Subsections

• 2.4.1 Inhibitory Synapses
• 2.4.2 Excitatory Synapses

2.4 Synapses

So far we have encountered two classes of ion channel, namely voltage-activated and calcium-activated ion channels. A third type of ion channel we have to deal with is that of transmitter-activated ion channels involved in synaptic transmission. Activation of a presynaptic neuron results in a release of neurotransmitters into the synaptic cleft. The transmitter molecules diffuse to the other side of the cleft and activate receptors that are located in the postsynaptic membrane. So-called ionotropic receptors have a direct influence on the state of an associated ion channel whereas metabotropic receptors control the state of the ion channel by means of a biochemical cascade of g-proteins and second messengers. In any case the activation of the receptor results in the opening of certain ion channels and, thus, in an excitatory or inhibitory postsynaptic current (EPSC or IPSC).

Instead of developing a mathematical model of the transmitter concentration in the synaptic cleft we try to keep things simple and describe transmitter-activated ion channels as an explicitely time-dependent conductivity g_syn(t) that will open whenever a presynaptic spike arrives. The current that passes through these channels depends, as usual, on the difference of its reversal potential E_syn and the actual value of the membrane potential,

I_syn(t) = g_syn(t) (u - E_syn) . (2.19)

The parameter E_syn and the function g_syn(t) can be used to characterize different types of synapse. Typically, a superposition of exponentials is used for g_syn(t). For inhibitory synapses E_syn equals the reversal potential of potassium ions (about -75 mV), whereas for excitatory synapses E_syn $\approx$ 0.

2.4.1 Inhibitory Synapses

The effect of fast inhibitory neurons in the central nervous system of higher vertebrates is almost exclusively conveyed by a neuro-transmitter called $\gamma$-aminobutyric acid, or GABA for short. In addition to many different types of inhibitory interneurons, cerebellar Purkinje cells form a prominent example of projecting neurons that use GABA as their neuro-transmitter. These neurons synapse onto neurons in the deep cerebellar nuclei (DCN) and are particularly important for an understanding of cerebellar function.

The parameters that describe the conductivity of transmitter-activated ion channels at a certain synapse are chosen so as to mimic the time course and the amplitude of experimentally observed spontaneous postsynaptic currents. For example, the conductance $\bar{{g}}_{{\text{syn}}}^{}$(t) of inhibitory synapses in DCN neurons can be described by a simple exponential decay with a time constant of $\tau$ = 5 ms and an amplitude of $\bar{{g}}_{{\text{syn}}}^{}$ = 40 pS,

$$\sum_{f}^{} \bar{{g}}_{{\text{syn}}}^{} \tau \Theta$$ (2.20)

Here, t^(f) denotes the arrival time of a presynaptic action potential. The reversal potential is given by that of potassium ions, viz. E_syn = - 75 mV.

Of course, more attention can be payed to account for the details of synaptic transmission. In cerebellar granule cells, for example, inhibitory synapses are also GABAergic, but their postsynaptic current is made up of two different components. There is a fast component, that decays with a time constant of about 5 ms, and there is a component that is ten times slower. The underlying postsynaptic conductance is thus of the form

$$\sum_{f}^{} \left(\vphantom{ \bar{g}_{\text{fast}} \, {\text{e}}^{-(t-t^{(f)}... ... \bar{g}_{\text{slow}} \, {\text{e}}^{-(t-t^{(f)})/\tau_{\text{slow}}} }\right. \bar{{g}}_{{\text{fast}}}^{} \tau_{{\text{fast}}} \bar{{g}}_{{\text{slow}}}^{} \tau_{{\text{slow}}} \left.\vphantom{ \bar{g}_{\text{fast}} \, {\text{e}}^{-(t-t^{(f)}... ... \bar{g}_{\text{slow}} \, {\text{e}}^{-(t-t^{(f)})/\tau_{\text{slow}}} }\right) \Theta$$ (2.21)

2.4.2 Excitatory Synapses

Most, if not all, excitatory synapses in the vertebrate central nervous system rely on glutamate as their neurotransmitter. The postsynaptic receptors, however, can have very different pharmacological properties and often different types of glutamate receptors are present in a single synapse. These receptors can be classified by certain amino acids that may be selective agonists. Usually, NMDA (N-methyl-D-aspartate) and non-NMDA receptors are distinguished. The most prominent among the non-NMDA receptors are AMPA-receptors^2.1. Ion channels controlled by AMPA-receptors are characterized by a fast response to presynaptic spikes and a quickly decaying postsynaptic current. NMDA-receptor controlled channels are significantly slower and have additional interesting properties that are due to a voltage-dependent blocking by magnesium ions (Hille, 1992).

Excitatory synapses in cerebellar granule cells, for example, contain two different types of glutamate receptors, viz. AMPA- and NMDA-receptors. The time course of the postsynaptic conductivity caused by an activation of AMPA-receptors at time t = t^(f) can be described as follows,

$$\bar{{g}}_{{\text{AMPA}}}^{} \cal {N} \left[\vphantom{ {\text{e}}^{-(t-t^{(f)})/\tau_{\text{decay}}} - {\text{e}}^{-(t-t^{(f)})/\tau_{\text{rise}}} }\right. \tau_{{\text{decay}}} \tau_{{\text{rise}}} \left.\vphantom{ {\text{e}}^{-(t-t^{(f)})/\tau_{\text{decay}}} - {\text{e}}^{-(t-t^{(f)})/\tau_{\text{rise}}} }\right] \Theta$$ (2.22)

with rise time $\tau_{{\text{rise}}}^{}$ = 0.09 ms, decay time $\tau_{{\text{decay}}}^{}$ = 1.5 ms, and maximum conductance $\bar{{g}}_{{\text{AMPA}}}^{}$ = 720 pS; cf. (Gabbiani et al., 1994). The numerical constant $\cal {N}$ = 1.273 normalizes the maximum of the braced term to unity.

NMDA-receptor controlled channels exhibit a significantly richer repertoire of dynamic behavior because their state is not only controlled by the presence or absence of their agonist, but also by the membrane potential. The voltage dependence itself arises from the blocking of the channel by a common extracellular ion, Mg²⁺ (Hille, 1992). Unless Mg²⁺ is removed from the extracellular medium, the channels remain closed at the resting potential even in the presence of NMDA. If the membrane is depolarized beyond -50 mV, then the Mg²⁺-block is removed, the channel opens, and, in contrast to AMPA-controlled channels, stays open for 10 - 100 milliseconds. A simple ansatz that accounts for this additional voltage dependence of NMDA-controlled channels in cerebellar granule cells is

$$\begin{multline} g_{\text{NMDA}}(t) = \bar{g}_{\text{NMDA}} \cdot {\cal N} \c... ...a \, u} \, [\text{Mg}^{2+}]_{\text{o}}/\beta \right )^{-1} \,, \end{multline}$$

with $\tau_{{\text{rise}}}^{}$ = 3 ms, $\tau_{{\text{decay}}}^{}$ = 40 ms, $\cal {N}$ = 1.358, $\bar{{g}}_{{\text{NMDA}}}^{}$ = 1.2 nS, $\alpha$ = 0.062 mV^-1, $\beta$ = 3.57 mM, and the extracellular magnesium concentration [Mg²⁺]_o = 1.2 mM (Gabbiani et al., 1994).

A final remark on the role of NMDA-receptors in learning is in order. Though NMDA-controlled ion channels are permeable to sodium and potassium ions, their permeability to Ca²⁺ is even five or ten times larger. Calcium ions are known to play an important role in intracellular signaling and are probably also involved in long-term modifications of synaptic efficacy. Calcium influx through NMDA-controlled ion channels, however, is bound to the coincidence of presynaptic (NMDA release from presynaptic sites) and postsynaptic (removal of the Mg²⁺-block) activity. Hence, NMDA-receptors operate as a kind of a molecular coincidence detectors as they are required for a biochemical implementation of Hebb's learning rule; cf. Chapter 10.