A Canonical Model of Multistability and Scale-Invariance in Biological Systems

Abstract

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.

Cortical rich club regions can organize state-dependent functional network formation by engaging in oscillatory behavior

“…However, future work is needed to address the neural mechanism underlying these local dynamics. A candidate mechanism comes in the form of interactions between cortical and thalamic neuronal populations (Breakspear et al, 2006;Freyer et al, 2011Freyer et al, , 2012. A neural mass model implementing these interactions in the presence of state-dependent noise has recently been shown to account for known spontaneous transitions between two distinct modes of power in the alpha frequency band (Freyer et al, 2011;Freyer, Aquino, Robinson, Ritter, & Breakspear, 2009).…”

“…Neural mass models characterized by the normal form of a Hopf bifurcation had previously been shown to provide a good characterization of regional dynamics (Freyer et al, 2011;Freyer, Roberts, Ritter, & Breakspear, 2012). Briefly, the dynamic behavior of each region j was given by ( )…”

The Virtual Brain Integrates Computational Modeling and Multimodal Neuroimaging

“…Each resulting waveform is classified according to several criteria that discern specific dynamical regimes. This could be, for example, the dynamic behavior of the system's state variables such as fixed points or limit cycles-switching between the two has been shown, for example, for the posterior alpha-rhythm (Freyer et al, , 2011(Freyer et al, , 2012. Another feature could be the expression of certain spatial patterns such as resting-state networks (RSNs) known from fMRI (Greicius et al, 2003;Gusnard and Raichle, 2001;Mantini et al, 2007;Raichle et al, 2001) or characteristic spectral distributions (brain chords).…”

“…Signals and other data can often be naturally described as points and flows on a manifold; that is, they are manifold-valued (Perdikis et al, 2011). Several physiologically motivated models of neuronal activity indicate that neuronal activity patterns have an underlying structure that is inherently lower dimensional and contained on the surface (or tightly around the surface) of a low-dimensional manifold that is embedded within the higher-dimensional data space (Deco et al, 2010;Freyer et al, , 2011Freyer et al, , 2012. Therefore, we view manifolds as geometric representations of meaningful system dynamics.…”

Critical fluctuations and coupling of stochastic neural mass models

“…This allows testing of hypotheses about internal dynamical mechanisms (e.g. Freyer et al, 2012) and, through model inversion, the estimation of neural and connectivity parameters that cannot be observed directly . In contrast to modeling at the microscopic scale, where the range of dynamics of healthy neurons is known to include nonlinear behavior such as limit cycles, modeling at the larger scale of mesoscopic neural masses or neural fields often assumes that the dynamics at this scale operate close to a stable fixed point where input fluctuations result in only small and brief perturbations of the population state.…”