title: Subthreshold dynamics of a single neuron from a Hamiltonian perspective.
abstract: We use Hamilton's equations of classical mechanics to investigate the behavior of a cortical neuron on the approach to an action potential. We use a two-component dynamic model of a single neuron, due to Wilson, with added noise inputs. We derive a Lagrangian for the system, from which we construct Hamilton's equations. The conjugate momenta are found to be linear combinations of the noise input to the system. We use this approach to consider theoretically and computationally the most likely manner in which such a modeled neuron approaches a firing event. We find that the firing of a neuron is a result of a drop in inhibition, due to a temporary increase in negative bias of the mean noise input to the inhibitory control equation. Moreover, we demonstrate through theory and simulation that, on average, the bias in the noise increases in an exponential manner on the approach to an action potential. In the Hamiltonian description, an action potential can therefore be considered a result of the exponential growth of the conjugate momenta variables pulling the system away from its equilibrium state, into a nonlinear regime.
authors.affiliation: Department of Engineering, University of Waikato, Private Bag 3105, Hamilton 3240, New Zealand. m.wilson@waikato.ac.nz
authors.forename: M T
authors.initials: MT
authors.lastname: Wilson
authors.forename: D A
authors.initials: DA
authors.lastname: Steyn-Ross
mesh_terms: D000200:Action Potentials / Q000502:physiology; D000818:Animals; D055592:Biophysical Phenomena; D016014:Linear Models; D008564:Membrane Potentials / Q000502:physiology; D008959:Models, Neurological; D009474:Neurons / Q000502:physiology*; D017711:Nonlinear Dynamics; D013269:Stochastic Processes