Accuracy of neuronal interspike times calculated from a diffusion approximation

Tuckwell, Henry C.; Cope, Davis K.

doi:10.1016/0022-5193(80)90045-4

Cited by 42 publications

(19 citation statements)

References 16 publications

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“…The case in which the synaptic inputs are time-dependent (i.e., described by a temporally inhomogeneous Poisson processes), is reviewed in the accompanying paper (Burkitt 2006). This review builds upon, and is indebted to, earlier reviews of these models (Ricciardi 1977;Ricciardi and Sacerdote 1979;Tuckwell 1988b;Lánský and Sato 1999).…”

Section: Introductionmentioning

confidence: 94%

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A Review of the Integrate-and-fire Neuron Model: I. Homogeneous Synaptic Input

2006

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The integrate-and-fire neuron model is one of the most widely used models for analyzing the behavior of neural systems. It describes the membrane potential of a neuron in terms of the synaptic inputs and the injected current that it receives. An action potential (spike) is generated when the membrane potential reaches a threshold, but the actual changes associated with the membrane voltage and conductances driving the action potential do not form part of the model. The synaptic inputs to the neuron are considered to be stochastic and are described as a temporally homogeneous Poisson process. Methods and results for both current synapses and conductance synapses are examined in the diffusion approximation, where the individual contributions to the postsynaptic potential are small. The focus of this review is upon the mathematical techniques that give the time distribution of output spikes, namely stochastic differential equations and the Fokker-Planck equation. The integrate-and-fire neuron model has become established as a canonical model for the description of spiking neurons because it is capable of being analyzed mathematically while at the same time being sufficiently complex to capture many of the essential features of neural processing. A number of variations of the model are discussed, together with the relationship with the Hodgkin-Huxley neuron model and the comparison with electrophysiological data. A brief overview is given of two issues in neural information processing that the integrate-and-fire neuron model has contributed to - the irregular nature of spiking in cortical neurons and neural gain modulation.

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Section: Introductionmentioning

confidence: 94%

“…The diffusion models of neurons treat the membrane potential as a diffusion process, namely as a continuous-time Markov process with a continuous path (Tuckwell 1988b). The stochastic nature of the synaptic input may be described by modeling the membrane potential, Eq.…”

Section: Diffusion Modelsmentioning

confidence: 99%

A Review of the Integrate-and-fire Neuron Model: I. Homogeneous Synaptic Input

2006

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“…In 1967, Stein [7] generalized his model to handle a distribution of post-synaptic potential amplitudes and then approximated the solution using the Monte-Carlo technique. Other methods for obtaining approximate solutions have since been developed by Tuckwell and Cope [6], Tuckwell and Richter [1] and Wilbur and Rinzel [17]. In the paper by Lange and Miura [8], the authors considered if in addition, there are inputs that can be modelled as a Wiener process, then the problem for determining the expected ÿrst-exit time for the generation of action potentials in nerve cells by random synaptic inputs in the dendrites, can be formulated as the general boundary-value problem for the linear second-order di erential-di erence equation…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis of boundary‐value problems for singularly perturbed differential‐difference equations: small shifts of mixed type with rapid oscillations

Kadalbajoo

Sharma

2004

Commun. Numer. Meth. Engng.

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SUMMARYWe study the boundary-value problems for singularly perturbed di erential-di erence equations with small shifts. Similar boundary-value problems are associated with expected ÿrst-exit time problems of the membrane potential in models for activity of neurons (SIAM J. Appl. Math. 1994; 54:249-283; 42:502-531; 45:687-734) and in variational problems in control theory. In this paper, we present a numerical method to solve boundary-value problems for a singularly perturbed di erentialdi erence equation of mixed type, i.e. which contains both type of terms having negative shifts as well as positive shifts, and consider the case in which the solution of the problem exhibits rapid oscillations. The stability and convergence analysis of the method is given. The e ect of small shift on the oscillatory solution is shown by considering the numerical experiments. The numerical results for several test examples demonstrate the e ciency of the method.

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“…The approximated solutions to this model was given by the same author using Monte Carlo techniques. Tuckwell and Richter [6], Tuckwell and Cope [7] and Wilbur and Rinzel [8] used different methods for obtaining the solutions to Stein's model. Lange and Miura [9,10,11] have also analyzed the layer behavior of DDEs in a series of papers.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Singularly Perturbed Differential-Difference Equations with Dual Layer

Sirisha¹,

Reddy²

2014

AJAMS

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In this paper, we discuss the numerical solution of singularly perturbed differential-difference equations exhibiting dual layer behavior. First the second order singularly perturbed differential-difference equation is replaced by an asymptotically equivalent second order singularly perturbed ordinary differential equation. Then, second order stable central difference scheme has been applied to get a three term recurrence relation which is easily solved by Discrete Invariant Imbedding Algorithm. Some numerical examples have been considered to validate the computational efficiency of the proposed numerical scheme. To analyze the effect of the parameters on the solutions, the numerical solutions have also been plotted using graphs. The error bound and convergence of the method have also been established.

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Accuracy of neuronal interspike times calculated from a diffusion approximation

Cited by 42 publications

References 16 publications

A Review of the Integrate-and-fire Neuron Model: I. Homogeneous Synaptic Input

A Review of the Integrate-and-fire Neuron Model: I. Homogeneous Synaptic Input

Numerical analysis of boundary‐value problems for singularly perturbed differential‐difference equations: small shifts of mixed type with rapid oscillations

Numerical Solution of Singularly Perturbed Differential-Difference Equations with Dual Layer

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