Mathematical Conditions for Brain Stability

We investigate the stability of a dynamic brain model using conventional techniques of linearization of the state space equations of motion for two nearby solutions followed by solution via expansion in the eigenvectors of the Jacobian matrix with eigenvalues λ1, λ2, ..., λn. The fundamental stability condition for brains is then |λk| ≅ 1. By equating terms in two versions of the characteristic polynomial of the Jacobian we obtain a set of equations relating principle minors of the Jacobian to the fundamental symmetric polynomials formed from the eigenvalues. These equations may be considered as constraints on the Jacobian since the eigenvalues are constrained by the stability conditions. In turn, these equations may be regarded as constraints on brain structure since the Jacobian embodies the dynamical structure of the brain model.