Formation of spiral wave in Hodgkin-Huxley neuron networks with Gamma-distributed synaptic input

We adopt Gamma renewal process to describe the synaptic input of Hodgkin-Huxley neurons on a square lattice with no-flux boundary. Based on the eigenfunction expansion of Laplace operator, we show that the Hopf bifurcation point of the lattice is coincident with that of a single neuron model. By center manifold theorem and normal form theory, we prove that the Hopf bifurcation is subcritical. And then, we exhibit the effect of Gamma noise on the formation of spiral wave for the first time. It is revealed that the occurrence and elimination of the spiral wave can be controlled by adjusting the inhibitory-excitatory ratio and noise parameters. Since spiral waves may contribute to both normal cortical and pathological activities, the present investigation should be helpful for understanding the functional role of general neural noise.