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Some models of action potential generation in neurons like the Fitzhugh–Nagumo and the Morris–Lecar model are given by slow–fast differential equations. We outline a general theory allowing us to quantify the effect of noise on such equations. The method combines local analyses around stable and unstable equilibria, and around bifurcation points. We discuss in particular two different mechanisms of excitability, which lead to different types of interspike statistics.

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.

The genes that contribute to specific cortical excitability phenotypes are difficult to predict. But one group of genes expressed in central neurons can be operationally defined and constitute a category of central concern: excitability genes. These genes initiate and maintain the critical capacity for voltage-dependent membrane behavior and synaptic transmission in specific brain networks, and participate directly in their ability to display rapid neuromodulation, synchronization, and long-term retrieval of information. It is from within this group that the genetic elements most directly responsible for the heritable component of human cognitive abilities will ultimately be found. This chapter discusses the genetics of circuit assembly and genetic strategies in cerebral excitability analysis.

The essence of neuronal function is to generate outputs, usually in the form of action potentials, in response to inputs, in the form of synaptic potentials. The morphology and membrane properties of dendrites are extremely important determinants of this input-output transformation. This chapter discusses the many factors affecting the dendritic integration of synaptic potentials, including the site of action potential initiation and dendritic excitability.

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 examines whether motor imagery may facilitate corticospinal activity in the areas corresponding to the muscles involved in the imagined movement. It notes that neuroimaging data provide evidence that the increase in brain activity is specific to the representation of the body part whose movement is imagined. It demonstrates that the facilitation of corticospinal excitability during motor imagery is associated with specific reductions in intracortical inhibition.

One of the consequences of the historiographic tradition that has considered the Galvani-Volta controversy as due to an irresolvable opposition between a medical and physical paradigm has been a great disregard of the interest of Volta in electrophysiological phenomena. In contrast to the received views, Volta was indeed very interested in biological and medical research and, in his experiments, he obtained important results on the mechanisms of electrical excitability of nerves and muscles. These results anticipate some aspect of the modern understanding of electrophysiological phenomena.

This chapter illustrates model-based approaches to the study of the neural basis of cognitive control. It presents recent studies that investigate whether the predictive information theoretic models provide a good index of cortical excitability (CSE) in simple choice reaction-time tasks, whether the brain can be used to explain neural responses related to the learning of contextual uncertainty, and whether one can apply a model-based approach to predict electrophysiological response related to contextual surprise.

Emotional disorders in children with ADHD are often difficult to detect. The emotional problems that occur in children with ADHD fall into three categories: emotional characteristics of ADHD (low frustration tolerance, preoccupation, thrill-seeking, dysthymia, and overexcitability), reactive emotions to having ADHD, and coexisting emotional disorders (depression, anxiety, obsessive-compulsive disorder, and bipolar disorder). Inefficient inhibitory processes in the brain are the basis of these emotional difficulties. It is essential to be aware of the frequent occurrence of emotional disorders in children with ADHD. This chapter discusses emotional disorders in ADHD, including the emotional characteristics of ADHD, reactive emotions, and coexisting emotional disorders.

Spines are specialized cellular compartments found at high density on the dendrites of many neurons. Glutamatergic synapses are made onto spines, which house much of the machinery needed to read out synaptic activity. By providing a diffusionally isolated signaling compartment, spines allow neighboring synapses to operate independently. Stimulus-evoked increases in spine calcium trigger many forms of synaptic plasticity and can influence local excitability and morphology. This chapter focuses on the mechanisms of calcium handling and signaling within individual spines.

This chapter tells the story of the discovery of the reticular activating system. At the same time, the chapter traces various attempts to address the larger question of “waking” the cortex and bringing it to a state of consciousness. It turns to two scientists, Horace Magoun and Giuseppe Moruzzi, both of whom conducted experiments to explore the possible effects on the cerebral cortex of stimulating the brain stem. Since the brain’s reticular formation ended just below the thalamus on either side, it was logical to see if it might alter cortical excitability. The chapter shows how Magoun and Moruzzi came to the conclusion that, through its action on the excitability of the cortex, the reticular formation could control the wakefulness of the brain.

The previous chapter provided a detailed description of the currents underlying the generation and propagation of action potentials in the squid giant axon. The Hodgkin-Huxley (1952d) model captures these events in terms of the dynamical behavior of four variables: the membrane potential and three state variables determining the state of the fast sodium and the delayed potassium conductances. This quantitative, conductance-based formalism reproduces the physiological data remarkably well and has been extremely fertile in terms of providing a mathematical framework for modeling neuronal excitability throughout the animal kingdom (for the current state of the art, see McKenna, Davis, and Zornetzer, 1992; Bower and Beeman, 1998; Koch and Segev, 1998). Collectively, these models express the complex dynamical behaviors observed experimentally, including pulse generation and threshold behavior, adaptation, bursting, bistability, plateau potentials, hysteresis, and many more. However, these models are difficult to construct and require detailed knowledge of the kinetics of the individual ionic currents. The large number of associated activation and inactivation functions and other parameters usually obscures the contributions of particular features (e.g., the activation range of the sodium activation particle) toward the observed dynamic phenomena. Even after many years of experience in recording from neurons or modeling them, it is a dicey business predicting the effect that varying one parameter, say, the amplitude of the calcium-dependent slow potassium current (Chap. 9), has on the overall behavior of the model. This precludes the development of insight and intuition, since the numerical complexity of these models prevents one from understanding which important features in the model are responsible for a particular phenomenon and which are irrelevant. Qualitative models of neuronal excitability, capturing some of the topological aspects of neuronal dynamics but at a much reduced complexity, can be very helpful in this regard, since they highlight the crucial features responsible for a particular behavior. By topological aspects we mean those properties that remain unchanged in spite of quantitative changes in the underlying system. These typically include the existence of stable solutions and their basins of attraction, limit cycles, bistability, and the existence of strange attractors.

There is no doubt that the phenomenon of chemical oscillation—the periodic or nearly periodic temporal variation of concentrations in a reacting system— provided the initial impetus for the development of nonlinear chemical dynamics, and has continued to be the most thoroughly studied of the phenomena that constitute this rich field. In our opening chapter, we alluded to the early skepticism that experimental observations of chemical oscillation engendered. We also noted that the first chemical oscillators were discovered accidentally, by researchers looking for other phenomena. It is relatively easy to understand intuitively why a typical physical oscillator, like a spring, should behave in a periodic fashion. It is considerably more difficult for most of us to see how a chemical reaction might undergo oscillation. As a result, the thought of building a physical oscillator seems far more reasonable than the notion of designing an oscillatory chemical reaction. In this chapter, we will examine how chemical oscillation can arise, in general, and how it is possible to create chemical reactions that are likely to show oscillatory behavior. In the next chapter, we will discuss how to take a chemical oscillator apart and analyze why it oscillates—the question of mechanism. We also look in detail there at the mechanisms of several oscillating reactions. In order to gain some insight into how oscillation might arise in a chemical system, we shall consider a very simple and general model for a reaction involving two concentrations, u and v. Two independent concentration variables is the smallest number that can generate oscillatory behavior in a chemical system. The basic idea, however, is applicable to many-variable systems, because the essential features of the dynamics are often controlled by a small number of variables, and the other variables simply follow the behavior of the key species.