Control of continuous dynamical systems modeling physiological states

Kesmia, Mounira; Boughaba, Soraya; Jacquir, Sabir

doi:10.1016/j.chaos.2020.109805

Cited by 3 publications

(2 citation statements)

References 33 publications

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“…The mathematical modeling of epileptic seizures appearing in small neural populations can follow a few alternative ways: modeling of individual cells and their interaction vs. modeling groups and clusters on neurons (Kesmia et al, 2020).…”

Section: Introductionmentioning

confidence: 99%

Statistical Model for Excitation and Hypersynchronization in the Small Neural Populations

Borisenok

2022

EJOSAT

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The mathematical modeling of epileptic seizures appearing in small neural populations can follow a few alternative ways: modeling of individual cells and their interaction vs. modeling groups and clusters on neurons. The purpose of this work is invention of a novel continuous (population-based) model for the appearance of the hyper-synchronized firing cells of the epileptiform type. In the same time, we use here the master equations based on the transition probabilities among different states of the cell excitation and hyper-synchronization. We developed an ODE model combining the dynamical equations for different sub-populations (unexcited, excited, and, as our novelty, hypersynchronized). Our model may serve as a simple but powerful tool to analyze the appearance and development of epileptiform dynamics in artificial neural networks. It can cover different cases of microepilepsy, and also may open the gate for studying drug-resistant epilepsy regime. Our dynamical set can be extended with the control inputs mimicking the external perturbations of the neural clusters with the electrical or optogenetic signals. In this case, the set of control algorithms can be applied to detect and suppress the epileptiform dynamics. Thus, the dynamic processes of epilepsy in small neural populations do not demand necessary the development of detailed models for individual neurons. Even the ‘averaged’ dynamical set for the unexcited, excited and hypersynchronized sub-populations can serve as an efficient tool for investigation and numerical simulations of microscopic seizures.

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Section: Introductionmentioning

confidence: 99%

Statistical Model for Excitation and Hypersynchronization in the Small Neural Populations

Borisenok

2022

EJOSAT

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“…Chaotic dynamics are commonplace, almost default in any complex physical or biological system. This includes scientific areas as diverse as neuroscience (Durstewitz & Gabriel, 2007;van Vreeswijk & Sompolinsky, 1996), physiology (Kesmia et al, 2020), geophysics (Sivakumar, 2004), climate systems (Tziperman et al, 1997), astrophysics (Laskar & Robutel, 1993), ecology (Duarte et al, 2010), chemical reactions (Field & Györgyi, 1993), cell (Olsen & Degn, 1977) or population (May, 1987) biology. Chaotic phenomena are also crucial for the understanding of societal and epidemiological processes, such as the spread of diseases (Mangiarotti et al, 2020), or in economics (Faggini, 2014).…”

Section: Introductionmentioning

confidence: 99%

On the difficulty of learning chaotic dynamics with RNNs

Mikhaeil¹,

Monfared²,

Durstewitz³

2021

Preprint

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Recurrent neural networks (RNNs) are wide-spread machine learning tools for modeling sequential and time series data. They are notoriously hard to train because their loss gradients backpropagated in time tend to saturate or diverge during training. This is known as the exploding and vanishing gradient problem. Previous solutions to this issue either built on rather complicated, purpose-engineered architectures with gated memory buffers, or -more recently -imposed constraints that ensure convergence to a fixed point or restrict (the eigenspectrum of) the recurrence matrix. Such constraints, however, convey severe limitations on the expressivity of the RNN. Essential intrinsic dynamics such as multistability or chaos are disabled. This is inherently at disaccord with the chaotic nature of many, if not most, time series encountered in nature and society. Here we offer a comprehensive theoretical treatment of this problem by relating the loss gradients during RNN training to the Lyapunov spectrum of RNN-generated orbits. We mathematically prove that RNNs producing stable equilibrium or cyclic behavior have bounded gradients, whereas the gradients of RNNs with chaotic dynamics always diverge. Based on these analyses and insights, we offer an effective yet simple training technique for chaotic data and guidance on how to choose relevant hyperparameters according to the Lyapunov spectrum.

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