Neuron channels

This page provides details for the full range of channel types that are available in an Axon neuron to drive biologically based and functionally important behavior in specific neuron types. Every neuron uses the basic excitatory, inhibitory, and leak channels discussed in detail in the neuron and neuron electrophysiology pages, and some of the following channels are used in most other neurons, while some are only used in specific neuron types where they are particularly important.

These biologically grounded channels provide accurate fits to the detailed electrophysiological properties of real neurons, based on the sources listed below. Although this results in a large number of parameters relative to the units in abstract neural networks, we almost never change these parameters from their default values, unless there is a clear biological or functional motivation to do so. Furthermore, extensive testing across a wide range of models has shown that these biologically grounded mechanisms, and parameter values, actually produce the best functional results.

The source code for these channels is in the axon chans directory. Most of these channels use exponential integration governed by time constant parameters shown in Table 1 below. The implementation of several of these channels comes from standard biophysically detailed models such as Migliore et al. (1999), Poirazi et al. (2003), and Urakubo et al. (2008). See also Brette et al. (2007) and the NEST model directory for documented examples, including: AdEx, Traub HH. The Brian Examples contain full easy-to-read equations for various standard models, including Brunel & Wang, 2001. Also see Wikipedia: Biological neuron model for a nice overview, and ModelDB Currents, ModelDB Current Search, and IonChannelGeneology for standardized lists of currents included in biophysical models made in NEURON and related software.

Parameters

The default parameters for each of the channel types covered here are shown in the following tables, using the standard units.

The exponential integration time constant parameters for the various channels are shown in Table 1.

The above Table 2 are the conductance scaling factors for each channel. These are not in the same units as the overall g-bar factors that convert the roughly normalized time-varying conductance values into the final nS conductance values, because we add these channels to the overall time-varying Ge(t) or Gk(t) conductances that are then multiplied by the g-bar factors in the end (which are set to 100 nS by default). Furthermore, these scaling factors multiply values that are not themselves normalized quantities, so they are not directly interpretable, and are included here just to document the full set of standard parameters. See the following table for values that give a better sense of overall relative magnitudes of conductances.

The above Table 3 shows the typical maximum conductances for each type of channel, including all the various factors that go into computing the final time-varying conductance values. These are all approximate values and depend on various network level parameters and dynamics (especially the AMPA, GABA-A inhibition, and NMDA), so they are just for getting an overall sense of relative magnitude.

AMPA

The AMPA (α-Amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid) channel is the primary excitatory synaptic input channel, discussed extensively in neuron. The channel is opened by the binding of the neurotransmitter glutamate, which causes the channel structure to twist open, allowing primarily \(Na^+\) ions to enter the cell, causing the electrical potential to increase (excitation).

The AMPA receptor conductance can be modeled using the double-exponential function, where t is the time since the binding of glutamate to the receptor:

\[ g(t) = e^{-t / \tau_r} - e^{-t / \tau_d} \]

\(\tau_r\) is a fast rise time constant for the increase in conductance when the glutamate first binds to the AMPA receptor (less than 1 ms according to Hestrin et al., 1990), and \(\tau_d\) is a slower decay time constant reflecting the inactivation of the AMPA receptor over time, estimated at 4.4 ms by Hestrin et al. (1990).

The alpha function as introduced by Rall (1967) has also been used to model relatively fast conductances, using a single time constant:

\[ g(t) = \frac{t}{\tau} e^{-t / \tau} \]

In the Axon model we use a time step of 1 ms for integrating all of the neuron level equations, so the relatively fast rise time constant happens too quickly to be of relevance. Thus, the AMPA conductance increases discretely in the 1 ms time step, and we use a single exponential decay function with a time constant of 5 ms, which can be computed using simple online exponential integration:

\[ g_{ampa}(t) = g_{e-raw}(t) + g_{ampa}(t-1) \left(1 - \frac{1}{\tau_d} \right) \]

As with all channels, this conductance then drives a corresponding current as a function of the reversal potential for AMPA (\(E_{ampa}\)), which is estimated at 0 mV:

\[ I_{ampa} = g_{ampa} \left(E_{ampa} - V_m\right) \]

GABA-A

The GABA-A channel is the standard inhibitory synaptic input channel discussed in neuron and inhibition. It is opened by the binding of the GABA (gamma-aminobutyric acid) neurotransmitter, released by special populations of inhibitory interneurons. It primarily allows negatively charged chloride ions \(Cl^-\) to flow into the cell, which act to keep the electrical potential negative.

We model GABA-A conductances in the same way as AMPA, with a single exponential decay function (Eq 3, using a decay time constant \(\tau_d\) of 7 ms Xiang et al., 1998. The reversal potential for GABA-A (\(E_{gaba_a}\)) is -75 mV.

K Leak

The \(K^+\) leak channel is always open, so it just has a constant conductance parameterized by \(\overline{g}_{gaba_a}\) and a reversal potential (\(E_{leak}\)) of -75 mV.

NMDA

pl := &chanplots.NMDAPlot{}
root, _ := tensorfs.NewDir("Root")
br := egui.NewGUIBody(b, pl, root, "NMDA", "NMDA Channel", "NMDA Channel equations")
pl.Config(root, br.Tabs)
br.FinalizeGUI(false)
br.Splits.Styler(func(s *styles.Style) {
	s.Min.Y.Em(25)
})
pl.GVRun()

NMDA (N-methyl-D-aspartate) channels are found throughout the brain, and play a critical role in learning as captured in the kinase algorithm used in Axon. The opening of NMDA channels is typically blocked by positively charged magnesium ions (\(Mg^{++}\)) when the membrane potential is close to the resting potential. The removal of this block as a result of membrane depolarization above this resting potential (known as an outward rectification, because the \(Mg^{++}\) ions are on the outside of the cell) is one of two key functional features of this channel. This unblocking dynamic is important both for learning and for its important contributions to the activation dynamics of the neuron.

The other critical functional property of the NMDA channel is that it also requires glutamate neurotransmitter binding to open, in addition to \(Mg^{++}\) unblocking. Thus, unlike many other channels that just have one (or none) gating factors, NMDA requires both of these factors. Furthermore, the \(Mg^{++}\) factor is a function of the postsynaptic activity (depolarization), while glutamate is released by the presynaptic neuron. Thus, the NMDA is in a unique position to respond to the conjunction of both pre and postsynaptic activity.

To see the unblocking in action, press the Plot 1:GV run button above, which generates a conductance (g) (and current I) vs. voltage (v) plot using the parameters shown to the left of the plot, assuming that there is plenty of glutamate around so that factor is not relevant. As the voltage increases above the -90 hyperpolarized starting point, the conductance steadily rises, reflecting the progressively increased likelihood that the \(Mg^{++}\) ions will not be blocking the channel opening. The reversal potential for the NMDA channel is around 0 mV, so as the voltage approaches this point, the net force pulling the ions through the channel gets progressively weaker (as explained by the tug-of-war analogy in neuron), so the current I decreases as the voltage approaches 0.

NMDA channels mostly allow calcium ions (\(Ca^{++}\)) to flow into the cell, and the learning effects of this channel are due to the ability of calcium to trigger various postsynaptic chemical reactions (see synaptic plasticity). The activation effects are due to positive charges on this ion, which therefore has a net excitatory (depolarizing) effect on the cell.

Plot 1:Time run shows the other critical feature of the NMDA channel, which is that the \(\tau_d\) decay time constant parameter (see Eq 3) is much longer than most other channels, on the order of 100 ms or more (like AMPA, NMDA has a sufficiently fast rise time that it can be ignored). This relatively long time constant is critical for the activation contributions of the NMDA channel, because it creates a stable activation pattern over time (see that page for more discussion and a demonstration).

The equation we use for the voltage-gated conductance is due to Jahr & Stevens, 1990, and is used in the widely cited Brunel & Wang, 2001 model:

\[ g_{nmda}(t) = \frac{g_{e-raw}(t) \overline{g}_{nmda}}{1 + \frac{[Mg^{++}]}{3.57} e^{-0.062 V_d}} - \frac{1}{\tau_d} g_{nmda}(t-1) \]

where \(\overline{g}_{nmda}\) is an NMDA-specific multiplier (see Table 2), and \([Mg^{++}]\) is the extracellular magnesium concentration, which is typically between 1 and 1.5, and \(g_{e-raw}(t)\) is the raw excitatory synaptic input described in Computing input conductances. We use the dendritic Vm membrane potential value \(V_d\) in this equation, because NMDA receptors are located primarily on input-receiving dendrites (see neuron dendrites for details).

This function is of a sigmoidal, “S-shaped” form, increasing to an asymptotic value as the voltage increases; the relevant portion of this function is on the left side of the sigmoid as you can see in Plot 1:GV run. The decrease that we see in I as voltage increases is due to the standard tug-of-war Ohm’s law multiplier that applies to all channel conductances (Eq 4).

GABA-B

pl := &chanplots.GABABPlot{}
root, _ := tensorfs.NewDir("Root")
br := egui.NewGUIBody(b, pl, root, "GABA-B", "GABA-B Channel", "GABA-B Channel equations")
pl.Config(root, br.Tabs)
br.FinalizeGUI(false)
br.Splits.Styler(func(s *styles.Style) {
	s.Min.Y.Em(25)
})
pl.GVRun()

The GABA-B channel has a much slower decay time constant compared to the standard GABA-A inhibitory channel, because it is coupled to the GIRK channel: G-protein coupled inwardly rectifying potassium (K) channel. It is ubiquitous in the brain, and is likely essential for basic neural function. The inward rectification is caused by a \(Mg^{++}\) ion block from the inside of the neuron, which means that these channels are most open when the neuron is hyperpolarized (inactive), and thus it serves to keep inactive neurons inactive. This is complementary to the effect of NMDA channels, and Sanders et al. (2013) emphasized the synergistic nature of these two channels in producing stable activation patterns: NMDA keeps active neurons active, while GABA-B keeps inactive neurons inactive.

Press the Plot 2:GV run button above to see the conductance (g) (and current I) vs. voltage (v) plot using the parameters shown to the left of the plot. In comparison with the comparable NMDA plot, you can see that GABA-B and NMDA are mirror-images of each other. Furthermore, the I value plotted here is the absolute value (positive) whereas the actual current has the opposite sign as NMDA, due to its reversal potential \(E_{gaba_b}\) being that of potassium (-90 mV).

The implementation of the GABA-B channel is based on Sanders et al. (2013) and Thomson & Destexhe (1999), with the following sigmoidal voltage-gated conductance function from Yamada et al. (1998):

\[ g_{gaba_b}(V) = \overline{g}_{gaba_b} \frac{1}{1 + e^{0.1(V-E_{gaba_b}+10)}} \]

There is an additional sigmoidal function needed for computing the time dynamics of the GABA-B conductance as a function of the GABA binding from inhibitory input spikes over time (in Figure 16 of Thomson & Destexhe, 1999):

\[ X = \frac{1}{1 + e^{-(s - 7.1) / 1.4}} \]

Where s is the rate of spiking over the recent time window, which we compute based on the \(g_i\) inhibition factor from the inhibition function used in Axon, and X drives increases in GABA-B activation (labeled M) according to the following double-exponential update equations with separate rise (\(\tau_r\)) and decay (\(\tau_d\)) factors (45 and 50 ms respectively, which fit the timecourse data from Thomson & Destexhe, 1999 well; Table 1):

\[ M(t) = \frac{1}{\tau_r} \left( \left[(\tau_d / \tau_r)^{(\tau_r / (\tau_d - \tau_r))} \right] X(t) - M(t-1) \right) \]

\[ X(t) = \left( \frac{1}{1 + e^{-(s - 7.1) / 1.4}} - X(t-1) \right) - \tau_d X(t-1) \]

The final GABA-B conductance is a product of the M activation factor above in Eq 8 and the voltage gating factor shown in Eq 6:

\[ g_{gaba_b}(t) = g_{gaba_b}(V) M(t) \]

Do Plot 2:Time run to see these time dynamics play out over a 500 ms window with a pulse of input at the start.

Kir

pl := &chanplots.KirPlot{}
root, _ := tensorfs.NewDir("Root")
br := egui.NewGUIBody(b, pl, root, "Kir", "Kir channel", "Kir channel equations")
pl.Config(root, br.Tabs)
br.FinalizeGUI(false)
br.Splits.Styler(func(s *styles.Style) {
	s.Min.Y.Em(25)
})
pl.GVRun()

The Kir channel that GABA-B couples to is also available as a separate standalone channel, using the same form of inwardly rectifying voltage dependence, but in a persistently active form. These channels are not used by default, but are critical for certain cell types. The inward rectification results in an additional threshold-like bistable behavior, where neurons that are not activated much above the resting potential get additional leak conductance from this Kir channel, but once they start to get activated, the Kir is inactivated and they can then fire more easily. This is critical for the behavior of medium spiny neurons (MSNs) in the striatum of the basal ganglia, which tend to be very quiet, but then fire in discrete bursts when sufficiently activated.

We use equations from Lindroos et al. (2018) parameterized for MSN neurons, with an activating factor M, which can be seen in Plot 3:GV run and Plot 3:Time run:

\[ M_{\infty} = \frac{1}{1 + e^{(V+102)/13}} \]

\[ A = 0.1 e^{-(V+60)/14} \]

\[ B = \frac{0.27}{1 + e^{-(V+31)/23}} \]

\[ \tau = \frac{1}{A + B} \]

\[ M(t) = M(t-1) + \frac{1}{3 \tau(t)} \left(M_{\infty} - M(t-1) \right) \]

\[ g_{kir} = \overline{g}_{kir} M(t) \]

VGCC

Voltage gated calcium channels (VGCC) are similar to NMDA channels in that their conductance to \(Ca^{++}\) has a voltage dependency, but they do not have a neurotransmitter binding property, and their voltage dependency is typically at higher threshold than NMDA (and is not caused by \(Mg^{++}\) block). Due to this higher threshold, the VGCC channels are typically only open during backpropagating action potentials (see neuron dendrites for details), and thus they provide a calcium signal that is closely tied to postsynaptic spiking. We leverage this property in the kinase algorithm learning rule. Because VGCCs also close very quickly once the spike is over, they do not have a big impact on activation dynamics — they are mostly important for learning.

There are a large number of VGCCs types (Dolphin, 2018; Cain & Snutch, 2012) denoted by letters in descending order of the voltage threshold for activation: L, PQ, N, R, T, which have corresponding Ca_v names: Ca_v1.1, 1.2, 1.3. 1.4 are all L type, 2.1, 2.2, 2.3 are PQ, N, and R, respectively, and T type (low threshold) comprise 3.1, 3.2, and 3.3. Each channel is characterized by the voltage dependency and inactivation functions.

• The L type is the classic “VGCC” in dendritic spines in pyramidal cells, which we plot below.

• PQ and R are specific to cerebellum.

• The T type is the most important for low frequency oscillations, and is absent in pyramidal neurons outside of the 5IB layer 5 neurons, which are the primary bursting type. It is most important for subcortical neurons, such as in TRN. See Destexhe et al, 1998 model in BRIAN for an implementation.

pl := &chanplots.VGCCPlot{}
root, _ := tensorfs.NewDir("Root")
br := egui.NewGUIBody(b, pl, root, "VGCC", "VGCC Channel", "VGCC L-type channel equations")
pl.Config(root, br.Tabs)
br.FinalizeGUI(false)
br.Splits.Styler(func(s *styles.Style) {
	s.Min.Y.Em(25)
})
pl.GVRun()

Our implementation of the L-type VGCC is based on Urakubo et al., 2008, using source code available at this link.

First, there is a temporally invariant aspect of the voltage gating defined by a sigmoidal function similar to those seen above:

\[ g_{vgcc}(V) = -\overline{g}_{vgcc} \frac{1}{1 - e^{0.0756 V}} \]

And there are two additional opponent gating factors denoted M (activating) and H (inactivating) that have a strong time dependency, and sigmoidal driving functions as follows:

\[ M_{max}(V) = \frac{1}{1 + e^{-(V + 37)}} \]

\[ H_{max}(V) = \frac{1}{1 + e^{2(V + 41)}} \]

The update equations just move toward these max values with associated time constants:

\[ M(t) = M(t-1) + \frac{1}{3.6} \left( M_{max}(V) - M(t-1) \right) \]

\[ H(t) = H(t-1) + \frac{1}{29} \left( H_{max}(V) - H(t-1) \right) \]

The final conductance over time reflects the activation vs. inactivating binding sites in a 3 to 1 ratio:

\[ g_{vgcc}(t) = g_{vgcc}(V) M^3(t) H(t) \]

To see the static voltage-gated sigmoidal functions, do Plot 4:GV run. To see the response of the M and H channels to discrete spiking inputs, do Plot 4:Time run. In both cases you will need to deselect variables to be able to see the values with smaller ranges. You should observe that the M activating channel rises up quickly at every action potential, and drops quickly back down, consistent with its 3.6 ms time constant. By contrast, the H inactivating factor builds up over time and slowly decreases the overall conductance value.

A-type K

pl := &chanplots.AKPlot{}
root, _ := tensorfs.NewDir("Root")
br := egui.NewGUIBody(b, pl, root, "AK", "A-type K Channel", "A-type K channel equations")
pl.Config(root, br.Tabs)
br.FinalizeGUI(false)
br.Splits.Styler(func(s *styles.Style) {
	s.Min.Y.Em(25)
})
pl.GVRun()

The A-type K channel is voltage-gated with maximal activation around -37 mV (Hoffman et al., 1997). It is particularly important for counteracting the excitatory effects of the VGCC L-type channels (with which they are co-localized) which can otherwise drive runaway excitatory currents. Think of it as an “emergency brake” and is needed for this reason whenever adding VGCC to a model.

It has two state variables, M (V-gated opening) and H (V-gated closing), which integrate with fast and slow time constants, respectively. H relatively quickly hits an asymptotic level of inactivation for sustained activity patterns, so we can actually ignore it with minimal consequences, and also simplify some of the faster time dynamics because we are not implementing the fast Hodgkin-Huxley spiking channels. Thus, in our simulations we just use a single sigmoidal function for the M activating component, with parameters that fit the rising portion of the more complex function:

\[ M = \frac{0.076}{1 + e^{0.075(V + 2)}} \]

This function is missing the declining values beyond the -37 mV peak, but given our spiking dynamics and 1 ms time step of integration, this is not relevant. The comparison between this simplified conductance and the full model from Hoffman et al. (1997) and Migliore et al. (1999) can be seen in the Plot 5:GV run plot. You can also see the time dynamics in the Plot 5:Time run plot.

Adaptation channels

There are a number of different channels that drive an adaptation effect in neurons (also known as accommodation or neural fatigue), over different time scales and in response to different activity signals. These channels cause neurons to decrease their responsiveness over time, as a function of how active they have been, which results in an overall suppression of responses to ongoing activity patterns, and a relative enhancement to novel ones (i.e., a novelty filter). All of these channels conduct \(K^+\) ions, like the leak channel, and we add their conductances together in an overall Gk(t) value.

Functionally, these are also known as afterhyperpolarization (AHP) channels, because they cause neurons to become refractory (less responsive) to further excitatory inputs for different time windows, from fast (fAHP; 2-5 ms), to medium (mAHP; 50-100 ms), and slow (sAHP; 0.1-2 s).

Biologically, there are a great variety of K channels, which are typically composed from 4 different α (alpha) subunits, of which there are voltage-dependent (Kv), inward rectifier (Kir), calcium (KCa), and sodium (KNa) types. Benda et al. (2010) provide a detailed computational comparison of several of these different types, in relation to another adaptation mechanism based on the Na channels that drive the Hodgkin-Huxley action potential spiking itself. They discovered important qualitative differences between the K-based adaptation channels and the spiking threshold adaptation.

M-type channels

There are a large number of different K channels that were historically called M-type due to their muscarinic acetylcholine (ACh) response in the bullfrog sympathetic ganglion cells. These are now classified as Kv7 KCNQ channels, which have been identified throughout the brain, with responsiveness to a wide range of different neurotransmitters and other factors (Greene & Hoshi, 2017). One well-characterized M-type channel is a major contributor to the mAHP K current, which we use in Axon as described below. It is voltage sensitive, but starts to open at low voltages (-60 mV), and can be closed by different neurotransmitters or other factors. In general it takes a while to activate, with a time constant of around 50 ms or so, and it also deactivates on that same timescale.

Functionally, the neurotransmitter binding to these M-type channels typically turns them off, not on, producing a disinhibitory effect which is fundamentally modulatory or multiplicative in nature: removing extra K leak conductance does not directly activate the neuron, but it makes it more responsive to other inputs, and less susceptible to adaptation effects.

mAHP

pl := &chanplots.MahpPlot{}
root, _ := tensorfs.NewDir("Root")
br := egui.NewGUIBody(b, pl, root, "mAHP", "mAHP Channel", "mAHP M-type equations")
pl.Config(root, br.Tabs)
br.FinalizeGUI(false)
br.Splits.Styler(func(s *styles.Style) {
	s.Min.Y.Em(25)
})
pl.GVRun()

Relative to the KNA channels described below, which respond to \(Na^+\) influx from spikes, the broadly tuned voltage sensitivity of the M-type mAHP channel produces a stronger anticipatory conductance prior to the spike. Thus, it will “head off” incipient spikes in a way that the KNa channels do not.

The original characterization of the M-type current in most models derives from Gutfreund et al. (1995), as implemented in NEURON by Mainen & Sejnowski (1996), see these ModelDB entries: 2488, and 181967 from Cutsuridis & Poirazi, 2015, and ICGeneology for the widespread use of this code.

There is a voltage gating factor N (often labeled M for other channels) which has an asymptotic drive value (\(N_{\infty}\)) and a time-dependent time constant \(\tau\) which are both composed from two sigmoidal functions of potential V, centered at -30 mV with a slope of 9 mV:

\[ V_o = V + 30 \]

\[ A = \frac{V_o}{\tau_{max} \left(1 - e^{-V_o/9} \right)} \]

\[ B = \frac{-V_o}{\tau_{max} \left(1 - e^{V_o/9} \right)} \]

\[ N_{\infty} = \frac{A}{A + B} \]

\[ \tau = \frac{1}{A + B} \]

\[ N(t) = N(t-1) + \frac{1}{\tau} \left( N_{\infty} - N(t-1) \right) \]

\[ g_{mahp} = \overline{g}_{mahp} 2.3^{(37-23)/10} N(t) \]

You can see these functions in Plot 6:GV run, and the time-course dynamics in Plot 6:Time run.

sAHP

pl := &chanplots.SahpPlot{}
root, _ := tensorfs.NewDir("Root")
br := egui.NewGUIBody(b, pl, root, "sAHP", "sAHP Channel", "sAHP M-type equations")
pl.Config(root, br.Tabs)
br.FinalizeGUI(false)
br.Splits.Styler(func(s *styles.Style) {
	s.Min.Y.Em(25)
})
pl.GCaRun()

It is difficult to identify the origin of a slow, long-lasting sAHP current, which has been observed in hippocampal and other neurons (Larsson, 2013). It appears to be yet another modulator on the M-type channels driven by calcium sensor pathways that have longer time constants. There is more research to be done here, but we can safely use a mechanism that takes a long time to build up before activating the K+ channels, and then takes a long time to decay as well.

The above equations (Eq 21) are used for sAHP, driven by a normalized integrated Ca value, with an offset of 0.8 and slope of 0.02. Unlike mAHP which is updated at the standard 1 ms time step, we update sAHP at the theta cycle interval (every 200 ms), which automatically extends the temporal dynamics. Plot 7:G ca run and Plot 7:Time run show you the functions.

KNa

Another source of adaptation are a family of sodium (Na) gated potassium (K) currents, KNa. As neurons spike, driving an influx of Na, this activates the K channels. Although synaptic Na from AMPA channels could also activate such channels, various considerations suggest that these channels are localized around the axon hillock in the soma and respond mostly to spiking-generated Na (Sanchez-Vives et al., 2000; Wang et al., 2003). Thus, contrasting with the voltage-gated mAHP M-type channel above, these KNa channels respond to above-theshold spiking, and are therefore more “reactive” to activity rather than anticipatory. Multiple different time constants have been identified, including Slick (medium), and Slack (slow) (Kaczmarek, 2013; Kohn, 2007; Sanchez-Vives et al., 2000; Wang et al., 2003; Benda et al., 2010).

Due to the discrete nature of spiking and our 1 ms time scale of updating, we can use a simplified exponential update dynamic as a function of spiking:

	if spike {
		gKNa += 1/Rise * (Max - gKNa)
	} else {
		gKNa -= 1/Decay * gKNa
	}

SKCa

pl := &chanplots.SKCaPlot{}
root, _ := tensorfs.NewDir("Root")
br := egui.NewGUIBody(b, pl, root, "SKCa", "SKCa Channel", "SKCa equations")
pl.Config(root, br.Tabs)
br.FinalizeGUI(false)
br.Splits.Styler(func(s *styles.Style) {
	s.Min.Y.Em(25)
})
pl.GCaRun()

There are two major types of Ca-gated K channels: “small” K (SK, SKCa) and “big” K (BK, BKCa). BK channels are high conductance with fast dynamics, and play a role in shaping the action potential, which we therefore ignore due to our use of the AdEx approximation and 1 ms time step (and they are not widely implemented in biophysical models according to ModelDB).

The SK channel is not turned on by default, but is critical for our model of the subthalamic nucleus (STN), which exhibits a distinct pausing pattern of activity after an initial burst of firing, which the SKCa contributes to (Hallworth et al., 2003). SKCa is activated by intracellular Ca stores that are released during the initial burst of firing, and the subsequent pause in firing induced by the additional K conductance allows these Ca to be re-buffered, thereby deactivating the SKCa current.

Our implementation is based on Fujita et al. (2012), which is in turn based on Günay et al. (2008), using a simple Hill equation which takes the form of:

\[ Y = \frac{X}{X + C_{50}} = \frac{\frac{X}{C_{50}}}{1 + \frac{X}{C_{50}}} \]

where \(C_{50}\) is the concentration at which the value is at 50%. This Hill equation, raised to various power factors reflecting a requirement for multiple elements to be in the same place and state, can be used to model many different chemical processes, and is used behind the scenes to derive various equations used above. A different sigmoidal exponential equation was given in Gillies & Willshaw (2006) in a model of the subthalamic nucleus (STN) cell. Adelman et al. (2012) and Dwivedi & Bhalla (2021) give an activation time constant of 5-15 ms and decay constant of around 30 ms for the SKCa.

As usual, we use an M activation factor, governed by the asymptotic Hill equation, operating on a normalized intracellular Ca concentration factor \([Ca]_i\):

\[ M_{\infty} = \frac{([Ca]_i / C_{50})^4}{1 + ([Ca]_i / C_{50})^4} \]

If \(M_{\infty}\) is greater than the current M activation value, M increases with a rise time constant of 15 ms, and otherwise it decays with a time constant of 30 ms. See Plot 8:G ca run and Plot 8:Time run for plots.