SignificanceMany fundamental neural computations from normalization to rhythm generation emerge from the same cortical hardware, but they often require dedicated models to explain each phenomenon. Recently, the stabilized supralinear network (SSN) model has been used to explain a variety of nonlinear integration phenomena such as normalization, surround suppression, and contrast invariance. However, cortical circuits are also capable of implementing working memory and oscillations which are often associated with distinct model classes. Here, we show that the SSN motif can serve as a universal circuit model that is sufficient to support not only stimulus integration phenomena but also persistent states, self-sustained network-wide oscillations along with two coexisting stable states that have been linked with working memory.

Proteins drive the function of neuronal synapses. The synapses are distributed throughout the dendritic arbor, often hundreds of micrometers away from the soma. It is still unclear how somatic and dendritic sources of proteins shape protein distribution and respectively contribute to local protein changes during synaptic plasticity. Here, we present a unique computational framework describing for a given protein species the dendritic distribution of the mRNA and the corresponding protein in a dendrite. Using CaMKIIa as a test case, our model reveals the key role active transport plays in the maintenance of dendritic mRNA and protein levels and predicts the short and long timescales of protein dynamics. Our model reveals the fundamental role of mRNA localization and dendritic mRNA translation in synaptic maintenance and plasticity in distal compartments. We developed a web application for neuroscientists to explore the dynamics of the mRNA or protein of interest.

Identifying the regime in which the cortical microcircuit operates is a prerequisite to determine the mechanisms that mediate its response to stimulus. Classic modeling work has started to characterize this regime through the study of perturbations, but an encompassing perspective that links the full ensemble of the network’s response to appropriate descriptors of the cortical operating regime is still lacking. Here we develop a class of mathematically tractable models that exactly describe the modulation of the distribution of cell-type-specific calcium-imaging activity with the contrast of a visual stimulus. The model’s fit recovers signatures of the connectivity structure found in mouse visual cortex. Analysis of this structure subsequently reveal parameter-independent relations between the responses of different cell types to perturbations and each interneuron’s role in circuit-stabilization. Leveraging recent theoretical approaches, we derive explicit expressions for the distribution of responses to partial perturbations which reveal a novel, counter-intuitive effect in the sign of response functions.

We consider a mathematical model, which describes piezoelectric material behavior. This model is similar to models of plasticity theory. However, piezoelectric models describe coupled mechanical and electrical material behavior. Therefore they contain additional nonlinearities in the piezoelectric tensor and in the enthalpy function, which is non quadratic. These nonlinearities cause difficulties in the proof of existence theorems. Under the assumption that the piezoelectric tensor is constant (i.e. independent of P ), we show how the system of model equations can be reduced to a doubly nonlinear evolution equation of the form zt ∈ G(−Mz − Φ(z) + f ), which contains a composition of two monotone operators. The monotone mapping G is a subdifferential of the indicator function of some convex set while the second monotone mapping Φ is the Nemyckii operator of a monotone function. We prove existence of strong solutions, if Φ is replaced by a regularization. If in addition Φ is Lipschitz continuous we can show that the solution is unique.

In this work we study the solvability of the initial boundary value problems, which model a quasi-static nonlinear behavior of ferroelectric materials. Similar to the metal plasticity the energy functional of a ferroelectric material can be additively decomposed into reversible and remanent parts. The remanent part associated with the remanent state of the material is assumed to be a convex non-quadratic function f of internal variables. In this work we introduce the notion of the measure-valued solutions for the ferroelectric models and show their existence in the rate-dependent case assuming the coercivity of the function f . Regularizing the energy functional by a quadratic positive definite term, which can be viewed as hardening, we show the existence of measure-valued solutions for the rate-independent and rate-dependent problems avoiding the coercivity assumption on f .