Rhythms of Life

Introduction

It is widely recognized that the birth of modern science dates to the moment when Galileo first timed physical processes taking place in space. In biology, it is only recently that scientists have felt the need for experimental and mathematical methods describing the development of living organisms in terms of dynamic processes. When we analyze the development of self-replicators, we see that they develop their characteristic patterns and self-similarity through processes that resemble a set of oscillators, operating on different time scales. The periodic behavior of these large scale processes and the relations between them both depend on information processing and on local activity. The end result is the steadily increasing complexity we observe in all biological development processes. Physiological rhythms are essential to the life of the organism. Some are maintained for its whole life and even a brief interruption signifies death. Others only operate for short periods of time - some under the control of the organism, some not. Rhythms interact with each other and with the external environment. In most cases, abnormal and new rhythms are associated with serious pathology. To understand the typical rhythms of living organisms, we have to integrate mathematics and physiology. One important approach is non-linear dynamics. Non-linear dynamics allow us to study biological observables as a function of time. Time series for physiological variables show four distinct classes of dynamic behavior: steady state, oscillations, chaos and noise. These are the same behaviors we observe in 2-D CAs, as described in previous chapters. “Steady state” is a way of describing homeostasis - the tendency of biological organisms to maintain the stability of their internal environment (for example, in terms of the presence of specific molecules). It also refers to a solution for an equation that is a constant. More technically, homeostasis is the ability of a system or living organism to adjust its internal environment to maintain a stable equilibrium, thanks to regulatory processes that come into operation when external conditions change. The investigation of the processes that maintain physiological variables within a narrow range of values is a lively area of physiological research. Many physiological phenomena - such as the beating and electrical activity of the heart - are approximately periodic. Other cyclical behaviors - such as breathing, sleep, the menstrual cycle - are equally familiar. Less well-known oscillatory behaviors include insulin release, release of male and female sex hormones, peristaltic waves in the intestine, and the electrical activity of the cortex and the autonomous nervous system. Each of these physiological oscillations is associated with a periodic solution to a mathematical equation. Of course, the phenomena we observe in physiology are highly variable; we never observe a perfect steady state or a perfectly regular oscillation. Even in systems we class as stationary or periodic, there are always fluctuations around the stable value or the regular cycle. There are also systems whose behavior is so irregular as to escape classification. For instance, blood pressure tends to be maintained at a constant value, but is nonetheless subject to fluctuations, reflecting the activity and emotional state of the organism. What is more, some physiological rhythms influence (and are influenced by) other rhythms. In arrhythmia, for instance, heart beat accelerates during inhalation. This is a relatively simple phenomenon. When we measure the electrical activity of the brain with an electroencephalogram, the situation becomes more complex. The instrument measures mean electrical activity in localized areas of the cortex and shows how this activity fluctuates over time. In many cases the fluctuations are fairly irregular. Understanding this irregularity is a difficult task. Mathematics has two distinct methods for studying irregularities in physiological rhythms. The first is to consider them as noise - the result of random processes such as the opening and closing of ion channels in neurons or in heart cells. The second is to apply the concept of chaos - an explanation for apparently random fluctuations in deterministic systems. We can observe chaos even in the absence of noise from the environment. Associated concepts include sensitivity to initial conditions and perturbations, strange attractors, multiple routes to chaos and the fractal boundaries of basins of attraction (for a review see Bilotta et al., 2007). Attempts to describe physiological rhythms in mathematical terms have led to the development of several models, including those incorporated in modern cardiac pacemakers. These models take account of a wide set of dynamic processes, including the processes that regulate ion exchange in neurons, and many others. These models are highly sensitive to perturbations and can generate chaotic behavior. In 1914, Brown (1914) demonstrated that cats have a Central Pattern Generator (CPG) that continues to generate rhythmic motor activity, even after afferent nerves have been cut. Even today we do not have a clear understanding of the mechanisms driving the CPG. It has been suggested that the observed behavior is the result of a cycle of mutual inhibition. It is known, furthermore, that negative and mixed feedback processes in systems, subject to interruption, can give to both chaotic and oscillatory behavior. In this chapter, we will propose an alternative interpretation based on our own observations of 1-D and 2-D CAs. The interpretation we propose is based on insights into the general patterns of development and the rhythms that these systems display. In chapter 15, we will see how we can translate these patterns into other languages - such as sound and music. In what follows, therefore, we will analyze the rhythms associated with the development of individual CAs and with the longer timescales on which CAs evolve. In both cases, what we will be talking about are “rhythms of growth.”