Stimulus-locked traveling waves and breathers in an excitatory neural network

We analyze the existence and stability of stimulus-locked traveling waves in a one-dimensional synaptically coupled excitatory neural network. The network is modeled in terms of a non-local integro-differential equation, in which the integral kernel represents the spatial distribution of synaptic weights and the output firing rate of a neuron is taken to be a Heaviside function of activity. Given an inhomogeneous moving input of amplitude $I_0$ and velocity $v$, we derive conditions for the existence of stimulus--locked waves by working in the moving frame of the input. We use this to construct existence tongues in $(v,I_0)$-parameter space whose tips at $I_0 =0$ correspond to the intrinsic waves of the homogeneous network. We then determine the linear stability of stimulus-locked waves within the tongues by constructing the associated Evans function and numerically calculating its zeros as a function of network parameters. We show that, as the input amplitude is reduced, a stimulus-locked wave within the tongue of an unstable intrinsic wave can undergo a Hopf bifurcation leading to the emergence of either a traveling breather or a traveling pulse emitter.


University of Utah | Department of Mathematics |
bressloff@math.utah.edu
Aug 2001.