Model response to prolonged stimuli

In order to analyze how neurons respond to prolonged stimulation, we examined the temporal evolution of activity-dependent changes in transmembrane ionic gradients and assessed their impact on ongoing neuronal activity. To this end, we implemented a single-neuron, conductance-based model including dynamic ion concentrations (detailed in the Materials and methods section). Ionic gradients determine the equilibrium (Nernst) potentials that in turn influence the driving forces of spike-generating ionic currents. Accumulation of ions over time consequently modifies the Nernst potentials as well as the spike generating currents and therefore also spike generation. A regulation of concentration gradients ([Na⁺] and [K⁺]) is mediated by the Na-K-ATPase pump: an electrogenic active-transporter whose activity intensifies when [Na⁺]_i accumulates. Due to its electrogenic nature (changing the net charges across the membrane), activity of the Na-K-ATPase pump affects the membrane potential.

Noise-free analysis.

First, we investigated the response of the model to a step input current—a typical protocol in patch-clamp experiments. Stimulation of the model for almost 10 seconds (Fig 1A) led to an accumulation of intracellular sodium, [Na⁺]_i, as well as an increase in extracellular potassium, [K⁺]_o (Fig 1B). The concentration changes resulted from the prolonged spiking activity and were indeed substantial enough to alter features of the generated action potentials during the duration of the stimulation protocol. Three major changes that have often been reported in experiments were observed in the model: (a) the emergence of a slow after-hyperpolarization (AHP), (b) adaptation (i.e., a reduction) of spike frequency, and (c) reduction of spike amplitudes (Fig 1A).

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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).

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a) The slow AHP became visible when the stimulus was set back to baseline and neuronal spiking stopped (Fig 1A). The slow AHP resulted from the hyperpolarising Na-K-ATPase pump current: Na-K-ATPase pump activity was enhanced by the action-potential-driven rise in intracellular sodium concentration because [Na⁺]_i accumulation increases the pump activity. When the stimulation ended, the neuron stopped firing and the membrane potential hyperpolarized with respect to the original resting membrane potential due to the transient change in the Na-K-ATPase pump current. As ongoing Na-K-ATPase activity progressively lowered the intracellular sodium concentration back to baseline levels, the hyperpolarization slowly diminished (Fig 1A).

b) Spike frequency adaptation, evident in Fig 1A, also resulted from the activity-dependent increase in Na-K-ATPase current, which effectively reduced the net excitatory drive of the neuron. The model does not contain adaptation currents besides the Na-K-ATPase pump (e.g., M-currents). Note that the pump current used in the model is only sensitive to Na⁺, which idealizes the α3 isoform of the Na-K-ATPase. The pump α3 isoform is negligibly sensitive to K⁺ and the membrane potential (V), but highly sensitive to changes in intracellular sodium concentration over the ranges simulated in this study [17, 18]. The impact of the pump current on spike-frequency adaptation was, however, preserved in models of other pump isoforms.

c) In our model, reductions in spike amplitude were directly related to intracellular sodium accumulation, see Fig 1C. Activity-dependent reduction of action potential amplitude has previously been attributed primarily to Na⁺-channel inactivation during prolonged stimulation [13, 14, 19]. Our simulations, however, demonstrate that the time course of amplitude reduction mirrors the drop in sodium reversal potential (see Fig 1A), which is related to the time course of sodium accumulation (see Fig 1B). The Na⁺-channel inactivation was more than an order of magnitude faster than the timescale of spike-amplitude reduction; the slow spike amplitude decay was modulated by intracellular sodium. A dynamical system’s perspective of this finding and an experimental confirmation are presented in the next sections.

Analysis in the presence of noise.

The results so far reflect idealized model responses in the absence of noise. To include the stochasticity of synaptic inputs that is typical for many neurons in the central nervous system, we next added coloured noise with wide sense stationary statistics to the input current—a useful exercise that reveals an interesting property in the response that was masked in the noise-free case discussed above.

Stimulating the model again with a step current in the presence of an additional colored noise component (Fig 2), the model neuron’s response during the first second was comparable to the noise-free case presented before: [Na⁺]_i and [K⁺]_o accumulated (resulting in changes in the reversal potentials E_Na and E_K); spike frequency adaptation, and spike-amplitude reduction were observed (compare to Fig 1A). Surprisingly, after the first second of stimulation, the response exhibited a sudden transition from regular spiking to an intermittently-interrupted, burst-like firing mode. Note that the stimulus statistics are in a wide-sense stationary such that there was no qualitative change in the stimulus during the simulation duration. This means that the qualitative switch in the firing pattern must arise from a bifurcation in the neuron’s dynamics. The switch in firing pattern occurred 1.2 seconds after stimulus onset, a time scale that largely exceeds the time scale associated with the dynamics of spike-generating conductances (which are about two orders of magnitude faster). Yet this time scale matches the time scale of changes in ionic concentrations, suggesting that the switch is causally related to the ion accumulation. Ion accumulation influences spike generation by changing ionic reversal potentials and engaging the electrogenic Na-K-ATPase.

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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 (n_K). 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).

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Consequences of extracellular potassium accumulation

Dynamical systems analysis.

We analyzed the dynamics of the fast system (i.e., the neuron model with fixed ionic concentrations) for different values of extracellular potassium. Fig 3 shows the resulting two-parameter bifurcation diagram, which depicts the dynamical state as a function of extracellular potassium concentration and size of the applied input current (for details see the Materials and methods section). Four different dynamical regimes can be found: a silent subthreshold state, a regularly spiking state, a bistable state, and a silent state of depolarization block (when the model is depolarized so strongly that spiking cannot occur any more).

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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.

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Let’s look at the diagram (Fig 3) in more detail, starting at lower extracellular potassium values (i.e., the bottom of the diagram). Depending on the input strength, the system here either remains subthreshold or exhibits regular firing. The transition to spiking corresponds to a SNIC bifurcation (See Supplementary material S1 Fig). When elevating the levels of extracellular potassium (to ∼12 mM), the situation changes. Here, an additional (bistable) region appears between the subthreshold and the regular spiking areas. The transition is marked by a codimension-two bifurcation called a saddle-node loop (SNL) [22]. The width of the bistable region increases for higher values of extracellular potassium concentration (dashed lines in Fig 3). The firing threshold corresponds to the left border of the bistable region. The transition to spiking now corresponds to a HOM bifurcation. In the bistable zone, in the presence of noise, intermittently-interrupted firing can be observed. Moreover, at elevated extracellular potassium values, the depolarization block (as the name suggests, usually occurring at very large depolarization levels) can be observed at progressively lower input currents. At very high extracellular potassium values, it directly borders the bistable zone.

The full system (with variable concentrations and pump activity), as analyzed in Fig 2, “lives” in the bottom part of the bifurcation diagram (Fig 3) at the onset of the stimulation, as here the values of extracellular potassium are moderate. Over time, extracellular potassium accumulates and the switching point to HOM dynamics is passed. Here, the bistable range is entered and the burst-like, intermittently interrupted firing mode can be observed in Fig 2 due to the presence of noise. The diagram shows that extracellular potassium is the bifurcating parameter that leads to the qualitative switches in spiking.

Experimental manipulation of extracellular potassium.

To experimentally test whether elevated levels of extracellular potassium can induce HOM dynamics of action potential generation, a verification of the model-predicted intermittently-interrupted burst-like firing mode suggests itself. In vitro, activity-driven accumulation of extracellular potassium is difficult to reproduce due to the continuously perfused bathing solution that constrains extracellular ion concentrations. We therefore recorded current-induced activity in mouse cortical pyramidal neurons exposed to different fixed concentrations of extracellular potassium. Action potentials were induced by constant-current stimulation in baseline conditions (3 mM extracellular potassium), and after increasing the concentration of extracellular potassium to 10 or 12 mM (see Materials and methods section). Neurons were stimulated with somatic current injection sufficient to maintain the membrane potential close to spiking threshold (see first panel of Fig 4), which in terms of dynamical system analysis, is close to limit cycle onset and for HOM dynamics, also to the bistable region, see Fig 3.

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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.

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Our experimental results support the model prediction portrayed in Fig 3, in which an increase in extracellular potassium concentration switches the spike generating mechanism. When extracelluar potassium is low (3 mM), the neuron shows very rhythmic (regular) action potential generation over time (see Fig 4 upper panel). In contrast, when extracellular potassium is increased to 10 or 12 mM, action potential generation in the same neuron becomes irregular (see Fig 4 lower panel). In 7 out of 8 neurons tested, we observed an increase in spiking irregularity when potassium levels were increased from 3 mM to 12 mM (see supplementary material Figs F and G in S3 Text). In 3 out of 10 neurons tested, we observed an increase in spiking irregularity when potassium levels were increased from 3 mM to 10 mM (see supplementary material Fig I in S3 Text), confirming a crucial prediction of the model. To rule out that irregularity came from network effects, we repeated the experiment while blocking synaptic input, and in 8 out of 12 neurons we observed an increase in spiking irregularity (Fig H in S3 Text). We hypothesize that not all neurons exhibited the same qualitative behavior because the distance to the switching point depends on other parameters as well [22] and hence is likely to be variable across cells [23]. It remains to be determined whether the activity of a single neuron generates a sufficient increase in [K⁺]_o to induce the SNL bifurcation. Presumably such transitions would be more likely when there are multiple neurons activated simultaneously, which would generate a larger [K⁺] flux to the extracellular space.

Consequences of intracellular sodium accumulation

The dynamical system analysis described above provides a mechanistic explanation for the emergence of intermittently-interrupted firing. However, the only concentration we varied for the analysis was that of potassium ions. Yet long periods of spiking accumulate both extracellular potassium and intracellular sodium ions. In the following section, we therefore describe physiological features that are altered by sodium accumulation. The space of sodium concentrations is explored to determine whether our results described above regarding extracellular potassium accumulation hold up under conditions of parallel intracellular sodium accumulation.

Sodium accumulation shapes two main properties of spike generation: spike amplitude and spiking threshold, which are determinant features for information transmission and encoding, respectively.

Intracellular-sodium-dependent spike amplitude reduction.

As outlined above, action potential amplitude is reduced as intracellular sodium accumulates during spiking (Figs 1 and 2), reducing E_Na and hence the driving force for sodium current. This effect is also reflected in the phase portraits (Fig 2C). The height of the stable limit cycle is squeezed during stimulation, correlating with the E_Na reduction (Fig 1A). We tested this model prediction in mouse cortical neurons in vitro during extended periods of current-induced spiking.

Extended activation of rodent cortical neurons led to a slow spike amplitude reduction (see Supplementary material Fig D in S2 Text). Rodent cortical neurons were activated for 40 seconds using short (2 ms) depolarizing current pulses (3 nA) generated at 40 Hz. These neurons exhibited a slow and progressive reduction in spike amplitude that was best fit by a double exponential decay with an average fast time constant (τ_fast) of 480 ms and an average slow time constant (τ_slow) of 17.7 s (n = 50) (see supplementary material Fig E and Table A in S2 Text for details). The observed slow spike amplitude decay fits the prediction of the model, given that sodium accumulation occurs on the order of seconds, and the faster time scale coincides with the previously reported effects of sodium inactivation [13, 24].

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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).

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More than one process contributes to spike amplitude decay: a fast process (sodium channel inactivation) and a slow process (sodium accumulation). In order to disentangle the contribution of the two, we punctuated prolonged somatic current application with progressively-longer (100 to 1000 ms) breaks sufficient to reset (deinactivate) sodium channels, but not long enough for the Na-K-ATPase to clear activity-dependent increases in intracellular sodium (Fig 5). We observed a progressive reduction in action potential amplitudes that was not rescued by hyperpolarizations as long as one second. Further, the slow time constant of spike amplitude decay was found to coincide with that measured in the previous protocol, τ_slow = 15.7s (Fig E, and Tables A and B in S2 Text). These observations suggest that the slow component of amplitude decay cannot result from sodium channel inactivation, and instead is likely driven by intracellular sodium accumulation and the resulting decrease in E_Na.

Intracellular sodium accumulation shifts the spiking threshold in the model.

Given that an accumulation of intracellular sodium ([Na⁺]_i) is likely to affect spike generation, we next systematically evaluated the dynamical regimes identified in the bifurcation diagram (Fig 3) at different, fixed [Na⁺]_i concentrations.

We found an identical bifurcation structure and splitting into different dynamical regimes for a wide range of [Na⁺]_i levels, with the exception of a shift towards higher input currents with larger [Na⁺]_i (Fig 6). In other words, as [Na⁺]_i accumulates, the spiking threshold is shifted to higher inputs. This shift can be attributed to the dependence of the Na-K pump on [Na⁺]_i; accumulation of [Na⁺]_i thus strengthens the hyperpolarizing pump current, counteracting the input current and reducing the net excitatory drive. Consequently, the bistable region was shifted along the current axis. Neither a significant change in the area of the bistable region, nor in the location of the transition point towards bistability (i.e., the SNL bifurcation) on the [K⁺]_o axis were observed.

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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.

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Consequences of simultaneous [Na⁺]_i and [K⁺]_o changes

The effect of ionic concentrations on neuronal voltage dynamics unfolds via changes in the respective reversal potentials, E_Na and E_K. Fixing the input current, we can summarize how the spiking regime depends on the concentrations ([Na⁺]_i and [K⁺]_o) in a plot that depicts the spiking regime (reached in the steady-state of the fast subsystem) as a function of the two corresponding reversal potentials (Fig 7). The regime was determined via the phase plane of each system (Fig 3B–3D) and can be classified as bistable, regularly spiking, or stable-resting (i.e., either subthreshold or in depolarization block). To relate the reduced fast subsystem with the complete system including slow concentration dynamics, three example trajectories of the complete system at different initial conditions in the space of reversal potentials are shown on top of the steady states of the fast subsystem. Each trajectory represents the evolution of ionic concentrations during 10 seconds of stimulation with a fixed current in the presence of noise (as in Fig 2). The corresponding voltage traces are presented for comparison (Fig 7B–7D). Traces that started at a higher firing rate (and, consequently, were accompanied by larger changes of ionic concentrations) moved farther than the ones that started out at a lower rate.

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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 E_K = −55.1mV and E_Na = 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 E_K = −63.9mV and E_Na = 75.7mV displayed in A. B. Membrane potential trace of the trajectory with initial conditions E_K = −74.2mV and E_Na = 69.3mV displayed in A.

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Which spiking regime a model neuron enters during stimulation can be read off the corresponding trajectory in the complete system. The trajectory depends on the initial ionic concentrations at stimulation onset. A neuron starting with a very low E_K yet high E_Na tends to move from regular spiking either to a resting state or to a lower firing rate within the regularly spiking regime (yellow trajectory in Fig 7D). Biophysically, spike-frequency adaptation results from the activity of the electrogenic pump, which generates a progressively larger hyperpolarizing current as levels of intracellular sodium increase. Trajectories initialized at low E_K do not reach the bistable region. They tend to a quiescent mode, remaining close to the border to regular firing. If the initial E_K is more elevated, however, a neuron that starts in the regularly spiking regime can reach the bistable region (orange trajectory in Fig 7C, similar to the example trace in Fig 2). Very high initial extracellular potassium concentrations promote depolarization block, but, depending on initial conditions, the bistable regime may also be encountered as an intermediate state (magenta trajectory in Fig 7B).

The three example trajectories displayed in Fig 7, illustrate that neurons with identical ion channels and stimulation can generate extremely different responses depending on the extracellular environment. Recent spiking activity of neurons alters their response, even when stimulation is unchanged; the rate of change of ionic concentrations strongly depends on neuronal firing rate. Consequently, neurons receiving strong and prolonged stimulation are more likely to experience dynamical regime changes due to ionic accumulation than neurons with weaker stimulation.

Switching to the HOM firing regime

We demonstrate that neuron models that start out with SNIC dynamics (i.e., the classical type I dynamics that has been thought to underlie the firing of most cells with a smooth onset of firing at threshold) can flip to HOM dynamics only seconds after the onset of the spike-inducing stimulation. This transition most obviously manifests in the spiking pattern, which turns from a regular firing mode to an intermittently interrupted one. Our bifurcation analysis shows that an accumulation of extracellular potassium drives this change, instantiating a bistability of the membrane potential. This finding is consistent with previous modelling work that studied the effect of extracellular potassium concentration changes [8, 25–27].

In this particular setting, an increase in the Nernst potential for potassium (E_K) and the Nernst potential for the leak (E_L) by constraining the magnitude of the potassium (I_K) and leak (I_L) currents, respectively, shift the spike (limit cycle orbit) towards more depolarized membrane potentials and farther from the resting state (stable node). When both attractors are sufficiently far from each other, both can coexist. The stable node does not lose its stability due to the hyperpolarizing current that is received by the neuron model, either via an external input (Fig 3), or via the hyperpolarizing pump current (Fig 6). Interestingly, other parameters such as temperature and capacitance can also induce the same bistability reported here [22] via a different biophysical mechanism, yet an identical bifurcation structure.

In the presence of a fluctuating input (be it noise or signal), the bistability renders neurons susceptible to switches between the two stable states, giving rise to an irregular, intermittently interrupted firing pattern of short firing phases and pauses of different durations. It has been shown that combining multistable systems, similar to the one reported here, with noise can have unexpected information transmission consequences such as “inverse stochastic resonance” [28], as well as drastic alteration in spiking statistics [29].

Long periods of silence that can be prominent in this mode resemble the ones observed during deterministic bursting reported by [8, 9, 27, 30–33]. In contrast to the deterministic bursting reported by the previous authors, however, the burst-like firing we describe here is driven by the input fluctuations and the bistable state. Thus, neurons in an environment with high extracellular potassium concentration, promoting HOM dynamics, may be more sensitive to input variability.

The SNIC and the HOM regimes yield very different neuronal encoding properties. For instance, the relationship between the input current and the neuronal firing rate (i.e., the gain) depends on the dynamical regime. The gain function of a neuron in the SNIC regime is continuous, the firing rate of such neuron is a continuous function of the input current. The gain function of a neuron in the HOM regime is discontinuous when irregular input is injected, meaning that the firing rate does not smoothly increase as a function of the input current but transitions from no spikes to high frequency spiking abruptly. The phase response curve (PRC), which captures the temporal sensitivity to inputs, also differs between the two regimes. In the SNIC regime, neurons display symmetric PRCs. As PRC symmetry predicts the synchronization of the neurons in the network [22, 34, 35], the switch in firing regime and the underlying bifurcation must also impact the propensity of the neuron to synchronize with other cells in its local network and beyond. Encoding capabilities (such as the profile of frequencies transmitted) are likely to be affected, as they also depend on the PRC characteristics.

Interestingly, intracellular sodium accumulation only quantitatively modulates the qualitative change in spiking regime, underlining the importance of potassium accumulation in this process. The effect of extracellular potassium on neuronal activity has been widely studied [36–38]. Experimental observations have found extracellular potassium dependent bursting [38] and its influence on other dynamical features [25, 26, 39]. While a bistability has been previously observed “in passing” [25, 26], we report a systematic effect and provide mechanistic explanations for the activity-driven changes.

Interpreting these results, we speculate that activity-dependent extracellular potassium accumulation can contribute to, or even induce, epileptiform activity as it has been proposed before [32]. Furthermore, epileptiform activity can be induced by manipulating extracellular potassium concentrations [40]. Both the bursting nature of HOM dynamics [41, 42], as well as their comparatively high susceptability to synchronization in inhibitory networks because of the HOM-characteristic PRC [22], favour synchronized, hyperexcitable states. In vivo, Singer and Lux observed that extracellular potassium accumulates in the visual cortex when a rapidly changing visual stimulus is presented to the cat’s retina [4]. Remarkably, similar visual stimuli elicit reflex seizures in 4–7% of human epilepsy patients [43]. Reflex seizures could be promoted by extracellular potassium accumulation occurring throughout the visual cortical region that is activated by visual stimulation.

Indeed, the observed drastic consequences of potassium accumulation might occur more frequently in vivo than in vitro. In vitro, extracellular potassium concentrations are clamped. The tissue is perfused with a solution that has a fixed [K⁺]_o concentration, analogue to an infinite buffer. In vivo, however, extracellular potassium concentration undergoes stimulus-induced changes. In the cat visual cortex, for instance, extracellular potassium accumulates when a graded stimulus is presented to the cat’s retina [4]. Regarding the universality of the dynamical changes described here, we expect these to generalize beyond the specific model choice of this study. Specifically, the bistability of conductance based models arises from a slow-down of hyperpolarization, which pushes the limit cycle trajectory to approach the saddle node through a very attractive path (i.e., a strong manifold). This feature can be expected in any neuron model that starts out with SNIC dynamics and ubiquitously favours a switch to HOM dynamics [22, 44]. The exact reversal potential at which the bistability is induced could be shifted by other parameters such as the leak conductance (refer to Fig A in S1 Text). Thus, the exact switching point between the two dynamical regimes can vary between neurons, as their properties are diverse [23, 45]. Therefore, we expect that the exact location of the switching point (i.e., the potassium concentration at which the switch is to be expected) will depend on cellular characteristics, both in neuron models as well as in neurons in vivo. Along these lines, milder extracellular potassium accumulations could suffice to induce the transition in cells with a lower critical value, singling out cells with a higher likelihood to switch their dynamics.

Attenuation of the spike amplitude

Next to the very promiment change in spiking regime, accumulation of ions are also reflected in the shape of action-potentials, namely the reduction of their peak amplitudes. Such attenuation is a regular feature observed in electrophysiological recordings. It has, however, been previously attributed to inactivation of sodium channels [14, 24]. Our data now suggest that activity-induced changes in the sodium reversal potential contribute substantially to the attenuation of spike peaks, especially during long periods of activity, as they far outlast the effects of inactivation. Our deinactivation experiments with hyperpolarizing current steps support this hypothesis and confirm that the larger and slower component of spike amplitude reduction persist even when sodium channel inactivation is largely diminished. Moreover, the timescales over which E_Na and peak amplitudes are reduced are close to identical for long recordings.

Concentration-change induced spike-frequency adaptation

Spike-frequency adaptation resulted from an activity-dependent increase in the hyperpolarizing sodium pump current (again mediated by sodium accumulation). This observation was previously reported for leech mechanoreceptor neurons [46] as well as for rodent cortical neurons [5]. The later study [5] demonstrated not only that the pump current produces a slow afterhyperpolarization (AHP) as a consequence of neural activity, but that its time-course mirrors the time-course of intracellular sodium decay. Results from our model are consistent with this finding.

Interestingly, for both spike amplitude attenuation and spike frequency adaptation, ion-channel-mediated equivalent effects on short timescales are well known. The dynamics of concentrations seem to smoothly extend these effects in time.

Limitations

We note that our model does not consider extracellular uptake of potassium by glial cells. The latter maintain the ionic homeostasis of the extracellular environment and serve as extracellular potassium buffers [47]. Experimental work Dallerac et al. [48] has shown that glial buffering of extracellular potassium saturates when changes are relatively fast. Yet the presence of glial cells in vivo is likely to slow the timescale of potassium accumulation. Another aspect that we did not include in the model is spatial ionic diffusion, which implies the assumption of a periodic boundary, meaning that all the neighboring neurons are undergoing the same transition in a synchronized fashion. Thus, including spatial diffusion would slow down and probably also shorten the periods of high accumulation. The effects described here are expected to hold on the order of seconds after activity onset, but given the lack of homeostatic mechanisms, the model does not account for ionic concentration recovery. The predictions presented here can be expected to arise most prominently in pathological conditions, be it glial dysfunction, injury, or energy-deprivation that impairs the pumps and thus facilitates accumulation of both extracellular potassium and intracellular sodium.

Ethics statement

Physiological experiments were approved by the Institutional Animal Care and Use Committee of Dartmouth College.

Computational model

Our goal is to understand the effects of ionic concentration dynamics on the excitability of neurons. Two ingredients are needed: an excitable system (capable of generating spikes) for which we use the Traub-Miles formulation [50], and a description for the slow ionic concentration dynamics. For the excitable system we chose the Traub-Miles model because it is one of the simplest mammalian models with type I dynamics (which display continuous and smooth FI curves). A summary scheme of the spike kinetics (Eqs (1)–(4)) and the concentration dynamics (Eqs (9)–(11)) is depicted in Fig 8. The action potential dynamics at a membrane is governed by a current balance equation involving the following ionic currents, (1) (2) (3) (4) (5) (6)

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Fig 8. Summary of the dynamics represented in the model.

The figure illustrates the flux of ions through the fast sodium (I_Na), the delayed rectifier I_K spike generating currents, the Na-K-ATPase pump, and through the neuronal membrane (I_L). 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).

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The leak potential (E_L) is described as an approximation of the Goldman–Hodgkin–Katz (GHK) equation for the membrane potential; (7)

Given that we don’t consider chloride dynamics, . Where , and . (8)

The pump model in Eq (8) constitutes a homeostatic mechanism that counteracts the movement of ions due to chemical gradients during neuronal spiking activity. Specifically, the sodium potassium pump (Na-K-ATPase) pumps 3 sodium ions out of the cell, while 2 potassium ions enter the cell every pump cycle. The pump is represented by sigmoidal function of the intracellular Na⁺ concentration in Eq (8) [31, 51]. I_maxp is the maximum pump rate which was chosen such that the neuron does not run into depolarization block for a broad range of stimulus intensities. Here, the pump’s dependence on potassium and on voltage is ignored. This is an analog of the α3 isoform of the Na-K-ATPase, which reacts very strongly to intracellular sodium changes, but is rather insensitive to potassium in the range we study ([K⁺]_o 4–20 mM) [17].k_Na is the sodium sensitivity of the pump to sodium, and [Na]_s is the target sodium concentration. Both values were set to 0.1/mM and 20 mM respectively, to match the range of highest pump α3 affinity to intracellular sodium [18].

Other modelling studies use pump models that resemble more closely the Na-K-ATPase α2 isoform [6, 11, 12]. The main difference between both isoforms is that α2 has a much higher sensitivity to extracellular potassium and membrane potential than α3 isoform [18]. Developmental studies of isoform expression in the brain have shown that neuronal cells from newborn mice express mainly α2, whereas adult animals express mainly α3 [52]. And given that our our physiological experiments were performed in neurons from adult mice, we found α3 isoform more appropriate.

To understand how our pump choice determines the qualitative results presented here, we performed a bifurcation analysis to the model described above with a pump expression that resembles pump isoform α2, refer to Fig C in S1 Text. The main difference between the pump expression in Eq 8 and the one used to resemble α2 (Fig C in S1 Text), is the pump sensitivity to extracellular potassium. Fig C in S1 Text illustrates how the pump sensitivity to [K⁺]_o shifts the spiking threshold towards higher input currents as [K⁺]_o accumulates.

The concentration dynamics is influenced by the transmembrane currents due to ion channels and the pump as follows. (9) (10) (11) Where ρ represents the surface to volume area, and here is set to 4000/cm [31], is set to 0.2 (note that this is a very small value mainly because we are not including extracellular regulation mechanisms, thus we assume glial cells, blood vessels, myelin, and other structures are part of the extracellular volume. Previous experimental measurements yield a of approximately 0.4 [53], however we decreased it to compensate for the lack of other homeostatic mechanisms and allow reproducibility of experimental observations. All expressions and parameters used in the simulations can be found in Tables C-F in S4 Text.

Experimental protocol

Physiological experiments were approved by the Institutional Animal Care and Use Committee of Dartmouth College. Female and male adult (3- to 4-month-old) C57BL/6J mice were bred in facilities accredited by the Association for Assessment and Accreditation of Laboratory Animal Care and maintained on a 12h-12h light-dark cycle with continuous free access to food and water.

On the day of experiments, mice were anesthetized with vaporized isoflurane and decapitated, with brains rapidly removed into an artificial cerebral spinal fluid (aCSF) composed of (in mM): 125 NaCl, 25 NaHCO₃, 3 KCl, 1.25 NaH₂PO₄, 0.5 CaCl₂, 6 MgCl₂ and 25 glucose (saturated with 95% O2–5% CO₂). Coronal brain slices (250 μm thick) of the frontal cortex were cut using a Leica VT 1200 slicer and stored in a holding chamber filled with aCSF containing 2 mM CaCl₂ and 1 mM MgCl₂. Slices were maintained in the holding chamber for 45 minutes at 35°C, and then at room temperature (∼25°C) until use in experiments.

Slices were transferred to a recording chamber on a fixed-stage microscope (Olympus), and continuously perfused (∼ 7 ml/min) with oxygenated aCSF heated to 35–36°C. Layer 5 pyramidal neurons in the prelimbic cortex were visually targeted using a 60X water-immersion objective, and whole-cell recordings made with patch pipettes (5–7 MΩ) filled with a solution containing the following (in mM): 135 potassium gluconate, 2 NaCl, 2 MgCl₂, 10 HEPES, 3 Na₂ATP and 0.3 Na₂GTP, pH 7.2 with KOH. Data were acquired using a BVC-700 amplifier (Dagan Corporation) connected to a HEKA 8+8 digitizer or an ITC-18-USB digitizer driven by AxoGraph software (AxoGraph Scientific; RRID: SCR—014284). Membrane potentials were sampled at 50 to 100 kHz and filtered at 5 or 10 kHz. Voltage measurements were corrected for a +12 mV liquid junction potential. Concentrations of KCl (3, 10, or 12 mM) and NaCl (125, 118, or 116 mM, respectively) were adjusted as indicated to test the impact of extracellular potassium concentration on action potential dynamics. In some experiments fast synaptic transmission was blocked with continuous bath-application of the AMPA receptor blocker DNQX (25 μM; Tocris), the NMDA receptor blocker D-AP5 (25 μM; Tocris), and the GABA_A receptor blocker picrotoxin (50 μM; Tocris).

Data analysis

Data analysis was done in python.