Dynamics of neural fields with exponential temporal kernel

• DOI: 10.48550/arxiv.1908.06324
• Publication Date: 2019-01-01
• Authors: Elham Shamsara; Marius E. Yamakou; Fatihcan M. Atay; Jürgen Jost
• Keywords: Exponential polynomial; Pitchfork bifurcation; Kernel (algebra); Constant (computer programming); Exponential decay

We consider the standard neural field equation with an exponential temporal kernel. analyze time-independent (static) and time-dependent (dynamic) bifurcations of equilibrium solution emerging spatiotemporal wave patterns. show that kernel does not allow static such as saddle-node, pitchfork, in particular, Turing bifurcations. However, possesses important property it takes into account finite memory past activities neurons, which Green's function not. Through a dynamic bifurcation analysis, we give explicit conditions. Hopf lead to temporally non-constant, but spatially constant solutions, Turing-Hopf generate non-constant traveling waves. Bifurcation parameters are coefficient kernel, transmission speed signals, time delay rate synapses, ratio excitatory inhibitory synaptic weights.