Dynamics of neural fields with exponential temporal kernel

Various experimental methods of recording the activity of brain tissue in vitro and in vivo demonstrate the existence of traveling waves. Neural field theory offers a theoretical framework within which such phenomena can be studied. The question then is to identify the structural assumptions and the parameter regimes for the emergence of traveling waves in neural fields. In this paper, 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 Spatio-temporal 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, in contrast to the Green's function used by Atay and Hutt (SIAM J. Appl. Math. 65: 644-666, 2004). However, the exponential temporal kernel possesses the important property that it takes into account the finite memory of past activities of neurons, which the Green's function does not. Through a dynamic bifurcation analysis, we give explicit Hopf (temporally non-constant, but spatially constant solutions) and Turing-Hopf (spatially and temporally non-constant solutions, in particular traveling waves) bifurcation conditions on the parameter space which consists of 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.Comment: 25 pages, 8 Figures, 44 Reference

EuCARE-POSTCOVID Study: a multicentre cohort study on long-term post-COVID-19 manifestations

BACKGROUND: Post-COVID-19 condition refers to persistent or new onset symptoms occurring three months after acute COVID-19, which are unrelated to alternative diagnoses. Symptoms include fatigue, breathlessness, palpitations, pain, concentration difficulties ("brain fog"), sleep disorders, and anxiety/depression. The prevalence of post-COVID-19 condition ranges widely across studies, affecting 10-20% of patients and reaching 50-60% in certain cohorts, while the associated risk factors remain poorly understood. METHODS: This multicentre cohort study, both retrospective and prospective, aims to assess the incidence and risk factors of post-COVID-19 condition in a cohort of recovered patients. Secondary objectives include evaluating the association between circulating SARS-CoV-2 variants and the risk of post-COVID-19 condition, as well as assessing long-term residual organ damage (lung, heart, central nervous system, peripheral nervous system) in relation to patient characteristics and virology (variant and viral load during the acute phase). Participants will include hospitalised and outpatient COVID-19 patients diagnosed between 01/03/2020 and 01/02/2025 from 8 participating centres. A control group will consist of hospitalised patients with respiratory infections other than COVID-19 during the same period. Patients will be followed up at the post-COVID-19 clinic of each centre at 2-3, 6-9, and 12-15 months after clinical recovery. Routine blood exams will be conducted, and patients will complete questionnaires to assess persisting symptoms, fatigue, dyspnoea, quality of life, disability, anxiety and depression, and post-traumatic stress disorders. DISCUSSION: This study aims to understand post-COVID-19 syndrome's incidence and predictors by comparing pandemic waves, utilising retrospective and prospective data. Gender association, especially the potential higher prevalence in females, will be investigated. Symptom tracking via questionnaires and scales will monitor duration and evolution. Questionnaires will also collect data on vaccination, reinfections, and new health issues. Biological samples will enable future studies on post-COVID-19 sequelae mechanisms, including inflammation, immune dysregulation, and viral reservoirs. TRIAL REGISTRATION: This study has been registered with ClinicalTrials.gov under the identifier NCT05531773

The recombination equation for interval partitions

Baake M, Shamsara E. The recombination equation for interval partitions. Monatshefte für Mathematik. 2017;182(2):243-269.The general deterministic recombination equation in continuous time is analysed for various lattices, with special emphasis on the lattice of interval (or ordered) partitions. Based on the recently constructed (Baake et al. in Discr Cont Dynam Syst 36:63-95, 2016) general solution for the lattice of all partitions, the corresponding solution for interval partitions is derived and analysed in detail. We focus our attention on the recursive structure of the solution and its decay rates, and also discuss the solution in the degenerate cases, where it comprises products of monomials with exponentially decaying factors. This can be understood via the Markov generator of the underlying partitioning process that was recently identified. We use interval partitions to gain insight into the structure of the solution, while our general framework works for arbitrary lattices