The past decade has seen growing support for the critical brain hypothesis, i.e., the possibility that the brain could operate at or very near a critical state between two different dynamical regimes. Such critical states are well-studied in different disciplines, therefore there is potential for a continued transfer of knowledge. Here, I revisit foundations of bifurcation theory, the mathematical theory of transitions. While the mathematics is well-known it's transfer to neural dynamics leads to new insights and hypothesis.
Figure 1: A pitchfork bifurcation, the source of criticality in the Ising model, is structurally unstable. An arbitrarily small modification that violates the symmetry assumption turns it into a fold bifurcation.