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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.

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)
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References:

[1] Stein, R. B., A theoretical analysis of neuronal variability, Biophys. J., 5, 173-194 (1965)
[2] Lange, C. G.; Miura, R. M., Singular perturbation analysis of boundary-value problems for differential difference equations. V. Small shifts with layer behavior, SIAM J. Appl. Math., 54, 249-272 (1994) · Zbl 0796.34049
[3] Lange, C. G.; Miura, R. M., Singular perturbation analysis of boundary-value problems for differential difference equations. VI. Small shifts with rapid oscillations, SIAM J. Appl. Math., 54, 273-283 (1994) · Zbl 0796.34050
[4] Segundo, J. P.; Perkel, D. H.; Wyman, H.; Hegstad, H.; Moore, G. P., Input-output relations in computer-simulated nerve cell: Influence of the statistical properties, strength, number and inter-dependence of excitatory pre-dependence of excitatory pre-synaptic terminals, Kybernetik, 4, 157-171 (1968)
[5] Fienberg, S. E., Stochastic models for a single neuron firing trains: A survey, Biometrics, 30, 399-427 (1974) · Zbl 0286.92003
[6] Holden, A. V., Models of the Stochastic Activity of Neurons (1976), Springer-Verlag: Springer-Verlag New York
[7] Stein, R. B., Some models of neuronal variability, Biophys. J., 7, 37-68 (1967)
[8] Johannesma, P. I.M., Diffusion models of the stochastic activity activity of neurons, (Caianello, E. R., Neural Networks. Neural Networks, Proceedings of the School on Neural Networks, Ravello, 1967 (1968), Springer-Verlag: Springer-Verlag New York), 116-144
[9] Tuckwell, H. C., Synaptic transmission in a model for stochastic neural activity, J. Theor. Biol., 77, 65-81 (1979)
[10] Gluss, B., A model for neuron firing with exponential decay of potential resulting in diffusion equations for probability density, Bull. Math. Biophys., 29, 223-243 (1967)
[11] Roy, B. K.; Smith, D. R., Analysis of the exponential decay model of the neuron showing frequency threshold effects, Bull. Math. Biophys., 31, 341-357 (1969) · Zbl 0172.45402
[12] Tuckwell, H. C.; Cope, D. K., Accuracy of neuronal interspike times calculated from a diffusion approximation, J. Theor. Biol., 377-387 (1980)
[13] Lange, C. G.; Miura, R. M., Singular perturbation analysis of boundary-value problems for differential difference equations, SIAM J. Appl. Math., 42, 502-531 (1982) · Zbl 0515.34058
[14] Lange, C. G.; Miura, R. M., Singular perturbation analysis of boundary-value problems for differential difference equations. II. Rapid oscillations and resonances, SIAM J. Appl. Math., 45, 687-707 (1985) · Zbl 0623.34050
[15] Lange, C. G.; Miura, R. M., Singular perturbation analysis of boundary-value problems for differential difference equations. III. Turning point problems, SIAM J. Appl. Math., 45, 708-734 (1985) · Zbl 0623.34051
[16] Lange, C. G.; Miura, R. M., Singular perturbation analysis of boundary-value problems for differential difference equations. IV. A nonlinear example with layer behavior, Stud. Appl. Math., 84, 231-273 (1991) · Zbl 0725.34064
[17] de G. Allen, D. N.; Southwell, R. V., Relaxation methods applied to determine the motion, in 2D, of a viscous fluid past a fixed cylinder, Quart. J. Mech. Appl. Math., 8, 129-145 (1955) · Zbl 0064.19802
[18] Doolan, E. P.; Miller, J. J.H.; Schilder, W. H.A., Uniform Numerical Methods for Problems with Initial and Boundary Layers (1980), Boole: Boole Dublin · Zbl 0459.65058
[19] Miller, J. J.H.; O’Riordan, E.; Shishkin, G. I., Fitted Numerical Methods for Singular Perturbation Problems (1996), World Scientific · Zbl 0945.65521
[20] Farrell, P. A.; O’Riordan, E.; Miller, J. J.H.; Shishkin, G. I., Parameter-uniform fitted mesh method for quasilinear differential equations with boundary layers, Comput. Methods Appl. Math., 1, 154-172 (2001) · Zbl 0977.34015
[21] Kellogg, R. B.; Tsan, A., Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comput., 32, 1025-1039 (1978) · Zbl 0418.65040
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