Expansions in series of non-orthogonal functions

Churchill, R. V.

doi:10.1090/s0002-9904-1942-07628-2

Cited by 23 publications

(16 citation statements)

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“…Due to the boundary conditions at the soma (x = 0), X n (x) are not orthogonal under the standard L 2 inner product on 0 ≤ x ≤ L (unless γ = 0). Therefore, we define a modified L 2 inner product under which X n (x) are orthogonal [7,10] …”

Section: Nonspike Solutionmentioning

confidence: 99%

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Bistability in a Leaky Integrate-and-Fire Neuron with a Passive Dendrite

Schwemmer¹,

Lewis²

2012

SIAM J. Appl. Dyn. Syst.

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Abstract. We examine the influence of dendritic load on the firing dynamics of a spatially extended leaky integrate-and-fire (LIF) neuron that explicitly includes spiking dynamics. We obtain an exact analytical solution for this model and use it to derive a return map that completely captures the dynamics of the system. Using the map, we find that dendritic properties can significantly change the firing dynamics of the system. Under certain conditions, the addition of the dendrite can change the LIF model from type 1 excitability to type 2 excitability and induce bistability between periodic firing and the quiescent state. We identify the mechanism that causes the periodic behavior in the bistable regime as somatodendritic ping-pong. Furthermore, we use the return map to fully explore the model parameter space in order to find regions where this bistable behavior occurs. We then give physical interpretations of the dependence of the bistable behavior on model parameters. Finally, we demonstrate that the simpler two-compartment model displays qualitatively similar dynamics to the more complicated ball-and-stick model. 1. Introduction. Neurons are spatially extensive and heterogeneous. They typically consist of a dendritic tree, a soma (cell body), and an axon. The type of model one uses to represent a neuron depends upon a balance between mathematical tractability and biological realism and on the issue being addressed. A common technique in neuronal modeling is to represent the neuron as a single-compartment object that ignores the spatial anatomy of the cell, e.g., [11,25,26]. Although this simplification allows for greater mathematical tractability and computational efficiency, many neurons are not electrotonically compact. Therefore, singlecompartment models cannot be expected to capture the full spectrum of electrical behavior of neurons. Dendrites can have substantial effects on the dynamics of individual neurons. For example, the architecture and ionic channel density of a dendritic tree can alter the firing pattern and encoding properties of a neuronal oscillator [19,21,24], while the additional electrical load due to the dendrite can alter firing frequency [36,38]. For a full understanding of this behavior, more detailed models are required.

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Section: Nonspike Solutionmentioning

confidence: 99%

“…Note that, like the spiking case, the series representation of the solution of the nonspiking portion of the model is of the form of an eigenfunction expansion and is also guaranteed to converge [7].…”

Section: Nonspike Solutionmentioning

confidence: 99%

Bistability in a Leaky Integrate-and-Fire Neuron with a Passive Dendrite

Schwemmer¹,

Lewis²

2012

SIAM J. Appl. Dyn. Syst.

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“…A direct but important observation is the fact that eigenfunctions X k , corresponding to different eigenvalues λ k , are not orthogonal with respect to the standard scalar product in L 0, 1 2 ( ) but are orthogonal with respect to the following modified scalar product [23] u…”

Section: Classical Description Of the Model: Solving The Field Equationsmentioning

confidence: 99%

Quantization of scalar fields coupled to point masses

Juárez-Aubry

Margalef-Bentabol

Villaseñor

2015

Class. Quantum Grav.

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We study the Fock quantization of a compound classical system consisting of point masses and a scalar field. We consider the Hamiltonian formulation of the model by using the geometric constraint algorithm of Gotay, Nester and Hinds. By relying on thisHamiltonian description, we characterize in a precise way the real Hilbert space of classical solutions to the equations of motion and use it to rigorously construct the Fock space of the system. We finally discuss the structure of this space, in particular the impossibility of writing it in a natural way as a tensor product of Hilbert spaces associated with the point masses and the field, respectively.

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“…The corresponding eigenfunctions U n = U (x, ω n ) = A n u n (x) are orthogonal with respect to an inner product defined in terms of a Lebesgue-Stieltjes integral [64] (see also Appendix and [69,73,74]). In other words,…”

Section: A Single Branch With Smooth Taperingmentioning

confidence: 99%

“…(see the methods in [27,64]). Substitution of equation (46) into equation (44) and changing the order of summation and integration result in…”

Section: A Single Branch With Smooth Taperingmentioning

confidence: 99%

A Graphical Approach to a Model of a Neuronal Tree with a Variable Diameter

2014

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Tree-like structures are ubiquitous in nature. In particular, neuronal axons and dendrites have tree-like geometries that mediate electrical signaling within and between cells. Electrical activity in neuronal trees is typically modeled using coupled cable equations on multi-compartment representations, where each compartment represents a small segment of the neuronal membrane. The geometry of each compartment is usually defined as a cylinder or, at best, a surface of revolution based on a linear approximation of the radial change in the neurite. The resulting geometry of the model neuron is coarse, with non-smooth or even discontinuous jumps at the boundaries between compartments. We propose a hyperbolic approximation to model the geometry of neurite compartments, a branched, multi-compartment extension, and a simple graphical approach to calculate steady-state solutions of an associated system of coupled cable equations. A simple case of transient solutions is also briefly discussed.

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Expansions in series of non-orthogonal functions

Cited by 23 publications

References 0 publications

Bistability in a Leaky Integrate-and-Fire Neuron with a Passive Dendrite

Bistability in a Leaky Integrate-and-Fire Neuron with a Passive Dendrite

Quantization of scalar fields coupled to point masses

A Graphical Approach to a Model of a Neuronal Tree with a Variable Diameter

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