Consider a living organism or an artificial mechanism, which we shall refer to for the moment as a system, aiming to perform optimally in an uncertain environment. Despite the fact that the environment may be uncertain, we will suppose that we know the structure of the physical laws of the environment determining plausible motions of the system. Suppose that we even know what the system's action might be and assume that criteria of optimality according to which the system must determine its actions are available. Would we be able to decide a priori which particular action a system must execute or how it should adjust itself in order to maintain its behavior at the optimum?

Depending on the language describing the system's behavior, environment, and uncertainties a number of theoretical frameworks can be employed to find an answer to this non-trivial question. If the available information about the system is limited to a statistical description of the events and their likelihoods are known, then a good methodological candidate is the theory of statistical decision making. On the other hand, if the more sophisticated and involved apparatus of stochastic calculus is used to formalize the behavior of a system in an uncertain environment then a reasonable way to approach the analysis of such an object is to employ the theory of stochastic control and regulation. Despite these differences in how the behavior of a system may be described in various settings, there is a fundamental similarity in the corresponding theoretical frameworks.

In this chapter we provide analysis tools for dynamical systems described as input–output and input–state mappings (or simply operators) in the corresponding spaces. Such a description is advantageous and natural when mathematical models of the systems are vaguely known and uncertain. We will see that the basic properties of these input–output and input–state mappings (such as boundedness and continuity) constitute important information for our understanding of the various ways in which an adaptation can be organized in these systems.

In particular, we will see that some basic stability notions (Lyapunov stability of invariant sets (LaSalle and Lefschetz 1961), stability of solutions in the sense of Lyapunov, and input-to-state, input-to-output, output-to-state, and input–output stability (Zames 1966)) are equivalent to continuity of a certain mapping characterizing the dynamics of the system (Theorems 4.1 and 4.3).

As we have said earlier, real physical systems, however, are not always stable and hence their input–output and input–state characterizations are not always continuous. Moreover, the target dynamics of these systems should not necessarily admit continuous input–output or input–state description. Indeed, continuity of a mapping S at a given point u0 in essence reflects the fact that the value of the mapping S(u), can be made arbitrarily close to S(u0), provided that u remains sufficiently close to u0. In reality this is a quite idealistic picture, and in most cases such infinitesimal closeness is not needed.

In this chapter we provide an overview of mathematical formulations of the problem of adaptation in dynamical systems. Adaptation is considered here as a special regulatory process that emerges as a response of a physical system to changes in the environment. In order to understand the notion of adaptation the terms “regulation,” “response,” and “environment” need to be given some physical sense and precise mathematical definitions. Instead of inventing a new language of our own, we adopt these terms from the language of mathematical control theory. It is clear that this adoption of terms will create a certain bias in our approach. On the other hand, it will allow us to operate with rigorously defined and established objects that have already passed the test of time.

We start with a retrospective overview and analysis of main ideas in the existing literature on adaptation in the domain of control. These ideas gave rise to distinct mathematical statements of the problem of adaptation. We will discuss their strengths and limitations with respect to the demands of new real-world applications in biology, neuroscience, and engineering. As a result of the analysis, we will formulate a list of features that a modern theory of adaptation must inherit, and propose a program that will allow us to contribute to the development of such a theory in the following chapters.

Logical principles of adaptation

The theory of adaptive regulation, as a set of notions, methods, and tools for systematic study of systems adjusting their properties in response to changes in the environment, has a long history of development.

Skeletal muscle contraction is ultimately initiated by activity in the nervous system. Muscle receives both sensory and motor nerve fibres. The sensory nerves convey information about the state of the muscle to the nervous system. This includes information about muscle length detected by the muscle spindles and tension detected by the Golgi tendon organs. There are also a variety of free nerve endings in the muscle tissue, some of which convey sensations of pain. Of the motor fibres in mammals, the γ-motoneurons provide a separate motor nerve supply for the muscle fibres of the muscle spindles. However, the bulk of the muscle fibres are supplied by the α-motoneurons. Each α-motoneuron innervates a number of muscle fibres, from less than ten in the extraocular muscles, which move the eyeball in its socket, to over a thousand in a large limb muscle. The complex of one motoneuron plus the muscle fibres which it innervates is called a motor unit. Since they are all activated by the same nerve cell, all the muscle fibres in a single motor unit contract at the same time. However, muscle fibres belonging to different motor units may well contract at different times. Thus, most mammalian muscle fibres are contacted by a single nerve terminal, although sometimes there may be two terminals originating from the same nerve axon. Muscle fibres of this type are known as twitch fibres, since they respond to nervous stimulation with a rapid twitch.

The processes described in the preceding chapters culminate in the generation of mechanical activity by skeletal muscle. These mechanical properties of muscles are readily investigated using isolated muscle or nerve–muscle preparations such as the gastrocnemius or sartorius nerve–muscle preparations in the frog. Experiments on large mammalian muscles require an intact blood supply, in which case the experiments must be performed on an anaesthetized animal, with the nerve supply to the muscle cut and its tendon dissected free and attached to some recording device. The muscle is excited by applying a brief pulse of stimulating current to its nerve or directly to the muscle itself.

Isometric and isotonic contractions

When muscles contract they exert a force on their attachments (this force is equal to the tension in the muscle) and they shorten if they are permitted to do so. Hence we can measure two different variables during the contraction of a muscle: its length and its tension. Most often one of these two is maintained constant during the contraction. In isometric contractions the muscle is not allowed to shorten (its length is held constant) and the tension it produces is measured. In an isotonic contraction the load on the muscle (which is equal to the tension in the muscle) is maintained constant and its shortening is measured.

An isometric recording device has to be stiff, so that it does not in fact allow the muscle to shorten appreciably while the force is being measured.

Electrophysiological recording methods

Although the nervous impulse is accompanied by effects that can under especially favourable conditions be detected with radioactive tracers, or by optical and thermal techniques, electrical recording methods normally provide much the most sensitive and convenient approach. A brief account is therefore necessary of some of the technical problems that arise in making good measurements both of steady electrical potentials and rapidly changing ones.

In order to record the potential difference between two points, electrodes connected to a suitable amplifier and recording system must be placed at each of them. If the investigation is only concerned with action potentials, fine platinum or tungsten wires can serve as electrodes, but any bare metal surface has the disadvantage of becoming polarized by the passage of electric current into or out of the solution with which it is in contact. When, therefore, the magnitude of the steady potential at the electrode tip is to be measured, non-polarizable or reversible electrodes must be used, for which the unavoidable contact potential between the metal and the solution is both small and constant. The simplest type of reversible electrode is provided by coating a silver wire electrolytically with AgCl, but for the most accurate measurements calomel (Hg/HgCl2) half-cells are best employed.

When the potential inside a cell is to be recorded, the electrode has to be very well insulated except at its tip, and so fine that it can penetrate the cell membrane with a minimum of damage and without giving rise to electrical leaks.

Both voltage-gated and ligand-gated ion channels are large protein molecules, as is the sodium pump Na,K-ATPase. In recent years the primary structure of a number of them has been determined, and by combining this information with the biophysical evidence, major advances have been made in our understanding of how they work at the molecular and submolecular levels.

cDNA sequencing studies

A protein consists of a long chain built up of 20 different amino acids (Table 5.1), folded on itself in a rather complicated way. Its properties depend critically on the arrangement of the folds, which is determined by the exact order in which its constituent amino acids are strung together. This in turn is specified by the sequence of the nucleotide bases that make up the DNA molecules which constitute the genetic material of the cell. There are only four different bases, and each of the 20 amino acids corresponds according to a universally obeyed triplet code to a specific group of three of them. The information embodied in the base sequence of a DNA molecule is transcribed on to an intermediary messenger RNA, and is then translated during the synthesis of the protein to yield the correct sequence of amino acids. Rapid sequencing methods for nucleotides were perfected by Sanger and his colleagues, and modern recombinant DNA technology makes possible the cloning of DNA so that the quantity required for the determination can be prepared from a single gene.

Structure of the cell membrane

All living cells are surrounded by a plasma membrane composed of lipids and proteins, whose main function is to control the passage of substances into and out of the cell. In general, the role of the lipids is to furnish a continuous matrix that is impermeable even to the smallest ions, in which proteins are embedded to provide selective pathways for the transport of ions and organic molecules both down and against the prevailing gradients of chemical activity. The ease with which a molecule can cross a cell membrane depends to some extent on its size, but more importantly on its charge and lipid solubility. Hence the lipid matrix can exclude completely all large water-soluble molecules and also small charged molecules and ions, but is permeable to water and small uncharged molecules like urea. The nature of the transport pathways is dependent on the specific function of the cell under consideration. In the case of nerve and muscle, the pathways that are functionally important in connection with the conduction mechanism are: (1) the voltage-sensitive sodium and potassium channels peculiar to electrically excitable membranes, (2) the ligand-gated channels at synapses that transfer excitation onwards from the nerve terminal, and (3) the ubiquitous sodium pump, which is responsible in all types of cell for the extrusion of sodium ions from the interior.

The essential feature of membrane lipids that enables them to provide a structure with electrically insulating properties, i.e. to act as a barrier to the free passage of ions, is their possession of hydrophilic (polar) head groups and hydrophobic (non-polar) tails.

Skeletal muscles are the engines of the body. They account for over a quarter of its weight and the major part of its energy expenditure. They are attached to the bones of the skeleton and so serve to produce movements or exert forces. Hence they are central to such activities as voluntary movement, maintenance of posture, breathing, eating, directing the gaze and producing gestures and facial expressions. Skeletal muscles are activated by motoneurons, as we have seen in previous chapters. Their cells are elongate and multi-nuclear and the contractile material within them shows cross-striations. Hence skeletal muscle is a form of striated muscle. In contrast, cardiac and smooth muscles have cells with single nuclei, and smooth muscles are not striated; we shall examine their properties in Chapters 12 and 13.

Anatomy

Skeletal muscle fibres are multi-nucleate cells formed by fusion of elongated uni-nucleate cells called myoblasts whose respective nuclei become arranged around the edge of the fibre. Mature fibres may be as long as the muscle (Figure 9.1) of which they form part, and 10 to 100 µm in diameter. Bundles of muscle fibres are surrounded by a further sheet of connective tissue, the perimysium, and the whole muscle is contained within an outer sheet of tough connective tissue, the epimysium. These connective tissue sheets are continuous with the insertions and tendons which serve to attach the muscles to the skeleton. An excellent blood supply provides a network of blood capillaries between individual fibres.