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Variability is inherent in neurons. To account for variability we have to make use of stochastic models. We will take a look at this biologically more rigorous approach by studying the fundamental signal of our brain’s neurons: the action potential and the voltage-gated ion channels mediating it. We will discuss how to model and simulate the action potential stochastically. We review the methods and show that classic stochastic approximation methods fail at capturing important properties of the highly nonlinear action potential mechanism, making the use of accurate models and simulation methods essential for understanding the neural code. We will review what stochastic modelling has taught us about the function, structure, and limits of action potential signalling in neurons, the most surprising insight being that stochastic effects of individual signalling molecules become relevant for whole-cell behaviour. We suggest that most of the experimentally observed neuronal variability can be explained from the bottom-up as generated by molecular sources of thermodynamic noise.

Ion channels are at the heart of generating electrical activity of neurons and coupling it to neurotransmitter release. They comprise a superfamily of transmembrane proteins that form pores through plasma membranes, enabling ions to pass with high efficiency. This chapter reviews the central role of ion channels in the generation and regulation of electrical activity of dopamine neurons. It focuses on midbrain dopamine neurons located in the nuclei substantia nigra (SN, A9) and the adjacent ventral tegmental area (VTA, A10).

This chapter describes the main structural determinants regulating the function of some voltage-gated Na⁺, K⁺, and Ca⁺⁺ channels and their fundamental role in cell excitability. Voltage-dependent Na⁺, K⁺, and Ca⁺⁺ channels are generally closed at the resting membrane potential of nerve cells, which is approximately -60 to -70 mV. However, the transmembrane potential of neurons undergoes continuous changes that are caused by incoming stimuli. In particular, depolarizing inputs trigger the opening of voltage-gated channels, which allow the flow of electrical signals throughout the nervous system. By contrast, membrane repolarization closes these channels and terminates the propagation of the impulse.

Ion channels are cellular proteins that conduct the movement of ions from one side of a membrane to the other. The resultant changes in local ion concentrations and electrical field play pivotal roles in physiological processes, as wide ranging as cell to cell communication, cell proliferation and secretion. This chapter provides a brief historical perspective then introduces the basic theory, terminology, and generic structural and functional features of ion channels. In addition, an overview of relevant ion channel methodologies is provided. The chapter aims to set the scene and to equip the non-specialist reader with sufficient background and understanding to comprehend and enjoy subsequent chapters which provide a more detailed analysis of channel families and individual channels.

Invented by Bert Sakmann and Erwin Neher during the 1970s, the patch-clamp recording technique aids scientists in examining the functions of individual protein molecules that form ion channels in cell membranes. The importance of this outstanding contribution, which has revolutionized neurophysiology and greatly augmented understanding of cell membranes, is underscored by the fact that Sakmann and Neher received the Nobel Prize in 1991. The fundamental finding arising from patch-clamp experiments is that currents through cell membranes pass through protein channels. This chapter provides a brief introduction to the patch-clamp technique and presents the fundamental equations that govern currents through individual ion channels. The electrophysiology of three types of ion channels is described: nongated chloride channels, voltage-gated sodium and potassium channels, and ligand-gated acetylcholine channels.

The Patch model illustrates results obtained from patch-clamp experiments. This chapter describes the elementary equations for currents through individual ion channels, based on Ohm's law. Because there are no time-dependent variables, no numerical integration is required.

This chapter is an introduction to modelling stochastically gating ion channels using continuous-time discrete-state Markov chains. Analytical and numerical methods are presented for determining steady-state statistics of single channel gating, including the stationary distribution and open and closed dwell times. Model reduction techniques such as fast–slow analysis and state lumping are discussed as well as Gillespie’s method for simulating stochastically gating ion channels. Techniques for the estimation of model parameters and identification of model topology are briefly discussed, as well as the thermodynamic requirements that constrain the selection of rate constants. Approaches for modelling clusters of interacting ion channels using Markov chains are also summarized. Our presentation is restricted to Markov chain models of intracellular calcium release sites where clusters of calcium release channels are coupled via changes in the local calcium concentration and exhibit stochastic calcium excitability reminiscent of calcium puffs and sparks. Representative release site simulations are presented showing how phenomena such as allosteric coupling and calcium-dependent inactivation, in addition to calcium-dependent activation, affect the generation and termination of calcium puffs and sparks. The chapter concludes by considering the state space explosion that occurs as more channels are included in Markov chain models of calcium release sites. Techniques used to mitigate against this state space explosion are discussed, including the use of Kronecker representations and mean-field approximations.

In the soma-dendrite region of a motoneurone, a spike is succeeded by a long-lasting afterhyperpolarization (AHP), which is important for the characteristics of maintained repetitive spike discharges. Such discharges may be evoked by steady depolarizing (inward) currents of appropriate intensity, and the relation between spike-frequency and current may have a steeper slope for high than for lower frequencies (secondary and primary range). During stimulation with steady current, there is a rapid initial and a slower late phase of spike-frequency adaptation. Potential-dependent ion channels may give rise to persistent inward currrents (PICs) which add to excitatory synaptic currents driving a discharge. Motoneurones are unusually well known with regard to the matching between their electrophysiological activation properties (e.g., range of spike frequencies, duration of AHP; input resistance, membrane resistivity, threshold current for activation) and the functional characteristics of their main target cells (the skeletal muscle fibres).

Fast synaptic transmission is crucial for real-time functioning of the brain. All the receptor molecules that mediate fast transmission events are also ligand-gated ion channels, i.e., they are ion channels that undergo allosteric structural changes on binding a particular neurotransmitter molecule, resulting in the opening of the channel, the entry of selected ions into the neuron and subsequent signalling events. Their primary function is to receive signal input at postsynaptic membranes, where some also play central roles in synaptic plasticity. However, they are also found in postsynaptic membranes outside synapses, and in presynaptic terminals, where they are involved in the control of transmitter release. This chapter presents an overview of current knowledge of the molecular biology of these receptors.

Bertil Hille helped establish the concept of ion channels as membrane proteins forming gated aqueous pores. He showed that Na⁺ and K⁺ channels of axons can be distinguished by drugs such as tetrodotoxin and tetraethyl ammonium ion, and that their ionic selectivity can be understood by a limiting pore size, the selectivity filter, and by movements of ions through a series of saturable sites. He showed that local anesthetics enter Na⁺ channels in a state-dependent manner. In later studies of modulation of ion channels by G-protein coupled receptors he distinguished two new signaling pathways. A fast, pertussis toxin sensitive pathway turned on inward rectifier K⁺ channels and turned off Ca²⁺ channels by G protein G(( subunits. A slow, pertussis toxin insensitive pathway turned off some K⁺ and Ca²⁺ channels by depleting the plasma membrane phosphoinositide PIP₂. Hille wrote the widely used textbook, Ion Channels of Excitable Membranes.

The Patch model simulates the dynamics of a small piece of cell membrane as observed with the patch-clamp technique. Either individual channels in a small patch of membrane or the macroscopic current observed with the whole cell patch clamp may be simulated. Channels are viewed as all-or-none conductances that are gated by voltage (sodium and potassium channels), by a messenger ligand (ACh channel), or not gated (chloride channel). The experimenter can specify the channel type and the number of channels found in the membrane patch. Properties that are simulated include the random opening of channels, the gating of the channels by membrane potential, the voltage dependence of the currents through the channels, the dependence of single channel currents on channel conductance, and the summation of microscopic currents from many individual, randomly gated channels to yield a noisy macroscopic current.

Ionotropic glutamate receptors (iGluRs) are the principal mediators of fast excitatory transmission in the central nervous system. These receptors were originally distinguished by their specific binding of and responses to agonists such as N-methyl-D-aspartate (NMDA), quisqualate/α-amino-3-hydroxy-5-methyl-4-isoxazole-4-propionate (AMPA), and kainate, thus defining three subfamilies. More recent molecular biological studies have basically confirmed that the principal receptor types fall into these three main classes on the basis of similarities in amino acid sequence, but they also have indicated that each subfamily comprises more than one gene and, as a result of posttranscriptional modifications, many more receptor protein subunits. In addition, researchers have identified a further subgroup in vertebrates (the orphan δ receptors, δ1 and δ2) and another subfamily, the kainate binding proteins, in non-mammalian vertebrates. This chapter discusses the general characteristics of ionotropic glutamate receptors, posttranscriptional modifications, alternative splicing, RNA editing, ligand-binding site of iGluRs, mechanism of channel gating of iGluRs, and the ion channel and carboxyl-terminal domain of glutamate receptors.

The processes within a neuron that are subject to modulation include changes in amplitude or kinetics of the ion channels, the insertion or removal of ion channel proteins from the membrane, changes in the types of ion channels expressed or their localization within the neuron, and changes in release of neurotransmitter from the synaptic terminal. One relatively recently recognized feature of signaling pathways in neurons is that minute-to-minute variations in ion channel properties can be brought about by changing the physical association of the ion channel with its modulating elements. Finally, the activation of biochemical pathways that signal to the nucleus can produce long-term modulation of neuronal excitability by increasing or decreasing the synthesis of proteins required for ion channel expression and function. These mechanisms provide the means whereby one neuron can alter the properties of another neuron and are thus crucial for plasticity observed in the nervous system.

This chapter focuses on the types and distributions of the main voltage-gated ion channels presently known to exist within the dendrites of CA1 pyramidal neurons, neocortical layer Vm pyramidal neurons, cerebellar Purkinje cells, and olfactory bulb mitral cells. This is a rapidly expanding field, and the focus of this chapter is intentionally limited to these four distinctly different types of central neurons because a substantial amount of high-quality information is available concerning their dendritic voltage-gated channels, and because they express a wide range of different types of dendritic electrogenesis. The chapter begins with a short survey of the known dendritic voltage-gated ion channel types and their modulation. A table comparing the physiologically relevant biophysical properties and some pharmacology of these ion channels is also included.

Metabotropic glutamate receptors (mGluRs) are a distinct family of excitatory amino acid receptors. Unlike the ionotropic glutamate receptors (iGluRs), which constitute cation-specific ion channels and mediate fast excitatory synaptic responses, the more recently characterized mGluRs are coupled to a variety of signal transduction pathways via guanine nucleotide binding proteins (G proteins), They produce alterations in intracellular second messengers, affect ion channels, generate relatively slow synaptic responses, and modulate synaptic transmission. In addition, recent observations indicate that G protein-coupled receptors, including mGluRs, are key components in multiprotein signaling assemblies that facilitate interactions with iGluRs and protein kinase cascades, such as the mitogen-activated protein kinase (MAPK) pathway. The prevalence of glutamate as a neurotransmitter, in combination with the widespread distribution of mGluRs, points to this system as a major modulator of second messengers in the mammalian central nervous system. This chapter discusses the molecular structure of mGluRs, along with their distribution, desensitization, effects on neurotransmission and ion channels, regulation of plasma membrane ion channels and intracellular calcium stores by Group I mGluRs, and pharmacology.

Potassium channels are one of the largest ion channel gene families. They are present in nearly all living organisms from bacteria through to man. There are a number of separate families of pore-forming subunit (ex subunit) characterized by distinct biophysical properties and often interacting with unique modulatory proteins (α subunits) distinct from the pore-forming subunit. This chapter discusses what is known of the assembly and targeting of these complexes. To state the obvious, there have also been enormous advances in other aspects of these channels. Issues such as the molecular basis of ion conduction and gating are dealt with briefly in relation to how they impinge on the theme of the chapter.