Dynamical neural systems are invariably infinite-dimensional, and understanding these complex biological systems requires techniques beyond the geometrical methods of Poincare and Lyapunov. Cable properties of single neurons are ubiquitous to all biological neuronal networks, and they define a system of higher-dimensional space. The current trends of importing artificial neural network methodologies to model neuronal networks, in which cable properties are filtered through compartments, foreshadow the differences between finite and infinite-dimensional (higher-order) dynamical systems. differences
Yet their exclusion from theoretical studies precludes our understanding of the roles
roles spatially distributed ionic channels, second-messengers, and intra and extra-
synaptic connectivity patterns play in the dynamics of neuronal network function.
This complexity ultimately undermines the dynamic states of all higher-order dynami-
cal systems. In particular, complex or irregular dynamics result from a vast number
of degrees of freedom of the dynamical system. What are the techniques for grasping such higher-order dynamical systems? This call for papers specifically aimed at mathematicians interested in exploring the dynamical states of neuronal networks in their natural environment by utilizing theories underlying complex dynamics in higher-dimensional space associated with multi-scale neuroscience.
