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Clinical thermal ablation outcomes display significant variability that classical bio-heat models cannot fully explain. One reason may lie in the fractal architecture of biological tissues, which has been identified as a robust biomarker…
We studied the impact of a dynamical threshold on the f-I curve-the relationship between the input and the firing rate of a neuron-in the presence of background synaptic inputs. First, we found that, while the leaky integrate-and-fire model…
Multi-system interaction is an important and difficult problem in physics. Motivated by the experimental result of an electronic circuit element "Fractor", we introduce the concept of dynamic-order fractional dynamic system, in which the…
Many systems of partial differential equations have been proposed as simplified representations of complex collective behaviours in large networks of neurons. In this survey, we briefly discuss their derivations and then review the…
Training dynamics during grokking concentrate along a small number of dominant update directions -- the spectral edge -- which reliably distinguishes grokking from non-grokking regimes. We show that standard mechanistic interpretability…
Celebrated fluctuation-dissipation theorem (FDT) linking the response function to time dependent correlations of observables measured in the reference unperturbed state is one of the central results in equilibrium statistical mechanics. In…
Recurrent neural networks are powerful tools for understanding and modeling computation and representation by populations of neurons. Continuous-variable or "rate" model networks have been analyzed and applied extensively for these…
In a series of two papers, we investigate the mechanisms by which complex oscillations are generated in a class of nonlinear dynamical systems with resets modeling the voltage and adaptation of neurons. This first paper presents…
The behaviour of neurons under the influence of periodic external input has been modelled very successfully by circle maps. The aim of this note is to extend certain aspects of this analysis to a much more general class of forcing…
We consider the fragmentation process with mass loss and discuss self-similar properties of the arising structure both in time and space, focusing on dimensional analysis. This exhibits a spectrum of mass exponents $\theta$, whose exact…
How neurons process their inputs crucially determines the dynamics of biological and artificial neural networks. In such neural and neural-like systems, synaptic input is typically considered to be merely transmitted linearly or sublinearly…
We introduce a nonlinear modification of the classical Hawkes process, which allows inhibitory couplings between units without restrictions. The resulting system of interacting point processes provides a useful mathematical model for…
Chemical and electrical synapses shape the dynamics of neuronal networks. Numerous theoretical studies have investigated how each of these types of synapses contributes to the generation of neuronal oscillations, but their combined effect…
Model-based reinforcement learning is a powerful tool, but collecting data to fit an accurate model of the system can be costly. Exploring an unknown environment in a sample-efficient manner is hence of great importance. However, the…
Motivated by modern observational studies, we introduce a class of functional models that expands nested and crossed designs. These models account for the natural inheritance of correlation structure from sampling design in studies where…
We set up a system of Caputo-type fractional differential equations for a reduced-order model known as the {\em denatured} Morris-Lecar (dML) neurons. This neuron model has a structural similarity to a FitzHugh-Nagumo type system. We…
Federated learning on connected electric vehicles (BEVs) faces severe instability due to intermittent connectivity, time-varying client participation, and pronounced client-to-client variation induced by diverse operating conditions.…
Fractional-order differentiation has many characteristics different from integer-order differentiation. These characteristics can be applied to the optimization algorithms of artificial neural networks to obtain better results. However, due…
Over the last decade, it has been demonstrated that many systems in science and engineering can be modeled more accurately by fractional-order than integer-order derivatives, and many methods are developed to solve the problem of fractional…
We are witnessing the birth of a new variety of pharmacokinetics where non-integer-order differential equations are employed to study the time course of drugs in the body: this is dubbed "fractional pharmacokinetics." The presence of…