation variable n are defined with the following equations: (3) (4) (5) Within the framework of the Wang-Buzsáki model, the activation variable of the transient sodium current (m) is considered to evolve rapidly and is thus replaced with its corresponding steady-state function.

This approach serves as a means to simplify the WB model, transitioning it from a four-dimensional system to a three-dimensional one.

The Wang-Buzsáki model also has a leak current, which is described below: (6) All of the symbols and constants of this section are described in Table 1.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Nomenclature.

https://doi.org/10.1371/journal.pcbi.1011751.t001

Slow-fast method

The deterministic bursting introduced in this paper results from the interaction between dynamics on different timescales. First, the fast spiking timescale mostly determined by gating variables of channels and the membrane time constant (Eqs 3, 4 and 8). Second, the slower timescale of the concentration dynamics. The slow dynamics is represented by variation in [K⁺]_out (Eq 9).

To gain a better understanding of our system’s behaviour, we analysed our neuron model using the method of timescale separation [9,26]. For this, one needs to assume that slow and fast timescales are sufficiently separated. As a result, the fast subsystem perceives the slow subsystem ([K⁺]_out) dynamics as a constant parameter. On the other hand, the oscillations of the spiking dynamics are so fast for [K⁺]_out dynamics that it only senses the average changes in the fast subsystem. By using this strategy, we can consider the fast and slow subsystems separately.

For the fast subsystem, we treat [K⁺]_out in the system of equations including Eqs 3, 4 and 8, as a bifurcation parameter. Then with the help of numerical continuation [81], we produce the bifurcation diagram of the fast subsystem. This is how we generated Figs 3, 5C, 5D, 6A, 7B and S1.

To analyse the slow subsystem, we apply a numerical averaging method to obtain the reduced slow subsystem [9,18]. The time evolution of the slow subsystem ([K⁺]_out) is given by Eq 9. Actually, the slow subsystem does not respond to the fast changes in the fast subsystem during the spiking phase but only perceives the average effect of each action potential. Therefore, for each desirable [K⁺]_out, we can average the terms that relate to the fast subsystem (I_P and I_K in Eq 9) over one full action potential limit cycle, as shown below: (14) where τ is the period of the fast subsystem’s limit cycle and the subscript "fast" shows that I_P and I_K belong to the fast subsystem in specific [K⁺]_out.

For each spiking branch of the fast subsystem, we discretize the [K⁺]_out range that the branch covers and compute the [K⁺]_out derivative with the averaging method (Eq 14) for each [K⁺]_out value. This allows us to calculate the reduced system for each spiking branch of the fast subsystem.

When the fast subsystem is quiescent and on its stable fixed point, to determine the slow subsystem dynamics, it is enough to take, at the desired [K⁺]_out, the fast subsystem’s fixed-point value of I_P and I_K and put it in Eq 9.

The result of the averaging method as reduced slow subsystem time derivative diagrams are shown in Figs 4, 6B, 6C and 7C.

The averaging method usually breaks down if the time scale difference between the slow and fast subsystems becomes too small. This situation can arise near the fast subsystem bifurcations, such as in the vicinity of the HOM bifurcation, where the spiking frequency is significantly reduced [26]. Hence, in these situations, we validate our results with numerical simulations to ensure the accuracy of our results.

Numerical implementation

We performed numerical simulations using Python 3 [82] and the Brian 2 package [83]. The Runge-Kutta fourth-order method was utilized with time steps of 0.005 ms, and in certain instances, adaptive time steps were employed. In case of fixed time step method, we verified the results with smaller time steps, which yielded consistent outcomes. Additionally, we carried out the continuation of the bifurcation diagrams using XPPAUT [81] and Auto-07p [84].