A thermodynamic description for physiological transmembrane transport

Work required for transmembrane molecular fluxes

Consider a system consisting of a biological membrane surrounded by two aqueous compartments (e.g. extracellular and intracellular). Assume, to start with, that the compartments contain molecules of a single type s (e.g. Na⁺, K⁺, glucose), possibly in different concentrations. Let ∆G_s be the energy required for the transport of the molecules across the membrane in a specific direction (e.g. inside to outside). To write an expression for ∆G_s it is necessary to take the direction of motion of the s-molecules into account. To do so, label the extracellular and intracellular compartments as 0 and 1, respectively, and let c_s and d_s ∈ {0, 1} represent the source and the destination compartments for the transport of the s-molecules. The pair (c_s, d_s )=(0,1) represents inward transport and the pair (c_s, d_s )=(1,0) represents outward transport. The work required to transport n_s molecules of type s from compartment c_s to compartment d_s can then be written as

(Aidley, 1998; Blaustein et al., 2004; De Weer et al., 1988) where T, k, q, z_s, [s]₀, and [s]₁ represent the absolute temperature (°K), Boltzmann's constant (1.381 ×10⁻²³ Joules/^oKelvin), the elementary charge (1.60217733 ×10⁻¹⁹ Coulombs/molecule), the valence, the extracellular, and the intracellular concentrations for the molecules of type s, respectively. The variable v represents the transmembrane potential. Two particular cases are worth noticing. First, if s is an ion, then z_s ≠ 0 and Equation (1) becomes

where v_s is the Nernst potential for the s-molecules¹ (Nernst, 1888). Second, if the s-molecules are not charged, then z_s = 0 and the work required to move the s-type molecules from c_s to d_s simplifies to

If ∆G_s < 0, then the molecules can be transported passively (e.g. electrodiffusion), decreasing the electrochemical gradient for s across the membrane. In contrast, if ∆G_s > 0, the transmembrane transport of s from c_s to d_s is not thermodynamically favourable, which means the transport from c_s to d_s requires energy that is not available in the electrochemical gradient for s (active transport). As a consequence, active transport of s would increase the driving force for the motion of s across the membrane.

Joint transmembrane transport of different types of molecules

To find an expression that describes a more general transport mechanism, assume that transport takes place as single events in which molecules of m different types move in parallel, or possibly sequentially (e.g. first Na⁺, then K⁺), across the membrane. Let S be a set that represents the types of molecules that are jointly transported in a single event. For instance, for Na⁺-H⁺ exchangers, S = {Na⁺, H⁺}, with m = 2. The energy required to transport these molecules is the sum of the energies required to transport each type of molecule in S. In other words,

As before, transport is thermodynamically favourable when ∆G_S ≤ 0. If not, extra energy is required. To distinguish between these two cases, define the total energy ∆G of the transport mechanism, possibly including an extra source of energy, as

where δ_Ext = 1 if ∆G_S > 0, and 0 otherwise. If ∆G_S > 0, then the energy from ATP hydrolysis or any other sources represented by δ_Ext∆G_Ext should be negative and larger in size in comparison to ∆G_S, so that ∆G ≤ 0, making the transport thermodynamically possible. In particular, for ATP-driven transport, the extra energy supplied by hydrolysis of ATP (Tanford, 1981; De Weer et al., 1988) is

with [ATP]₁, [ADP]₁, and [P_i]₁ representing, respectively, the intracellular concentrations of adenosine triphosphate (ATP), adenosine diphosphate (ADP), and inorganic phosphate (P_i) (De Weer et al., 1988). The potential v_ATP ≈ −450 mV (Endresen et al., 2000), but could vary depending on the intracellular amounts of ATP, ADP, and P_i (De Weer et al., 1988). The first term in Equation (6) represents the "standard" change in Gibbs free energy for the hydrolysis of ATP². Similar expressions could be derived for active transport driven by light, or other sources of energy. The concentrations of ATP, ADP, and P_i are assumed to be constant in most models presented here, but it should be noted that such concentrations are not necessarily constant, and in fact, may vary a lot in some cases, as it is the case for skeletal muscle (Wackerhage et al., 1998).

Flux due to transmembrane transport

The formulation in Equation (5) can be combined with Equation (1) to derive a generalised expression for flux and model different known mechanisms of physiological transmembrane transport, possibly combining the transport of different molecules simultaneously (e.g. Na-H exchange). In this case, the forward direction of the transport would be described by the combined forward transport of each of the different molecules under consideration. For instance, the source and target compartments for Na⁺ and Ca²⁺ are different in Na-Ca exchangers. The stoichiometry for the transport mediated by Na-Ca exchangers in the forward direction involves three Na⁺ molecules moving inward (along their electrochemical gradient) in exchange for one Ca²⁺ molecule moving outward (against their electrochemical gradient) (Mullins, 1979; Venetucci et al., 2007).

Let α and β be the flux rates in the forward and backward directions, in units of molecules per ms per µm⁻². These rates depend, a priori, on the energy required for the transport of the molecules in S. The net flux rate associated to the net transmembrane transport, can then be written as

How do α and β depend on ∆G? The steady state relationship between the energy ∆G and the the forward and backward flow rates, hereby represented by α and β, can be expressed by means of a Boltzmann distribution (Boltzmann, 1868) as

Assuming that α and β are continuous functions, the rates α and β can be rewritten as

where r and b represent the rate at which the transport takes place (molecules per ms per µm⁻²) and the bias of the transport in one of the two directions (b = 1/2 means the transport is symmetrical relative to the point at which ∆G=0). Note that the functional form of the α and β in Equations (9) are similar to those by Butler (1924); Erdey-Grúz & Volmer (1930). Also, notice that the steady state relationship between α and β in Equation (8) can be obtained from Equations (9), for any r and any b. However, it should be the case that r and b vary in specific ranges depending on the physicochemical characteristics of the pore through which molecules cross the membrane, and in general, on the transport mechanism. As already mentioned, the rate r should be larger for electrodiffusive transport mediated by ion channels, in comparison to the slower transport rates for facilitated diffusion and active transport meditated by carrier proteins and pumps. The rate r may depend on temperature (Sen & Widdas, 1962), the transmembrane potential (Starace et al., 1997), the concentrations inside and outside of the membrane (Yue et al., 1990), and other factors (Novák & Tyson, 2008). If the parameter b ∈ [0, 1], then b∆G and (b − 1)∆G have opposite signs and can be thought of as the energies required to the transport of the molecules in S in the forward and backward directions, respectively, with b biasing the transport in the forward direction when close to 1, and in the backward direction when close to 0 (Figure 1).

Figure 1. Fluxes biased in the target→source (backward, b=0.1, black dashed line), source→target (forward, b=0.9, red dash-dot line), or showing no rectification (b=0.5, blue solid line).

The energy ∆G_S necessary can be found on the right axis if the transport is primary active, and on the left axis when transport is secondary active or passive. Extra energy from ATP hydrolysis or other sources added to ∆G_S shift the total energy to the left axis and make the transport thermodynamically possible. See Table 1 for examples.

The flux can then be written explicitly combining Equation (7) and Equation (9) to obtain,

Taking the above observations into account, it is possible to combine Equation (4) and Equation (5), to write an expression similar to Equation (8) for the steady state balance between the forward and backward transport of all the molecules in S (see Table 1 for examples of different transport mechanisms with their energies and total charge movements).

Flux and current

After substitution of the formulas for ∆G from Equation (4) and Equation (5) into Equation (10), the flux rate resulting from simultaneously transporting molecules in S across the membrane can be written explicitly as

where v_T = kT/q and

represents the net number of charges moved across the membrane. Note that v_Ext should be v_ATP for ATPases. The first, more complex, form of the flux in Equation (11) could be useful when working with models for which changes in the concentrations of different molecules are relevant.

If the transport is electrogenic, then the product qη (in Coulombs) represents the net charge moved across the membrane, relative to the extracellular compartment. Non electrogenic transport yields η = 0, which means the flow does not depend on the transmembrane potential, and

If only ions are involved in the transport, the flux simplifies to

where

The quantity v_o/η can be thought of as a reversal potential. If η < 0, then positive charge is transported inward, or negative charge is transported outward. In contrast, η > 0 means that positive charge is transported outward or negative charge transported inward. For instance, inward electrodiffusion of single Na⁺ ions gives η = −1, which can be thought of as loosing one positive charge from the extracellular compartment in each transport event.

Transmembrane current. A flux that results in electrogenic transport (Equation (11) and Equation (14)) can be converted to current density after multiplication by qη. In short form,

with qr in Amperes/m² or equivalent units.

Substitution of Equation (11) or Equation (14) into Equation (16) yields a general formula for the current generated by transmembrane ionic flux (Figure 2), that uses the same functional form for channels (protein or lipid) and pumps. Recall that Equation (16) can also be written explicitly in terms of the transmembrane concentrations of one or more of the ions involved using Equation (11). It is possible to derive expressions for r that take into account biophysical variables like temperature and the shape and length of the pore through which the molecules cross (Endresen et al., 2000; Pickard, 1969).

Figure 2. Currents mediated by K-electrodiffusion for b_K ∈ {0.1, 0.5, 0.9} using the general formulation (solid lines) and its cubic approximation (dashed lines).

Inward rectification occurs for b_K < 1/2 and outward rectification for b_K > 1/2 and qr_KN_K = 1.

Special cases and examples

A number of nontrivial and important properties of transmembrane ionic currents, including rectification, are also described by Equation (16). Also, different models for current already in the literature can be obtained by making approximations or setting particular cases from Equation (16). Examples include electrodiffusive currents that result from integration of the Nernst-Planck equation along the length of membrane pore (Jacquez & Schultz, 1974; Johnston et al., 1995; Pickard, 1969). Of particular interest, conductance-based currents are linear approximations of the formulation (16), around the reversal potential for the current.

Lower order approximations to the general formulation and conductance based models. The general current from Equation (16) can be rewritten as a series around the reversal potential v_o to obtain useful approximations with possible computational or analytical advantages. To do so, rewrite equation Equation (16) using Taylor’s theorem (Courant & John, 2012; Spivak, 2018) to obtain

Conductance-based currents (Hodgkin & Huxley, 1952) can then be obtained as linear approximations of the general current around v_o,

where g = η²qr/v_T is in units of nS/µm². For instance, the linear approximation for the current through open sodium channels around v_Na in Equation (18) gives g_Na = qr_Na/v_T, and v_o/η_Na = v_Na, with η_Na = −1, so that i_Na ≈ g_Na(v − v_Na).

Note that if the truncation order is larger than 1, then approximations from Equation (17) include the rectification parameter b. One already known consequence is that conductance-based currents do not capture rectification.

The Goldman constant field approximation is a particular case of the general formulation. The Goldman-Hodgkin-Katz equation describing the transmembrane current carried by ions of type x is given by

where x_j, j ∈ {0, 1}, represent the extra and intracellular concentrations of an ion of type x.

Multiplying the numerator and denominator of Equation (19) by $x_1^{b-1}{x}_0^b\exp [(b-1)z_{x}v/v_T]$ , and algebraically rearranging terms yields,

where

is an amplitude term that can be approximated by a constant (Endresen et al., 2000; Herrera-Valdez & Lega, 2011; Nonner & Eisenberg, 1998; Nonner et al., 1998, Equation 25 and Equation 26).

Asymmetric bidirectional flow causes rectification. If the flux of molecules across a membrane is mediated by proteins, it can be biased in either the outward or the inward direction (Hollmann et al., 1991; Riedelsberger et al., 2015). This was first called ”anomalous rectification” by Katz (1949), who noticed that K⁺ flows through muscle membranes more easily in the inward, than in the outward direction (Adrian, 1969; Armstrong & Binstock, 1965). It was later found that some K⁺ channels display the bias in the opposite direction (Woodbury, 1971). The former type of K⁺ current rectification is called inward, and the latter outward (Figure 2).

Rectification is thus a bias in one of the two directions of transport. The type of rectification (inward or outward) depends on the molecules being transported and on the structure of the proteins mediating the transport. Flux rectification is not only displayed by ions, as shown by molecules like glucose, which may cross membranes via GLUT transporters bidirectionally and asymmetrically, even if the glucose concentration is balanced across the membrane (Lowe & Walmsley, 1986).

Rectification can be described by setting b in Equation (11) to values different from 1/2 and it becomes more pronounced as b is closer to 0 or 1. These values represent biases in the transport toward the source, or the target compartment, respectively. As a consequence, rectification yields an asymmetry in the graph of α − β as a function of ∆G (Figure 1). For electrogenic transport, rectification can be thought of as an asymmetric relationship between current flow and voltage, with respect to the reversal potential v_o. The particular case b = 1/2 (non rectifying) yields a functional form for current similar to that proposed by Pickard (1969), and later reproduced by (Endresen et al., 2000), namely

From here on, subscripts will be used to represent different transport mechanisms. For instance, the current for a Na-Ca pump will be written as i_NaCa.

Electrodiffusion of K⁺ through channels (η = 1 and v_o = v_K), is outward for v > v_K, and inward for v < v_K. The K⁺ current through the open pore is therefore

Current flow through inward rectifier K-channels (Riedelsberger et al., 2015) can be fit to values of b_K < 1/2. For instance,

describes a current generated by a K⁺ flow that is limited in the outward direction, similar to the currents described originally by Katz (1949). Analogously, b_K > 1/2 limits the inward flow. For example, the current

describes outward rectification (Riedelsberger et al., 2015).

Based on the work of Riedelsberger et al. (2015) on K⁺ channels, inward (outward) rectification arises when the S4 segment in K⁺ channels is located in the inner (outer) portion of the membrane. These two general configurations can be thought of in terms of ranges for the parameter b_K, namely, b_K < 1/2 for inward, and b_K > 1/2 for outward rectification (Figure 2).

In general, ion channels are typically formed by different subunits, that combined produce structural changes that may result in rectified flows. For instance, non-NMDA glutamatergic receptors that can be activated by kainic acid and α-amino-3-hydroxy-5-methyl-4-isoxazole propionic acid (AMPA), display different permeabilities to Na⁺, K⁺, and Ca²⁺ depending on the subunits that form the receptor (Hollmann et al., 1991). In particular, the Ca²⁺ currents recorded in oocytes injected with combinations of GluR1 and GluR3 cRNA have different steady state amplitudes and show different levels of rectification (Figure 3).

Figure 3. Currents mediated by AMPA-Kainate receptors with different ionic permeabilities (different reversal potentials) and different degrees of rectification caused by differences in the sub-unit composition of the receptors.

Currents were recorded from oocytes injected with GluR3 cRNA (blue), or a combination of GluR1 and GluR3 cRNA (orange), in Ca²⁺ ringer solution after activation by AMPA. The curves were fit with(v_o, b, rq) = (−30, 0.45, 21), for GluR3 and (v_o, b, rq) = (−35, 0.35, 20), for GluR1+GluR3. Data was digitised from Figure 3B in the article by Hollmann et al. (1991) (Table 2).

Primary active transport. The Na-K ATPase is a primary active transporter that uses the energy from the hydrolysis of one molecule of ATP for the uphill transport of 3 Na⁺ ions outward and 2 K⁺ ions inward in each cycle (η_NaK = 1), respectively (De Weer et al., 1988). The reversal potential v_NaK = v_ATP + 3v_Na − 2v_K (Table 1), in a single transport event (Chapman, 1973; Gadsby et al., 1985; Garrahan & Glynn, 1967; Post & Jolly, 1957). As a consequence, the transport kinetics of the Na-K ATPase reverse for potentials smaller than v_NaK (De Weer et al., 1988).

The current-voltage relationships recorded from Na-K ATPases in guinea pig ventricular cells are shaped as hyperbolic sines (Gadsby et al., 1985). Those currents would be fit with b_NaK ≈1/2, yielding

In response to steroids like strophandin, the voltage-dependence of the Na-K ATPase current has been reported to show a plateau as v increases past the reversal potential for the current (Nakao & Gadsby, 1989). In such cases, the Na-K ATPase current can be assumed to be inwardly rectifying and fit with values of b_NaK ≈ 0, so that,

or alternatively,

The rectification for the Na-K pump ATPase has also been reported to occur in small neurons of the dorsal root ganglion in rats (Hamada et al., 2003). The alternative expression (28) also explains qualitatively different behaviors of the Na-K current as a function of the transmembrane concentrations of Na⁺ and K⁺. For instance, if either [Na]₁ or [K]₀ increase and v > v_NaK, then the amplitude of i_NaK would increase at a smaller rate of change in comparison to when v < v_NaK, which grows exponentially in size. This is also in line with reports of non significant changes in the transport by Na-K ATPases in response to elevated intracellular Na⁺ during heart failure (Despa et al., 2002), in which the transmembrane potential is likely to be depolarised.

Secondary active transport. An example of a pump that mediates secondary active transport is the Na-Ca exchanger, which takes 3 Na⁺ ions from the extracellular compartment in exchange for one intracellular Ca²⁺ ion in forward mode (Pitts, 1979; Reeves & Hale, 1984). The reversal potential for the current is v_NaCa = 2v_Ca − 3v_Na, with η_NaCa = −1. Assuming b_NaCa = 1/2, the Na-Ca current can be rewritten as

The driving force v − v_NaCa could reverse in sign with large enough increases in the intracellular concentration of Ca²⁺, or in the membrane potential. As a result, the current could have a dual contribution to the change in transmembrane potential, as predicted by some theoretical models of cardiac pacemaker activity (Rasmusson et al., 1990a; Rasmusson et al., 1990b).

Electrodiffusive transport. Consider transmembrane electrodiffusive transport of a single ionic type x, with z_x and v_x representing the valence and the Nernst potential for ions of type x, respectively. In this case, the reversal potential satisfies

which means that η_x can be factorised in the argument for the exponential functions and the general expression (16) can be rewritten as

In the absence of rectification (b_x = 0.5) the formula simplifies to

since hyperbolic sines change signs if their arguments do. For calcium channels,

See Jacquez & Schultz (1974); Pickard (1969) Table 1 for other examples.

The applicability of the general formulations described above is illustrated next with models of cardiac and neuronal membrane potential.

Cardiac pacemaking in the sinoatrial node

The pacemaking dynamics of cells in the rabbit sinoatrial node (Figure 4) can be modelled as low dimensional dynamical systems based on the assumption that v changes as a function of a combination of channel-mediated electrodiffusion and pumping mechanisms involving Ca²⁺, K⁺, and Na⁺ (Herrera-Valdez & Lega, 2011; Herrera-Valdez, 2014). Explicitly, assume that Ca²⁺ transport is mediated by L-type Ca²⁺ channels (Mangoni et al., 2003) and Na⁺-Ca²⁺ exchangers (Sanders et al., 2006). K⁺ transport is mediated by delayed-rectifier voltage-activated channels (Shibasaki, 1987), and Na⁺-K⁺ ATPases (Herrera-Valdez & Lega, 2011; Herrera-Valdez, 2014).

Figure 4. Central sinoatrial node pacemaking dynamics using the system (35)–(42).

A. Transmembrane potential (gray) and the reversal potential v_NaCa (blue) as a function of time. B. Dynamics of large calcium (blue) and potassium (orange) currents in pA. The total current shown by a gray line labelled as i_m. C. Currents mediated by NaK ATPases (orange) and Na-Ca exchangers (blue), respectively. Notice the carrier mediated currents (C) are about one order of magnitude smaller than those mediated by channels shown (B).

The temporal evolution for v can then be described by

where w represents the proportion of activated K⁺ channels (and also, the proportion of inactivated Ca²⁺ channels) (Av-Ron et al., 1991; Herrera-Valdez & Lega, 2011). The variable c represents the intracellular Ca²⁺ concentration. The transmembrane currents (normalised by the membrane capacitance), are given by

where φ_x is a difference of exponential functions as defined in Equation (16), with x ∈ {NaK, NaCa, KD, CaL}.

The steady state for the activation of voltage-dependent channels in the model is described by a generic function of the form

with labels m and w used for L-type Ca²⁺ and delayed rectifier K⁺ channels, respectively. F_u is increasing as a function of v, with a graph of sigmoidal shape (Herrera-Valdez & Lega, 2011). The parameters g_u and v_u represent the steepness and the half-activation potential, respectively. The voltage-dependent activation for the L-type Ca²⁺ channels (Equation (39)) is assumed to be fast (quasi-stedy state), so it is described by the function F_m(v) (Av-Ron et al., 1991; Herrera-Valdez & Lega, 2011; Rinzel & Ermentrout, 1989).

The activation of K⁺ currents recorded in voltage-clamp experiments often displays sigmoidal time courses that resemble logistic population dynamics (Covarrubias et al., 1991; Hodgkin & Huxley, 1952; Tsunoda & Salkoff, 1995). The dynamics of the proportion of activated K⁺ channels, w, evolve according to

where F_w and R_w respectively representing the voltage-dependent steady state and rate (1/ms) for the opening of K_d channels (Willms et al., 1999).

The rate of activation for K⁺ channels is a voltage-dependent function of the form

where b_w represents a bias in the conformational change underlying channel activation. The function R_w has the shape of a hyperbolic cosine when b_w is 1/2.

During pacemaking oscillations, the intracellular Ca²⁺ concentration may change 10-fold or more (Rasmusson et al., 1990a; Rasmusson et al., 1990b). Therefore, the system includes an equation for the dynamics for c, assuming linear convergence to a steady state c_∞ in the absence of Ca²⁺ fluxes, and an increase proportional to the total transport of Ca²⁺ ions through L-type channels and Na⁺-Ca²⁺ exchangers (Figure 4). Explicitly,

where k_c (µM/mV) represents the impact of the transmembrane Ca²⁺ fluxes on the intracellular Ca²⁺ concentration. The term −r_c c can be thought of as a generic form of buffering and other transport mechanisms contributing to decrease the intracellular Ca²⁺ concentration. The minus sign in front of k_c accounts for the fact that the sign of the J_CaL is negative. The sign in front of J_NaCa is because the forward flux of Ca²⁺ mediated by the Na-Ca exchanger is opposite to that of electrodiffusive Ca²⁺. The transmembrane concentrations of Na⁺ and K⁺ across the membrane are assumed to change negligibly (Herrera-Valdez & Lega, 2011; Rasmusson et al., 1990a; Rasmusson et al., 1990b).

The solutions of Equations (35)–(43) with parameters as in Table 3 reproduce important features of the membrane dynamics observed in the rabbit’s central sinoatrial node, including the period (ca. 400 ms), amplitude (ca. 70 mV), and the maximum ∂_tv (<10 V/s) of the action potentials (Zhang et al., 2000).

A number of interesting features of ionic fluxes can be observed from closer examination of the solutions of Equation (35)–Equation (43). First, the Na-Ca current reverses during pacemaking, as v = v_NaCa (Figure 4A, blue line). Between the initial depolarisation and until the maximum downstroke rate, approximately, v_NaCa < v, which means that J_NaCa > 0. Then, Ca²⁺ extrusion by the Na-Ca exchanger occurs only for a brief period of time during the downstroke and also after each action potential (Figure 4C, blue line). Second, as previously reported in different studies involving spiking dynamics, the time course of the Ca²⁺ current shows a partial inactivation with a double peak (Figure 4B, blue line) around a local minimum (Carter & Bean, 2009; Rasmusson et al., 1990a; Rasmusson et al., 1990b), and in agreement with data from voltage-clamp experiments (Mangoni et al., 2006). A number of models have been constructed in attempts to reproduce the double activation by making extra assumptions about gating (Rasmusson et al., 1990a; Rasmusson et al., 1990b). For instance, some models include a second activation variable, or multiple terms in the steady state gating, or in the time constant for activation or inactivation. However, the explanation for the double peak can be much simpler. The calcium current J_CaL is a negative-valued, non monotonic function for v < v_Ca, which can be thought of as a product of a amplitude term that includes gating and the function φ_CaL. The normalised current J_CaL has a local minimum (maximum current amplitude) around -10 mV (Figure 4B, blue line and Figure 5A, blue line), after which the current decreases, reaching a local maximum as the total current passes through zero, at the peak of the action potential around 10 mV (Figure 4B, where ∂_t v = 0). The first peak for the Ca²⁺ current occurs when v reaches the maximum depolarisation rate (Figure 5B). As v increases (e.g. upstroke of the action potential). The second peak for the current occurs as the membrane potential decreases, and passes again through the region where the maximal current occurs (local minimum for J_Ca). The two local minima for J_CaL occur at different amplitudes because of the difference in the evolution of v during the upstroke and the downstroke of the action potential (Figure 5A, blue line, and Figure 5B, where ∂_t v = 0). It is important to remark that the dual role played by w is not the cause of the double activation. This is illustrated by analysing the behaviour of a non-inactivating J_CaL without the inactivation component (Figure 5A, gray line). The double activation can also be observed in models in which the activation of K⁺ channels and the inactivation of Ca²⁺ or Na⁺ channels are represented by different variables (Rasmusson et al., 1990a) and in dynamic voltage clamp experiments on neurons in which there are transient and persistent sodium channels (Carter & Bean, 2009).

Figure 5. Dynamics of the calcium current and double activation during pacemaking oscillations.

A. Behaviour of the inactivating L-type Ca²⁺ current with respect to the transmembrane potential (blue line) and a non-inactivating current (gray line). Notice that two points with locally maximum current amplitude occur during the action potential. The local minimum with larger current values occurs during the upstroke of the action potential. The local minimum with smaller current values occurs during the downstroke of the action potentials. (B) Calcium current as a function of the time-dependent change in v. The maximum rate of change for v occurs when the calcium current reaches its maximum amplitude.

The double peak in the Ca²⁺ current reflects on the intracellular Ca²⁺ concentration (Figure 6, gray line), and by extension, on the Nernst potential for Ca²⁺ (Figure 6, blue line), which display two increasing phases and two decreasing phases, respectively. The first and faster phase in both cases occur during the initial activation of the L-type channels. The second phase occurs during the downstroke, as second peak of the Ca²⁺ current occurs. As a consequence, the reversal potential for the Na-Ca exchanger, v_NaCa = 2v_Ca − 3v_Na (Figure 6, orange line) also has two phases, this time increasing. Increasing the intracellular Ca²⁺ (Figure 6, gray line) concentration decreases the Nernst potential for Ca²⁺, and vice versa. By extension, v_NaCa, becomes larger when c increases. Ca²⁺ enters the cell in exchange for Na⁺ that moves out when v > v_NaCa, during most of the increasing phase and the initial depolarisation phase of the action potential (blue lines in Figure 4A and C, and Figure 6).

Figure 6. Calcium dynamics during pacemaking.

Time courses of the intracellular calcium concentration (gray, left axis), the Nernst potential for Ca²⁺ (orange, right axis), and the reversal potential for the Na-Ca exchanger (blue, right axis). Notice the two phases of calcium increase that occur in agreement with the double peak observed in the calcium current (see Figure 4B, blue trace).

Fast spiking interneuron dynamics

To construct of a simple model for the dynamics of a fast spiking (FS) striatal interneuron, assume the transmembrane potential depends on three currents respectively mediated by Na-K pumps, non-inactivating K⁺ channels, and Na⁺ channels with transient dynamics, with voltage-dependent gating in both channels. It is also assumed that the proportion of activated K⁺ is represented by a variable w ∈ [0, 1], which also represents the proportion of inactivated Na⁺ channels (Av-Ron et al., 1991; Rinzel, 1985). That is, 1 − w represents the proportion of non-inactivated Na⁺ channels. Since w models the activation of a population of channels, it makes sense assume that its dynamics follow a logistic scheme, without adding extra powers to w. This also follows the experimentally observed dynamics which have been reported repeatedly, including those of delayed-rectifier K⁺ currents recorded in voltage clamp (see for instance Figure 3 in Hodgkin & Huxley, 1952).

Explicitly, the dynamics for the FS-interneuron membrane can be described by a system with two coupled differential equations of the form

The activation rate for K⁺ channels depends is a voltage-dependent functions R_w and F_w as defined for the cardiac pacemaking model. It is also assumed that the activation of sodium channels is at a quasi steady state as a function of v.

Striatal FS interneurons display maximum ∂_tv between 100 and 200 V/s. In current clamp mode, most neurons are silent, and show transitions between rest and repetitive spiking at a rheobase current of approximately 90 pA, with initial firing rates between 50 and 60 Hz and a delay to first spike in the transition that decreases as the stimulus amplitude increases (Figure 7, parameters in Table 4).

Figure 7. Rest to spiking transitions of FS interneuron under current clamp.

The traces show responses to current-clamp stimulation of different amplitudes. The transition between rest and spiking with a rheobase occurs between 40 and 50 pA, as shown for some FS neurons in the mouse striatum (Orduz et al., 2013). The traces correspond to stimulation amplitudes of 0 (gray dots), 40 (black dashed line), 50 (blue), and 80 pA (gray). Parameters can be found in Table 4.

To include these properties into the model, the membrane capacitance was specified first, then the maximum ∂_tv was adjusted by fitting the parameter a_NaT, and then the contributions for the K⁺ channels and the Na-K ATPase are set to obtain spiking and fit the rheobase. The model in Equations (44)–(45) reproduces dynamics observable in fast spiking neurons in CA1 (Erisir et al., 1999) or in the striatum (Orduz et al., 2013; Tepper et al., 2010).

Even though FS-interneuron spiking is conditional to the reception of input, the dynamics of the system can be thought of as qualitatively similar to those observe in the SAN. Within the time of a single spike, and by extension during the repetitive spiking regime, if v increases, w also increases, but at a slower rate in comparison to v. This is because the activation w is always moving toward its steady state value, which increases as v increases. Once w increases, the Na⁺ current tends to decrease and the K⁺ current tends to increase, thereby causing a decrease in v. The slower dynamics in w relative to those in v capture the delay between the amplification caused by the Na⁺ current and the recovery caused by the negative feedback of the K⁺ current. The current mediated by Na/K-ATPase acts as an extra attracting force toward v_NaK that increases nonlinearly as the distance between v and v_NaK increases.