arXiv subjects

M. A. Aziz-Alaoui

Publications and source records attributed to M. A. Aziz-Alaoui.

15 recordsLinked to original sources

arXiv2026

Physics-constrained inference of somatic dynamics from dendritic recordings with sparse somatic supervision in weakly coupled two-compartment neuron model

Somatic membrane potential is the primary determinant of neuronal output, yet it remains inaccessible in many experimental setups where only dendritic recordings are available. Reconstructing somatic dynamics from distal measurements is a challenging inverse problem, particularly when the soma and dendrites are weakly coupled, as dendritic signals represent a filtered and attenuated version of somatic activity. To address this, we use a physics-informed neural network (PINN) constrained by a two-compartment Hodgkin--Huxley model. The network is trained on dense dendritic voltage recordings and the known injected current, complemented by a small number of somatic voltage samples (at most 5\% of the time points), and it reconstructs the full somatic trajectory while adjusting a selected set of somatic maximal conductances. On synthetic data from four stimulation protocols, we quantify how the reconstruction depends on the amount of somatic supervision. Without somatic samples, the present formulation returns a smooth trajectory in which the action potentials are absent and the subthreshold level is biased, with a root-mean-square error of 10--14~mV; 1\% of the somatic time points is enough to recover every spike; and 5\% brings the root-mean-square error to about 2~mV, spikes included, with relative errors below 0.1\% on the sodium and delayed-rectifier conductances. We also compare the PINN with an unscented Kalman filter constrained by the same model and assimilating the same observations, and we assess robustness to measurement noise and to random initialization. The reconstructions reported here therefore rely on sparse somatic anchoring in addition to the dendritic recordings. The results delimit what can be inferred in this synthetic, weakly supervised two-compartment setting and identify the amount of somatic information required by the present formulation.

q-bio.NC↗

arXiv2026

A dynamical system approach to modeling neural network activity in Drosophila orientation

We introduce and analyze a class of neural network models motivated by the Drosophila central complex nervous system, designed to capture the emergence and dynamics of orientation-selective activity bumps. Starting from a biologically inspired ring-connectivity model, we derive a simplified reduced model of recurrent neural activity that supports stable, localized patterns encoding angular position during the fly's flight orientation. We first study the deterministic dynamics and identify parameter regimes ensuring existence and global stability of bump solutions. We then extend the framework to a stochastic setting, incorporating both additive Brownian noise and a Markovian switching mechanism representing time-varying external cues. The resulting system is a switching diffusion with piecewise linear drift, for which we establish well-posedness, characterize the infinitesimal generator, and prove the existence of an invariant measure. Numerical simulations in low and high dimensions illustrate the robustness of the bump attractor under noise and switching stimuli, as well as the convergence toward the predicted stationary states. These results provide a mathematically tractable framework for understanding how population activity in the insect central complex encodes heading direction in the presence of variability.

math.DS↗

arXiv2024

Analysis of a spatio-temporal advection-diffusion model for human behaviors during a catastrophic event

In this work, using the theory of first-order macroscopic crowd models, we introduce a compartmental advection-diffusion model, describing the spatio-temporal dynamics of a population in different human behaviors (alert, panic and control) during a catastrophic event. For this model, we prove the local existence, uniqueness and regularity of a solution, as well as the positivity and $L^1$--boundedness of this solution. Then, in order to study the spatio-temporal propagation of these behavioral reactions within a population during a catastrophic event, we present several numerical simulations for different evacuation scenarios.

math.AP↗

arXiv2023

Consensus dynamics and coherence in hierarchical small-world networks

The hierarchical small-world network is a real-world network. It models well the benefit transmission web of the pyramid selling in China and many other countries. In this paper, by applying the spectral graph theory, we study three important aspects of the consensus problem in the hierarchical small-world network: convergence speed, communication time-delay robustness, and network coherence. Firstly, we explicitly determine the Laplacian eigenvalues of the hierarchical small-world network by making use of its treelike structure. Secondly, we find that the consensus algorithm on the hierarchical small-world network converges faster than that on some well-studied sparse networks, but is less robust to time delay. The closed-form of the first-order and the second-order network coherence are also derived. Our result shows that the hierarchical small-world network has an optimal structure of noisy consensus dynamics. Therefore, we provide a positive answer to two open questions of Yi \emph{et al}. Finally, we argue that some network structure characteristics, such as large maximum degree, small average path length, and large vertex and edge connectivity, are responsible for the strong robustness with respect to external perturbations.

stat.ME↗

arXiv2022

The generation of diverse traveling pulses and its solution scheme in an excitable slow-fast dynamics

In this paper, we report on the generation and propagation of traveling pulses in a homogeneous network of diffusively coupled, excitable, slow-fast dynamical neurons. The spatially extended system is modelled using the nearest neighbor coupling theory, in which the diffusion part measures the spatial distribution of the coupling topology. We derive analytically the conditions for traveling wave profiles that allow the construction of the shape of traveling nerve impulses. The analytical and numerical results are used to explore the nature of the propagating pulses. The symmetric or asymmetric nature of the traveling pulses is characterized and the wave velocity is derived as a function of system parameters. Moreover, we present our results for an extended excitable medium by considering a slow-fast biophysical model with a homogeneous, diffusive coupling that can exhibit various traveling pulses. The appearance of series of pulses is an interesting phenomenon from biophysical and dynamical perspective. Varying the perturbation and coupling parameters, we observe the propagation of activities with various amplitude modulations and transition phases of different wave profiles that affect the speed of the pulses in certain parameter regimes. We observe different types of traveling pulses, such as envelope solitons and multi-bump solutions and show how system parameters and the coupling play a major role in the formation of different traveling pulses. Finally, we obtain the conditions for stable and unstable plane waves.

nlin.PS↗

arXiv2022

Non trivial dynamics in the FizHugh-Rinzel model and non-homogeneous oscillatory-excitable reaction-diffusions systems

In this article, we discuss the dynamics of the 3-dimensional FitzHugh-Rinzel (FHR) model and a class of non-homogeneous FitzHugh-Nagumo (Nh-FHN) Reaction-Diffusion systems. The Nh-FHN models can be used to generate relevant wave propagation phenomena in Neuroscience context. This gives raise locally to complex dynamics such as canards, Mixed Mode Oscillations, Hopf-Bifurcations some of which can be observed in the FHR model.

nlin.PS↗

arXiv2021

Emergent Properties in a V1-Inspired Network of Hodgkin-Huxley Neurons

This article is devoted to the theoretical and numerical analysis of a network of excitatory and inhibitory neurons of Hodgkin-Huxley (HH) type, for which the topology is inspired by that of a single local layer of visual cortex V1. Our model is related to the recent CSY model and therefore differs from other classical models in the field. It combines a driven stochastic drive -- which may be interpreted as an ambient drive for each neuron -- with recurrent inputs resulting from the network activity. After a review of the dynamics of a single HH equation for both the deterministic and the stochastically driven case, we proceed to an analysis of the network. This analysis reveals emergent properties of the system such as partial synchronization and synchronization (defined here as a state of the network for which all the neurons spike within a short interval of time), correlation between excitatory and inhibitory conductances, and oscillations in the gamma-band frequency. The collective behavior enumerated herein is observed when the input-amplitude parameter $S^{EE}$ measuring excitatory-to-excitatory coupling (recurrent excitation) increases to within a certain range. Of note, our work indicates a distinct mechanism for obtaining the emergent properties, some of which have been classically observed. As a consequence our article contributes to the understanding of how assemblies of inhibitory and excitatory cells interact together to produce rhythms in the network. It also aims to bring problems from neuroscience to the realm of mathematics, where they can be analyzed rigorously.

q-bio.NC↗

arXiv2020

On a coupled time-dependent SIR models fitting with New York and New-Jersey states COVID-19 data

This article describes a simple Susceptible Infected Recovered (SIR) model fitting with COVID-19 data for the month of march 2020 in New York (NY) state. The model is a classical SIR, but is non-autonomous; the rate of susceptible people becoming infected is adjusted over time in order to fit the available data. The death rate is also secondarily adjusted. Our fitting is made under the assumption that due to limiting number of tests, a large part of the infected population has not been tested positive. In the last part, we extend the model to take into account the daily fluxes between New Jersey (NJ) and NY states and fit the data for both states. Our simple model fits the available data, and illustrates typical dynamics of the disease: exponential increase, apex and decrease. The model highlights a decrease in the transmission rate over the period which gives a quantitative illustration about how lockdown policies reduce the spread of the pandemic. The coupled model with NY and NJ states shows a wave in NJ following the NY wave, illustrating the mechanism of spread from one attractive hot spot to its neighbor. }

q-bio.PE↗

arXiv2017

Propagation of bursting oscillations in coupled Hodgkin-Huxley Reaction-Diffusion systems

We consider networks of reaction-diffusion systems of Hodgkin-Huxley type. We give a general mathematical framework, in which we prove existence and unicity of solutions as well as existence of invariant regions and of the attractor. Then, we illustrate some relevant numerical examples and exhibit bifurcation phenomena and propagation of bursting oscillations through one and two coupled systems.

math.DS↗

arXiv2017

Canard Phenomenon in a modified Slow-Fast Leslie-Gower and Holling type scheme model

Geometrical Singular Perturbation Theory has been successful to investigate a broad range of biological problems with different time scales. The aim of this paper is to apply this theory to a predator-prey model of modified Leslie-Gower type for which we consider that prey reproduces mush faster than predators. This naturally leads to introduce a small parameter $ε$ which gives rise to a slow-fast system. This system has a special folded singularity which has not been analyzed in the classical work of Krupa-Szmolyan. We use the blow-up technique to visualize the behavior near this fold point $P$. Outside of this region the dynamics are given by classical singular perturbation theory. This allows to quantify geometrically the attractive limit-cycle with an error of $O(ε$) and shows that it exhibits the \textit{canard} phenomenon while crossing $P$.

math.DS↗

arXiv2015

Attractor and synchronization for a complex network of reaction-diffusion systems of FitzHugh-Nagumo type

We focus on the long time behavior of complex networks of reaction-diffusion (RD) systems. We prove the existence of the global attractor and a $L^{\infty}$-bound for a network of $n$ RD systems with $d$ variables each. This allows us to prove the identical synchronization for general class of networks and establish the existence of a coupling strength threshold value that ensures such a synchronization. Then, we apply these results to some particular networks with different structures (i.e. different topologies) and perform numerical simulations. We found out theoretical and numerical heuristic laws for the minimal strengh coupling needed for synchronization relatively to the number of nodes and the network topology, and discuss the link between spatial dimension and synchronization.

math.DS↗

arXiv2014

Emergence of Cooperation in Non-scale-free Networks

Evolutionary game theory is one of the key paradigms behind many scientific disciplines from science to engineering. Previous studies proposed a strategy updating mechanism, which successfully demonstrated that the scale-free network can provide a framework for the emergence of cooperation. Instead, individuals in random graphs and small-world networks do not favor cooperation under this updating rule. However, a recent empirical result shows the heterogeneous networks do not promote cooperation when humans play a Prisoner's Dilemma. In this paper, we propose a strategy updating rule with payoff memory. We observe that the random graphs and small-world networks can provide even better frameworks for cooperation than the scale-free networks in this scenario. Our observations suggest that the degree heterogeneity may be neither a sufficient condition nor a necessary condition for the widespread cooperation in complex networks. Also, the topological structures are not sufficed to determine the level of cooperation in complex networks.

physics.soc-ph↗

arXiv2014

Fence-sitters Protect Cooperation in Complex Networks

Evolutionary game theory is one of the key paradigms behind many scientific disciplines from science to engineering. In complex networks, because of the difficulty of formulating the replicator dynamics, most of previous studies are confined to a numerical level. In this paper, we introduce a vectorial formulation to derive three classes of individuals' payoff analytically. The three classes are pure cooperators, pure defectors, and fence-sitters. Here, fence-sitters are the individuals who change their strategies at least once in the strategy evolutionary process. As a general approach, our vectorial formalization can be applied to all the two-strategies games. To clarify the function of the fence-sitters, we define a parameter, payoff memory, as the number of rounds that the individuals' payoffs are aggregated. We observe that the payoff memory can control the fence-sitters' effects and the level of cooperation efficiently. Our results indicate that the fence-sitters' role is nontrivial in the complex topologies, which protects cooperation in an indirect way. Our results may provide a better understanding of the composition of cooperators in a circumstance where the temptation to defect is larger.

physics.soc-ph↗

arXiv2014

Weakly coupled two slow- two fast systems, folded node and mixed mode oscillationsM

We study Mixed Mode Oscillations (MMOs) in systems of two weakly coupled slow/fast oscillators. We focus on the existence and properties of a folded singularity called FSN II that allows the emergence of MMOs in the presence of a suitable global return mechanism. As FSN II corresponds to a transcritical bifurcation for a desingularized reduced system, we prove that, under certain non-degeneracy conditions, such a transcritical bifurcation exists. We then apply this result to the case of two coupled systems of FitzHugh- Nagumo type. This leads to a non trivial condition on the coupling that enables the existence of MMOs.

math.DS↗

arXiv2012

Effective Fokker-Planck Equation for Birhythmic Modified van der Pol Oscillator

We present an explicit solution based on the phase-amplitude approximation of the Fokker-Planck equation associated with the Langevin equation of the birhythmic modified van der Pol system. The solution enables us to derive probability distributions analytically as well as the activation energies associated to switching between the coexisting different attractors that characterize the birhythmic system. Comparing analytical and numerical results we find good agreement when the frequencies of both attractors are equal, while the predictions of the analytic estimates deteriorate when the two frequencies depart. Under the effect of noise the two states that characterize the birhythmic system can merge, inasmuch as the parameter plane of the birhythmic solutions is found to shrink when the noise intensity increases. The solution of the Fokker-Planck equation shows that in the birhythmic region, the two attractors are characterized by very different probabilities of finding the system in such a state. The probability becomes comparable only for a narrow range of the control parameters, thus the two limit cycles have properties in close analogy with the thermodynamic phases.

nlin.CD↗