Eli Shlizerman


Neural Activity Measures and Their Dynamics

Eli Shlizerman, Konrad Schroder and J. Nathan Kutz

Download the paper published in SIAM J. on App. Math. (SIAP)


  • The collective behavior of a population is often determined by the individual dynamics.
  • Interactions within the population modify the dynamics even of a completely synchronized population.
  • Weak pairwise connections can still modify population dynamics.

  • We resolve the dynamics of a projection of the network (Activity Measure). The projection can be the mean over all neurons, higher order moment or orthogonal decomposition.
  • Formulation

  • Neurons' dynamics are described by ODEs separated to self dynamics and pairwise interactions.
  • Self dynamics are very general and can be modeled by nonlinear equations with several gating variables: Hodgkin-Huxley, Morris-Lecar, FitzHugh-Nagumo and other reductions.
  • Pairwise interactions can be also very general. In the examples here they are simplified to analytic and only in the voltage variable.
  • Reduction

  • When there is a "synchronous" attractor, such that assymptotically each neuron's trajectory is described by a trajectory on that attractor, evolution equations of the activity measure are simplified.
  • Here we demonstrate these equations for the mean and identical neurons.

  • Computational examples

    Identical FitzhHugh-Nagumo neurons

  • Differnet interactions (here linear and bilinear) result with various "effective" collective dynamics and the evolution equations of the activity measure describe the dynamics.

  • Almost identical FitzhHugh-Nagumo neurons

  • Once synchronization occurs the evolution equations of the activity measure describe the dynamics.
  • Department of Applied Mathematics, University of Washington, Guggenheim Hall #414, Box 352420, Seattle, WA 98195-2420 USA
    Email 'info' (at amath.washington.edu) Phone 206-543-5493 Fax 206-685-1440