10.1 Introduction
Voltage sensitive ion channels underly the information processing capabilities of nerve cells (Koch & Segev, 1998; Dayan & Abbott, 2001; Hille, 2001). Ion channels are integral membrane proteins which depending on conformation can pass ionic currents and thus induce dynamic changes in transmembrane potentials (Hille, 2001). Neural and muscle cells use voltage sensitive ion channels as the fundamental nonlinear elements for electrical signaling. In these cells pulse-like electrical signals called action potentials (APs) are induced by an avalanche-like opening of channels.
Biophysical models for AP generation almost universally assume that indi-vidual channels open and close statistically independently and are coupled only through the transmembrane voltage. However, biological ion channels for a va-riety of physiologically important ions have been found capable of cooperative gating when clustered (Schindler, 1984; Saito et al., 1988; Undrovinas et al., 1992; Marx et al., 1998; Molina et al., 2006). In cooperative gating the states of individual channels are not independent but coordinated such that the opening of one channel increases the opening probability of neighboring channels. Examples of cooperative gating have been found in Na+channels (Undrovinas et al., 1992), K+channels (Molina et al., 2006),Ca2+channels (Marx et al., 1998) and in neuro-transmitter receptors (Schindler, 1984). Fig. 10.1 shows the structure of clustered Na+and Ca2+channels from electron microscopy. Intriguingly, patch recordings of such channels exhibit synchronized openings of double or triple channels.
Cooperative gating of ion channels has been proposed to represent a general capability of proteins to undergo so called ‘conformational spread’ (Bray & Duke, 2004). Recently cooperative gating of Na+channels has been hypothesized to underly the observed rapid onset of APs in cortical neurons and to strongly
influence the coding properties of cortical neurons (Naundorf et al., 2006).
100 nm 50 nm
(a)
(d)
500 ms4 pA 100 ms5 pA
(c)
(b)
Figure 10.1: Channel clustering and cooperative gating in Na+(a,b) and Ca2+channels (c,d). (a) Freeze-fracture electron microscopy reveals clustering of membrane particles in cardiac myocytes after ischaemia (Post et al., 1985).
(b) Inside-out patch recordings of such cells showing simultaneous openings of pairs and triples of sodium channels (Undrovinas et al., 1992). (c) Transmission EM shows a dense crystalline array of RyRCa2+release channels in sarcoplasmic reticulum membrane (Saito et al., 1988). (d) Current traces through pairs and tripples of such channels exhibiting synchonized opening and closing (Marx et al., 1998). Dotted lines in (b, d) indicate single channel current steps.
Here we examine the dynamical and functional consequences of channel co-operativity in a conductance based model of neuronal AP generation in which a fraction p of sodium channels exhibit cooperative gating. The model is con-structed such that the strength of inter-channel coupling is quantified in voltage units and can be continuously varied between statistical independence and strong cooperativity.
We examine activation kinetics and AP waveforms predicted by the model for the entire range of cooperative channel fractions and coupling strengths. For strong cooperativity AP onsets become very rapid and for a small fraction of strongly cooperative channels APs exhibit a pronounced biphasic waveform often observed in nerve cells of the central nervous system (Eccles et al., 1958; Bean,
2007). We point out that in this regime the AP onset is triggered by simulta-neous opening of the cooperative channel fraction. We calculate the threshold for this synchronized opening and show that it depends on the fraction of non-inactivated sodium channels. While increasing cooperativity lowers the threshold, the amount of threshold variability resulting from time varying levels of channel inactivation is largely insensitive to the strength of cooperativity.
To assess the functional impact of sodium channel cooperativity we character-ize the ability of the neuronal firing rate to follow high frequency fluctuation in input current. Our results demonstrate that strongly cooperative sodium chan-nel gating can boost the spike encoding of rapidly varying signals even if they represent only a small fraction of all sodium channels.
10.2 AP Generator with Channel Cooperativity
Modeling Cooperative Gating of Sodium Channels
To model cooperative gating of Na+channels, we assume that a channel in the cooperative population is coupled toK neighboring channels such that the open-ing of each neighbor increases the probability of the channel to open. Usopen-ing an activation variable m(t) this is most simply realized by a kinetics ofm as
τm(V) ˙mJ(t) =m∞ V(t) +KJ mJ(t)x
h
−mJ(t). (10.1) Here m∞(V) is the steady state activation curve of individual channels, τm(V) is the activation time constant,h is the available fraction,(mJ(t))xh is the open probability so thatKh(mJ(t))x is the expected number of open neighbors. J is a coupling constant in units of mV that measures the strength of coupling by the voltage shift that would increase the open probability of an isolated channel by the same amount. Eq. (10.1) represents the mean field approximation of cooper-ative channel gating among a coupled population in which opening of individual channels is modeled as a Makov process (Naundorf et al., 2006). In the limit of J = 0, Eq. (10.1) reduces to the classical case of independent channel activation.
We used this approach to examine the predicted signature of channel coop-erativity on the activation of a voltage clamped population of sodium channels.
Assuming a fixed available fraction H0 = 1, very short activation time constant τm(V) and a Bolzmannian single gate activation curve
m∞(V) =
The fraction of open channels mJ∞(V) after a voltage step called the collective activation curve satisfies the self-consistent equation
mJ∞(V) = m∞ V +KJ mJ(t)x
H0
. (10.3)
For the following analysis we take x= 1 as suggested by recent in vitro record-ings of Na+currents in cortical neurons (Baranauskas & Martina, 2006), where the activation time course was best fitted by a linear rising mono-exponential function.