A theoretical connection between the Noisy Leaky integrate-and-fire and the escape rate models: The non-autonomous case

Finding a mathematical model that incorporates various stochastic aspects of neural dynamics has proven to be a continuous challenge. Among the different approaches, the noisy leaky integrate-and-fire and the escape rate models are probably the most popular. These two models are generally thought to express different noise action over the neural cell. In this paper we investigate the link between the two formalisms in the case of a neuron subject to a time dependent input. To this aim, we introduce a new general stochastic framework. As we shall prove, our general framework entails the two already existing ones. Our results have theoretical implications since they offer a general view upon the two stochastic processes mostly used in neuroscience, upon the way they can be linked, and explain their observed statistical similarity.

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Nonlinear Compartment Models with Time-Dependent Parameters

Mathematics ◽

10.3390/math9141657 ◽

2021 ◽

Vol 9 (14) ◽

pp. 1657

Author(s):

Jochen Merker ◽

Benjamin Kunsch ◽

Gregor Schuldt

Keyword(s):

Stable Equilibrium ◽

Compartment Model ◽

Basins Of Attraction ◽

Dynamical Behaviour ◽

Time Dependent ◽

Compartment Models ◽

Autonomous Case ◽

Hyperbolic Equilibrium ◽

Interior Points ◽

Two Parameter

A nonlinear compartment model generates a semi-process on a simplex and may have an arbitrarily complex dynamical behaviour in the interior of the simplex. Nonetheless, in applications nonlinear compartment models often have a unique asymptotically stable equilibrium attracting all interior points. Further, the convergence to this equilibrium is often wave-like and related to slow dynamics near a second hyperbolic equilibrium on the boundary. We discuss a generic two-parameter bifurcation of this equilibrium at a corner of the simplex, which leads to such dynamics, and explain the wave-like convergence as an artifact of a non-smooth nearby system in C0-topology, where the second equilibrium on the boundary attracts an open interior set of the simplex. As such nearby idealized systems have two disjoint basins of attraction, they are able to show rate-induced tipping in the non-autonomous case of time-dependent parameters, and induce phenomena in the original systems like, e.g., avoiding a wave by quickly varying parameters. Thus, this article reports a quite unexpected path, how rate-induced tipping can occur in nonlinear compartment models.

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Modal Analysis of the Axial Compressor Blade: Advanced Time-Dependent Electronic Interferometry and Finite Element Method

International Journal of Turbo and Jet Engines ◽

10.1515/tjj-2020-0014 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Mykhaylo Tkach ◽

Serhii Morhun ◽

Yuri Zolotoy ◽

Irina Zhuk

Keyword(s):

Experimental Data ◽

Mathematical Model ◽

Finite Element ◽

High Reliability ◽

Axial Compressor ◽

Time Dependent ◽

Experimental Setup ◽

Vibration Modes ◽

Compressor Blades ◽

Isoparametric Finite Element

AbstractNatural frequencies and vibration modes of axial compressor blades are investigated. A refined mathematical model based on the usage of an eight-nodal curvilinear isoparametric finite element was applied. The verification of the model is carried out by finding the frequencies and vibration modes of a smooth cylindrical shell and comparing them with experimental data. A high-precision experimental setup based on an advanced method of time-dependent electronic interferometry was developed for this aim. Thus, the objective of the study is to verify the adequacy of the refined mathematical model by means of the advanced time-dependent electronic interferometry experimental method. The divergence of the results of frequency measurements between numerical calculations and experimental data does not exceed 5 % that indicates the adequacy and high reliability of the developed mathematical model. The developed mathematical model and experimental setup can be used later in the study of blades with more complex geometric and strength characteristics or in cases when the real boundary conditions or mechanical characteristics of material are uncertain.

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Noise in Integrate-and-Fire Neurons: From Stochastic Input to Escape Rates

Neural Computation ◽

10.1162/089976600300015835 ◽

2000 ◽

Vol 12 (2) ◽

pp. 367-384 ◽

Cited By ~ 89

Author(s):

Hans E. Plesser ◽

Wulfram Gerstner

Keyword(s):

Membrane Potential ◽

Diffusion Approximation ◽

Hazard Function ◽

Time Dependent ◽

Integrate And Fire ◽

Probability Current ◽

Periodic Input ◽

Stochastic Input ◽

Free Membrane ◽

Noise Models

We analyze the effect of noise in integrate-and-fire neurons driven by time-dependent input and compare the diffusion approximation for the membrane potential to escape noise. It is shown that for time-dependent subthreshold input, diffusive noise can be replaced by escape noise with a hazard function that has a gaussian dependence on the distance between the (noise-free) membrane voltage and threshold. The approximation is improved if we add to the hazard function a probability current proportional to the derivative of the voltage. Stochastic resonance in response to periodic input occurs in both noise models and exhibits similar characteristics.

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Time-dependent, two-dimensional mathematical model for simulating the hydraulic, thermal, and water quality characteristics in shallow water bodies

10.2172/4283817 ◽

1974 ◽

Author(s):

Moshe Siman-Tov

Keyword(s):

Mathematical Model ◽

Water Quality ◽

Shallow Water ◽

Quality Characteristics ◽

Water Bodies ◽

Time Dependent ◽

Two Dimensional

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A frequency-resolved mutual information rate and its application to neural systems

Journal of Neurophysiology ◽

10.1152/jn.00354.2014 ◽

2015 ◽

Vol 113 (5) ◽

pp. 1342-1357 ◽

Cited By ~ 10

Author(s):

Davide Bernardi ◽

Benjamin Lindner

Keyword(s):

Mutual Information ◽

Information Filtering ◽

Coherence Function ◽

Time Dependent ◽

Information Rate ◽

Qualitative Behavior ◽

Theoretical Problem ◽

Neural Firing ◽

Integrate And Fire ◽

Model Free

The encoding and processing of time-dependent signals into sequences of action potentials of sensory neurons is still a challenging theoretical problem. Although, with some effort, it is possible to quantify the flow of information in the model-free framework of Shannon's information theory, this yields just a single number, the mutual information rate. This rate does not indicate which aspects of the stimulus are encoded. Several studies have identified mechanisms at the cellular and network level leading to low- or high-pass filtering of information, i.e., the selective coding of slow or fast stimulus components. However, these findings rely on an approximation, specifically, on the qualitative behavior of the coherence function, an approximate frequency-resolved measure of information flow, whose quality is generally unknown. Here, we develop an assumption-free method to measure a frequency-resolved information rate about a time-dependent Gaussian stimulus. We demonstrate its application for three paradigmatic descriptions of neural firing: an inhomogeneous Poisson process that carries a signal in its instantaneous firing rate; an integrator neuron (stochastic integrate-and-fire model) driven by a time-dependent stimulus; and the synchronous spikes fired by two commonly driven integrator neurons. In agreement with previous coherence-based estimates, we find that Poisson and integrate-and-fire neurons are broadband and low-pass filters of information, respectively. The band-pass information filtering observed in the coherence of synchronous spikes is confirmed by our frequency-resolved information measure in some but not all parameter configurations. Our results also explicitly show how the response-response coherence can fail as an upper bound on the information rate.

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Noise-Suppressing Neural Dynamics for Time-Dependent Constrained Nonlinear Optimization With Applications

IEEE Transactions on Systems Man and Cybernetics Systems ◽

10.1109/tsmc.2021.3138550 ◽

2022 ◽

pp. 1-12

Author(s):

Lin Wei ◽

Long Jin ◽

Xin Luo

Keyword(s):

Nonlinear Optimization ◽

Time Dependent ◽

Neural Dynamics ◽

Constrained Nonlinear Optimization

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Influence of Autapses on Synchronization in Neural Networks With Chemical Synapses

Frontiers in Systems Neuroscience ◽

10.3389/fnsys.2020.604563 ◽

2020 ◽

Vol 14 ◽

Author(s):

Paulo R. Protachevicz ◽

Kelly C. Iarosz ◽

Iberê L. Caldas ◽

Chris G. Antonopoulos ◽

Antonio M. Batista ◽

...

Keyword(s):

Neural Networks ◽

Random Network ◽

Inhibitory Neurons ◽

Neural Dynamics ◽

Excitatory Synapses ◽

Closed Loops ◽

Integrate And Fire ◽

Network Topologies ◽

Chemical Synapses ◽

Synchronous Behavior

A great deal of research has been devoted on the investigation of neural dynamics in various network topologies. However, only a few studies have focused on the influence of autapses, synapses from a neuron onto itself via closed loops, on neural synchronization. Here, we build a random network with adaptive exponential integrate-and-fire neurons coupled with chemical synapses, equipped with autapses, to study the effect of the latter on synchronous behavior. We consider time delay in the conductance of the pre-synaptic neuron for excitatory and inhibitory connections. Interestingly, in neural networks consisting of both excitatory and inhibitory neurons, we uncover that synchronous behavior depends on their synapse type. Our results provide evidence on the synchronous and desynchronous activities that emerge in random neural networks with chemical, inhibitory and excitatory synapses where neurons are equipped with autapses.

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An Efficient Approach to Define the Input Stimuli to Suppress Epileptic Seizures Described by the Epileptor Model

International Journal of Neural Systems ◽

10.1142/s0129065720500628 ◽

2020 ◽

Vol 30 (11) ◽

pp. 2050062

Author(s):

João Angelo Ferres Brogin ◽

Jean Faber ◽

Douglas Domingues Bueno

Keyword(s):

Nonlinear Dynamics ◽

Mathematical Model ◽

Exact Solution ◽

Feedback Control ◽

Epileptic Seizures ◽

Neural Dynamics ◽

Practical Application ◽

Control Gain ◽

The World ◽

The Mathematical Model

Epilepsy affects about 70 million people in the world. Every year, approximately 2.4 million people are diagnosed with epilepsy, two-thirds of them will not know the etiology of their disease, and 1% of these individuals will decease as a consequence of it. Due to the inherent complexity of predicting and explaining it, the mathematical model Epileptor was recently developed to reproduce seizure-like events, also providing insights to improve the understanding of the neural dynamics in the interictal and ictal periods, although the physics behind each parameter and variable of the model is not fully established in the literature. This paper introduces an approach to design a feedback-based controller for suppressing epileptic seizures described by Epileptor. Our work establishes how the nonlinear dynamics of this disorder can be written in terms of a combination of linear sub-models employing an exact solution. Additionally, we show how a feedback control gain can be computed to suppress seizures, as well as how specific shapes applied as input stimuli for this purpose can be obtained. The practical application of the approach is discussed and the results show that the proposed technique is promising for developing controllers in this field.

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