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Biological neural networks are notoriously hard to model due to their stochastic behavior and high dimensionality. We tackle this problem by constructing a dynamical model of both the expectations and covariances of the fractions of active…
We consider a new class of non Markovian processes with a countable number of interacting components, both in discrete and continuous time. Each component is represented by a point process indicating if it has a spike or not at a given…
We address the problem of identifying functional interactions among stochastic neurons with variable-length memory from their spiking activity. The neuronal network is modeled by a stochastic system of interacting point processes with…
We consider a slow-fast stochastic process where the slow component is a jump process on a measurable index set whose transition rates depend on the position of the fast component. Between the jumps, the fast component evolves according to…
Motivated by a general principle governing regulation mechanisms in biological cells, we investigate a general interaction scheme between different populations of particles and specific particles, referred to as agents. Assuming that each…
Plasticity is one of the most important properties of the nervous system, which enables animals to adjust their behavior to the ever-changing external environment. Changes in synaptic efficacy between neurons constitute one of the major…
We present a general approach to study a class of random growth models in $n$-dimensional Euclidean space. These models are designed to capture basic growth features which are expected to manifest at the mesoscopic level for several…
In this paper we address the question of statistical model selection for a class of stochastic models of biological neural nets. Models in this class are systems of interacting chains with memory of variable length. Each chain describes the…
Synaptic plasticity depends on the interaction between electrical activity in neurons and the synaptic proteome, the collection of over 1000 proteins in the post-synaptic density (PSD) of synapses. To construct models of synaptic plasticity…
Recent physiological measurements have provided clear evidence about scale-free avalanche brain activity and EEG spectra, feeding the classical enigma of how such a chaotic system can ever learn or respond in a controlled and reproducible…
We consider a simple Markovian class of the stochastic Wilson-Cowan type models of neuronal network dynamics, which incorporates stochastic delay caused by the existence of a refractory period of neurons. From the point of view of the…
We present a probabilistic model for stochastic iterative algorithms with the use case of optimization algorithms in mind. Based on this model, we present PAC-Bayesian generalization bounds for functions that are defined on the trajectory…
To gain a deeper understanding of the behavior and learning dynamics of (deep) artificial neural networks, it is valuable to employ mathematical abstractions and models. These tools provide a simplified perspective on network performance…
We consider the averaging principle for deterministic or stochastic systems with a fast stochastic component (family of continuous-time Markov chains depending on the state of the system as a parameter). We show that, due to bifurcations in…
Markovian population models are suitable abstractions to describe well-mixed interacting particle systems in situation where stochastic fluctuations are significant due to the involvement of low copy particles. In molecular biology,…
A generalized version of the Wilson-Cowan (WC) model is proposed which accounts for the evolution of the synaptic resources. Adiabatic elimination of the fast variables is performed to yield a simplified framework for the coupled…
Continuous adaptation allows survival in an ever-changing world. Adjustments in the synaptic coupling strength between neurons are essential for this capability, setting us apart from simpler, hard-wired organisms. How these changes can be…
We consider a noise driven network of integrate-and-fire neurons. The network evolves as result of the activities of the neurons following spike-timing-dependent plasticity rules. We apply a self-consistent mean-field theory to the system…
The dynamics of local cortical networks are irregular, but correlated. Dynamic excitatory--inhibitory balance is a plausible mechanism that generates such irregular activity, but it remains unclear how balance is achieved and maintained in…
This paper focuses on stochastic partial differential equations (SPDEs) under two-time-scale formulation. Distinct from the work in the existing literature, the systems are driven by $\alpha$-stable processes with $\alpha \in(1,2)$. In…