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Applications of the Fokker-Planck Equation in Computational and Cognitive Neuroscience.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Applications of the Fokker-Planck Equation in Computational and Cognitive Neuroscience./
Author:	Vellmer, Sebastian.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2020,
Description:	156 p.
Notes:	Source: Dissertations Abstracts International, Volume: 82-03, Section: B.
Contained By:	Dissertations Abstracts International82-03B.
Subject:	Neurosciences. -
Online resource:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28118847
ISBN:	9798662563353

Applications of the Fokker-Planck Equation in Computational and Cognitive Neuroscience.
Vellmer, Sebastian.

Applications of the Fokker-Planck Equation in Computational and Cognitive Neuroscience. - Ann Arbor : ProQuest Dissertations & Theses, 2020 - 156 p.

Source: Dissertations Abstracts International, Volume: 82-03, Section: B.

Thesis (Ph.D.)--Humboldt Universitaet zu Berlin (Germany), 2020.

This item must not be sold to any third party vendors.

This thesis is concerned with the calculation of statistics, in particular the power spectra, of point processes generated by stochastic multidimensional integrate-and-fire (IF) neurons, networks of IF neurons and decision-making models from the corresponding Fokker-Planck equations. In the brain, information is encoded by sequences of action potentials, the spike trains that are emitted by its most important elements, the neurons. In studies that focus on spike timing and not on the detailed shape of action potentials, IF neurons that drastically simplify the spike generation have become the standard model since they are easy to use and even provide analytical insights. Stochastic IF neurons are particularly advantageous when studying the influence of noise on the firing variability and information transmission. One-dimensional IF neurons do not suffice to accurately model neural dynamics in many situations. However, the extension towards multiple dimensions yields realistic subthreshold and spiking behavior at the price of growing complexity. The first part of this work develops a theory of spike-train power spectra for stochastic, multidimensional IF neurons. From the corresponding Fokker-Planck equation, a set of partial differential equations is derived that describes the stationary probability density, the firing rate and the spiketrain power spectrum. The equations are solved numerically by a finite-difference method for three special cases. The effect of temporally correlated input fluctuations, so-called colored noise, is studied by means of a one-dimensional Markovian embedding that generates either high-pass-filtered (cyan or white-minus-red) or low-pass-filtered (white-plus-red) noise as input for a leaky IF (LIF) neuron. The theory is also applied to white-noise-driven exponential IF neurons with adaptation and, as a three-dimensional example, to a LIF neuron driven by narrow-band noise. Many examples of solutions are presented and compared to simulations in order to test the theory, to display the variety of spectra and to gain a better understanding of the influence of neural features on the spike-train statistics. Furthermore, a set of equations is derived and tested to calculate the Pade approximation of the spectrum at zero frequency yielding an analytical function that is accurate at low frequencies and also matches the high-frequency limit. In the second part of this work, a mean-field theory of large and sparsely connected homogeneous networks of LIF neurons is developed that takes into account the self-consistent temporal correlations of spike trains. Neural input, as the sum of many independent spike trains, is approximated by colored Gaussian noise that is generated by a multidimensional Ornstein-Uhlenbeck process (OUP) yielding a multidimensional IF neuron. In contrast to the first part, the coefficients of the OUP are initially unknown but determined by the self-consistency condition and define the solution of the theory. Since finite-dimensional OUPs cannot exhibit arbitrary power spectra, approximations are introduced and solved up to two-dimensional OUPs for one network. An alternative approach is used to explore heterogeneous networks in which the distribution of power spectra is self consistent. An iterative scheme, initially introduced for homogeneous networks, is extended to determine the distribution of spectra for networks with only distributed numbers of presynaptic neurons and, additionally, distributed synaptic weights. In the third part, the theoretical framework of the Fokker-Planck equation is applied to the problem of binary decisions from diffusion-decision models (DDM). The theory considers the temporal statistics of decisions in situations in which a subject performs consecutive trials of a decisionmaking experiment. The decision trains are introduced in which spikes at the decision times capture the experimental results and encode correct and incorrect decisions by their signs. For the analytically tractable DDM, a Wiener process within two boundaries, the statistics of the decision trains including the decision rates, the distributions of inter-decision intervals and the power spectra are calculated from the corresponding Fokker-Planck equation. Nonlinear DDMs arise as the approximation of decision-making processes implemented in competing populations of spiking neurons. For these models, the threshold-integration method, an efficient numerical procedure that was originally introduced for IF neurons, is generalized to solve the corresponding Fokker-Planck equations and determine the decision-train statistics and the linear response of the decision rates.

ISBN: 9798662563353Subjects--Topical Terms:

588700
Neurosciences.
Subjects--Index Terms:

Fokker-Planck equation

Applications of the Fokker-Planck Equation in Computational and Cognitive Neuroscience.
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500 $a Source: Dissertations Abstracts International, Volume: 82-03, Section: B.
500 $a Advisor: Lindner , Benjamin.
502 $a Thesis (Ph.D.)--Humboldt Universitaet zu Berlin (Germany), 2020.
506 $a This item must not be sold to any third party vendors.
520 $a This thesis is concerned with the calculation of statistics, in particular the power spectra, of point processes generated by stochastic multidimensional integrate-and-fire (IF) neurons, networks of IF neurons and decision-making models from the corresponding Fokker-Planck equations. In the brain, information is encoded by sequences of action potentials, the spike trains that are emitted by its most important elements, the neurons. In studies that focus on spike timing and not on the detailed shape of action potentials, IF neurons that drastically simplify the spike generation have become the standard model since they are easy to use and even provide analytical insights. Stochastic IF neurons are particularly advantageous when studying the influence of noise on the firing variability and information transmission. One-dimensional IF neurons do not suffice to accurately model neural dynamics in many situations. However, the extension towards multiple dimensions yields realistic subthreshold and spiking behavior at the price of growing complexity. The first part of this work develops a theory of spike-train power spectra for stochastic, multidimensional IF neurons. From the corresponding Fokker-Planck equation, a set of partial differential equations is derived that describes the stationary probability density, the firing rate and the spiketrain power spectrum. The equations are solved numerically by a finite-difference method for three special cases. The effect of temporally correlated input fluctuations, so-called colored noise, is studied by means of a one-dimensional Markovian embedding that generates either high-pass-filtered (cyan or white-minus-red) or low-pass-filtered (white-plus-red) noise as input for a leaky IF (LIF) neuron. The theory is also applied to white-noise-driven exponential IF neurons with adaptation and, as a three-dimensional example, to a LIF neuron driven by narrow-band noise. Many examples of solutions are presented and compared to simulations in order to test the theory, to display the variety of spectra and to gain a better understanding of the influence of neural features on the spike-train statistics. Furthermore, a set of equations is derived and tested to calculate the Pade approximation of the spectrum at zero frequency yielding an analytical function that is accurate at low frequencies and also matches the high-frequency limit. In the second part of this work, a mean-field theory of large and sparsely connected homogeneous networks of LIF neurons is developed that takes into account the self-consistent temporal correlations of spike trains. Neural input, as the sum of many independent spike trains, is approximated by colored Gaussian noise that is generated by a multidimensional Ornstein-Uhlenbeck process (OUP) yielding a multidimensional IF neuron. In contrast to the first part, the coefficients of the OUP are initially unknown but determined by the self-consistency condition and define the solution of the theory. Since finite-dimensional OUPs cannot exhibit arbitrary power spectra, approximations are introduced and solved up to two-dimensional OUPs for one network. An alternative approach is used to explore heterogeneous networks in which the distribution of power spectra is self consistent. An iterative scheme, initially introduced for homogeneous networks, is extended to determine the distribution of spectra for networks with only distributed numbers of presynaptic neurons and, additionally, distributed synaptic weights. In the third part, the theoretical framework of the Fokker-Planck equation is applied to the problem of binary decisions from diffusion-decision models (DDM). The theory considers the temporal statistics of decisions in situations in which a subject performs consecutive trials of a decisionmaking experiment. The decision trains are introduced in which spikes at the decision times capture the experimental results and encode correct and incorrect decisions by their signs. For the analytically tractable DDM, a Wiener process within two boundaries, the statistics of the decision trains including the decision rates, the distributions of inter-decision intervals and the power spectra are calculated from the corresponding Fokker-Planck equation. Nonlinear DDMs arise as the approximation of decision-making processes implemented in competing populations of spiking neurons. For these models, the threshold-integration method, an efficient numerical procedure that was originally introduced for IF neurons, is generalized to solve the corresponding Fokker-Planck equations and determine the decision-train statistics and the linear response of the decision rates.
520 $a In dieser Arbeit werden mithilfe der Fokker-Planck-Gleichung die Statistiken, vor allem die Leistungsspektren, von Punktprozessen berechnet, die von mehrdimensionalen Integratorneuronen [Engl. integrate-and-fire (IF) neuron],Netzwerken von IF Neuronen und Entscheidungsfindungsmodellen erzeugt werden.Im Gehirn werden Informationen durch Pulszuge von Aktionspotentialen kodiert.IF Neurone mit radikal vereinfachter Erzeugung von Aktionspotentialen haben sich in Studien die auf Pulszeiten fokussiert sind als Standardmodelle etabliert. Eindimensionale IF Modelle konnen jedoch beobachtetes Pulsverhalten oft nicht beschreiben und mussen dazu erweitert werden. Im erste Teil dieser Arbeit wird eine Theorie zur Berechnung der Pulszugleistungsspektren von stochastischen, multidimensionalen IF Neuronen entwickelt. Ausgehend von der zugehorigen Fokker-Planck-Gleichung werden partiellen Differentialgleichung abgeleitet, deren Losung sowohl die stationare Wahrscheinlichkeitsverteilung und Feuerrate, als auch das Pulszugleistungsspektrum beschreibt.Im zweiten Teil wird eine Theorie fur grose, sparlich verbundene und homogene Netzwerke aus IF Neuronen entwickelt, in der berucksichtigt wird, dass die zeitlichen Korrelationen von Pulszugen selbstkonsistent sind. Neuronale Eingangstrome werden durch farbiges Gaussches Rauschen modelliert, das von einem mehrdimensionalen Ornstein-Uhlenbeck Prozess (OUP) erzeugt wird. Die Koeffizienten des OUP sind vorerst unbekannt und sind als Losung der Theorie definiert. Um heterogene Netzwerke zu untersuchen, wird eine iterative Methode erweitert.Im dritten Teil wird die Fokker-Planck-Gleichung auf Binarentscheidungen von Diffusionsentscheidungsmodellen [Engl. diffusion-decision models (DDM)] angewendet. Explizite Gleichungen fur die Entscheidungszugstatistiken werden fur den einfachsten und analytisch losbaren Fall von der Fokker-Planck-Gleichung hergeleitet. Fur nichtliniear Modelle wird die Schwellwertintegrationsmethode erweitert.
590 $a School code: 5416.
650 4 $a Neurosciences. $3 588700
650 4 $a Theoretical physics. $3 2144760
650 4 $a Cognitive psychology. $3 523881
653 $a Fokker-Planck equation
653 $a Computational
653 $a Cognitive
653 $a Neuroscience
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710 2 $a Humboldt Universitaet zu Berlin (Germany). $3 3480542
773 0 $t Dissertations Abstracts International $g 82-03B.
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791 $a Ph.D.
792 $a 2020
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856 4 0 $u https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28118847