Dynamics of neural fields with exponential temporal kernel

• Published in: Theory in biosciences = Theorie in den Biowissenschaften Vol. 143; no. 2; pp. 107 - 122
• Main Authors: Shamsara, Elham, Yamakou, Marius E., Atay, Fatihcan M., Jost, Jürgen
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2024; Springer Nature B.V
• Subjects: Action Potentials - physiology; Algorithms; Animals; Bioinformatics; Biomedical and Life Sciences; Complex Systems; Computer Simulation; Connectivity; Evolutionary Biology; Humans; Investigations; Life Sciences; Mathematical and Computational Biology; Models, Neurological; Nerve Net - physiology; Neurons - physiology; Partial differential equations; Philosophy of Biology; Propagation; Synapses - physiology; Synaptic strength; Synaptic Transmission; System theory; Theoretical Ecology/Statistics; Time Factors; Leakage; Exponential temporal kernel; Neural fields; Spatiotemporal patterns; Transmission delays; Bifurcation analysis
• ISSN: 1431-7613; 1611-7530
• DOI: 10.1007/s12064-024-00414-7

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.