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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\u2019s 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\u2013Hopf 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.<\/jats:p>","DOI":"10.1007\/s12064-024-00414-7","type":"journal-article","created":{"date-parts":[[2024,3,9]],"date-time":"2024-03-09T12:02:01Z","timestamp":1709985721000},"page":"107-122","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Dynamics of neural fields with exponential temporal kernel"],"prefix":"10.1007","volume":"143","author":[{"given":"Elham","family":"Shamsara","sequence":"first","affiliation":[]},{"given":"Marius E.","family":"Yamakou","sequence":"additional","affiliation":[]},{"given":"Fatihcan M.","family":"Atay","sequence":"additional","affiliation":[]},{"given":"J\u00fcrgen","family":"Jost","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,3,9]]},"reference":[{"key":"414_CR1","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1177\/1073858408317066","volume":"14","author":"JY Wu","year":"2008","unstructured":"Wu JY, Huang XY, Zhang C (2008) Propagating waves of activity in the neocortex: what they are, what they do. 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