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Since the pioneering works of Lapicque [17] and of Hodgkin and Huxley [16], several types of models have been addressed to describe the evolution in time of the potential of the membrane of a neuron. In this note, we investigate a connected version of N neurons obeying the leaky integrate and fire model, previously introduced in [1–3,6,7,15,18,19,22]. As a main feature, neurons interact with one another in a mean field instantaneous way. Due to the instantaneity of the interactions, singularities may emerge in a finite time. For instance, the solution of the corresponding Fokker-Planck equation describing the collective behavior of the potentials of the neurons in the limit N ⟶ ∞ may degenerate and cease to exist in any standard sense after a finite time. Here we focus out on a variant of this model when the interactions between the neurons are also subjected to random synaptic weights. As a typical instance, we address the case when the connection graph is the realization of an Erdös-Renyi graph. After a brief introduction of the model, we collect several theoretical results on the behavior of the solution. In a last step, we provide an algorithm for simulating a network of this type with a possibly large value of N.

The growth and invasion of solid tumors are associated with changes in their viscoelastic properties, influenced by both internal cellular factors and physical forces in the tumor microenvironment. Due to the lack of a comprehensive investigation of tumor tissue viscoelasticity, the relationship between such physical properties and cancer malignancy remains poorly understood. Here, the viscoelastic properties of breast cancer spheroids, 3D (in vitro) tumor models, are studied in relation to their metastatic potentials by imposing controlled, dynamic compression within a microfluidic constriction, and subsequently monitoring the relaxation of the imposed deformation. By adopting a modified Maxwell model to extract viscoelastic properties from the compression data, the benign (MCF-10A) spheroids are found to have higher bulk elastic modulus and viscosity compared to malignant spheroids (MCF-7 and MDA-MB-231). The relaxation is characterized by two timescales, captured by a double exponential fitting function, which reveals a similar fast rebound for MCF-7 and MCF-10A. Both the malignant spheroids exhibit similar long-term relaxation and display residual deformation. However, they differ significantly in morphology, particularly in intercellular movements. These differences between malignant spheroids are demonstrated to be linked to their cytoskeletal organization, by microscopic imaging of F-actin within the spheroids, together with cell-cell adhesion strength., (© 2024 The Author(s). Advanced Healthcare Materials published by Wiley‐VCH GmbH.)

Long-range dependence (LRD) has been observed in a variety of phenomena in nature, and for several years also in the spiking activity of neurons. Often, this is interpreted as originating from a non-Markovian system. Here we show that a purely Markovian integrate-and-fire (IF) model, with a noisy slow adaptation term, can generate interspike intervals (ISIs) that appear as having LRD. However a proper analysis shows that this is not the case asymptotically. For comparison, we also consider a new model of individual IF neuron with fractional (non-Markovian) noise. The correlations of its spike trains are studied and proven to have LRD, unlike classical IF models. On the other hand, to correctly measure long-range dependence, it is usually necessary to know if the data are stationary. Thus, a methodology to evaluate stationarity of the ISIs is presented and applied to the various IF models. We explain that Markovian IF models may seem to have LRD because of non-stationarities.