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Cognitive Neurodynamics

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03-11-2021 | Research Article

Autonomous learning of nonlocal stochastic neuron dynamics

Authors: Tyler E. Maltba, Hongli Zhao, Daniel M. Tartakovsky

Published in: Cognitive Neurodynamics | Issue 3/2022

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Abstract

Neuronal dynamics is driven by externally imposed or internally generated random excitations/noise, and is often described by systems of random or stochastic ordinary differential equations. Such systems admit a distribution of solutions, which is (partially) characterized by the single-time joint probability density function (PDF) of system states. It can be used to calculate such information-theoretic quantities as the mutual information between the stochastic stimulus and various internal states of the neuron (e.g., membrane potential), as well as various spiking statistics. When random excitations are modeled as Gaussian white noise, the joint PDF of neuron states satisfies exactly a Fokker-Planck equation. However, most biologically plausible noise sources are correlated (colored). In this case, the resulting PDF equations require a closure approximation. We propose two methods for closing such equations: a modified nonlocal large-eddy-diffusivity closure and a data-driven closure relying on sparse regression to learn relevant features. The closures are tested for the stochastic non-spiking leaky integrate-and-fire and FitzHugh-Nagumo (FHN) neurons driven by sine-Wiener noise. Mutual information and total correlation between the random stimulus and the internal states of the neuron are calculated for the FHN neuron.

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Due to the large number of these channels, channel noise is often ignored by arguing that fluctuations average out (Yamakou et al. 2019). However, channel noise in and of itself can change the behavior of neurons (White et al. 2000; Zhou et al. 2020), hence, it is important to study its effects on a neuron’s dynamics.

In "Stochastic NS-LIF model" Section , the solution to (1) is positive, so the domain is reduced to (0, L] with a homogeneous boundary condition at \(X=L\) and a reflective (i.e., zero-flux) boundary condition at \(X=0\).

A FHN neuron is considered to be in the excitable regime when starting in the basin of attraction of a unique stable fixed point, a large excursion, or spike, occurs in phase space and then permanently returns to the fixed point (Izhikevich 2007).

We found the semi-local closure (8) to provide only minor improvements in accuracy relative to the closure (9), while being significantly more involved computationally. Consequently, only the numerical results for the semi-local closure (9) are presented therein.

Although the accuracy of the local and semi-local closures is nearly identical for the stochastic NS-LIF model (24), this is generally not the case, as seen in "Stochastic FSH model" Section.

To be concrete, we represented \(\beta _i(t)\) (\(i=1,2\)) with the first ten Legendre polynomials.

A neuron is considered to be in the excitable regime when starting in the basin of attraction of a unique stable fixed point, a large excursion, or spike, occurs in the phase space and then permanently returns to the fixed point (Izhikevich 2007).

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Title: Autonomous learning of nonlocal stochastic neuron dynamics
Authors: Tyler E. Maltba
Hongli Zhao
Daniel M. Tartakovsky
Publication date: 03-11-2021
Publisher: Springer Netherlands
Published in: Cognitive Neurodynamics / Issue 3/2022
Print ISSN: 1871-4080
Electronic ISSN: 1871-4099
DOI: https://doi.org/10.1007/s11571-021-09731-9

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