Elham Shamsara, Marius E Yamakou, Fatihcan M Atay, Jürgen Jost
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Dynamics of neural fields with exponential temporal kernel.
We consider the standard neural field equation with an exponential temporal kernel. We analyze the time-independent (static) and time-dependent (dynamic) bifurcations of the equilibrium solution and the emerging spatiotemporal wave patterns. We show that an exponential temporal kernel does not allow static bifurcations such as saddle-node, pitchfork, and in particular, static Turing bifurcations. However, the exponential temporal kernel possesses the important property that it takes into account the finite memory of past activities of neurons, which Green's function does not. Through a dynamic bifurcation analysis, we give explicit bifurcation conditions. Hopf bifurcations lead to temporally non-constant, but spatially constant solutions, but Turing-Hopf bifurcations generate spatially and temporally non-constant solutions, in particular, traveling waves. Bifurcation parameters are the coefficient of the exponential temporal kernel, the transmission speed of neural signals, the time delay rate of synapses, and the ratio of excitatory to inhibitory synaptic weights.
期刊介绍:
Theory in Biosciences focuses on new concepts in theoretical biology. It also includes analytical and modelling approaches as well as philosophical and historical issues. Central topics are:
Artificial Life;
Bioinformatics with a focus on novel methods, phenomena, and interpretations;
Bioinspired Modeling;
Complexity, Robustness, and Resilience;
Embodied Cognition;
Evolutionary Biology;
Evo-Devo;
Game Theoretic Modeling;
Genetics;
History of Biology;
Language Evolution;
Mathematical Biology;
Origin of Life;
Philosophy of Biology;
Population Biology;
Systems Biology;
Theoretical Ecology;
Theoretical Molecular Biology;
Theoretical Neuroscience & Cognition.