Two Continuous Waves Sources Are Oscillating Exaclty in Phase

J Physiol. 2018 Oct 15; 596(20): 4819–4829.

Phase waves and trigger waves: emergent properties of oscillating and excitable networks in the gut

Sean P. Parsons

¹ Farncombe Family Digestive Health Research Institute, Department of Medicine, McMaster University, Hamilton, ON, Canada

Jan D. Huizinga

¹ Farncombe Family Digestive Health Research Institute, Department of Medicine, McMaster University, Hamilton, ON, Canada

Received 2018 Jul 11; Accepted 2018 Jul 18.

Abstract

The gut is enmeshed by a number of cellular networks, but there is only a limited understanding of how these networks generate the complex patterns of activity that drive gut contractile functions. Here we review two fundamental types of cell behaviour, excitable and oscillating, and the patterns that networks of such cells generate, trigger waves and phase waves, respectively. We use both the language of biophysics and the theory of nonlinear dynamics to define these behaviours and understand how they generate patterns. Based on this we look for evidence of trigger and phase waves in the gut, including some of our recent work on the small intestine.

Keywords: ICC, Interstitial cells of Cajal, non‐linear dynamics, pacemaker networks, trigger waves, phase waves, Gastrointestinal Motility

Introduction

The gut's muscular wall contracts in waves from the stomach to the colon. When directed toward the anus, waves push digested food to its final destination and when directed alternately they churn, so aiding absorption and bacterial homeostasis. The waves are generated by networks of cells of various kinds, primarily muscle, neurons and the interstitial cells of Cajal (ICC). The waves can be of two fundamental types, either trigger waves or phase waves (Bordiougov & Engel, 2006), corresponding to two distinct types of cellular behaviour, excitable and oscillating, respectively. Traditionally all gut waves were thought to be trigger waves, but in the 1960s it was suggested that there are phase waves in the small intestine (Bortoff, 1961). There has been some controversy over this suggestion (Duthie, 1974; Publicover & Sanders, 1989; Daniel et al. 1994; Tse et al. 2016). We believe that much of the controversy results from a confusion as to what trigger and phase waves are and the terminology surrounding them. It is our primary aim to clear up this confusion by a review of the theoretical concepts underpinning trigger and phase waves. Based on this we then examine some examples of waves in the gut where the designation as phase or trigger is fairly straightforward. We do not attempt a critical review of the whole gut wave literature. There are many excellent reviews of various aspects of this literature such as modelling (Lees‐Green et al. 2011), molecular mechanisms (Tse et al. 2016), phase waves (van Helden & Imtiaz, 2003) and arrhythmia (O'Grady et al. 2014).

Networks and waves

In common with cardiac and skeletal muscle, contraction in the gut is the end point of a sequence of events called excitation–contraction coupling (Sanders, 2008). A small decrease (depolarisation) in the negativity of the voltage (electrical potential) across the muscle cell's membrane triggers voltage‐sensitive ion channels in that membrane to open, and so further depolarise the cell (Fig.1 E). This depolarisation‐triggered depolarisation is called excitation and the cell is said to be excitable. Some of the voltage‐sensitive channels admit calcium ions (Ca²⁺) and as these enter the cytoplasm they cause a rise in Ca²⁺ concentration, either directly or by activating release of further Ca²⁺ from stores within the cytoplasm. Ca²⁺ causes contraction by activation of myosin–actin cross‐bridges in the muscle cell.

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Cell behaviour and wave formation

Cell behaviour can be visualised and classified with phase portraits (A–C). The portrait is a graph of the time‐dependent variables of the cell, one axis per variable – here we use membrane potential and activation of an ion channel as examples. As the values of these state variables change they trace out a trajectory (red) through the state space of the graph. In response to a depolarising stimulus, the trajectory of a non‐excitable cell (A) makes a short excursion before quickly returning to its equilibrium point (black filled circle), here at −60 mV as an example of a typical 'resting' membrane potential. In contrast, an excitable cell (B) makes a long excursion before returning to equilibrium. In an oscillating cell (C) the stable equilibrium is replaced by an unstable equilibrium (open circle) around which it circles indefinitely. When cells are uncoupled a depolarising stimulus in one cell is not transmitted to the other (D and E) and oscillating cells do not synchronise (F). In contrast, in a syncytium of cells electrically coupled by gap junctions (G–I, grey channels) a depolarising stimulus in one cell decays exponentially to its neighbour (G) and in an excitable cell this causes excitation with a lag (H). Oscillating cells synchronise with a lag dependent on their uncoupled frequency difference (I). In a chain of coupled non‐excitable cells, a depolarising stimulus decays exponentially over space and time (J). In a chain of coupled excitable cells, excitation travels outward from the stimulus as a trigger wave (K). In a chain of coupled oscillating cells the synchronised oscillations form rhythmic phase waves (L) with an apparent velocity dependent on the natural frequency (lag) gradient along the chain. These waves do not 'propagate' in the sense that trigger waves do.

The cytoplasms of neighbouring muscle cells are connected to each other by ionic channels called gap junctions. A network of cells (of any type) thus connected is called a syncytium. When current is injected from an electrode into one cell, it flows out through the cell's gap junctions to the rest of the syncytium. In a syncytium of non‐excitable cells the current density, and so depolarisation, decays exponentially from the injected cell (Fig.1 G). This behaviour is described by the cable equation, first written down by William Thomson, Lord Kelvin in the mid‐nineteenth century to describe current decay along an oceanic telegraph cable (Rall, 1977).

The cable equation is mathematically equivalent to the diffusion and heat equations that describe the exponential decay of concentration and temperature from a source of molecules or heat, respectively (Rall, 1977; Keener, 2002). In the syncytium the current or depolarisation can be thought of as diffusing away from the injection point or from any cell that is depolarised (Fig.1 J). In a syncytium of excitable cells the diffusing depolarisation causes sequential excitation of one cell after another (Fig.1 H) and so a wave of excitation spreads through the syncytium (Fig.1 K). Any type of wave that propagates by diffusion of some x coupled to its autocatalytic activation or production (x makes more x) is called a trigger wave, regenerative wave or reaction–diffusion wave (Bordiougov & Engel, 2006). x can be depolarisation, as in the syncytium, or some molecule, as in a chemical wave.

A different class of wave occurs when the membrane potential of each cell in the syncytium oscillates autonomously. Electrical oscillations can arise in cells from feedback loops between voltage‐ and calcium‐dependent ion channels on the cell membrane and release of calcium ions (Ca²⁺) from stores within the cell (Berridge, 2008). Membrane potential and cytosolic Ca²⁺ oscillate together. This has been called the 'membrane clock, Ca²⁺ clock paradigm' (Yaniv et al. 2015). Though each cell in the syncytium oscillates autonomously, current diffusion between cells encourages their oscillations to synchronise, to minimise the difference in their cycle position or phase (Fig.1 I) (Strogatz & Stewart, 1993). The synchronised oscillations form rhythmic waves, a wave train (Fig.1 L), but the waves do not actually 'propagate'. They are called phase waves, pseudowaves or kinematic waves (Winfree, 1980; Bordiougov & Engel, 2006). Synchronisation will not necessarily be total, phase difference will not go to zero, and the velocity of the phase wave is the inverse of this phase lag.

In the 1990s it was found that many gut wave patterns disappear when a class of cells, the interstitial cells of Cajal (ICC), are prevented from developing (Ward et al. 1994; Huizinga et al. 1995). ICC form several distinct syncytia that cover the length and circumference of the gut (Huizinga et al. 2009; Sanders et al. 2016). The syncytia are named by the depth at which they occur in the muscle wall – serosal ICC (the outer surface), myenteric plexus ICC (between the longitudinal and circular muscle layers) and so on. ICC form gap junctions with the muscle cells and so oscillations or excitations in the ICC are transmitted to the muscle and can be recorded from there. In the last few years another type of syncytial cell has been discovered in the gut, the fibroblast‐like PDGFRα⁺ cell (Sanders et al. 2016).

Models and coupling

Since Newton it has been a prime objective of the physical sciences to create models of the world in the form of the differential equation,

u is a quantity of interest that changes over time, t. du/dt is the rate of change of u, its differential with respect to time. f(u) is a function of u, any mathematical expression that contains u. u is called a state variable – its value at any time is the 'state' of the model. A model may consist of a number of state variables (u ₁, u ₂, etc.) each with its differential equation, a function of one or more of the other state variables.

$\begin{matrix} \frac{d u_{1}}{d t} & = & f_{1} (u_{1}, u_{2}, \dots etc .) \\ \frac{d u_{2}}{d t} & = & f_{2} (u_{1}, u_{2}, \dots etc .) \\ \dots etc . \end{matrix}$

(2)

This is called a system of differential equations. 'The system' is synonymous with 'the model' and the number of differential equations (state variables) is its size.

To produce a readout of the state variable(s) over time, u(t), the differential equations are integrated or solved. This can be done on paper, using tricks of mathematics to find a mathematical expression for u(t) (an analytical solution), or by a computer that iteratively calculates the value of du for tiny intervals of time, dt (a numerical solution). If f(u) is linear (it contains no powers or products of the state variables) it can be solved analytically by standard methods. If it is nonlinear it has to be solved numerically, except in some rare cases.

Equation (2) is a system of ordinary differential equations (ODEs) because the only differential (du) is with respect to time. If f(u) includes a spatial differential du/dx, where x is a spatial dimension, the equation is a partial differential equation (PDE) and by historical convention one replaces the d with ∂,

In the PDE, space is represented by a continuous variable (x) or variables (for more than one dimension). Alternatively, space can be represented by m discrete points arranged as a lattice (an array or network, etc.). Each point is modeled by n ODEs and so the lattice is m × n in size. Any PDE can be translated into an equivalent lattice of ODEs by replacing the spatial differential with a term that approximates it, a procedure known as the finite difference method. For instance, the cable equation was written by Kelvin as a PDE (Rall, 1977; Keener, 2002),

V is the voltage, λ is the length constant and τ is the time constant. This is equivalent to the ODE system,

$\begin{matrix} \frac{d V_{i}}{d t} & = & \frac{1}{τ} [λ^{2} (V_{i + 1} + V_{i - 1} - 2 V_{i}) / Δ x^{2} - V_{i}] \\ i & = & 1 \dots m \end{matrix}$

(5)

i is the index of a point along x and Δx is the distance between points. The spatial differential of the PDE has been replaced by (V_i ₊₁ +V_i ₋₁ − 2V_i )/Δx ². This is the coupling term and when used is known as diffusive coupling (reflecting the equivalence to the diffusion equation) or resistive coupling. A lattice of 100 points will give approximately the same solution as one of 1000 points if Δx is adjusted to give the same total length. Therefore each point does not have to represent a single cell but a stretch of syncytium. Three‐dimensional versions of the cable equation are often called core conductor models and thus 'cable theory' and 'core conductor theory' are somewhat synonymous.

A wealth of mathematical models of gut waves have been published since the 1960s (Lees‐Green et al. 2011; Corrias et al. 2013). Almost all have employed diffusive coupling. It has been common to model gut syncytia as a one‐dimensional lattice or chain of point ODE systems representing the gut's long axis (e.g. Aliev et al. 2000; Parsons & Huizinga, 2016). Chains are useful because their solution is faster than a 3‐D model and can be condensed into a space–time image where the spatial axis is distance along the chain (Parsons & Huizinga, 2016). The space–time image is both visually digestible and convenient for publication. Also it can be directly compared to experimental diameter maps (DMaps), space–time images of diameter (and thereby contraction) along the length of a piece of gut. The gut is stretched out in a bath of saline and videoed, from which video the intestine's diameter is measured at each pixel. DMaps have become a mainstay of the gut field and have revealed a rich menagerie of waves and other spatiotemporal patterns (Hennig, 2016). In recent years computing power has made it possible to represent the gut as an anatomical model with circumference and thickness (Corrias et al. 2013). These models are PDEs solved by the finite element method.

Physics and toys

As we progress from the fundamental models of physics to the models of biology two things happen. First, we stop thinking of those models as 'reality' or 'laws of nature'. Instead they are just an abstraction or simplification of some 'reality' we cannot quite get at with our current knowledge. Second, this reality gets more diverse and multifarious in the ways that the fundamental laws are expressed and combined in diverse biological objects. These two facts bring a tension into the way we model biology. On the one hand we may wish to approach the reality of a particular biological object as close as we can. We include the maximal number of state variables and parameters relevant to the object and insist that these are always physical quantities with SI units. This can be called the physical approach.

On the other hand we might not be interested in modelling a particular object, but a general class of objects. We step back from reality and use models that express some principle or 'law' that, while not necessarily expressible in SI units, is nevertheless observed in all the particular objects of interest. The number of state variables and parameters is kept minimal. This approach is called by physicists, without any derogatory implication, the toy model. Toy models are valuable in that they allow us to understand and study 'emergent laws', general principles that emerge from physical laws but are not necessarily transparent in complex physical models. Once we understand the toy models we have a strong light with which to inspect physical models.

The distinction between physical and toy models is primarily of intention rather than form. Experimental and theoretical knowledge is always limited, and so physical models will usually contain terms without SI units or a basis in physical law. Nevertheless the intention is to approach some physical reality. When experimental knowledge is poor, it may be more productive to take the opposite tack to toy models. Some of the earliest models of gut waves in the 1970s were resolutely toy. They were motivated by a need to understand the emergent principles responsible for certain wave phenomena exhibited by the small intestine (see section on 'Oscillating cells' below). Also, though the outlines of excitation–contraction and gap junction coupling were established at the time, knowledge of the particular cells, ion channels and other molecules involved was fragmentary. As experimental knowledge of the gut has improved, physical models have become more prevalent. Many systems have been devised to model membrane–calcium clocks in ICC and smooth muscle cells (Lees‐Green et al. 2011). Their state variables naturally include membrane potential and cytosolic Ca²⁺ concentration, but also activation states of various channels, concentrations of cytosolic second messengers and so on.

Nonlinear behaviour

Linear differential equations such as the cable equation cannot produce excitable or oscillatory behaviour in u. They usually only produce exponential behaviour and so it is not surprising that they are infrequent in nature. The physicist Stanislaw Ulam memorably expressed this as 'using a term like nonlinear science is like referring to the bulk of zoology as the study of non‐elephant animals' (Campbell et al. 1985). Nevertheless many fundamental physical laws are expressed as linear differential equations and often appear embedded within nonlinear systems. For example Hodgkin and Huxley's Nobel prize‐winning model of trigger waves in the squid axon (Hodgkin & Huxley, 1952) has at its centre the cable equation with the right‐most V replaced by a sum of membrane channel currents,

$\frac{\partial V}{\partial t} = \frac{1}{c} [g_{c} \frac{\partial^{2} V}{\partial x^{2}} - \sum_{j} g_{j} n_{j}^{p} (V - e_{j})]$

(6)

c is the membrane capacitance and g _c is the axon's cytoplasmic conductance (for comparison to eqn (4) τ =c ×r _m and λ² =g _c ×r _m, where r _m is the membrane resistance). There are three currents (j = 3) each with its own conductance (g), activation or inactivation (n) and reversal potential (e). Though physical channels and ions are not modelled, the currents are given ionic names according to their reversal potential: sodium, potassium and a non‐specific (e = 0 mV) 'leak'. The activation/inactivation of the sodium and potassium currents are state variables with nonlinear ODEs and thus the system is nonlinear. Hodgkin and Huxley's model is the ur‐model of biological waves. With g _c as the gap junction conductance, eqn (6) (in either its PDE or ODE form) is the template of almost all physical models of syncytia. The number of currents, their reversal potentials (ionic types) and activation/inactivation state variables vary whilst other state variables can be added for cytosolic calcium and intracellular ion channels (Lees‐Green et al. 2011; Noble et al. 2012).

As the parameter values of a nonlinear system are changed, its qualitative behaviour can change suddenly, from non‐excitable to excitable or oscillatory. This kind of qualitative change in behaviour is known as bifurcation and all nonlinear models have a large number of bifurcations and behaviours (Strogatz, 2015). The Hodgkin–Huxley model can be excitable, oscillatory and everything in between (Fitzhugh, 1961; Keener & Sneyd, 2009). This plasticity is not always appreciated. Biologists often talk of 'models of excitability' or 'oscillatory models' as if a different model is required for each type of behaviour. A good example is the Van der Pol system and its close relative the Fitzhugh–Nagumo (Fitzhugh, 1961; Keener & Sneyd, 2009). These are highly popular 'relaxation oscillator models'. In a relaxation oscillation a state variable changes slowly during one part of the cycle, fast in the other. The opposite of this is the harmonic oscillation where the state variable traces a sine wave. Relaxation oscillation is brought about by a separation of time scales between state variables, such that one state variable changes much faster or slower than the other (Fitzhugh, 1961; Keener & Sneyd, 2009). In the Van der Pol model this is achieved by the μ parameter,

v is the voltage and w is the 'recovery' variable. As μ increases from unity, the rate of change of w slows relative to v, and the oscillations go from being harmonic to relaxation. As other parameters are changed the oscillations disappear and the system can be excitable or non‐excitable. Nevertheless Van der Pol emphasised the relaxation oscillation behaviour of his system (van der Pol, 1926) and so it has become virtually synonymous with 'relaxation oscillator'.

State space

How to comprehend the rich behavioural repertoire of nonlinear systems? As most nonlinear systems cannot be studied analytically, a semi‐qualitative and pictorial theory of their behaviour has been developed called nonlinear dynamics (Campbell et al. 1985; Izhikevich, 2007; Strogatz, 2015). The central object of nonlinear dynamics is the state space. Each dimension of state space is an axis representing the value of one state variable. Thus there are as many dimensions as there are ODEs in the system. The simplest thing to do with state space is to plot onto it a read‐out (numerical solution) of the state variables, creating a single line, a trajectory, that winds through the space (Fig.1 A–C). The plot is called a phase portrait as another name for state space is the phase space. This 'phase' has nothing to do with oscillation phase but is rather a historical vestige (Nolte, 2010). The phase portrait allows a condensed assessment of how one state variable relates to another and for one numerical solution to be compared against another.

Each point in state space corresponds to a set of values of the state variables and the ODEs give the corresponding values of du/dt. The latter can be represented as an arrow or vector in state space that points to where a trajectory will go next and, by the length of the arrow, how fast. A number of such arrows arrayed over state space is a vector field. The vector field allows one to see how any trajectory will behave, starting from any point in state space. In addition to the vectors, one can also plot points in state space where all the du/dt are zero; the vector has zero length. These are equilibrium points where the trajectory stops forever. The point can be stable (black filled circle, Fig.1 A and B), where trajectories tend toward it from the surrounding space, or unstable, where trajectories pull away from it unless they are exactly on it. These are like the bottom of a hollow or top of a hill, respectively. By extension the vector field as a whole can be thought of as a topographical map of the dynamic landscape of state space. There can be closed loops in state space where the trajectory will go around in circles indefinitely following a series of tail‐to‐end vectors (Fig.1 C). The loop is called a limit cycle because surrounding trajectories fall onto it in the limit of time and it describes an oscillation in the state variables. The limit cycle must have an unstable equilibrium within it (open circle, Fig.1 C), just as a moat must circumscribe higher land.

As the system's parameters are changed, equilibrium points or limit cycles appear or disappear, merge or separate, change character from stable to unstable. The qualitative behaviour of the system then changes and this is the bifurcation. The boundaries of bifurcations in parameter space, the parameter equivalent of state space, can be discovered by the technique of continuation (which cannot be explained without a lot of maths), allowing one a global view of the system's behaviour.

Despite its possibilities, nonlinear dynamics has had little traction in the gut field. There is in fact just one solitary paper, a study of a physical model of ICC (Imtiaz et al. 2006). This is unfortunate because one can give very concrete definitions of 'excitable' and 'oscillatory' in terms of nonlinear dynamics and these can help one understand the general properties of trigger waves and phase waves, and models of them.

Excitable cells

A trajectory can be 'kicked' away from a stable equilibrium point by an externally applied pulse or some noise (Lindner et al. 2004; Izhikevich, 2007). Depending on the vector field, the trajectory might immediately return to the equilibrium point (Fig.1 A) or it may make a large excursion through state space before returning (Fig.1 B). The latter can occur when the system is near to a bifurcation that births a limit cycle, when the vector field circles around some point, but no actual closed trajectory (limit cycle) is yet present. From the point of view of nonlinear dynamics this is the definition of an excitable system (Lindner et al. 2004; Izhikevich, 2007).

Trigger waves occur when one excitable point system (cell) kicks its neighbour, which then kicks its neighbour, and so on (Lindner et al. 2004). In a lattice of uniform points the wave will travel with constant velocity. But how is the wave initiated in the first place? There could be a patch of oscillating cells (a 'pacemaker') that excites the surrounding excitable tissue, much like the sinoatrial node in the heart. Alternatively, the excitable cells could be noisy, the little kicks are generated internally. In this case trigger waves will propagate out at random points and times in a V (see Fig. 67 of Lindner et al. 2004) and the trigger wave rhythmicity, defined as the uniformity of intervals between waves, is very low. Put more quantitatively, their temporal correlation, or temporal coherence as it is called in physics, is low.

Temporally incoherent, stochastic V‐waves have been observed throughout the gut and are usually called a ripple pattern (D'Antona et al. 2001; Roberts et al. 2007, 2010; Burns et al. 2009; Janssen et al. 2009; Hennig et al. 2010). It appears that ripples can be generated by the muscle syncytia alone as they occur in mice lacking ICC and where the enteric nervous system is either absent by mutation or its activity has been blocked by drugs (Roberts et al. 2007, 2010; Burns et al. 2009). In electrical recordings under these conditions, ICC‐generated slow waves are absent (Ward et al. 1995; Der‐Silaphet et al. 1998). Instead there are shorter depolarisations called spikes. The spikes are only partially rhythmic, a little coherent – there are frequent long intervals interspersed between trains of shorter intervals (Ward et al. 1995; Der‐Silaphet et al. 1998).

Spikes and ripples can be activated by stretch of the muscle (e.g. Bulbring & Kuriyama, 1963). The frequency and temporal coherence of the spikes increases with stretch. This is accompanied by some depolarisation, thus membrane potential could be a bifurcation parameter that changes the pattern. In a lattice of excitable point systems temporal coherence depends on the amount of noise (Lindner et al. 2004). As noise increases, so does coherence, but only up to a point, beyond which it decreases again. As of yet no one has looked for such stochastic coherence in the gut.

Oscillating cells

When a stimulus pulse kicks a trajectory from a limit cycle, the path (and time) the trajectory takes to return to the cycle is entirely dependent on the vector field. When it returns, the trajectory will probably be at a different point on the cycle from where it would have been had it not been kicked – it will be phase shifted. The shift can be measured experimentally from the shift in the timing of an identifiable point in the cycle (for example a maximum depolarisation) in response to a kick (for example an electrical pulse). The shift is quantified as the phase response curve (PRC), a graph of shift as a function of the time in the cycle when the kick occurred (see Fig. 1 of Parsons & Huizinga, 2016).

PRCs have been measured for a diverse range of biological oscillators, from metabolic and gene‐regulation pathways with oscillating substrates, to flashing fireflies, to rhythmically firing neurons (Pavlidis, 1973; Winfree, 1980). These PRCs invariably have the same shape: first a period of zero shift (refractory period) followed by negative shift (phase delay) and then finally positive shift (phase advance). At first sight this homogeneity is remarkable. However, there is a clear adaptive (evolutionary) reason why these diverse oscillators should converge on the same PRC: synchrony (Strogatz & Stewart, 1993). The PRC tells us how oscillators will interact with one another when coupled. In this case the kick comes from another oscillator when it 'fires', for example the depolarisation phase of an oscillating cell. If two oscillators fire close to each other, they will have no mutual effect (they are refractory); if oscillator A fires just before B is due to, B will fire earlier (phase advance); but if A fires some time before B, B will fire later (phase delay) so that it is nearer to the next firing of A. Either way, the phase difference between A and B decreases, they synchronise, and so a network of oscillators will oscillate together and form a phase wave (Fig.1 I and L).

For over a century it has been observed that the rhythmic contractions of the small intestine decrease in frequency down its length (Alvarez, 1914). As technology improved it became clear that the contractions were associated with depolarisations called slow waves and that their frequency often decreased in sudden steps between regions of fixed frequency (plateaus). Also, waves usually moved down the frequency gradient toward the colon. In 1961 it was suggested that these three phenomena – a frequency gradient, plateaus and colon‐directed waves – could be explained by a chain of coupled oscillators (Bortoff, 1961). In isolation an oscillator oscillates at its natural frequency (ω). In a chain of oscillators with a gradient in ω, synchronisation will not be total. The lower ω oscillator will lag behind and so the phase wave will appear to travel down the ω gradient. Also high ω oscillators will continually phase advance their lower ω neighbours, so 'pulling up' their frequency (Paciorek, 1965; Pikovsky et al. 2001). With strong enough coupling, higher frequency oscillators pull up the frequency of the oscillators down the chain into a plateau of fixed frequency. Pulling only fails when the frequency difference between the plateau and an oscillator's ω becomes too large. At this point there is a step decrease in frequency and the beginning of a new plateau.

Bortoff's hypothesis was supported by a number of models through the 1970s that showed that diffusively coupled chains of limit cycle systems produced frequency plateaus and phase waves that 'travelled' down the ω gradient (see Introduction of Parsons & Huizinga, 2016). These included the Van der Pol model but also more physical systems like the Hodgkin–Huxley. The models inspired experiments that further supported the theory, such as ligating the intestine to induce a new plateau (Diamant & Bortoff, 1969; Sarna et al. 1972; Lammers & Stephen, 2008). At the time the oscillators were presumed to be muscle cells but in the 1990s it became evident that they were the ICC of the myenteric plexus (Huizinga et al. 1995; Ward et al. 1995).

Recently we have re‐examined coupled oscillators in the mouse small intestine. We examined the effects of the gap junction blocker carbenoxolone on patterns in diameter maps and modelled these effects with a chain of coupled phase oscillators (Parsons & Huizinga, 2015, 2016; Wei et al. 2017). We have measured the small intestine's PRC, which has the typical refractory–delay–advance shape (Parsons & Huizinga, 2017). We have shown that frequency plateaus are composed of discrete waves of interval increase (Parsons & Huizinga, 2018). Though contractions are an output of the muscle rather than the underlying ICC electrical oscillations, the tight gap junction coupling between the two ensures the former is a good assay of the latter. If coupling between them is reduced this may not be the case, and indeed we have found that some phenomena observed at high concentrations of carbenoxolone can be modelled by weakened interaction between muscle and ICC (authors' unpublished observations).

It is the totality of a system that determines its vector field and therefore its synchronisation properties if it has a limit cycle. There is no way to answer the question of what system components are more or less important for synchronisation, other than by seeing how synchrony or PRC shape change in response to various manipulations of those components or parameter changes in a model. We are not aware of anyone who has done this systematically for a physiological system, experimental or model. Two studies looked at the difference in PRC shape and synchrony between some standard biophysical models (the Hodgkin–Huxley, Connor and Morris–Lecar) but they did not examine the role of particular parameters (Hansel et al. 1995; Ermentrout, 1996). In a calcium–membrane clock model of an ICC, van Helden and Imtiaz found that the rate of inositol 1,4,5‐trisphosphate synthesis had dramatic effects on PRC shape and synchrony (Imtiaz et al. 2006). They have published a series of papers on the nonlinear dynamics of the model, its trajectories and bifurcations (Imtiaz et al. 2006, 2007, 2010). Also they have studied phase waves and synchronisation experimentally in gastric and lymphatic muscle (van Helden & Imtiaz, 2003; Imtiaz et al. 2007). They gave an elegant demonstration of desynchronisation between gastric slow waves, locally decoupled by perfusion of caffeine (van Helden & Imtiaz, 2003).

Conflation of concepts

The enthusiasm for coupled oscillators in the gut faded quickly after the 1970s. Apart from our recent work and that of Imtiaz and van Helden, studies where coupled oscillator theory has been front and centre in the explanation of gut waves have been rare in the last 35 years (Aliev et al. 2000). A reason for coupled oscillator theory's failure to take off may be the paradigmatic, textbook status of the Hodgkin–Huxley model. Though the model can display oscillatory behaviour and physiologists revere it (with good reason) and learn it as a matter of course during their undergraduate education, it is rarely presented as other than a 'model of regenerative [i.e. trigger] waves'. Thus physiologists, whilst entirely comfortable with the idea of trigger waves, might have found the theory of coupled oscillators unfamiliar, obtuse or exotic even, when it was first introduced to them in the 1970s. It is therefore unfortunate that three critical reviews of the time did little to clarify coupled oscillators to the wider field (Duthie, 1974; Publicover & Sanders, 1989; Daniel et al. 1994). All three present a case of 'relaxation oscillators' against 'cable theory' (or 'core conductor theory') as opposing explanations of waves in the gut. This is explicit in their titles and section headings. In their texts one finds that they are not just (or at all) talking about relaxation behaviour but are in fact conflating 'relaxation oscillators' with coupled oscillators in general. This conflation no doubt originates with most of the models of the 1970s being of the Van der Pol type. Similarly they conflate cable/core‐conductor theory with 'excitation', which is wrong. These conflations have propagated to the present: a recent review (Tse et al. 2016) presents the case of 'core conductor' against coupled oscillator theories.

These conflations muddle the discourse to such an extent that there can be no real, sensible debate or understanding. The concept of coupled oscillators is questioned in general because specific models with particular parameters do not reproduce the waveform seen experimentally (Publicover & Sanders, 1989). Gap junction coupling between cells (cable theory) is seen as all that is needed to support excitation and thereby exclude any need for coupled oscillators (Publicover & Sanders, 1989) when, as we have pointed out, diffusive coupling is necessary for both trigger waves and phase waves but does not in itself generate either excitatory or oscillatory behaviour. Waves propagate from a pacemaker site due to its 'entrainment' of the surrounding gut, but that propagation is nevertheless thought to be by excitation (a trigger wave) in the context of which entrainment has no meaning (e.g. Koh et al. 2003; Lammers & Stephen, 2008). Another critique of the 'relaxation oscillator' models was that they were not physical (Publicover & Sanders, 1989), but there should always be a place for toy models given the emergent behaviour of nonlinear systems. We still do not have a full grasp on either the emergent principles or biophysical details of gut waves.

Waves and the clinic

Advances in the clinic are revolutionising our knowledge of gut waves in human physiology and pathophysiology. Chief among these advances has been high‐resolution manometry (HRM). Just as with the video data used to create DMaps, data from multiple pressure channels (up to 84 in the state‐of‐the‐art) spaced along a catheter can be plotted as spatiotemporal maps (often called Clouse or topography plots). These show a large range of wave phenomena that can be related to different functions. Over the last decade HRM has become the diagnostic gold standard for oesophageal motility disorders, enshrined in the wave pattern‐based Chicago classification (Kahrilas et al. 2015). HRM is increasingly applied to the colon and gastro‐duodenum (Dinning et al. 2015). The continence mechanism of the colon includes trains of retrograde waves, assumed to be orchestrated by ICC (Lin et al. 2017). Pan‐colonic, highly rhythmic, simultaneous pressure waves are associated with gas expulsion (Chen et al. 2017). In the small intestine dislocations occur during the phase III wave train of the migrating motor complex (Siwiec & Wo, 2015).

High spatial resolution arrays of recording electrodes applied to the surface of the small intestine and stomach have revealed re‐entrant spiral waves, the poster child of cardiac arrhythmia (Lammers, 2013; O'Grady et al. 2014; Lammers, 2015; Du et al. 2017). These waves circle (spiral) continuously around a point which may be fixed or not and can be induced by temporary or chronic blocks in coupling. In principle and in models, spiral waves can occur in both excitable and oscillatory syncytia. In the heart they can occur purely in the excitatory regime, where they are initiated by some transient block (for instance by focal electrical stimulation) but can also be induced when muscle cells become oscillatory (for example due to cytosolic calcium overload) (Qu et al. 2014).

Conclusion: the super network

The gut is a super‐network, a massive syncytium composed of a plethora of sub‐networks of excitable and oscillating cells: ICC, neurones and muscle cells. Trigger and phase waves occur in the sub‐networks of excitable and oscillatory cells, respectively. Undoubtedly other patterns can emerge from the interaction between excitable and oscillatory cells and cells can change behaviour, from excitable to oscillatory and everything in between, depending on physiological conditions. We have barely begun to examine these excitatory–oscillatory interactions and modulations but understanding them and the behaviour they produce in the super‐network will be impossible without both experiment and nonlinear models.

Additional information

Competing interests

None declared.

Author contributions

Both S.P.P. and J.D.H. contributed to the conception, writing and editing of this review. Both authors approved the final version of the manuscript and agree to be accountable for all aspects of the work. All persons designated as authors qualify for authorship, and all those who qualify for authorship are listed.

Funding

This work was funded by the Canadian Institutes of Health Research Grant 12874 and 20006288, as well as a Natural Sciences and Engineering Research Council Grant 386877 and 20006550 to J.D.H. S.P.P. was supported in part by a research scholarship from the Farncombe Family Digestive Health Research Institute.

Biographies

•

Jan Huizinga is a Professor of Medicine at McMaster University and a faculty member at the School of Biomedical Engineering. His interest lies in understanding control mechanisms of gastrointestinal motility with a focus on the role of interstitial cells of Cajal as pacemaker cells in the intestine and colon.

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Sean Parsons has worked in the fields of smooth and cardiac muscle physiology, focusing on the cellular physiology of their pacemaker cells. He has trained in the labs of Thomas Bolton (University of London, UK; PhD), Kenton Sanders (University of Nevada, USA) and Edward Lakatta (National Institutes on Ageing, USA) and is presently a Research Scientist in the lab of Jan Huizinga.

Notes

Edited by: Ole Petersen and Fernando Santana

This is an Editor's Choice article from the 15 October 2018 issue.

References

Aliev RR, Richards W & Wikswo JP (2000). A simple nonlinear model of electrical activity in the intestine. J Theor Biol 204, 21–28. [PubMed] [Google Scholar]
Alvarez WC (1914). Functional variations in contractions of different parts of the small intestine. Am J Physiol 35, 177–193. [Google Scholar]
Berridge MJ (2008). Smooth muscle cell calcium activation mechanisms. J Physiol 586, 5047–5061. [PMC free article] [PubMed] [Google Scholar]
Bordiougov G & Engel H (2006). From trigger to phase waves and back again. Physica D 215, 25–37. [Google Scholar]
Bortoff A (1961). Electrical activity of intestine recorded with pressure electrode. Am J Physiol 201, 209–212. [Google Scholar]
Bulbring E & Kuriyama H (1963). The effect of adrenaline on the smooth muscle of guinea‐pig taenia coli in relation to the degree of stretch. J Physiol 169, 198–212. [PMC free article] [PubMed] [Google Scholar]
Burns AJ, Roberts RR, Bornstein JC & Young HM (2009). Development of the enteric nervous system and its role in intestinal motility during fetal and early postnatal stages. Semin Pediatr Surg 18, 196–205. [PubMed] [Google Scholar]
Campbell D, Crutchfield J, Farmer D & Jen E (1985). Experimental mathematics: the role of computation in nonlinear science. Commun ACM 28, 374–384. [Google Scholar]
Chen JH, Yu Y, Yang Z, Yu WZ, Chen WL, Yu H, Kim MJ, Huang M, Tan S, Luo H, Chen J, Chen JD & Huizinga JD (2017). Intraluminal pressure patterns in the human colon assessed by high‐resolution manometry. Sci Rep 7, 41436. [PMC free article] [PubMed] [Google Scholar]
Corrias A, Du P & Buist ML (2013). Modeling tissue electrophysiology in the GI tract: past, present and future In New Advances in Gastrointestinal Research, ed. Cheng LK, Pullan AJ. & Farrugia G. Springer, Heidelberg. [Google Scholar]
Daniel EE, Bardakjian BL, Huizinga JD & Diamant NE (1994). Relaxation oscillator and core conductor models are needed for understanding of GI electrical activities. Am J Physiol Gastrointest Liver Physiol 266, G339–G349. [PubMed] [Google Scholar]
D'Antona G, Hennig GW, Costa M, Humphreys CM & Brookes SJ (2001). Analysis of motor patterns in the isolated guinea‐pig large intestine by spatio‐temporal maps. Neurogastroenterol Motil 13, 483–492. [PubMed] [Google Scholar]
Der‐Silaphet T, Malysz J, Hagel S, Larry AA & Huizinga JD (1998). Interstitial cells of Cajal direct normal propulsive contractile activity in the mouse small intestine. Gastroenterology 114, 724–736. [PubMed] [Google Scholar]
Diamant NE & Bortoff A (1969). Effects of transection on the intestinal slow‐wave frequency gradient. Am J Physiol 216, 734–743. [PubMed] [Google Scholar]
Dinning PG, Carrington EV & Scott SM (2015). The use of colonic and anorectal high‐resolution manometry and its place in clinical work and in research. Neurogastroenterol Motil 27, 1693–1708. [PubMed] [Google Scholar]
Du P, Calder S, Angeli TR, Sathar S, Paskaranandavadivel N, O'Grady G & Cheng LK (2017). Progress in mathematical modeling of gastrointestinal slow wave abnormalities. Front Physiol 8, 1136. [PMC free article] [PubMed] [Google Scholar]
Duthie HL (1974). Electrical activity of gastrointestinal smooth muscle. Gut 15, 669–681. [PMC free article] [PubMed] [Google Scholar]
Ermentrout B (1996). Type I membranes, phase resetting curves, and synchrony. Neural Comput 8, 979–1001. [PubMed] [Google Scholar]
Fitzhugh R (1961). Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1, 445–466. [PMC free article] [PubMed] [Google Scholar]
Hansel D, Mato G & Meunier C (1995). Synchrony in excitatory neural networks. Neural Comput 7, 307–337. [PubMed] [Google Scholar]
Hennig GW (2016). Spatio‐temporal mapping and the enteric nervous system In The Enteric Nervous System: 30 Years Later, ed. Brierley S. & Costa M, pp. 31–42. Springer International Publishing, Cham. [Google Scholar]
Hennig GW, Gregory S, Brookes SJ & Costa M (2010). Non‐peristaltic patterns of motor activity in the guinea‐pig proximal colon. Neurogastroenterol Motil 22, e207–217. [PubMed] [Google Scholar]
Hodgkin AL & Huxley AF (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117, 500–544. [PMC free article] [PubMed] [Google Scholar]
Huizinga JD, Thuneberg L, Kluppel M, Malysz J, Mikkelsen HB & Bernstein A (1995). W/kit gene required for interstitial cells of Cajal and for intestinal pacemaker activity. Nature 373, 347–349. [PubMed] [Google Scholar]
Huizinga JD, Zarate N & Farrugia G (2009). Physiology, injury, and recovery of interstitial cells of Cajal: basic and clinical science. Gastroenterology 137, 1548–1556. [PMC free article] [PubMed] [Google Scholar]
Imtiaz MS, Katnik CP, Smith DW & van Helden DF (2006). Role of voltage‐dependent modulation of store Ca²⁺ release in synchronization of Ca²⁺ oscillations. Biophys J 90, 1–23. [PMC free article] [PubMed] [Google Scholar]
Imtiaz MS, von der Weid PY & van Helden DF (2010). Synchronization of Ca²⁺ oscillations: a coupled oscillator‐based mechanism in smooth muscle. FEBS J 277, 278–285. [PubMed] [Google Scholar]
Imtiaz MS, Zhao J, Hosaka K, von der Weid PY, Crowe M & van Helden DF (2007). Pacemaking through Ca²⁺ stores interacting as coupled oscillators via membrane depolarization. Biophys J 92, 3843–3861. [PMC free article] [PubMed] [Google Scholar]
Izhikevich EM (2007). Dynamical Systems in Neuroscience. MIT Press, Cambridge, MA, USA. [Google Scholar]
Janssen PW, Lentle RG, Hulls C, Ravindran V & Amerah AM (2009). Spatiotemporal mapping of the motility of the isolated chicken caecum. J Comp Physiol B 179, 593–604. [PubMed] [Google Scholar]
Kahrilas PJ, Bredenoord AJ, Fox M, Gyawali CP, Roman S, Smout AJ & Pandolfino JE; International High Resolution Manometry Working Group (2015). The Chicago Classification of esophageal motility disorders, v3.0. Neurogastroenterol Motil 27, 160–174. [PMC free article] [PubMed] [Google Scholar]
Keener J (2002). Spatial modeling In Computational Cell Biology, ed. Fall CP, Maryland ES, Wagner JM. & Tyson JJ, pp. 171–197. Springer, New York. [Google Scholar]
Keener J & Sneyd J (2009). Excitability In Mathematical Physiology I: Cellular Physiology, pp. 195–228. Springer, New York. [Google Scholar]
Koh SD, Ward SM, Ordog T, Sanders KM & Horowitz B (2003). Conductances responsible for slow wave generation and propagation in interstitial cells of Cajal. Curr Opin Pharmacol 3, 579–582. [PubMed] [Google Scholar]
Lammers WJEP (2013). Arrhythmias in the gut. Neurogastroenterol Motil 25, 353–357. [PubMed] [Google Scholar]
Lammers WJEP (2015). Normal and abnormal electrical propagation in the small intestine. Acta Physiol (Oxf) 213, 349–359. [PubMed] [Google Scholar]
Lammers WJEP & Stephen B (2008). Origin and propagation of individual slow waves along the intact feline small intestine. Exp Physiol 93, 334–346. [PubMed] [Google Scholar]
Lees‐Green R, Du P, O'Grady G, Beyder A, Farrugia G & Pullan AJ (2011). Biophysically based modeling of the interstitial cells of Cajal: current status and future perspectives. Front Physiol 2, 29. [PMC free article] [PubMed] [Google Scholar]
Lin AY, Du P, Dinning PG, Arkwright JW, Kamp JP, Cheng LK, Bissett IP & O'Grady G (2017). High‐resolution anatomic correlation of cyclic motor patterns in the human colon: Evidence of a rectosigmoid brake. Am J Physiol Gastrointest Liver Physiol 312, G508–G515. [PMC free article] [PubMed] [Google Scholar]
Lindner B, Garcia‐Ojalvo J, Neiman A & Schimansky‐Geier L (2004). Effects of noise in excitable systems. Phys Rep 392, 321–424. [Google Scholar]
Noble D, Garny A & Noble PJ (2012). How the Hodgkin–Huxley equations inspired the Cardiac Physiome Project. J Physiol 590, 2613–2628. [PMC free article] [PubMed] [Google Scholar]
Nolte DD (2010). The tangled tale of phase space. Physics Today April, 33–38. [Google Scholar]
O'Grady G, Wang TH, Du P, Angeli T, Lammers WJ & Cheng LK (2014). Recent progress in gastric arrhythmia: pathophysiology, clinical significance and future horizons. Clin Exp Pharmacol Physiol 41, 854–862. [PMC free article] [PubMed] [Google Scholar]
Paciorek LJ (1965). Injection locking of oscillators. Proc IEEE 53, 1723–1727. [Google Scholar]
Parsons SP & Huizinga JD (2015). Effects of gap junction inhibition on contraction waves in the murine small intestine in relation to coupled oscillator theory. Am J Physiol Gastrointest Liver Physiol 308, G287–G297. [PMC free article] [PubMed] [Google Scholar]
Parsons SP & Huizinga JD (2016). Spatial noise in coupling strength and natural frequency within a pacemaker network; consequences for development of intestinal motor patterns according to a weakly coupled phase oscillator model. Front Neurosci 10, 19. [PMC free article] [PubMed] [Google Scholar]
Parsons SP & Huizinga JD (2017). The phase response and state space of slow wave contractions in the small intestine. Exp Physiol 102, 1118–1132. [PubMed] [Google Scholar]
Parsons SP & Huizinga JD (2018). Slow wave contraction frequency plateaus in the small intestine are composed of discrete waves of interval increase associated with dislocations. Exp Physiol 103, 1087–1100. [PubMed] [Google Scholar]
Pavlidis T (1973). Biological Oscillators: Their Mathematical Analysis. Academic Press, New York. [Google Scholar]
Pikovsky A, Rosenblum M & Kurths J (2001). Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge. [Google Scholar]
Publicover NG & Sanders KM (1989). Are relaxation oscillators an appropriate model of gastrointestinal electrical activity? Am J Physiol Gastrointest Liver Physiol 256, G265–G274. [PubMed] [Google Scholar]
Qu Z, Hu G, Garfinkel A & Weiss JN (2014). Nonlinear and stochastic dynamics in the heart. Phys Rep 543, 61–162. [PMC free article] [PubMed] [Google Scholar]
Rall W (1977). Core conductor theory and cable properties of neurons In Handbook of Physiology, The Nervous System, Cellular Biology of Neurons , pp. 39–97. American Physiological Society, Bethesda, MD, USA. [Google Scholar]
Roberts RR, Ellis M, Gwynne R, Bergner AJ, Lewis M, Beckett EA, Bornstein JC & Young HM (2010). The first intestinal motility patterns in fetal mice are not mediated by neurons or interstitial cells of Cajal. J Physiol 588, 1153–1169. [PMC free article] [PubMed] [Google Scholar]
Roberts RR, Murphy JF, Young HM & Bornstein JC (2007). Development of colonic motility in the neonatal mouse – studies using spatiotemporal maps. Am J Physiol Gastrointest Liver Physiol 292, G930–G938. [PubMed] [Google Scholar]
Sanders KM (2008). Regulation of smooth muscle excitation and contraction. Neurogastroenterol Motil 20 (Suppl. 1), 39–53. [PMC free article] [PubMed] [Google Scholar]
Sanders KM, Kito Y, Hwang SJ & Ward SM (2016). Regulation of gastrointestinal smooth muscle function by interstitial cells. Physiology (Bethesda) 31, 316–326. [PMC free article] [PubMed] [Google Scholar]
Sarna SK, Daniel EE & Kingma YJ (1972). Effects of partial cuts on gastric electrical control activity and its computer model. Am J Physiol 223, 332–340. [PubMed] [Google Scholar]
Siwiec RM & Wo JM (2015). Chronic intestinal pseudo‐obstruction In Handbook of Gastrointestinal Motility and Functional Disorders, ed. Rao SSC, Parkman HC. & McCallum RW. Slack Inc., Thorofare, NJ, USA. [Google Scholar]
Strogatz SH (2015). Nonlinear Dynamics and Chaos. Westview Press, Boulder, CO, USA. [Google Scholar]
Strogatz SH & Stewart I (1993). Coupled oscillators and biological synchronization. Sci Am 269, 102–109. [PubMed] [Google Scholar]
Tse G, Lai ET, Yeo JM, Tse V & Wong SH (2016). Mechanisms of electrical activation and conduction in the gastrointestinal system: lessons from cardiac electrophysiology. Front Physiol 7, 182. [PMC free article] [PubMed] [Google Scholar]
van der Pol B (1926). On "relaxation‐oscillations". Philos Mag 2, 978–992. [Google Scholar]
van Helden DF & Imtiaz MS (2003). Ca²⁺ phase waves: a basis for cellular pacemaking and long‐range synchronicity in the guinea‐pig gastric pylorus. J Physiol 548, 271–296. [PMC free article] [PubMed] [Google Scholar]
Ward SM, Burns AJ, Torihashi S, Harney SC & Sanders KM (1995). Impaired development of interstitial cells and intestinal electrical rhythmicity in steel mutants. Am J Physiol 269, C1577–C1585. [PubMed] [Google Scholar]
Ward SM, Burns AJ, Torihashi S & Sanders KM. (1994). Mutation of the proto‐oncogene c‐kit blocks development of interstitial cells and electrical rhythmicity in murine intestine. J Physiol 480, 91–97. [PMC free article] [PubMed] [Google Scholar]
Wei R, Parsons SP & Huizinga JD (2017). Network properties of interstitial cells of Cajal affect intestinal pacemaker activity and motor patterns, according to a mathematical model of weakly coupled oscillators. Exp Physiol 102, 329–346. [PubMed] [Google Scholar]
Winfree AT (1980). The Geometry of Biological Time. Springer‐Verlag, Heidelberg. [Google Scholar]
Yaniv Y, Lakatta EG & Maltsev VA (2015). From two competing oscillators to one coupled‐clock pacemaker cell system. Front Physiol 6, 28. [PMC free article] [PubMed] [Google Scholar]

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