A Canonical Model of Multistability and Scale-Invariance in Biological Systems
Multistability and scale-invariant fluctuations occur in a wide variety of biological organisms from bacteria to humans as well as financial, chemical and complex physical systems. Multistability refers to noise driven switches between multiple weakly stable states. Scale-invariant fluctuations arise when there is an approximately constant ratio between the mean and standard deviation of a system's fluctuations. Both are an important property of human perception, movement, decision making and computation and they occur together in the human alpha rhythm, imparting it with complex dynamical behavior. Here, we elucidate their fundamental dynamical mechanisms in a canonical model of nonlinear bifurcations under stochastic fluctuations. We find that the co-occurrence of multistability and scale-invariant fluctuations mandates two important dynamical properties: Multistability arises in the presence of a subcritical Hopf bifurcation, which generates co-existing attractors, whilst the introduction of multiplicative (state-dependent) noise ensures that as the system jumps between these attractors, fluctuations remain in constant proportion to their mean and their temporal statistics become long-tailed. The simple algebraic construction of this model affords a systematic analysis of the contribution of stochastic and nonlinear processes to cortical rhythms, complementing a recently proposed biophysical model. Similar dynamics also occur in a kinetic model of gene regulation, suggesting universality across a broad class of biological phenomena. Biological systems are able to adapt to rapidly and widely changing environments. Many biological organisms employ two distinct mechanisms that improve their survival in these circumstances: Firstly they exhibit rapid, qualitative changes in their internal dynamics; secondly they possess the ability to respond to change that is not absolute, but scales in proportion to the underlying intensity of the environment. In this paper, we study a simple class of noisy, dynamical systems that mathematically represent a very broad range of more complex models. We hence show how a combination of nonlinear instabilities and state-dependent noise in this model is able to unify these two apparently distinct biological phenomena. To illustrate its unifying potential, this simple model is applied to two very distinct biological processes – the spontaneous activity of the human cortex (i.e. when subjects are at rest), and genetic regulation in a bacteriophage. We also provide proof of principle that our model can be inverted from empirical data, allowing estimation of the parameters that express the nonlinear and stochastic influences at play in the underlying system.