Encyclopedia of Computational Neuroscience

Living Edition
| Editors: Dieter Jaeger, Ranu Jung

Kinetic Models of Postsynaptic Currents

  • Alain DestexheEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4614-7320-6_355-1

Keywords

NMDA Receptor Reversal Potential Kinetic Scheme Maximal Conductance Synaptic Current 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Definition

Modeling synaptic currents and conductances is a central aspect of network simulations. The type of model for synaptic currents depends on the receptor type present in the synapse, as well as if one needs to include mechanisms such as the saturation of successive synaptic events and synaptic depression or facilitation. Kinetic models can provide a way to model these interactions in a compact form and can be analytic in some cases, leading to very fast algorithms to simulate synaptic interactions.

Detailed Description

Introduction

Synaptic currents and conductances represent the most common type of interaction between neurons, and they must be simulated in neuronal networks using the most efficient model as possible. It was shown previously that simplified two-state kinetic models can be used to simulate postsynaptic currents with a reasonable degree of accuracy (Destexhe et al. 1994a, 1998), as an alternative to the more complex Markov models. This approach is similar in spirit to the classic Hodgkin and Huxley (1952) model of action potentials, in which the voltage-dependent currents are also described by two-state kinetic models (reviewed in Destexhe et al. 1994b). We overview here these kinetic models and their fitting to experimental recordings.

Two-State Kinetic Models

The essential properties of ion channel activation can be captured by simplified kinetic models with just two states. The simplest kinetic models for the gating of different classes of ion channels are illustrated in Table 1. For synaptic currents (Destexhe et al. 1994a, 1998), as for voltage-dependent currents (Destexhe et al. 1994b), simplified kinetic models provide an efficient way to incorporate their basic properties, such as the time course of rise and decay and their summation behavior, in simulations that do not require a great level of detail. Typical examples of this kind are simulations of networks of neurons where the most salient features of ion channel interactions must be represented with maximal computational efficiency.
Table 1

Two-state kinetic models of different classes of ion channels. Voltage-dependent channels: the channel is assumed to have opened (O) and closed (C) states modulated by voltage-dependent transition rates (α and β). Calcium-dependent channels: in this case, the opening of the channel depends on the binding of one or several intracellular Ca2+ ions (Ca i ). Transmitter-gated channels: molecules of neurotransmitter (T) are released transiently and bind to the channel, leading to its opening. Second messenger-gated channels: in this case, the opening of the channel is provided by the binding of one or several intracellular second messengers (G). Kinetic equations allow to describe all these processes, which underlie electrophysiological properties and synaptic interactions, using the same formalism (Destexhe et al. 1994b)

Voltage-dependent gating (Hodgkin-Huxley)

\( C\underset{\upbeta \left(\mathrm{V}\right)}{\overset{\upalpha \left(\mathrm{V}\right)}{\rightleftarrows }}O \)

Calcium-dependent gating

\( C+n\kern0.5em C{a}_i\underset{\upbeta}{\overset{\upalpha}{\rightleftarrows }}O \)

Transmitter gating

\( C+n\kern0.5em T\underset{\upbeta}{\overset{\upalpha}{\rightleftarrows }}O \)

Second messenger gating

\( C+n\kern0.5em G\underset{\upbeta}{\overset{\upalpha}{\rightleftarrows }}O \)

In synaptic currents, however, one must specify the time course of the transmitter concentration (T in Table 1). The simple model for transmitter time course is a square pulse, based on experiments in excised membrane patches which showed that 1 ms pulses of 1 mM glutamate reproduced PSCs that were quite similar to those recorded in the intact synapse (Hestrin 1992; Colquhoun et al. 1992; Standley et al. 1993). The simplest model is to assume that the transmitter, either glutamate or GABA, is released according to a pulse when an action potential invades the presynaptic terminal. Then, a two-state (open/closed) kinetic scheme (as in Table 1), combined with such a pulse of transmitter, can be solved analytically (Destexhe et al. 1994a). The same approach also yields simplified algorithms for three-state and higher schemes (Destexhe et al. 1994b). As a consequence, extremely fast algorithms can be used to simulate most types of synaptic receptors because no differential equation must be solved for synaptic currents (Destexhe et al. 1994a).

AMPA-Kainate Receptors. The simplest model that approximates the kinetics of the fast AMPA/kainate type of glutamate receptors can be represented by the two-state diagram:
$$ C+T\underset{\upbeta}{\overset{\upalpha}{\rightleftarrows }}O, $$
(1)
where α and β are voltage-independent forward and backward rate constants. If r is defined as the fraction of the receptors in the open state, it is then described by the following first-order kinetic equation:
$$ \frac{ dr}{ dt}=\upalpha \left[T\right]\kern0.5em \left(1-r\right)-\upbeta r, $$
(2)
and the postsynaptic current I AMPA is given by:
$$ {I}_{AMPA}={\overline{g}}_{AMPA}\kern0.5em r\left(V-{E}_{AMPA}\right), $$
(3)
where AMPA is the maximal conductance, E AMPA is the reversal potential, and V is the postsynaptic membrane potential.
The best fit of this kinetic scheme to whole-cell recorded AMPA/kainate currents (Fig. 1a) gave α = 1.1 × 106 M −1 s −1 and β = 190 s −1 with E AMPA  = 0 mV (Destexhe et al. 1998).
Fig. 1

Best fits of simplified kinetic models to averaged postsynaptic currents obtained from whole-cell recordings. (a). AMPA/kainate-mediated currents (Obtained from Xiang et al. (1992); recorded at 31 ºC). (b). NMDA-mediated currents (Obtained from Hessler et al. (1993); recorded at 22–25 ºC in Mg2+-free solution). (c). GABA A -mediated currents. (d). GABA B -mediated currents (cd recorded at 33–35 ºC by Otis et al. 1992, 1993). For all graphs, the averaged recording of the synaptic current (noisy trace) is represented with the best fit obtained using the models (continuous trace). Transmitter time course was a pulse of 1 mM and 1 ms duration in all cases (ad: Modified from Destexhe et al. (1994b, 1996, 1998))

NMDA Receptors. A unique property of NMDA receptors is that they are voltage dependent and activate only for depolarized membrane potentials. The activation properties NMDA-mediated postsynaptic currents can be represented with a two-state model similar to AMPA/kainate receptors, with a slower time course and a voltage-dependent term. The voltage dependence of NMDA receptor channels is due to its sensitivity to block by physiological concentrations of Mg2+ (Nowak et al. 1984; Jahr and Stevens 1990a, b). The Mg2+ block is voltage dependent, allowing NMDA receptor channels to conduct ions only when depolarized. The necessity of both presynaptic and postsynaptic gating conditions (presynaptic neurotransmitter and postsynaptic depolarization) makes the NMDA receptor a molecular coincidence detector. Furthermore, NMDA currents are carried partly by Ca2+ ions, which may also play a role in triggering intracellular biochemical cascades.

Using the same scheme as Eqs. 1 and 2, the postsynaptic current is given by:
$$ {I}_{NMDA}={\overline{g}}_{NMDA}\kern0.5em B(V)\kern0.5em r\kern0.5em \left(V-{E}_{NMDA}\right), $$
(4)
where NMDA is the maximal conductance, E NMDA is the reversal potential, and B(V) represents the magnesium block of the channel. The magnesium block of the NMDA receptor channel is an extremely fast process compared to the other kinetics of the receptor (Jahr and Stevens 1990a, b). The block can therefore be accurately modeled as an instantaneous function of voltage (Jahr and Stevens 1990b):
$$ B(V)=\frac{1}{1+ \exp \left(-0.062\kern0.5em V\right)\kern0.5em {\left[{\mathrm{Mg}}^{2+}\right]}_o/3.57}, $$
(5)
where [Mg 2+] o is the external magnesium concentration (1–2 mM in physiological conditions).

The best fit of this kinetic scheme to whole-cell recorded NMDA currents (Fig. 1b) gave α = 7.2 × 104 M −1 s −1 and β = 6.6 s −1 with E NMDA  = 0 mV (Destexhe et al. 1998).

GABA A Receptors. GABA A receptors can also be represented by the scheme in Eqs. 1 and 2, with the postsynaptic current given by:
$$ {I}_{GAB{A}_A}={\overline{g}}_{GAB{A}_A}\kern0.5em r\left(V-{E}_{GAB{A}_A}\right), $$
(6)
where \( {\overline{g}}_{GAB{A}_A} \) is the maximal conductance and \( {E}_{GAB{A}_A} \) is the reversal potential.

The best fit of this kinetic scheme to whole-cell recorded GABA A currents (Fig. 1c) gave α = 5 × 106 M −1 s −1 and β = 180 s −1 with \( {E}_{GAB{A}_A}=-80\kern0.5em mV \) (Destexhe et al. 1998).

GABA B Receptors and Neuromodulators. In the three types of synaptic receptors discussed above, the receptor and ion channel are both part of the same protein complex, and these receptors are called “ionotropic.” In contrast, other classes of synaptic response are mediated by an ion channel that is not directly coupled to a receptor but rather is activated (or deactivated) by an intracellular “second messenger” that is produced when neurotransmitter binds to a separate receptor molecule. These so-called “metabotropic” synaptic responses generally affect K + channels but can also affect other channels or biochemical processes. This is the case for GABA B receptors, whose response is mediated by K + channels that are activated by G-proteins (Dutar and Nicoll 1988).

One of the fundamental properties of GABA B receptors is that they are strongly dependent on the pattern of presynaptic spikes. There are no GABA B -mediated responses for single or isolated presynaptic spikes, but GABA B responses appear with intense presynaptic activity such as typically with bursts of presynaptic spikes (Dutar and Nicoll 1988; Davies et al. 1990; Kim et al. 1997). It was shown that such properties can be due to nonlinear properties in the activation kinetics of these receptors (Destexhe and Sejnowski 1995), a hypothesis which was later confirmed by paired recordings of neurons connected with GABA B receptors (Kim et al. 1997; Thomson and Destexhe 1999).

These nonlinear activation properties of GABA B responses, unfortunately, cannot be handled correctly by a two-state model. The simplest model of GABA B -mediated currents has two variables and was obtained from a more complex model (Destexhe and Sejnowski 1995) and is given by:
$$ \begin{array}{c}\frac{ dr}{ dt}={K}_1\kern0.5em \left[T\right]\kern0.5em \left(1-r\right)-{K}_2\kern0.5em r\\ {}\frac{ ds}{ dt}={K}_3\kern0.5em r-{K}_4\kern0.5em s,\\ {}{I}_{GAB{A}_B}={\overline{g}}_{GAB{A}_B}\frac{s^n}{s^n+{K}_d}\left(V-{E}_K\right)\end{array} $$
(7)
where r is the fraction of receptor in the activated (transmitter-bound) state, s is the second messenger (G-protein) concentration, \( {\overline{g}}_{GAB{A}_B}=1 \)nS is the maximal conductance of K + channels, E K  = −95 mV is the potassium reversal potential, and K 1K 4 are rate constants. Fitting of this model to whole-cell recorded GABA B currents (Fig. 1d) gave the following values: K d  = 100 μM 4, K 1 = 9 × 104 M −1 s −1, K 2 = 1.2 s −1, K 3 = 180 s −1, and K 4 = 34 s −1 with n = 4 binding sites (Destexhe et al. 1998). This model was found to correctly capture the dynamical properties of GABA B responses in thalamic slices (see Destexhe et al. 1996).

Many other types of neurotransmitters are involved in metabotropic responses on specific receptor types. This is the case for glutamate (through metabotropic receptors), acetylcholine (through muscarinic receptors), noradrenaline, serotonin, dopamine, histamine, opioids, and others. These neurotransmitter systems have been shown to mediate slow intracellular responses via the intracellular activation of G-proteins, which may affect ionic currents as well as the metabolism of the cell. As with GABA acting on GABA B receptors, the main electrophysiological target of many neuromodulators is to open or close K + channels (reviewed in McCormick 1992). The model of GABA B responses outlined above could thus be used to model these currents, with rate constants adjusted to fit the time courses reported for the particular responses (see details in Destexhe et al. 1994b).

Discussion

In this chapter, we reviewed simple models for the main types of synaptic receptors found in the nervous system, namely, the glutamate AMPA and NMDA receptors and the GABAergic GABA A and GABA B receptors. It is important to keep in mind that the models shown here are among the simplest models that capture the time course and kinetics of these synaptic receptors, while there exist a wide variety of more complex models (see Destexhe et al. 1998 for an overview of simple and complex models). The main target of simple kinetic models is to be included in network simulations, where computational efficiency is an essential concern. With pulses of transmitted, it is possible to solve these models and obtain an analytic expression for the postsynaptic conductance (Destexhe et al. 1994a). These models are therefore very fast to simulate since no differential equation must be added to model synaptic interactions.

References

  1. Colquhoun D, Jonas P, Sakmann B (1992) Action of brief pulses of glutamate on AMPA/kainate receptors in patches from different neurons of rat hippocampal slices. J Physiol 458:261–287PubMedCentralPubMedGoogle Scholar
  2. Davies CH, Davies SN, Collingridge GL (1990) Paired-pulse depression of monosynaptic GABA-mediated inhibitory postsynaptic responses in rat hippocampus. J Physiol 424:513–531PubMedCentralPubMedGoogle Scholar
  3. Destexhe A, Sejnowski TJ (1995) G-protein activation kinetics and spill-over of GABA may account for differences between inhibitory responses in the hippocampus and thalamus. Proc Natl Acad Sci USA 92:9515–9519PubMedCentralPubMedCrossRefGoogle Scholar
  4. Destexhe A, Mainen Z, Sejnowski TJ (1994a) An efficient method for computing synaptic conductances based on a kinetic model of receptor binding. Neural Comput 6:14–18CrossRefGoogle Scholar
  5. Destexhe A, Mainen Z, Sejnowski TJ (1994b) Synthesis of models for excitable membranes, synaptic transmission and neuromodulation using a common kinetic formalism. J Comput Neurosci 1:195–230PubMedCrossRefGoogle Scholar
  6. Destexhe A, Bal T, McCormick DA, Sejnowski TJ (1996) Ionic mechanisms underlying synchronized oscillations and propagating waves in a model of ferret thalamic slices. J Neurophysiol 76:2049–2070PubMedGoogle Scholar
  7. Destexhe A, Mainen M, Sejnowski TJ (1998) Kinetic models of synaptic transmission. In: Koch C, Segev I (eds) Methods in neuronal modeling, 2nd edn. MIT Press, Cambridge, MA, pp 1–26Google Scholar
  8. Dutar P, Nicoll RA (1988) A physiological role for GABAB receptors in the central nervous system. Nature 332:156–158PubMedCrossRefGoogle Scholar
  9. Hessler NA, Shirke AM, Malinow R (1993) The probability of transmitter release at a mammalian central synapse. Nature 366:569–572PubMedCrossRefGoogle Scholar
  10. Hestrin S (1992) Activation and desensitization of glutamate-activated channels mediating fast excitatory synaptic currents in the visual cortex. Neuron 9:991–999PubMedCrossRefGoogle Scholar
  11. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500–544PubMedCentralPubMedGoogle Scholar
  12. Jahr CE, Stevens CF (1990a) A quantitative description of NMDA receptor-channel kinetic behavior. J Neurosci 10:1830–1837PubMedGoogle Scholar
  13. Jahr CE, Stevens CF (1990b) Voltage dependence of NMDA-activated macroscopic conductances predicted by single-channel kinetics. J Neurosci 10:3178–3182PubMedGoogle Scholar
  14. Kim U, Sanches-Vives MV, McCormick DA (1997) Functional dynamics of GABAergic inhibition in the thalamus. Science 278:130–134PubMedCrossRefGoogle Scholar
  15. McCormick DA (1992) Neurotransmitter actions in the thalamus and cerebral cortex and their role in neuromodulation of thalamocortical activity. Prog Neurobiol 39:337–388PubMedCrossRefGoogle Scholar
  16. Nowak L, Bregestovski P, Ascher P, Herbet A, Prochiantz A (1984) Magnesium gates glutamate-activated channels in mouse central neurones. Nature 307:462–465PubMedCrossRefGoogle Scholar
  17. Otis TS, De Koninck Y, Mody I (1992) Whole-cell recordings of evoked and spontaneous GABAB responses in hippocampal slices. Pharmacol Commun 2:75–83Google Scholar
  18. Otis TS, De Koninck Y, Mody I (1993) Characterization of synaptically elicited GABAB responses using patch-clamp recordings in rat hippocampal slices. J Physiol 463:391–407PubMedCentralPubMedGoogle Scholar
  19. Standley C, Ramsey RL, Usherwood PNR (1993) Gating kinetics of the quisqualate-sensitive glutamate receptor of locust muscle studied using agonist concentration jumps and computer simulations. Biophys J 65:1379–1386PubMedCentralPubMedCrossRefGoogle Scholar
  20. Thomson AM, Destexhe A (1999) Dual intracellular recordings and computational models of slow IPSPs in rat neocortical and hippocampal slices. Neuroscience 92:1193–1215PubMedCrossRefGoogle Scholar
  21. Xiang Z, Greenwood AC, Brown T (1992) Measurement and analysis of hippocampal mossy-fiber synapses (abstract). Soc Neurosci Abstr 18:1350Google Scholar

Further Reading

  1. Destexhe A, Sejnowski TJ (2001) Thalamocortical assemblies. Oxford University Press, Oxford, UKGoogle Scholar
  2. Koch C, Segev I (eds) (1988) Methods in neuronal modeling, 2nd edn. MIT Press, Cambridge, MAGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.UNIC (Unit of Neuroscience Information and Complexity)CNRS (Centre national de la recherche scientifique)Gif-sur-YvetteFrance