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**Numerical treatment of a mathematical model arising from a model of neuronal variability.**
*(English)*
Zbl 1062.92012

Summary: We describe a numerical approach based on finite difference methods to solve a mathematical model arising from a model of neuronal variability. The mathematical modelling of the determination of the expected time for generation of action potentials in nerve cells by random synaptic inputs in dendrites includes a general boundary-value problem for singularly perturbed differential-difference equations with small shifts. In the numerical treatment for such type of boundary-value problems, first we use Taylor approximation to tackle the terms containing small shifts which converts it to a boundary-value problem for singularly perturbed differential equations. A rigorous analysis is carried out to obtain priori estimates on the solution of the problem and its derivatives up to third order. Then a parameter uniform difference scheme is constructed to solve the boundary-value problem so obtained.

A parameter uniform error estimate for the numerical scheme so constructed is established. Though the convergence of the difference scheme is almost linear its beauty is that it converges independently of the singular perturbation parameter, i.e., the numerical scheme converges for each value of the singular perturbation parameter (however small it may be it remains positive). Several test examples are solved to demonstrate the efficiency of the numerical scheme presented and to show the effect of small shifts on the solution behavior.

A parameter uniform error estimate for the numerical scheme so constructed is established. Though the convergence of the difference scheme is almost linear its beauty is that it converges independently of the singular perturbation parameter, i.e., the numerical scheme converges for each value of the singular perturbation parameter (however small it may be it remains positive). Several test examples are solved to demonstrate the efficiency of the numerical scheme presented and to show the effect of small shifts on the solution behavior.

### MSC:

92C20 | Neural biology |

65L99 | Numerical methods for ordinary differential equations |

34K60 | Qualitative investigation and simulation of models involving functional-differential equations |

65Q05 | Numerical methods for functional equations (MSC2000) |

### Keywords:

Singular perturbations; Action potential; Fitted mesh; Differential-difference equation; Positive shift; Negative shift; Boundary layer
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\textit{M. K. Kadalbajoo} and \textit{K. K. Sharma}, J. Math. Anal. Appl. 307, No. 2, 606--627 (2005; Zbl 1062.92012)

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### References:

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