The authors have declared that no competing interests exist.
Conceived and designed the experiments: MDF MJW DAP JF. Performed the experiments: MDF MJW. Analyzed the data: MDF MJW. Contributed reagents/materials/analysis tools: MDF MJW. Wrote the paper: MDF. Read and amended manuscript: MJW JF.
The cerebellum coordinates the execution and adaptation of motor behaviours
We propose that the Na+/K+ pump controls intrinsic Purkinje cell activity. The Na+/K+ pump uses the energy of one ATP molecule to exchange three intracellular Na+ ions for two extracellular K+ ions
GABAergic stellate cells make inhibitory synaptic contacts upon the dendrites of Purkinje cells
The Purkinje cell model can replicate the
Our experiments with rats were conducted in accordance with the UK Animals (Scientific Procedures) Act 1986 and associated guidelines. We had approval to conduct these experiments from the Biological Ethics Sub-Committee, of the Research Ethics Committee, at the University of Warwick. This committee scrutinizes in accordance with the appropriate guidance for such committees from the Home Office, the RSCPA and the Laboratory Animal Science Association (LASA). This committee ensures that staff and students are trained and experienced, and that the potential benefits of the research outweigh the effects on the animals concerned. The committee is committed to the promotion of the 3Rs (reduction, refinement and replacement). At the University of Warwick, highly qualified and experienced veterinary staff are actively involved in (1) the ethical scrutiny of research (2) the welfare, care and husbandry of animals and (3) provide advice and support to all staff and students involved in research using animals.
Simulations were performed with the NEURON 5.6 simulator
The model incorporates twenty-one gated ion channels with descriptions established from the literature. The soma has highly TEA sensitive (IK_fast), moderately TEA sensitive (IK_mid) and TEA insensitive (IK_slow) voltage-gated K+ currents, a BK voltage-and-Ca2+-gated K+ current (IBK), a resurgent Na+ current (INa-R), a P-type Ca2+ current (ICaP), a hyperpolarization activated cation current (IH), a leak current (IL) and an intracellular Ca2+ dynamics abstraction - all sourced from Khaliq et al.
Current | Soma | Dendrite |
Resurgent Na+ | 156 | 0 |
P-type Ca2+ | 0.52 | 1.6 |
T-type Ca2+ | 0 | 0.6 |
E-type Ca2+ | 0 | 3.2 |
A-type K+ | 0 | 32 |
D-type K+ | 0 | 36 |
M-type K+ | 0 | 0.004 |
Delayed rectifier K+ | 0 | 0.24 |
Bk K+ | 72.8 | 60 |
SK K+ | 10 | 0 |
K2 K+ | 0 | 0.16 |
Kv1.2 K+ | 0 | 1 |
Highly TEA sensitive K+ | 41.6 | 0 |
Moderately TEA sensitive K+ | 20.8 | 0 |
TEA insenstitive K+ | 41.6 | 0 |
hyperpolarization activated cation, Ih | 1.04 | 0.29 |
Leak | 0.1 | 0.08 |
The somatic Na+/K+ pump (density = dspump, 1 mA/cm2) transports 3 Na+ out (ispump_Na) for every 2 K+ in (ispump_K)
Extracellular K+ concentration ([K+]o) to the dendritic compartments is initiated at 2 mM and then changes in time
A “ceiling” for extracellular K+ accumulation is set physiologically by the glial buffer system
Formula
The model dendrites have two different Na+/K+ pump mechanisms. One has already been described (
Convention permits inward (depolarizing) currents to be denoted negative and outward (repolarising) currents to be denoted positive
The model's four different Na+/K+ pump equations are each valid and founded in previously published Na+/K+ pump descriptions. They capture different aspects of Na+/K+ pumping. The first captures [Na+]i and voltage dependency (
Ouabain irreversibly blocks the Na+/K+ pump
Catch coding, which is code of the form: if (x<0) [x = 0], is used to prevent negative values of (dxpump,. gxpump, x = s,d) from occurring.
To illustrate the sensitivity of the model to the KNa, KK and τ parameters a variant of the model was constructed in which they were modified. KNa was changed from 40 mM to 10.5 mM, KK from 2.245 mM to 50 mM and
GABAergic stellate inputs make inhibitory synaptic contacts upon the model dendrites; two inputs to every smooth dendrite compartment and one input to every spiny dendrite compartment
All parameters were established by prior literature (as referenced) except the twenty-one current densities, the Na+/K+ pump and Na+/Ca2+ exchanger densities, the synaptic weight,
To generate a simpler version of the described Purkinje cell model, the dendritic arbour was collapsed into fewer compartments with a reduction algorithm that conserves axial resistance (Ra)
Following the methodology of Destexhe et al.
We used this “Ra conservation” algorithm to collapse the 1089 compartments of the full Purkinje cell model into the 41 compartments of a reduced model (1 soma, 40 dendritic compartments). The reduced model's 40 dendritic compartments were allocated smooth and spiny (20 smooth, 20 spiny) in an arbitrary mirroring of the full model (85 smooth, 1003 spiny). Note that in our approach, smooth and spiny dendrites are not distinguished from one another by the actual modeling of dendritic spines but by an abstraction, with a larger specific membrane capacitance (
Compartment | Length (µm) | Diameter (µm) |
1 | 20.545455 | 3.3166248 |
2 | 19.4 | 2.236068 |
3 | 18 | 1.7320508 |
4 | 17.571429 | 2.6457513 |
5 | 8.5306122 | 3.3045423 |
6 | 13.344828 | 2.6305893 |
7 | 12.567568 | 2.8213472 |
8 | 16.9 | 3.1622777 |
9 | 11 | 3.3166248 |
10 | 10.352941 | 3.4467376 |
11 | 12.732394 | 3.9849718 |
12 | 12.5 | 3.7309516 |
13 | 10.475728 | 4.7791213 |
14 | 15.361446 | 4.3405069 |
15 | 11.986486 | 3.9899875 |
16 | 12.692308 | 5.5497748 |
17 | 10.326667 | 5.9665736 |
18 | 9.6402116 | 6.9079664 |
19 | 13.5625 | 6.2289646 |
20 | 9.8686831 | 6.5415792 |
21 | 10.347368 | 6.6932802 |
22 | 8.5744681 | 7.5232971 |
23 | 10.075188 | 8.2267855 |
24 | 9.5446429 | 7.3972968 |
25 | 8.6412429 | 9.1389277 |
26 | 8.8216783 | 8.5135187 |
27 | 8.1569732 | 9.4462488 |
28 | 6.1189159 | 8.5064444 |
29 | 7.8575581 | 10.055844 |
30 | 6.7650755 | 10.560032 |
31 | 6.892365 | 10.617743 |
32 | 6.5219251 | 11.041739 |
33 | 7.6343948 | 11.421714 |
34 | 7.9040284 | 10.478619 |
35 | 10.028048 | 10.878798 |
36 | 18.067147 | 9.2108216 |
37 | 17.821097 | 8.0038358 |
38 | 57.640576 | 7.3301444 |
39 | 24 | 3.5777088 |
40 | 18 | 4 |
Compartment 1 is linked to the somatic compartment.
The
We strove for the smallest number of compartments, but through testing we found that 41 (1 soma, 40 dendrite) was the smallest number that could behave equivalently to the full 1089 compartments. This is because the collapsing algorithm does not conserve dendritic length fully, and Purkinje functioning requires a degree of uncoupling (distance) between somatic and dendritic events. 41 compartments is the smallest number that confers the sufficient distance (uncoupling) with a dendritic length of ∼502 µm. Without this sufficiency of length, for instance if the reduced model is the full model transposed by the algorithm to just 3 compartments (dendritic length of ∼129 µm), the tonic mode is absent in the trimodal firing pattern i.e. the model cannot replicate the Purkinje cell's trimodal firing pattern. By contrast, the reduced model of 41 compartments can replicate this activity motif.
We confirmed that the problem with the 3 compartment model was a lack of dendritic distance, rather than a lack of dendritic load, because increasing dendritic load could not enable a compartment number lower than 41. The “rallbranch” variable is unique to the NEURON simulator
With another round of simplification, we produced a 5 compartment model that could reproduce the behavior of the full 1089 compartment model. Firstly, we collapsed the dendritic tree of the full 1089 compartment model to just 3 compartments (using the “Ra conservation” algorithm). We then overcome the dendritic distance issue by linking these 3 dendritic compartments to the somatic compartment through a coupling compartment that had a length of our setting, which we ensured was sufficient. This coupling compartment had the same passive and conductance properties as the dendritic compartments and, indeed, can just be thought of as component to the reduced model's dendritic tree. The coupling compartment and the next dendritic compartment along were set as smooth dendrites (
Compartment | Length (µm) | Diameter (µm) |
1 | 400 | 3 |
2 | 16.11816 | 18.5088 |
3 | 95.16785 | 7.947963 |
4 | 18 | 4 |
Compartment 1 is linked to the somatic compartment.
Given that this 5 compartment model has some dimensions of our setting, its line of sight to the full model and the Biology is distorted. But at this cost it runs much faster, with 26 seconds of CPU time required for 1 second of simulation (Intel Pentium PC). This is ∼51* faster than the full model and ∼3* faster than the 41 compartment model.
In the
Parasagittal slices of cerebellum (250 µm) were prepared from male Wistar rats, at postnatal days 28–35 (P28–35), with methods based on
Individual slices were viewed on a Zeiss FS Axioskop microscope with a 40× water immersion objective and Nomarski differential interference optics, at a total magnification of 640×. Slices were maintained at 30–32°C and continuously perfused (1–5 ml min−1) with a CSF, which was bubbled with 95% O2 and 5% CO2. Whole-cell patch-clamp recordings were made from visualized Purkinje cells using an EPC 8 amplifier (Heka, Digitimer, Welwyn Garden City, UK) controlled via a Digidata 1322a interface (Axon Instruments INC, Foster City CA, USA) using Clampex (v 9, Axon Instruments). Patch-pipettes (thick-walled borosilicate glass, Harvard Apparatus, Edenbridge, UK) were fire-polished, and had resistances of 1.5–4 MΩ when filled with an intracellular solution containing (mM): 135 K gluconate, 7 NaCl, 10 HEPES, 0.5 EGTA, 2 Na2-ATP 0.3 Na2-GTP and 10 mM Na phosphocreatine (adjusted to pH 7.2 with KOH and osmolarity adjusted to 300 mOSM with sucrose). Aliquots of intracellular solution were stored frozen at −20°C and thawed on the day of recording.
Drugs were dissolved at 1–100 mM in deionised water: bicuculline methiodide (Sigma), TTX (tetrodotoxin, Advent Scientific) and ouabain (Sigma). Aliquots of these stock solutions were stored frozen at −20°C, thawed and diluted in perfusion medium on the day of recording. All drugs were bath applied.
In the absence of synaptic input, the model Purkinje cell fires spontaneously in a repeating trimodal pattern that consists of tonic spiking (
There is some uncertainty as to whether the trimodal firing pattern is intrinsically generated or a function of neuromodulatory input
The model soma can fire Na+ spikes (
The model's relationship between dendritic spikes and trimodal bursting is in alignment with experimental data, where a similar stereotypical burst waveform has been observed
The model dendrites have an intrinsic capacity to fire Ca2+ spikes. Their low-threshold T-type and Class-E voltage-gated Ca2+ channels open near the resting potential and they depolarize the membrane potential to activate high-threshold P-type voltage-gated Ca2+ channels. These P-type Ca2+ channels then produce Ca2+ spikes, which are repolarized by K+ flow through BK-type K+ channels (in alignment with Miyasho et al.
This system is gated by dendritic Kv1.2 voltage-gated K+ channels. Kv1.2 channels generate hyperpolarizing current which clamps dendritic excitability and prevents Ca2+ spike generation, thus allowing the tonic mode of firing. However, the power of this excitability clamp diminishes with time because extracellular K+ accumulates (
All panels plot over the same period of time.
Kv1.2 is low-voltage gated. In addition to Kv1.2, the model has other K+ currents/channels in its dendrites. However, these are largely uninvolved in the clamping of dendritic excitation as, unlike Kv1.2, they are high-voltage gated and not open at the relevant potentials. The D-type and A-type K+ currents are low-voltage gated but their involvement is limited because they inactivate quickly. By contrast, the Kv1.2 current is non-inactivating, which is why its current persists long enough to be tempered by slow ion relaxation processes. Indeed, experiment has shown the Kv1.2 current to be non-inactivating in the Purkinje cell (McKay, 2005).
The control of the tonic to burst transition by Kv1.2 channels enables the model to replicate an experimental investigation in which Kv1.2 channel block dramatically shortened the tonic phase within the trimodal pattern (
The Na+/K+ pump hyperpolarizes the membrane potential with a stoichiometry of three internal Na+ ions exchanged for every two external K+ ions (
The trimodal repeat length is the duration of a single repeat of the trimodal pattern. It is described as a constant for an individual Purkinje cell but can reportedly vary from 20 seconds to 20 minutes between different cells
We present durations of the tonic mode, bursting mode, firing mode (tonic+bursting), quiescent mode and trimodal repeat length (tonic+bursting+quiescent) for different model settings.
The Na+/K+ pump's affinity for internal Na+ (KNa model parameter) sets the “firing length”, which is the combined duration of the tonic and bursting modes. The lower KNa, the smaller the quantity of intracellular Na+ that is needed to stimulate the Na+/K+ pump to a level of pumping that can hyperpolarize the cell to silence, which equates to a shorter period of Na+ entry i.e. a shorter “firing length” (
The Na+/K+ pump's affinity for external K+ (KK model parameter) sets the duration of tonic firing. The higher KK, the shorter the tonic mode (
Intracellular Na+ dynamics (encoded by the
KNa, KK and
The model cell has a morphology reconstructed from a Purkinje cell in an adult rat cerebellum (
The model has a soma capable of spontaneously firing Na+ spikes and a dendritic tree capable of spontaneously firing Ca2+ spikes. Thus, the model's dendritic spiking is not reliant upon an excitatory drive from the soma. This enables the model to replicate the persistence of dendritic Ca2+ spikes when Na+ channels are blocked (by TTX;
In the presence of TTX, somatic activity is bimodal with periods of Ca2+ spike activity alternating with periods of quiescence
Some Purkinje cells express the trimodal firing pattern without pharmacological manipulation while others express it only upon the condition that GABAergic synaptic inputs are blocked
GABAergic stellate cells make inhibitory synaptic contacts upon the dendrites of Purkinje cells
Left panels are somatic membrane potential and right panels are dendritic membrane potential (vs. Time). By referring to the dendritic membrane potential one can distinguish bursting from tonic firing at the soma. Because bursting (unlike tonic firing) is co-incidental with dendritic spiking.
Some Purkinje cells intrinsically fire in this repeating bimodal pattern, in the absence of synaptic input
An intrinsically bimodal model cell can be switched to trimodal firing by blocking the Kv1.2 current (setting Kv1.2 channel density to 0) (
A model cell firing in the trimodal pattern can be switched into the bimodal pattern by removing its dendritic P-type Ca2+ channel complement (
Without synaptic input, not all Purkinje cells fire spontaneously - some are quiescent
As an objective assay of the model's validity we investigated whether it could predict the Purkinje cell response to the application of ouabain. This is an experimental test that the model has not been specifically tuned to capture and the ability of a model to predict/fit data not used in determining its parameters is an independent measure of how well the model approximates reality. Ouabain inhibits the Na+/K+ pump irreversibly
Panel
In the Purkinje cell response to ouabain (real or simulated), the duration of the trimodal pattern's silent periods became shorter until firing was continuous (
In this matching experimental and model response to Na+/K+ pump block, the loss of the quiescent mode indicates that the Na+/K+ pump generates trimodal quiescence. The loss of the tonic mode indicates that the Na+/K+ pump controls the tonic to burst transition through its setting of extracellular [K+], which controls a Kv1.2 channel “gate” to bursting. The depolarization block demonstrates that the Na+/K+ pump is required to counterbalance a strong and depolarizing force. Indeed, the pump has been reported to offset a depolarizing tetrodotoxin (TTX) resistant Na+ entry in the Purkinje cell
Unfortunately, the ouabain response of the real and model Purkinje cell was not in complete alignment. The model's deflections in the membrane potential, furrowing the depolarization block, are smaller than those observed experimentally. Also, the model's timescale of response to ouabain (∼10 seconds,
In experiments with a lower ouabain concentration (1.5 µM) the same sequence of events occurred but took longer (
Ouabain block of the Na+/K+ pump also imposes a switch in behaviour for Purkinje cells that have intact inhibitory synaptic inputs (no bicuculline applied) and which fire in a repeating bimodal pattern of tonic spiking and quiescence. Ouabain (2.5 µM) initially switches their firing from bimodal to trimodal, before a switch to continual bursting and an eventual depolarization block (
Panels
In the presence of TTX (1 µM), the Purkinje cell expresses a repeating bimodal pattern of Ca2+ spike activity and quiescence
This investigation valuably reconciles the divergent Purkinje behaviours observed in different
To date, the trimodal firing pattern has only been observed
The model shows that the trimodal pattern of firing is regulated by the extracellular K+ concentration. Neighboring Purkinje cells share an immediate extracellular milieu, which raises the possibility that adjacent or nearby Purkinje cells can signal to each other through the extracellular K+ concentration variable. This communication could be modulated by glial cells, which buffer K+
The trimodal pattern's quiescent mode can span seconds or minutes
The reader must be aware that Na+/K+ pump block might eradicate quiescence in the trimodal and bimodal firing patterns, not because the Na+/K+ pump generates quiescence (per se), but because this block causes depolarization which compromises the activity of the true entity responsible. This is actually a problem inherent to many pharmacological block experiments that seek to parse the function of individual neuronal currents. By knocking out a current, the voltage trajectory is changed and in interpretation one is then unsure as to whether functional changes are due to that loss or due to changes in another current(s), which is regulated by the membrane potential. We did control experiments where we injected hyperpolarizing current to counter the ouabain induced depolarization (
Different cell types can have different response profiles to a Na+/K+ pump block. With hypoglossal motoneurons of the rat, “bath application of a 4–20 µM ouabain solution produced a partial block of Na+/K+ pump activity” and produced “no significant change in either the initial, early, or late phases of spike-frequency adaptation” i.e. 4–20 µM ouabain application did not cease or significantly alter the firing rate of rat motorneuons
Pharmacological block of the Na+/K+ pump has been shown to promote firing and prevent the generation of quiescent periods in canine coronary sinus fibers. This result is interpreted as evidence that the Na+/K+ pump is the generative mechanism to these quiescent periods
Before we considered the Na+/K+ pump as the drive to trimodal quiescence, we entertained the possibility that a Ca2+-activated K+ current might be responsible. However, we discounted the SK K+ current because the trimodal pattern's quiescent mode can still occur when this current is blocked by apamin
The detailed biophysical model replicates Purkinje cell firing patterns with the mechanisms that we hypothesise to be responsible in real Purkinje cells. And these predictions are forceful given the model's level of detail, with its faithfully reconstructed morphology and current, synapse and pump equations predominantly parameterized to experimental data. The model's mechanism of generating the trimodal firing pattern is endorsed by endowing the model with the ability to replicate a wealth of other experimentally observed Purkinje cell operating states: quiescence, the intrinsic bimodal firing pattern (
The reduced models run faster than the full Purkinje cell model, yet faithfully reproduce its electrical behaviour. This similarity, between the outputs of the full model (1089 compartments) and its derivatives (the 41 compartment and 5 compartment models), serves to illustrate the reproducibility (semi-quantitative) of the computational experiments performed with the full model.
The mechanisms that we propose to drive the Purkinje cell's trimodal pattern of firing have been observed in other classes of neuron. Extracellular K+ has been experimentally observed to accumulate during sustained neural activity
CF input has been observed to toggle the cell between a tonic firing state and a quiescent state. So, state transitions occur at the frequency of CF input (∼1 Hz) and tonic firing periods of ∼1 s alternate with quiescent periods of ∼1 s (