Synonyms

Numerical continuation; Parameter studies; Path-following; Qualitative analysis

Definition

Bifurcation theory refers to the study of qualitative changes to the state of a system as a parameter is varied. It can be applied to steady state systems, or to dynamical systems and can be understood best at the level of a mathematical model, although recent techniques allow the method to be applied to experiments with feedback control. Typically the theory is applied to a continuous model, but can also be used in discrete models and mathematics, difference equations. There are dedicated numerical implementations of bifurcation theory using path-following, or numerical continuation. There is a distinction between a local bifurcation, which can be understood in terms of a change to the number or stability of simple steady states, and a global bifurcation, which cannot. Often global bifurcations cause catastrophic changes to the attractor of the system. Typical local examples are the Hopf bifurcation which leads to the onset of oscillation and the saddle-node bifurcation where a stable steady state is created or destroyed, often leading to bistability.

Characteristics

Once a model of a biological system has been constructed and, consequently, the number of parameters in the model carefully defined and evaluated – perhaps using optimization and parameter estimation – one may wonder how the long-term behavior of a dynamical system changes when these parameters are changed. This question is the basis of bifurcation theory and it underlies a qualitative understanding of many biological processes and transitions, such as the onset of oscillation, switching, morphogenesis, multi-stability, emergence, and localization.

Bifurcation theory can be applied to a wide variety of deterministic models and processes, including partial differential equation (PDE) models and dynamical systems theory, delay differential equations but for simplicity this entry shall consider only the context of dynamical systems theory, ordinary differential equations. Specifically, consider such a parametrized ODE model written in state-space form:

$$ x = f\left( {x,\rm A} \right), $$
(1)

where x ɛ Rn is the set of states of the system, λ ɛ Rp is a parameter set, and a dot denotes differentiation with respect to time.

In contrast with many equations arising in physics, engineering, and economics, a key feature of most biological models of the form (Danino et al. 2010) is that they are nonlinear, essentially because of large deformations, excitability, thresholding and interaction processes governed by the law of mass action. Nonlinear systems can have multiple stable steady states, or attractors, that can have many different stable regions, or basins of attraction in state space. They can also often support limit cycle periodic oscillations. Moreover, as a parameter is varied, a new attractor can emerge out of “thin air,” typically when an unstable state gains local stability. For this reason, usual numerical methods and computer simulation for ordinary differential equations (Partial Differntial Equations, Numerical Methods and Simulations; Ordinary Differential Equation (ODE), Model) can be highly unreliable in understanding the true dynamics of the system, if the method is based on simulation from fixed initial conditions. Bifurcation theory can be especially useful in this context as it enables the tracing of paths of unstable states as parameters vary and hence determine precisely the transitions (or “bifurcation points”) at which qualitatively distinct stable behavior emerges.

The codimension of a bifurcation is defined as the number of parameters required to observe a bifurcation in a structurally stable way. So a codimension-one bifurcation can be observed at an isolated value of a single parameter, whereas a codimension-two bifurcation would typically only be seen at an isolated point in a two-parameter diagram. There is a distinction drawn between local bifurcations that can be understood in terms of loss of stability of a simple state such as an equilibrium or a limit cycle and global bifurcations that cannot. Often global bifurcations involve rearrangements of stable and unstable manifolds of other simple states, such as in homoclinic bifurcations.

A codimension-one bifurcation can often be represented in a bifurcation diagram that depicts a measure, or norm, of a system state against a single parameter; see Fig. 1 for examples. Codimension-one bifurcations can also be used to divide regions in a parameter plain in which qualitatively distinct bifurcations can occur.

Dynamical Systems Theory, Bifurcation Analysis, Fig. 1
figure 427

Schematic bifurcation diagrams depicting codimension-one local bifurcations. (a) An s-shaped fold featuring two saddle-node bifurcations (at parameter values λ1 and λ2) and a consequent parameter interval between these two values in which bistability is observed. In this and subsequent panels, solid lines represent paths of stable steady states and dashed lines unstable steady states. Also ||x|| represents a characteristic norm or scalar measure of the vector state x. (b) A supercritical pitchfork bifurcation (upper plot) and a transcritical bifurcation (lower plot). (c) A supercritical Hopf bifurcation (upper plot) and a subcritical Hopf bifurcation (lower plot). Here, a curve composed of solid circles represents a path of stable limit cycle oscillations, whereas a curve of open circles represents a path of unstable limit cycles. (d) A representation of a supercritical Hopf bifurcation in state and parameter space

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A comprehensive treatment of local bifurcations of codimension-one and two, and many examples of global bifurcations can be found in Kuznetsov (2004). That book also contains analytical and numerical techniques for analyzing bifurcations in practical examples, principally using the theory of center manifolds and normal forms. That theory can be seen as a counterpart to more traditional dynamical systems theory, asymptotics, and singular perturbations. The qualitative geometry or topology of bifurcations is also stressed by Shilnikov et al. (2001). Many examples in biological systems, especially in reaction-diffusion-advection equations are found in Murray (2007), and applications to cell biology in Fall et al. (2002). A more elementary introduction to bifurcation theory and nonlinear dynamical systems theory, stability analysis in general can be found in Strogatz (1994). For more on numerical techniques for performing parameter continuation and bifurcation analysis, see Krauskopf et al. (2007).

Rather than reiterate this theory, this entry shall try to give a qualitative flavor to how bifurcation theory can underlie several key phenomena in biological systems. Specifically treated are threshold behavior, oscillation, bi- and multi-stability, synchronization and emergence of collective behavior, before some final remarks on bifurcation theory applied directly to experiments.

Threshold

It is common in biological systems for a threshold concentration of some chemical signal to be there in order for certain behavior to be triggered. Thresholds can often be understood in terms of the phenomenon of excitability, which in itself is a property of a dynamical system with two timescales, where a transient pushes the system beyond the region where a large excitation occurs. In systems that reach steady state though, the variation of a parameter can cause a global bifurcation that makes the large excitation occur.

Examples:

  • Control of calcium oscillations, see for example (Sneed and Keener 2008)

  • Cell cycle model analysis, bifurcation theory, which is a canonical example of the practical applicability of bifurcation theory in explaining how biological decisions occur as emergent bifurcation events upon integrating the various environmental and internal parameters

Oscillation

Periodic oscillations are important in biology, appearing in diverse areas such as Circadian rhythms, metabolic networks and their evolution, heart beats, cell signaling, etc. Oscillations can be described by simple harmonic motion, governed by second-order linear ODEs, but such descriptions suffer from several serious limitations. First, they are not robust, as a small amount of damping destroys the oscillation. Second, oscillation amplitude and phase depends on the initial condition. Finally, the frequency is typically not adaptable. In contrast, stable limit cycles of nonlinear models are robust, because following a small perturbation away from the cycle, the system will return to the cycle by itself. Also, if the dynamics changes a little, a limit cycle will still exist, close to the original one.

The canonical way to generate oscillation from a system that is otherwise at rest is via a Hopf bifurcation. These oscillatory instabilities can occur in two ways (see Fig. 1c), either as a supercritical bifurcation in which a stable limit cycle is created at small amplitude, or as a subcritical bifurcation, often accompanied by bistability which would cause the jump to a fully formed large-amplitude attractor.

Examples:

Bistability and Multi-stability

Bistability refers to the existence of two distinct attractors to which the system may evolve given different initial states or perturbations. A typical bifurcation scenario in which bistability occurs is via a pair of saddle-node bifurcations that are connected in an S-shaped fold, see Fig. 1a. Such folded structures are often seen as part of the the unfolding of a codimension-two cusp bifurcation point (see Fig. 2a, b) which is one of the elementary catastrophes of singularity theory.

Dynamical Systems Theory, Bifurcation Analysis, Fig. 2
figure 428

Examples of two-parameter bifurcation diagrams indicating parameter regions in which qualitatively distinct behavior is observed. (a) A cusp bifurcation point linking two curves of saddle-node bifurcations. (b) A representation of the cusp in 3D showing a norm of the solution state on the vertical axis. (c) Codimension-two bifurcations in a simple model for plateau bursting in excitable systems that arises in the unfolding of a certain degenerate codimension-three bifurcation point; see (Golubitsky et al. 2001). Here CP represents a cusp bifurcation point, TB a Takens-Bogdanov point (where Hopf and saddle-node combine), DH a degenarate Hopf bifurcation (a transition between super and subcritical cases), SNIPER represents a Saddle Node of Infinite PERiod bifurcation point (a kind of global bifurcation), SN a saddle node of equilibria, SNP a saddle node of periodic orbits, HB a Hopf bifurcation, and HC a homoclinic bifurcation. (d) Two-parameter bifurcation diagram of an open-cell model for calcium oscillations (Sneed and Keener 2008). The lines (for three different values of a parameter δ) represent Hopf bifurcation curves in the parameter plane and separate regions in which oscillations do and do not exist

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Bistability also often accompanies subcritical bifurcations, where an extra fold causes the unstable bifurcating branch to turn around, become stable, and coexist with the primary branch, again see Fig. 1.

Multi-stability refers the situation when there are more than two competing attractors, each with distinct basins of attraction, which can occur via a sequence of bifurcations, or directly via a global bifurcation, such as that caused by a Shilnikov-type homoclinic bifurcation.

Examples:

  • The Toggle switch (Gardner et al. 2000) is a synthetic biology construct that matches what is believed to be a common network motif in systems biology. It is composed of two genes that mutually repress each other, which causes bistability between steady states in which either one gene or the other is expressed at high levels.

  • Multi-stability is also seen in unusual cell-division phenotypes in cell cycle model analysis, bifurcation theory and in recent systems biology models of tumors as competing attracting states in cancer dynamics (Huang et al. 2009).

Synchronization and Emergence of Behavior

Many biological systems composed of near identical cells, components, or organisms are known to undergo common collective dynamics. The simplest such state is that of synchronization, where each state oscillates, typically periodically perfectly in time with the others. This is an example of a symmetric state of a dynamical system and the onset or loss of synchronicity can be understood as a form of symmetry breaking bifurcation. Other collective states include spatially localized patterns of either steady state or dynamic behavior. For more on the bifurcation theory of pattern formation systems, see (Hoyle 2006).

Example:

  • Collective oscillations in proliferating bacterial population (Danino et al. 2010).

Experimental Bifurcation Theory

Recently, the possibility of performing bifurcation analysis directly in feedback controlled experiments has raised (Sieber et al. 2008). This poses the possibility of direct intervention in vitro or in vivo in order to analyze, or indeed to influence and control bifurcations to desirable or undesirable states as an external or internal parameter is varied. Such technology is likely to have a significant impact on synthetic biology and personalized medicine.

Cross-References

Attractor

Bifurcation

Bifurcation, Supercritical and Subcritical

Cell Cycle Model Analysis, Bifurcation Theory

Circadian Rhythm

Continuous Model

Differential-Difference Equations

Dynamical Systems Theory, Asymptotics and Singular Perturbations

Dynamical Systems Theory, Delay Differential Equations

Feedback Regulation

Gene Regulatory Networks

Goodwin Oscillator

Hopf Bifurcation

Law of Mass Action

Limit Cycle

Metabolic Networks, Evolution

Network Motif

Partial Differntial Equations, Numerical Methods and Simulations

Optimization and Parameter Estimation, Genetic Algorithms

Ordinary Differential Equation (ODE)

Oscillation Amplitude

Partial Differential Equation (PDE), Models

Periodic Oscillation

Perturbation

Reaction-Diffusion-Advection Equation

Repressilator and Oscillating Network

Saddle-Node Bifurcation

Stability

Stability, States and Regions

Steady State